Science, Faculty of
Chemistry, Department of
DSpace
UBCV
Scheiber, Hayden Owen
Doctor of Philosophy - PhD
University of British Columbia
The lithium halide salts may at first appear to be simple chemical systems. However, previous research indicates lithium halides have complex and poorly understood crystallization pathways from aqueous solutions. While lithium halides exist in the rocksalt crystal structure under ambient conditions, common lithium halide classical force fields more often predict wurtzite as the stable structure. This failure severely limits their application in molecular simulations of crystallization. Research in this thesis focuses on presenting new results and computational methodology to better understand the structural chemistry of both lithium halides and alkali halides in general. Density functional theory together with classical force fields are employed to examine the relative stability of candidate crystal structures for lithium halides and produce accurate theoretical reference data. Dispersion interactions are shown to play a key role in the stability of rocksalt over closely competing crystal structures. Classical models can be corrected in their structural predictions by scaling up the strength of dispersion, indicating a pathway towards better lithium halide force fields. Convolutional neural networks are used for the structural classification of simulated alkali halides. The neural networks are trained on a large data set generated from molecular dynamics simulations of alkali halides across a range of temperatures. Time convolution filters out short-lived structural fluctuations. The structure classifiers are shown to be accurate in bulk phase simulations, then demonstrated on crystallization of model alkali halide systems from the melt. The neural network classifiers are implemented in a melting point calculation algorithm for model binary salts. Finite size effects are characterized, then melting points of alkali halides using common rigid-ion interaction potentials are calculated and discussed. The methods developed throughout the research are employed for the optimization of pairwise lithium halide force fields, fitted to reference data using a reinforcement learning approach, Bayesian optimization. Limitations on the Coulomb Lennard-Jones potential form are uncovered, which do not appear to hold for the more flexible Coulomb Buckingham potential form. By introducing advanced computational methodology, this research reveals the inherent structural complexity of lithium halides and emphasizes the importance of considering structural landscapes during classical forcefields construction for molecular simulation.
Lithium Halide Structural ChemistryComputational Analysis with Machine Learning, Quantum Chemistry, andMolecular DynamicsbyHayden Owen ScheiberA THESIS SUBMITTED IN PARTIAL FULFILLMENTOF THE REQUIREMENTS FOR THE DEGREE OFDoctor of PhilosophyinTHE FACULTY OF GRADUATE AND POSTDOCTORAL STUDIES(Chemistry)The University of British Columbia(Vancouver)December 2022© Hayden Owen Scheiber, 2022The following individuals certify that they have read, and recommend to the Faculty of Graduateand Postdoctoral Studies for acceptance, the thesis entitled:Lithium Halide Structural ChemistryComputational Analysis with Machine Learning, Quantum Chemistry, and Molecular Dynamicssubmitted by Hayden Owen Scheiber in partial fulfillment of the requirements for the degree ofDoctor of Philosophy in Chemistry.Examining Committee:Gren N. Patey, Professor, Chemistry, UBCSupervisorKeng Chou, Professor, Chemistry, UBCSupervisory Committee MemberMark Thachuk, Associate Professor, Chemistry, UBCSupervisory Committee MemberYan Alexander Wang, Associate Professor, Chemistry, UBCSupervisory Committee MemberEd R. Grant, Professor, Chemistry, UBCUniversity ExaminerJoerg Rottler, Professor, Physics and Astronomy, UBCUniversity ExaminerRichard Bowles, Professor, Chemistry, University of SaskatchewanExternal ExamineriiAbstractThe lithium halide salts may at first appear to be simple chemical systems. However, previousresearch indicates lithium halides have complex and poorly understood crystallization pathwaysfrom aqueous solutions. While lithium halides exist in the rocksalt crystal structure under ambientconditions, common lithium halide classical force fields more often predict wurtzite as the stablestructure. This failure severely limits their application in molecular simulations of crystallization.Research in this thesis focuses on presenting new results and computational methodology to betterunderstand the structural chemistry of both lithium halides and alkali halides in general.Density functional theory together with classical force fields are employed to examine therelative stability of candidate crystal structures for lithium halides and produce accurate theoreticalreference data. Dispersion interactions are shown to play a key role in the stability of rocksaltover closely competing crystal structures. Classical models can be corrected in their structuralpredictions by scaling up the strength of dispersion, indicating a pathway towards better lithiumhalide force fields.Convolutional neural networks are used for the structural classification of simulated alkalihalides. The neural networks are trained on a large data set generated from molecular dynamicssimulations of alkali halides across a range of temperatures. Time convolution filters out short-livedstructural fluctuations. The structure classifiers are shown to be accurate in bulk phase simulations,then demonstrated on crystallization of model alkali halide systems from the melt.The neural network classifiers are implemented in a melting point calculation algorithm formodel binary salts. Finite size effects are characterized, then melting points of alkali halides usingcommon rigid-ion interaction potentials are calculated and discussed.The methods developed throughout the research are employed for the optimization of pair-wise lithium halide force fields, fitted to reference data using a reinforcement learning approach,Bayesian optimization. Limitations on the Coulomb Lennard-Jones potential form are uncovered,which do not appear to hold for the more flexible Coulomb Buckingham potential form.iiiBy introducing advanced computational methodology, this research reveals the inherent struc-tural complexity of lithium halides and emphasizes the importance of considering structural land-scapes during classical forcefields construction for molecular simulation.ivLay SummaryLithium halides, defined by a small positively charged lithium ion and a large negatively chargedhalide ion, form a series of four salts with interesting properties. Although chemically nonreactive,the experimentally observed crystal structure of the lithium halides exists due to a fine balance be-tween competing forces at the atomic scale. This means that accurately simulating lithium halidesin computer simulations is challenging compared with similar salts, such as sodium chloride (tablesalt).This thesis focuses on the exploration of lithium halide salts at the atomic scale with computersimulation. Several new computational protocols, involving applications of machine learning, aredeveloped to aid in this investigation. Many of the methods employed in this work are generalizableto the simulation of other chemical systems, offering the potential of new insights. This investi-gation emphasizes the importance of considering alternative crystal structures when exploring theprocess of crystal formation with computer simulation.vPrefaceThe research chapters in this thesis are based on work that is either already published, or will bepublished in the near future, in peer-reviewed scientific journals.A modified version of Chapter 3 was published by H. O. Scheiber and G. N. Patey, Analysisof the relative stability of lithium halide crystal structures: Density functional theory and classicalmodels, The Journal of Chemical Physics 154, 184507 (2021). A modified version of Chapter 4 andChapter 5, along with the supplementary videos included with this thesis, was published by H. O.Scheiber and G. N. Patey, Binary salt structure classification with convolutional neural networks:Application to crystal nucleation and melting point calculations, The Journal of Chemical Physics157, 204108 (2022). The research covered in Chapter 6 is currently being prepared for publication.For all work covered in this thesis, including the above manuscripts, the primary researchwas performed by me alone. This includes the generation of all original data, the data analyses,production of all figures, development of new computational methodology, and writing of initialmanuscript drafts. For all work in this thesis, G. N. Patey contributed by helping to identify researchquestions, design the research work, form hypotheses, and edit the final manuscripts. Chapter 6also includes contributions from Z. Deng, M. Thachuk, and R. V. Krems in the form of fruitfuldiscussions that helped to design the research work and form hypotheses.viTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiLay Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiiiList of Supplementary Material . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xviiList of Abbreviations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xviiiAcknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxivDedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxv1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Lithium Halide Structural Chemistry . . . . . . . . . . . . . . . . . . . . . . . . . 21.2 Phases and Classical Nucleation Theory . . . . . . . . . . . . . . . . . . . . . . . 31.3 Machine Learning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.4 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 Models and Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12vii2.2 Time Integration: Generating an MD Trajectory . . . . . . . . . . . . . . . . . . . 132.3 Interaction Potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.4 Periodic Boundaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192.5 Tail Corrections and Ewald Summation . . . . . . . . . . . . . . . . . . . . . . . 212.6 Temperature and Pressure Control . . . . . . . . . . . . . . . . . . . . . . . . . . 242.7 Molecular Dynamics Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . 272.7.1 Crystal Minimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272.7.2 Rapid Relaxation and Production Simulations . . . . . . . . . . . . . . . . 282.8 System Initiation and Minimization . . . . . . . . . . . . . . . . . . . . . . . . . 282.8.1 Solid Crystals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282.8.2 Molten Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292.8.3 Robust Minimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303 Analysis of the Relative Stability of Lithium Halide Crystal Structures: DensityFunctional Theory and Classical Models . . . . . . . . . . . . . . . . . . . . . . . . . 323.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323.1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323.1.2 Quantum Chemistry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333.1.3 Overview of Research . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383.2 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393.2.1 Recalculation of Experimental Lattice Energies . . . . . . . . . . . . . . . 393.2.2 Candidate Crystal Structures . . . . . . . . . . . . . . . . . . . . . . . . . 493.2.3 Hartree-Fock and Density Functional Theory Calculations . . . . . . . . . 503.2.4 Empirical Models and Classical Calculations . . . . . . . . . . . . . . . . 533.2.5 Crystal Structure Optimization . . . . . . . . . . . . . . . . . . . . . . . . 533.2.6 Thermal Corrections to DFT Calculations . . . . . . . . . . . . . . . . . . 543.3 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 553.3.1 Basis Set Influence on Incompleteness Errors . . . . . . . . . . . . . . . . 553.3.2 Level of Theory: Quantum and Classical Models for LiX Crystals . . . . . 573.3.3 Role of Dispersion in LiX Structure Determination . . . . . . . . . . . . . 673.3.4 Stability of Alternative Crystal Structures . . . . . . . . . . . . . . . . . . 743.3.5 Thermochemical Considerations . . . . . . . . . . . . . . . . . . . . . . . 773.4 Summary and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79viii4 Binary Salt Structure Classification with Convolutional Neural Networks . . . . . . 824.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 824.1.1 Machine Learning for Structure Classification . . . . . . . . . . . . . . . . 834.1.2 Neural Networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 834.2 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 854.2.1 Molecular Dynamics Simulations . . . . . . . . . . . . . . . . . . . . . . 854.2.2 Neural Network Inputs . . . . . . . . . . . . . . . . . . . . . . . . . . . . 864.2.3 Neural Network Training Data . . . . . . . . . . . . . . . . . . . . . . . . 904.2.4 Neural Network Architecture . . . . . . . . . . . . . . . . . . . . . . . . . 914.2.5 Neural Network Training . . . . . . . . . . . . . . . . . . . . . . . . . . . 924.3 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 944.3.1 Neural Network Classifier Accuracy . . . . . . . . . . . . . . . . . . . . . 944.3.2 Alkali Halide Crystal Nucleation . . . . . . . . . . . . . . . . . . . . . . . 1004.4 Summary and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1055 Automated Melting Point Calculations . . . . . . . . . . . . . . . . . . . . . . . . . . 1085.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1085.2 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1105.2.1 Overview and Main Algorithm . . . . . . . . . . . . . . . . . . . . . . . . 1105.2.2 Preparing the Liquid-Solid Interface . . . . . . . . . . . . . . . . . . . . . 1125.2.3 Checking for Melting or Freezing . . . . . . . . . . . . . . . . . . . . . . 1155.2.4 Narrowing the Temperature Bracket . . . . . . . . . . . . . . . . . . . . . 1185.3 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1215.3.1 System Size Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1215.3.2 Melting Points of Common Alkali Halide Models . . . . . . . . . . . . . . 1255.4 Summary and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1306 Bayesian Optimization of Lithium Halide Force Field Parameters . . . . . . . . . . 1326.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1326.1.1 Bayesian Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1336.2 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1356.2.1 Bayesian Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1366.2.2 Objective Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1386.2.3 Parameter Domain and Constraints . . . . . . . . . . . . . . . . . . . . . 143ix6.3 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1496.3.1 Coulomb Lennard-Jones Model . . . . . . . . . . . . . . . . . . . . . . . 1496.3.2 Coulomb Buckingham Model . . . . . . . . . . . . . . . . . . . . . . . . 1556.4 Summary and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1607 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1647.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1647.2 Future Directions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170A General Supporting Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188A.1 Classical Interaction Potential Parameters . . . . . . . . . . . . . . . . . . . . . . 189A.2 Crystallographic Information . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190B Supporting Materials for Chapter 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . 198B.1 Quantum Mechanical Calculation Details . . . . . . . . . . . . . . . . . . . . . . 198B.2 Technical Points on Basis Set Errors . . . . . . . . . . . . . . . . . . . . . . . . . 203B.3 Thermal Corrections to Experimental Lattice Parameters . . . . . . . . . . . . . . 207B.4 Technical Points on Dispersion Corrections . . . . . . . . . . . . . . . . . . . . . 208B.5 Classical Models vs DFT at High Pressure . . . . . . . . . . . . . . . . . . . . . . 212B.6 Scalar Relativistic Effects on Heavy Lithium Halides . . . . . . . . . . . . . . . . 214C Supporting Materials for Chapter 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . 221C.1 Rigid-Ion Models: Crystal Energies at (T = 0, P = 0) . . . . . . . . . . . . . . . . 221D Supporting Materials for Chapter 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . 230D.1 Systematic MP Uncertainty in the Thermodynamic Limit . . . . . . . . . . . . . . 230D.2 Melting Point Distributions - Unfiltered Data . . . . . . . . . . . . . . . . . . . . 234E Supporting Materials for Chapter 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . 235E.1 Choice of Covariance and Acquisition Functions . . . . . . . . . . . . . . . . . . 235E.1.1 Acquisition Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235E.1.2 Covariance Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237E.2 Distributions of Thermal Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242E.3 Coulomb Lennard-Jones Model: Additional Data . . . . . . . . . . . . . . . . . . 243xList of TablesTable 3.1 Recalculated experimental lattice energies: comparison with literature . . . . . 41Table 3.2 Born-Haber cycle: thermodynamic properties . . . . . . . . . . . . . . . . . . 42Table 3.3 Born-Haber cycle: experimental data . . . . . . . . . . . . . . . . . . . . . . . 43Table 3.4 Lattice energy calculation method 1: experimental data . . . . . . . . . . . . . 47Table 3.5 Lattice energy calculation method 2: experimental data . . . . . . . . . . . . . 48Table 3.6 Experimental lattice energy comparison: three methods . . . . . . . . . . . . . 49Table 3.7 List of quantum-mechanical theories used . . . . . . . . . . . . . . . . . . . . 52Table 3.8 Relative crystal energies of LiX and NaCl: highest accuracy DFT results . . . . 75Table 4.1 Summary of the neural network structure classifiers . . . . . . . . . . . . . . . 95Table 5.1 Calculated melting points of for common alkali halide rigid-ion models . . . . . 127Table 6.1 Summary of target experimental data . . . . . . . . . . . . . . . . . . . . . . . 139Table 6.2 Summary of target DFT data . . . . . . . . . . . . . . . . . . . . . . . . . . . 140Table 6.3 Parameter domains for Bayesian optimization . . . . . . . . . . . . . . . . . . 148Table A.1 TF interaction potential parameters . . . . . . . . . . . . . . . . . . . . . . . . 189Table A.2 JC and SD interaction potential parameters . . . . . . . . . . . . . . . . . . . . 190Table A.3 Rocksalt crystallographic information . . . . . . . . . . . . . . . . . . . . . . . 191Table A.4 Wurtzite crystallographic information . . . . . . . . . . . . . . . . . . . . . . . 192Table A.5 β -BeO crystallographic information . . . . . . . . . . . . . . . . . . . . . . . 193Table A.6 CsCl crystallographic information . . . . . . . . . . . . . . . . . . . . . . . . . 194Table A.7 5-5 crystallographic information . . . . . . . . . . . . . . . . . . . . . . . . . 195Table A.8 NiAs and antiNiAs crystallographic information . . . . . . . . . . . . . . . . . 196Table A.9 Sphalerite crystallographic information . . . . . . . . . . . . . . . . . . . . . . 197xiTable B.1 LiX basis set superposition error estimates from the gCP method . . . . . . . . 204Table B.2 Experimental LiX and NaCl lattice parameters . . . . . . . . . . . . . . . . . . 208Table B.3 LiX and NaCl rocksalt dispersion energy contributions: comparison of models . 210Table B.4 LiX dispersion coefficients: comparison of models . . . . . . . . . . . . . . . . 211Table B.5 Rocksalt vs wurtzite geometric dispersion favouring measures . . . . . . . . . . 213Table C.1 Energy-minimized alkali halide lattice energies: TF model . . . . . . . . . . . . 223Table C.2 Energy-minimized alkali halide lattice energies: JC(SPC/E) model . . . . . . . 225Table C.3 Energy-minimized alkali halide lattice energies: JC(TIP3P) model . . . . . . . 227Table C.4 Energy-minimized alkali halide lattice energies: JC(TIP4PEW) model . . . . . . 229Table C.5 Energy-minimized NaCl lattice energies: SD model . . . . . . . . . . . . . . . 229xiiList of FiguresFigure 1.1 Visualization of classical nucleation theory . . . . . . . . . . . . . . . . . . . 5Figure 1.2 Overview of machine learning methods . . . . . . . . . . . . . . . . . . . . . 8Figure 2.1 Lennard-Jones interaction potential . . . . . . . . . . . . . . . . . . . . . . . 15Figure 2.2 Buckingham interaction potentials . . . . . . . . . . . . . . . . . . . . . . . . 17Figure 2.3 Visualization of error function and its complement . . . . . . . . . . . . . . . 23Figure 2.4 Force field augmentation for robust minimization . . . . . . . . . . . . . . . . 31Figure 3.1 Visualization of the Born-Haber cycle . . . . . . . . . . . . . . . . . . . . . . 41Figure 3.2 Diagram of lattice energy vs Born-Haber enthalpy . . . . . . . . . . . . . . . 45Figure 3.3 EL(Rocksalt)−EL(Wurtzite): LiX basis sets . . . . . . . . . . . . . . . . . . 56Figure 3.4 EL(Rocksalt)−EL(Wurtzite): uncorrected XC functionals . . . . . . . . . . . 58Figure 3.5 EL(Rocksalt)−EL(Wurtzite): D3/D4-corrected XC functionals . . . . . . . . 59Figure 3.6 EL(Rocksalt)−EL(Wurtzite): XC functionals with Sapporo basis sets . . . . . 60Figure 3.7 EL(Theory) - EL(Exp.): D3/D4-corrected XC functionals . . . . . . . . . . . . 61Figure 3.8 EL(Theory) - EL(Exp.): uncorrected XC functionals . . . . . . . . . . . . . . . 62Figure 3.9 EL(Theory) - EL(Exp.): XC functionals with Sapporo basis sets . . . . . . . . 63Figure 3.10 Error in a: D3/D4-corrected XC functionals . . . . . . . . . . . . . . . . . . . 64Figure 3.11 Error in a: uncorrected XC functionals . . . . . . . . . . . . . . . . . . . . . . 65Figure 3.12 Error in a: XC functionals with Sapporo basis sets . . . . . . . . . . . . . . . 66Figure 3.13 Dispersion energy for LiX: rocksalt vs wurtzite . . . . . . . . . . . . . . . . . 69Figure 3.14 Dispersion scaling for the JC(SPC/E) model . . . . . . . . . . . . . . . . . . . 70Figure 3.15 Dispersion scaling for the TF model . . . . . . . . . . . . . . . . . . . . . . . 71Figure 3.16 Differences in free energy contributions vs T : LiX rocksalt vs wurtzite . . . . . 78Figure 3.17 Differences in free energy contributions vs T : LiX rocksalt vs NiAs . . . . . . 79xiiiFigure 4.1 Architecture of a neural network structure classifier . . . . . . . . . . . . . . . 93Figure 4.2 Confusion matrix of a neural network structure classifier . . . . . . . . . . . . 96Figure 4.3 Pairwise correlation plots of Set A neural network inputs . . . . . . . . . . . . 97Figure 4.4 Pairwise correlation plots of Set N neural network inputs . . . . . . . . . . . . 98Figure 4.5 Pairwise correlation plots of Set V neural network inputs . . . . . . . . . . . . 99Figure 4.6 Comparison of four neural network classifiers . . . . . . . . . . . . . . . . . . 101Figure 4.7 Comparison of all neural network classifiers . . . . . . . . . . . . . . . . . . . 102Figure 4.8 Visualization of NaCl/JC(SPC/E) crystallization with a neural network classifier 104Figure 4.9 Visualization of LiBr/TF crystallization with a neural network classifier . . . . 106Figure 5.1 Overview of the melting point algorithm . . . . . . . . . . . . . . . . . . . . . 111Figure 5.2 An initial geometry for liquid/solid coexistence . . . . . . . . . . . . . . . . . 112Figure 5.3 Procedure for generating solid/liquid interfaces . . . . . . . . . . . . . . . . . 114Figure 5.4 Overview of the “melting point check” subroutine . . . . . . . . . . . . . . . . 116Figure 5.5 Overview of the melting point bracket reduction algorithm . . . . . . . . . . . 119Figure 5.6 Temperature trajectory for an example melting point calculation . . . . . . . . 120Figure 5.7 Demonstration of finite size effects on calculated melting points . . . . . . . . 123Figure 5.8 Temperature distributions of TUB, TUB, and indeterminate points: 11664 ions . 124Figure 5.9 Temperature distributions of TUB, TUB, and indeterminate points: 2000 ions . . 125Figure 5.10 Comparison of calculated melting points for alkali halide rigid-ion models . . . 128Figure 5.11 Molar enthalpy of LiBr/JC(SPC/E): rocksalt vs wurtzite . . . . . . . . . . . . 129Figure 6.1 Bayesian optimization 1D example . . . . . . . . . . . . . . . . . . . . . . . 134Figure 6.2 Overview of Bayesian optimization methodology . . . . . . . . . . . . . . . . 137Figure 6.3 Overview of the objective function used for fitting force field parameters . . . . 142Figure 6.4 Methodology for calculating molten LiX thermal properties . . . . . . . . . . 144Figure 6.5 Methodology for calculating solid LiX thermal properties . . . . . . . . . . . 145Figure 6.6 Example of feasible points within force field parameter space . . . . . . . . . 147Figure 6.7 LiX CLJ force fields optimized to T = 0 properties: target errors . . . . . . . . 151Figure 6.8 LiX CLJ force fields optimized to thermal properties: target errors . . . . . . . 153Figure 6.9 LiX CLJ fitted force fields: potential energy curves . . . . . . . . . . . . . . . 154Figure 6.10 LiX CLJ fitted force fields: comparison with a literature model . . . . . . . . . 155Figure 6.11 LiX CBH force fields (γi j > 7) optimized to T = 0 properties: target errors . . 157Figure 6.12 LiX CBH force fields (γi j > 7) optimized to thermal properties: target errors . . 158xivFigure 6.13 LiX CBH force fields (γi j < 6) optimized to T = 0 properties: target errors . . 160Figure 6.14 LiX CBH force fields (γi j < 6) optimized to multiple properties: target errors . 161Figure A.1 Rocksalt crystal structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191Figure A.2 Wurtzite crystal structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192Figure A.3 β -BeO crystal structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193Figure A.4 CsCl crystal structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194Figure A.5 5-5 crystal structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195Figure A.6 NiAs crystal structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196Figure A.7 Sphalerite crystal structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197Figure B.1 EL(Rocksalt) - EL(Wurtzite): D3(BJ)/gCP corrected XC functionals . . . . . . 205Figure B.2 Error in a: D3(BJ)/gCP corrected XC functionals . . . . . . . . . . . . . . . . 206Figure B.3 EL(Theory) - EL(Exp.): D3(BJ)/gCP corrected XC functionals . . . . . . . . . 207Figure B.4 Thermal corrections to experimental LiX lattice parameters . . . . . . . . . . . 208Figure B.5 Dispersion scaling for the JC(SPC/E) model: Li+-X− only . . . . . . . . . . . 212Figure B.6 Dispersion scaling for the TF model: Li+-X− only . . . . . . . . . . . . . . . 214Figure B.7 Dispersion scaling for the JC(SPC/E) model: JC X−-X− only . . . . . . . . . 215Figure B.8 Dispersion scaling for the TF model: X−-X− only . . . . . . . . . . . . . . . 216Figure B.9 Rocksalt and wurtzite potential energy curves: quantum vs classical models . . 217Figure B.10 EL(Rocksalt) - EL(Wurtzite): scalar relativistic effects on LiBr and LiI . . . . . 218Figure B.11 EL(Theory) - EL(Exp.): scalar relativistic effects on LiBr and LiI . . . . . . . . 219Figure B.12 Error in a: Scalar relativistic effects on LiBr and LiI . . . . . . . . . . . . . . 220Figure C.1 Relative crystal energies of alkali halides: TF model . . . . . . . . . . . . . . 222Figure C.2 Relative crystal energies of alkali halides: JC(SPC/E) model . . . . . . . . . . 224Figure C.3 Relative crystal energies of alkali halides: JC(TIP3P) model . . . . . . . . . . 226Figure C.4 Relative crystal energies of alkali halides: JC(TIP4PEW) model . . . . . . . . . 228Figure C.5 Relative crystal energies of NaCl: SD model . . . . . . . . . . . . . . . . . . 228Figure D.1 Probability distribution of TUB and TUB in the thermodynamic limit . . . . . . 233Figure D.2 Probability distribution of Tm in the thermodynamic limit . . . . . . . . . . . . 233Figure D.3 Melting point finite size effects: full data set . . . . . . . . . . . . . . . . . . . 234Figure E.1 Comparison of acquisition functions: objective function traces . . . . . . . . . 236Figure E.2 Comparison of acquisition functions: detailed final results . . . . . . . . . . . 237xvFigure E.3 Comparison of covariance functions: objective function traces . . . . . . . . . 238Figure E.4 Comparison of covariance functions: detailed final results . . . . . . . . . . . 239Figure E.5 Comparison of Mate´rn covariance functions: objective function traces . . . . . 240Figure E.6 Comparison of Mate´rn covariance functions: detailed final results . . . . . . . 241Figure E.7 Example distributions of calculated thermal properties . . . . . . . . . . . . . 242Figure E.8 LiX CLJ forcefields optimized to T = 0 rocksalt properties: target errors . . . . 243Figure E.9 LiX CLJ forcefields optimized with fractional charge: target errors . . . . . . . 244Figure E.10 LiX CLJ forcefields optimized without combining rules: target errors . . . . . 245xviList of Supplementary Material1. Video of LiBr/TF crystallization into the wurtzite structure, classified by the 10-1a convolu-tional neural network classifier introduced in Chapter 4.2. Video of NaCl/JC(SPC/E) crystallization into the rocksalt structure, classified by the 10-1aconvolutional neural network classifier introduced in Chapter 4.xviiList of AbbreviationsAIMD Ab Initio Molecular DynamicsARD Automatic Relevance DeterminationB3LYP Becke three-parameter exchange functional combined with the correlationfunctional of Lee, Yang, and ParrBH Buckingham, a functional form for the van der Waals part of pairwiseforce fields that can be employed in alkali halde simulationsBSIE Basis Set Incompleteness ErrorBSSE Basis Set Superposition ErrorCBH Coulomb Buckingham, a functional form for pairwise force fields that canbe employed in alkali halde simulationsCBHM Coulomb Born Huggins Meyer, a functional form for pairwise force fieldsoften employed in alkali halde simulationsCHA Chemical Hamiltonian ApproachCLJ Coulomb Lennard-Jones, a functional form for pairwise force fields oftenemployed in alkali halde simulationsCNA Common Neighbour AnalysisCNN Convolutional Neural NetworkCNT Classical Nucleation TheoryxviiiCSP Centrosymmetry ParameterD3(0) Third generation dispersion corrections with short-range damping to zeroD3(BJ) Third generation dispersion corrections with Becke-Johnson rationaldampingD4 Fourth generation dispersion corrections, includes Becke-Johnson rationaldampingDFT Density Functional TheoryDKH Douglas–Kroll–Hess theory, a relativistic transformation of quantumequations that can recover scalar relativistic effectsDKH3 Douglas–Kroll–Hess theory taken to third orderELU Exponential Linear Unit, an activation function commonly used in neuralnetworksFCC Face Centered CubicGAPW Gaussian Augmented Plane Waves, an all-electron quantum calculationmethod used in the CP2K softwareGGA Generalized Gradient approximation, a class of exchange-correlationfunctionalsGP Gaussian Process, a generalization of Gaussian distributions to functionsGPR Gaussian Process RegressionGPW Gaussian Plane Waves, an pseudopotential-based quantum calculationmethod used in the CP2K softwareGROMACS Groningen Machine for Chemical SimulationsHCP Hexagonal Closed PackedHF Hartree-Fock, a quantum chemistry methodxixHSE06 Exchange-correlation functional of Heyd, Scuseria, and Ernzerhof from2006HSEsol Exchange-correlation functional of Heyd, Scuseria, and Ernzerhof,reparamerized for solidsID Identification, a binary input into neural networksJC Joung and Cheatham, authors of a popular set of Coulomb Lennard-Jonesforce fields for alkali halide simulationsKS Kohn-Sham, a quantum chemistry methodLCB Lower Confidence Bound, an acquisition function used in BayesianoptimizationLDA Local density approximation, a class of exchange-correlation functionalsLSDA Local spin density approximation, a class of exchange-correlationfunctionalsLT-ABD Low Temperature Atomic Beam Deposition, an experimental techniqueLiX Lithium halides (X = {F, Cl, Br, I})M062X Reparamertized Minnesota 2006 hybrid exchange-correlation functionalwith twice the exact exchange contribution of M06-HFM06L Reparamertized Minnesota 2006 meta-GGA exchange-correlationfunctional for main-group chemistry and transition metal chemistryM06 Minnesota 2006 exchange-correlation functionalMD Molecular DynamicsML Machine LearningMP Melting PointMSM Minkowski Structure Metric, an order parameter used for crystal structureanalysisxxNN Neural NetworkNaN Not a Number, often used as a placeholder an unknown or undefinednumber in computingNPT Constant particle Number, Pressure, and Temperature. An ensemblechoice for molecular dynamics simulationsNVE Constant particle Number, Volume, and Energy. An ensemble choice formolecular dynamics simulationsNVT Constant particle Number, Volume, and Temperature. An ensemblechoice for molecular dynamics simulationsOVITO Open Visualization ToolPBC Periodic Boundary ConditionsPBEsol Perdew, Burke, and Ernzerhof, an exchange-correlation functionalreparamerized for solidsPBE Perdew, Burke, and Ernzerhof, a popular exchange-correlation functionalPDF Probability Distribution FunctionPW1PW A hybrid exchange correlation functional that combines Perdew andWang’s 1991 exchange functional with Perdew and Wang’s 1992correlation functionalPW Perdew and Wang’s 1991 exchange functional or Perdew and Wang’s1992 correlation functionalQHA Quasi-Harmonic Approximation, a method for approximating thermalcorrections to crystals in quantum chemistry calculationsReLU Rectified Linear Unit, an activation function commonly used in neuralnetworksSANN Solid-Angle Based Nearest-Neighbor, a near neighbour selectionalgorithmxxiSCAN Strongly Constrained and Appropriately Normed Semilocal DensityFunctional, a popular exchange-correlation functionalSCF Self-Consistent FieldSD Smith and Dang, authors of a force field for NaCl simulationsSG4 Semiclassical GGA at fourth order, an exchange-correlation functionalSPC/E Extended simple point charge model, a classical force field forsimulations of waterSPME Smooth Particle-Mesh Ewald, a method for efficiently calculating periodiclong range Coulombic interactions in molecular dynamics simulationsSVWN Slater exchange functional with the Vosko-Wilk-Nusair correlationfunctionalSapporo-DKH3-QZP See Sapporo-QZP, but optimized for third order Douglas–Kroll–HesstheorySapporo-DKH3-TZP See Sapporo-QZP, but triple-zeta quality and optimized for third orderDouglas–Kroll–Hess theorySapporo-QZP Quadruple-zeta quality contracted basis sets with polarization functionsfrom the Koga research group in Hokkaido University, Sapporo, JapanSapporo-TZP See Sapporo-QZP, but triple-zeta qualitySoftmax An activation function commonly used in neural networksTDSE Time-Dependent Schro¨dinger EquationTF Tosi and Fumi, authors of a set of Coulomb-Born-Huggins-Meyer forcefields for alkali halide simulationsTIP3P Transferable Intermolecular Potential with Three Points, a force field usedfor simulations of waterTIP4P/EW Transferable Intermolecular Potential with Four Points adjusted for Ewaldsummation, a force field used for simulations of waterxxiiTISE Time-Independent Schro¨dinger EquationTMTPSS Tao and Mo 2016 exchange functional with Tao, Perdew, Staroverov, andScuseria correlation functionalTM Tao and Mo, authors of a 2016 exchange functionalTPSS Tao, Perdew, Staroverov, and Scuseria, authors of a 2003 correlationfunctionalXC Exchange-Correlation, a functional used in Density Functional Theory ofwhich the exact form is not knownmeta-GGA Meta Generalized Gradient approximation, a class ofexchange-correlation functionalsoptB88 Optimized Becke-88 exchange-correlation functionalpob-TZVP The Peintinger, Oliveira, and Bredow Triple-ζ split Valence qualitysolid-state basis set with added Polarization functionsrVV10L Revised non-local dispersion functional of Vydrova and van Voorhis from2010, reparameterized for layered systemsrVV10m Revised non-local dispersion functional of Vydrova and van Voorhis from2010, reparameterized for solidsrVV10 Revised non-local dispersion functional of Vydrova and van Voorhis from2010rev-vdW-DF2 Revised van der Waal second generation Density FunctionalvdW Van der Waals, often referring to intermolecular forces. May refer to adispersion correction scheme of the same namevdW2 Second generation van der Waals, a dispersion correction schemexxiiiAcknowledgmentsI wish to thank my family and friends for their support throughout my PhD studies. In particular,my parents for their unconditional love and support. I also wish to thank NSERC for their generousscholarship, the Digital Research Alliance of Canada (formerly Compute Canada), Westgrid, andthe university of British Columbia for their facilities and financial support. More than anyone else,I wish to thank my advisor Gren Patey for his unending encouragement, sage advice, and selflessguidance throughout my PhD.xxivFor AnnaxxvChapter 1IntroductionThe imagination of nature is far, far greater than the imagination of man.— Richard P. Feynman (1955)The nature of scientific discovery can be conceptualized as an expanding sphere of knowledge,where each layer of the sphere necessarily builds upon the work of the previous. Rather thansimply narrowing the space of the unknown, scientific discoveries, while expanding our domain ofknowledge, also expand the horizon of our imaginations, opening up exciting new questions notpreviously conceivable. For example, there exists a vast set of chemically interesting questions thathas only expanded in scope as our collective knowledge of chemistry has developed. Many of thesequestions are extremely difficult to explore with currently available experimental methods. Oftenthe experiments needed to answer such questions can take years to design and properly execute. Ina subset of these, the experimental difficulty arises due to the tiny length or rapid time scales of thephenomena under investigation. Answering this subset of questions is the major focus of chemicalsimulation.Computer simulation of chemical systems is a field that has flourished in recent decades dueto a series of compounding factors. The theoretical framework and computational methodologycontinuously developed over the last hundred years or more, combined with the relatively recentrapid advancement in computational capacity, has allowed computer simulation of chemical sys-tems, and its associated analysis, to reach a new level of scientific possibility. We are entering anage where computational chemists can investigate chemically complex systems of unprecedentedscale1,2 with ever-increasing efficiency and accuracy3,4 in the unexplored domain where physicalexperiments are most difficult to perform.11.1 Lithium Halide Structural ChemistryThis research was originally motivated by recent efforts aimed at understanding the microscopicmechanisms by which crystals transition from pure liquids and solutions.5,6 Molecular dynamics(MD) simulations can provide detailed insight into crystal birth and growth on short time and smalllength scales that are not easily probed by experiment.7–11In order to carry out realistic simulations of crystal nucleation, it is essential to have reasonablyaccurate models of the system that, at the very least, predict the experimentally observed crystalstructures. Recent work has shown10 that this is not the case for common lithium halide LiX (X =F, Cl, Br, I) models. Lattice energy calculations at (T = 0, P = 0), together with MD simulationsat finite temperatures, demonstrated that two models commonly used to simulate alkali halides,the Tosi-Fumi12,13 (TF) and Joung Cheatham14 (JC) models, do not meet this minimum condi-tion for all LiX. The TF model incorrectly predicts that the hexagonal wurtzite structure is themost energetically stable crystal form for all four lithium halides. This is a serious problem be-cause experimentally all LiX have cubic rocksalt crystal structures, while wurtzite-like metastablestructures have been experimentally reported for LiCl, LiBr, and LiI.15–17 Similarly, the best per-forming JC LiX model predicts that wurtzite is more stable in the case of LiBr and LiI, althoughit correctly predicts the rocksalt crystal structure for LiF and LiCl. The developers of these forcefields likely did not realize that incorrect lower energy crystal structures existed for their lithiumhalide models. In any case, only the JC model for LiF and LiCl meets the most basic requirement(correct crystal structure) for nucleation studies. These otherwise carefully constructed LiX mod-els demonstrate the importance of explicitly considering multiple possible crystal structures whenconstructing force fields intended for use in solid state simulations.More recently, a polarizable model for the alkali halides was developed by Walz et al.18 Whilethis new force field corrects the problem of erroneous LiX crystal structures associated with theJC and TF models, the force field still substantially overestimates the melting points of lithiumhalides, despite the increased computational cost associated with explicit polarizability.19 Part ofthe investigation cataloged in this thesis is the reparameterization of simple non-polarizable LiXforce fields intended for use in large-scale MD simulation of solids and liquids. To avoid the samemistakes that plague the JC and TF models of LiX, the parameter fitting procedure employed in thiswork explicitly ensures that rocksalt is maintained as the correct thermodynamically stable crystalstructure at (T = 0, P = 0). In addition to targeting crystal properties the melting points of thelithium halides are also targeted during LiX force field parameter fitting, although this is acheivedindirectly through the fusion enthalpy and other high temperature thermodynamic properties. The2technique employed for parameter optimization, Bayesian optimization,20,21 is a sophisticated re-inforcement learning approach. Such advanced data-driven methods are gaining traction in themodel-building world22,23 as they promise a robust and more efficient framework for parameteroptimization. At the same time, supervised machine learning is also being applied to generatecomputationally efficient force fields outside of the traditional predefined functional forms.24Previous theoretical calculations of alkali halide crystal properties25–27 have proven extremelyhelpful as a starting point in this thesis. From these earlier works I obtained the seven plausibleLiX crystal structures considered in this thesis: rocksalt, wurtzite, NiAs, sphalerite, 5-5, β -BeO,and CsCl. AntiNiAs, which is the NiAs structure with anions and cations exchanged, is also con-sidered. Through an extensive theoretical study of the alkali halides over a range of pressures,Cˇancˇarevic´ et al.27 reported that at least one lithium halide salt is thermodynamically stable in oneof these solid phases within some range of pressures. Crystallographic details of the candidate LiXstructures are given in the Appendix, Section A.2. While the crystal structure classes of the lithiumhalide salts given by Cˇancˇarevic´ et al. have proven useful, the quantitative results for energies andlattice parameters are not as accurate as can be acheived through modern density functional theory(DFT) methods. As such, a considerable amount of this thesis is spent improving the accuracyof quantum mechanical LiX crystal calculations using state-of-the-art periodic density functionaltheory methods. In particular, dispersion corrections are shown to be extremely important in gov-erning the relative energies of LiX crystal structures. These new results extend the work of earliersolid state LiX quantum mechanical calculations.1.2 Phases and Classical Nucleation TheoryLithium halides, as with all physical matter, can exist in multiple phases corresponding to differentmicroscopic arrangements of atoms, structural symmetries, densities, components, and constraintson internal movement. One can conceptualize the relative stability of phases by imagining thefree energy of each phase as a continuous function of some appropriate thermodynamic variables(e.g. temperature and pressure). Each phase of matter varies with the chosen thermodynamic vari-ables differently, and the phase with lowest free energy for a given set of conditions constitutes thethermodynamically stable phase. By tracking the relative free energies of all conceivable phasesover a range of thermodynamic variables, one can imagine building up a free energy “landscape”of phases,26,28 wherein the free energies of all possible phases vary relative to each other. Whileany particular set of thermodynamic variables usually affords a single thermodynamically stable3phase,† the transition between phases does not proceed instantaneously once the thermodynami-cally stable phase shifts. Indeed, metastable phases can persist indefinitely as long as the molarfree energy barrier to phase transition is much larger than the average kinetic energy per mole.The process of crossing the free energy barrier to transition from a metastable phase to a stableone begins with nucleation, although the stable phase is not necessarily the phase that initiallynucleates (e.g. solid-solid phase transitions may evolve through a liquid-like intermediate). Properexperimental measurement of nucleation in crystal formation is challenging because crystal growthrates can vary substantially as the size of the nucleus increases, and only larger crystals can beeasily observed. The process of nucleation in many cases is approximately described by classicalnucleation theory (CNT).5 Essentially, as shown in Figure 1.1, CNT posits that the initial stages of aphase change such as crystallization can be viewed as a competition between two free energy terms.The first is a destabilizing term stemming from the surface tension at the border of the growingcluster, proportional to the crystallite surface area. While CNT does not necessarily require aspherical nucleus, one is generally assumed in practice for mathematical convenience. The secondterm is the internal free energy, which is stabilizing whenever a phase change is thermodynamicallyfavourable (i.e. whenever the nucleating phase is metastable). This stabilizing term is proportionalto the crystallite volume and only begins to dominate over the destabilizing surface term once thecrystallite reaches a critical size r∗. Under CNT, the free energy associated with the internal termis usually assumed equivalent to the corresponding bulk phase free energy difference. Therefore,any free energy shift in the growing cluster due to internal structures that differ from the bulk areneglected. CNT therefore predicts that the relative free energy of a spherical nucleus ∆G, underconstant pressure and temperature conditions, is given by∆G(r) = 4pir2σ +43pir3ρ∆g, (1.1)where r is the radius of a growing cluster, 4pir2 is its surface area, σ is the (intensive) surfacetension of the interface, 43pir3 is the cluster volume, ρ is the number density, and ∆g is the chemicalpotential difference between the stable and metastable phases. The critical nucleus size can bereached through random fluctuations in a homogeneous medium, as shown in Figure 1.1 but, morecommonly in nature, the relatively large barrier to homogeneous nucleation can be sidestepped viaheterogeneous nucleation. Under heterogeneous nucleation, the processes is catalyzed by reducingthe destabilizing surface area term, often by simply lowering the surface area to volume ratio of the†The exceptions, of course, are phase boundaries. These are the crossover points between phase stability withrespect to thermodynamic variables.4Figure 1.1: Relative free energy ∆G vs cluster radius r, including an energy decompositioninto surface and bulk terms. The inset is a visualization of an example nucleation pro-cess: crystalline NaCl nucleating from the surrounding melt (liquid atoms are partiallyhidden to emphasize the growing crystal). This visualization was generated from an MDtrajectory discussed in Chapter 4.cluster.CNT gives an attractively simple but often incomplete picture of nucleation. The motivationfor the research undertaken in this thesis partially originated from the desire to better understandthe nucleation mechanism of lithium halides salts from supersaturated aqueous solutions. CNTpredicts that nucleus formation is a single step process but, in the case of lithium halides, previouswork showed this may not to be the case.9,11 Strong ion hydration may impose a second freeenergy barrier to LiX nucleation, which constitutes a “non-classical” nucleation pathway worthexploring. The characteristics of this additional free energy barrier are still not well understood.Additionally, while most simple salts increase in solubility at higher temperature, high pressureaqueous solutions of LiF decrease in solubility above 475 K.29 The mechanism for this phenomenais currently unknown, but is likely related to the very strong ion hydration of the relatively smallLi+ and F− ions. A complete mechanistic description of nucleation beyond CNT would haveprofound implications on a wide range of fields. Such fields include: chemical engineering, where5nucleation is often undesirable; biochemistry, where protein crystallization is extremely importantfor X-ray characterization; and climate science, where water nucleation plays a critical role.However, as mentioned above, to confidently explore the mechanisms of LiX nucleation viaMD simulation, accurate LiX models are needed. The relatively large system sizes and time scalesrequired to simulate the phenomena likely preclude on-the-fly calculation of quantum mechanicalinteractions due to the high computational cost. Polarizable models can likely be used, but thesetoo may be unnecessarily computationally costly. As such, my research has focused on exploringand developing simple rigid-ion LiX force fields, as well as advanced computational methods to beused for exploring the open questions in LiX nucleation and solution chemistry.One aspect of particular difficulty in the analysis of molecular dynamics simulations is thecharacterization of local intermolecular structures (i.e. phases) as they evolve in time. In particu-lar, the liquid-like atoms and various solid structures that may self-assemble over the course of anucleation simulation are not trivial to accurately track. Typically, such intermolecular structure istracked via an order parameter that depends on localized intermolecular properties. Many such or-der parameters have been developed, some are intended for very specific applications while othersare designed for broader application in structural analyses of atomistic simulations. One of the old-est and most commonly employed order parameters is the bond orientational order parameters ofSteinhart et al.,30 which are discussed in detail in Chapter 4. Another early approach, the commonneighbor analysis (CNA),31,32 is an order parameter based on the graphs of local bond connectivitywhich extends earlier work by Blaisten-Farojas and Andersen.33 CNA has been employed to detectdefects in simple crystalline systems,34,35 while developments to CNA have improved the selectionof local bond networks.34,36,37 A related approach, the bond angle distribution method,38 generatesorder parameters constructed from the distribution of bond angles formed by pairs of neighborsaround a central atom. The centrosymmetry parameter (CSP),39 which measures deviations fromperfect centrosymmetry, is used for tracking defects in centrosymmetric crystal structures. A recentmodification to CSP, the minimum-weight matching CSP, ensures that CSP neighbor relationshipsare reciprocal.37 The common neighbor parameter40 combines aspects of both CNA and CSP,aiming to unite the generality of CNA with the simplicity of CSP. The topological cluster classifi-cation41 approach is used to classify the arrangements of atoms in clusters. More recently, severalorder parameters have been introduced based on local topological properties.42–46 A recent reviewof order parameters is available at Ref. 47.While the above order parameters can prove useful in roughly characterizing the boundaries ofbetween certain classes of structure, they can fail to accurately predict local structure at high tem-peratures where large molecular motions temporarily disrupt intermolecular structure. This can be6particularly troublesome for tracking high temperature simulations of lithium halide crystallizationfrom the melt. Furthermore, prior to the work reported in Chapter 4, no known order parameter wascapable of distinguishing between all candidate LiX structures discussed in Section 1.1. As such,a promising alternative to the order parameters given above, specific to binary salt simulations, ispresented in this thesis. My approach can be thought of as a meta-order parameter; it is based onconvolutional neural networks using multiple Steinhart order parameters30 as inputs. By averaginglocal order parameters locally in both space and time, it offers extremely high accuracy, particu-larly in high temperature situations where other order methods struggle. The methodology itself isgenerally applicable, and could be extended for use in simulations of other molecular systems.Melting points are a key property calculable from molecular simulation that strongly dependon the accuracy of force fields. While experimental melting points of chemical systems are readilyavailable, the theoretical calculation of melting points is not a trivial task.48 Previous work in cal-culating the melting points of alkali halides indicates that the simplest route to melting points is viadirect coexistence simulations,19,48,49 yet such calculations still involve many repeated molecularsimulations that are tedious to perform by hand. In this thesis, melting points calculations are usedto validate new force fields. To achieve this, I introduce robust methodology for automated meltingpoint calculations that employs the aforementioned convolutional neural network structure classi-fiers as an order parameter to tract the fraction of solid versus liquid in coexistence simulations.While proving reliable as a global order parameter at high temperatures, these classifiers are alsoused to ensure that the correct crystal structure is maintained in the solid, an novel aspect of meltingpoint calculations that improves robustness.1.3 Machine LearningMachine learning is a umbrella term for a broad set of statistical techniques for producing data-driven models, and it can be considered a form of artificial intelligence.50 Simply put, machinelearning is about making predictions based on data. Data can come from physical equations, sim-ulations, experiments, or any other source. In recent years, the combination of improved compu-tational resources, breakthroughs in theory, and greatly improved user accessibility has usheredin a renaissance of machine learning in many scientific disciplines, particularly in those prone togenerating large data sets. One such discipline is computational chemistry, but also the broaderfield of chemistry. Now, more than ever before, researchers are applying sophisticated data-drivenmethodology and algorithms to solve complex research problems.51,52Machine learning can be broken down into three broad categories (Figure 1.2). These are7Machine Learning Supervised Learning Unsupervised Learning Reinforcement Learning Classification Regression Clustering Dimensionality Reduction Bayesian Optimization Monte Carlo Methods Deep Q Neural Network Q-Learning State-Action-Reward-State-Action (SARSA) Proximal Policy Optimization Policy Gradient Temporal Difference K-Nearest Neighbours Regression Neural Network Regression Decision Tree Regression Support Vector Regression Random Forest Regression Locally Weighted Scatterplot Smoothing Multivariate Adaptive Regression Splines Neural Network Classification Random Forest Classification Support Vector Machines Decision Tree Classification Naïve Bayes Logistic Regression K-Nearest Neighbours Classification Gradient Boosted Trees Gaussian Mixture Models K-Means Density-Based Spatial Clus-tering of Applications with Noise (DBSCAN) Heirarchical Agglomerative Clustering Linear Discriminant Analysis Principle Component Analysis Locally Linear Embedding Isomap Embedding t-Distributed Stochastic Neighbour Embedding Uniform Manifold Approximaion and Projection Multidimensional Scaling Figure 1.2: An overview of some common machine learning methods, sorted by category.The methods highlighted in red, neural network classification and Bayesian optimiza-tion, are used in this research.supervised learning, unsupervised learning, and reinforcement learning. Each category has dozensof methods associated with it. I give a handful of examples in Figure 1.2, but this is by no meansan exhaustive list. New machine learning methodology is currently produced at a rapid rate, asthese methods have become ubiquitous in practical applications throughout science, engineering,and even day-to-day life.8Supervised learning is, broadly speaking, the use of labeled training data (i.e. training data inthe form of input/output pairs) to construct statistical models for classifying input data or predictingoutcomes. Neural networks, a sophisticated class of flexible machine learning methods defined bytheir layered interconnected structure, are often used in supervised learning tasks. In Chapter 4, Iapply neural networks in a supervised learning application for classifying the local phase of binarysalt ions in molecular dynamics trajectories.Unsupervised learning is a class of approaches defined by the use of unlabeled training data (i.e.input data given without corresponding outputs). The goal of unsupervised learning techniquesis, broadly speaking, to find patterns in data. One such use is to find clusters of data within ahigh-dimensional input space. Another application is a type of data compression, where a high-dimensional input space is optimally represented in a lower-dimensional space based on patternsof variation within a set of data.Reinforcement learning can be viewed as a form of function optimization or automated search-ing. In reinforcement learning, an objective function is defined and an algorithm is tasked withoptimizing it. The machine performs repeated objective function evaluations within an a prede-fined domain to update its internal state based on the output(s) of the objective function. The ideaof reinforcement learning is to use the limited set of data available to the machine to efficientlyprogress towards optimizing the objective function. Bayesian optimization, a technique applied inChapter 6, is a form of reinforcement learning. In my application, the predefined domain is a set ofparameters that characterize the shape of an interaction potential to be used in molecular simula-tion. The machine performs a complex and computationally expensive set of molecular simulationsfor set of each input parameters. The output is an objective function that quantifies the differencebetween the simulation outcomes (e.g. lattice energies, lattice parameters, enthalpies of fusion,etc.) and known experimental or theoretical values. In this way, reinforcement learning is used toefficiently fit a set of empirical parameters to optimally match experimental and theoretical data.1.4 OutlineThe remainder of this thesis is broken up into six chapters, plus additional appendices. In Chapter 2I describe the models, methods, and algorithms common to subsequent chapters. These are mostlyaspects of molecular dynamics simulations. Methods specific to each research project are discussedin the corresponding chapter introductions.In Chapter 3, I describe my efforts to obtain an accurate quantum mechanical description oflithium halide crystal structures with the goal of better understanding the structural chemistry of9lithium halides. A suite of classical, density functional theory, and Hartree-Fock calculations areemployed to explore the relative lattice energies of candidate lithium halide crystal structures. Icompare various basis sets and exchange-correlation functionals in an effort to reproduce the avail-able experimental data as accurately as possible. This research shows that the lithium halides haveuniquely complex structural chemistry among the alkali halides due to the close proximity of theircation/anion effective radius ratios to the tetrahedral-octahedral boundary. As such, the lithiumhalides have several competing crystal structures under ambient conditions. The octahedral rock-salt crystal structure, found experimentally for all LiX, is energetically favoured over the tetrahedralwurtzite structure in LiCl, LiBr, and LiI due to attractive anion-anion dispersion interactions thatraise the effective cation/anion radius ratios. As a result of this work, high quality reference datafor LiX crystal properties is generated, including absolute lattice energies, lattice parameters, andthe relative lattice energies between candidate crystal structures for all LiX. The data generated andlessons learned are employed throughout the remainder of my thesis research.For analyzing the detailed evolution of phase changes in MD simulations of lithium halides,it was necessary to develop an accurate order parameter to track the local intermolecular structureof ions at each step of the nucleation. In Chapter 4, I apply convolutional neural networks tothe problem of structure classification in binary salt (including LiX) MD simulations. The neuralnetworks are trained on millions of data points from multiple simulations of alkali halide salts in theeight candidate crystal structures, as well as the liquid phase. The classifiers are robust to thermalfluctuations in structure due to time convolution and a neighbour averaging technique applied tothe input data. I demonstrate the classifiers with two MD simulations of molten alkali halide saltscrystallizing into different crystal structures.In Chapter 5, I apply the neural network classifiers constructed in Chapter 4 to build an au-tomated melting point calculation algorithm for binary salts. Calculating melting points of modelchemicals using MD simulations is a difficult task, but melting points are an excellent measureof model quality in both the liquid and solid phase. The robust methodology developed in thischapter allows for the calculation of melting points for model alkali halides to arbitrary accuracyby simulating a series of liquid/solid interfaces that systematically and efficiently approach thethermodynamic melting point of the system. Since the neural network classifiers are capable ofdifferentiating between eight crystal structures and the liquid phase, the algorithm automaticallychecks that the solid remains as the crystal structure of interest. I carefully quantify the precisionof melting point calculations due to finite size effects, through statistical analysis of repeated calcu-lations at two system sizes. This analysis appears to be the first quantitative measurement of finitesize effects on the melting point distribution of model systems at fixed pressure. Apart from deter-10mining the precision of melting point estimates, these distributions are an interesting demonstrationof finite size thermodynamics, and show how the thermodynamic limit is approached. The melt-ing point algorithm is validated by calculating the melting points of several alkali halide modelscommonly employed for molecular dynamics simulations, comparing results with literature valueswhere available.In Chapter 6, I focus on parameterizing new lithium halide models for molecular dynamicssimulations with the aim of avoiding the issues present in the JC and TF LiX models. I fit themodels to a wide array of experimental data, but also to the LiX crystal reference data producedin Chapter 3 through quantum mechanical calculations. I cast the fitting of model parameters asa reinforcement learning problem, wherein molecular dynamics simulations are performed at eachiteration of a machine learning optimization method. Using the MD simulation results and targetdata, an objective function is constructed that quantifies how close a model constructed from inputparameters is to the set of targets. The fitting procedure is performing using Bayesian optimization.This method constructs a data-driven model of the objective function by interpolating betweenmeasured points, efficiently optimizing the objective function. Through these efforts, I show thatpairwise classical models of LiX that simultaneously fit important physical properties, such as thecrystal structure, lattice energies, and melting point, are possible. These models can be used toobtain more accurate molecular dynamics simulations of lithium halides than any other pairwiserigid-ion models currently available.I close in Chapter 7 with a summary of my thesis research in its totality, plus some additionalremarks including possible future directions for this thread of research.11Chapter 2Models and Methods2.1 OverviewA general theme of my research is the adaptation of computational and statistical methods to studythe structural chemistry of lithium halides. These include well established theoretical chemistrymethods such as density functional theory (Chapter 3) and molecular dynamics, as well as convo-lutional neural networks and multi-level optimization schemes based on reinforcement learning. Ihave also developed my own computational methods, models, and algorithms to help in this explo-ration. Due to the variety of methods applied in my research, a significant portion of the thesis willbe dedicated to the discussion of methodology. The methodology unique to each research projectis introduced in the corresponding chapters (3 to 6). Here, I discuss the methods common to allresearch projects, which are primarily aspects of MD simulation.MD simulation can be conceptualized as a computational microscope for examining the dy-namical and statistical behaviour of model molecular systems. Typically, atomic nuclei are treatedas point masses. The method requires some initial microstate be given (atomic positions and ve-locities) plus a method for calculating interaction energies and generating forces between atoms.The generation of interaction energies and forces are the most computationally demanding aspectsof any MD paradigm. In ab initio molecular dynamics (AIMD), forces are calculated directly fromquantum mechanical first principles, typically via DFT. This increases the accuracy and predic-tive power of MD, but limits the size and time scales accessible because quantum calculations arecomputationally expensive and typically scale unfavourably with system size. Classical molec-ular dynamics simplifies the calculation of interatomic forces by approximating the interactionsbetween atoms with an empirical force field. This approximation, although significant, greatly in-12creases the reach of MD, allowing for the simulation of large physical or biological systems suchas nanomaterials or proteins for up to microseconds of simulation time.In the remainder of the chapter, I provide specific details on aspects of MD simulations relevantto all research in this dissertation. First I discuss the specifics of time evolution in MD (Section 2.2),followed by interaction potentials (Section 2.3), then periodic boundary conditions (Section 2.4),tail corrections, and Ewald summation (Section 2.5). I then discuss the methods employed for tem-perature and pressure control of MD simulations (Section 2.6). Finally, I give some MD simulationparameters used throughout my research (Section 2.7) and the methodology employed to initializesystems (Section 2.8).2.2 Time Integration: Generating an MD TrajectoryComputationally, MD is nothing more than the numerical integration of a large set of coupled 3-dimensional (3D) classical equations of motion through time, using finite time steps. MD beginswith a set of initial conditions for all N interaction sites (typically atoms) within a system: {ri},{r˙i}, and {mi} which are the positions, velocities, and masses of all sites, respectively. For eachsite i at each time step, total forces fi are calculated from the sum of derivatives of interactions withall other sites,fi =−N∑j 6=i∇ui j(ri j) (2.1)where ui j is the interaction potential between site i and j, ri j is the vector that points from site ito j, and ∇ is the gradient operator with respect to position. If ui j are spherically symmetric, eachcomponent of the sum is oriented along the corresponding ri j. The sum is formally over all atomsin the system, but is usually truncated to include only atoms closer than some cutoff distance rc.Once the total force vector is known, the acceleration of site i (r¨i) can be calculated by Newton’ssecond law,r¨i = fi/mi. (2.2)Once the acceleration is calculated, velocities and positions can be updated over a small prede-termined time step. The time step must be shorter than the time scale of the fastest molecularmotions to reduce energy drift and numerical errors that can lead to the simulation undergoinghighly unrealistic dynamics. There are several methods (called integrators) available to update thepositions and velocities of atomic positions once the accelerations are known. In all MD simula-tions employed in this thesis, the leapfrog integrator is used.53 In leapfrog integration, positions13and velocities of each site i are updated at alternating half time steps (∆t/2) usingri(t+∆t) = ri(t)+ r˙i(t+∆t/2)∆t, (2.3)r˙i(t+∆t/2) = r˙i(t−∆t/2)+ r¨i(t)∆t, (2.4)where t is the current time step and r˙i is the velocity. The main advantage of the leapfrog methodfor MD simulation is its time symmetry, which implies proper energy conservation for sufficientlysmall time steps. If velocities are required at integer time steps, they can be calculated from themean of particle velocities at the half steps before and after each integer time step. Once positionsand velocities are updated to the next time point, the entire process of calculating forces and accel-erations then updating velocities and positions is iterated at the new time point. Over the course ofan MD simulation, the process is iterated many times to obtain a trajectory.2.3 Interaction PotentialsAll MD simulations in this research involve alkali halide salts modeled using three classes of pair-wise additive rigid-ion interaction potentials. The first is the Coulomb-Lennard-Jones (CLJ) po-tential uCLJi j , used in the JC14 and Smith Dang54 (SD) models, which takes the formuCLJi j (ri j) =14piε0qiq jri j+4εi j[(σi jri j)12−(σi jri j)6], (2.5)where ri j is the distance between atoms i and j, ε0 is the vacuum permittivity, qi is the chargeof atom i, εi j is the Lennard-Jones energy parameter between atoms i and j, and σi j is the sizeparameter. The first term of the CLJ model, proportional to r−1i j , is the Coulombic interactionbetween two point particles i and j. Because the Coulombic term is a long range interaction† witha relatively large prefactor, it accounts for the majority of the lattice energy in alkali halide systems.Coulombic energy is positive for cation-cation or anion-anion interactions and negative for cation-anion interactions. The remaining Lennard-Jones term, visualized in Figure 2.1, models electron-electron repulsion at short range through a term proportional to r−12i j plus the attractive Londondispersion interaction through a term proportional to r−6i j . The mathematical form of the dispersioninteraction is derived from the leading term in a multipole expansion of the interaction betweentwo spherical, polarizable quantum-mechanical harmonic oscillators,55 modeling the correlateddynamical movement of electrons in atoms. On the other hand, the r−12i j repulsive term form is not†Long range interactions decay algebraically with a power smaller than the spatial dimension.14Figure 2.1: The plot of a general Lennard-Jones interaction potential versus distance r withsize parameter σ and energy parameter ε .physically motivated, and was chosen for computational efficiency (r−12i j being the square of r−6i j ).Often, as with the JC models for alkali halides or SD model for NaCl, the energy and lengthparameters of CLJ models are only explicitly defined for like-like interactions. In the JC and SDmodels, metal-halide cross parameters are constructed via Lorentz-Berthelot combining rules,56,57σi j =σii+σ j j2, (2.6)εi j =√εiiε j j. (2.7)The JC models for alkali halides were designed for use in simulations of aqueous systems. Each al-kali halide has three sets of JC parameters for use with different water models: simple point chargeextended (SPC/E),58 transferable intermolecular potential with 3 points (TIP3P),59 and transfer-able intermolecular potential with 4 points for Ewald methods (TIP4PEW).60 The parameters of theJC potential were fit to solvation free energies, radial distribution functions, solvated ion-water in-teraction energies, rocksalt lattice energies, and rocksalt lattice parameters at 298 K. The SD modelparameters for NaCl were fit to experimental gas-phase binding enthalpies of ion-water clusters and15solvation enthalpies of ionic solutions.The second class of interaction potentials employed in this research are the Coulomb-Buckingham(CBH) potentials,61uCBHi j (ri j) =14piε0qiq jri j+Ai je−ri j/ρi j −Ci jr6i j. (2.8)In CBH models, an exponential function, which has some theoretical justification,61,62 serves asthe repulsive term. The Buckingham (BH) part of the CBH potential (uBHi j ) can be reformulated63in terms of a size parameter R0,i j, energy parameter εi j, and a unitless shape parameter γi j asuBHi j (ri j) = εi j(6γi j−6 exp(γi j(1− ri jR0,i j))− γi jγi j−6(R0,i jri j)6), (2.9)whereρi j =R0,i jγi j, (2.10)Ai j =6εi jγi j−6 exp(γi j) , (2.11)Ci j =εi jγi jR60,i jγi j−6 . (2.12)In this form, the Buckingham potential has three separate parameter domains with physically plau-sible forms, separated by an undefined critical point at γi j = 6 and a well-defined crossover point atγi j = 7. The first domain is γi j = (0,6) with εi j ≤ 0, the second domain is γi j = (6,7) with εi j ≥ 0,and the third domain is γi j = (7,∞) with εi j ≥ 0. A plot of the three BH potential domains and thecrossover point at γi j = 7 are given in Figure 2.2. Casting the Buckingham potential in the form ofEq. (2.9) has the advantage of separate length and energy scale parameters, which allows for thedefinition of cross-parameter combining rules analogous to those of the Lennard-Jones potentialR0,i j =R0,ii+R0, j j2, (2.13)εi j =√εiiε j j, (2.14)γ = γi j = γii = γ j j, (2.15)where the final combining rule ensures the validity of the other two. The value of γi j gives thesteepness of the repulsive wall and the corresponding narrowness of the interaction well; a larger γ16Figure 2.2: Plots of the Buckingham interaction potential over four ranges of γ versus dis-tance r with positive size parameter R0 and energy parameter ε . Line shades correspondto different values of γ . a) Darker blue corresponds to γ→ 6 from below. As γ→ 6, theinteraction minimum depth increases without bound; up to γ = 5.9 is shown. b) Darkershades correspond to γ → 6 from above. Once again, the interaction minima increaseswithout bound as γ → 6; γ ≥ 6.1 is visualized. c) At the γ = 7 crossover point, the re-pulsive wall and attractive well merge such that R∗ = R0 and ε∗ =−ε . d) As γ→∞, therepulsive wall height increases without bound; γ ≤ 8.5 is shown. R∗ and ε∗ correspondto the well position when γ < 7 and the peak position when γ > 7.corresponds to steeper walls and narrower wells.Since the exponential repulsive term of the Buckingham potential rises to a constant at ri j = 0,while the dispersion term falls without bound, both the Buckingham and Born-Huggins-Meyer po-tentials12,13 (discussed below) give an unphysical attractive interaction beyond the initial repulsivewall. In appropriately prepared MD simulations, this is not an issue as atoms do not typically over-come the repulsive barrier to sample the attractive interaction on the other side. In the Buckinghamparameter domain 0 < γi j < 6 with εi j ≤ 0 (Figure 2.2a), −εi j defines the height of the repulsivewall above zero and R0,i j gives the peak position. In this domain, the depth (ε∗i j) and position (R∗i j)of the attractive well do not have closed-form solutions, but depend on γi j, εi j, and R0,i j. In theparameter domain 6 < γi j < 7 with εi j ≥ 0 (Figure 2.2b), −εi j gives the repulsive wall value at its17peak and R0,i j gives its position. In this parameter domain, the repulsive wall drops below 0, withthe attractive minima placed lower, and −εi j defines the energy of the peak. When 7≤ γi j <∞ andε ≥ 0 (Figure 2.2d), R0,i j and −εi j instead define the position and depth of the interaction minima,while the height and position of the repulsive wall do not have closed-form solutions.The three distinct domains of γ for the BH potential all correspond to physically reasonablepotential forms. All correctly decay proportional to r−6 for large r, and all contain an exponentialrepulsive wall at close range. When Mason originally introduced the Eq. (2.9) form of the Buck-ingham potential,63 only the 7≤ γi j < ∞ and ε ≥ 0 domain was considered. In this γ domain, R0,i jand εi j give the position of the attractive well, which is a useful feature. However, since γ controlsthe repulsive wall steepness, 7≤ γi j < ∞ may be physically incorrect for systems characterized byrelatively soft close range repulsive interactions. Hence all three domains of γ should be consideredwhen parameterizing Buckingham potentials.The final class of interaction potentials used in my research are the Coulomb-Born-Huggins-Meyer (CBHM) potentials, often used for the alkali halides with a set of parameters collected byTosi and Fumi.12,13 The CBHM pairwise potentials are similar to the CBH potentials and take theformuCBHMi j (ri j) =14piε0qiq jri j+Ai je−ri j/ρi j −Ci jr6i j− Di jr8i j, (2.16)with parameters Ai j, Ci j, Di j, and ρi j needed for each pairwise interaction. CBHM potentials differfrom CBH potentials in that two dispersion terms are included, representing fluctuating dipole-dipole (∝ r−6i j ) and dipole-quadrupole (∝ r−8i j ) interactions. The r−8i j term adjusts the total potentialat closer range than the r−6i j term, and is generally parameterized to have a relatively small effect.CBHM have the drawback of increased complexity, as an additional set of Di j parameters arerequired. Several investigations have shown that r−8i j dispersion terms may be important to achievechemical accuracy in quantum-mechanical calculations, particularly where dispersion interactionsare the strongest intermolecular interaction.64,65 However, classical simulations are generally ofonly moderate accuracy compared to their quantum counterparts, and the loss of energy in CBHpotentials due to the missing r−8i j dispersion terms may be absorbed into a reparameterization ofthe r−6i j term without significant loss of accuracy.The TF parameterization of the CBHM model for alkali halides was fit to experimental solid-state properties. The repulsive parameters (Ai j and ρi j) were optimized to reproduce the vibrationalHildebrand equation of state for the solid,66dELdV=βT TαV, (2.17)18where EL is the lattice energy, βT is the isothermal compressibility and αV is the coefficient ofvolumetric expansion. The parameterization also fit to the volume derivative of Eq. (2.17) at fixedpressure, while assuming a rocksalt crystal structure. By fitting this equation of state, together withthe dispersion parameters, the TF model reproduces the experimental rocksalt lattice energies andlattice parameters for all alkali halides under ambient conditions. The dispersion parameters of theTF model (Ci j and Di j) were calculated by Meyer using ultraviolet absorption data of alkali halidesolids.67In the TF models, only one exponential parameter ρi j is used for all interactions of a given salt,while no other combining rules are employed. Instead, all other parameters are given explicitly.TF potential parameters are defined for all alkali halides that have rocksalt crystal structures underambient experimental conditions. However, ρi j parameters are missing for CsCl, CsBr, and CsIsalts, which experimentally occupy body-centered cubic crystal structures under ambient condi-tions, commonly called the CsCl structure (see the Appendix, Section A.2). For these three salts,all other force field parameters are given in the original TF papers. To employ TF-like CBHMinteraction potentials for these cesium salts, I estimated the parameters ρCsCl, ρCsBr, and ρCsI basedon the published value of ρCsF and the trends in rubidium halide ρi j parameters given in the originalTF papers. This was done by simply taking the numerical differences in ρi j between subsequentrubidium halides from RbF to RbI and adding these differences to the cesium halide ρi j, startingfrom the published value of ρCsF. Note that CsF experimentally occupies a rocksalt structure whilethe other cesium halides occupy a CsCl structure. I make no claims about the validity of these pa-rameter estimates; they are used only in my work for demonstration and statistical model training.Note that the resulting TF-like models for CsCl, CsBr, and CsI do not give the correct lowest energystructure at (T = 0, P = 0), favouring rocksalt over CsCl (rocksalt is also favoured for CsF withthe TF model). However, these interaction potentials yield metastable CsCl structures that closelymatch experimental melting points (see Chapter 5). All literature interaction potential parametersemployed in this work are reproduced in the Appendix, Section A.1, for reference.2.4 Periodic BoundariesAll MD simulations in this work employ periodic boundary conditions (PBC), which apply at theboundaries of simulation cells. PBC are an approximation to a bulk system by application of aninfinitely repeating simulation cell. Under PBC, an atom or molecule that passes through anyedge of the simulation cell reappears on the opposite side of the cell with its velocity maintained.PBC eliminate the outer system boundary, thereby removing any associated surface effects, but19do not eliminate all finite size effects. Such effects remain because the motion of periodic atomsthroughout the system are identical. Therefore, any collective system motions or structures thatmay arise at length scales larger than the central simulation cell are not included.Since, in principle, there are an infinite number of interactions between particle i and the pe-riodic copies of particle j, a method to truncate the infinite set of interactions is required. Theminimum image convention is common condition applied in MD simulations. This conventionmaintains that each individual interaction site in an MD simulation only interacts through short-range interactions with the nearest periodic image of any other site. In MD simulations of solidcrystals, one generally employs simulation cells that match the natural periodicity of the crystal,repeated to generate a suitable supercell. For example, a crystal with unit cell vectors a, b, c canbe replicated along each vector to produce a supercell of size A×B×C = la×mb×nc, where l,m, and n are integers. When working with general triclinic simulation cells, applying the minimumimage convention exactly can be complicated.68 Instead, an approximate method can be appliedwith a restriction to ensures its validity. The Cartesian coordinates of atoms in the simulation cellare transformed to fractional coordinates with respect to the supercell vectors (A, B, C) using theinverse of the basis transformation matrix Tˆ given byTˆ =A Bcosγ C cosβ0 Bsinγ C(cosα− cosβ cosγ)/sinγ0 0√C2−C2 cos2β −C2(cosα− cosβ cosγ)2/sin2 γ , (2.18)where A, B, and C are the magnitudes of the supercell vectors while α , β , and γ are the cell anglesbetween BC, AC, and AB, respectively. Fractional coordinates for each atom i, si, are calculatedbysi = Tˆ−1ri. (2.19)Once transformed into the fractional coordinate basis, atoms that pass through the boundary ofthe simulation cell during MD simulation are brought back into the cell by applying the modulo1 function to each component of their fractional coordinates. Similarly, the smallest possible iondistance between i and the periodic replicates of j can be easily computed bysi j = (si− s j) mod 1, (2.20)where the mod 1 operator is understood to apply to each component of the fractional coordinatevector si− s j independently. Once any fractional coordinate vector si is computed, it can be easily20transformed back into the corresponding Cartesian coordinate vector r viari = Tˆ si. (2.21)Use of this method to enforce the minimum image convention and PBC requires that a sphericalpotential cutoff is used which is not greater than half the smallest perpendicular width of the simu-lation cell.68 All simulations performed in this research are constructed to satisfy this condition. Asdiscussed in the proceeding section, the minimum image convention does not apply to long rangeinteractions such as the Coulomb force. Long range interactions are instead treated with Ewaldsummation.53,692.5 Tail Corrections and Ewald SummationA common method for approximating interactions in a periodically repeating simulation cell isto truncate the maximum interaction distance at some cutoff distance rc. All interactions withseparations further than rc are then excluded. rc can be tuned to trade accuracy for computationalcost. However, a systematic error in the total potential energy U total results if the interaction is notstrictly zero for distances greater than rc. To correct for the error in the total energy for sphericallysymmetric potentials, one simple approach is to compute an approximate tail correctionU total =∑i< jui j (ri j)+Nρ2∫ ∞rcu(r)4pir2dr, (2.22)where the first term is a sum over all interactions between atoms with separation distance ri j < rcand the second is the tail correction. Here, N is the number of atoms in the simulation cell, ρ is theaverage number density and u(r) is the average potential. For example, in a system with multipleatoms, the dispersion energy can be tail-corrected by defining u(r) = 〈C6〉r6 where〈C6〉= 2N(N−1)N∑iN∑j>iC6,i j. (2.23)C6,i j is the dispersion coefficient between atom i and j. This tail correction scheme relies on theassumption that the radial distribution function g(r)≈ 1 for r > rc and that the system is isotropic.Inspection of Eq. (2.22) suggests that this integral is only convergent if u(r) decays faster than afunction of inverse distance to a power that is larger than three, the number of spatial dimensions.Long range interactions are those that do not have a convergent tail correction, such as the Coulom-21bic interaction. Because they decay so slowly, truncation of Coulombic interactions leads to verylarge systemic errors for any reasonable rc, so these tail corrections cannot simply be ignored. Onesolution for periodic systems is to calculate the full Coulombic energy using Ewald summation.The total interaction energy of a 3D periodic system of point charges isE =14piε0∑n12N∑i=1N∑′j=1qiq j∣∣ri j +n∣∣ , (2.24)where n = n1a+n2b+n3c are vectors that point to the origins of periodically repeating unit cells(ni are integers), and the first sum is over the infinite set of periodic images. The factor of 12eliminates double counting and ′ indicates that the innermost sum does not include j = i if n = 0.It can be shown that this sum is conditionally convergent70 and, when summed convergently, thatit converges slowly. Ewald summation converts this conditionally convergent sum in real spaceinto two quickly and absolutely convergent sums, one in real space and one in reciprocal space.Conceptually, this is achieved by breaking up the Coulomb potential into a sum of two functions;one short range function, and one slowly varying function that extends to long range. The twofunctions of choice are the error functionerf(r) =2√pi∫ r0e−t2dt, (2.25)and its complementerfc(r) = 1− erf(r), (2.26)which can be used to express the r-dependence of the Coulomb potential as1r=1+ erf(κr)− erf(κr)r=erfc(κr)r+erf(κr)r, (2.27)where κ is a parameter that scales the domain of the error function and its complement as shownon the left side of Figure 2.3. Therefore, κ is chosen to set the relative strength of the short rangevs long range potential at the cutoff. With the separation given in Eq. (2.27), the total Coulombicenergy UCoul of a periodic system of point charges becomesUCoul =12N∑i(uSRi +uLRi)+N∑iuselfi (2.28)22Figure 2.3: (Left side) Plot of the error function and complementary error function as a func-tion of r. (Right side) How the error function and its complement can separate 1/r intoa short range and long range component.where the sums are over all N charges in the periodic box, uSRi is the short range real space contri-bution to the energy from charge i, uLRi is the long range reciprocal space contribution, and uselfi is aself-interaction correction necessarily introduced by application of the reciprocal space summation.These terms can be written as71uSRi =14piε0∑nN∑′jqiq j∣∣ri j +n∣∣ erfc(κ ∣∣ri j +n∣∣) , (2.29)uLRi =1Vε0 ∑k 6=0∑jqiq j|k|2 eik·ri j e−|k|24κ2 , (2.30)uselfi =−κ√piq2i4piε0. (2.31)Since the real space sum in Eq. (2.29) is now quickly converging, only a few terms in the sum overn need to be included. Likewise, the sum over reciprocal lattice vectors k in Eq. (2.30) convergesover a relatively small number of terms. As written, Eq. (2.28) implicitly assumes that the systemis surrounded by a perfect conducting boundary.72 If the periodic box has a dipole moment, andone wishes to surround the periodic system with a dielectric boundary (of dielectric constant εr) at23infinity, the an additional dipole term Udipole must be added to UCoul,Udipole =12ε01(2εr +1)V∣∣∣∣∣ N∑i qiri∣∣∣∣∣2. (2.32)Formally, Ewald summation is only applicable to neural systems. Straightforward application ofEq. (2.28) to systems with a net charge implicitly neutralizes the system by introducing a uniformbackground charge.73Ewald summation, as originally formulated, has rather unfavourable scaling with system size,O(N2) where N is the number of point charges. This scaling can be reduced to O(N log(N)) usingthe particle mesh Ewald technique,74 which applies fast Fourier transforms to speed up the recipro-cal space summation. All MD simulations presented in this work employ the smooth particle-MeshEwald (SPME) technique,75 which reformulates the reciprocal sum of particle mesh Ewald usingB-spline interpolation for further increased efficiency.2.6 Temperature and Pressure ControlGenerating a trajectory using the equations of motion described in Section 2.2 corresponds to aconstant particle number, volume, and energy (NVE) simulation. Such a situation is not typicallyencountered in experiments, as these are usually performed at constant temperature and pressure.Therefore, if one wishes to compare simulation with experiment, fixing the temperature and pres-sure of the simulation is much more practical. Maintaining temperature in simulations can beachieved by application of a thermostat, while pressure can be controlled by a barostat. I will nowintroduce and discuss the thermostat and barostats employed in this research.In MD simulations temperature (T ) can be calculated from the equipartition theorem,T =2EKkBF, (2.33)where EK is the total kinetic energy, kB is the Boltzmann constant, and F is the total number ofdegrees of freedom (3N− 6 for a system without constraints). Hence, to control the temperatureof a molecular system, one must regulate the kinetic energy of the particles. There are severalthermostats available that can do this, but not all of them produce energy fluctuations correspondingto the ensemble of choice. The canonical velocity rescaling thermostat76 was designed specificallyto produce the correct energy fluctuations and maintain ergodicity in MD simulations. It works byslowly and stochastically rescaling the velocities of all particles in the MD simulation to a target24kinetic energy sampled from the canonical ensemble distribution. At each time step, the totalkinetic energy EK is updated towards the target kinetic energy E ′K according toEK(t+∆t) = EK(t)+∆EK (2.34)∆EK =(E ′K−EK(t)) ∆tτT+2√E ′KEK(t)FdW√τT(2.35)where dW is a random Gaussian-distributed displacement with zero mean and unit variance, and τTis a time constant that controls the strength of the system-thermostat coupling. All MD simulationsin this thesis employ the canonical velocity rescaling thermostat to control temperature.The instantaneous pressure tensor (P) in a system with total volume V can be calculated fromthe difference between the total kinetic energy EK and the virial† Ξ due to intermolecular forces,P =2V(EK−Ξ) , (2.36)P =2V(EK +12N∑i=1N∑j>ifi j⊗ ri j), (2.37)where the sum is over outer products between the atomic forces (fi j = −∇ui j(ri j)) and relativeposition vectors ri j of all particle pairs. For an isotropic system, the scalar pressure P is computedas the average of the pressure tensor trace,P = tr(P)/3, (2.38)P =23EKV+13VN∑i=1N∑j>ifi j · ri j. (2.39)To control the pressure of an MD simulation, assuming fixed cell angles, one can rescale the sim-ulation cell vectors by a factor ηq (where q = {a,b,c}) at regular intervals of the time step nPwhile maintaining the fractional coordinates of all particles. For a target pressure P′, the Berendsenbarostat77 calculates ηq asηq(t) = 1− βqnP∆t3τP(P′−Pq(t)), (2.40)where ∆t is the time step, Pq(t) is the q-diagonal component of the pressure tensor at time t, βq isthe isothermal compressibility in direction q, and τP is a parameter that controls the strength of the†The expression for the virial comes from the virial theorem, 53 and is the contribution to the total pressure fromintermolecular forces.25system-barostat coupling. The ratio of βq/τP determines the rate of system relaxation towards thetarget pressure in the q direction. For an isotropic barostat, ηa = ηb = ηc. Although the Berendsenbarostat does not produce correct volume or pressure fluctuations,78 it is very efficient and robustfor relaxing systems to the target pressure. Hence, I employ it in many MD simulations for initialsystem relaxation.To achieve proper volume and pressure fluctuations characteristic of the isobaric-isoenthalpic(NPT) ensemble, the Parrinello-Rahman barostat can be employed.79 This barostat introducesequations of motion that evolve the simulation cell vectors in time such that the correct densityfluctuations are reproduced. To construct the Parrinello-Rahman barostat, begin by defining a ma-trix h of the simulation cell vectors,h =a1 a2 a3b1 b2 b3c1 c2 c3 . (2.41)The Hamiltonian H for the new equations of motion is then given byH = EK +Epot+12∑i, jMi j(dhi jdt)2+PV (2.42)where EK and Epot are the normal kinetic and potential energies of the system. The componentshi j are those given in Eq. (2.41), and Mi j are elements of a mass parameter matrix that determinesthe strength of coupling. The third term in Eq. (2.42) represents the kinetic energy of a fictitiouspiston acting on the simulation cell vectors, while the fourth term is its potential energy. Under theParrinello-Rahman barostat, particle fractional coordinates are maintained while the matrix of cellvectors evolves according tod2hdt2=V M−1σ (P−Pref) , (2.43)whereσ = (b× c,c×a,a×b) (2.44)is a matrix containing reciprocal lattice vectors. In GROMACS, the components of M−1 are givenby (M−1)i j =4pi2βi j3τ2pL, (2.45)where L is the largest matrix element of h, βi j are a matrix of compressibilities (some of which may26be set to zero), and τp is the coupling constant. In practice, the Parrinello-Rahman barostat can bedifficult to apply when a system is far from its target pressure. If the compressibilities are too large,the barostat is prone to producing large resonating volume fluctuations that can eventually causethe simulation cell to erroneously explode or implode. Hence, the Parrinello-Rahman barostat isemployed in my research only after initial relaxation using the Berendsen barostat.2.7 Molecular Dynamics SimulationsAll MD simulations presented and discussed in this research were performed using GROMACS(versions 2018 or 2019),80,81 employing the CLJ, CBHM, or CBH interaction potentials for alkalihalides described in Section 2.3. All simulations employ periodic boundary conditions and the fastSmooth Particle-Mesh Ewald (SPME) technique75 with interpolation order 4 (cubic) to account forlong range Coulombic interactions, adopting the implicit conducting boundary condition at infinitydiscussed in Section 2.5. The calculation of van der Waals (vdW) energies (i.e. the non-Coulombicterms of the JC and TF models) are performed only in direct space. The same cutoff radius rc isused for both the vdW interactions and for the real space part of the Ewald calculation. The vdWpotential is shifted up by its value at rc to maintain a continuous potential energy curve. Dispersiontail corrections are employed with the CLJ or CBH models, but not for CBHM models. No validdispersion tail correction is available with potentials of the CBHM form in GROMACS.† All alkalihalide MD simulations in this research employ a 1 fs time step unless otherwise indicated.2.7.1 Crystal MinimizationTo obtain geometry optimized crystals, a minimization scheme is employed. The details of thisscheme are given in Section 2.8.1. Crystal minimization calculations use rc = 2 nm and a Fouriergrid spacing of 0.1 nm. The κ parameter discussed in Section 2.5 is set to 10−7, which, along withthe large vdW cutoff and tight Fourier grid spacing, was necessary to ensure smoothness in thenumerical first derivative of the total potential with respect to changes in crystal lattice parameters.Potential energy smoothness is essential for geometry optimization via numerical gradient-basedapproaches. For crystal minimization, the minimum number of primitive unit cells are used toconstruct a supercell with all perpendicular walls are at least twice rc.†GROMACS employs a dispersion tail correction that assumes dispersion is given by a single interaction termproportional to r−6 while the CBHM potential also has a dispersion term proportional to r−8.272.7.2 Rapid Relaxation and Production SimulationsIn this work, all MD simulations (excluding energy minimization calculations) are performed withconstant particle number, pressure, and temperature. Two classes of NPT simulations are em-ployed: relaxation and production simulations. The relaxation simulations are designed to effi-ciently thermalize the system while bringing it rapidly to its natural volume at a given temperatureand pressure. Relaxation simulations can also be used to compute equilibrium properties, but theydo not achieve proper equilibrium fluctuations. These simulations employ the Berendsen baro-stat77 along with the canonical velocity-rescale thermostat.76 Both the thermostat and barostat arecoupled very strongly to the system, using 1 fs time constants. Relaxation simulations employ a10−7 bar−1 compressibility which is a trade off between the speed of relaxation and the amplitudeof system volume fluctuations.Production simulations are used primarily in the simulation of liquid-solid coexistence, crystalnucleation, and the calculation of diffusion constants. Production simulations are initiated from thefinal microstate of a relaxation simulation. They employ the Parrinello-Rahman barostat79 alongwith the canonical velocity-rescale thermostat,76 using barostat and thermostat coupling constantsof τP = 1 ps and τT = 0.2 ps, respectively. Production simulations of alkali halides employ barostatcompressibilities matching the inverse experimental bulk moduli of the solid alkali halides.82 OtherMD parameters were set specific to the type of research performed, and these are discussed in themethods sections of Chapter 3 to Chapter 6.2.8 System Initiation and MinimizationThroughout the research presented in this dissertation, initial positions and velocities of alkalihalide salt systems (in particular, lithium halides) are generated for MD simulation. These ini-tial microstates correspond to either solid alkali halide crystals or molten salts, typically at somespecified temperature. In all cases, initial particle velocities are generated by randomly samplinga Maxwell-Boltzmann velocity distribution at the specified initial temperature. The generation ofinitial geometries is more involved, and described separately for solid crystals and molten salts.2.8.1 Solid CrystalsAll solid alkali halide crystals used in this work begin with the generation of a periodic crystalstructure based on a unit cell. The unit cells of all solids explored in this work are given in theAppendix, Section A.2. The lattice parameters of unit cells corresponding to the experimentalcrystal structure (rocksalt for all alkali halides except CsCl, CsBr, and CsI) are taken from experi-28ment.83 For other unit cells, the initial lattice parameters (a, b, and c) are estimated by setting thebond length to that of the experimental structure, while maintaining the symmetry of the ideal unitcell. For MD simulations, the unit cell is replicated in all directions to generate a supercell thatis at least twice as large as the interaction potential cutoff in all directions, ensuring the minimumimage convention is satisfied.Once the crystal supercell is initiated, its lattice energy is minimized. For optimization ofcrystals using classical interaction potentials, I constructed a software interface to connect GRO-MACS energy calculations to the MATLAB optimization toolbox.84 Lattice parameters were en-ergy minimized using the gradient-based nonlinear optimization function fmincon, where eachenergy is computed by a GROMACS single-point (static) calculation and returned to the optimizer.The “active set” algorithm is employed within fmincon.85 Numerical gradients of crystal energywith respect to lattice parameters are computed by two-point finite differences. After optimiza-tion of lattice parameters, if desired, fractional coordinates are optimized by direct application ofthe GROMACS energy minimization subroutine with fixed volume, using settings as described inSection 2.7. This two-step optimization scheme, if enabled, is iterated until both lattice parametersand fractional coordinates are converged between subsequent iterations. Convergence in latticeparameter optimizations is obtained once the maximum absolute value of any numerical gradientcomponent is less than 10−3 kJmol−1 nm−1 or the change in energy between optimization steps isless than 10−6 kJmol−1. The intervening GROMACS energy minimization stages were consideredconverged when the maximum force on any atom was less than 10−3 kJmol−1 nm−1. Final conver-gence between the alternating minimization stages was reached when the energy change betweeniterations was less than 10−6 kJmol−1. The geometry optimization of alkali halide crystals forquantum-mechanical calculations, employed in Chapter 3, is discussed separately in Section 3.2.5.After energy minimization of the solid crystal structure is complete, if an MD simulation isto be performed, the system is relaxed to the temperature and pressure of interest prior to theproduction simulation. This is achieved by performing a rapid relaxation simulation, using the MDparameters described in Section 2.7.2.2.8.2 Molten SystemsIn this research, molten systems are always generated after the energy-minimized solid crystalstructure is already known. The optimized solid provides an estimate of the liquid density prior toconstruction of the system. I use an initial liquid density that is 80 % of the minimized rocksalt soliddensity. An empty simulation cell is first generated, usually cubic in shape, although it may also29be constructed to match the a and b lattice vectors of a solid unit cell. The empty simulation cell isusually constructed such that the minimum distance across the cell is at least twice the interactionpotential cutoff distance, plus a 20 % buffer in case the volume decreases. For example, if theinteraction potential cutoff is 1.4 nm, then the minimum simulation cell is 1.68×1.68×1.68 nm or4.74 nm3. Once the empty simulation cell is generated, it is randomly filled with an equal numberof cations and anions using the GROMACS insert-molecules subroutine, such that the finalnumber of ions gives the required density. The insert-molecules subroutine avoids placingions in very close proximity. After the ions are randomly placed into the simulation cell, the systemis energy minimized at fixed volume using the GROMACS native energy minimization subroutine.This energy minimization proceeds for only 1000 iterations to avoid over minimizing the liquidinto a solid-like structure. After minimization, the liquid is relaxed to the desired temperature andpressure using a short MD simulation with the relaxation parameters described in Section 2.7.2.2.8.3 Robust MinimizationIf an interaction potential with a divergence at small separation distance is employed (such as theTF model for alkali halides), there is a possibility that at least one pair of ions will be placed tooclose together prior to the minimization stage of the solid or the liquid, leading to overlappingatoms if normal energy minimization proceeds. To prevent this, all energy minimization stagesfor classical models automatically augment the interaction potentials to remove any divergenceat close separation. As demonstrated in Figure 2.4, this is achieved by numerically solving forthe inflection point rinfl of any interaction potential that occurs prior to a repulsive wall peak. Atseparation distances closer than this inflection point, all attractive non-Coulombic (i.e. dispersion)and repulsive terms are set to their value at the inflection point, and an additional repulsive termui j,aug is added,ui j,aug = Baug(1r12i j− 1r12infl), (2.46)where Baug is a proportionality constant,Baug =−dui j(rinfl)drr13infl12, (2.47)that insures the potential derivative is continuous at rinfl. The exact functional form of ui j,aug is notparticularly important, as long as the potential and its derivative are continuous, and the repulsivewall is strictly increasing beyond rinfl. The second term of Eq. (2.46) simply shifts ui j,aug to zero at30Figure 2.4: Demonstration of force field augmentation to prevent divergence issues at closeseparation distance ri j. Plotted in black is the total pairwise interaction potential be-tween Na+ - Cl+ for the TF model, ui j, which produces an unphysical attractive forcebeyond the repulsive wall peak. The red circle is the numerical inflection point of therepulsive wall peak. The dashed red line is the augmented interaction potential, whichsmoothly connects to the original potential, augmenting it only beyond the inflectionpoint without a discontinuity in the potential or its derivative.rinfl. Note that the augmented force fields are only used during minimization, and are not employedin the relaxation or production MD simulations.31Chapter 3Analysis of the Relative Stability ofLithium Halide Crystal Structures:Density Functional Theory andClassical ModelsThe research presented in this chapter has been published in the Journal of Chemical Physics withcoauthor G. N. Patey.863.1 Introduction3.1.1 MotivationDue to the limitations of current rigid-ion classical models for LiX salts (Section 1.1), better effec-tive potentials are necessary to explore their crystal nucleation from melts or aqueous solutions. Inthis chapter, I use a series of Hartree-Fock (HF) and DFT calculations to probe the landscape ofLiX crystal structures. By exploring the energy and free energy landscapes of these salts, I con-struct reliable reference states for model building, tease apart the delicate factors that determinecrystal structure stability, and explore regions of geometrical space that currently evade the reachof experiment. One important finding from this project is that dispersion interactions have an im-portant role in determining the relative stability of different possible crystal structures for LiX. Thisinsight suggests a simple modification of both the TF and JC forces fields, such that at least the32correct rocksalt structure can be reached.The quantum mechanical study of simple ionic salts has a history that reaches back to the 1950sand earlier,87–97 where early pioneers struggled to contend with the complexities of HF calculationsapplied in periodic systems. These early works modeled lithium halides with a small quantumsystem embedded into a lattice of point charges. More recently, computational and theoreticaladvances have allowed researchers to quantum-mechanically simulate bulk phase lithium halidesunder periodic boundary conditions.26,27,98,99 In the first of these studies98 the authors examinedlighter alkali halides at the HF level, assuming rocksalt crystal geometries. They found reasonableagreement with experimental observables from pure HF results, but noted that better agreementcould be reached with pairwise additive empirical dispersion corrections. The previous work mostrelevant to this chapter comes from Cˇancˇarevic´ et al.,27 where they examined a wide range ofpossible structure candidates for all alkali halides at several levels of quantum mechanical theory(DFT and HF). Their calculations employed double-ζ quality basis sets for lithium, and eithertriple-ζ all-electron basis sets (F and Cl) or effective core potentials plus double-ζ quality basissets (Br and I) for the halides. This chapter extends the work of Cˇancˇarevic´ et al., employing largerbasis sets and more modern exchange-correlation (XC) functionals. Additionally, Cˇancˇarevic´ etal. did not examine the effect of dispersion corrections on the relative energies of LiX crystalstructures, which is a central focus of this chapter. In the next section, the equations of ground stateDFT, which the chapter depends heavily on, are reviewed and discussed.3.1.2 Quantum ChemistryNewtonian physics, at the heart of classical MD simulations, approximates the interactions betweenatoms and molecules via application of an effective interaction potential, the form of which istypically fitted to some combination of experimental data and quantum-mechanical calculations.Classical simulation is used to model physical systems due to its relatively low computationalcost, and often works well enough for modeling properties of bulk systems. In particular, classicalsimulation works well for heavy atoms or molecules at temperatures approaching the classical limit(around 300 K), where quantum effects are less apparent. However, the accuracy of predictionsfrom classical simulation hinges on the accuracy of the underlying interaction potential, limitingit’s predictive power. In cases where classical simulation fails to recover known experimentalor theoretical results, quantum chemistry methods can be applied to either improve the classicalinteraction potentials or to directly simulate interactions of the chemical system.100 The primarydownside of applied quantum calculations is their high computational cost compared with classical33calculations.The mathematical machinery for quantum chemistry calculations derives from the time-dependentSchro¨dinger equation (TDSE),101 from which the time-independent version follows by applicationof a time-independent Hamiltonian Ĥ,−}22m∇2ψ(r)+V (r)ψ(r)︸ ︷︷ ︸Ĥψ(r)= Eψ(r). (3.1)Here, r is a position vector of a quantum particle, } is the reduced Planck constant, ∇2 is theLaplacian, ψ(r) is the particle wavefunction (a solution to the equation) and E is the total energy.Note that Eq. (3.1) is already an approximation to the exact answer even for time-independent Ĥ, asit neglects relativistic effects. Such effects become important in the core electrons of heavy atoms,where the expectation value of the electron’s speed approaches the speed of light.The Born-Oppenheimer ApproximationAn “exact” non-relativistic quantum-mechanical approach to simulating a chemical system in-volves treating both the atomic nuclei and electrons as quantum particles, then solving the completeTDSE for that system. However, such an approach is not usually pursued due to the high compu-tational complexity. Instead, one can make use of the large mass of nuclei relative to electrons.Since particle speed is inversely proportional to the square root of mass, electrons move quicklyrelative to the much heavier atomic nuclei such that they can be treated as continuously occupyingstationary states given parametrically by nuclear positions. The Born-Oppenheimer approximationis a formalization of this idea.102The Born-Oppenheimer approximation begins formally from the time-independent Schro¨dingerequation (TISE) for a chemical system of M atomic nuclei and N electrons. To simplify the equa-tions moving forward, Hartree atomic units are used such that the elementary charge e, electronmass me, reduced Planck constant }, and Coulomb constant ke are all chosen to be unity. Withthese units, the Hamiltonian of the chemical system can be writtenĤ =−12N∑i=1∇2i −12M∑A=11MA∇2A+N∑i=1N∑j>i1ri j+M∑A=1M∑B>AZAZBrAB−N∑i=1M∑A=1ZAriA, (3.2)where lowercase subscripts and summation indices refer to electrons while corresponding upper-case symbols refer to nuclei. The first term of Eq. (3.2) is the kinetic energy operator of the34electrons, the second is the kinetic energy operator of the nuclei, the third is the electron-electroninteraction (given by Coulomb’s law), the fourth is the nuclear-nuclear interaction, and the fifth isthe electron-nuclear interaction. MA refers to the mass of atomic nucleus A, ZA is its charge, ri j isthe distance between electron i and j, rAB is the distance between nuclei A and B, and riA is the dis-tance between electron i and nucleus A. After application of the Born-Oppenheimer approximation,the TISE for the arbitrary chemical system is reduced to((− 12N∑i=1∇2i +N∑i=1N∑j>i1ri j−N∑i=1M∑A=1ZAriA)︸ ︷︷ ︸ĤelecΨelec)Ψnucl+( M∑A=1M∑B>AZAZBrAB)ΨelecΨnucl− 12M∑A=11MA(∇2AΨnucl)Ψelec = EΨelecΨnucl,(3.3)where the electronic (Ψelec) and nuclear (Ψnucl) wavefunctions are treated as separable. The firstterm is known as the electronic problem, with corresponding electronic Hamiltonian Ĥelec, andcan be solved separately with the nuclear degrees of freedom treated parametrically to yield theelectronic energy εelec. Formally, the nuclear problem is solved after the electronic subproblem by(− 12M∑A=11MA∇2A+M∑A=1M∑B>AZAZBrAB+ εelec︸ ︷︷ ︸Potential energy landscape for nuclear motion)Ψnucl = EΨnucl. (3.4)However, for many chemical properties of interest, the application of quantum mechanics to atomicnuclei is unnecessary to achieve sufficient accuracy. Instead, the nuclei can be treated classicallyas point masses in such calculations. The electronic problem, on the other hand, must be solvedquantum-mechanically to achieve even reasonable accuracy for most chemical properties. Solvingthe electronic problem cannot be done analytically in all but the simplest of cases, so in practicethe electronic subproblem is solved by numerical methods such as DFT, the HF theory, or a moresophisticated post-HF method.103 In this research project, I applied both the HF method and DFTto solve electronic problems for bulk lithium halide systems.35Density Functional Theory: MotivationDFT is a formally exact reformulation of quantum mechanics with the aim of reducing the com-putational complexity involved in solving the electronic problem for the many-body electronicwavefunction. The method starts with the electronic problem,ĤelecΨelec(rN ;{RM})= εelecΨelec (rN ;{RM}) . (3.5)The above notation indicates that the electronic wavefunction depends explicitly on the N electronicpositions (rN) but only parametrically on the M nuclear positions (RM). The electronic HamiltonianisĤelec =−12N∑i=1∇2i +N∑i=1N∑j>i1ri j−N∑i=1( M∑A=1ZAriA)︸ ︷︷ ︸V (ri), (3.6)where V (ri) is the external potential on electron i due to interactions with all nuclei. The premiseof DFT is that one can work with the 3-dimensional electron density, ρ(r),† rather than the 3N-dimensional many-electron wavefunction Ψelec(rN ;{RM}). In doing so, the computational com-plexity of the electronic problem can be greatly reduced. Although originally proposed by Thomasand Fermi,104,105 DFT became theoretically important after the premise was formally proven exactfor non-degenerate ground state systems by the existence theorems of Hohenberg and Kohn.106The first Hohenberg and Kohn theorem proves that, up to an additive constant, there is a one-to-one correspondence between the external potential of a chemical system (i.e. the positions ofatomic nuclei) and the electron density of that chemical system. This unique electron density isalso associated with a unique Hamiltonian, which uniquely determines the ground state energyand wavefunction. This first theorem was later extended to apply to degenerate systems,107 time-dependent (excited) states,108 and systems in strong magnetic fields.109 The second Hohenberg andKohn theorem proves that the global minimum of the universal functional for the total electronicenergy (E[ρ(r)], given by the first theorem) corresponds to the true ground state energy, and itsinput corresponds to the true ground state electron density. Together, these two theorems are suf-ficient to show that the external potential V (r) is uniquely determined by the ground state electrondensity ρ(r), meaning it is possible in principle to recast quantum mechanics in terms of the elec-tron density ρ(r) rather than the all-electron wavefunction. This reduces the electronic problem†For systems with unpaired electrons, one must instead work with a 6-dimensional spin density.36from 3N to 3 variables (or 6 variables if up and down electron spin densities must be separatelyconsidered).Density Functional Theory: The Kohn-Sham MethodThe Kohn-Sham (KS) method110 is a technique for solving the ground electronic state of a chemicalsystem that employs DFT. The KS method is by far the most widely used DFT methodology forsolving the electronic problem. In KS DFT, the electronic problem is recast into a single-electronproblem, where each electron experiences an effective potential Veff(ri) from all other electronsand nuclei that is corrected for many-body and dynamical effects. This is in contrast to numericalwavefunction methods, such as the Hartree-Fock method,111–114 that solve the electronic problemby approximating the wavefunction directly. The KS method begins by imagining a fictitiousreference system of non-interacting electrons,Ĥeff =N∑i=1(−12∇2i +Veff(ri)), (3.7)Ĥeff =N∑i=1hˆ(ri), (3.8)where hˆ(ri) is the one-electron energy operator for the fictitious system. This leads to a verysimplified one-electron eigenvalue problem,110hˆ(ri)ψi(ri) = εiψi(ri), (3.9)where ψi(ri) are one-electron wavefunctions and εi are their associated energies. The key is tochoose Veff(ri), the effective potential of the non-interacting fictitious system, such that the electrondensity is equal to that of the true fully-interacting system of interest. To enforce the Pauli exclusionprinciple, the total Kohn–Sham wavefunction is then constructed from a Slater determinant of one-electron wavefunctions ψi(ri) that are solutions to Eq. (3.9). The mean-field effective potentialcreated by the nuclei and other electrons acting dynamically on the electron of interest, Veff(ri),can be expressed as a sum of three termsVeff(r) =Vext(r)+∫ ρ(r′)|r− r′|dr′+VXC[ρ(r)]. (3.10)37Here, Vext(r) is the external potential due to the electron-nuclei interactions, the second term is themean-field Coulombic electron-electron repulsion, and the final term VXC[ρ(r)] = δExc[ρ(r)]δρ(r) is theexchange-correlation potential (Exc is the total exchange-correlation energy). The electron densityis given byρ(r) =N∑i=1ψ∗i (r)ψi(r), (3.11)where the one-electron orbitals are given by solutions to Eq. (3.9).The XC potential contains corrections on the effective potential due to electron exchange anti-symmetry (the Pauli exclusion principle) as well as corrections due to dynamical electron-electroninteractions (correlations) that are not properly captured by the mean-field Coulombic term, suchas London dispersion. While DFT is formally exact, the exact form of the XC functional is notknown. However, many approximations to the XC term exist in a “ladder” of increasing com-plexity referred to as Jacob’s ladder of density functional approximations.115 Some modern XCfunctionals on the higher rungs of Jacob’s ladder are very accurate over a wide range of chemicalsystems, but come at a high computational cost. Part of the “art” of DFT calculations is finding thebest choice of XC functional for the chemical system of interest.1163.1.3 Overview of ResearchIn the remainder of this chapter, I report the results of periodic DFT calculations, using a rangeof XC functionals, that give the relative energies of seven lithium halide candidate structures. Ifind that the TMTPSS-rVV10 method,117,118 combined with a quadruple-ζ Gaussian basis set foranions and a triple-ζ basis set for lithium, gives the most accurate rocksalt lattice energies and lat-tice parameters, compared with experiment. I use this method, and others, to calculate the relativeenergies of metastable lithium halide crystal structures to high accuracy. Using both DFT and clas-sical models, I show that dispersion interactions play a central role in determining the low-energycrystal structure of lithium halides. When the dispersion interactions are explicitly modeled inDFT calculations, rocksalt is favoured over wurtzite for all lithium halides. Using dispersion-freeHF calculations and geometric arguments I show that, if not for the dispersion interaction betweenanions in the lithium halides, the hexagonal wurtzite crystal structure would likely be more ther-modynamically stable than rocksalt for LiCl, LiBr, and LiI. Notably, I also find that my highestaccuracy calculations give the NiAs crystal structure of LiI very close in lattice energy to rocksalt(|∆EL|< 1 kJmol−1). This finding calls into question why the NiAs crystal structure of LiI has not38yet been observed experimentally. Finally, I show that pairwise classical models for LiX can bemade to favour rocksalt simply by scaling up the dispersion interaction term of the force field, atthe cost of a slightly increased error in the rocksalt lattice parameter and lattice energy.The rest of this paper is divided into three parts. In Section 3.2 I discuss the methods employed,beginning with the recalculation of experimental lattice energies, followed by additional crystalstructure candidates, then quantum mechanical calculations, classical calculations, crystal structureoptimization, and finally thermal corrections to DFT calculations. Section 3.3 is a discussion ofmy results divided into five subsections: basis set errors in quantum mechanical calculations; theimportance of the level of theory in calculations; the role of dispersion in LiX crystal structuredetermination; a general discussion of the stability of alternative LiX crystal structures; and lastlya discussion of thermal and entropic contributions to crystal free energies. Section 3.4 gives asummary of my primary results and conclusions for this chapter.3.2 Methods3.2.1 Recalculation of Experimental Lattice EnergiesThe idea of lattice energy (EL) as originally presented by Born119 is a semi-classical concept, con-ceived during the infancy of quantum mechanics. Lattice energy was invented as a tool to makestability comparisons between salts. The precise conventional interpretation of lattice energy issomewhat ambiguous as it neglects the concept of zero-point vibrational energy. An educationalarticle by Jenkins120 discusses the calculation of experimental lattice energies from the correspond-ing Born-Haber cycle enthalpies. In this thesis, I use the following definition of lattice energyEL ≡UFixed, Crys(0 K,0 bar)−UIons(0 K,0 bar), (3.12)where UFixed, Crys(0 K,0 bar) refers to the potential energy of a hypothetical crystal with its ionsfixed exactly on their lattice sites and UIons(0 K,0 bar) refers to the potential energy of infinitely sep-arated stationary ions. This definition is computationally convenient because UFixed, Crys(0 K,0 bar)and UIons(0 K,0 bar) are both easily calculable from electronic structure methods that use theBorn-Oppenheimer approximation. It is more computationally demanding to calculate zero-pointvibrational energies for crystals, which is typically approximated via a harmonic model.121,122Eq. (3.12)) is the same definition of lattice energy used by Jenkins,120 and captures the spirit oflattice energy as the underlying potential energy of a crystal. Note that it is possible for zero point39and thermal vibrational energy to affect the relative free energies of crystal structures. I examinedthis possibility using a harmonic model and found that, although LiX bulk phase vibrational ener-gies can be significant, they are of similar magnitude for all crystal structures considered, and donot alter their relative energetic stability in all but one case. I calculate that the NiAs and rocksaltcrystal structures of LiI are extremely close in lattice energy, so I cannot rule out the possibility ofvibrational energy playing a role in the stabilization of rocksalt over other LiI crystal structures.See Section 3.3.5 for further discussion of vibrational corrections.The closest one can get to experimental lattice energies, which are used for comparison here,are those calculated via thermodynamic corrections to the Born-Haber cycle (Figure 3.1).120,123,124In preliminary research, I found that the typically reported experimental lattice energies123 forlithium halides are given without error estimates. Furthermore, these lattice energies were calcu-lated by a series of approximations which are not necessarily accurate nor needed, especially forlighter salts such as LiF. Better experimental lattice energies can be calculated for the LiX saltsfrom thermodynamic data available in the literature.To address the issue of unreported error and unnecessary approximations in experimental latticeenergies, I calculated LiX lattice energies from experimental data with improved accuracy, startingwith enthalpies obtained via the Born-Haber cycle. NaCl is also considered as a reference. I noticedthat in order to generate a fully experimental calculation of lattice energies as defined above, onlyone piece of the required information was missing: the vibrational zero-point energies of LiX salts.I calculated these theoretically by employing a harmonic model.121,122 Details on the harmonicvibrational calculations are given in Section 3.2.6. I observe that the usual approximate method forlattice energy calculation123 from experimental measurements benefits from a cancellation of errorsbetween overestimated thermal vibrational energy and neglected zero-point vibrational energy toyield a surprising level of accuracy for lithium halide salts. However, I made small adjustmentsto the lighter LiX salts, since these have large vibrational zero point energies that are not capturedby the approximate method. The subsection below gives mathematical details on the method usedto accurately recalculate experimental lattice energies. A comparison of the literature values andnewly calculated lattice energies, including error estimates, is given in Table 3.1.Mathematical Details: Recalculation of Experimental Lattice EnergiesHere I derive an expression for experimental lattice energy in terms of observable properties. I be-gin by defining lattice energy (EL) as in Eq. (3.12). Unfortunately, the values of UFixed, Crys(0 K,0 bar)and UIons(0 K,0 bar) are not directly measurable by experiment. The Born-Haber cycle120,123,12440H0Figure 3.1: A visualization of the Born-Haber cycle, used to calculate LiX Born-Haber en-thalpies, ∆H◦BH(LiX(s),T ′), from experimentally measurable quantities. The definitionsof quantities shown in this plot are given in the caption of Table 3.2.LiF LiCl LiBr LiI NaClLiterature EL 1049 864 820 764 790Recalculated EL 1054.2±1.3 865.4±1.5 821.3±1.1 764.5±1.1 791.3±0.8Table 3.1: Previously reported123 and recalculated (this work) experimental lattice ener-gies for the lithium halides and NaCl, including estimated uncertainties. All units arekJmol−1. Error estimates for experimental and calculated vibrational energies are notavailable; these were assumed exact for the purpose of uncertainty propagation.41is very useful for resolving this, and can be summarized (for lithium halide (LiX) salts) by the fivemeasurable steps in Table 3.2 (which are visualized in Figure 3.1).Reaction Experimental MeasurementLi(s)+ 12 X2(g)→ LiX(s) ∆H◦f (LiX(s),T ′)Li(g)→ Li(s) −∆H◦f (Li(g),T ′)X(g)→ 12 X2(g) −∆H◦f (X(g),T ′)Li+(g)+ e−→ Li(g) −EIE(Li(g))X−(g)→ X(g)+ e− −EEA(X(g))Table 3.2: The five steps involved in the Born-Haber cycle for lithium halides. T ′ refers toT = 298.15 K; X is the atomic halide species of interest; X2 is its molecular form; ∆H◦fis a species’ standard formation enthalpy; EIE is ionization energy; and EEA is electronaffinity.If one sums together the reactions in Table 3.2, the result is the desired chemical reactiondefining ELLi+(g)+X−(g)→ LiX(s). (3.13)However, a simple summation of the enthalpy and energy terms of the Born-Haber cycle does notresult in the lattice energy. Instead, the summation contains a series of energy and enthalpy contri-butions which I denote the Born-Haber enthalpy (∆H◦BH). Each of the energy and enthalpy valuesin Table 3.2 have been measured and made available in the literature with a reported uncertainty.See Table 3.3 for a breakdown of the experimental values and their sources.At this stage it is worth pointing out that there are no significant rotational degrees of freedomfor non-molecular crystals, atoms, or ions. Furthermore, many other partitions of energy — suchas electronic and nuclear energies — are essentially unaffected by thermal fluctuations at 298.15 K.Hence, I assume that only potential energy, translational energy, vibrational energy, and pressure-volume work contribute to the enthalpy differences in Table 3.2. Breaking down the enthalpies and42∆H◦f (MX(s),T ′) Ref ∆H◦f (M(g),T ′) RefLiF -616.9(8) 125 (P. 1065) 159(1) 126 (P. 26)LiCl -408(1) 125 (P. 777) 159(1) 126 (P. 26)LiBr -350.9(4) 125 (P. 447) 159(1) 126 (P. 26)LiI -270.1(4) 125 (P. 1394) 159(1) 126 (P. 26)NaCl -411.1(3) 125 (P. 789) 107.5(7) 126 (P. 27)EIE(M(g)) Ref EEA(X(g)) RefLiF 520.221948(2) 127 -328.1649(2) 128LiCl 520.221948(2) 127 -348.575(3) 129LiBr 520.221948(2) 127 -324.5369(2) 130LiI 520.221948(2) 127 -295.1531(3) 131NaCl 495.84554(2) 132 -348.575(3) 129∆H◦f (X(g),T ′) RefLiF 79.4(3) 126 (P. 21)LiCl 121.301(8) 126 (P. 21)LiBr 111.9(1) 126 (P. 22)LiI 106.76(4) 126 (P. 22)NaCl 121.301(8) 126 (P. 21)Table 3.3: Summary of the experimental values required to calculate ∆H◦BH at 298.15 K forlithium halides and NaCl. Here M is either Li or Na depending on the salt of interest.All values are reported in kJmol−1, and uncertainties in the final digit are reported inbrackets.energies of Table 3.2 into their components yields the following series of equations:∆H◦f (LiX(s),T′) =[U◦+E◦Vib(T′)+P◦V (T ′)](LiX(s))−[U◦+E◦Vib(T′)+P◦V (T ′)](Li(s))− 12[U◦+E◦Vib(T′)+E◦Trans(T′)+P◦V (T ′)](X2(g));(3.14)43−∆H◦f (Li(g),T ′) =[U◦+E◦Vib(T′)+P◦V (T ′)](Li(s))−[U◦+E◦Trans(T′)+P◦V (T ′)](Li(g));(3.15)−∆H◦f (X(g),T ′) =12[U◦+E◦Vib(T′)+E◦Trans(T′)+P◦V (T ′)](X2(g))−[U◦+E◦Trans(T′)+P◦V (T ′)](X(g));(3.16)−EIE(Li(g)) =U(Li(g),0 bar)−U(Li+(g),0 bar)−Ee− ; (3.17)−EEA(X(g)) =U(X(g),0 bar)+Ee−−U(X−(g),0 bar). (3.18)A full summation of the terms from the Born-Haber cycle yields (after cancellations),∆H◦BH(LiX(s),T′)≡ ∆H◦f (LiX(s),T ′)−∆H◦f (Li(g),T ′)−∆H◦f (X(g),T ′)−EIE −EEA=[U◦+E◦Vib(T′)+P◦V (T ′)](LiX(s))+(U(0 bar)−U◦)(Li(g))+ (U(0 bar)−U◦)(X(g))−U(Li+(g),0 bar)−U(X−(g),0 bar)−[E◦Trans(T′)+P◦V (T ′)](Li(g))−[E◦Trans(T′)+P◦V (T ′)](X(g)).(3.19)To proceed further, I make the assumption that for all species of interest,U◦ =U(0 bar)≡U. (3.20)For the gaseous species, this is simply the ideal gas assumption. It is justified for all species bythe relative weakness of inter-ionic interactions at standard pressure. Under extreme pressures, thepotential energy of both the salt and the atoms/ions becomes significantly raised through repulsiveinteractions. I attempted to calculate the ambient pressure-induced potential energy correction of44VEE°ZP-Vib(LiX(s))P°V(LiX(s),T') + E°Therm-Vib(LiX(s),T')P°V(Li(g)+X(g),T') + E°Trans(Li(g)+X(g),T')Crystal at Standard Pressure and T'Ions with Corrections for P° and T'Motionless Separated IonsIons Fixed on Lattice SitesEL0∆H°BHCrystal at 0 KFigure 3.2: A diagram illustrating the thermodynamic terms associated with the calculationof lattice energies for LiX crystals.lithium halide crystals (through quantum mechanical geometry optimization including an externalstress of 1 bar), but was unable to detect any modification to the structure or energy using fairlytight convergence thresholds. With the above approximation, and the lattice energy definition givenin Eq. (3.12), Eq. (3.19) can be arranged to yieldEL(LiX(s)) = ∆H◦BH(LiX(s),T′)−[E◦Vib(T′)+P◦V (T ′)](LiX(s))+[E◦Trans(T′)+P◦V (T ′)](Li(g))+[E◦Trans(T′)+P◦V (T ′)](X(g)).(3.21)To proceed, experimental values must be acquired for [E◦Vib(T′)+P◦V (T ′)](LiX(s)), [E◦Trans(T′)+P◦V (T ′)](Li(g)), and [E◦Trans(T′)+P◦V (T ′)](X(g)) (see Figure 3.2). There are at least two methodsof producing experimental values for these terms. A third method, based on ideal gas and hightemperature approximations, is often used in the literature123 and will also be discussed.Method OneTo begin with, one can neglect the P◦V (T ′) term of the salt. This is justified since P◦V (T ′)(LiX(s))45is on the order of 10−3 kJmol−1, below the uncertainty threshold of other experimental measure-ments. Next, based on my assumption of energy partitioning at T ′, it is easy to see that[H◦(T ′)−H◦(0 K)](Li(g)) = [E◦Trans(T ′)+P◦V (T ′)](Li(g)) (3.22)and[H◦(T ′)−H◦(0 K)](X(g)) = [E◦Trans+P◦V (T ′)](X(g)) (3.23)since other enthalpy contributions are not significantly activated for these species. The above termsare available with uncertainties,126 calculated via integration of the experimental constant-pressureheat capacities of each species from 0 to T ′.The remaining E◦Vib(LiX(s),T′) term can be broken down into thermal and zero-point contribu-tions,E◦Vib(LiX(s),T′) = E◦T herm−Vib(LiX(s),T′)+E◦ZP−Vib(LiX(s)). (3.24)Although the zero-point vibrational energy cannot easily be measured experimentally for crystals— and is not currently available for lithium halides or NaCl — the thermal contribution to the vi-brational energy is measurable for non-molecular salts by integration of the heat capacities. Basedon the same reasoning as for the gaseous atoms,E◦T herm−Vib(LiX(s),T′) = [H◦(T ′)−H◦(0 K)](LiX(s)). (3.25)Experimental data for [H◦(T ′)−H◦(0 K)] is available for LiF, LiCl, and NaCl without reporteduncertainties.125 Since no uncertainties were available, values were assumed exact for my erroranalysis. [H◦(T ′)−H◦(0 K)] experimental data is not currently available for LiBr or LiI.In order to estimate E◦T herm−Vib for LiBr and LiI, I utilized a DFT-based harmonic-vibrationalmodule available in CRYSTAL17.121,122,133,134 This approach also yields calculated zero pointenergies, which are not available experimentally for any salts under consideration here. To gatherzero-point vibrational energies, I computed a vibrational analysis for all five salts of interest. Thiswas achieved by generating a 4×4×4 (primitive rhombohedral) supercell of each lithium halide inthe rocksalt crystal structure, containing 128 atoms total. The input geometry of the seed unit cellwas taken from the output of full geometry optimization calculations with the same XC functional,dispersion correction, basis set, and other calculation parameters. Uncertainties are difficult to46[E◦Trans+P◦V ](M(g),T ′) Ref. [E◦Trans+P◦V ](X(g),T ′) Ref.LiF 6.197(1) 126 (P. 26) 6.518(1) 126 (P. 21)LiCl 6.197(1) 126 (P. 26) 6.272(1) 126 (P. 21)LiBr 6.197(1) 126 (P. 26) 6.197(1) 126 (P. 22)LiI 6.197(1) 126 (P. 26) 6.197(1) 126 (P. 22)NaCl 6.197(1) 126 (P. 27) 6.272(1) 126 (P. 21)E◦T herm−Vib(MX(s),T′) Ref. E‡T herm−Vib(MX(s),T′) E‡ZP−Vib(MX(s))LiF 6.473 125 (P. 1065) 5.7598 12.7972LiCl 9.305 125 (P. 777) 8.2656 8.0395LiBr 9.4540 6.4499LiI 10.0404 5.6391NaCl 10.611 125 (P. 789) 9.6829 5.9239Table 3.4: Summary of the experimental and calculated values required to calculate latticeenergy from ∆H◦BH via method 1 for lithium halides and NaCl. All values are reportedin kJmol−1, and uncertainties in the final digit are reported in brackets where available.Here M refers to either Li or Na depending on the salt. ‡Refers to values calculated via aharmonic-crystal model with zero external pressure at the PBE-D3(BJ)/pob-TZVP levelof theory.ascertain for purely theoretical calculations, so errors were assumed zero for these data points forthe purposes of uncertainty propagation. For more details on these calculations, see Sections 3.2.3and 3.2.6. All of the experimental and theoretical data required to calculate lattice energies bymethod one is made available in Table 3.4.Method TwoThis method makes use of a different version of the Born-Haber cycle, starting from Eq. (3.19)as before, but replacing ∆H◦f measured at 298.15 K with derived values of ∆H◦f at 0 K. Suchvalues are produced by integration of the constant-pressure heat capacities Cp,i with respect to thereference enthalpy T ′,∆H◦f (0 K) = ∆H◦f (T′)−∫ T ′0∑iνiCp,i(T )dT, (3.26)where the sum is over all species in the formation reaction, and νi is their (signed) stoichiometriccoefficients. If such values are available for both the salt and reference atoms, Eq. (3.21) can be47∆H◦f (MX(s)) Ref. ∆H◦f (M(g)) Ref. ∆H◦f (X(g)) Ref.LiF -614.4(8) 125 (P. 1065) 158(1) 125 (P. 1497) 77.3(3) 125 (P. 1051)LiCl -408(1) 125 (P. 777) 158(1) 125 (P. 1497) 119.621(6) 125 (P. 737)LiBr 158(1) 125 (P. 1497) 117.92(6) 125 (P. 429)LiI 158(1) 125 (P. 1497) 107.16(4) 125 (P. 1387)NaCl -410.7(3) 125 (P. 789) 107.5(7) 125 (P. 1641) 119.621(6) 125 (P. 737)Table 3.5: Summary of T = 0 K experimental data points required to calculate lattice energythrough method 2. All values are reported in kJmol−1, and uncertainties in the final digitare reported in brackets. Here M refers to either Li or Na depending on the salt of interest.reduced toEL = ∆H◦BH(0 K)−E◦ZP−Vib(LiX(s)), (3.27)with other terms becoming zero or negligible. The necessary experimental values of ∆H◦f (0 K) areavailable125 for all species under consideration here except for LiBr and LiI, and are reproduced inTable 3.5 with uncertainties for convenience.Method ThreeMethod three is the simplest and least accurate method for calculating lattice energies fromlattice enthalpies. This method is often cited as the experimental lattice energy.120,123 Lattice en-thalpies are still calculated from Eq. (3.19). However, starting from Eq. (3.19) the thermal termsare simply approximated from simple models.As before, the P◦V (T ′)(LiX(s)) term is neglected. Next, the gaseous P◦V (T ′) terms are bothapproximated as ideal gases viaP◦V (Li(g),T ′) = P◦V (X(g),T ′)≈ RT ′. (3.28)Now the atom translational energy E◦Trans for both atomic species is approximated from the equipar-tition theorem asE◦Trans(Li(g),T′) = E◦Trans(X(g),T′)≈ 32RT (3.29)All that remains is the vibrational energy of the crystal E◦Vib(LiX(s),T′). An approximation to thetotal vibrational energy is derived from either the Debye or Einstein models for a crystal in the high48LiF LiCl LiBr LiI NaClMethod One 1054.2±1.3 865.4±1.5 821.3±1.1 764.5±1.1 791.3±0.8Method Two 1054.2±1.3 865.4±1.5 791.0±0.8Method Three 1050 863 820 764 790Table 3.6: A summary of experimental lattice energies for the lithium halides and NaCl, in-cluding uncertainties calculated via the three methods discussed here. Units are kJmol−1.temperature regime (classical limit)E◦Vib(LiX(s),T′)≈ 3nRT ′ = 6RT ′, (3.30)where n is the number of atoms per formula unit. It is expected that light salts such as LiF do notexist in the high temperature vibrational regime under ambient conditions, so this approximationovershoots the vibrational energy of such species. In addition, the classical approximation excludesthe zero-point vibrational energy, which is species- and structure-dependent but not expected to begreater than approximately 10 kJmol−1 for ionic salts.135 Fortunately, the overshooting of thermalvibrational energy combined with the exclusion of zero-point vibrational energy tends to mostlycancel these two errors. Overall, the approximate method used to calculate the lattice energy oflithium halide salts through the Born-Haber cycle reduces toEL ≈ ∆H◦BH(T ′)+RT ′. (3.31)A comparison of lattice energies calculated via the three methods discussed here are listed in Ta-ble 3.6.3.2.2 Candidate Crystal StructuresIn this chapter, I examine how modifications to interionic potentials affect the relative energeticsof LiX crystal structures. In doing so, it is important to not only examine the most commonpolymorphs, rocksalt and wurtzite, but to also consider a wider landscape of energetically low-lyingcandidate crystal structures. Assessing the energies of alternative crystal structures is necessary toavoid designing new LiX force fields that can give incorrect lowest-energy crystal structures, asis the case for the TF and JC models. In this regard, the earlier work of Jansen and co-workershas proven indispensable.25–28 Ref. 25 is a theoretical analysis of alternative crystal structuresfor several simple ionic salts, including LiF and NaCl. The paper introduced a computational49technique for the efficient exploration of possible polymorphs in simple ionic salts, and reportedthe 5-5 candidate crystal structure for the first time.When Cˇancˇarevic´ et al.27 applied the global crystal structure search algorithm to the alkalihalides at pressures from -160 to 1600 GPa, they found dozens of candidate crystal structures. Fromthose candidates, I chose to examine seven that are relatively low energy for the lithium halides:rocksalt, sphalerite, wurtzite, 5-5, NiAs, β -BeO, and CsCl. I selected these crystal structuresbecause Cˇancˇarevic´ et al. reported them as most stable for at least one lithium halide at one ormore of the pressures examined in their study. I used these structures to explore the possibility of analternative (i.e., not rocksalt) global minimum of lattice energy by performing calculations of LiXand NaCl at all levels of theory discussed below. As an additional reference point, I also calculatedthe lattice energies of the antiNiAs crystal structure of LiX, which is constructed by switchingthe anion and cation positions of NiAs. See Section A.2 of the Appendix for crystallographicinformation on each of the candidate crystal structures.3.2.3 Hartree-Fock and Density Functional Theory CalculationsFor the HF and DFT calculations reported here, I used two software packages: CRYSTAL17133,134and CP2K version 7.1.136 CRYSTAL17 is a quantum chemistry program designed for periodicelectronic structure calculations which uses periodic Gaussian basis functions. CRYSTAL17 in-cludes many tools for calculating crystal properties, but does not yet implement the most mod-ern XC functionals and dispersion corrections. To implement more modern DFT methods andcorrections I used CP2K, a general open source quantum chemistry package. CP2K enables pe-riodic DFT calculations with Dion and co-workers’ non-local vdW-DF137 correction, vdW-DFversion 2 by Hamanda,138 the revised139 version of the non-local dispersion functional of Vydrovaand van Voorhis140 (rVV10), scalar relativistic corrections with the Douglas–Kroll–Hess (DKH)transformation,141–143 and an interface to Libxc144 providing access to hundreds of modern XCfunctionals. CP2K uses periodic Gaussian basis sets, which are expanded in plane waves in inter-stitial regions between atoms by the Gaussian Plane Waves (GPW) or Gaussian Augmented PlaneWaves (GAPW) methods. All CP2K calculations reported here employed the all-electron GAPWmethod.136 For all-electron calculations of LiBr crystals, LiI crystals, Br− ions, and I− ions usingCP2K, I introduced scalar relativistic corrections with a third order DKH transformation and anappropriately optimized basis set.For most crystalline calculations I adopted the Peintinger, Oliveira, and Bredow triple-ζ splitvalence quality solid-state basis set with added polarization functions, pob-TZVP.145,146 I chose50this basis set due to its consistently high quality and portability for crystalline systems. pob-TZVP,along with its revised version,147 is a high quality atom-centered basis set designed for crystallinecalculations that is available for all atoms under consideration here. Due to its robust design, CP2Kis less sensitive to numerical basis set overlap issues that prevent the use of larger basis sets inCRYSTAL17. Hence, I used either pob-TZVP or the larger Sapporo-QZP148–155 and Sapporo-DKH3-QZP156 basis sets for the anions in CP2K calculations, while keeping the pob-TZVP basisset for cations. Basis sets with diffuse functions were used for calculations of isolated ion energies.See the Appendix, Section B.1 for complete details on the basis sets used here.To establish the basis set dependence of my results, I used a series of variable quality basissets which I refer to as the “CRYSTAL” basis sets. These are a collection of crystalline basissets which, before pob-TZVP was released, were the highest quality solid state basis sets for LiXcalculations built into the CRYSTAL code: see Refs. 157 (Li), 158 (Na), 159 (F), 160 (Cl), and 97(Br and I). I also compared the above with calculations involving the STO-3G (Slater Type Orbital- 3 Gaussians) basis sets, which are minimum-quality molecular basis sets: see Refs. 161 (Li andF), 162 (Na and Cl), 163 (Br), and 164 (I).It is difficult to determine in advance which particular choice of DFT XC functional worksbest for modeling a system of interest. Hence, I ran DFT calculations of LiX using many differ-ent XC functionals from each rung of “Jacob’s Ladder”165 except the double-hybrid level whichincorporates a costly post-HF/KS calculation of correlation energy, requiring correlated basis setsand pseudopotentials to remain feasible for solids.166 In particular, I compare pure HF and DFTcalculations with XC functionals from the local density approximation (LDA), generalized gradi-ent approximation (GGA), meta-GGA approximation, and global/Coulomb-screened hybrid levelof theory. The choice of XC functionals to test was partly based on the availability of dispersioncorrections. Table 3.7 gives a summary of XC functionals and dispersion corrections used in thisresearch project.In an attempt to compensate for generally poor representation of medium-to-long-range dy-namic electron correlation in most DFT methods,116 I tested three classes of dispersion correc-tions. The first class of dispersion correction comes from Grimme and co-workers. I testedGrimme’s third generation dispersion64,184 with Becke-Johnson (BJ) rational damping185,186 (re-ferred to as D3(BJ)) or zero damping (referred to as D3(0)) as well as the recently developed D4dispersion.187–189 I opted not to use the optional D3 3-body terms as they are not well tested,184and were not found to make significant energy contributions in preliminary tests I performed. D4dispersion corrections were applied with 3-body terms included as recommended by the authors,although the effect of 3-body terms is small: less than 0.1 % change to the lattice parameter and51Name Level of Theory Dispersion Correction(s) Ref.HF Mean field ab inito D3(BJ), D4 113LSDA (SVWN) DFT LDA D3(0) 167, 168PBE DFT GGA D3(BJ), D4, rVV10L 169optB88 DFT GGA vdW 170PBEsol DFT GGA rVV10 171SG4 DFT GGA rVV10m 172, 173rev-vdW-DF2 DFT GGA vdW2 138TMTPSS DFT Meta-GGA rVV10L 117, 118SCAN DFT Meta-GGA rVV10 174, 175M06L DFT Meta-GGA D3(0), D4 176B3LYP DFT GGA Global Hybrid D3(BJ), D4 177, 178, 179PW1PW DFT GGA Global Hybrid D3(BJ), D4 180HSE06 DFT GGA Screened Hybrid D3(BJ) 181HSEsol DFT GGA Screened Hybrid D3(BJ) 182M062X DFT Meta-GGA Global Hybrid D3(0), D4 183M06 DFT Meta-GGA Global Hybrid D3(0), D4 183Table 3.7: List of the quantum mechanical theories tested here. Column three gives thedispersion corrections used with each theory in this work.less than 3 kJmol−1 shift in lattice energy in all cases, compared to D4 two-body terms alone. Notethat D4 dispersion is only implemented with BJ-type rational damping.The second class of dispersion correction tested is the non-local vdW correlation functional ofDion et al.137 as well as it’s second generation.138 These dispersion corrections are parameter-free,and based on a non-local kernel of the electron density. The vdW corrections are implementedalongside an LDA correlation functional, as they are designed to capture all non-local correlationinteractions. The final class of dispersion corrections tested here is the non-local rVV10139 cor-rection. rVV10 is similar to the vdW correction in that it is implemented as a non-local kernel onthe electron density. However, much like Grimme’s dispersion corrections, rVV10 includes globalfitting parameters for each functional it’s paired with, allowing for correction of missing medium-to-long range dispersion without double counting the correlation interaction at short distances,where semi-local correlation functionals are accurate. For more complete detailed information onall aspects of HF and DFT calculations performed in this study, refer to the Appendix, Section B.1.523.2.4 Empirical Models and Classical CalculationsTwo popular, non-polarizable LiX empirical force fields are the JC14 and TF12,13 models. Bothmodels are defined as pairwise additive effective potentials and introduced in more detail in Sec-tion 2.3. In this project, I examined all three JC models, defined with respect to the followingwater models: SPC/E;58 TIP3P;59 and TIP4PEW.60 However, I primarily focus on the JC modelassociated with the SPC/E water model as the other two models perform generally worse for LiXlattice energy and structure prediction. In this project, all crystal calculations involving empiricalmodels employed the calculation parameters as described in Section 2.7 with GROMACS version2018.803.2.5 Crystal Structure OptimizationThe energy minimization of crystal structures is key to computing their ground state properties. Allcrystal structures reported in this work were geometry optimized. The crystal optimization for theclassical and quantum calculations was achieved differently. GROMACS does not have a built-inmethod for optimizing lattice parameters, whereas the software used for the quantum calculationsdoes. Hence, for optimization of crystals using classical models, I used the minimization schemedescribed in Section 2.8.1. For energy minimization calculations with CRYSTAL17 or CP2K, Iemployed the geometry optimization subroutines native to the corresponding software package.For most energy minimization calculations, unit cell parameters constrained by the symmetry ofthe structure’s space group were held fixed. Loosening of the symmetry constraint (i.e. placingthe unit cells in space group P1) did not significantly change the resulting optimized structures orlattice energies.DFT and HF CalculationsAll HF and DFT calculations employed geometry optimization subroutines from the respectivecodes. For geometry optimization calculations employing CRYSTAL17, five convergence criteriawere set. The first criterion is the maximum change in energy between optimization steps, set to10−11 EH . The second criterion is the maximum root-mean-squared of the gradient with respectto the change in atomic positions or lattice vectors, set to 3×10−5 EH/a0. The third criterion isthe maximum largest component of the gradient, set to 4.5×10−5 EH/a0. The fourth criterion isthe maximum root-mean-squared of the estimated displacements, set to 1.2×10−4 a0. Finally, thefifth criterion is the absolute value of the maximum estimated displacement, set to 1.8×10−4 a0.All thresholds were set very tight in order to allow for subsequent vibrational analysis calculations.53CP2K requires five thresholds for geometry optimization of crystals: maximum force, root-mean-squared force, maximum displacement, root-mean-squared displacement, and pressure. I set theseto to the defaults provided by CP2K version 7.1, as I did not perform vibrational analysis on thecrystal structures produced from CP2K calculations.3.2.6 Thermal Corrections to DFT CalculationsI ran harmonic121,122 and quasi-harmonic190–193 vibrational analysis calculations with input of fullyoptimized geometries from CRYSTAL17 geometry optimization calculations of the same level oftheory and structure, as described above. In order to calculate thermodynamic properties, phononmodes were calculated by construction of supercells made from up to 4×4×4 primitive unit cellsfor harmonic calculations and 2×2×2 primitive unit cells for quasi-harmonic calculations.My largest vibrational analysis calculations were performed using the PBE XC functional169and pob-TZVP basis set.145,146 PBE was chosen mainly due to its reasonable accuracy and lowercomputational cost for large supercells, allowing me to model a larger number of phonon modes.I compared harmonic vibrational calculations with smaller supercells using two additional XCfunctionals (PW1PW,180 and B3LYP)177–179 and found that the resulting zero-point and ambient-temperature vibrational energies were consistent to within 0.5 kJmol−1 of the PBE result for alllithium halides and NaCl. I ran all vibrational analysis calculations with D3(BJ) corrections. Itested rocksalt supercells from 1× 1× 1 to 4× 4× 4 with the PBE XC functional, and foundthat harmonic calculated zero-point and finite-temperature vibrational energies shifted less than0.1 kJmol−1 between 3× 3× 3 and 4× 4× 4 supercells for calculations of each LiX and NaClsalt. Therefore, I consider the 4×4×4 calculations sufficiently converged with respect to supercellsize. Harmonic calculations of ET herm−Vib and EZP−Vib were performed with zero external pressure.Based on geometry optimization calculations performed at 1 bar, any ambient pressure correctionto the vibrational energies of lithium halides is expected to be negligible. I found that QHA-calculated rocksalt lattice parameter thermal expansions were converged with respect to supercellsize by supercells of 2×2×2 primitive unit cells. In my calculations, I modified the default QHAsettings to set the volume range from 97 % to 115 % in 7 steps, allowing for more robust modelingof thermal expansion at 298 K. Other QHA calculation parameters were left to their CRYSTAL17defaults.543.3 Results and Discussion3.3.1 Basis Set Influence on Incompleteness ErrorsTo measure the importance of basis set completeness for LiX crystal structure prediction, I testedfour basis sets on their ability to correctly predict the lowest energy crystal structure of the lithiumhalides. NaCl was also tested as a reference. Figure 3.3 shows the lattice energy difference be-tween (fully optimized) rocksalt and wurtzite structures for four basis sets. Two different XCfunctionals are compared to help asses possible errors introduced by the theory itself: that ofPerdew, Burke, and Ernzerhof (PBE)169 and the modified Perdew-Wang (PW) hybrid XC func-tional (PW1PW).180 No dispersion corrections are included in the results presented in Figure 3.3.The pob-TZVP/ECPS28MDF-VTZ basis set refers to DFT/pob-TZVP calculations where the io-dine valence basis set was replaced with the Stuttgart/Cologne ECP28MDF-VTZ basis set.194 Thisiodine basis set has the same pseudopotential as pob-TZVP, but a larger valence basis set opti-mized for small molecules in post-HF calculations. Only the pob-TZVP basis set predicts thecorrect rocksalt crystal structure for all four lithium halide salts, highlighting the importance ofbasis set completeness for solid state LiX structure modeling. In contrast, the lowest energy crystalstructure of NaCl (rocksalt) is independent of the tested basis sets.Basis set errors, which are always due to basis set incompleteness, can be broken down into twocategories. Basis set superposition error (BSSE) is a source of error unique to quantum mechani-cal calculations involving localized basis functions centered on multiple atomic centers.195 BSSEemerges from the added variational freedom of molecular orbitals associated with bonded systemsof atom-centered basis functions compared with dissociated fragments. As fragments are broughttogether, the basis associated with each fragment becomes more complete by overlap with the basisfunctions of the nearby fragment, which tends to artificially lower the energy of more condensedstates. Hence, use of a typical geometry optimization algorithm can lead to a non-physical short-ening of bond lengths (and unit cell vectors in crystalline calculations). BSSE also increases bondstrength (i.e. lowers bond energy) when the same basis set is employed on both the bonded systemand dissociated fragments. Basis set errors other than BSSE are generally classified as basis setincompleteness errors (BSIE).Establishing the effect of BSSE and BSIE on calculations of ionic solids is not straightforward.The most common approach for recovering BSSE energy in molecular calculations is Boys andBernardi’s counterpoise method.196 However, this approach is ill-suited for ionic solids as it relieson a similarity in electronic structure between isolated ions and in-crystal ions. The physical extent55Figure 3.3: The difference (∆EL = EL(Rocksalt)−EL(Wurtzite)) in lattice energy betweenfully optimized rocksalt and wurtzite crystal structures using basis sets of various qual-ity. Experimental observation suggests that all lithium halides should have negative ∆ELvalues here, which is only exhibited in all LiX with the pob-TZVP basis set.of valence electron density for anions in a crystal field is much more constrained than for freeanions, whose density extends far into space. This difference is why diffuse basis functions arerequired for isolated anions, but should not be used for in-crystal anions. Another notable approach,which corrects for BSSE by omitting the terms in the Hamiltonian that cause it, is the ChemicalHamiltonian Approach (CHA).197 Unfortunately, CHA is not implemented in the software chosenfor this project.A promising BSSE correction scheme for solids is Grimme’s empirical gCP correction.198,199Presently gCP is not well tested on inorganic crystalline systems. After extensive testing of the gCPmethod in combination with the pob-TZVP basis set and D3 dispersion corrections for the lithiumhalides, I found that gCP corrections tend to reduce the accuracy of lattice energies comparedwith D3 corrections alone. They also reduce the accuracy of lattice parameters for the LiX saltscompared to calculations performed without D3 nor gCP. gCP corrections also lead to incorrectpredictions of the lowest-energy crystal structure for LiF for certain levels of theory, presumablyby lowering the effective cation/anion radius ratio. Since a primary focus of this research is toestablish accurate energy differences between LiX crystal structures, I conclude that gCP should56not be applied with pob-TZVP basis sets in LiX crystal calculations. A more complete discussionof my calculations involving gCP is available in the Appendix, Section B.2.Having exhausted other available avenues of BSSE correction, I return to increasing the sizeof the basis set to improve accuracy. I calculated a set of static crystal properties (lattice energies,lattice parameters) for the lithium halides with the large Sapporo-QZP basis set on the anions atseveral levels of theory. The pob-TZVP was retained for cations as replacement with the largerSapporo-QZP does not significantly change the resulting energies. These results are discussed indetail in the following section.3.3.2 Level of Theory: Quantum and Classical Models for LiX CrystalsIn total, I tested fifteen DFT XC functionals plus HF with various dispersion corrections on allfour LiX salts (see Table 3.7) and seven structure candidates. At most levels of theory, the secondmost energetically favourable crystal structure behind rocksalt is either NiAs, wurtzite, 5-5, or aspace-filling hexagonal structure intermediate between ideal wurtzite and ideal 5-5.In Figure 3.4, I compare differences in calculated lattice energies between fully optimizedrocksalt and wurtzite LiX crystals at various levels of theory using the pob-TZVP basis set. Theexact experimental values of these differences are currently unknown, I only know that they shouldbe negative (i.e., the rocksalt structure is more energetically favourable). Clearly, the level ofquantum mechanical theory is somewhat less important for structure prediction of LiX crystalsthan the choice of a complete basis set, although some methods fail to predict the experimentalstructure. In particular, both uncorrected HF and B3LYP calculations predict wurtzite as the lowestenergy crystal structure of LiCl, LiBr, and LiI. Figure 3.5 makes the equivalent comparison withD3 or D4 dispersion corrections included. I observe that the difference in lattice energy betweenrocksalt and wurtzite is always increased after D3 or D4 correction with re-optimization, but theeffect is smaller with D4 than D3. The energy difference between rocksalt and wurtzite is alsomore consistent across the different XC functionals with either dispersion correction. These resultsare the first indication that the relatively weak dispersion interaction plays a critical role in LiXstructure determination, which is discussed in detail below.Another set of results, presented in Figure 3.6, shows that non-local, density-based dispersioncorrections (of vdW- and rVV10-type) also perform well for lithium halide structure prediction.Additionally, Figure 3.6 shows how basis set errors contribute to the difference between rocksaltand wurtzite energies at the triple-ζ level, by comparison with results using a quadruple-ζ qualitybasis set on the anion. Note that results for LiBr and LiI with PBE in Figure 3.6 differ from57Figure 3.4: The difference (∆EL = EL(Rocksalt)−EL(Wurtzite)) in lattice energy betweenfully optimized rocksalt and wurtzite crystal structures for various quantum mechanicalcalculations. The pob-TZVP basis set is used in all cases. Dispersion corrections arenot included here. Negative values favour rocksalt, in agreement with experiment.Figure 3.4 and 3.5, as third order DKH transformations (with DKH-consistent basis sets) are used toaccount for scalar relativistic effects rather than the core electron pseudopotential incorporated intopob-TZVP. Figure 3.6 shows a trend from LiF to LiI: differences in energy between rocksalt andwurtzite decrease but remain in favour of rocksalt. The TMTPSS-rVV10 results, which are the mostaccurate results compared to experiment (see below), give a clear example of this phenomenon.The calculation results shown in Figure 3.6 with PBE, PBE-D3, SCAN, and SCAN-rVV10functionals are comparable to results from Yang et al. published in a recent report.200 In somecases, Yang et al.’s calculations of relative LiX rocksalt-wurtzite energies qualitatively differ frommine with the same XC functional. In contrast to my results, they found that the PBE functionalfavours wurtzite for LiF, PBE-D3 favours wurtzite for LiCl and LiI, SCAN favours wurtzite forLiBr, and SCAN-rVV10 favours wurtzite for LiI. There are several differences between their cal-culation parameters and mine. Most critically, Yang et al. employ pseudopotentials on all atomsand a plane wave basis with a relatively low energy cutoff (38 Ry). In contrast, my best calcula-tions are all-electron with large quadruple-ζ Gaussian basis sets and third order DKH correctionsfor LiBr and LiI. Based on the strong basis set dependence of LiX rocksalt-wurtzite energy differ-58Figure 3.5: The difference (∆EL = EL(Rocksalt)−EL(Wurtzite)) in lattice energy betweenfully optimized rocksalt and wurtzite crystal structures for various quantum mechanicalcalculations, with dispersion corrections included. D3 dispersion corrections are indi-cated by solid bars while D4 dispersion corrections are indicated with hatched bars. BJdamping is used in all cases, except for M06-D3 where zero damping is used, as BJparameters are not available. Note that D4 dispersion parameters are not available forthe HSE06 and HSEsol XC functionals. The pob-TZVP basis set is used in all cases.Negative values favour rocksalt, in agreement with experiment.ences shown in Figs. 3.3 and 3.6, I deduce that the results of Yang et al. likely differ from mine dueto pseudopotential or basis set incompleteness errors. Yang et al. did not report predicted latticeenergies or lattice parameters, making further comparison difficult.For calculations of rocksalt lattice energies, the TMTPSS-rVV10L/pob-TZVP/Sapporo-QZPlevel of theory is the most accurate method compared to experiment with an average absoluteerror of 0.47 % across all lithium halides, followed closely by PW1PW-D3(BJ)/pob-TZVP withan average absolute error of 0.59 %. HSEsol-D3(BJ)/pob-TZVP and HSE06-D3(BJ)/pob-TZVPalso performed very well, with average errors of 0.64 % and 0.80 %, respectively. These resultscan be contrasted to the corresponding average absolute errors without dispersion correction: 5.70,4.34, and 5.99 % for PW1PW/pob-TZVP, HSEsol/pob-TZVP, and HSE06/pob-TZVP, respectively.Figure 3.7 shows the relative lattice energy error (with respect to experiment) of various dispersion-corrected quantum mechanical calculations. Figure 3.7 can be compared with Figure 3.8 to show59Figure 3.6: The difference (∆EL = EL(Rocksalt)−EL(Wurtzite)) in lattice energy betweenfully optimized rocksalt and wurtzite crystal structures for various quantum mechanicalcalculations. Use of the pob-TZVP basis set (or Sapporo-DKH3-TZP for Br and I) isindicated by solid bars, while hatched bars indicate the larger Sapporo-QZP (or Sapporo-DKH3-QZP for Br and I) basis set on the anions. Scalar relativistic effects are includedby a third order DKH transformation for all LiBr and LiI calculations. Negative valuesfavour rocksalt, in agreement with experiment.the relative error before and after dispersion correction (where available) for particular levels oftheory. The D4 dispersion correction performs worse for lattice energies than the D3 correctionin all cases except HF. In Figure 3.9 I show the relative error in lattice energy for the set of XCfunctionals with vdW- and rVV10-type dispersion corrections (and some without, for comparison).Overall, my results indicate that an accurate description of the dispersion interaction is importantfor lattice energy calculations of lithium halides.Finally, I compare the experimental rocksalt lattice parameter, a, to those predicted by geome-try optimization of rocksalt unit cells at various levels of theory. There are two sets of experimentaldata points I can compare with. The first is the lattice parameters measured at 298 K83 and thesecond are lattice parameters corrected for thermal contraction at 0 K by an integration of theexperimental linear thermal expansion coefficients (see Section B.3 of the Appendix for details).Comparisons between theoretical lattice parameters calculated from geometry optimization arecompared to the experimental lattice parameters corrected to 0 K. I also compare rocksalt lattice60Figure 3.7: The relative error ([EL(Theory, Rocksalt)−EL(Experiment)]/|EL(Experiment)|)in lattice energy for various dispersion-corrected quantum mechanical calculations. D3dispersion corrections are indicated by solid bars while D4 dispersion corrections areindicated with hatched bars. BJ damping is used in all cases, except for M06-D3 wherezero damping is used, as BJ parameters are not available. Note that D4 dispersionparameters are not available for the HSE06 and HSEsol XC functionals. The pob-TZVPbasis set is used in all cases for the calculation of crystal energies, the def2-TZVPD basisset is used for the calculation of Li+, F−, Cl−, and Br− isolated ion energies, while thebasis set from Section S8.4 of the SM is used for isolated I− energies. Theoretical latticeenergies are those calculated from optimized rocksalt structures.parameters measured at room temperature by calculating thermal expansion at 298 K theoreti-cally from the quasi-harmonic approximation190–193 (QHA) with a supercell of 2×2×2 primitiverocksalt unit cells. Note that expansion of the crystal lattice due to vibrational zero-point motion(zero-point vibrational expansion) is not accounted for by integration of linear thermal expansioncoefficients. I expect zero-point vibrational expansion is a relatively small effect as it arises fromanharmonicity in the potential energy of phonon modes near their minima. However, zero-pointvibrational expansion should be larger for the lighter salts with higher energy phonons.The QHA-predicted lattice parameter expansion from 0 to 298 K (including zero-point expan-sion) is theory-dependent, but is around 1 – 1.5 % for most levels of theory examined, with morevariable expansions predicted at the M06/pob-TZVP level of theory (0.3 – 4.1 %). Figure 3.1061Figure 3.8: The relative error ([EL(Theory, Rocksalt)−EL(Experiment)]/|EL(Experiment)|)in lattice energy for various quantum mechanical calculations without dispersion or ba-sis set superposition error corrections. The pob-TZVP basis set is used in all cases forthe calculation of crystal energies, the def2-TZVPD basis set is used for the calculationof Li+, F−, Cl−, and Br− isolated ion energies, while the basis set from Ref. 86 is usedfor isolated I− energies. Theoretical lattice energies are those calculated from optimizedrocksalt structures.shows the relative errors in a (i.e., (acalculated−aexperiment)/aexperiment) for several dispersion-correctedquantum calculations. Figure 3.10 also includes lattice parameter comparisons with experiment at298 K by application of QHA corrections. These comparisons are overlaid on the 0 K compar-isons (hatched bars). D4 dispersion is not yet compatible with QHA in CRYSTAL17, so onlyfixed-lattice results are available in those cases. Figure 3.10 shows that QHA performs well forapproximating thermal expansion of LiX rocksalt lattice parameters, compared with experiment,in most cases. At most levels of theory tested, thermal expansion corrections from QHA give morepositive relative errors in a than the comparisons at 0 K. I view this primarily as a manifestationof missing zero-point expansion in fixed-lattice structures, which is present in 0 K experimentallattice parameters. Zero-point expansion is captured along with thermal expansion by QHA. Assuch, the gap between the solid and hatched bars in Figure 3.10 gives an estimate of the zero-pointvibrational expansion as a fraction of the total lattice parameter, which is larger in the lighter LiXsalts, as expected.62Figure 3.9: The relative error ([EL(Theory, Rocksalt)−EL(Experiment)]/|EL(Experiment)|)in lattice energy for various quantum mechanical calculations. Use of the pob-TZVPbasis set (or Sapporo-DKH3-TZP for Br and I) for the crystal energies is indicated bysolid bars, while hatched bars indicate the larger Sapporo-QZP (or Sapporo-DKH3-QZP for Br and I) basis set on the anions. The def2-TZVPD basis set for isolated ionscorresponding to pob-TZVP, while diffuse augmented Sapporo basis sets are otherwiseused. Scalar relativistic effects are included by a third order DKH transformation forall LiBr and LiI calculations. Theoretical lattice energies are those calculated fromoptimized rocksalt structures.Figure 3.10 can be compared with Figure 3.11, which gives the equivalent comparisons withoutdispersion corrections. Since dispersion is an attractive interaction, dispersion corrections alwayslead to a contraction of the unit cell parameters. In Figure 3.12 I also show the relative error inrocksalt lattice parameter a, computed without QHA and compared to experimental results cor-rected to 0 K. Figure 3.12 includes a comparison between DFT calculations employing a triple-ζ(solid bars) and a quadruple-ζ (hatched bars) anion basis set. The XC functionals used in Fig-ure 3.12 include several with non-local, density-dependent dispersion corrections (i.e. vdW andrVV10 corrections). Overall, the best performing calculation for LiX rocksalt lattice parameters isthe TMTPSS-rVV10L/pob-TZVP/Sapporo-QZP level of theory, with an average absolute error ofonly 0.054 % across all lithium halides. TMTPSS-rVV10/pob-TZVP, optB88-vdW/pob-TZVP, andPBEsol/pob-TZVP also perform exceptionally well, achieving average absolute errors of 0.36 %,63Figure 3.10: The relative error ((acalculated− aexperiment)/aexperiment) in calculated lattice pa-rameter a for various dispersion-corrected quantum mechanical calculations. BJ damp-ing is used in all cases, except for M06-D3 where zero damping is used, as BJ param-eters are not available. The pob-TZVP basis set is used in all cases. Theoretical latticeparameters are those from fully optimized rocksalt unit cells. Overlaid hatched barsare relative errors in calculated rocksalt lattice parameter a where the theoretical cal-culation has been corrected for thermal expansion at 298.15 K via the Quasi-HarmonicApproximation (not available for D4-corrected calculations).0.43 %, and 0.44 % across the lithium halides, respectively.Some recent publications assert that the lowest energy crystal structure of both LiBr and LiIis experimentally wurtzite at (T = 0,P = 0).200,201 However, a careful reading of the full body ofexperimental evidence indicates that rocksalt is almost certainly the stable (T = 0,P = 0) structurein both cases. The wurtzite crystal structure of LiBr was reported in 2008 by Liebold-Ribeiro andcoworkers16 while hexagonal LiI (most likely wurtzite) was reported by Ru¨hl in 1956,202 Wasser-man et al. in 1998,203 and Cˇancˇarevic´ et al. in 2005.204 Both Liebold-Ribeiro et al. and Cˇancˇarevic´et al. used the low temperature atomic beam deposition (LT-ABD) technique to deposit hot gaseousLiBr or LiI onto a chilled substrate under a range of temperatures and pressures. Liebold-Ribeiro etal. found that high fractions of LiBr nucleate into wurtzite under specific experimental conditions,with (Tsubstrate = 213 K,P = 2×10−4 mbar) most optimal. Similarly, Cˇancˇarevic´ et al. found thatLiI nucleates into wurtzite below 273 K using LT-ABD. Heating wurtzite crystals of LiBr or LiI64Figure 3.11: The relative error ((acalculated− aexperiment)/aexperiment) in calculated lattice pa-rameter a for various quantum mechanical calculations. The pob-TZVP basis set isused in all cases. Theoretical lattice parameters are those from fully optimized rock-salt unit cells. Overlaid hatched bars are relative errors in calculated rocksalt latticeparameter a where the theoretical calculation has been corrected for thermal expansionat 298.15 K via the Quasi-Harmonic Approximation.to room temperature brings about an irreversible phase change to rocksalt, indicating that rocksaltis the thermodynamically stable structure under ambient conditions for both. How then is wurtzitecrystallization favoured experimentally by low substrate temperatures?Crucially, Liebold-Ribeiro et al. note that LT-ABD “favours low density modifications, irre-spective of whether they are the thermodynamically most stable phase.”16 In LT-ABD experiments,the substrate temperature controls not only the final crystal temperature, but also the cooling rate ofthe crystallizing system. When the substrate is sufficiently cold, the cooling rate of the depositedmaterial is rapid, and crystallization is kinetically controlled. Rapid cooling favours the crystal-lization of less dense phases as low density nuclei face a smaller entropic barrier from the lowdensity amorphous phase, an example of Ostwald’s rule of stages.205 Since wurtzite is less densethan rocksalt, wurtzite nucleates first and is kinetically selected under rapid cooling. With sub-strate temperatures much colder than 213 K, Liebold-Ribeiro et al. find that LiBr again nucleatespreferentially into rocksalt (Cˇancˇarevic´ et al do not report this for LiI). In this temperature regime,cooling is so rapid that the amorphous phase is likely supercooled in place before given a chance65Figure 3.12: The relative error ((acalculated− aexperiment)/aexperiment) in calculated lattice pa-rameter a for various quantum mechanical calculations. Use of the pob-TZVP basis set(or Sapporo-TZP for I) for the crystal energies is indicated by solid bars, while hatchedbars indicate the larger Sapporo-QZP basis set on the anions. Theoretical lattice pa-rameters are those from fully optimized rocksalt unit cells.to nucleate; the entropy barrier to both rocksalt and wurtzite is then greatly reduced. Presumably,LiBr rocksalt has a lower enthalpy barrier to nucleation than wurtzite and eventually crystallizes.The lower rocksalt enthalpy barrier may not apply for LiI. There are several other reported exam-ples of deposition experiments producing metastable crystal structures206–210 including, notably,wurtzite LiCl15 which to my knowledge has no debate regarding its (T = 0,P = 0) crystal struc-ture (rocksalt). The highest quality calculations reported here (which best match the experimentallattice energies and rocksalt lattice parameters) support the preceding interpretation of the experi-mental evidence. Neither LiBr nor LiI are thermodynamically stable as wurtzite at (T = 0,P = 0).LiBr is rocksalt while LiI has both rocksalt and NiAs crystal structures in very close competition.In summary, I find that the TMTPSS-rVV10L/pob-TZVP/Sapporo-QZP method (where pob-TZVP is used for lithium ions and Sapporo-QZP for halides) gives both LiX lattice energies andlattice constants in excellent agreement with experiment. This method predicts that rocksalt in thelowest energy crystal structure for LiX, except for the notable case of LiI. The TMTPSS-rVV10Llevel of theory combines the 2016 meta-GGA exchange functional of Tao and Mo117 (TM), the662003 meta-GGA correlation functional from Tao, Perdew, Staroverov, and Scuseria211,212 (TPSS),and a recently parameterized version of Vydrov and van Voorhis’ efficient density-dependent dis-persion correction140 optimized TMTPSS calculations of layered materials, called rVV10L.118 Myresults indicate that TMTPSS-rVV10L performs very well for ionic lithium halide systems, despitebeing designed for layered van der Waals solids. TMTPSS-rVV10L/pob-TZVP/Sapporo-QZP andother high accuracy calculations performed here show that the (frozen lattice) NiAs crystal struc-ture of LiI is lower in energy than rocksalt by about 0.5 kJmol−1. It is possible that this theoreticalprediction reflects reality, but the NiAs crystal structure has not yet been reported experimentallyfor LiI. There may be higher entropic or enthalpic barriers to NiAs structure nucleation comparedto rocksalt, and one could explore this question with molecular dynamics simulations. My resultsalso indicate that despite expected improvements of D4 over D3 for condensed ionic systems,189D3 dispersion corrections tend to yield better energetics than D4 for the lithium halides with thepob-TZVP basis set, possibly due to cancellation of errors (see Section 3.3.3).3.3.3 Role of Dispersion in LiX Structure DeterminationAlthough the strength of dispersion interactions is far weaker than the static Coulombic interactionsfound in ionic salts such as the lithium halides, my calculations show that differences in latticeenergy between distinct LiX crystal structures depend quite strongly on the subtleties of theseinteractions. In the previous section, I presented evidence that dispersion corrections added toDFT calculations improve structure determination and energetics for LiX salts. To further analyzethe effect of the dispersion interaction on relative crystal structure energies in classical models, Iintroduce a simple dispersion scale factor SD into the JC force fieldui j(ri j) =14piε0qiq jri j+4εi j[(σi jri j)12−SD(σi jri j )6], (3.32)which may apply to any or all pair interactions. I also use a similar factor to the TF model, scalingboth dispersion terms for selected interactions,ui j(ri j) =14piε0qiq jri j+Bi je−αi jri j −SD(Ci jr6i j+Di jr8i j). (3.33)I then set SD to zero for all pair interactions and recalculated the optimized crystal structuresand lattice energies of the lithium halides. Analyzing the effect of dispersion with DFT calcula-tions was performed analogously (although less completely) by switching off and on dispersion67corrections, with full re-optimization of the crystal. Thus, the effect of dispersion on lattice en-ergy is derived not only from the dispersion interaction itself, but also from the change in energyfrom other interactions after structural relaxation. Figure 3.13 illustrates the differences in latticeenergies (termed ∆Edispersion) where∆Edispersion = EL, disp on−EL, disp off. (3.34)Here, EL, disp on and EL, disp off refers to the lattice energy of optimized crystal structures with dis-persion turned on and off, respectively. For classical models this is done by setting SD = 1 andSD = 0, respectively, for all pairwise interactions. For quantum mechanical models, this refers toenabling and disabling (respectively) dispersion corrections (i.e. D3, vdW, or rVV10). In Fig-ure 3.13 I show the effects on both rocksalt and wurtzite energies at several levels of theory, bothDFT and classical. Since dispersion is a purely attractive interaction, the energy is lowered in allcases. However, for each dispersion model tested, the LiX rocksalt crystal structure is energeticallyfavoured more than wurtzite by the effects of dispersion (Fig 3.13). Although the size of the dis-persion correction varies with the level of theory (DFT correlation functionals usually have someclose-range dispersion built in), it always favours rocksalt over wurtzite. The dispersion energyitself is quite small compared to the total lattice energy of the crystal, but large enough to be impor-tant in lithium halides since the difference in energy between rocksalt and low-lying, metastablestructures is typically smaller than the total dispersion energy.LiX Dispersion in Classical ModelsTo pin down the mechanism of differing rocksalt vs wurtzite responses to dispersion in pair-wise classical models, I scaled the dispersion strength SD from 0 to 5 for all pair interactionsof all lithium halides. I did this for all seven candidate structures, with both the JC-SPC/E andTF models. At each point, I minimized the structures with respect to lattice parameters, hold-ing the ideal fractional coordinates and the cell symmetry fixed. I plot lattice energy differences(∆EL = Erocksalt−Eother structure) as functions of SD in Figs. 3.14 and 3.15, for the JC-SPC/E and TFmodels, respectively. A negative slope indicates that rocksalt is favoured by increased dispersionstrength, while a positive slope indicates the other crystal structure is favoured. These plots extendthe findings in Figure 3.13, showing that rocksalt is energetically favoured in classical models bydispersion versus the other candidate structures, except NiAs and (with the TF potential) CsCl.To aid in understanding the effect of dispersion on LiX crystal structures, it is instructive toexamine the relative energies of LiX crystal structures using dispersion-free models. This is simple68Figure 3.13: Differences (∆Edispersion = Eopt, disp on−Eopt, disp off) in lattice energy betweenfully-optimized crystal structures before (Eopt, disp off) and after (Eopt, disp on) dispersioncorrections are added for a selection of models. For DFT calculations (PBE, PW1PW,and B3LYP), dispersion was added by application of D3(BJ) corrections. For empiricalmodels (JC-SPC/E and TF), the pairwise r−6 and r−8 dispersion interactions wereremoved and restored to calculate the effect of dispersion.enough to do using empirical models, as I need only to scale SD to zero for all pair interactions.Removing dispersion raises the energy of all LiX crystal structures and expands each of theircoordination spheres slightly. The resulting dispersion-free model is comparable to a “hard” ionicsphere model, the lowest energy structures of which are strictly governed by the conventional (andanalytically provable) radius ratio rules.213 If one takes Shannon’s “effective ionic radii”,214 thecation/anion radius ratios for the series LiF through LiI are 0.571, 0.420, 0.388, and 0.345, whereasNaCl has a radius ratio of 0.564. The radius ratio rules give tetrahedrally coordinated structures(e.g. wurtzite) for cation/anion ratios between 0.255 and 0.414 while those between 0.414 and0.732 favour octahedral structures (e.g. rocksalt). Based on Shannon’s effective ionic radii, I wouldexpect the rocksalt crystal structure for LiF and LiCl, but wurtzite for LiBr and LiI. In Figs. 3.14and 3.15, the left side of each panel shows the relative energies of each lithium halide structurecompared with rocksalt in dispersion-free classical models. The dispersion-free JC-SPC/E modelis in agreement with the radius ratio rules: wurtzite is most stable for LiBr and LiI. On the otherhand, the dispersion-free TF model predicts wurtzite for all LiX, suggesting an adjustment to the69Figure 3.14: The JC(SPC/E) model: dispersion scaling factor SD applied to all dispersion in-teractions versus the difference in lattice energy (∆EL =EL(Rocksalt)−EL(Other)) be-tween cell-minimized rocksalt (EL(Rocksalt)) and six other (EL(Other)) cell-optimizedcrystal structures with fixed ideal fractional coordinates.TF repulsive parameters for (at least) LiF and LiCl may improve the model for structure prediction.Generally speaking, the reason that theoretical LiX models often give the incorrect crystal structure(especially for LiCl through LiI) is the small ionic radius of Li+ compared with its halide counter-anion.For the pairwise classical dispersion models considered here, there are several mechanismsthrough which LiX crystal structures could be affected by relatively short-range dispersion forces.One such mechanism is the interaction between each ion and its first coordination shell of nearestneighbors, i.e. the Li+-X−-type dispersion forces. These can further be broken down (in the TFmodel) into dispersion terms proportional to r−6LiX or r−8LiX . Another possible mechanism is throughinteractions between ions and their second coordination shell of like-charged ions, i.e. the Li+-Li+-and X−-X−-type dispersion forces. However, Table B.3 in the Appendix shows that the Li+-Li+dispersion contributions are too small to affect the energetics of lithium halides in both the JC andTF models. Furthermore, through close examination of JC and TF pair potentials, I’ve observedthat all dispersion interactions beyond the second coordination shell, taken together, only accountfor between 10 – 18 % of the total dispersion energy. Therefore, dispersion interactions beyond70Figure 3.15: TF model: dispersion scaling factor SD applied to all dispersion interactionsversus the difference in lattice energy (∆EL =EL(Rocksalt)−EL(Other)) between cell-minimized rocksalt (EL(Rocksalt)) and six other (EL(Other)) cell-optimized crystalstructures with fixed (ideal) fractional coordinates.the second shell can play only a minor role in structure determination, if any. In essence, theprimary mechanisms through which dispersion influences LiX energetics is via first shell Li+-X−and second shell X−-X− dispersion interactions.In Section B.4 of the Appendix, I examine the effect of each mechanism on LiX salt energeticsin detail. I find that, although the magnitude of Li+-X−-type and X−-X−-type dispersion forcescan be similar, only the X−-X−-type dispersion forces significantly favour rocksalt over wurtzite.LiX crystal structures have a direct relationship between the first shell coordination number andthe average Li+-X− bond length: structures with low first coordination numbers (such as wurtzite,sphalerite, and β -BeO) have shorter Li+-X− bond lengths while more coordinated structures (suchas rocksalt, NiAs, and CsCl) have longer bond lengths. This trade-off can be derived directlyfrom analysis of pairwise models for simple ionic salts.215 Due to the curvatures of the disper-sion function, increasing the strength of dispersion interactions by scaling C6 and C8 dispersioncoefficients more strongly affects structures with shorter bond lengths. However, this favouring ofshorter bond lengths is counteracted by the higher multiplicity of more coordinated structures withlonger bond lengths. These two factors tend to cancel each other in the first coordination shell of71LiX structures, such that Li+-X−-type dispersion forces have little net effect on the relative en-ergies of different crystal structures. There is no such correlation between second shell distanceand second shell coordination number. Hence, the relatively strong second-shell X−-X− dispersionforces have the greatest impact on relative LiX crystal energies. Another valid interpretation of theabove phenomenon comes from the perspective of effective radius ratios. Increasing the strength ofX−-X−-type dispersion forces has the effect of pulling together X− ions, shortening their effectiveradius without effecting Li+ ions. This has the effect of decreasing the effective cation/anion radiusratio, which will tend to favour rocksalt.The technical discussion in Section B.4 of the Appendix offers some insight into how the JCand TF models can be improved for LiX structure prediction. The JC-SPC/E model partitionsa large portion of the LiX total dispersion energy into Li+-X−-type interactions, with relativelylittle in X−-X−-type interactions. It should be understood that the JC models are effective pair-wise potentials which are simply fit to experimental data points; the fitting data did not includeion polarizabilities or dispersion forces. Hence, it is not particularly surprising that the JC-SPC/Emodel generates an unphysical96,97 partitioning of dispersion energy. The JC model also lacks adipole-quadrupole dispersion term (proportional to 1/r8), making direct comparison to the otherdispersion models less useful. If one wishes to improve the LiX structure prediction capabilities ofthe JC-SPC/E model, which fails for LiBr and LiI, then my above discussion suggests that simplyscaling up the X−-X−-type dispersion interactions will tend to favour rocksalt. However, doingso will necessarily shorten unit cell parameters and lower the lattice energy, pushing these valuesslightly away from agreement with experiment. For example, scaling up the I−-I−-type dispersionof the JC-SPC/E model by 2.4 is enough to flip the lowest energy structure of LiI from wurtzite torocksalt. At the same time, such a scaling lowers the rocksalt lattice energy from−761.78 kJmol−1to −787.13 kJmol−1, whereas the experimental LiI lattice energy is −764.5(11) kJmol−1. Simul-taneously, the rocksalt lattice parameter decreases by 1.4 %.The dispersion parameters of the TF model, unlike the JC model, are based on theoretical cal-culations of in-crystal ion polarizabilities. The TF dispersion coefficients are likely to be morephysically realistic than JC. In particular, the TF Li+-X−-type dispersion interactions are muchweaker than in the JC model. For the TF model, my results show that LiX structure predictioncan be corrected by a scaling of the X−-X−-type dispersion by 3.1, 1.3, 1.2, and 2.0 for LiF, LiCl,LiBr, and LiI, respectively. However, a further revision of the TF repulsive parameters is likelyrequired to correct the LiX structure prediction problem while maintaining agreement with otherexperimental results, such as rocksalt lattice parameters and lattice energies. For example, scalingup the TF X−-X−-type dispersion by the minimum amount necessary to correct the structure pre-72diction problem in LiF lowers the resulting rocksalt lattice energy from−1043 to−1075 kJmol−1,compared to the experimental value of−1054.2 kJmol−1. At the same time, the LiF rocksalt latticeconstant a shrinks by approximately 3 %.LiX Dispersion in Quantum Mechanical CalculationsA quantum analogue to neglecting dispersion in a classical model can be made by use of the HFmethod, which fails to include all dynamic Coulombic correlation by construction. Since disper-sion originates from fluctuating electron correlations, pure HF can be considered a dispersion-free,quantum-mechanical model (note that intraionic electron correlation and static correlation are alsomissing). Examination of Fig. 3.4 shows that HF fails to predict the experimental low energy crys-tal structure of LiCl through LiI to an increasing degree, congruent with a picture where the crystalstructure is determined primarily by radius ratio rules. The same behaviour is observed for pureDFT calculations with the B3LYP functional, which is known to yield a very poor description ofdispersion.216 Turning on D3/D4 dispersion corrections (Fig. 3.5) with HF or B3LYP causes boththeories to favour the rocksalt crystal structure for all lithium halides, which has the same effect asscaling up the pairwise dispersion terms in classical models.The D3 dispersion model applied to LiX salts is unlikely to be physically realistic due to itsrelatively large Li+-Li+- and Li+-X−-type dispersion energies. The apparent overestimation ofLi+-related dispersion energies is due to a lack of reference species with comparable electronicstructure in the D3 reference molecule set. When modeling LiX solids with D3, Li+ polarizabilitiesare extrapolated by comparison to LiH, the bond of which is likely much more covalent than thein-crystal Li+ cation.92,217,218 The polarizability of lithium in particular is very sensitive to itscharge state, shifting by three orders of magnitude (164.2 to 0.194 a.u.) on the path from neutralto its gas phase cationic form.219,220 In contrast, the polarizability of sodium shifts by two ordersof magnitude going from the neutral atom to cation (162.7 to 1.001 a.u.), and the larger alkalimetals show even smaller polarizability shifts. Hence, the polarizability of lithium is uniquelysensitive to its specific electronic structure, making an accurate reference state in D3-like schemesexceptionally important. We might expect that the polarizability of lithium in isolated LiH is a poorchoice for the description of in-crystal Li+-Li+ and Li+-X− dispersion coefficients, as is the casein D3.The D4 dispersion model is expected to have improved partitioning of dispersion energies dueto its explicit charge dependence and expanded reference molecule set, although this is difficult toprove. A comparison of the D3 and D4 dispersion coefficients (Table B.3 in the Appendix) shows73that D4 has larger X−-X−-type dispersion coefficients, but smaller Li+-Li+- and Li+-X−-typecoefficients, which are more in line with previous calculations of in-crystal Li+-Li+ and Li+-X−dispersion coefficients.92,93,96,97,221 In light of this, the fact that my results (at the triple-zeta level ofbasis set completeness) indicate D3-corrected models generally predict LiX lattice energies closerto experiment than D4 is unexpected. However, the issue is confounded by basis set errors (whichFig. 3.9 shows are significant for pob-TZVP in-crystal anions), errors associated with the level oftheory, and the empirically fit damping parameters of both the D3 and D4 models. Indeed, I expectthe D4 model has more physically realistic dispersion coefficients, but performs worse than D3 inmy calculations of total lattice energy simply because the larger overall dispersion correction givenby D3 cancels other errors.The discussion of non-local density-based vdW137 and rVV10139 dispersion corrections is lessstraightforward, as these corrections cannot be broken down easily into their component interac-tions. My results indicate that rVV10 dispersion corrections in conjunction with a suitable corre-lation functional can attain extremely accurate LiX crystal properties, without the need for exactexchange (hybrid DFT) or more costly post-HF correlation (double-hybrid DFT). The vdW cor-rection is at a disadvantage to rVV10 as it requires an LDA correlation functional, and thereforemisses out on the close range accuracy of modern GGA and meta-GGA correlation functionals.The culmination of my results suggest that LiX crystal structures are determined primarily by ra-dius ratio rules using the “effective ionic radii” of Shannon.214 The cation-anion radius ratios fromthese radii place the heavier lithium halides very near the border between tetrahedral and octahe-dral coordination, slightly favouring tetrahedral for both LiBr and LiI. However, the ionic radii oflarge highly-polarizable anions (especially Br− and I−) are not perfectly well defined. Their ionicradii change slightly due to the forces present in their chemical environment. In particular, second-shell anion-anion dispersion forces tend to “shrink” their effective radii in LiX crystals, raising thecation-anion radius ratio enough to favour the octahedrally-coordinated rocksalt crystal structure.3.3.4 Stability of Alternative Crystal StructuresIn this section I discuss the energetics of possible LiX crystal structures beyond rocksalt andwurtzite (see the Appendix, Section A.2, for a description of all structures considered). A sum-mary of the lattice energies of the four lowest-energy candidate structures relative to rocksalt forthe lithium halides and NaCl, calculated at the TMTPSS-rVV10L/pob-TZVP/Sapporo-QZP levelof theory, is given in Table 3.8.As discussed above, the dispersion interaction plays a major role in the selection of rocksalt or74Wurtzite 5-5 NiAs SphaleriteLiF 17.31 12.62 9.43 19.13LiCl 9.53 10.99 2.93 10.51LiBr 7.64 10.70 1.55 8.25LiI 4.68 9.57 -0.43 5.05NaCl 28.68 17.41 5.41 29.88Table 3.8: Difference in lattice energy between fully optimized low energy crystal struc-tures (Eother) and fully optimized rocksalt (Erocksalt) at the TMTPSS-rVV10L/pob-TZVP/Sapporo-QZP level of theory: Eother−Erocksalt. All units are kJmol−1. For LiBrand LiI, third a order DKH transformation was used to account for scalar relativisticeffects, along with the corresponding Sapporo-DKH3-QZP basis set on Br and I.wurtzite for the lithium halides. In fact, the general trends that apply between dispersion energeticsof the rocksalt and wurtzite crystal structures also apply between rocksalt and other tetrahedrallycoordinated structures: wurtzite, β -BeO, and sphalerite. Of the three tetrahedrally coordinatedstructures examined here, wurtzite is almost always found to be the lowest energy for all lithiumhalides. The low-density β -BeO structure in particular is only expected to come into play energet-ically at large negative pressures27 or high temperatures.The 5-5 crystal structure, first reported by Scho¨n and Jansen,25 is penta-coordinated with a trig-onal bipyramidal geometry (see the Appendix, Figure A.5). The 5-5 first coordination shell Li+-X−distance is intermediate between rocksalt and the tetrahedrally coordinated structures. Dispersionfavours rocksalt over 5-5 for the same reasons that dispersion favours rocksalt over tetrahedrallycoordinated structures, although to a lesser extent. At the TMTPSS-rVV10L/pob-TZVP/Sapporo-QZP level of theory, the 5-5 crystal structure is lower in energy than wurtzite for LiF but wurtziteis lower energy than 5-5 for LiCl, LiBr, and LiI due to the increasing anion radius that tends tofavour tetrahedral structures. For some DFT methods and classical models tested here (JC-SPC/E,B3LYP-D3, LSDA, and M06L) wurtzite was not found to be a local energy minimum in the land-scape of LiF structures, with geometry optimizations converging to the 5-5 structure instead, againlikely due to the radius ratio of LiF favouring higher coordinated structures. For all other DFTmethods tested here, the optimized LiF “wurtzite” structure was not near to ideal wurtzite, butintermediate between 5-5 and wurtzite. Note that the detected wurtzite/5-5 intermediate structuremay simply be an artifact of a flat potential energy surface between wurtzite and 5-5 combined withfinite convergence criteria. In all of my DFT calculations the fully minimized LiF 5-5 structure islower in energy than any wurtzite-like structure, though rocksalt is the lowest. The three JC models75also follow this trend. It is only the TF model that gives LiF wurtzite as lower in energy than both5-5 and rocksalt. This suggests a critical issue with the repulsive parts of the TF potential for LiF,leading to improper relative Li+ and F− ion sizes which select wurtzite via radius ratio rules.It is somewhat interesting that, despite being quite a poor model for lithium halide crystalstructure prediction, I found that the TF model does perform better than JC at high pressures.Since TF has a more physical functional form in it’s repulsive interaction, it predicts more accuratechanges in energy due to changes in volume than the JC models do. This digression is discussedfurther in Section B.5 of the Appendix.The NiAs crystal structure (see the Appendix, Figure A.6) is similar to rocksalt in many ways.Both have first coordination shells of 6 and second X−-X− coordination shells of 12. NiAs differsfrom rocksalt in that the second (like-charge) coordination shell of the cations has a coordinationnumber of 8 (composed of two subshells). In contrast, rocksalt has a first Li+-Li+ coordinationshell of 12, and Li+-Li+ distances that are longer than the Li+-Li+ distances in the NiAs structure.Since rocksalt and NiAs are so similar in the arrangement of their first Li+-X− and second X−-X− coordination shells, only the very weak Li+-Li+-type dispersion interactions can favour onestructure over the other. In both the JC and TF models, I found that increasing the strength ofLi+-Li+-type dispersion favours NiAs over rocksalt for all LiX salts. However, rocksalt is slightlylower energy than NiAs at almost all levels of theory for all LiX except LiI. Notably, the LiI NiAscrystal structure is very close in energy to LiI rocksalt, 0.43 kJmol−1 lower as calculated with thebest level of theory. Its unclear whether or not the NiAs crystal structure truly has a lower latticeenergy than rocksalt for LiI, as I cannot verify that my calculations attain sub-kJmol−1 accuracy.In a later chapter, I also examine the relative energy of the antiNiAs crystal structure, where cationsand anions are switched roles. In general, antiNiAs is a much higher energy structure than NiAsand rocksalt for all LiX.The final structure I consider is the CsCl crystal structure. This structure has a first coordina-tion number of 8 with a longer first shell Li+-X− distance than any other candidate structure. Dueto radius ratio rule considerations, the CsCl structure is much higher energy than rocksalt for alllithium halides, even for dispersion-free models. I find that first shell dispersion generally favoursrocksalt over CsCl, while second shell dispersion tends to favour CsCl, especially the term propor-tional to r−8. However, since radius ratio considerations strongly favour rocksalt over CsCl for allLiX salts, the CsCl structure is by far the least stable candidate structure examined here for all LiXsalts. Only at very high pressure do I expect the CsCl structure to play a significant role in LiXstructural chemistry.763.3.5 Thermochemical ConsiderationsTo further examine the relative stability of LiX crystal structures, I explored the possibility thatthe LiX rocksalt crystal structures might be favoured over wurtzite (or visa versa) by temperature-dependent, vibrational enthalpic or entropic effects. To determine the effect of vibrational energyand entropy on the relative free energies of LiX crystal structures, I calculated vibrational frequen-cies for rocksalt and wurtzite using the CRYSTAL17 harmonic vibrational module.121,122 Sincethese calculations require large supercells to reach convergence with respect to vibrational energyper unit cell, I calculated them at the PBE-D3(BJ)/pob-TZVP level of theory.In Figure 3.16, I plot the calculated ∆H, −T∆S, and ∆G between rocksalt and wurtzite for thelithium halides as a function of temperature under ambient pressure (negative energy differencesfavour rocksalt). An equivalent comparison between rocksalt and NiAs is given in Figure 3.17. Ifind that rocksalt and wurtzite have very similar vibrational entropies, as calculated via the har-monic model. It appears that wurtzite is slightly favoured by vibrational entropy at increasedtemperature (likely due to its lower density), but this difference is not enough to tip the free energyin favour of wurtzite for any lithium halide salt at any temperature below the melting point. I alsonote that the Li+ “cage rattling” phonons of rocksalt are lower energy and become exited at lowertemperature than the equivalent phonons of wurtzite. This small difference in vibrational energyis detectable in calculations at low temperature, where the greater zero-point vibrational energy ofwurtzite leads to a dip in the enthalpy difference of rocksalt and wurtzite between 0 and 150 Kvisible in Figure 3.16. This difference in phonon energy was previously noted in experiments ofLiI thin films, which take on a wurtzite crystal structure.203From Figure 3.17, it appears that LiI rocksalt has a slight vibrational entropy advantage over theNiAs crystal structure at finite temperature, which overcomes an even smaller vibrational energyadvantage of the NiAs structure. Overall, I calculate that LiI rocksalt is very slightly favouredover NiAs by the combined effects of vibrational energy and entropy. However, NiAs is slightlyfavoured by lattice energy at my best level of theory. Since the difference in free energy betweenLiI rocksalt and NiAs is so small, one may also need to take into account configurational entropy toexplain why only rocksalt is observed in experiment.222–224 Although the information required todescribe both rocksalt and NiAs unit cells is equal (both have two ions in the asymmetric unit), thenumber of symmetry elements of rocksalt is larger than NiAs. This implies a greater informationalentropy in rocksalt, which may present a larger entropy barrier to nucleation for NiAs than rocksalt.Since iodine is a heavy atom, the effects of spin-orbit coupling may also play a role in LiI structuredetermination. Recent theoretical work shows that spin-orbit coupling plays an important role77Figure 3.16: Differences in calculated thermodynamic properties (∆X = XRocksalt−XWurtzite,where X = G, H, −T S) between rocksalt (XRocksalt) and wurtzite (XWurtzite) for lithiumhalide salts as a function of temperature (T ), calculated along the P = 1 atm isobar bya quantum harmonic model. Calculated at the PBE-D3(BJ)/pob-TZVP level of theorywith 4× 4× 4 supercells (rocksalt, 54 ion) or 2× 2× 2 supercells (wurtzite, 32 ions)of primitive unit cells with PBE-D3(BJ)/pob-TZVP optimized geometries.in selecting the crystal structure of lead.225 Spin-orbit coupling is captured approximately by thepseudopotential used in my DFT calculations with the pob-TZVP basis set (where rocksalt is thelowest energy LiI crystal structure), but spin-orbit coupling is neglected in my scalar relativistic(third order DKH) all-electron calculations with the larger Sapporo-DKH3-QZP basis set, whichmatch best with experiment.I examined the other candidate structures and found that they are similarly affected by vibra-tional entropy, in some cases a slight shift in free energy towards rock salt is observed, while inothers there is a slight shift away. In no case did I find that vibrational energy or entropy ef-fects are enough to tip another structure to lower free energy than rocksalt before the LiX meltingpoints. In summary, harmonic calculations suggest that vibrational effects do not significantly im-pact differences in free energy between LiX crystal structures. Free energy differences betweenLiX structures are almost entirely determined by relative lattice energies.78Figure 3.17: Differences in calculated thermodynamic properties (∆X = XRocksalt− XNiAs,where X = G, H, −T S) between rocksalt (XRocksalt) and wurtzite (XNiAs) for lithiumhalide salts as a function of temperature (T ), calculated along the P = 1 atm isobar bya quantum harmonic model. Calculated at the PBE-D3(BJ)/pob-TZVP level of theorywith 4× 4× 4 supercells (rocksalt, 54 ion) or 2× 2× 2 supercells (NiAs, 32 ions) ofprimitive unit cells with PBE-D3(BJ)/pob-TZVP optimized geometries.3.4 Summary and ConclusionsIn this chapter I employ high accuracy DFT calculations together with empirical classical modelsto investigate the relative stability of possible crystal structures for lithium halides. Unlike otheralkali halides, LiX salts pose a particular problem for theory and modeling mainly because, from aradius ratio rule perspective (charged hard sphere ions), they tend to sit near the rocksalt/wurtziteboundary. This is particularly true for the larger halide ions, and means that I would expect relativecrystal stabilities to be sensitive to inaccuracies in quantum mechanical calculations, and to detailsof empirical force fields. Examples are provided by the classical TF and JC models, which, apartfrom two exceptions in the case of JC, incorrectly predict that wurtzite is more stable than rocksaltfor the lithium halides. I identify and discuss the important factors that determine the relativeenergetics of LiX crystal structures.I examined seven possible crystal structures, comparing results of different density functionals,basis sets, and dispersion corrections. The energy differences between rocksalt and competingcrystal structures (especially wurtzite, 5-5, and NiAs) can be rather small (. 10 kJmol−1), hence79very accurate calculations are required. I illustrate the importance of using complete basis setstogether with dispersion corrections in order to obtain the correct, lowest-energy crystal structurefor all lithium halides. Comparing with experimental lattice energies (as recalculated in this paper) Ishow that good agreement is obtained using the TMTPSS-rVV10L/pob-TZVP/Sapporo-QZP levelof theory. Furthermore, this level of theory gives excellent geometric parameters for LiX rocksaltcrystals. I note that for the lithium halides, gCP BSSE corrections do not systematically improveagreement between calculations and experiment for any DFT-D3/pob-TZVP level of theory that Iconsidered. However, a significant change in LiCl, LiBr, and LiI energies and lattice parametersare observed when switching the anion basis set from pob-TZVP to Sapporo-QZP, indicating thatpob-TZVP is still far from complete for the in-crystal description of Cl, Br, and I anions.My calculations show that dispersion interactions play a very important role in LiX structuredetermination. Dispersion-free HF calculations, in agreement with the conventional radius ratiorules, suggest that LiCl, LiBr, and LiI would exist as wurtzite crystals were it not for dispersion,while LiF would be rocksalt. Additionally, I observe that the addition of a dispersion correctionfavours the rocksalt structure at all levels of theory. I applied both D3 and D4 dispersion correctionsto the LiX salts, and found that D3(BJ) corrections give lattice energies in better agreement withexperiment. However, based on how these corrections are derived, I would expect the D4 pairinteractions (especially the two involving lithium) to be more realistic for the crystal environment.Possibly, the better agreement with experiment found for D3(BJ) is due to a fortuitous cancellationof errors. I also applied vdW and rVV10 dispersion corrections to several modern XC functionals,finding that these density-dependent dispersion corrections perform extremely well for the lithiumhalides, especially rVV10 when combined with modern correlation functionals.To further probe exactly how dispersion favours rocksalt over wurtzite, I used basic geometricarguments to reveal that X−-X− dispersion interactions involving the second anion coordinationshell are the most important cause of dispersion interactions favouring rocksalt over wurtzite forLiX salts.Informed by my observations of the importance of dispersion, I demonstrated that both the TFand JC models can be amended to predict the correct rocksalt crystal structure by simply scalingup the dispersion interactions, especially the halide-halide term. This does come at a cost becausesimple scaling without further potential adjustment leads to less accurate lattice energies and latticeparameters. Nevertheless, this points the way towards better models, and for some basic nucleationstudies simply scaling to get the correct structure is likely sufficient.I examined the relative stability of alternative LiX crystal structures compared with rocksaltand found that, at the TMTPSS-rVV10L/pob-TZVP/Sapporo-QZP level of theory, the first four80crystal structures in order of energetic stability are: rocksalt, NiAs, 5-5, and wurtzite for LiF. ForLiCl and LiBr, the relative energy order switches to rocksalt, NiAs, wurtzite, and sphalerite, with5-5 fifth lowest in energy. For LiI the order remains similar, but NiAs is slightly favoured overrocksalt. These changes in structural ordering likely reflect the increasing weight of the radiusratio rules, which favour tetrahedrally coordinated structures, versus dispersion interactions, whichfavour higher coordination numbers.Finally, I employed a harmonic model to examine the influence of vibrational energy and en-tropy on the stability of LiX crystals. I find that entropy slightly favours wurtzite, likely due to itslower density. However, at least at the harmonic level of approximation, the entropic contributionis not large enough to make wurtzite more stable than rocksalt for any salt, at a temperature belowthe melting point. In the case of LiI, vibrational effects appear to very slightly favour rocksalt overNiAs, but these effects alone are not enough to tip the total free energy significantly in favour ofrocksalt. Further investigation is required to determine why the NiAs crystal structure has not beenobserved experimentally for LiI.81Chapter 4Binary Salt Structure Classification withConvolutional Neural NetworksThe research presented in this chapter has been published in the Journal of Chemical Physics withcoauthor G. N. Patey.2264.1 IntroductionThe accurate identification of local geometric structure in molecular simulations is an importantand nontrivial problem, made particularly challenging by the thermal motion of particles. Theneed for order parameters that allow researchers to reliably distinguish between different phases orpolymorphs with high spatial resolution is apparent in, for example, nucleation research whereinthe precise mechanics of a structural phase change are often of central interest.6–9,227–234 Histor-ically, one of the most common order parameters for molecular simulation analyses are the bondorientational order parameters introduced by Steinhardt et al.30 These order parameters, which playa central role in the present work, are scalars constructed from spherical harmonic expansions ofbond vectors of near neighbor atoms surrounding a reference atom. Steinhardt order parametersencode local structural information and are fast to compute, yet incorporate rotational, transla-tional, and near neighbour permutational invariance into their mathematical form. These crucialproperties, combined with their sensitivity to structural changes, account for their widespread usein the analyses of molecular simulations. However, the Steinhardt order parameters are not unique.Many alternatives have been constructed, each with their own strengths and weaknesses.47824.1.1 Machine Learning for Structure ClassificationIn the last decade, machine learning (ML) structural classifiers for molecular simulation have ex-ploded in popularity. The application of ML to problems in molecular simulation was exploredby Behler and Parrinello235 employing trained neural networks (NNs) to generate DFT potentialenergy surfaces using symmetry functions as a basis. These symmetry functions encode variouslocal aspects of intermolecular structure, such as bond lengths and angles. Geiger and Dellagoapplied the symmetry functions as input for NN-based structure classifiers,236 which were trainedon example structures in monatomic Lennard-Jones systems and simulated water. Phillips andVoth applied unsupervised ML techniques, which can help identify patterns in complex unlabeleddata, to discover and classify crystal phases in simulations.237 Since these initial efforts, the fieldhas rapidly grown. There have been many developments in NNs applied to intermolecular struc-ture classification, including applications to hard spheres,238,239 monatomic systems with Yukawapotentials,240 simple polymers,241 water,242 glassy systems,243 colloidal systems,244 mesophasesystems,245 and polymorphic liquids with core-softened potentials.246 Supervised ML techniquesbesides standard NNs (commonly called feedforward NNs) have also been applied to intermolec-ular structural analysis, including the quantification of particle “softness” in glassy materials withsupport vector machines,247–249 the classification of crystal structures with convolutional NNs ap-plied to diffraction pattern images,250 liquid crystal polymer characterization with the random for-est technique,251 the analysis of colloidal systems with image-recognizing convolutional NNs,252and the analysis of amorphous material with message passing NNs.253In conjunction with the explosion of supervised ML applications, many unsupervised learn-ing approaches have been developed. These approaches allow the automatic detection of hiddenintermolecular structure without prior knowledge. Since they do not require prior knowledge, un-supervised ML analysis techniques are often designed to be broadly applicable. These techniqueshave provided insight into colloidal systems,254–256 hard spheres and cubes,256 supercooled liq-uids,257–259 elemental Zr in both liquid and crystal states,260 pure and two-component metallicsystems,261 and hard disks.262 Unsupervised ML approaches have also been successfully used toexplore topological phase transitions,263,264 exotic phases in correlated many-body systems,265 anddefects in two-dimensional liquid crystal phases.2664.1.2 Neural NetworksIn this chapter, I apply supervised ML via one dimensional (1D) convolutional neural networks(CNNs) to the problem of crystal structure classification in binary salts by convolving structural83information encoded in Steinhardt order parameters over time. This represents the first instance,of which I am aware, where order parameters are computed with an explicit time dependence inthe input, although a time dependence has been used to generate training labels elsewhere.247,248Central to the present work is the concept of NNs. NNs are flexible statistical models267 wellsuited for nonlinear regression or multiclass classification tasks, particularly when a large quantityof training data is available.The classification capacity of NNs emerges from their highly interconnected, hierarchical struc-ture. The initial layers of an NN detect patterns in input data, while deeper layers combine and buildupon these patterns to detect ever more complex features. Sufficiently large NNs can be trained toaccurately classify arbitrarily complex differences in input data given a large and robust trainingdata set. The ability of NNs to perform classification on data that is not linearly separable withintheir input space is due to the nonlinearity of their internal mathematical operations. Nonlinear-ity is introduced to NNs through their activation functions. These can be visualized as functionscentered on the nodes of NN layers; they are differentiable switch-like functions (e.g. the logisticfunction) that take weighted sums of outputs from previous connected layers as input, plus a bias,and pass their function output on to nodes in the next layer. The weights and biases are all trainableparameters. NN training is nothing more than a high dimensional minimization of a differentiableloss function, such as the categorical cross-entropy function used here, with respect to all NN pa-rameters. Total categorical cross-entropy H is a continuous measure of classification accuracy; itis defined as the sum over Ndata labeled training data points,H =Ndata∑i=1M∑j=1−yi j log(pi j) , (4.1)where the inner sum is over the M classes, and yi j is a binary indicator that vanishes for all classesexcept the labeled class j given to the data point i, where yi j = 1. pi j is the output probability ofdata point i on class j, as predicted by the NN undergoing training. NN training is made com-putationally feasible by the analytical calculation of the loss function gradient with respect to thetrainable parameters in the network. This high dimensional gradient calculation is known as back-propagation due to a layered use of the chain rule, applied backward through the network layersfrom output to input.Most of the NNs presented here contain 1D CNN architecture,268 where localized structuralinformation about ions in molecular dynamics trajectories is temporally convolved to improve clas-sification accuracy (discussed in Section 4.2.4). In the continuous mathematical case, the output84of a convolution of two functions is a third function. It is the overlap integral as one function isreflected and shifted over the other along its input domain, as a function of the input variable shift.NN convolution is analogous, but instead operates between two discrete numerical objects: a con-volutional kernel (consisting of trainable parameters) and input data, replacing the integral with asum. The output of this operation is a convolutional filter: a vector, matrix, or higher order tensordepending on the dimensionality of the convolutional kernel and input data. The convolutional fil-ters are mathematically recombined in subsequent NN layers as patterns in the data are extracted.In my application, convolutions act as moving time averages that filter out noise caused by randomthermal fluctuations.Two-dimensional (2D) CNNs are commonly applied to image classification,250,252,253 where2D kernels are convolved over input images to extract patterns. These patterns are often furtherconvolved in hidden convolutional layers to generate more complex and specific features used fordownstream classification. 1D CNNs are more commonly applied to find patterns in time seriesdata. In my case, 1D convolutional layers are used to convolve a set of kernels with a time seriesof multidimensional inputs that contain information about the angular orientation of near neighborbonds. The particular number and size of kernels used in a convolutional layer, and thereforethe number and size of filters produced by that layer, are selected as part of the NN architecture.Typically, NN architecture is chosen to optimize classification accuracy or some related heuristic,such as categorical cross-entropy, given a set of training data and a training scheme.The remainder of this chapter is organized into three sections. In Section 4.2, I detail themethods employed in this project, including the molecular dynamics methodology (4.2.1), theNN inputs (4.2.2), the NN training data (4.2.3), the NN architecture (4.2.4), and the NN trainingprocedure (4.2.5). Section 4.3 includes the results and discussion; I discuss the accuracy and utilityof my NN classifiers (4.3.1) and demonstrate the classifiers on two alkali halide simulations ofhomogeneous crystal nucleation (4.3.2). Finally, I summarize my work in Section 4.4 and proposesome further possible applications for the NN classifiers.4.2 Methods4.2.1 Molecular Dynamics SimulationsAll MD simulations presented or discussed in this chapter were performed using GROMACS ver-sion 2019.6,80,81 employing the JC and TF interaction potentials introduced in Section 2.3 withthe methodology given in Sections 2.7 and 2.8. In liquid-solid coexistence simulations, a semi-85isotropic barostat (for rocksalt/CsCl solids) or anisotropic barostat with fixed cell angle (for othersolids) was employed. For single phase solid simulations, the number of independent barostatdegrees of freedom was set to match the number of independent lattice vectors in the unit cell.Simulations beginning as liquids employed isotropic barostats.The barostat and thermostat time constants used in this project were selected by sweepingthrough a series of 12 sets of NaCl trial melting point (MP) calculations (see Chapter 5 for details)with varying time constants, using the JC(SPC/E) interaction potential and 2,000 ions. Each trialset involved 10 repeat calculations with different initial conditions (positions, velocities, and initialtemperatures) but otherwise identical calculation parameters. No statistically significant depen-dence on the simulated NaCl melting temperature was found for velocity-rescale thermostat timeconstants between τT = 0.01−5 ps, nor for the Parrinello-Rahman barostat time constants betweenτP = 0.1−2 ps.For simulations using JC models I set both the short-range Coulombic and non-Coulombicinteraction cutoff at 1.40 nm, shifted the non-Coulombic potential to zero at the cutoff, and includeda dispersion tail correction to the energy and pressure.269 No valid long-range dispersion correctionis available with potentials of the CBHM form in GROMACS. Hence, I ran these simulationswith an increased 1.90 nm non-Coulombic and short-range Coulombic cutoff, shifted the non-Coulombic potential to 0 at the cutoff, and excluded dispersion tail corrections. Further increasingthe cutoff beyond 1.90 nm affects the total energy by less than 0.1 kJmol−1.4.2.2 Neural Network InputsThe structure classification algorithms presented here take MD simulation trajectories of alkalihalides as input and classify the structure of each atom at every requested time step into 1 of 8possible crystal structures (given in the Appendix, Section A.2) or a liquid phase based on the localspatial arrangement of atoms surrounding each reference atom, with local structural informationpossibly convolved over time. The necessary input information for classification is encoded in aspecific set of Steinhardt order parameters chosen to balance classification accuracy with computa-tional cost. I also include a binary identification (ID) input labeling each atom as metal (0) or halide(1). The ID input helps the classifiers to better distinguish between (at least) the NiAs structure andthe antiNiAs structure.Steinhardt order parameters are a popular and convenient choice of features for structure classi-fication. They can be conceptualized as rotationally invariant moments in a multipole expansion ofthe near neighbor “bond” vector distribution on a unit sphere.30,270 They are defined with respect86to a reference atom i and exist in increasing orders, the quadratic Steinhardt order parameters beingQ`(i)≡(4pi2`+1`∑m=−`|Qm` (i)|2)1/2, (4.2)whereQm` (i) =1Nb(i)∑j∈Nb(i)Y m` (θ(rˆi j),φ(rˆi j)), (4.3)and Y m` are spherical harmonics of degree ` and order m (where ` is a whole number and−`≤ m≤ `is an integer). θ(rˆi j) and φ(rˆi j) are the polar and azimuth angles, respectively, of a unit vector rˆi jdirected from the reference atom i and to a neighbor j. The angles are defined with respect to anarbitrary coordinate system. Nb(i) is a suitable set of near neighbor bond vectors for atom i toaverage over, containing Nb(i) near neighbors, which must be carefully selected.The cubic Steinhardt order parameters areW`(i)≡∑m1+m2+m3=0(` ` `m1 m2 m3)Qm1` (i)Qm2` (i)Qm3` (i)(`∑m=−`∣∣Qm` (i)∣∣2)3/2, (4.4)where the numerator coefficients are Wigner 3 j-symbols. The sum in the numerator of Eq. (4.4)is over all valid triplet permutations of m that sum to zero. Both Q` and W` are defined such thatthey are invariant to rotation, translation, or inversion of the coordinate system for even values of`. Additionally, all order parameters have a similar magnitude, which improves the speed of neuralnetwork training because the gradients of NN parameters computed during backpropagation areproportional to the magnitude of inputs.271 When inputs are at different scales, the NN parametersconnected to large inputs are updated much faster than those connected so small inputs, leading toan imbalanced and inefficient learning process. In principle, there are an infinite set of Steinhardtorder parameters to choose from. Since Steinhardt order parameters are constructed from samplingspherical harmonics over a given set of near neighbor bond angles, they vary predictably for sets ofbond angles that differ in orientational overlap with the set of spherical harmonics of a given degree.Hence, if two crystal structures differ, then in principle a Steinhardt order parameter exists that candistinguish this difference. Combining multiple order parameters into a structure classificationscheme increases robustness against thermal noise and allows one to classify multiple classes of87structures simultaneously.In Eq. (4.3), Qm` (i) is calculated over some suitable set of near neighbor bonds Nb(i). Theselected set has a large impact on the resulting distribution of Q` and W` for a given structure.270The two simplest methods for selecting near neighbors are to use either a constant cutoff rc, or tofind a fixed set of n nearest neighbors surrounding each atom (i.e. Nb(i) = n for all i). The constantcutoff approach yields very different sets of neighbors for different pressures, temperatures, andalkali halides with a given rc, a highly undesirable feature. The use of a constant neighbor number,although requiring a parameter n, does not suffer from this density dependence. Mickel et al.270have criticized the use of a fixed Nb as asymmetric (i.e. the neighbors of atom i do not necessar-ily contain atom i in their neighborhoods) and yielding discontinuous Steinhardt order parameterswhen small changes in the relative distance between atoms results in new definitions of near neigh-bor sets. They introduce a neighborhood selection method based on a Voronoi tessellation, whereeach neighbor in the calculation of Qm` (i) is weighted by its relative Voronoi facet area with i. Theresulting Steinhardt order parameters are called Minkowski structure metrics (MSM). The MSMapproach to neighbor selection is appealing; it has the advantage of being both symmetric and freefrom discontinuities in Qm` (i). However, the computational cost of generating the neighbor listsand Voronoi facet areas used to calculate MSM is significantly higher than for fixed near neigh-bors. A similar, lower cost neighbor construction method, the solid-angle based nearest-neighboralgorithm (SANN),272 also provides some advantages over fixed Nb neighborhoods in that it givesmore meaningful order parameters near density inhomogeneities, but it too suffers from the samediscontinuities and asymmetrical behaviour as fixed Nb neighborhoods.The discontinuities and asymmetry inherent to employing a simple fixed Nb neighborhood def-inition may present obstacles to the manual construction of a structure classification scheme, butas shown in Table 4.1, I find that NN-based structure classifiers with fixed Nb in the order pa-rameters used as inputs can be trained to accurately classify local structural environments. I findmarginal accuracy gains in employing NN classifiers trained on MSM inputs versus the correspond-ing Steinhardt order parameters defined with a fixed Nb = 18 neighborhood when time convolutionis not used. When time convolution is employed, MSM typically perform slightly worse than fixedNb Steinhardt order parameters. Note that Nb(i) for MSM vary between atoms in a system; forsimulated alkali halides I observed average MSM neighborhood sizes of 15 – 18 near neighbors,depending on the system phase. I found that the accuracy of structure classification is highly de-pendent on the number of near neighbors included in Eq. (4.3), so even if a Voronoi tessellationneighborhood strategy offers some theoretical improvement over other near neighbor definitions,using more near neighbors than the average Voronoi neighborhood can overcome any accuracy88advantage it offers. So why use Nb = 18, rather than a higher number? Firstly, smaller values ofNb are less computationally costly. Secondly, my intention is to produce an accurate local structureclassification scheme with as small a neighborhood as possible, as this allows for a high resolutiondefinition of the boundaries between phases. All crystal structures considered here have at most 18neighbors in their first and second coordination shells combined. A minimum of two coordinationshells are required to distinguish (for example) an ideal rocksalt structure from an ideal NiAs struc-ture, because each appears locally octahedral if only the first shell is considered. Hence, Nb = 18is a natural choice to ensure good separability between all crystal structures.Classification accuracy can be improved by increasing the number of near neighbors† or, simi-larly, by averaging each Qm` from Eq. (4.3) over its set of near neighbors,273Q¯m` (i) =1Nb(i)+1∑j∈{ i,Nb(i)}Qm` ( j). (4.5)These Q¯m` are then used in place of Qm` (i) in Eq. (4.2) and Eq. (4.4). This averaging strategyincurs only a slight increase in computational cost, as the expensive generation of neighbor listsis unaffected and no additional Qm` need be calculated. This strategy implicitly expands the set ofnear neighbors, as information about near neighbors of near neighbors is included in Q¯m` . Hence,the locality of a structure classification scheme employing Q¯m` vs Qm` is reduced, which may beundesirable if high resolution structural details are of interest.In total, I generated three sets of structure classifiers: Set N, Set A, and Set V, all of whichemploy the same set of Steinhardt order parameters, but differ in the application of these order pa-rameters. Set N is the least computationally demanding, as it employs Steinhardt order parametersdefined with the set of 18 nearest neighbors. Set A uses the set of 18 nearest neighbors, but em-ploys the Q¯m` neighbor averaging of Eq. (4.5). Set V employs the MSM described above, withoutQ¯m` averaging, where neighbor weights are proportional to the relative Voronoi facet area such thatQm` (i) = ∑j∈Nb(i)Ai jAiY m` (θ(rˆi j),φ(rˆi j)), (4.6)where Ai j is the Voronoi facet area between atom reference atom i and neighbor j, and Ai is thetotal surface area of the Voronoi cell surrounding i. Based on a balance between computational costand classification accuracy, I use W` and Q` with `= {2,4,6,8} plus the atom ID, totaling 9 inputs.†I trained several NNs on order parameters calculated with neighbor sizes up to Nb(i) = 18 and saw steady improve-ment in accuracy.89One practical reason to include both Q` and W` of the same ` in a set of inputs is that both can besimultaneously calculated with the highly efficient and parallelized Freud python library.274 Calcu-lation of neighbor lists and the Steinhardt order parameters is the most computationally expensivestep of my structure classification system, and W` can practically be calculated for free from theQ`. I found that inclusion of W` in the NN inputs improves classification accuracy versus Q` alone.4.2.3 Neural Network Training DataTo train the binary salt structure classification NNs described above, I generated a huge quantityof data in the form of MD trajectories labeled by their phase. MD simulations of all lithiumhalides and NaCl in the nine bulk phases of interest using both JC(SPC/E) and TF models wereperformed. Since these salt/model combinations are unstable in the CsCl crystal structure, I addedsimulations of bulk CsCl-type crystals using the JC(SPC/E) and TF models for the CsCl salt, whichhave metastable CsCl structures at high temperatures (see Section 5.3.2). Each training trajectorycontained a minimum of 5,000 ions, with the exact number depending on cell shape and density(e.g. LiF simulations in some phases included up to 11,616 ions to accommodate a simulation cellsize that was more than twice the interaction potential cutoff distance). Trajectory frames weresaved at 1 ps intervals.To generate an unbiased set of training data, 2 million training points were randomly and uni-formly sampled for each of the nine alkali halide phases considered (these are detailed in the Ap-pendix, Section A.2). In this context, a training point includes the full set of inputs (possiblyincluding multiple time points) necessary to generate one classification output. Half of the train-ing points from each phase were randomly placed into a validation data set which was not usedfor training. Hence, each neural network was trained on 9 million data points and validated onanother 9 million. Increasing the training and validation data sets to 18 million data points eachoffered no improvement in overall accuracy on the validation set versus training on 9 million datapoints. Hence, the NN classification accuracies appear saturated with respect to their training dataat 9 million training points. Any residual classification error is likely attributable to the overlapof classes in input space, which primarily originates from thermal noise in the MD trajectoriesused for training and validation, particularly at high temperature. Convolving over more trajectoryframes reduces classification error as this allows the NN to filter out thermal fluctuations.To diversify training inputs and improve robustness in the structure classifiers, in addition tousing multiple alkali halide salts, simulations employed multiple interaction potentials for a givensalt. This is based on the assumption that different interaction potentials and salts tend to explore90different local structural fluctuations, even within the same phase. I initialized all training trajec-tories as either a randomly generated liquid, equilibrated to the starting temperature and pressure,or one of eight ideal energy-minimized crystal structures: rocksalt, wurtzite, β -BeO, sphalerite,5-5, NiAs, antiNiAs,* or CsCl. Crystallographic information about these crystal structures, alongwith a discussion of the crystal energy minimization scheme, is available in a previous publication(particularly in the supplementary materials).86 Solid and liquid simulations were initialized at 0 Kand 2200 K, respectively. All training simulations employed simulated annealing with a 100 Kns−1heating or cooling rate for solids and liquids, respectively. The solids were heated continuously to2200 K and the liquids were cooled to 0 K. Since these training simulations were not at equilib-rium due to changing temperatures, the quickly relaxing Berendsen barostat77 was employed (witha 1 ps time constant) to hold the pressure fixed, rather than the Parrinello-Rahman thermostat usedin other simulations. The Berendsen barostat does not generate the correct volume fluctuations ofthe NPT ensemble,78 but these are not important for generating training geometries.All training simulations produced a phase change (melting or freezing) at some point duringthe trajectory. Since all training and validation data are labeled by their initial phase, it is necessaryto crop the training trajectories to include only data prior to any phase change. All trajectory framesfollowing phase changes were excluded from the pool of training data, plus a 100 ps buffer prior tothe phase change. Phase changes were detected through rapid volume and potential energy changesin the trajectories. After trajectory cropping, a set of labeled training trajectories remained, eachcontaining a single alkali halide salt in a specific phase. Each training trajectory was designed tosample the maximum range of temperatures over which each salt/model/structure remained stablein simulations. In total, 9.27 billion usable labeled data points were generated.4.2.4 Neural Network ArchitectureSince physical systems at finite temperature have inherent fluctuations, local atom arrangementsand their associated order parameters are buffeted by random noise. For example, in a simulation ofa globally rocksalt structure, an atom may be randomly placed into a position that leaves it lookinglocally liquid-like for a time step, but is quickly returned to rocksalt thereafter. To help reduce thenoise caused by these short-time thermal fluctuations, I created a set of CNN structure classifiers,wherein local structural information is convolved over predefined periods of time ranging from 2 psto 50 ps. The Steinhardt order parameters provide local spatial information that the CNN convolvesin time. The result is a classifier with increased robustness compared to one that has no temporal*AntiNiAs is NiAs with the metal and halide positions reversed. The other crystal structures studied here aresymmetric to this operation.91width, assuming that filtering out rapid structural fluctuations is desirable. The averaging out ofshort-lived structures may not always be desirable, so I also created NN classifiers that classifystructures based on single time frame inputs. These can be used to detect, for example, fluctuatingcrystal-like structures in molten alkali halide simulations.All neural networks were constructed with TensorFlow Keras version 2.7275–277 and their ex-act architecture was optimized with a Bayesian optimization scheme278 intended to maximize themodel accuracy for a given sample of input data. In searching for a suitable network architecture,I varied the number of dense layers, the number of nodes per layer, the activation functions, thenumber of convolutional filters, and the convolutional kernel size. The resulting classifiers typi-cally include 1 convolutional layer, followed by two fully connected dense network layers, and afinal classification layer, plus a Softmax function layer which normalizes the range of each NNclassification output to [0,1] and the sum of all outputs to 1. The Softmax layer simply allows thefinal NN output vector to be interpreted as a set of inferred classification probabilities, one for eachstructure. In Figure 4.1 I show the architecture of one of my most accurate CNN classifiers. In thispaper, I name the classifiers by their convolutional time width and time step in ps, while classifierswithout time convolution are given the name “0-0”. Classifiers from Set A and Set V are also givena suffix “a” or “v”, respectively. For example, the CNN classifier from Set A that convolves over a10 ps time window with 1 ps time steps is given the name “10-1a”.For any of my NN classifiers, the class (i.e. structure) with the highest probability in the outputlayer can be straightforwardly assigned to each reference atom. However, the output classificationprobabilities also give the opportunity for further analysis. For example, atoms with a maximumclassification probability in any single class below some threshold can be classified as “interfacial”,since they are typically long-lived only at the interface between different phases. It is also useful tovisualize crystallization with atom transparency given by a function of their liquid class probabili-ties. For most visualizations presented here, I set the atom transparency to a (increasing) sigmoidfunction of the liquid class probability. This choice emphasizes crystal-like and interfacial atomswithout completely hiding ions with high liquid probability.4.2.5 Neural Network TrainingNeural networks were trained with Adam stochastic gradient descent optimization281 as imple-mented in Keras, using categorical cross-entropy as the loss function. Initially, I used small batchesof input data and a high learning rate (optimization step size) to train my networks. As trainingproceeded, the batch size was increased and the learning rate reduced until the loss function was9211 sequential trajectory frames with 1 ps time step Input layer Output: 11 x 9 9 features kernel size: 6 Convolutional layer ELU activation Output: 30 x 6 Dense layer 1 ReLU activation Output: 300 x 1 Flatten Output: 180 x 1 Dense layer 2 ReLU activation Output: 50 x 1 Classification layer Linear activation Output: 9 x 1 softmax function Probability output layer Softmax activation Output: 9 x 1 reference atom and 18 neighbors calculate features Rocksalt Liquid Wurtzite 5-5 NiAs Sphalerite β-BeO AntiNiAs CsCl ID ID ID ID ID ID ID ID ID ID ID Q2(t1) Q2(t2) Q2(t3) Q2(t4) Q2(t5) Q2(t6) Q2(t7) Q2(t8) Q2(t9) Q2(t11) Q2(t10) W2(t1) W2(t2) W2(t3) W2(t4) W2(t5) W2(t6) W2(t7) W2(t8) W2(t9) W2(t11) W2(t10) Q4(t1) Q4(t2) Q4(t3) Q4(t4) Q4(t5) Q4(t6) Q4(t7) Q4(t8) Q4(t9) Q4(t11) Q4(t10) W4(t1) W4(t2) W4(t3) W4(t4) W4(t5) W4(t6) W4(t7) W4(t8) W4(t9) W4(t11) W4(t10) Q6(t1) Q6(t2) Q6(t3) Q6(t4) Q6(t5) Q6(t6) Q6(t7) Q6(t8) Q6(t9) Q6(t11) Q6(t10) W6(t1) W6(t2) W6(t3) W6(t6) W6(t5) W6(t6) W6(t7) W6(t8) W6(t9) W6(t11) W6(t10) Q8(t1) Q8(t2) Q8(t3) Q8(t4) Q8(t5) Q8(t6) Q8(t7) Q8(t8) Q8(t9) Q8(t11) Q8(t10) W8(t1) W8(t2) W8(t3) W8(t4) W8(t5) W8(t6) W8(t7) W8(t8) W8(t9) W8(t10) W8(t11) 11 time steps Figure 4.1: The architecture of the NN classifiers 10-1, 10-1a, and 10-1v in Table 4.1 whichdiffer in their inputs (see Section 4.2.2). Trainable weights between dense layers arerepresented by black connections while biases are represented by red connections. Asubset of connections between the input, convolutional, and flatten layers are indicatedby shaded fields with colors to aid in visualization. Exponential linear unit (ELU),rectified linear unit (ReLU),279 linear, and softmax are types of activation functions.28093minimized as much as possible. Training continued until classification accuracy converged to 10−4between training epochs (i.e. complete passes of all training data through the optimization scheme).Various regularization conditions were employed, including NN node dropouts, to prevent overfit-ting to training data. Overfitting was monitored by comparing the training set accuracy versus thevalidation set accuracy (overfitting produces higher accuracy on the training set than the validationset). Within each set of NNs, seven neural network classifiers were created for differing purposes:six are CNNs, differing in the choice of trajectory frames used for time convolution, while the re-maining one is a non-convolutional NN, which classifies structures without convolving over a timewindow. Table 4.1 in Section 4.3.1 gives a summary of all 21 structure classification NNs.4.3 Results and DiscussionI next discuss the utility of our NN binary salt structure classifiers. Section 4.3.1 compares andcontrasts the statistical accuracy of the classifiers on a validation data set, and provides an exampleof how one can filter out structural fluctuations from MD trajectories of binary salts using variousNN classifiers. Section 4.3.2 demonstrates the capacity of my NN classifiers for following the crys-tallization of alkali halides into different crystal structures from melts. Our classification approachaccurately detects crystallites of different structures from their initial homogeneous nucleation inthe melt through to complete saturation of the simulation cell with crystalline ions, correctly iden-tifying interfacial ions along the way.4.3.1 Neural Network Classifier AccuracyTwenty-one NN binary salt structure classifiers are developed, each capable of distinguishing thelocal structure of an ion in a binary salt simulation as one of nine structures, as discussed in Sec-tion 4.2.3. The NN classifiers, partitioned into three sets of seven, are summarized in Table 4.1.As measured on an validation data set of 9 million points not used in training (1 million valida-tion points of each class) at temperatures between 0 and 2200 K, the classifiers vary in accuracybetween 91.61 % and 99.99 %. In Figure 4.2 I give the confusion matrix for my most accurateCNN classifier from Set N. Confusion matrices provide detailed information about the source ofinaccuracies in a predictive classification model. Diagonal elements are predictions that matchthe given classification label. As one might expect, the liquid phase is a relatively high source ofclassification inaccuracy for the more spatially localized classifiers of Set N and Set V, as liquidions often undergo local crystal-like fluctuations (particularly at temperatures near or below theMP). For example, classifier 10-1 incorrectly classifies liquid as the 5-5 crystal structure 0.05 % of94Set N Accuracy [%] Set A Accuracy [%] Set V Accuracy [%] tconv tstep No. Frames0-0 91.61 0-0a 99.58 0-0v 92.50 0 - 12-1 98.37 2-1a 99.94 2-1v 98.49 2 1 34-1 99.41 4-1a 99.97 4-1v 99.23 4 1 510-1 99.90 10-1a 99.99 10-1v 99.74 10 1 1110-5 98.35 10-5a 99.95 10-5v 98.42 10 5 320-5 99.33 20-5a 99.97 20-5v 99.25 20 5 550-5 99.90 50-5a 99.99 50-5v 99.82 50 5 11Table 4.1: Summary of the NN classifiers. Set N classifiers are those with order parametersdefined by the set of 18 nearest neighbors. Set A classifiers employ order parametersdefined by the set of 18 nearest neighbors, where the Qm` are averaged over the set of nearneighbors,273 as in Eq. (4.5). Set V classifiers use Voronoi tessellation to construct andweight the set of near neighbors, as in Eq. (4.6).270 All classifiers include an atom ID asan input, followed by Q` and W` pairs with ` = {2, 4, 6, 8}. tconv is the total time windowof convolution in ps, tstep is the time between convolved trajectory frames in ps, and “No.Frames” indicates the total number of trajectory frames convolved over. Classifiers arelabeled with tconv-tstep, and an additional suffix “a” or “v” is given to Set A and Set Vclassifiers, respectively.the time. Notably, Set A classifiers, which include structural information from the near neighborsof near neighbors, work exceptionally well for classifying liquid ions. Set A classifiers filter outsmall structural fluctuations detectable with Set N or Set V classifiers. Although performing muchbetter in most cases, Set A classifiers perform slightly worse than Set N or Set V classifiers indistinguishing NiAs from antiNiAs structures. This is reflected in the Steinhardt order parameterpair correlation plots given in Figures 4.4, 4.3, and 4.5. In these figures, off-diagonal elements arecovariance plots for pairs of features, while diagonal elements are kernel density estimate plots offeature variance. As can be seen in Figure 4.3, NiAs and antiNiAs appear to strongly overlap in theSet A input space but less completely in the Set N (Figure 4.4) and Set V (Figure 4.5) input spaces.All classifying NNs are useful for probing mechanistic details of various alkali halide phenomena,but the exact requirements of structural classification may vary.The calculation of neighbor lists and order parameters is the computational bottleneck in allstructure classifiers (the class inference step has negligible cost). One might expect, therefore,that the computational cost of classifier 10-1 is about 11 times higher than 0-0. However, this95LiquidRocksaltWurtzite 5-5NiAsSphaleriteβ-BeOAntiNiAsCsClPredictionLiquidRocksaltWurtzite5-5NiAsSphaleriteβ-BeOAntiNiAsCsClActual99.79 0.01 0.06 0.05 0.00 0.02 0.03 0.00 0.000.02 99.88 0.00 0.01 0.06 0.00 0.00 0.05 0.000.06 0.00 99.80 0.05 0.00 0.00 0.06 0.00 0.000.07 0.01 0.09 99.89 0.00 0.00 0.00 0.00 0.000.01 0.05 0.00 0.00 99.91 0.00 0.00 0.00 0.010.02 0.00 0.00 0.00 0.00 99.98 0.00 0.00 0.000.03 0.00 0.05 0.00 0.00 0.00 99.91 0.00 0.000.00 0.04 0.00 0.00 0.00 0.00 0.00 99.93 0.010.00 0.01 0.00 0.00 0.02 0.00 0.00 0.02 99.98020406080100Fraction[%]Figure 4.2: The confusion matrix for NN classifier 10-1, with 99.90 % total classificationaccuracy on the validation set averaged across all classes. All numbers are percentages,and each column is normalized to 100 %.is not necessarily the case as the convolutional time window of each time step usually overlapswith previous steps such that most order parameters can be reused and need not be recalculated.In general, classifiers in Set V have approximately five to six times higher computational costthan those of Set N or A due to the computationally demanding Voronoi tessellation step. Set Vclassifiers are therefore generally inferior to other available structure classifiers for the systemsconsidered here, as their computational cost is higher and their accuracy is generally similar orlower, as shown in Table 4.1. Note that the computational cost of a typical analysis with even themost costly structure classifier is very low compared to the cost of generating MD trajectories. Thisrelatively low cost is made possible in part by the highly efficient and parallelized Freud pythonlibrary274 which is used for building neighbor lists and calculating order parameters.96Figure 4.3: Pairwise correlation relationships for Set A classifier order parameters, used asinput features. The features are Steinhart order parameters calculated using the 18 near-est neighbors, with additional Qm` averaging.97Figure 4.4: Pairwise correlation relationships for Set N classifier order parameters used asNN inputs, grouped by class. This and other pairwise correlation plots are all based on50000 randomly-sampled data training points. The features are Steinhart order parame-ters calculated using the 18 nearest neighbors, without Qm` averaging.98Figure 4.5: Pairwise correlation relationships for Set V classifier order parameters, used asinput features. The features are Steinhart order parameters with sets of near neigh-bors constructed from Voronoi tessellation, with neighbor weights given by the relativeVoronoi facet area.270 Qm` is not averaged over the set of near neighbors.994.3.2 Alkali Halide Crystal NucleationI now analyze two example simulations of homogeneous alkali halide crystal nucleation. The firstis a simulation of NaCl/JC(SPC/E) while the second is LiBr/TF, both with N = 20,000 ions. TheNaCl/JC(SPC/E) system was chosen as it strongly favours the rocksalt crystal structure and pro-vides a quintessential example of nucleation and crystallization into the crystal structure shared bymost alkali halides under ambient conditions. The LiBr/TF system was chosen as it was been pre-viously shown to favour the wurtzite crystal structure due to defects in the LiBr/TF interaction po-tential.10,86 Experimentally, LiBr occupies a rocksalt crystal structure while wurtzite is metastableat low temperatures.16 Both simulations are performed at constant pressure with P = 1 bar andbegin as a randomly generated liquid, followed by an equilibration period at 1200 K and 1400 Kfor NaCl/JC(SPC/E) and LiBr/TF, respectively. The equilibration is then followed by cooling at50 Kns−1, maintaining fixed pressure. In the NaCl/JC(SPC/E) simulation, nucleation is detectedafter 6200 ps of cooling (890 K, whereas the MP of NaCl/JC(SPC/E) is 1290 K). Crystal growthproceeds rapidly, filling the simulation cell approximately 125 ps after nucleation. In the LiBr/TFsimulation, the initial stages of nucleation are detected after 17440 ps (528 K, whereas the MP ofLiBr/TF is 819 K, see Section 5.3.2 for a discussion of the melting points of common alkali halideforce fields). Crystal growth proceeds somewhat slower in the LiBr/TF simulation due to the lowernucleation temperature, filling the simulation cell approximately 200 ps after nucleation.Figure 4.6 gives a demonstration of structure analysis, from a single frame of the NaCl/JC(SPC/E)simulation described above, with four different NN classifiers. Small structural fluctuations can bedetected by the Set N or Set V classifiers, which are higher resolution than Set A classifiers. Set Aclassifiers detect only relatively large structural fluctuations. This difference in spatial resolution isdemonstrated clearly in Figure 4.6 by the comparison of structural classification from NN 0-0 vs0-0a (or 10-1 vs 10-1a). Short-lived structural fluctuations can also be filtered out of an analysisby application of time convolution with an appropriate time window. This idea is demonstrated inFigure 4.6 by comparison of the analyses from NN 0-0a vs 10-1a (or 0-0 vs 10-1). In general, clas-sifiers with longer time convolution windows filter out structural fluctuations with longer lifetimes.In the Figure 4.7 I provide a visual comparison of all 21 structure classifiers applied to a time frameof the same simulation.In Figures 4.8 and 4.9 I give structural analyses of simulated homogeneous liquid-to-solid crys-tallization, described above, with NN classifier 10-1a. In both analyses, I assign ions to the classwith the highest inferred probability given by the NN classifier. Weakly classified ions, those witha highest classification probability below some threshold, are labeled as interfacial. The threshold100Filter Short-Lived Fluctuations with Time Convolution Trajectory Snapshot Instantaneous Med to Large Crystallites NN 0-0 NN 0-0a Instantaneous Small to Large Crystallites Filter Short-Lived Fluctuations with Time Convolution Figure 4.6: A demonstration of four NN classifiers, applied to the same frame trajectorysnapshot, intended for tracking structural fluctuations of different sizes and lifetimes.The central visualization shows the trajectory snapshot prior to classification. The fourouter visualizations show ions classified into the class with highest probability given bythe indicated NN classifier, with classification color codes given in the legend. Ion trans-parency is set by a monotonic function of liquid-class probability, as provided by the NNclassifier. All five visualizations show the same frame of a NaCl/JC(SPC/E) simulation(N = 20,000 ions) undergoing homogeneous crystal nucleation into a rocksalt structurefrom a supercooled liquid phase.101Set N Set A Set V 0-0 2-1 4-1 10-1 10-5 20-5 50-5 0-0v 2-1v 4-1v 10-1v 10-5v 20-5v 50-5v 0-0a 2-1a 4-1a 10-1a 10-5a 20-5a 50-5a Figure 4.7: Comparison of all 21 NN classifiers applied to an example NaCl crystallizationfrom the melt using the JC(SPC/E) model.14 Each trajectory snapshot is the same frame,classified by the corresponding NN classifier, with ion transparency set to a sigmoidfunction of the liquid classification probability. Ions are colored by their most probablestructure classification, with the colors given in the legend.102is arbitrary, with results depending both on the system and classifier. Typically, ions are weaklyclassified at the interfaces between phases and within single phases during short-lived fluctua-tions. Increasing the time convolution window reduces the number of fluctuating weakly classifiedions within bulk phases, but tends not to remove weakly classified ions at the boundaries betweenphases. In Figure 4.8, ions with a classification probability below 90 % in any class are labeled asinterfacial, while in Figure 4.9, the threshold is set to 75 %. Videos of these two examples of thecrystal nucleation and growth, classified by NN 10-1a, are provided as part of the SupplementaryMaterials.As can be seen in Figure 4.8, very small crystallites and those with lifetimes less than 10 psare largely filtered out by application of NN 10-1a. Still, as can be seen in Figure 4.8 (a), severalsubcritical nuclei can be observed as they fade in and out of existence over time. According toclassical nucleation theory,5 these crystallites do not reach the critical size required for irreversiblecrystal growth. Irreversible crystal growth appears to follow only after a critical nucleus of approx-imately 20 ions is reached. This cluster size is comparable to critical nuclei observed in previouswork on simulated NaCl crystallization from supersaturated aqueous solutions,9 which reportedirreversible crystal growth proceeding from highly crystalline NaCl clusters once they surpassedapproximately 20-30 ions in size.The simulation trajectory illustrated in Figure 4.9 is for the LiBr/TF system, for which wurtziteas the lowest energy crystal structure at (T = 0, P = 0),10 as demonstrated in Section 3.3. In thiscase, crystallization into a wurtzite crystal structure is observed. Ions classified as 5-5, β -BeO, orsphalerite (all of which are similar in their first coordination shell to wurtzite) appear as well, butthis does not imply a multiphase crystal. The ions classified as 5-5, β -BeO, and sphalerite mostlyappear, along with interfacial atoms, at the interface of the growing wurtzite crystal. Later in thesimulation, 5-5 and β -BeO ions appear in disordered regions of the simulation cell, possibly causedby the misalignment of the simulation cell with the primary wurtzite crystal.For LiBr/TF, crystal growth proceeds more slowly than in the NaCl/JC(SPC/E) example. Thisis primarily due to slower diffusion in the molten LiBr, which is at a lower temperature duringthe wurtzite crystal growth. The diffusion constant of liquid NaCl/JC(SPC/E) (averaged over bothcations and anions) prior to nucleation at 890 K is D = 2.0×10−5 cm2/s while the correspondingdiffusion constant of LiBr/TF prior to nucleation at 528 K is D = 5.7×10−6 cm2/s.An interesting difference between the NaCl/JC(SPC/E) and LiBr/TF nucleation simulations isthe NN classification of fluctuating crystallite structures in the liquid and those at the solid-liquidinterfaces. With NaCl/JC(SPC/E), the transient structures are octahedral (rocksalt, NiAs, antiNiAs)or trigonal bipyramidal (5-5), while the LiBr/TF liquid fluctuates into tetrahedral (wurtzite, β -BeO,103a) b) Figure 4.8: a) Visualization of NaCl/JC(SPC/E) crystallization into rocksalt from the melt(N = 20,000 ions). Snapshots are separated by 10 ps. Each ion is classified with NN10-1a based on the highest classification probability; ions with probability less than90 % in any single class are classified as interfacial. Each ion’s transparency is givenby a monotonic function of its liquid classification probability. b) The mole fractionof each class plotted over time. Inset is the same plot, magnified in the y-axis. Thevisualization in a occurs between 95 – 175 ps in b. Classification color codes are givenin the legend.104sphalerite) or trigonal bipyramidal (5-5) structures. Some of the long-lived subcritical nuclei whichfollow these structual trends can be seen in the liquid surrounding the developing crystals in Figures4.8 (a) and 4.9 (a). At the rocksalt-liquid interface of NaCl/JC(SPC/E), atoms classified as NiAs,5-5, and antiNiAs are most common. These are low-energy metastable bulk crystal structures, somay appear as accessible fluctuations at the semi-disordered rocksalt-liquid interface. Additionally,defect sites between rocksalt crystals produce planar zones of long-lived NiAs-classified ions. Atthe wurtzite-liquid interface of LiBr/TF, the appearance of low-energy crystal structures is similarto the NaCl/JC(SPC/E) case, except that rocksalt-like fluctuations are not common. Rocksalt is adense low-energy structure that might be disfavoured by entropy in the less dense LiBr/TF system.Additionally, 5-5 ions are more common at the wurtzite-liquid interface than would be expectedby bulk lattice energy considerations alone, as the rocksalt crystal structure is lower in energy than5-5 for LiBr/TF (see the Appendix, Figure C.1). It appears that 5-5 structures may play a role inreducing wurtzite-liquid interfacial tension.4.4 Summary and ConclusionsI report twenty-one trained NNs, divided into three sets, which (in most cases) contain 1D CNNarchitecture. The NNs are trained to accurately classify simple binary salts into the liquid phase orone of eight solid crystal structures: rocksalt, wurtzite, CsCl, 5-5, sphalerite, NiAs, AntiNiAs, andβ -BeO. The NN classifiers are extensively trained on data from multiple simulated alkali halidephases at temperatures between 0 – 2200 K. The classifiers take as input information about thelocal orientation of near neighbor bonds surrounding each reference atom, plus an atom identityparameter (which is 0 for metal ions or 1 for halide ions). The local bond orientational informationis encoded in eight Steinhardt order parameters,30 whose near neighborhood set, near neighborweighting, and Qm` averaging vary between the three NN classifier sets. The output of the NNclassifiers is a set of classification probabilities for the reference atom. The structure classifiers areshown to be highly accurate (up to 99.99 %) for classifying atoms in bulk alkali halide phases, evenat high temperatures.I demonstrate that NN structure classification accuracy can be significantly improved by ap-plication of time convolution. CNN classifiers improve accuracy by filtering out rapid structuralfluctuations occurring at finite temperatures. CNN classifiers with larger time windows filter outlonger lived structural fluctuations. This allows one to select an appropriate time convolution win-dow that attenuates structural fluctuations shorter than some desired lifetime. Additionally, I showthat NN output classification probabilities can be employed to track interfacial ions associated with105a) b) Figure 4.9: a) Visualization of LiBr/TF crystallization into wurtzite from the melt (N =20,000 ions). Snapshots are separated by 18 ps. Each ion is classified with NN 10-1a based on the highest classification probability; ions with probability less than 75 %in any single class are classified as interfacial. Each ion’s transparency is given by amonotonic function of its liquid classification probability. b) The mole fraction of eachclass plotted over time. Inset is the same plot, magnified in the y-axis. The visualizationin a occurs between 36 – 180 ps in b. Classification color codes are given in the legend.106nucleating crystal structures, by identifying weakly classified ions (those with maximal classifica-tion probability below a threshold) as interfacial.To demonstrate the utility of my most accurate NN structure classifier, I applied it to MDsimulations of the homogeneous nucleation of NaCl/JC(SPC/E) and LiBr/TF. I discuss these sim-ulations, showing that they produce different crystal structures. NaCl/JC(SPC/E) crystallizes intorocksalt while LiBr/TF crystallizes into wurtzite. Experimentally, both salts occupy a rocksaltstructure under ambient conditions. LiBr/TF crystallizes into the wurtzite structure due to a defectin the LiBr/TF interaction potential, but the simulation does provides insight into wurtzite crystal-lization, which occurs experimentally for LiBr under certain conditions.16 The NN classifier alsodetects subcritical nuclei that fluctuate in and out of existence in the supercooled liquid phases.The structures of these subcritical nuclei differ for the two MD trajectories considered, reflectingthe structural difference in the nucleating crystals.The NN structure classification framework presented here has many actual and potential usesin applications beyond what is discussed in this chapter, and a few of these will be introduced inChapters 5 and 6. I also believe that my NN classifiers will perform well for analyzing structuresin binary salt crystal nucleation from supersaturated solutions, just as they do for crystal nucleationfrom the melt. My NN classification algorithms may also be useful for characterizing the struc-tures of binary salts beyond the alkali halides that they were trained on, as well as for analyzingsimulations involving mixtures of binary salts.107Chapter 5Automated Melting Point CalculationsThe research presented in this chapter has been published in the Journal of Chemical Physics withcoauthor G. N. Patey.2265.1 IntroductionThe melting point of a model crystal is a key parameter for judging the accuracy and transferabilityof the force field employed. It is a widely available experimental observable, yet computationally,the MP is extremely sensitive to force field details, and depends on thermal properties of boththe solid and liquid phases. In Chapter 6, I use MP calculations to judge the utility and accuracyof many newly parameterized classical LiX force fields. As such, it is important to develop andcharacterize an automated method for efficiently obtaining accurate MPs of simulated alkali halidesystems. The method introduced in this chapter can automatically and robustly calculate simulatedalkali halide MPs to high accuracy, although the computational cost raises steeply with the desiredaccuracy.The calculation of an MP at a specified pressure using MD simulations may at first seem to bea relatively straightforward task. The naive approach is to start with a energy-minimized periodicsolid at 0 K and a specified pressure, then raise the thermostat temperature steadily until the crystalstructure disintegrates, yielding the apparent melting temperature Tm. However, this approach failsto account for hysteresis due to the kinetic free energy barrier of homogeneous nucleation, whichcan be very large. Hysteresis effects allow for both metastable liquids or solids to persist for longperiods well below or above the thermodynamic Tm of a system, respectively.282 Rather than thesystem Tm, the naive single-phase heating or cooling approach yields thermal limits of stability.108These stability limits are the maximum or minimum temperature at which a metastable solid orliquid can exist, respectively, before the barrier of nucleation to the thermodynamically stable phasebecomes small enough to be overcome by thermal motion within simulation timescales. The upperand lower limits of stability can be used to estimate Tm via the hysteresis method,283Tm ≈ T++T−−√T+T−, (5.1)where T+ is the high temperature solid limit of stability, and T− is the low temperature liquid limitof stability. However, this approach assumes that the free energy barrier to melting and freezing isidentical, which is not necessarily true for the alkali halides.A 2012 article by Zhang and Maginn284 gives a detailed review of the methods used for calcu-lating the MPs of model systems with MD simulation. These can be generally classified into twocategories: direct simulation and thermodynamic integration methods. In general, direct simulationapproaches are simpler than thermodynamic integration methods, although direct simulation maybe less accurate in some cases. Previous work on model alkali halides has shown that at least onetype of direct simulation method, the liquid-solid coexistence approach, gives equivalent resultsto more complicated thermodynamic integration methods for calculating alkali halide MPs.19,48,49The liquid-solid coexistence approach for MP calculation involves the direct NPT simulation of asolid-liquid interface with the goal of finding a temperature and pressure at which the solid and liq-uid remain in dynamic equilibrium, such that the chemical potentials of both phases are matched.By introducing a solid-liquid interface, the free energy barrier to both melting and freezing is elim-inated in many systems. The absence of large barriers to attachment and detachment is a necessarycondition for the method to work in practical simulations. If barriers to ion attachment and detach-ment persist once an interface is introduced, then the method cannot be employed to calculate MPswith any degree of precision.In this chapter, I introduce an automated liquid-solid coexistence method for calculating theMPs of alkali halides. The approach involves repeated simulations of liquid-solid interfaces thatconverge toward the model system’s MP. The method I introduce employs the binary salt NNstructure classifiers introduced in Chapter 4 as an accurate and robust solid-liquid global phaseclassifier. Use of this NN-based classifier has the added benefit of checking that the correct solidcrystal structure is maintained.1095.2 MethodsIn this chapter, I give results generated from many alkali halide MD simulations, all of which aretaken at constant particle number, pressure, and temperature. I employ two of the interactionspotentials discussed in Section 2.3; the JC14 and the TF12,13 models. The MD simulation detailsand methodology employed here are given in Sections 2.7 and 2.8.5.2.1 Overview and Main AlgorithmI implemented one of my classification NNs introduced in Chapter 4 at the core of an automatedMP calculation algorithm for model alkali halides, an overview of which is given in Figure 5.1. Theidea behind the algorithm is to systematically sample a series of temperatures using the direct co-existence approach.48 My algorithm works by fixing pressure and directly simulating systems witha solid-liquid interface at specific temperatures to narrow a melting/freezing temperature bracketuntil below a desired threshold.Besides the MD simulation parameters, the MP calculation algorithm itself takes several keyparameters as input: a system and associated interaction potential (e.g. NaCl with JC(SPC/E)potential); the initial temperature T0; the crystal structure (i.e. unit cell) of the solid phase; astopping criteria ∆Tmin that halts the program when TUB− TLB ≤ ∆Tmin where TUB is the lowestdetected melting T and TLB the highest detected freezing T ; the maximum simulation time allowedto detect a melting or freezing event tmax; the minimum change in the mole fraction of liquid,∆x∗liq, required for the algorithm to report a melting or freezing event; and the minimum number ofparticles N (the algorithm may increase N, as explained below).The MP calculation algorithm is designed to calculate MPs efficiently and automatically. Itsimulates a series of planar liquid-solid interfaces at temperatures that systematically approach theMP of the model system. The procedure is analogous to a direct search line minimization of anarbitrary 1D function where no derivative information is available. It begins with an MD simulationof time tmax (I use tmax = 5 ns) at a temperature T ∗ (T ∗ is initialized to T0). The algorithm checksthe sample for melting or freezing, employing a NN classifier at checkpoint intervals of tc (I chosetc = 25 ps). Hence, the MP calculation procedure is efficient even if the initial guess temperatureT0 is far from the system MP, as the algorithm will often only proceed for a small number ofcheckpoints before detecting a melting or freezing event and moving to the next temperature.For the MP calculations reported here, I chose systems with a minimum of 10,816 ions ini-tialized in a 1:1 ratio of solid to liquid particles. The height (c) to width (a, b) ratio of the systemboundaries is approximately 2:1 (a visualization of an example initial system setup is shown in110Initialize: T* = T0 TLB = 0 TUB = ∞ result = [ ] Melt: update T* = T* ˗ Tstep Freeze: update T* = T* + Tstep (TUB ˗ TLB) ≤ ΔTmin? No Yes Tm = (TUB + TLB)/2 Return Tm (TUB ˗ TLB) ≤ ΔTmin? Yes Both freezing and melting points found? No Yes Bracket Reduction Algorithm No Bracket Reduction Algorithm Tstep = γTstep Indeterminate point found? Update: TLB TUB result No: System Melted or Froze at T* Did System Freeze or Melt? Yes Check for melting or freezing, at T* Example of an indeterminate point: NaCl/JC (SPC/E) at 1289.5 K There are three possibilities for a melting-point simulation that can be reported: indeterminate (0), melting (1), or freezing (2). An indeterminate outcome is where neither melting nor freezing has occurred by tmax. Figure 5.1: Overview of the melting point algorithm. T ∗ is the current temperature to betested; TLB is the current lower bound, which is updated by freezing events; TUB is thecurrent upper bound, which is updated by melting events; Tstep is the temperature stepsize; γ is the step size multiplier; “result” is an expanding list of recorded outcomesfrom each temperature tested; ∆Tmin is the desired melting/freezing bracket; and Tm isthe final reported melting point.111b c Figure 5.2: An initial geometry for MP calculations. View looking down the a-axis of a fullyrelaxed rocksalt-liquid interfacial geometry (N = 11664 ions, P = 1 bar, T = 1289.5 K)for the NaCl/JC(SPC/E) model (MP is 1289.9(1) K for this system). For visualizationof the structure, each atom is classified by NN 0-0a, where ions with less than 99 %classification probability are reclassified as interfacial. This and all other trajectorysnapshots in this dissertation are visualized with Open Visualization Tool (OVITO).285Figure 5.2). The solid and liquid phases are both arranged into periodically repeating slabs alongthe ab plane (the c dimension is also periodic). The cell angles of the simulation box are givenby the angles of the solid unit cell. The (001) plane of the solid (with respect to the conventionalunit cell) is initially exposed to the liquid in all cases. For example, a typical rocksalt structure MPcalculation in this work employs a slab consisting of 9×9×9 conventional unit cells, where eachunit cell contains 8 ions, placed below a liquid layer of 5,832 ions for a total of 11,664 ions. Fordense alkali halides such as LiF, the exact number of particles required may increase well above theminimum depending on the density, unit cell angles, and the interaction potential cutoff distance,such that the shortest distance across the simulation cell is more than twice the interaction potentialcutoff.†5.2.2 Preparing the Liquid-Solid InterfaceIn order to check for melting or freezing, the MP calculation algorithm requires that two-phasesystems be repeatedly initialized, in an appropriate manner, to perform interfacial liquid-solid MDsimulations at each temperature. This preparation stage is delicate because some interaction po-tentials diverge to negative infinity at zero ion separation, such as the TF models. These require†This is necessary to ensure the minimum image convention is maintained, which is enforced by GROMACS whenperiodic boundary conditions are employed.112a robust initialization where no atoms are placed too close. Figure 5.2 shows an example rock-salt/liquid interfacial system for the NaCl/JC(SPC/E) model, as produced by the subroutine usedto prepare relaxed solid-liquid interfaces.The MP calculation algorithm begins each direct coexistence simulation with a 1:1 mole ratio offully relaxed solid and liquid, giving the system maximum room to change phase in either direction,while minimizing any thin-layer finite size effects. The system preparation subroutine is designedto prepare such states, and it is called to initialize every simulation at each new temperature overthe course of the MP calculation. An overview of the system preparation procedure is shownin Figure 5.3; preparing a fully relaxed two-phase system for MD simulation is not as trivial asit may first appear. However, I have automated this process and optimized it for alkali halides.One key parameter that can be difficult to gauge without prior knowledge is the initial density ofthe liquid prior to relaxation. If the initial density is set too low, the simulated liquid may formlarge voids in the simulation cell. If the initial liquid density is set too high, the atoms may beforced too close together, giving the system an extreme initial pressure that causes a rapid outwardexpansion of the simulation cell. In my calculations, I use an initial liquid density 80 % that of theminimized solid at (T = 0,P= 0), and found that this generally worked without issue in most cases.However, I include a check in the algorithm to catch exploding simulation cells, automatically re-initializing the initial liquid microstate with a reduced density in case of system blow-up. If thesystem preparation procedure outlined in Figure 5.3 is used for more complex systems with slowerdynamics, the relaxation times and initial liquid densities would likely need to be modified.The liquid-solid interface preparation subroutine employs three energy minimization stages:energy minimization of the ideal solid (as discussed in Section 2.8.1), the random liquid (Sec-tion 2.8.2), and atoms near the liquid-solid interface. Atoms near the liquid-solid interface areminimized as in the random liquid, but with most of the system held fixed. Robust minimization,as described in Section 2.8.3 is used when the TF model is employed.There are three stages of system preparation that employ rapid relaxation as described in Sec-tion 2.7.2. Rapid relaxation increases computational efficiency and, during the final stage of systempreparation, prevents drift of the system liquid/solid ratio prior to the production MD simulation.Although not shown in Figure 5.3, after rapid relaxation of both the solid and liquid, the subroutinescans the relaxation trajectories for phase changes using an NN classifier. If the liquid crystal-lizes during this relaxation stage, reaching xliq ≤ 1−∆x∗liq, where xliq is the fraction of liquid inthe simulation as classified by the NN (expected to be near unity for true liquids) and ∆x∗liq is astopping parameter discussed in Section 5.2.3, the subroutine reports back a freezing event to themain program without proceeding to a liquid-solid coexistence simulation. Likewise, if the solid113Build small solid supercell n(a0, b0, c0) Minimize perfect solid at T = 0 K Rapid-relax solid with NPT at T = T* for 15 ps Potential Volume Average Obtain (a0, b0, c0) Thermal solid Nsol = N/2 Matching liquid Box Nliq = N/2 add randomly Fix V; minimize liquid (avoids system blow-up) Rapid-relax liquid with NPzT at T = T* for 5 ps Potential Volume Merge relaxed liquid + thermal solid boxes Keep final geometry of relaxed liquid frame Minimize particles at interface Rapid-relax comb. system NPT at T* for 0.3 ps Potential Volume Keep final system state Figure 5.3: An overview of the procedure used to generate solid-liquid interfaces for MPcalculations. The “Minimize perfect solid at T = 0 K” subroutine is discussed in Sec-tion 3.2.5.disintegrates such that xref ≤ 1−∆x∗liq, where xref is the fraction of reference solid, the subroutinereports back a melting event without proceeding further. If in either case, xother ≥ ∆x∗liq wherexother = 1− xliq− xref, then a structure other than the requested reference structure has crystallized,in which case the algorithm can be set to abort the calculation. Alternatively, the algorithm can beset to treat alternate structure crystallization as freezing or melting events. If the reference soliddisintegrates, this can be considered melting. If the liquid crystallizes into any solid structure, itcan be considered freezing. Under this special setting, the algorithm will converge towards thethermal stability limit of the solid if liquid-solid coexistence is not possible for the reference crys-tal structure. Such coexistence is not possible in some situations when another crystal structure issignificantly more stable than the reference structure.After producing a relaxed liquid at the specified temperature T ∗, the system preparation sub-routine performs a final check to ensure the relaxed liquid is not an amorphous solid. The NN114classifiers alone cannot distinguish these two phases (they were not trained to classify amorphoussolids), so a 50 ps liquid simulation is performed, beginning from the relaxed liquid microstateusing production settings (see Section 2.7.2). From this simulation trajectory, the diffusion con-stant of the metal cation is calculated from its mean-squared displacement and compared with thecorresponding experimental diffusion in the molten salt at the experimental MP.286,287 If the modeldiffusion constant is less than 1/100 that of the experiment, the subroutine reports back a freezingevent without proceeding further. This situation is atypical, occasionally occurring far below theMP of the system, but necessary to check for. If amorphous solids are not otherwise detected, thealgorithm may declare an indeterminate event (see Section 5.2.3) for solid/amorphous coexistenceat low temperature, which usually does not undergo crystallization or melting.5.2.3 Checking for Melting or FreezingAt the core of the MP calculation algorithm is a subroutine for checking the behaviour of the liquid-solid interfacial system (melting, freezing, other structure, or indeterminate) at a given temperature.A graphical overview of the subroutine is given in Figure 5.4. This subroutine detects meltingor freezing events primarily by calculating the change in mole fraction of liquid in the system,compared with the initial value, as determined by the NN structure classifier,∆xliq(t) = xliq(t)− xliq(0), (5.2)where xliq(t) is the mole fraction of liquid-classified ions in the system at time t. The subroutinecan employ any NN classifier given in Table 4.1. Since the MP calculation requires only a globalorder parameter to track the fraction of liquid to solid in a simulation, the calculation is not verysensitive to the particular classifier. I chose classifier 0-0a as it is accurate and relatively low incomputational cost. Note that xliq(0) necessarily includes structural information at future times if atime-convolved NN is employed. Other researchers have used changes in potential energy,48 diffu-sion coefficient,19 or even the compressibility of the trajectory data file itself288 as order parametersin MP calculations. These order parameters may be complicated to employ in an automated MPdetection algorithm, as the correlation of these second-order effects with melting and freezing mustbe calibrated in some way, adding some additional complexity. NN classifiers offer a low computa-tional cost, high accuracy measure of the solid/liquid composition at any point in an MD trajectory.If at any checkpoint during the simulation, ∆xliq(t)≥ ∆x∗liq, the subroutine stops the simulationand records a melting event. Likewise, if ∆xliq(t)≤−∆x∗liq at any point, the MD simulation is halted115Prepare solid-liquid interface at T*Continue NPT simulation for tc ps. i++ Initialize i = 0 No Yes: Indeterminate result.append(0) Yes Yes: Melted, update result.append(1) Update: Update: Update list of checked T Return updated result list Example of freezing with three competing structures: LiBr/TF at 818.3 K (NN 20-5) No No: Froze, update result.append(2) Yes: System froze to non-reference structure. Calculate with NN model: No Compute stability limit? No: Abort Calculation Yes Relaxed 2-phase system prepared Figure 5.4: An overview of the subroutine, employed by the MP calculation algorithm, usedfor checking if a melting or freezing event has occurred at a specific temperature. Notshown here is a backup stopping criterion that halts the simulation before tmax if thetotal liquid fraction is above 95 % or below 5 %, returning a melting or freezing event,respectively.116and a freezing event is recorded. These, combined with a maximum time limit tmax, are the threenormal termini to each simulation within an MP calculation. At some temperatures far from the MP,one of the initial phases might be totally unstable such that the system decays rapidly into nearly100 % liquid or solid during the relaxation stage of the system preparation. When this occurs, astructure classifier finds the initial liquid fraction, xliq(0), near 0 or 100 %. Hence, a secondarystopping criterion halts any simulation within an MP calculation where the liquid fraction is above95 % (melting) or below 5 % (freezing). Additionally, if xother ≥ ∆x∗liq is detected at any point, thealgorithm will stop the entire calculation unless the stability limit option is enabled. If enabled,the calculation will continue as shown in Figure 5.4, converging towards the stability limit of thereference solid if liquid-solid coexistence is not possible.If the simulation proceeds for tmax with neither a melting nor freezing event detected, the algo-rithm stops the MD simulation and records an indeterminate outcome. As long as tmax is sufficientlylarge, indeterminate outcomes are expected only to occur rarely and very near the thermodynamicMP of the system. Choosing an appropriate tmax is therefore important; a larger tmax reduces thechance of an indeterminate outcome, allowing the algorithm to narrow the melting/freezing bracketfurther for a given system size, but increases computational cost. Furthermore, since melting ratesand freezing rates near the MP are not necessarily equal, indeterminate outcomes may bias a calcu-lation towards an inaccurate MP. For example, if freezing rates are slow compared to melting rates,simulations that would otherwise return a freezing event may instead return an indeterminate pointif tmax is too short for freezing to occur, while simulations yielding the more rapid melting eventsare not similarly affected. This would lead the MP algorithm to bias the resulting calculated MP(Tm) to lower temperatures. Hence, a sufficiently large tmax should be chosen to keep indeterminateevents rare.Choosing the ∆x∗liq parameter is also important. If ∆x∗liq is too small, random fluctuations cancause the algorithm to detect a melting or freezing event where one is not occurring. If ∆x∗liqis too large, the algorithm wastes simulation time waiting to detect a phase change, which canproceed slowly as the calculation approaches the thermodynamic MP. Larger ∆x∗liq also increasesthe probability of indeterminate outcomes and sets up a requirement that the system reaches astate with a thin liquid or solid layer before an outcome is declared. Thin layer states are bestavoided, as they may contribute to finite size effects due to their large surface area to volume ratio.In my calculations, I use ∆x∗liq = 25 %. Preliminary tests showed that ∆x∗liq = 25 % only slightlyreduced the MP variance of repeated calculations for a NaCl/JC(SPC/E) test system of 11,644 ionscompared with ∆x∗liq = 15 %. ∆x∗liq = 25 % also ensures that the fraction of liquid or solid doesnot become much less than 25 %, avoiding thin layers. Overall, ∆x∗liq = 25 % was found to be an117optimal choice of this parameter, balancing a reduction in finite-size effects (further discussed inSection 5.3.1) with reduced computational cost and systematic uncertainty.5.2.4 Narrowing the Temperature BracketThe MP algorithm works to narrow the range of possible MP temperatures down to a given re-quested temperature interval, ∆Tmin, by checking a series of temperatures to find melting, freezing,or indeterminate outcomes. The upper bound on the MP bracket is initially infinite, but is updatedby each newly detected melting event. Likewise, the lower bound on the MP bracket is initiallyzero but is updated by each newly detected freezing event. The MP calculation begins at a speci-fied temperature T0 (the experimental Tm is a suitable choice). The temperature is then increasedor decreased, depending on whether freezing or melting events, respectively, are detected. Thetemperature step between iterations is given by the initial step size and is updated by the step sizemultiplier (γ) after each iteration. The choice of initial step size and γ affect the rate of convergence;I used an initial step size of 10 K with γ = 2 for most calculations. As the algorithm proceeds, thelower bound (TLB) or upper bound (TUB) is updated, and the result of each simulation is appendedto a list. Once both a melting and freezing event are recorded, or an indeterminate point is found,the algorithm checks if the current melting/freezing bracket ∆T = TUB−TLB (which may includeone or both initial bounds) is less than or equal to the desired ∆Tmin. If it is, the calculation iscomplete and the average of the two bounds is taken as the estimated Tm. At this stage, if ∆T isgreater than ∆Tmin, a bracket reduction subroutine is called to reduce ∆T below ∆Tmin. This changein procedure accelerates calculation efficiency.When no indeterminate points have been found, the bracket reduction subroutine iterativelychecks the midpoint between the current TUB and TLB, updating these bounds as new freezing ormelting events are detected. If one or more indeterminate points are found, the subroutine in-stead checks temperatures near to, but outside, the range of temperatures covered by indeterminatepoints. For example, if a single indeterminate point is found the algorithm will next check temper-atures ∆Tmin/2 above or below the indeterminate point. In cases where two or more indeterminatepoints have been found, the melting/freezing bracket is calculated using∆T ′ = (TUB−TLB)− (T xmax−T xmin) , (5.3)where T xmax is the highest temperature indeterminate point, and Txmin is the lowest temperature inde-terminate point. This alternative definition ensures that ∆T ′ will eventually fall below ∆Tmin. ∆T ′redefines ∆T to exclude the indeterminate region between T xmax and T xmin, and the algorithm nar-118Any inde-terminate points? Current Bracket: (TUB ˗ TLB) Yes Txmin - TLB less than ΔTmin/2? Yes T* = Txmin ˗ ΔTmin/2 No Check for melting, freezing, or indeter-minate point at T* TUB - Txmax less than ΔTmin/2? No T* = Txmax + ΔTmin/2 Tm = (TUB + TLB)/2 No (TUB ˗ TLB) ≤ ΔTmin? Yes No T* = (TUB + TLB)/2 Yes Check for melting, freezing, or indeter-minate point at T* INFO • TUB is the current up-per bound on the melt-ing temperature. • TLB is the current low-er bound on the melt-ing temperature. • Txmin is the lowest T indeterminate point found so far. • Txmax is the highest T indeterminate point found so far. • ΔTmin is the stopping criterion. Figure 5.5: An overview of the subroutine used to systematically reduce the melting/freezingbracket. This subroutine is initialized by the MP calculation algorithm once both a melt-ing or freezing point has been found, or if an indeterminate point is identified (i.e. thealgorithm cannot conclude if the system is melting or freezing at a specific temperature).rows the two unknown regions of temperature adjacent to it. A graphical overview of the bracketreduction subroutine is given in Figure 5.5.The systematic error of any particular MP calculation is always given by the final TUB−TLBinterval. The algorithm aims to produce a final systematic error of ∆Tmin, or lower, but dependingon simulation outcomes, this target systematic uncertainty can be updated. If one or zero inde-terminate points are found, the calculation proceeds until TUB−TLB ≤ ∆Tmin, satisfying the ∆Tmintarget. Hence, in those cases, an upper bound on the systematic error can be fixed at ∆Tmin. Whenno indeterminate points are found, ∆T > ∆Tmin usually changes to ∆T < ∆Tmin in a single step,as explained in the Appendix, Section D.1. For cases with a single indeterminate point, the algo-rithm checks ∆Tmin/2 above and below the indeterminate point. This yields a final systematic error119Figure 5.6: The temperature trajectory from an example MP calculation of the Na-Cl/JC(SPC/E) system with N = 11,644 ions. The initial step size is 10 K, T0 = 1278 K,γ = 1, and ∆Tmin = 1 K. Blue circles represent detected freezing events, red circlesare melting events, and white circles are indeterminate points. The dashed green lineindicates the final Tm.of precisely ∆Tmin, centered around the indeterminate point, as long as no further indeterminatepoints are found. However, if two or more indeterminate points are found, the systematic uncer-tainty associated with the MP calculation necessarily increases from ∆Tmin to ∆Tmin+(T xmax−T xmin)to encompass both the indeterminate region, and the two unknown regions surrounding it. For theMP calculations reported here (∆Tmin = 1 K in all cases), the systematic uncertainty interval neverincreased beyond 4 K, and the majority of calculations finished with a systematic uncertainty in-terval of 1 K or lower. While all MP calculations produce a well-defined systematic uncertainty,the true calculation error can be much larger due to finite-size effects, as discussed below (see Sec-tion 5.3.1). A visualization of the temperature trajectory for an example MP calculation is givenin Figure 5.6. In this case, the bracket reduction algorithm is engaged after three liquid-solid sim-ulations, and two indeterminate points are found. The final systematic uncertainty interval for thisexample calculation is 1.5 K.1205.3 Results and DiscussionIn this section, I give the results of MP calculations using rigid-ion alkali halide interaction poten-tials. In Section 5.3.1 a finite-size effect that limits the accuracy of MP calculations is discussed.Afterwards, in Section 5.3.2, I present and discuss the results of accurate MP calculations on modelalkali halides employing commonly used rigid-ion interaction potentials. These MP results arecompared with corresponding values available in the literature.In Chapter 3 I demonstrated that many of the rigid-ion interaction potentials used for lithiumhalides predict incorrect low-energy crystal structures (wurtzite) at both 0 K and finite temperatures.In particular, LiX/TF for X = {F, Cl, Br, I}, LiX/JC(SPC/E) for X = {Br, I}, LiX/JC(TIP3P) forX = {Cl, Br, I}, and LiX/JC(TIP4PEW) for X = {Cl, Br, I} give wurtzite as the low-energy crystalstructure at T = 0 K. The corresponding interaction potentials for other alkali halides do not sufferfrom this issue, except for NaF/TF which also favours wurtzite, and all JC interaction potentials forCsCl, CsBr, and CsI, which incorrectly favour rocksalt over the experimental CsCl structure. The(T = 0, P = 0) energies of all alkali halides in all eight crystal structures discussed in this chapter,using the TF, JC, and SD interaction potentials, are reported in the Appendix, Section C.1. In thecase of LiBr/TF, wurtzite is the lowest energy crystal structure at low temperatures, beating outrocksalt by about 1.4 kJmol−1 at 0 K.86 As seen in Section 4.3.2, this small energy gap is enoughto favour crystallization into the wurtzite structure.The possibility of a thermodynamically favoured crystal structure other than rocksalt was notconsidered in previous MP calculations of the model alkali halides.19,48,49 As discussed below, aproper treatment of the crystal structure is very important for obtaining accurate MPs for modelalkali halides.5.3.1 System Size EffectsMP calculations have two main sources of error, systematic uncertainty and finite-size effects. Sys-tematic uncertainty, ∆Tm, is the uncertainty of each MP calculation attributable to the calculationmethodology itself. In each of my MP calculations, systematic uncertainty manifests as the final∆Tm = TUB−TLB interval. The key calculation parameters governing this source of uncertainty are∆Tmin, tmax, and ∆x∗liq, the effects of which are discussed in Sections 5.2.3 and 5.2.4. As the systembecomes very large, we approach the thermodynamic limit, and the systematic uncertainty becomesthe only source of uncertainty on Tm. In this limiting case, assuming tmax is sufficiently long, the fi-nal TUB and TLB of repeated MP calculations would form uniform (or possibly non-uniform, see theAppendix, Section D.1) non-overlapping distributions centered ∆Tm ≤ ∆Tmin apart, each of width121∆Tm, with the two distributions abutting without overlap at the exact thermodynamic MP. Since Iestimate Tm ≈ (TLB +TUB)/2, for sufficiently large systems, the TUB and TLB distributions wouldyield a corresponding uniform Tm distribution of width ±∆Tm/2 centered on the thermodynamicMP. See the the Appendix, Section D.1 for visualizations of these distributions.The exact value of ∆Tm, and the shape of the TUB, TLB, and Tm distributions for sufficiently largesystems depends on three calculation parameters: the initial step size, γ , and initial temperatureT0. For the present calculations, γ = 2 and the initial step size is 10 K. With these parameters,∆Tm = 0.625 K and, if T0 is chosen randomly, the distributions of TUB, TLB, and Tm are uniform(see Appendix, Section D.1). Other choices of γ and initial step size can yield somewhat differentdistributions, but all have ∆Tm ≤ ∆Tmin.Wales and Berry289,290 have considered the thermodynamics of finite systems and concludedthat, for finite systems, the solid-liquid coexistence line that occurs in the thermodynamic limitis replaced by a solid-liquid coexistence “band”. That is, for a given pressure there is no uniquemelting point, liquid-like and solid-like phases can coexist over a range of temperatures. This is amanifestation of the fact that finite size restricts fluctuations in the bulk and interfacial structures,and this influences the coexistence behaviour of finite systems. This clearly has implications formy MP calculations. In particular, if I start calculations with different initial microstates, I wouldexpect my algorithm to detect a range of melting points, which, depending on system size, mightextend beyond the systematic uncertainty discussed above. As the system size is increased, the MPdistribution due to finite-size effects will become narrower, approaching zero in the thermodynamiclimit.Variations in my Tm measurements beyond the quantifiable systematic uncertainty discussedabove can be attributed to finite-size effects. I performed an analysis of finite-size effects for theNaCl/JC(SPC/E) system using 100 repeat MP calculations for N = 2,000 and N = 11,644 ions,with ∆Tmin = 1 K. The resulting distributions of calculated Tm values are shown in Figure 5.7.To simplify the analysis, I excluded all Tm calculations where the systematic uncertainty increasedabove the target ∆Tmin = 1 K, due to indeterminate regions (see Section 5.2.4). This excluded nocalculations for the 2,000 ion system, but excluded 31 calculations for the 11,644 ion system.A plot of the full distributions including all 100 calculations from both sets is available in theAppendix, Section D.2, although these are essentially indistinguishable from Figure 5.7.All MP calculations for the same system size shown in Figure 5.7 differ only in the initial tem-perature T0 and the initial microstate, but yield a distribution of MPs that is clearly narrower forthe larger system. In both cases, the upper limit on the systematic error interval (∆Tmin = 1 K) issmaller than the widths of the full Tm distributions, indicating that finite-size effects are dominating1221280 1282 1284 1286 1288 1290 1292 1294 1296 1298 1300Tm [K]00.10.20.30.40.50.60.70.8;(T m)N = 2000PDF: 7 = 1289:8 K, < = 1:79 KN = 11664PDF: 7 = 1289:9 K, < = 0:49 KFigure 5.7: Demonstration of the system size effect on calculated MP precision. The MPprobability density, ρ(Tm), of repeated calculations for the NaCl/JC(SPC/E) system(rocksalt/liquid) using my MP calculation method. The N = 2,000 distribution showsthe results of 100 calculations, while the N = 11,644 distribution includes 69 calcula-tions. Each calculation includes a systematic uncertainty of no more than ±0.5 K. Toaid visualization, both distributions include a best-fit Gaussian probability distributionfunction (PDF) envelope with mean µ and standard deviation σ given in the legend. Theinitial temperatures (T0) given to the MP algorithm were randomly generated between1270 – 1310 K. All calculations use ∆x∗liq = 25 %, ∆Tmin = 1 K, and tmax = 5 ns.the MP uncertainty. Note that repeated MP calculations with different initial microstates, but iden-tical T0, also produce a similar size-dependent distribution of MPs (not shown). While the 2,000ion system yields a much wider distribution of MPs than the 11,644 ion system, both distributionsin Figure 5.7 have consistent mean values within statistical uncertainty, indicating finite-size ef-fects do not result in different average MPs for these different system sizes. In the N = 2,000 case,the complete range of calculated Tm covers almost 12 K. On the other hand, the finite-size effectfor the N = 11,644 system is much nearer to the systematic uncertainty. However, the existenceof tails in the distribution that exceed a 1 K interval implies that the systematic uncertainty is stillsmaller than the finite-size uncertainty. In the Figures 5.8 and 5.9, I give distributions of the finalTUB and TLB found in the MP calculations shown in Figure 5.7. These behave as expected, devi-ating from the thermodynamic limit described above. Since all Tm values in the distribution for1231280 1282 1284 1286 1288 1290 1292 1294 1296 1298 1300T [K]00.10.20.30.40.50.60.7;(T)T xTUBTLBFigure 5.8: Temperature distributions of the lowest T melting events (TUB), highest T freez-ing events (TLB), and all indeterminate points (T x) for the NaCl/JC(SPC/E)14 systemwith N = 11664 ions. The distributions are based on 100 repeat MP calculations withmy MP algorithm where the initial temperature given to the MP algorithm (T0) was ran-domly generated between 1270 – 1310 K. For all MP calculations, ∆x∗liq = 25 % and∆Tmin = 1 K.the N = 11,644 NaCl/JC(SPC/E) system lie within a 4 K interval (even when all 100 calculationsare included) such that any particular calculation is no more than 2 K from the mean, I estimatethe uncertainty of any NaCl/JC(SPC/E) MP calculation to be ±2 K at this system size. Repeatedsamples of calculated MPs for the NaCl/JC(SPC/E) system, using identical calculation parame-ters, are independent and identically distributed. However, the computed MPs are not necessarilynormally distributed. Hence, to estimate the MP mean and uncertainty for the N = 11,644 ionNaCl/JC(SPC/E) system, I computed a bias-corrected and accelerated bootstrap interval291 using10,000 bootstrap samples, including all 100 MP results in the analysis. This yields a mean MP of1289.95 K with 95 % confidence interval [1289.86, 1290.05] K.Figure 5.7 clearly demonstrates that simulating larger systems can produce more precise MPs,but the difference in computational cost between MP calculations with N = 2,000 vs N = 11,644ions is not trivial. An MP calculation involving 2,000 ions takes an average of 30 min to completeon 32 high performance compute cores, while a calculation with 11,644 ions averages around 12 h.1241280 1282 1284 1286 1288 1290 1292 1294 1296 1298 1300T [K]00.050.10.150.20.25;(T)T xTUBTLBFigure 5.9: Temperature distributions of the lowest T melting events (TUB), highest T freez-ing events (TLB), and all indeterminate points (T x) for the NaCl/JC(SPC/E)14 systemwith N = 2000 ions. The distributions are based on 100 repeat MP calculations withmy MP algorithm where the initial temperature given to the MP algorithm (T0) was ran-domly generated between 1270 – 1310 K. For all MP calculations, ∆x∗liq = 25 % and∆Tmin = 1 K. In this set of calculations, no T x were found.The MP results reported in Section 5.3.2 were obtained from single MP calculations with at least10,816 ions. Aside from the salt, interaction potential, and in some cases the solid crystal structure,these calculations have parameters identical to those of the N = 11,644 ion NaCl/JC(SPC/E) sys-tem. Therefore, it is reasonable to assume that the total uncertainty (including finite-size effects)of these MP calculations is similar to that obtained for the NaCl/JC(SPC/E) case with N = 11,644ions. Hence, I assign ±2 K estimated precision to all MP calculations reported in Section 5.3.2.5.3.2 Melting Points of Common Alkali Halide ModelsUsing my binary salt structure classifiers, I attempted MP calculations on all alkali halides (notincluding the francium halides) for several common interaction potentials. In addition to rock-salt, I calculated the MPs of several alkali halides with alternative crystal structures. From thesecalculations, I found three distinct outcomes. First are “true” MPs, calculated with the most sta-ble solid structure at the melting temperature. These are the highest MPs obtained for each given125model and usually, but not always, correspond to the stable solid at (T = 0, P = 0). The secondoutcome are MPs, obtained via solid/liquid coexistence, for metastable solids. These correspondto temperatures where the free energies of the metastable liquid and solid are equal. The thirdpossible outcome are crystal structures too unstable to establish solid/liquid coexistence, and hencedo not have well-defined MPs. In these cases, the metastable crystal structure decays and a morestable crystal structure grows before solid/liquid coexistence is observed. For these, I determinedthe highest temperature where the initial structure remained metastable, using my methodologydescribed in Section 5.2. MPs calculated for both stable and metastable solids, where liquid/solidcoexistence could be established, are plotted in Figure 5.10. In Table 5.1 I report the results of allattempted MP calculations, including cases where solid/liquid coexistence was not achieved. TheMPs of the most stable solids are given in bold, MPs for metastable solids are given in normal font,and the thermal stability limits for crystal structures where liquid/solid coexistence could not beachieved are given in italics. Not included in Table 5.1 is the calculated MP of NaCl/SD in therocksalt structure: 1333(2) K, compared with 1327(5) K in Ref 48.Many of the alkali halide MPs shown in Figure 5.10 have been previously calculated by oth-ers19,48,49,123 using different methods, but always employing the rocksalt crystal structure. Forsome of the models used in earlier calculations, rocksalt is not the most stable structure. I comparemy MP results with available literature values in Table 5.1. Guided by knowledge of the relativeenergies of (T = 0, P = 0) crystal structures of alkali halide models (Appendix, Section C.1), Icalculated wurtzite (or 5-5 where wurtzite is less stable than 5-5) MPs for the lithium halides andNaF/TF. I also attempted MP calculations with the CsCl-structure for CsCl, CsBr, and CsI mod-els. These three salts occupy CsCl crystal structures experimentally under ambient conditions,although the CsCl salt transitions to rocksalt above 742 K292 prior to melting at 919 K.123 I foundno previous reports of MP calculations with JC models for CsBr or CsI in the literature.One of the more surprising results of my MP calculations is for LiBr/JC(SPC/E). Crystal energyminimization calculations (Appendix, Table C.2) show that wurtzite is the energetically favouredcrystal structure of LiBr/JC(SPC/E) at (T = 0, P= 0) by 0.81 kJmol−1 over rocksalt, while experi-mentally rocksalt is thermodynamically stable at low temperature.16 Despite being higher in energyat T = 0, I find that the rocksalt crystal structure of LiBr/JC(SPC/E) has an MP 40 K higher than thewurtzite structure, indicating that rocksalt is more stable at high temperature. Close examination ofthe relative enthalpies of both structures at finite temperature and P = 1 bar, given in Figure 5.11,indicates that there is a crossover from wurtzite to rocksalt at approximately 800 K. Above thistemperature, the rocksalt crystal structure becomes enthalpically more stable than wurtzite, leadingto its higher melting point. The steeper enthalpy increase with temperature for wurtzite versus rock-126Exp. TF JC(SPC/E) JC(TIP3P) JC(TIP4PEW)Ref. 123 This work Ref. 48 Ref. 49 This work Ref. 48 This work Ref. 19 This workLiF 1121.4 340/876* 1010(15) 1623/1337† 1625/1370† 1630(25) 1562/1328†LiCl 883 778/834* 780(15) 777(3) 1224/1096† 982/1123* 990(25) 999/1079*LiBr 823 803/819* 1073/1033* 0/1088* 1060(25) 0/1058*LiI 742 0/680* 850/994* 0/998* 959(25) 0/983*NaF 1269 618/796* 611(15) 1508 1508 1516(25) 1468NaCl 1075.168 1081 1082(13) 1081(2) 1289.95 1285(5) 1314 1324(25) 1268NaBr 1020 1022 1022(15) 1250 1254 1270(25) 1214NaI 934 898 1166 1131 1134(25) 1106KF 1131 857 859(15) 1374 1366 1368(25) 1334KCl 1044 1038 1039(15) 1038(2) 1197 1225 1230(25) 1189KBr 1007 1043 1043(15) 1178 1198 1207(25) 1176KI 954 1004 1134 1128 1129(25) 1124RbF 1068 994 996(15) 1340 1339 1343(25) 1302RbCl 997 1091 1092(15) 1091(3) 1161 1190 1191(25) 1156RbBr 965 1048 1047(15) 1143 1166 1178(25) 1145RbI 929 1032 1103 1103 1107(25) 1100CsF 976 1252 1273 1320 1330(25) 1272CsCl 919 1179/898‡ 1119/93‡ 1152/817‡ 1155(25) 1117/610‡CsBr 909‡ 1124/903‡ 1094/0‡ 1125/721‡ 1104/0‡CsI 905‡ 1087/946‡ 1055/0‡ 1066/380‡ 1062/0‡Table 5.1: Alkali halide MPs, in K, calculated with TF and JC models at 1 bar, comparedwith experiment (Exp.) and literature values. MPs for JC and TF models from the liter-ature employ a rocksalt crystal structure and include the reported uncertainty of the lastdigit(s) in brackets. Rocksalt structure MPs or thermal stability limits are reported for allsalts (N = 11,644 ions), listed without reference marks. * Indicates the wurtzite crystalstructure (N = 10816 ions). † Indicates the 5-5 crystal structure (N = 11520 ions). ‡Indicates the CsCl (body-centered-cubic) crystal structure (N = 10976 ions). Numbersreported in bold are MPs of the most stable solid, while non-bold numbers indicate MPsof metastable solids. Numbers reported in italics are the thermal stability limits obtainedfor structures unable to achieve solid/liquid coexistence. Values of 0 indicate the crystalstructure is unstable at all temperatures for that system. These MPs include an estimated±2 K uncertainty except NaCl/JC(SPC/E), which has 95 % confidence interval [1289.86,1290.05] K based on 100 calculations.127LiF LiClLiBr LiI NaFNaClNaBr NaIKF KClKBr KI RbFRbClRbBr RbICsFCsClCsBr CsI40060080010001200140016001800T[K]ExperimentJC (TIP3P)JC (TIP4PEW)JC (SPC/E)SDTFFigure 5.10: Comparison of calculated MPs for all alkali halides (excluding franciumhalides) using the TF, JC, and SD interaction potentials with experiment. A mini-mum system size of N = 10,816 ions is used in all cases. Squares indicate MPs forthe rocksalt crystal structure, diamonds indicate wurtzite, triangles indicate 5-5, andcircles indicate CsCl. Dashed lines connect the MPs of the most thermally stable solidstructure for each model.salt implies a greater heat capacity for LiBr/JC(SPC/E) in the wurtzite crystal structure, which maybe attributable to the lower density of wurtzite. Heat capacity calculations of LiBr/JC(SPC/E),derived from enthalpy fluctuations over 2 ns at 800 K, yield Cp = 61.4 Jmol−1 K−1 for rocksalt(N = 5488 ions) and Cp = 63.3 Jmol−1 K−1 for wurtzite (N = 5408 ions).1280 200 400 600 800 1000 1200 1400T [K]-820-800-780-760-740-720-700H[kJmol!1 ]WurtziteRocksaltFigure 5.11: Molar enthalpy (H) vs temperature (T ) for simulations of the LiBr/JC(SPC/E)14system occupying both rocksalt and wurtzite crystal structures. Results are from twoNPT simulations of periodic bulk crystals (P= 1 bar) initialized at 0 K into ideal struc-tures with energy minimized lattice parameters. The crystals are heated at 50 Kns−1until 2200 K. The rocksalt simulation contains 5832 ions while the wurtzite simula-tion contains 5408 ions. Initially, the wurtzite crystal structure is lower in energy thanrocksalt by 1.05 kJmol−1. At approximately 800 K, the rocksalt crystal structure be-comes the lower energy structure. Both crystals remain metastable well above theirMPs (rocksalt Tm = 1073 K and wurtzite Tm = 1033 K) due to hysteresis. The wurtzitecrystal melts at approximately 1190 K while the rocksalt crystal melts at approximately1380 K. The rise in wurtzite enthalpy around 1050 K is associated with a solid-solidphase transition into the β -BeO crystal structure.There are three models where my calculated MPs disagree with MPs reported in the literature.These are LiF/TF,48 LiBr/JC(TIP3P),19 and LiI/JC(TIP3P).19. In all three cases, the reported MPsare for the rocksalt crystal structure. However, for all three models I found that the rocksalt structurebecame unstable at temperatures well below the reported MPs, and I was unable to obtain rocksaltMPs.For LiF/TF, the highest temperature attained before rocksalt spontaneously converted to wurtzite,which is more stable than rocksalt at 0 K (Appendix, Table C.1), was 340 K. For LiF/TF in the129wurtzite structure, I obtained an MP of 876(2) K, which is well below the value (1010(15) K)reported for rocksalt in Ref. 48. I do not know the origin of this discrepancy. A careful check ofmy calculations, including the interaction potentials, and other details of my methodology failed touncover any issues. I note that in Ref. 48, Monte Carlo calculations and Gibbs-Duhem integration,rather than direct coexistence, was used to obtain the MP. Yet, it is difficult to see how this couldgive rise to a rocksalt MP which is higher than the MP of the more stable wurtzite structure.For LiBr/JC(TIP3P) and LiI/JC(TIP3P), wurtzite is more stable than rocksalt at 0 K (Appendix,Table C.3), and for both salts I found that the rocksalt structure was completely unstable at finitetemperature, rapidly decaying into disordered wurtzite-like structures through liquid or amorphoussolid intermediate steps. For the stable wurtzite structures, I obtained MPs of 1088(2) K for Li-Br/JC(TIP3P), and 998(2) K LiI/JC(TIP3P), both of which differ from the values (1060(2) K and959(2) K, respectively) reported in Ref. 19. In Ref. 19, the authors noted that the structures ini-tially set up as rocksalt changed into some unspecified structure. My simulations suggest that thesestructures were disordered wurtzite-like solids. Therefore, I can attribute the discrepancies withmy results to the disordered wurtzite-like structure present in the earlier MP calculations. I wouldexpect disordered wurtzite-like solid structures to be less stable than wurtzite crystals initialized asperfect lattices, which explains why I obtain higher MPs.5.4 Summary and ConclusionsAs an application of the NN structure classifiers introduced in Chapter 4, I introduce an automatedMP calculation algorithm, based on direct MD simulation of solid-liquid coexistence. The algo-rithm employs a NN classifier at its core to detect melting or freezing. Using repeated MP calcula-tions with different initial conditions, I show that finite-size affects lead to a range of temperatureswhere both melting and freezing can occur. Under the framework of finite-cluster thermodynam-ics,289,290 I interpret this as a band of temperatures that allows for liquid-solid coexistence at a fixedpressure. Using the NaCl/JC(SPC/E) system as an example, I show that the coexistence band sets alimit on the precision of MP calculations, higher precision is achieved with increasing system size.Based on the MP distribution of obtained for NaCl/JC(SPC/E) with N = 11664 ions, I estimate theuncertainty of a single alkali halide MP calculation with approximately 10,000 ions to be ±2 K.I calculated the MPs of all alkali halides using several common rigid-ion interaction potentials,and find mostly good agreement with earlier MP calculations. Three cases of disagreement withthe literature values are identified and discussed. The first case of disagreement is the MP I obtainfor LiF/TF, compared with the MP reported by Aragones et al.48 In this case rocksalt, an unstable130crystal structure for the LiF/TF model, was reportedly used as the solid in the earlier MP calculationvia thermodynamic integration. I suspect that the unstable structure used in the MP calculation viathermodynamic integration lead to an erroneous MP. The other two cases of disagreement are forthe LiBr/JC(TIP3P) and LiI/JC(TIP3P) systems, where MPs are reported by Walz and van derSpoel.19 In these cases, I conclude that the disagreement with my results is likely due to the use ofrocksalt, which is unstable for these models, as the initial solid crystal structure in the liquid-soliddirect coexistence simulations of Walz and van der Spoel. This unstable structure possibly decayedinto a highly defective wurtzite-like crystal with a lower MP than the perfect wurtzite structureused in my simulations.In the next chapter, I apply my automated alkali halide MP calculations, along with the lessonslearned and methods developed in earlier chapters, to develop new lithium halide force fields forclassical MD simulation. The ability to efficiently calculate the MPs of arbitrary alkali halide in-teraction potentials allows me to rapidly characterize new force fields, aiding in their development.The aim of the force field development project is to fix the problems in the commonly used rigid-ion lithium halide force fields discussed in Chapter 3; the principle problem is their inability toobtain the correct experimental crystal structure.131Chapter 6Bayesian Optimization of LithiumHalide Force Field Parameters6.1 IntroductionAs the previous chapters have demonstrated, the accurate molecular dynamics simulation of lithiumhalides is a deceptively difficult problem. At first glance, the lithium halide salts appear to be aboutas simple a chemical system as one can conjure up. But, as pointed out in Chapter 3, the structurallandscape of lithium halides is complex. Due to the relative ion sizes of lithium versus its counteranion, these salts have multiple low energy crystal structures that compete to complicate theirstructural chemistry under ambient conditions. The relative energies of these crystal structures arehighly dependent on the details of the attractive and repulsive interactions between ions, particu-larly the strength of London dispersion (Chapter 3). Therefore, modeling the interaction potentialsof lithium halides through classical pairwise force fields proves to be a particularly difficult problemcompared with other alkali halides. Two commonly used classical force fields currently availablefor LiX, the JC and TF models, yield the incorrect lowest-energy crystal structure in many cases,particularly for LiBr and LiI. Even those force fields that do get the correct low-energy structure(e.g. LiF/JC(SPC/E) and LiCl/JC(SPC/E)) tend to yield incorrect relative energies for the compet-ing low-lying metastable structures. Furthermore, as demonstrated in Chapter 5, the JC and TF LiXforce fields either cannot achieve rocksalt/liquid coexistence, or yield rocksalt melting points thatare far from experiment. This indicates a recurring issue with the high temperature thermodynamicbehaviour of the solid phase, liquid phase, or both.132In this chapter, I use multiple LiX structural, energetic, and thermochemical data points fromexperiment, combined with the set of metastable crystal energies and geometries calculated fromhigh accuracy DFT in Chapter 3, as targets for a computationally efficient scheme to globally op-timize pairwise classical force field parameters for LiX MD simulations. The resulting force fieldsare validated by comparison of MD simulation results with a large set of available experimentaland theoretical data, including experimental melting points. The neural network classifiers intro-duced in Chapter 4 are employed throughout this chapter to analyze the phase of ions in LiX MDsimulations, while model melting points are calculated by leveraging the methodology describedin Chapter 5.6.1.1 Bayesian OptimizationBayesian optimization20,21 is a direct function optimization scheme classified as reinforcementlearning,293 that employs Bayesian inference to update a statistical model of some objective func-tion at each stage of an iterative procedure. The goal is to find the minimum value of the objectivefunction in as few iterations as possible by employing a surrogate model that is nonlinearly re-gressed to the currently available data (objective function input and output pairs). The scheme isbest employed to optimize low-dimensional objective functions that are very expensive to compute.This is because the point selection process of Bayesian optimization is much more computationallycostly than traditional optimizers, and scales in computational complexity cubically with the sizeof the data set. If the objective function is fast to compute, most of the computational expense willbe spent on point selection. High input dimensionality leads to the “curse of dimensionality”,294which is a problem faced by all optimization algorithms. In essence, as the number of dimensionsgrows the amount of data required to accurately model the input space grows exponentially.Bayesian optimization can handle stochastic noise in the objective function output, and doesnot require gradients. It works best over continuous domains, but can also be employed over cate-gorical domains.20 The surrogate model is typically constructed from Gaussian process regression(GPR),295,296 which is a non-parametric, nonlinear regression approach that leverages the com-plexity of the available data to determine the nature of the interpolation between known points.297In addition to approximating the objective function over some specified domain, GPR produces aGaussian distribution for each point in the domain, which includes some distribution width withstandard deviation σ . The distribution width decreases near measured points in the domain andincreases between points. The behaviour of the GPR surrogate model between measured points ispartly determined by the covariance function, also called the kernel, whose parameters are fit to133Figure 6.1: (Upper panel) An snapshot of Bayesian optimization applied to an arbitrarynoiseless 1-dimensional objective function, plotted in black. The Mate´rn 5/2 covariancefunction is used. The state of the Gaussian process surrogate model and its error barsat 1 standard deviation are visualized after five objective function evaluations. (Lowerpanel) The expected improvement acquisition function is plotted over the objective func-tion domain.the data as part of the regression process. Figure 6.1 gives a visualization of Bayesian optimiza-tion applied to an arbitrary continuous 1-dimensional objective function using the Mate´rn 5/2298covariance function.20After an initial stage of (typically random) objective function evaluations, each step of Bayesianoptimization begins with the construction of a Gaussian process surrogate model. After the surro-gate model is built, the model mean and uncertainty are combined in some manner, possibly withadditional information, to construct an acquisition function,296 an example of which is shown inthe lower panel of Figure 6.1. The purpose of the acquisition function is to guide the algorithmintelligently through parameter space. The acquisition function is tuned to balance explorationversus exploitation. A larger weight on exploration tends to direct the process towards unexploredregions of parameter space, while more weight on exploitation favours the further optimization ofrelatively optimal, well-explored regions of parameter space. Once the acquisition function is con-structed, its maximum value is obtained, yielding the next search point. Since acquisition functioncalls are quick to compute, the acquisition function can be optimized rapidly by traditional function134optimization methods. An appropriately constructed acquisition function will guide the Bayesianoptimization process to a mix of unexplored areas and semi-optimal explored areas. Bayesianoptimization is a global optimizer, meaning that it is typically used to find the global optimumof the objective function on the given domain. However, it can only be guaranteed to eventuallyfind this global minimum if exploration is given an appropriately high weighting in the acquisi-tion function. In any case, the rate of convergence towards the global optimum is dependent onmany factors, including the complexity of the data, the quality of the covariance function, and theacquisition function.The remainder of this chapter is arranged into 3 sections. Section 6.2 gives specific details onthe methodology employed in this work, including details on Bayesian optimization (6.2.1), theconstruction of an objective function for parameter fitting (6.2.2), plus the choice of parameter do-main and objective function constraints (6.2.3). Section 6.3 gives the results and a discussion ofthe Bayesian optimization scheme for fitting LiX force field parameters to experimental and theo-retical target data. The section is split into two subsections, discussion application to the CoulombLennard-Jones potential (6.3.1) and the Coulomb Buckingham potential (6.3.2). Section 6.4 is asummary and brief overview of the main findings, plus some discussion of future directions.6.2 MethodsThe research in this chapter involves the reparameterization of interaction potentials previouslydiscussed in Section 2.3, the CLJ and CBH functional forms. The CLJ form was selected asLennard-Jones approximations to vdW forces are the most ubiquitous functional form for non-bonded interaction potentials in classical MD simulations.299 Preliminary results indicated that theCLJ functional form could not be parameterized to adequately fit both low temperature crystaldata and high temperature molten salt data. Therefore, CBH, a pairwise interaction potential withmore flexibility in shape of the repulsive wall, is also chosen for parameterization to see if it canbetter fit the target data. The CBH functional form is similar to the CLJ functional form but,if combining rules are employed, contains one additional parameter γi j that adjusts the shape ofrepulsive interactions.To optimize the interaction potential parameters, MD simulations of single-phase bulk LiXsalts are employed. All simulations are performed under NPT conditions using the methodologydescribed in Section 2.7. The vdW cutoff radius and Ewald sum real space cutoff rc is set to1 nm for computational efficiency. Periodic boundary conditions (Section 2.4), tail correctionsto the vdW potential, and the smooth particle mesh Ewald method (Section 2.5) are employed.135Liquid and solid geometry initialization, including crystal minimization, are performed with themethodology described in Section 2.8.6.2.1 Bayesian OptimizationAn overview of the Bayesian optimization scheme employed is given in Figure 6.2. All Bayesianoptimization calculations were performed with the MATLAB bayesopt function from the Statis-tics and Machine Learning Toolbox,300 which was modified for use with covariance functions otherthan the default Mate´rn 5/2.298 Bayesian optimization begins with an initial uniform (or, in somecases, log transformed) random sample over the feasible parameter space. This initial sample isused to generate a Gaussian process posterior surrogate model through application of Bayes’ rule,assuming a uniformly zero prior distribution with unit variance for the objective function. Thenumber of initial random points to compute (NSample) was set to ten times the number of inputparameters (i.e. search dimensions) in all cases. Some preliminary Bayesian optimizations of CLJparameters (with mixing rules) were performed using various covariance and acquisition functions.These were set to optimize parameters towards a few LiBr crystal properties (rocksalt lattice en-ergy, relative wurtzite lattice energy, and lattice parameters) that are relatively quick to compute.A discussion of these preliminary results is available in the Appendix, Section E.1. Based on ananalysis of the preliminary results, I employed the following Bayesian optimization methodologyin all production calculations.All production Bayesian optimization calculations employ the Mate´rn-5/2 covariance func-tion298 with automatic relevance determination (ARD).301 ARD gives a separate length scale pa-rameter for each input dimension, improving the quality of the model regression when the objec-tive function varies differently along each input dimension. Each Bayesian optimization calculationproceeds for either 500 or 1000 iterations. Calculations targeting only crystal properties are faster tocompute, so these were given 1000 iterations while calculations targeting thermal properties weregiven 500 iterations. The first 300 (or 800) iterations employ the “expected improvement plus”acquisition function with exploration ratio parameter 2,302 while the latter 200 iterations employthe “lower confidence bound” acquisition function with exploration parameter κ = 1.20 Since MDsimulations are not necessarily reproducible for different initial conditions, objective function callsare not assumed deterministic (see the Appendix, Figure E.7). During Gaussian process regression,a quantity of additive Gaussian noise is fit as part of the overall surrogate model. After all Bayesianiterations are complete, the point corresponding to the lowest observed objective function call ispassed to the initial point of MATLAB’s patternsearch optimization algorithm303 with initial136Main user inputs • Salt: LiF, LiCl, LiBr, or LiI • Functional form of forcefield • Apply mixing rules? • Objective function targets and weights Other calculation parameters • Parameter exploration domain • Deterministic constraints • Unit cells of structures • Acquisition function and parameters • Covariance function • Number of iterations • Initial sample points (NSample) • ... Problem definition Uniformly sample NSample feasible points within parameter domain Initialize Bayesian optimization Gaussian Process Regression GPR parameters • Covariance function • Initial kernel parameters • … Set of sampled points {xi , f(xi)} Evaluate objective function Sample feasible points Next query point xi at acquisition function max Acquisition functions trade-off exploration with exploitation and avoid regions with high probability of returning an error upon evaluation. Objective function parameters • Crystal structures • Initial conditions • Convergence criteria • MD parameters • Weights and targets Reached NMax Samples? No Yes Return best point {f(x*), x*} i = i + 1 Surrogate models of objective function f *(x) and output errors η*(x) regressed to data; models include uncertainty σ*(x) Construct and maximize acquisition function from objective and error models Figure 6.2: Overview of the Bayesian optimization scheme employed here.mesh size of 0.1 along all dimensions. The patternsearch optimizer further locally optimizesthe best known point through a direct search algorithm. This local optimization proceeds until itsatisfies two convergence criteria, or reaches some number of iterations. The first criteria is that theloss converges to less than 10−6 between iterations. The second criteria is that the parameter stepsize converges to less than 10−3 between iterations. Assuming the level of objective function noiseis small, the true extent of objective function global convergence can be estimated by examiningreplicate Bayesian optimization calculations. Well converged calculations are expected to be foundnear the global minimum of the objective function, which is reproduced in well converged replicatecalculations. A significant scatter in the optimized objective function across replicate calculations,and corresponding scatter in properties of interest, indicates that the parameter optimization is notwell converged.Once the local optimization stage is complete, a final set of calculations is performed at theoptimal point to calculate all LiX properties of interest for the optimized model. This final set of137calculations includes a melting point calculation. All melting point calculations employ a minimumof 2,000 ions for their MD simulations, with ∆Tmin = 10 K and all other parameters as described inChapter 5. Hence, melting point calculations in this chapter include an estimated error of ±5 K.Bayesian optimization calculations are typically repeated five times, using different randominitial points, to check for consistency in results. Inconsistent results suggest that the calculationsare not converged, or that multiple local minima of similar depth exist for the given objectivefunction. The visualization given in Figure 6.2 corresponds to standard Bayesian optimizationmethodology. The methodology introduced in this chapter involves the development of an objectivefunction and judicious use of constraints. Once an appropriate objective function is constructed,standard Bayesian optimization is employed to minimize it.6.2.2 Objective FunctionThe problem addressed in this research can be conceptualized as follows. I wish to find the setof parameters for a pairwise LiX interaction potential that will minimize the error in several keyphysical properties of interest, such that these interaction potentials can be used in MD simulationsof LiX nucleation. A few structural and thermodynamic properties that should ideally match theknown experimental and theoretical values are selected. Most important is the correct lowest energycrystal structure at (T = 0, P = 0), rocksalt. Beyond the crystal structure, the following proper-ties which all have known experimental or theoretical values may also be important for producingaccurate LiX nucleation simulations: the lattice energy of rocksalt (EL), the lattice parameters ofrocksalt (a, b, c), the relative energies (∆EL =EL(other structure)−EL(rocksalt)) and lattice param-eters of other low-lying crystal structures within the structural landscape, the melting point (Tm),the diffusion coefficient of Li+ ions in the melt (Dliq,Li+), the molecular volume of the melt (Vliq),the molecular volume of the thermalized solid (Vsol), and the standard enthalpy of fusion (∆H◦fus =H◦liq (Tm)−H◦sol (Tm)). The molecular volume change of fusion, ∆Vfus = Vliq (Tm)−Vsol (Tm), canalso be targeted. All experimental LiX properties of interest used in this work are given in Ta-ble 6.1, while the theoretical values (the energies and lattice parameters of alternative LiX crystalstructures) are given in Table 6.2. Theoretical values are those obtained in Chapter 3 using highaccuracy DFT calculations employing the TMTPSS-rVV10L exchange-correlation potential andquadruple-ζ quality basis sets.If fitting the best set of LiX interaction potential parameters can be cast as an optimization138Property LiF LiCl LiBr LiI Units Ref.Rocksalt (a, b, c) 4.00514(5) 5.09631(4) 5.447(6) 5.957(7) A˚ 83Rocksalt V (T = 0) 16.0618(6) 33.0908(8) 40.4(1) 52.8(2) A˚3 83Rocksalt EL -1054.2(13) -865.4(15) -821.3(15) -764.5(13) kJmol−1 86Wurtzite (a, b) 3.831(1) 4.1250(5) 4.479(1) A˚ 15, 16, 204Wurtzite c 6.085(1) 6.6087(2) 7.254(1) A˚ 15, 16, 204Wurtzite V (T = 0) 38.682(2) 48.69(1) 63.02(4) A˚3 15, 16, 204Tm 1121.4(1) 883(1) 823(1) 742(1) K 123Vliq(Tm) 23.80 46.87 57.04 71.49 A˚3 123Vsol(Tm) 18.39 37.14 45.89 59.47 A˚3 83, 304∆Vfus 5.41 9.73 11.15 12.01 A˚3 123, 83, 304∆H◦fus 27.09 19.8 17.66 14.6 kJmol−1 123, 305Dliq,Li+(Tm) 1.22 1.06 1.19 1.10 10−4 cm2/s 286Table 6.1: Summary of experimental data used as targets for parameter fitting or modelvalidation. Experimental uncertainty in the last digit is given in brackets where avail-able. a, b, and c refer to lattice parameters corrected to 0 K by integration of lin-ear thermal expansion coefficients, as described in Section B.3. Experimental linearthermal expansion coefficients for LiX are taken from Ref. 306. I could find no re-ports of an experimental LiF wurtzite crystal structure in the literature, so theoreticaldata (Table 6.2) is used for LiF wurtzite lattice parameters. Tm refers to the experi-mental melting point, Vliq is the molecular volume of the liquid, Vsol is the molecularvolume of the solid, ∆Vfus = Vliq(Tm)−Vsol(Tm) is the volume change due to fusion,∆H◦fus = H◦liq(Tm)−H◦sol(Tm) is the fusion enthalpy, and Dliq,Li+ is the diffusion constantof lithium ions in the melt.problem,argminxf (x)subject to hi(x) = 0, i = 1, . . . ,m, (6.1)where f (x) is the objective function and hi(x) = 0 are a set of m deterministic constraints, then itcan be solved with Bayesian optimization. Hence, the objective function summarized in Figure 6.3is constructed. The idea is to take in a set of model parameters x as input, build the three LiX in-teraction potentials from those parameters (metal-metal, halide-halide, and metal-halide), performcrystal minimization and MD simulations using these interaction potentials, calculate all structuraland thermodynamic properties of interest {pi}, calculate the error of these properties with respect139Property Structure LiF LiCl LiBr LiIV (T = 0) Wurtzite 20.16NiAs 16.06 32.98 40.20 52.65Sphalerite 20.16 40.74 49.50 64.655-5 19.00 39.16 47.96 63.14AntiNiAs 16.72 35.20 43.04 56.68β -BeO 21.12 42.78 51.95 67.86CsCl 15.95 34.63 41.90 59.18EL Wurtzite -1036.89 -855.87 -813.66 -759.82NiAs -1044.77 -862.47 -819.75 -764.93Sphalerite -1035.07 -854.89 -813.05 -759.455-5 -1041.58 -854.41 -810.60 -754.93AntiNiAs -1031.91 -841.16 -797.49 -739.45β -BeO -1032.48 -852.14 -809.77 -756.24CsCl -998.47 -808.38 -767.57 -700.06∆EL Wurtzite 17.31 9.53 7.64 4.68NiAs 9.43 2.93 1.55 -0.43Sphalerite 19.13 10.51 8.25 5.055-5 12.62 10.99 10.70 9.57AntiNiAs 22.29 24.24 23.81 25.05β -BeO 21.72 13.26 11.53 8.26CsCl 55.73 57.02 53.73 64.44Table 6.2: Summary of theoretical data used as targets for parameter fitting or model val-idation, derived from the most accurate DFT results from Chapter 3. EL is the latticeenergy of a crystal structure while ∆EL refers to the difference in lattice energy betweenthe given crystal structure and that of rocksalt (positive values favour rocksalt). Sinceexperimental rocksalt lattice energies are used as target data, all DFT EL for other crystalstructures are shifted by the difference between the experimental and DFT rocksalt EL.This energy shift maintains the exact DFT ∆EL. Lattice energies and energy differenceshave units of kJmol−1 while molecular volumes V have units of A˚3per formula unit.140to known targets {p∗i }, and return a weighted sum of the squared errors†g(x) =∑iwi(pi(x)− p∗ip∗i)2. (6.2)Here, the sum is over all properties with non-zero weights, wi is the weight of property i, pi(x) is thecomputed value of property i for the interaction potential defined by x, and p∗i is its correspondingthe experimental or theoretical target. Each target property of interest p∗i is given its own weight wi,selected heuristically in advance based on the perceived importance of the corresponding property.The weighted error sum g(x) is transformed byf (x) = log(1+g(x)) (6.3)before returning to the optimizer. This has the effect of smoothing out rapid variations in theobjective function while maintaining f (x)≥ 0.Under certain circumstances, it becomes difficult or impossible to define an error with respectto properties of interest. There are three such cases, as shown in Figure 6.3, and these produce anundefined or “not a number” (NaN) output from the objective function. The crystal minimizationsubroutine, described in Section 2.8, is given certain constraints on the crystal volume and energy toprevent the unnecessary simulation of highly unrealistic systems; these are given in Section 6.2.3.If the objective function subroutine detects that one or more crystal constraints are not satisfied, theremainder of the objective function calculations are skipped and an undefined output is returnedimmediately. If all crystal constraints are satisfied, and properties of the molten salt are targeted,then an MD simulation of the molten salt at the experimental Tm proceeds. If the molten LiX systemspontaneously crystallizes during this simulation, or the simulation otherwise fails, the value ofVliq(Tm), Dliq,Li+ , and ∆H◦fus are not calculable for the given interaction potential parameters. Inthese cases, no further simulations are performed and an undefined output is returned. Finally, ifthe objective function subroutine has proceeded to the thermal solid calculation without trippingan error, and thermal solid properties are targeted, an MD simulation of a bulk periodic rocksaltcrystal is performed at Tm. If this simulation produces a phase change into another solid or themelt, or the calculation otherwise fails, then Vsol(Tm) and ∆H◦fus cannot be calculated. Again, anundefined output is returned. Undefined outputs are dealt with by bayesopt through an errormodel, as described in Section 6.2.3.†Squared errors yield a smoother objective function than the absolute value of the error, which has a discontinuousderivative at zero.141For each crystal structure i: Minimize perfect solid at T = 0 K Compute crystal properties {pij}: • (a, b, c)(i) • EL(i) • ΔEL(i) Objective function inputs • Model parameters x • Interaction potential • Target properties {p*} • Target weights {w} • MD parameters • Apply mixing rules? • ... Generate list of crystal structures needed to calculate target properties • Rocksalt, wurtzite, ... Save all crystal properties Targeting proper-ties of molten salt at Tm? Targeting proper-ties of solid salt at Tm? Yes No Simulate liquid at Tm Compute liquid properties {pk}: • Vliq(Tm) • Hliq(Tm) • Dliq,Li+(Tm) Save liquid properties Yes No Simulate rocksalt solid at Tm Compute solid properties {pk}: • Vsol(Tm) • Hsol(Tm) Compute liquid/solid difference properties {pk}: • ΔVfus(Tm) • ΔHfus(Tm) Phase Change? No Yes Return f(x) = NaN Return For each thermal property k: Phase Change? Yes No Targeting proper-ties of solid and liquid at Tm? Yes Save thermal solid properties Crystal constraints satisfied? Yes No For each crystal property j: Initialize objective function Figure 6.3: Outline of the objective function used. Tm refers to the experimental meltingpoint. f (x) is the final output objective function value while g(x) is an intermediatevalue used to generate it. A description of the other symbols is available in the captionsof Table 6.1 and 6.2.142The calculation of liquid properties for a given set of interaction potential parameters x is per-formed by a subroutine which is called during execution of the objective function, an overviewof which is given in Figure 6.4. The subroutine employs the MD methodology described in Sec-tion 2.8.2, requiring a minimum of 2000 ions, and checks for phase changes after an initial 25 psrelaxation simulation. A phase change is declared if the fraction of liquid classified atoms fallsto less than 85 %, as determined by the 0-0a neural network classifier (Chapter 4). To calculateDliq,Li+ , a dynamical property, a relatively long 200 ps simulation is performed using productionMD simulation parameters, starting from the final microstate of the rapid relaxation simulation. Aneural network structure classifier also checks the production simulation for phase changes. If thetotal liquid fraction remains above 85 %, all liquid properties of interest are calculated from theproduction simulation. If at any point a liquid simulation crashes (e.g. the simulation cell volumeexplodes), the simulation is automatically re-initialized with a time step that is 50 % shorter thanthe current time step (initially, the time step is 1 fs). This may be repeated until the time step fallsbelow 0.1 fs, after which a crashed simulation returns an error to the objective function, whichreturns an undefined output to the optimizer.The calculation of thermal solid properties is similar to the calculation of thermal liquid proper-ties. Solid properties are computed by MD simulation of a periodic rocksalt crystal of at least 2000atoms using the subroutine outlined in Figure 6.5. Unlike the liquid, no dynamic properties are cal-culated for the solid; only equilibrium properties are computed (Vsol(Tm) and Hsol(Tm)). Therefore,a rapid relaxation simulation with a relatively short 50 ps duration is employed for the thermalizedsolid. Only the final 25 ps of the simulation is used to calculate solid properties of interest. Aswith the liquid, a neural network classifier is employed to ensure that the rocksalt crystal remainsintact, requiring a mole fraction of at least 85 % rocksalt, otherwise a phase change flag is returned,leading to an undefined output in the objective function. The solid simulation subroutine employsthe same re-initiation procedure on failed simulations as described above for liquid simulations.6.2.3 Parameter Domain and ConstraintsTo implement a global optimization scheme such as Bayesian Optimization, a finite domain of inputparameters must be established beforehand; each input parameter is given its own range of possiblevalues. Within the total parameter space, it is also useful to constrain the domain into a feasibleregion using a constraint function, eliminating regions of the input space with known pathologicalproperties. In my case, the feasible region is determined by enforcing several deterministic con-straints on the shapes of interaction potentials. The constraints are designed to incorporate prior143Compute cell size L Randomly add N metal and halide ions Generate cubic box with side length L Estimate excluded radius of ions Fix V; energy minimize liquid (avoids blow-up) Rapid relax liquid with NPT at T = Tm for 25 ps Enthalpy Volume Check final frame for phase change with neural network classifier Return phase change = true No Yes Production liquid simulation with NPT at T = Tm for 200 ps Enthalpy Volume Use final microstate of rapid relax simulation as initial microstate Check final frame for phase change with neural network classifier Yes No Return phase change = false Initial density from crystal calculations Figure 6.4: The methodology used for calculating molten LiX thermal properties at the ex-perimental melting point Tm. All simulations are performed at P = 1 bar. N0 and V0are the number of formula units and volume of the energy minimized rocksalt unit cell.δ = 1.2 is a buffer on the initial volume to ensure all perpendicular cell distances re-main greater than 2rc if the cell volume decreases during simulation. Nmin = 1000 isthe minimum number of LiX formula units (i.e. half the total number of ions). Q = 0.8is a parameter used to estimate the initial density of the liquid relative to the energyminimized solid. During the random addition of metal and halide ions to the simulationcell, random ion placement is constrained such that each new ion is placed no closerthan 2rexcl. to any other ion.144Compute unit cell skew factors Compute supercell size while N < Nmin: Import energy-minimized primitive cell vectors (a, b, c) Generate na·nb·nc supercell Return phase change = true Check final frame for phase change with neural network classifier Yes No Return phase change = false Rapid relax solid with NPT at T = Tm for 50 ps Volume Enthalpy Compute averages over final 25 ps Figure 6.5: The methodology used for calculating solid LiX thermal properties at the experi-mental melting point Tm. All simulations are performed at P= 1 bar. δ = 1.2 is a bufferon the initial volume to ensure all perpendicular cell distances remain greater than 2rcif the cell volume decreases during simulation. (aˆ, bˆ, cˆ) are unit vectors that point alongthe primitive cell vectors, and indices on these refer to their Cartesian components. ξais the “skew factor” of cell vector a, which is 1 for a cubic cell. Skew factors are usedto ensure all perpendicular cell walls are separated by at least 2rc if δ = 1. N0 is thenumber of LiX formula units per primitive unit cell. La is the length of the supercell inthe a direction.145physical knowledge about the interaction potentials and reduce computational cost, thereby elim-inating pathological models without needing to waste resources simulating them. The constraintfunction constructs all three total LiX interaction potentials (metal-metal, halide-halide, and metal-halide) from the given input parameters and numerically checks for pathological behaviour. Thefeasibility requirements are left fairly loose, favouring freedom in the Bayesian optimization searchspace.On the lithium-halide cross interaction, the following constraints are employed. (1) The in-teraction must have a minimum, it cannot be entirely repulsive or attractive. (2) The interactionminima cannot be lower than−1000 kJmol−1, as lower values will not lead to a physically reason-able LiX model. (3) The repulsive wall must peak at a height of at least 300 kJmol−1 above theinteraction minima. This constraint only applies to the CBH functional form, as the CLJ model hasa monotonically increasing repulsive wall as r→ 0. Without a reasonably high wall, high velocityparticles can overcome the repulsive interaction and crash the simulation. (4) The interaction mini-mum should not be closer than 0.5 A˚ or further than 10 A˚. If the system is too dense, the simulationcell contains many interaction sites, rapidly driving up computational cost. On the other hand, avery low density system is not likely to be a useful LiX model.For the metal-metal and halide-halide interactions, slightly different constraints are employed.(1) The total interaction potentials must be entirely repulsive, with no interaction minima prior tothe repulsive wall. Note that dispersion can reduce the slope of the total repulsive interaction, butthe repulsive Coulombic interaction should not be overcome to produce an interaction minima. (2)The height of the total repulsive interaction potential must be at least 300 kJmol−1 above zero. (3)The repulsive interaction should not reach above 300 kJmol−1 at a distance closer than 0.5 A˚ orfurther than 10 A˚. (4) The anion radius should be larger than the cation. The two ion radii aredetermined by the σii parameters for CLJ potentials and r0,ii parameters for CBH potentials.With these constraints in place, it is possible to quickly and deterministically search for a pa-rameter domain that contains all possible feasible points through trial and error, as demonstratedin Figure 6.6. The figure shows the feasibility of 10,000 random search points within the domaingiven in Table 6.3 for the CBH potential (with combining rules and 7 < γi j < 20) projected ontothe εX−X− vs εLi+Li+ subdomain. Table 6.3 gives the parameter domains used in this work, foundby searching through parameter space for feasible regions. While the parameter domains of Ta-ble 6.3 may contain many infeasible points, only feasible points within the domain are consideredin Bayesian optimization. The acquisition function of Bayesian optimization is maximized onlywithin the feasible region of the domain, considerably reducing the volume of search space.In addition to the deterministic constraints described so far, there are additional constraints that146Figure 6.6: The feasibility of 10,000 uniformly sampled random search points within the5-dimensional domain spanned by the LiX CBH (7 ≤ γ ≤ 20) interaction potential pa-rameters with mixing rules enabled, projected onto the 2-dimensional r0,Li+Li+ versusr0,X−X− slice. A description of the CBH potential and its parameters is given in Sec-tion 2.3. Red circles are feasible points, while crosses are infeasible points. The unitsof both axes are nm.147Potential Parameter Lower Bound Upper Bound Units TransformCBH (γi j < 6) r0,Li+Li+ 0 0.2 nm normalr0,X−X− 0 0.4 nm normalεLi+Li+ -108 -50 kJmol−1 logεX−X− -106 -1 kJmol−1 logγi j 0 6 normalCBH (γi j > 7) r0,Li+Li+ 0 0.5 nm normalr0,X−X− 0.15 0.7 nm normalεLi+Li+ 0 400 kJmol−1 normalεX−X− 0 250 kJmol−1 normalγi j 7 20 normalCLJ σLi+Li+ 0.05 0.46 nm normalσX−X− 0.13 0.55 nm normalεLi+Li+ 0 600 kJmol−1 normalεX−X− 0 200 kJmol−1 normalTable 6.3: Parameter domains used in this chapter for Bayesian optimization calculations.Interaction potential parameters are described in Section 2.3. The Transform columnindicates whether log transformation was applied to the parameter domain. Log transfor-mation recasts the Bayesian optimization search into log space for the given parameter.For calculations without combining rules, metal-halide cross parameters are given thesame domains as halide-halide parameters. The CBH potential with γi j between 6 and 7is not considered in this work.apply directly to the crystal minimization and thermal calculations discussed in Section 6.2.2. Aspreviously mentioned, energy minimized crystals that do not satisfy the crystal constraints are as-signed an undefined output in the objective function. The crystal constraints are: (1) no minimizedlattice parameter may be shorter than 2 A˚ nor larger than 20 A˚, (2) no crystal volume may be lessthan 10 A˚3nor greater than 2000 A˚3per formula unit, and (3) no crystal lattice energy may belower than −2000 kJmol−1. Any thermal liquid or solid simulations that lead to a phase change orsimulation error are also assigned an undefined output.Undefined outputs in the objective function are treated separately by bayesopt. These areused to construct an error model η (x) of the parameter space, which is incorporated into theacquisition function. Input points that lead to well-defined outputs are assigned a value of −1 forthe purposes of error modeling, while inputs that yield undefined outputs are assigned +1. η (x)148is an additional Gaussian process surrogate model regressed to the error data over the parameterdomain. The acquisition function is then weighted by the probability of yielding a well-definedoutput at each feasible input point, computed using η (x).307 The error model prior is 0 everywhere,such that all feasible points within the parameter domain are assumed equally likely to yield well-defined outputs, unless nearby undefined objective function calls are found.6.3 Results and DiscussionIn this section, results of Bayesian optimization calculations applied to pairwise LiX force field pa-rameter optimization are discussed in two parts. Section 6.3.1 features the application of Bayesianoptimization to the CLJ potential, finding that the model is incapable of fitting both low T crystalstructure and high T thermodynamic data. In Section 6.3.2, similar calculations are applied to theCBH potential and the results are discussed. It appears that the added flexibility of the CBH poten-tial may be sufficient to overcome the systematic issues stemming from use of the CLJ functionalform.6.3.1 Coulomb Lennard-Jones ModelThe Coulomb Lennard-Jones model functional form is the most commonly employed model forthe interactions of alkali halides in MD simulations. However, as demonstrated in Chapter 3, awidely used set of CLJ parameterizations for LiX by Joung and Cheatham (the JC models)14 failto recover the correct crystal structure of the lithium halides more often than not across all threesets of models.† The JC models were fit to reproduce aqueous solution properties, but were alsodesigned to give the correct rocksalt lattice energy and crystal density at 300 K. To overcome theincorrect relative crystal energies common to many JC models of LiX, a reparameterization of theLiX CLJ model is attempted with experimental and theoretical crystal data (Table 6.1 and 6.2) astargets for the Bayesian optimization scheme described above, while explicitly ensuring the correctlowest energy crystal structure (rocksalt) is maintained. I begin with a fit similar to the JC model,targeting rocksalt lattice energies EL and lattice parameters a (as a stand-in for molecular volume)with equal weight and employing parameter combining rules for the cross interaction. The resultsof this preliminary fit are given in the Appendix, Figure E.8. Both LiX rocksalt lattice energies andlattice parameters can be fit fairly well to the target data, although the fit is progressively worse forthe lighter halides. Despite reproducing accurate rocksalt lattice energies and densities, fitting only†The JC models have three parameterization for each alkali halide: intended for use with the SPC/E, TIP3P, andTIP4P/EW water models.149to these two experimental crystal targets sometimes yields the incorrect wurtzite relative latticeenergy, reproducing the pathology of the JC models.Since preliminary results give reasonable fits to rocksalt crystal properties, the next step is toadd the relative wurtzite lattice energy as a target in the objective function. This was done withequal weighting to the rocksalt targets, allowing the Bayesian optimization to converge towards amore useful set of parameters that ensures the correct ordering of crystal energies. The results ofthis second fit, performed with five replicates for each LiX salt using different initial random searchpoints, are given in Figure 6.7. This figure gives the error in the final fitted force fields, not onlywith respect to the targets, but also for all other crystal and thermodynamic data of interest. Theresults are well converged; crystal properties are fit fairly well across the board, with most replicatecalculations yielding the same or very similar final results. In particular, the relative LiX wurtziteenergies are reproduced accurately. In all cases, the lowest energy crystal structure (rocksalt) iscorrectly reproduced. Overall, The fits are progressively better for larger anions. The errors on LiFcrystal properties are small but significant; the rocksalt lattice energy is approximately 30 kJmol−1(3 %) too low, while the molecular volume is 1.5 A˚3(9 %) too large. Across the board, tetrahedralcrystal structures (wurtzite, sphalerite, and β -BeO) fit the DFT ∆EL differences with respect torocksalt, but their molecular volumes are slightly too large. Note that molecular volumes V aregiven per formula unit.Perhaps most critically, the melting points of all fitted CLJ models given in Figure 6.7 are all atleast 328 K higher than experiment, with worsening Tm for the lighter LiX salts. A similar problemwas observed for all alkali halides employing the JC and SD54 interaction potentials (Chapter 5),indicating a systematic problem with the high temperature behaviour of all CLJ models fit to low-Tcrystal data. In an attempt to improve the CLJ fits given in Figure 6.7, I conducted two additionalsets of calculations. In the first set, the LiX ion charges are allowed to vary between 0.95 and1.0 by setting the ion charge as an additional parameter in the Bayesian optimization. Fractionalcharge has previously been suggested as an alternative to explicit polarization for improving someproperties of alkali halides.308–310 The results of these calculations are given in the Appendix, Fig-ure E.9. Although fits to the low-T crystal data are improved with fractional charge, the fits tomelting points and other high-T thermodynamic data are even worse. Clearly, fractional chargecannot save the CLJ interaction potential from its high temperature pathological behaviour. Thesecond additional set of calculations removed combining rules, increasing the number of indepen-dent force field parameters from 4 to 6. The results of these calculations are also given in theAppendix, Figure E.10. No improvement in fit is observed by removing the combining rules andallowing all three LiX interactions to vary independently.150Figure 6.7: A comparison of fitted LiX CLJ force field results to experimental and theoreticaldata of interest, including untargeted data. The plots in the top row give signed errorswith respect to the indicated experimental thermal property p− p∗, where p∗ is a targetproperty. Tm is the melting point, ∆H◦fus is the standard enthalpy of fusion, and ∆Vfus isthe change in molecular volume. Vliq(T ∗m) and Vsol(T ∗m) are the molecular liquid and solidvolumes at the experimental melting point, respectively. Dliq,Li+(T∗m) is the diffusioncoefficient of lithium ions in the liquid at the experimental melting point. EL is thelattice energy at T = 0, ∆EL is the lattice energy of a structure relative to rocksalt, andV is the molecular volume of a structure. An asterisk indicates a target value rather thana model value. The optimization employed (T = 0, P = 0) crystal properties as targets:EL(rocksalt), the rocksalt lattice parameter (as a stand-in for V ), and ∆EL(wurtzite), asindicated by red boxes in the figure. In most cases, a value near zero indicates a better fit.For ∆EL plots, better fits are toward the indicated DFT values. Force field parametersare obtained after 1000 Bayesian optimization iterations, followed by local objectivefunction optimization until convergence. The experimental and theoretical referencedata used here is given in Table 6.1 and 6.2, respectively.151One key thermodynamic property of the lithium halides are the melting points, which areknown to be extremely sensitive to force field details.19 While in principle it is possible to addmelting points as a direct target to the Bayesian optimization algorithm employed here, the highcomputational cost of the MP calculations developed in Chapter 5 makes this an impractical choice.Although attempting to fit the melting point directly is too computationally expensive, preliminarycalculations indicated that experimental ∆H◦fus and Dliq,Li+(Tm) targets could be used as an approx-imate stand in for Tm. Moreover, an exact fit to experimental Dliq,Li+(Tm) values does not appearto be necessary to target melting points. Since the neural network structure classifiers from Chap-ter 4 cannot differentiate between amorphous solid and liquid, a Dliq,Li+(Tm) value near experimentensures that the simulated liquid is not amorphous solid.With the above discussion in mind, another fivefold repeated set of Bayesian optimization cal-culations are performed using experimental values of ∆H◦fus and Dliq,Li+(Tm) as targets with equalweights. Crystal and thermodynamic properties calculated from the fitted force field parameters,compared with reference values, are shown in Figure 6.8. A significant spread in fusion enthalpiesfor the fitted models is obtained, possibly indicating some optimizations are not fully convergedafter 500 Bayesian optimization cycles plus an additional 200 local optimization cycles. Still, someobvious conclusions can be made. Although targeting the high-T thermal properties produced sev-eral LiX force fields with Tm much closer to experiment, all fitted force fields have extremely highmolecular volumes and lattice energies (i.e., the absolute EL are too small), both at low temperatureand at the experimental melting point. Additionally, almost all models obtain the incorrect lowestenergy crystal structure in all cases, either favouring wurtzite or CsCl. The single exception is aLiF model, indicated with a downward-facing triangle in Figure 6.8, that obtains a good fit to themelting point while maintaining rocksalt as the lowest energy structure. However, this model alsofails to obtain a reasonably accurate absolute lattice energy or molecular volume.Together, the results in Figure 6.7 and Figure 6.8 indicate that although the CLJ model can fitthe correct ordering of LiX crystal energies and rocksalt properties, the CLJ functional form lacksthe flexibility to obtain both low-T crystal and high-T thermodynamic properties simultaneously,even if fractional charge is allowed or additional flexibility is added by removing the mixing rules.An additional preliminary set of fivefold repeated CLJ parameter optimization calculations usingNaCl target data (not shown), with simultaneous targets on low T rocksalt crystal data and high Tthermodynamic data, was also attempted. The motivation for testing NaCl was twofold; NaCl hashigher accuracy experimental data available than LiX, but also less interference from the wurtzitecrystal structure. The energy gap between rocksalt and wurtzite is larger for NaCl than for the LiX.This set of NaCl optimizations failed to achieve simultaneous fits to both crystal properties and152Figure 6.8: A comparison of fitted LiX CLJ force field results to experimental and theoreticaldata of interest, including untargeted data. The plots and symbols are as describedin the caption of Figure 6.7. ∆H◦fus and Dliq,Li+(Tm) are targeted with equal weights,as indicated by red boxes in the figure. Force field parameters are obtained after 500Bayesian optimization iterations, followed by up to 200 iterations of local objectivefunction. The experimental and theoretical reference data used here is given in Table 6.1and 6.2, respectively. In some cases, untargeted thermal data could not be calculated fora given model due to a phase change in the corresponding simulation; these results arenot shown.melting points in any case.In Figure 6.9, the interaction potentials corresponding to the fitted LiX models in Figure 6.7 areplotted. Not surprisingly, the total potentials match the JC(SPC/E) corresponding potential energycurves fairly closely, with only subtle changes required to ensure the correct ordering of crystalstructures while maintaining the lattice energy and rocksalt molecular volume. In Figure 6.10, thefinal fitted parameters of these same models (including cross-parameters generated from combiningrules) are compared with those of the JC(SPC/E) model (note that the symbols match those ofFigure 6.7). Notably, there is a fairly wide spread in most of the fitted parameters, despite a fairlytight congruence in the crystal properties. This indicates there is a range of parameters that give153Figure 6.9: A comparison of CLJ LiX force fields to the JC model,14 given in black. Theupper row shows the Lennard-Jones part of the potential, while the lower row showsthe full pair potentials including the Coulombic part. In each case, five fitted modelsare plotted, indicated with coloured lines. Each set of parameters corresponds to threepotentials, which are plotted with matching colours within each axis. The identity of thethree potentials are given in the legend at the top.equivalent results. In particular, larger values of σi j are observed to pair with smaller values of εi j.It is not yet clear if some criterion can be used to select a superior set of parameters, but fittingto additional properties may help constrain the range of fitted parameters. In particular, the CsClcrystal structure relative energies vary substantially in Figure 6.7, indicating a potential additionaltarget. It may also be possible to fit the crystal properties, then use high temperature thermalproperties with a small weight to select a “best” set of fitted CLJ parameters that maintain theirgood fit to low-T crystal properties.In summary, the CLJ potential form can be non-uniquely fit to the most important low-temperatureLiX crystal properties of interest: the correct stable crystal structure (rocksalt), its lattice energy,and its molecular volume. The fits are fairly accurate, but are progressively worse for the smalleranion salts. Although the results fit the crystal properties well, melting points are much too high,indicating an overstabilization of the solid or understabilization of the liquid at high temperatures.Fitting the LiX melting points is possible, but produces CLJ models of LiX with low crystal densi-ties, weak lattice energies, and generally incorrect crystal structures. This raises the question: can154Figure 6.10: A comparison of fitted CLJ LiX force field parameters to the JC(SPC/E) LiXmodels, indicated by black crosses. A description of these force field parameters isgiven in Section 2.3.we do better with a different rigid-ion pairwise force field? Perhaps the slope of the Lennard-Jonesrepulsive wall (proportional to r−12) gives an unrealistic description of lithium halide interactionpotentials, especially at higher temperatures. In the next section, I address this question by fittingthe CBH potential to the same experimental and theoretical LiX data. This involves optimizing anadditional interaction potential shape parameter γi j (Section 2.3), increasing the size of the param-eter space and the complexity of the optimization.6.3.2 Coulomb Buckingham ModelIn efforts to generate simple pairwise force fields for the lithium halides, the CLJ potential wasfound lacking. As discussed in Section 6.3.1, it cannot be parameterized to reproduce high-Tthermodynamic results and low-T crystal properties of the lithium halides simultaneously. In aneffort to overcome this challenge, Bayesian optimization was applied to parameterizing the CBHmodel, with mixing rules as described in Section 2.3. However, the Buckingham force field issignificantly more complicated and prone to pathology than the Lennard-Jones potential. Thispathology stems from the exponential function used to represent close-range repulsive interactions,rather than the power law employed in the Lennard-Jones potential.155Although an exponential repulsive wall shape has more theoretical justification than a powerlaw at intermediate ranges,61 the exponential function raises to a constant as r → 0, while thedispersion term falls without bound. The Coulombic interaction terms are also unbounded as r→ 0.In the Lennard-Jones potential, the dispersion and Coulombic terms are overcome by the r−12repulsive interaction, which dominates at close range, such that the potential grows without boundas r→ 0. On the other hand, the total CBH potential contains an unphysical peak in the repulsivewall, followed by an unbounded close-range attractive interaction, as the dispersion term begins todominate at small r. As discussed in Section 6.2, the unphysical behaviour is not an issue if therepulsive wall is high enough to prevent atoms from overcoming it, and ions are never initializedwith relative positions closer than the repulsive peak. However, if the CBH repulsive barrier is toosmall, high velocity particles may overcome it leading to overlap of ions.Besides the inherent close-range pathology, the Buckingham potential written in r0,i j/εi j form†(Section 2.3) is complicated by multiple physically plausible domains of the shape parameter γi jthat are best examined independently: 0 < γi j < 6, 6 < γi j < 7, and 7 < γi j < ∞. Each of theseγi j domains produces different feasible regions for the remaining interaction parameters. In thefirst two γi j domains, R0,i j and εi j describe properties of the repulsive wall peak, while in the7< γi j <∞ domain, R0,i j and εi j describe properties of the interaction minima. This latter domain istherefore somewhat easier to work with, and was used as a starting point for Bayesian optimizationof CBH interaction parameters for LiX. However, as shown below, the 7 < γi j < ∞ turns out notto contain the most physically realistic set of parameters for the lithium halides found so far; thesewere found in the 0 < γi j < 6 domain. I have not specifically optimized CBH parameters withinthe 6 < γi j < 7 domain, but it is not expected to yield improvement over optimization within the0< γi j < 6 domain.γi j > 7 DomainAs with the CLJ model, an initial Bayesian optimization was performed on the CBH interactionpotential using only T = 0 crystal properties for 1000 Bayesian optimization cycles, followed bylocal optimization. Within the 7 < γi j < ∞ domain, only γi j values less than 20 were considered,as higher values clearly produce unrealistically narrow interaction potential shapes. The results ofthis initial fit, performed with five replicates for each LiX salt using different initial random searchpoints, are given in Figure 6.11. These results show a strikingly similar pattern to those observedfor the CLJ LiX parameter fitting, albeit LiX crystal parameters are fit with slightly better accuracy.†Recall that this form is used because combining rules similar to those defined for Lennard-Jones potentials can beemployed.156Figure 6.11: A comparison of fitted LiX CBH force field results (7< γi j < 20) to experimen-tal and theoretical data of interest, including untargeted data. The plots and symbolsare as described in the caption of Figure 6.7. The optimization employed (T = 0,P = 0) crystal properties as targets: EL(rocksalt), the rocksalt lattice parameter (as astand-in for V ), and ∆EL(wurtzite), as indicated by red boxes in the figure. Force fieldparameters are obtained after 1000 Bayesian optimization iterations, followed by localobjective function optimization until convergence. The experimental and theoreticalreference data used here is given in Table 6.1 and 6.2, respectively.Due to the added flexibility of the additional shape parameter γi j, properties not targeted by the fitare more widely dispersed. This is an indication that more properties can be targeted without lossof accuracy in the current targets.The next step was to target thermal properties in an attempt to reproduce accurate model melt-ing points, while still maintaining the correct order of rocksalt vs wurtzite lattice energy. There-fore, the next set of Bayesian optimizations for the CBH model parameters were set to target ∆H◦fus,Dliq,Li+(Tm), and the wurtzite ∆EL. The results of this optimization are given in Figure 6.12. Un-fortunately, the outcome is similar to that observed for the CLJ model, but with added pathology.Some fitted models have reasonable rocksalt melting points, but also favour either the wurtzite or157Figure 6.12: A comparison of fitted LiX CBH force field results (7< γi j < 20) to experimen-tal and theoretical data of interest, including untargeted data. The plots and symbols areas described in the caption of Figure 6.7. ∆H◦fus, Dliq,Li+(Tm), and the relative wurtziteEL are targeted, as indicated by red boxes in the figure. Force field parameters areobtained after 500 Bayesian optimization iterations, followed by up to 200 iterationsof local objective function. The experimental and theoretical reference data used hereis given in Table 6.1 and 6.2, respectively. In some cases, untargeted thermal datacould not be calculated for a given model due to a phase change in the correspondingsimulation; these results are not shown.CsCl crystal structure. Additionally, most models have rocksalt lattice energies and densities thatare quite far from experiment, indicating a systematic problem with the potentials that cannot beovercome through parameterization within this domain. As such, the 0< γi j < 6 parameter domainwas examined next. In general, this domain allows for much softer potentials. If the problem withthe potentials examined so far is in the steep slope of their repulsive walls, then smaller γi j valuesmay prove more physically realistic.158γi j < 6 DomainIts worth recalling that many TF models for all alkali halides, which have functional forms similarto the CBH interaction potential, are able to simultaneously produce accurate melting points andcrystal lattice energies for many of the alkali halides (Section 5.3.2). However, for LiX, all TFmodels yield incorrect ordering of crystal structure energies. Instead, these models favour wurtziteas the stable structure under ambient conditions. If one calculates the γi j of TF models for the alkalihalides, every value falls within the 0< γi j < 6 domain. This is a first hint that the γi j < 6 domaincould yield more physically reasonable results than the γi j > 7 domain. In any case, Bayesianoptimization was applied to this domain of the CBH parameter space as before, starting with T =0 crystal properties as targets. The results of this initial optimization are given in Figure 6.13.As before, fitting the CBH interaction potential parameters to crystal properties in this domainproduces a large spread in untargeted properties due to the added flexibility of the model comparedwith CLJ. Importantly, this domain of the CBH model appears to be able to fit targeted rocksalt T =0 crystal properties more closely than previous parameter optimizations. The final loss functionvalues (not shown) obtained for each salt after optimization vary widely between the replicatecalculations, indicating that complete convergence is not yet acheived. However, some replicatecalculations are more converged than others, as reflected in the relatively large spread of relativewurtzite energies obtained between replicate calculations. Although most models in Figure 6.13do not obtain accurate melting points, with one LiF model being an exception, the spread of Tmvalues indicates the possibility of adding high temperature thermal properties as a simultaneoustarget, which is attempted next.Picking the best set of targets for parameter fitting is a difficult decision. The results presentedso far suggest that not all LiX properties can be simultaneously fit to the reference data with perfectaccuracy using simple pairwise force fields (for example, the NiAs structure relative energies seemparticularly difficult to fit). One has to judiciously select the most important properties to target.Several combinations of targets were tried before arriving on the promising results shown in Fig-ure 6.14. Examining the final objective function values of replicates (not shown) suggests that thesecalculations are not yet well converged. However, Figure 6.14 shows melting points within 50 Kof experimental values are simultaneously fit to T = 0 crystal data within a reasonable accuracy.Crucially, most fitted models yield rocksalt as the low energy crystal structure, although some LiIand LiBr replicate calculations do not.While the results given in Figure 6.14 are considered preliminary, and can almost certainly befurther converged, they suggest that simple pairwise interaction potentials with combining rules can159Figure 6.13: A comparison of fitted LiX CBH force field results (0< γi j < 6) to experimentaland theoretical data of interest, including untargeted data. The plots and symbols are asdescribed in the caption of Figure 6.7. The optimization employed (T = 0, P= 0) crys-tal properties as targets: EL(rocksalt), the rocksalt lattice parameter (as a stand-in forV ), and ∆EL(wurtzite), as indicated by red boxes in the figure. Force field parametersare obtained after 1000 Bayesian optimization iterations, followed by local objectivefunction optimization until convergence. The experimental and theoretical referencedata used here is given in Table 6.1 and 6.2, respectively. In some cases, untargetedthermal data could not be calculated for a given model due to a phase change in thecorresponding simulation; these results are not shown.be made to produce models of LiX that simultaneously reproduce some critical low-T crystal data,high-T thermodynamic data, and the correct stable crystal structure (rocksalt). Further calculationswill be required to see how closely these fits can be made.6.4 Summary and ConclusionsAlthough popular pairwise interaction potentials exist in the literature for the lithium halides,12–14these LiX models tend to produce the incorrect lowest energy crystal structure, melting point, or160Figure 6.14: A comparison of fitted LiX CBH force field results (7< γi j < 20) to experimen-tal and theoretical data of interest, including untargeted data. The plots and symbolsare as described in the caption of Figure 6.7. Several key properties were targeted inthe objective function, as indicated by red boxes in the figure. The lithium diffusionconstant was targeted with a relatively small weight. Force field parameters are ob-tained after 500 Bayesian optimization iterations, followed by up to 200 iterations oflocal objective function. The experimental and theoretical reference data used here isgiven in Table 6.1 and 6.2, respectively.both. In this chapter, Bayesian optimization is applied to efficiently explore the parameter spaceof simple LiX pairwise interaction potentials. The aim is to find the limits of particular functionalforms and, if possible, produce a simple rigid-ion pairwise force field for the lithium halides thatavoids the issues found in the common literature models. I focus on two functional forms forthe LiX interaction potentials: the Coulomb-Lennard-Jones model and the Coulomb-Buckinghammodel, both of which can be defined with combing rules for the lithium-halide cross interactionparameters. The CLJ LiX model contains four optimizable parameters when combining rules areactive, while the CBH model contains five. The fifth parameter gives the CBH potential additionalflexibility in the shape of its repulsive interactions.161To apply Bayesian optimization as a parameter fitting technique, an objective function is con-structed that takes force field parameters as input, performs some set of crystal minimization calcu-lations and/or molecular dynamics simulations using the force field defined by the input parameters,and then outputs a measure of error for crystal and thermodynamic properties associated with theforce field. This error is calculated with respect to known experimental and theoretical target data.Since the objective function is slow to compute, nondeterministic, and does not produce derivativeswith respect to input parameters, Bayesian optimization is chosen as a suitable and efficient ma-chine learning based optimizer. The acquisition function and covariance function of the Bayesianoptimization scheme are selected through a series of preliminary tests. The parameter search spaceis greatly reduced by asserting that all force fields generated from input parameters satisfy a setof physically motivated feasibility constraints. The feasible parameter region produced from theseconstrains is used as a guide to set the search domain of each force field parameter. Only feasiblepoints within the full parameter domain are sampled.Once the methodology is established, it was first applied to the CLJ model in an effort to fit pa-rameters to crystal and thermodynamic LiX data. The results demonstrate that while the CoulombLennard-Jones functional form can be parameterized to reproduce either known low temperatureLiX crystal data or melting points, the functional form lacks the flexibility to fit both low andhigh temperature properties simultaneously. Allowing fractional charge can improve the fit to LiXcrystal data, but this further pushes the model melting points away from their experimental targetvalues. Disabling combing rules, which allows the cross interactions to vary independently fromthe lithium-lithium and halide-halide interactions, does not improve the fit. The parameter fittingcalculations are fivefold repeated, yielding five similarly optimized sets of parameters that all fit thetarget data well. Notably, the parameters corresponding to these ostensibly equivalent force fieldsvary quite dramatically, indicating that there exists a range of optimal force field parameters ratherthan a single set.Since application of parameter fitting to CLJ could not produce a satisfactory set of LiX models,I next fit CBH parameters to the same crystal data. The CBH functional form is more complicated,with three domains of the shape parameter γi j that are best explored separately due to their differingproperties. Two of the domains are examined in my calculations. Exploration of the γi j > 7 CBHdomain, where the repulsive interaction slope is relatively steep, yields fitted force fields compara-ble in quality to the fitted CLJ force fields already produced. However, the γi j < 6 region is muchmore fruitful. Preliminary calculations suggest that the CBH force field within this domain canbe fit to both low temperature crystal and high temperature thermodynamic data simultaneously,although more work is needed to verify the extent of this fit.162In future calculations, several aspects of the Bayesian optimization methodology employed sofar can be further improved. The most obvious target for improvement is the input force fieldparameter constraints, which have so far been set very loosely. Using the results generated sofar as a guide, the parameter search space can be greatly reduced by tightening constraints on theinteraction potentials. As fewer cycles will be wasted searching in unfruitful regions of parameterspace, tightening up constraints will allow the next set of parameter LiX optimization calculationsto converge much faster. The objective function targets and weights can also likely be tuned toproduce more desirable fitted potentials, and this will be the focus of future calculations. Finally,other functional forms of the pairwise potentials may be considered in future work. For example,allowing a reduction in the Lennard-Jones repulsive exponent from 12 may improve fits to LiXdata with the CLJ functional form.163Chapter 7Conclusions7.1 SummaryThe structural chemistry of the lithium halides is complex; several low energy metastable crystalstructures compete with the thermodynamically stable rocksalt crystal structure, and complicatethe behaviour of these ostensibly simple salts, particularly for the heavier lithium halides. Somepreliminary evidence suggests that, while many alkali halide salts undergo single-step classical nu-cleation behaviour5 when precipitating from supersaturated aqueous solutions, the lithium halidesdo not.11 The small lithium ion size may contribute to a second non-classical free energy barrier tonucleation from aqueous solution, but this barrier is not yet well characterized. A possibly relatedexperimental observation, wherein high-pressure aqueous solutions of LiF decrease in solubilityabove 475 K,29 is also not well understood. While experiments have difficulty dealing with thelength and time scales necessary to explore these phenomena, molecular dynamics simulationsare well suited to characterize them. However, the utility of molecular simulations hinges on theaccuracy of their underlying force fields. Ab initio molecular simulations are possible, and thesecan yield chemically accurate intermolecular interactions given an accurate enough level of theory.However, the relatively large system sizes and time scales required to accurately simulate the phe-nomena of interest likely precludes the use of ab initio interactions for the study of LiX nucleationfrom aqueous solution. On the other hand, the most commonly used classical pairwise rigid-ionpotentials for LiX simulations12–14 yield either incorrect crystal structures, inaccurate high tem-perature thermodynamic properties, or both. A recent polarizable model for all alkali halides cor-rects the crystal structures of lithium halides,18 but yields inaccurate melting points for the lithiumhalides and may be more computationally complex than necessary.19 The research communicated164in this thesis is focused on characterizing the complex structural chemistry of the lithium halidesusing a wide array of methodologies.In Chapter 2, I introduce the methodology common to all of the research conducted for thisthesis. Most of the common methods discussed in Chapter 2 involve various aspects of LiX molec-ular dynamics simulations, including the interaction potentials used and system initiation tacticsemployed. Methodology specific to particular research projects are given in the methods section ofthe corresponding chapters.Chapter 3 is a deep dive into the structural landscape of lithium halides using quantum chemi-cal calculations, primarily density functional theory. There are many aspects of practical quantumchemical calculations wherein the best choice of some calculation parameter is not immediatelyobvious. Much of this chapter outlines the efforts taken to ensure these quantum calculations are asaccurate as possible. In particular, the basis sets, exchange correlation functional, dispersion, andrelativistic corrections are all considered in detail. By comparing results of LiX bulk phase quan-tum chemical calculations with the available experimental data at T = 0 (rocksalt lattice energiesand thermally corrected lattice parameters), I converged to a set of density functional theory cal-culation parameters that best reproduce this experimental data. I use this calculation methodologyto compute the theoretical crystal properties for the lithium halides, including lattice energies andlattice parameters of seven crystal structures that constitute low energy crystals in the structurallandscape of at least one lithium halide salt. Notably, I find that the NiAs crystal structure of LiIis energetically in very close competition with rocksalt, yet the NiAs structure has not yet beenreported experimentally. Previous work has also noted the appearance of NiAs as a low energycrystal structure of LiI in quantum-mechanical calculations.204 The work in this chapter updates arelatively long history of solid state lithium halide crystal calculations with modern density functiontheory methods, yielding extremely high accuracy with respect to experimental observables.In addition to producing high accuracy reference data for lithium halide crystal structures,Chapter 3 outlines efforts taken to determine why commonly used classical force fields frequentlyyield incorrect crystal structures for lithium halides. I find that the strength of the anion-anion dis-persion interaction is particularly important for determining the relative energies of crystal struc-tures, a subtle finding that was not reported in previous lithium halide crystal calculations. Therocksalt crystal structure is favoured over wurtzite by increased dispersion strength. Without dis-persion, I conclude that the heavier lithium halides (LiBr and LiI) would exist as wurtzite crystalsunder ambient conditions. Finally, thermal free energy corrections are applied to the T = 0 crystalresults via harmonic121,122 and quasi-harmonic190–193 vibrational calculations. These calculationsare used to characterize the impact of thermal effects on the relative free energy landscape of the165lithium halides. I find that the impact of thermal corrections is small, and likely does not affect themost stable crystal structure of the lithium halides at any temperature below their melting points.In Chapter 4, I introduce a set of neural network structure classifiers developed for detectingthe local structural environment of binary salt ions (in particular, lithium halides) in molecular dy-namics simulations. These classifiers are trained and validated on millions of data points generatedfrom bulk phase alkali halide molecular dynamics simulations over a wide range of temperatures.Unlike any previously available order parameters, these classifiers are capable of accurately and si-multaneously distinguishing between nine local structural environments: liquid, rocksalt, wurtzite,NiAs, sphalerite, 5-5, antiNiAs, β -BeO, and CsCl. By leveraging uncertainty in the classificationprobabilities, I also demonstrate that the classifiers can detect a tenth local environment, interfa-cial. Convolutional neural network architecture is applied in many of the neural networks. Clas-sification accuracy is improved by time convolution, which is a method for averaging out thermalfluctuations that would otherwise disrupt the accuracy of instantaneous structure classification. Themost accurate neural network classifier is demonstrated on two molecular dynamics simulations oflithium halide homogeneous nucleation from the melt. Videos of the trajectories after process-ing from neural network classifiers, available as part of the supplementary materials, demonstratethe utility of the neural network classifiers for lithium halide nucleation research. This researchdemonstrates a powerful and generalizable supervised machine learning methodology that offersincredibly high accuracy for simultaneously classifying multiple local intermolecular structures inmolecular dynamics simulations at high temperatures, in contrast with conventional order param-eters that typically perform poorly at high temperature. While the concept of neural networks forstructure classification is not new, the specific application to alkali halide phases and the use oftime convolution is.Chapter 5 outlines another advancement in methodology for alkali halide research: an auto-mated melting point calculation algorithm for alkali halides. The algorithm employs a series ofdirect solid-liquid coexistence48 simulations that converge towards the thermodynamic meltingpoint of the model system. The overall methodology is general, but it is applied specifically to thealkali halides. At its core, the algorithm uses a neural network classifier from Chapter 4 as a globalorder parameter to accurately track the ratio of liquid to solid in direct coexistence simulations. Theneural network classifiers ensure that the correct solid crystal structure is maintained, in contrast toprevious melting point calculations.19,48,49 This feature is important because the calculated meltingtemperature is shown to be strongly dependent on the particular crystal structure employed for thesolid. To characterize the extent of finite size effects on melting point calculations, 100 replicatemelting point calculations with differing initial conditions are performed using a rigid-ion NaCl166model at two system sizes: 2,000 vs 11,644 ions. I find that the melting point precision, but notthe average, differs between the two system sizes. The smaller 2,000 ion system yields a naturaluncertainty in the calculated melting temperature of approximately ±6 K, in while an otherwiseidentical set of melting point calculations employing a 11,644 ion system achieves a finite-size un-certainty of±2 K. The uncertainty in finite size melting point calculations is linked to finite clusterthermodynamics described by Wales and Berry.289,290 As far as I can tell, this is the first character-ization of finite-size effects in melting point calculations on model systems. After characterizingthe precision of the melting point calculation algorithm, it is employed to compute relatively accu-rate melting points of several popular pairwise rigid-ion alkali halide models at P = 1 bar: thoseof Joung/Cheatham,14 Tosi/Fumi,12,13 and Smith/Dang.54 The melting point results are comparedwith literature values where available, finding agreement in most cases. Points of disagreementwith literature are discussed. Notably, the calculated melting points of all alkali halides employ-ing pairwise models based on the Coulomb Lennard-Jones functional form are much higher thanexperimental values, indicating a systematic issue with this class of force field.In Chapter 6, a reinforcement learning technique, Bayesian optimization, is employed to pa-rameterize pairwise rigid-ion lithium halide force fields. As established in Chapter 3, the structuralchemistry of lithium halides is uniquely complex among the alkali halides since the ratio of cationto anion size is near the border between tetrahedral and octahedral, as described by the radius ra-tio rules.213 Therefore, previously parameterized lithium halide models often obtain the incorrectlowest energy crystal structure. In this chapter, I develop a methodology to ensure that the correctcrystal structure is maintained during force field parameterization. This is acheived by adding therelative rocksalt/wurtzite energy as an explicit parameterization target, using data generated fromDFT calculations as a target. The fitting of force field parameters is cast as a general optimiza-tion problem by constructing an appropriate objective function. This objective function takes forcefield parameters as input, and outputs a number that encapsulates a measure of fitness with respectto a set of known target properties, such as lattice energies, molecular volumes, or enthalpies offunction. The objective function is slow to compute and generally not deterministic, because itperforms a series of computationally expensive crystal minimization calculations and moleculardynamics simulations using input parameters to construct a force field. The selection of targetproperties is found to be an important and highly nontrivial step in force field parameterization.Targets and their corresponding weights must be carefully selected to obtain the most useful LiXforce fields. Furthermore, the rate of convergence towards a fitted set of parameters can be slow,requiring hundreds of optimization cycles for good convergence. The choice of constraints andparameter search domains is important for reducing the volume of search space and accelerating167convergence. Overall, this methodology offers advantages over the traditional parameter fitting ap-proach employed by Joung and Cheatham14 in that it explores the full force field parameter spacerelatively efficiently and offers a framework for balancing the importance of target properties.Through a series of parameter optimizations, I conclude that the Coulomb Lennard-Jones func-tional form cannot be parameterized to fit low temperature lithium halide crystal properties and hightemperature thermodynamic properties simultaneously, although it can be fit to the low tempera-ture crystal data alone. I generate a set of lithium halide models that are similar to those of Joungand Cheatham, but give the correct low energy crystal structure for all lithium halides. In searchof a better overall set of pairwise lithium halide force fields, I parameterize the Coulomb Bucking-ham model using the same methodology. This force field has additional flexibility in the repulsiveinteraction. In general, the Coulomb Buckingham model is more difficult to parameterize due toinherent pathological behaviour at close range and an additional optimizable parameter. However,preliminary results suggest that the functional form includes the flexibility required to obtain fits toboth low temperature crystal targets and high temperature thermodynamic targets for the lithiumhalides, in contrast to the Coulomb Lennard-Jones model.7.2 Future DirectionsThe body of work outlined in this thesis gives an excellent starting point for further lithium halidenucleation research. However, the research discussed in Chapter 6 is not yet fully matured. Addi-tional model fitting calculations are required to determine the extent to which the Coulomb Buck-ingham functional form can be fit to the lithium halide target data of interest. The results so far arequite promising, indicating that a simple pairwise force field that attains both accurate low temper-ature crystal data and high temperature thermodynamic data is likely possible. Once the research isconcluded, a new set of simple lithium halide force fields for nucleation research will be availablethat likely obtains both reasonable crystal properties and melting points. This set of force fieldsmay prove indispensable in exploring the unanswered questions related to lithium halide solubilityand crystal nucleation from aqueous solutions outlined at the beginning of this chapter.An interesting question that arose from the research in Chapter 3 involves the NiAs crystalstructure for LiI. The highest accuracy DFT calculations of lithium halides performed so far indi-cate that the NiAs crystal structure of LiI is lower in energy than rocksalt by 0.43 kJmol−1, yet theNiAs crystal structure has not been reported experimentally for LiI, while metastable LiI wurtzitecrystals have.204 One avenue of future research is to generate a force field that reproduces the rela-tive energy gap between rocksalt and NiAs for LiI, and then study its nucleation dynamics through168molecular simulation. There may be a large free energy barrier to NiAs crystallization that is notpresent for rocksalt. The neural networks classifiers of Chapter 4 would prove extremely useful forthis research.Another possible direction for future research involves exploring the mystery of competitiverocksalt versus wurtzite crystallization observed experimentally for LiCl,15 LiBr,16 and LiI.204These experiments show that growth of wurtzite crystals can be favoured over the thermodynam-ically stable rocksalt structure under certain conditions, but the microscopic mechanism of this isnot well established. The experiments employ the “low temperature atomic beam deposition” tech-nique204 to deposit hot gaseous ions onto a chilled substrate. The current hypothesis is that, sincewurtzite is less dense than rocksalt, wurtzite can nucleate faster from the low density, low pressureinitial conditions employed in the experiments. If the cooling rate is fast enough, rocksalt crystalsdo not have a chance to nucleate, while the small wurtzite crystals that do immediately nucleatecan continue to grow over time. 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Frazier. arXiv preprint arXiv:1807.02811, 2018. → page 235187Appendix AGeneral Supporting Materials188A.1 Classical Interaction Potential Parametersρi j AMX AMM AXX CMX CMM CXX DMX DMM DXXLiF 0.299 22115.04 9553.47 40616.27 48.17713 4.396163 873.2104 36.13285 1.806642 1023.764LiCl 0.342 31327.14 4809.727 161885.1 120.4428 4.396163 6684.576 144.5314 1.806642 13429.37LiBr 0.353 36482.09 4145.154 254744.4 150.5535 4.396163 11140.96 198.7306 1.806642 25473.66LiI 0.430 15745.82 1811.326 108597.5 198.7306 4.396163 22763.69 319.1735 1.806642 63834.69NaF 0.330 25122.03 30557.68 19362.45 270.9963 101.172 993.6532 228.8414 48.17713 1204.428NaCl 0.317 121075.6 40870.57 336258.8 674.4798 101.172 6985.683 837.0776 48.17713 14031.59NaBr 0.340 98869.28 24805.36 369443.9 843.0997 101.172 11803.4 1144.207 48.17713 27099.63NaI 0.386 58972.18 10923.13 298482.2 1150.229 101.172 23606.79 1866.864 48.17713 66243.55KF 0.338 50507.98 146278.4 16349.75 1174.317 1463.38 1120.118 1264.65 1445.314 1324.871KCl 0.337 172444.2 150084.6 185751.4 2890.628 1463.38 7497.565 4396.163 1445.314 15055.35KBr 0.335 269108.7 158069.6 429515.3 3613.285 1463.38 12405.61 5961.919 1445.314 28304.06KI 0.355 270026.7 96636.87 707362.4 4938.156 1463.38 24269.23 9394.54 1445.314 68050.19RbF 0.328 93549.44 405668.4 20224.72 1866.864 3577.152 1138.185 2408.856 4938.156 1385.092RbCl 0.318 437206.7 549956.3 325849.2 4757.491 3577.152 7828.783 8069.669 4938.156 15657.57RbBr 0.335 389656.2 331402.5 429515.3 5961.919 3577.152 12947.6 10839.85 4938.156 29508.49RbI 0.337 647757.1 313282.2 1255625 8129.89 3577.152 25774.76 16861.99 4938.156 72265.69CsF 0.282 593302.9 5050732 65338.62 3131.513 9153.654 1150.229 4697.27 16741.55 1385.092CsCl 0.272 3851810 7909071 1758635 7768.562 9153.654 7768.562 15055.35 16741.55 15657.57CsBr 0.289 2965606 3758672 2193632 9816.09 9153.654 12887.38 20475.28 16741.55 29508.49CsI 0.291 5268394 3463423 7513152 13489.6 9153.654 25533.88 31315.13 16741.55 71663.48Table A.1: The TF interaction potential parameters12,13 used in this dissertation. SubscriptsMX , MM, and XX refer to metal-halide, metal-metal, and halide-halide interactions,respectively. Note that the exponential parameter ρi j is the same for all interactions of agiven salt. Units of ρi j are A˚, Ai j are kJmol−1, Ci j are A˚6kJ/mol, and Di j are A˚8kJ/mol.189JC (SPC/E) Li Na K Rb Cs F Cl Br Iσii 1.409 2.160 2.838 3.095 3.601 4.022 4.830 4.902 5.201εii 1.408897 1.475453 1.797887 1.862313 0.375960 0.030964 0.053492 0.112795 0.179010JC (TIP3P) Li Na K Rb Cs F Cl Br Iσii 1.826 2.439 3.038 3.230 3.521 4.103 4.478 4.647 5.096εii 0.117108 0.365846 0.810369 1.371607 1.700961 0.014075 0.148913 0.245414 0.224604JC (TIP4PEW) Li Na K Rb Cs F Cl Br Iσii 1.440 2.184 2.833 3.045 3.364 4.522 4.918 4.932 5.260εii 0.435087 0.704742 1.169282 1.812297 1.650303 0.006591 0.048792 0.127099 0.174507SD Na Clσii 2.350 4.400εii 0.543920 0.418400Table A.2: The Joung-Cheatham14 (JC) and Smith-Dang54 (SD) interaction potential param-eters used in this dissertation. Note that there are three sets of JC potential parametersdesigned for use with different water models. Metal-halide interaction parameters arecalculated via Lorentz-Berthelot combining rules.269 Units of σii are A˚, while εii arekJmol−1.A.2 Crystallographic InformationThis section provides crystallographic information about each of the eight structures examined inthis study, which might be useful for future calculations. In all cases except β -BeO, the structurespresented here are ideal. In the case of β -BeO, the experimental structure311 is presented.190Rocksalt (NaCl-Type)The rocksalt crystal structure can be thought of as interpenetrating face centered cubic (FCC) sub-structures of anions and cations.(a) Bonding (b) Unit CellFigure A.1: The rocksalt crystal structure with asymmetric ion sizes.Lattice Type: Face CenteredCrystal System: CubicSpace Group: Fm3¯m (#225)Unit Cell Parameters: a = b = cUnit Cell Angles: α = β = γ = 90°Nearest-neighbor Bond Length: 12 a1st Coordination: 6 (Octahedral)2nd Coordination: 12 (Cuboctahedral)Fractional Coordinates (Asymmetric Unit): Cation: ( 12 , 0, 0)Anion: (0, 0, 0)Table A.3: Crystallographic information for the rocksalt crystal structure.191Wurtzite (ZnS-Type)The wurtzite crystal structure can be thought of as interpenetrating hexagonal close-packed (HCP)substructures of anions and cations. The 1st and 2nd coordination shells become separated into 1,3and 6,6 subshells respectively when c 6=√83 a.(a) Bonding (b) Unit CellFigure A.2: The wurtzite crystal structure with asymmetric ion sizes.Lattice Type: PrimitiveCrystal System: HexagonalSpace Group: P63mc (#186)Unit Cell Parameters: a = bc =√83 a (Ideal wurtzite)Unit Cell Angles: α = β = 90°γ = 120°Nearest-neighbor Bond Length:√38 a (Ideal wurtzite)1st Coordination: 4 (Tetrahedral)2nd Coordination: 12 (Anticuboctahedral)Fractional Coordinates (Asymmetric Unit): Cation: ( 13 ,23 ,38 )Anion: ( 13 ,23 , 0)Table A.4: Crystallographic information for the wurtzite crystal structure.192β -BeO (β -Beryllia)β -BeO can be thought of as a expanded and warped wurtzite or skewed 5-5 type structure. Indeed,β -BeO is the high temperature form of BeO, which has a wurtzite crystal structure under ambientconditions. The first coordination shell of β -BeO is essentially the same for both cation and anion.The nearest like-like coordination shells of the cations and anions are different. The nearest cation-cation coordination shell is made up of three sub-shells: a single cation in the same a-b plane at ashort distance, followed by a subshell of two cations above and below along the c-axis. The lastsub-shell is at a further distance with a warped square-prismatic shape and a coordination numberof 8. The nearest anion-anion coordination shell is similar in structure but has a longer inner sub-shell distance and a shorter outer sub-shell distance. The middle anion-anion sub-shell has anidentical bond length to the cation-cation middle subshell.(a) Bonding (b) Unit CellFigure A.3: The β -BeO crystal structure with asymmetric ion sizes.Lattice Type: PrimitiveCrystal System: TetragonalSpace Group: P 42m nm (#136)Unit Cell Parameters: a = bc = 1√3a (Experimental β -BeO)Unit Cell Angles: α = β = γ = 90°Nearest-neighbor Bond Length: (√2x2+2y2−2x−2y+1)a1st Coordination: 4 (Warped Tetrahedral)2nd Coordination: 1+2+8Fractional Coordinates (Asymmetric Unit): Cation: (x, 1-x, 0)Anion: (y, y, 0)Table A.5: Crystallographic information for the β -BeO crystal structure. Experimentally311of x = 0.336 and y = 0.310 for β -BeO.193CsCl (Primitive Cubic)The CsCl crystal structure can be thought of as interpenetrating primitive cubic substructures ofanions and cations where the corner of one unit cell rests at the center of the other. Lithium halidesplaced in the CsCl structure have the longest nearest-neighbor bond lengths compared with theother candidate structures.(a) Bonding (b) Unit CellFigure A.4: The CsCl crystal structure with asymmetric ion sizes.Lattice Type: PrimitiveCrystal System: CubicSpace Group: Pm3¯m (#221)Unit Cell Parameters: a = b = cUnit Cell Angles: α = β = γ = 90°Nearest-neighbor Bond Length:√32 a1st Coordination: 8 (Square Prismatic)2nd Coordination: 6 (Octahedral)Fractional Coordinates (Asymmetric Unit): Cation: (0, 0, 0)Anion: ( 12 ,12 ,12 )Table A.6: Crystallographic information for the CsCl crystal structure.1945-5 (Hexagonal or Graphitic BN)The 5-5 crystal structure was first described in Ref. 25 and later named in Ref. 27. Ideal 5-5can be described with a smaller primitive hexagonal unit cell in the P63mc space group. The 5-5 structure is similar to graphite in structure, but with alternating chemical species around everyhexagonal ring. All five bond lengths are equal in an ideal 5-5 crystal structure. The nearest like-like coordination shell is made up of two subshells, each with coordination number 6. The firstsub-shell has a trigonal prismatic shape while the second subshell is a hexagonal disk in the b-c plane at a greater distance. These two subshells combine into a compressed anticuboctahedralgeometry.(a) Bonding (b) Unit CellFigure A.5: The 5-5 crystal structure with asymmetric ion sizes.Lattice Type: PrimitiveCrystal System: OrthorhombicSpace Group: Pnnm (#58)Unit Cell Parameters: b = 32 a (Ideal 5-5)c =√32 a (Ideal 5-5)Unit Cell Angles: α = β = γ = 90°Nearest-neighbor Bond Length: 12 a (Ideal 5-5)1st Coordination: 5 (Trigonal Bipyramidal)2nd Coordination: 6+6Fractional Coordinates (Asymmetric Unit): Cation: ( 14 ,16 ,12 )Anion: ( 14 ,13 , 0)Table A.7: Crystallographic information for the 5-5 crystal structure.195NiAs (Nickeline) and AntiNiAsNiAs has stacking order ABACABAC, with cations on the A sites and anions on B and C sites.c/a scales all bond lengths as it changes. The bonding environments of cations and anions aredifferent. In the first coordination shell, anions form a trigonal prismatic shell while cations havean octahedral environment. The second coordination shell of the anions is made up of two sub-shells: a hexagonal disk in the a-b plane followed by a trigonal prismatic subshell at a slightlyfurther distance. Together these subshells form a stretched anticuboctahedral shape. The secondcoordination shell of cations forms a compressed hexagonal bipyramidal shape: one linear subshellof coordination 2 directed along the c axis, plus a hexagonal disk of slightly longer bond length inthe a-b plane. The antiNiAs crystal structure is identical to NiAs, but with the anions and cationsswitched.(a) Bonding (b) Unit CellFigure A.6: The NiAs crystal structure with asymmetric ion sizes.Lattice Type: PrimitiveCrystal System: HexagonalSpace Group: P 63m mc (#194)Unit Cell Parameters: a = bc =√83 a (ideal)Unit Cell Angles: α = β = 90°γ = 120°Nearest-neighbor Bond Length:√a23 +c2161st Coordination: 6 (Octahedral/Trigonal Prismatic)2nd Coordination: 2+6/6+6Fractional Coordinates (Asymmetric Unit): Cation: (0, 0, 0)Anion: ( 13 ,23 ,14 )Table A.8: Crystallographic information for the NiAs and AntiNiAs crystal structures.196Sphalerite (Zincblende)The sphalerite crystal structure is similar to the diamond crystal structure, but with alternatingcations and anions. It is also very similar to the wurtzite crystal structure, but the second coor-dination shell of sphalerite is cuboctahedral, whereas wurtzite has an anticuboctahedral secondcoordination shell. As such, sphalerite has M-X-M-X dihedral angles of 60°, 180°, and 300° whilewurtzite has M-X-M-X dihedral angles of 0°, 120°, and 240°.(a) Bonding (b) Unit CellFigure A.7: The sphalerite crystal structure with asymmetric ion sizes.Lattice Type: Face CenteredCrystal System: CubicSpace Group: F4¯3m (#216)Unit Cell Parameters: a = b = cUnit Cell Angles: α = β = γ = 90°Nearest-neighbor Bond Length:√34 a1st Coordination: 4 (Tetrahedral)2nd Coordination: 12 (Cuboctahedral)Fractional Coordinates (Asymmetric Unit): Cation: (0, 0, 0)Anion: ( 14 ,14 ,14 )Table A.9: Crystallographic information for the sphalerite crystal structure.197Appendix BSupporting Materials for Chapter 3B.1 Quantum Mechanical Calculation DetailsDue to the definition of lattice energy, constructing a quantum mechanical system to approximatethe lattice energy is relatively simple. One first calculates the total energy per unit cell of theoptimized infinite crystalline system under the Born-Oppenheimer (BO) approximation Etotal, thenseparately the total energy of all isolated ions in the unit cellEions = NanionEanion+NcationEcation, (B.1)where Nanion and Ncation are the number per unit cell while Eanion and Ecation are the total energy ofthe lone anion and cation, respectively. The two energies are then subtracted to yieldEL =Etotal−EionsF, (B.2)where F is the number of formula units per unit cell, and EL is the resulting lattice energy performula unit. I adopted the definition of lattice energy specifying the potential energy of the fixedcrystal lattice, completely neglecting the zero-point vibrational energy.For quantum mechanical calculations of lattice energies, I employed different basis sets forthe condensed phases versus their isolated ions. The construction of Gaussian basis sets for non-metallic crystals such as LiX involves variational optimization of orbital exponents and contractioncoefficients to a crystalline environment while constraining the search space to prevent the intro-duction of diffuse functions.134,145,146 On the other hand, diffuse functions are included in atomic198and molecular basis sets to capture the tail end of electron density, which is particularly importantfor small anions. Inclusion of diffuse functions in the basis sets of electronic structure calcula-tions for periodic systems can cause quasi-linear dependencies between basis functions leading tosingular Hamiltonians in the Kohn-Sham or Hartree-Fock-Roothaan equations.134,145For most of the crystalline quantum mechanical calculations I performed with CRYSTAL17— except those used to compare basis set effects — I adopted pob-TZVP, a recently constructedtriple-ζ split valence quality crystalline basis set.145,146 The acronym means: Peintinger-Oliveira-Bredow (the authors) Triple-Zeta Valence with added Polarization functions. This basis set isavailable for all elements used in my study, and was optimized for solid state calculations throughKohn-Sham110 DFT calculations with the PW1PW global hybrid XC functional.180 The authorscreated a widely transferable crystalline basis set for all elements of consistent quality. The focuson transferability is important here due to the range of LiX bonding environments studied.I performed all CP2K calculations using the Quickstep module.136 Since the iodine pob-TZVPbasis set contains a pseudopotential incompatible with GAPW, I switched out the iodine basis set atthe triple-zeta level of completeness for the all-electron Sapporo-TZP149,153 and Sapporo-DKH3-TZP156 basis sets, applied without or with a third order DKH transformation,141–143 respectively.The Sapporo-DKH3-TZP basis set also replaced pob-TZVP for bromine in calculations run witha third order DKH correction. CP2K allows for large basis sets in condensed systems withoutnumerical singularity issues. This is possible due to CP2K’s robust self-consistent field (SCF)algorithm, involving the expansion of periodic Gaussian basis set in terms of orthogonal planewaves. Hence, I used the larger all-electron Sapporo-QZP148–155 and Sapporo-DKH3-QZP156 basissets for the anions in another set of CP2K calculations. I only used Sapporo-DKH3-QZP for Brand I anions in conjunction with a third order DKH relativistic Hamiltonian141–143 to recover scalarrelativistic effects in these heavy atoms. In all cases, I kept the pob-TZVP basis set for cationsas use of Sapporo-QZP for lithium or sodium lead to a massive increase in computational cost.Furthermore, when Sapporo-QZP was used on lithium or sodium the change in total energy wasnegligible compared with pob-TZVP, indicating that pob-TZVP is nearly complete for lithium ionsin LiX and sodium ions in NaCl.Since the electronic environment of isolated anions is so different from the crystal, it wasnecessary to use separate basis sets for crystal and ion calculations in order to achieve a similarbasis set completeness for both sets of calculations. Therefore, I matched the quality of the crystalbasis set pob-TZVP with the reference ion basis set of Li+, Na+, F−, Cl−, and Br− by choosingthe molecular triple-ζ split valence quality basis set upon which pob-TZVP is derived, namely,Karlsruhe’s diffuse augmented def2-TZVPD basis set.312,313 For isolated anions corresponding to199crystal calculations involving the Sapporo basis set, I used the diffuse-augmented versions of theSapporo basis sets.For calculations of I− reference ion energy with CRYSTAL17, the def2-TZVP312 basis setwas not usable because it is constructed around an older effective core potential (ECP) fromthe Stuttgart/Cologne group’s 2003 paper314 (referred to as ECP28oldMDF in their database).315Whereas the ECP used in the iodine pob-TZVP basis set (ECP28MDF) was released in a laterpaper.194 Identical ECPs in the basis set of the reference ion and the crystal are required to com-pute lattice energies. Hence, I created a new ad hock iodide basis set, constructed to match thepseudopotential and core basis functions of the pob-TZVP basis set (the EPC28MDF pseudopo-tential),194 but augmented with outer Gaussian functions from the def2-TZVPD iodine basis set.This basis set is available in the supplementary material of Ref 86. Note the computation of iso-lated ion energies is only for calculating lattice energies for direct comparison with experiment.The calculation of structural properties, differences in lattice energies between crystal structures,and vibrational energies does not depend on the isolated ion energies.Using CRYSTAL17, I performed several calculations with a larger basis set for iodine in boththe crystal and reference ion state to test the iodine basis set completeness of pob-TZVP. For theserevisited crystal calculations, I used the ECP28MDF-VTZ basis set194 (short for “Effective CorePotential for 28 electrons, Multi-electron Dirac-Fock fit with Valence of Triple-Zeta quality”). Thereference I− ion calculations in this case were performed with the diffuse-augmented version ofthis basis set called ECP28MDF-AVTZ194 (where the A is short for “Augmented”).For each crystal and level of theory examined in this study, I performed a full geometry opti-mization. These optimizations were performed first with respect to the unit cell parameters only,and then with respect to both unit cell parameters and fractional coordinates while maintainingthe full symmetry of the space group (i.e. cell angles, relationships between cell parameters, andsymmetry operations). I also ran geometry optimization calculations of all lithium halides in allcandidate structures without maintaining symmetry (by converting each unit cell to space groupP1) at several levels of theory and found no significant structural or energetic changes in any casecompared with symmetry-constrained optimizations. I took initial conditions for rocksalt unit cellsfrom experimental structures, whereas initial conditions for other candidate structures were con-structed to match the rocksalt LiX bond length.In an attempt to compensate for the lack of accurate medium-to-long range electron correlationin DFT, several approaches have been introduced.64 Two popular low-cost methods are Grimme’sD364,184–186 model with Becke-Johnson (BJ) rational damping and Grimme’s newly developedD4187–189 model. Both D3 and D4 add an additional dispersion energy to the total energy calculated200from DFT asE total = E DFT +E disp (B.3)where E DFT is the usual total energy of the system. In both D3(BJ) and D4 dispersion models asimplemented for periodic systems, the dispersion energy is calculated by a sum of two-body termsasE disp =−12N∑iN∑j∑n′(s6C6,i jR6vdW,i j + |ri j +n|6+ s8C8,i jR8vdW,i j + |ri j +n|8)(B.4)where i and j are indices of the N nuclei of the unit cell, n is a unit cell translation vector (thesum over these is infinite in principle). The summation of translation vectors excludes i = j whenn = 0, as indicated by the prime (′). ri j is the distance between nucleus i and j. sn are globalXC-dependent scaling parameters. s6 is generally kept equal to unity in order to maintain the ex-act long-range behaviour of dispersion, while s8 is fit empirically to each XC functional. Cn,i j arethe dispersion coefficients — representing transient dipole-dipole and dipole-quadrupole interac-tions — which are calculated from time-dependent DFT calculations of dynamical polarizabilities.These pre-calculated reference dispersion coefficients are mapped to the system under considera-tion, the exact procedure differs between D3 and D4. In D3, the mapping is based on calculated“fractional coordination numbers” of both atom i and j while D4 dispersion introduces an addi-tional atomic charge dependence, calculated by a classical model. In Eq. (B.4) RvdW,i j are thedamping radii, calculated from the two dispersion coefficients then modified by two global XC-dependent parameters. This type of damping acts to bring the dispersion energy to a constant valueat short distances where the XC functional more accurately describes dispersion, and circumventsthe unphysical close range repulsion found in zero-damped DFT-D3.64The calculation of fractional coordination numbers, as well as the unit cell translation vectorsummation in Eq. (B.4) are infinite in principle, so are cutoff at some selected radius. I used thedefault D3 and D4 cutoff radii of 21.2 A˚ for calculation of fractional coordination numbers and thedefault 50.2 A˚ for the dispersion summation in Eq. (B.4).I tested pob-TZVP/GGA parameterized geometrical counterpoise (gCP)198,199 corrections ona series of geometry optimization calculations of LiX crystals in an attempt to reduce basis setsuperposition errors (BSSE). The gCP method adds a pairwise, basis set and theory dependentexponential repulsive potential to account for the effects of BSSE, although its effectiveness foruse in correcting triple-ζ quality BSSE is not well established.For the quantum mechanical calculations performed with CRYSTAL17 in my research, self-consistent field (SCF) convergence criteria were set to 10−11 EH between SCF cycles in order to201ensure accurate force constants could be computed from second derivatives for vibrational analy-sis. This was relaxed to 10−8 EH for CP2K calculations. Periodic quantum mechanical calcula-tions require definition of a grid of points in reciprocal space for calculating the wavefunctions andenergies of an infinitely periodic lattice. In CRYSTAL17, these sample points are used to approx-imate integrals over the irreducible part of the first Brillouin zone (BZ) when calculating crystalorbital energies. In all of my calculations, this k-point grid was constructed by the Pack-Monkhorstmethod,316 where a commensurate grid of k-points is selected based on an input FS, generating anFS×FS×FS grid of k-points which span the first BZ. For CRYSTAL17, Only k-points within theirreducible BZ are selected as sample points for energy band calculations, whereas CP2K selectsthe entire BZ. Due to the symmetry properties of the first BZ, integration over the irreducible BZresults in a substantial reduction in the number of k-points that need to be sampled for a givenk-point density. For my calculations with CRYSTAL17, I chose a rather high value of FS = 12,which translates to 72 sample k-points for the rocksalt crystal structure and 133 sample k-pointsfor the wurtzite crystal structure.134 In preliminary studies I found this value sufficiently large suchthat total energies were reliably converged with respect to FS, independent of the particular lithiumhalide salt and structure. For calculations with CP2K, I found that in preliminary tests, a value ofFS = 6 was sufficient to converge the total energy of rocksalt to within 0.01 kJmol−1 of the FS = 5result. Hence FS = 6 was used in CP2K calculations.DFT calculations require a real space 3D grid for XC-functional integration. For DFT calcula-tions with CRYSTAL17, I used the largest predefined integration grid, called XXLGRID.134 CP2Kimplements a multi-grid system, defined in terms of the number of grids, a relative cutoff, and theplanewave cutoff for the finest grid. In preliminary tests, I found that crystal energies convergedwithin 0.01 kJmol−1 by selecting 6 grids, a 60 Ry relative cutoff, and a 400 Ry cutoff for the finestgrid. However, I selected a cutoff of 600 Ry for the finest grid in production calculations as this leadto a smoother potential energy surface for geometry optimization. Using the Gaussian AugmentedPlane Waves (GAPW) method also requires a local grid around the atomic centers for each atom,defined in terms of a radial and angular grid. After a convergence test, I settled on the followinglocal grids: an angular grid of 50 grid points, and a radial grid of 80 points for cations and 150points for anions.Formally, the Coulomb and exchange series summations of infinite crystals involve a sum overall unit cells in the lattice. This series involves the calculation of overlap integrals between a par-ticular atomic orbital and each of the other infinitely many atomic orbitals in the lattice. Thisoverlap quickly decays for increasingly distant unit cells. To approximate this infinite series, theCRYSTAL17 software package demands the selection of five tolerances which define when certain202overlap integrals between atomic orbitals are discarded or approximated.134 For all of my calcula-tions, I defined the following tolerances: 10−8 EH for the overlap threshold of Coulomb integrals;10−8 EH for the penetration threshold of Coulomb integrals; 10−8 EH for the overlap threshold ofHF exchange integrals; 10−8 EH for the threshold of the pseudo-overlap for the exchange seriesintegrals; and 10−16 EH for the threshold of the second pseudo-overlap for the exchange seriesintegrals. Reducing these tolerances further did not appreciably change the resulting energies ofany tested preliminary calculations. The corresponding setting in CP2K is the ‘EPS DEFAULT’parameter, which I set to a minimum of 10−10 EH. Smaller tolerances did not appreciable affect theresulting energies, but were sometimes used to ensure smooth geometry optimization.In some cases, CRYSTAL17 calculations had problems with SCF convergence. To resolvethese, I used a few convergence accelerating techniques. Typically, my calculations combined Fockmixing with a modified Broyden scheme317,318 using the following settings. The W0 parameterfrom Anderson’s paper318 was set to 10−4. The Fock mixing was set to 50% and maintained at50% once the Broyden method was activated after the second SCF iteration. This method provedsuccessful in most cases, but occasionally was not able to produce SCF convergence, such as certaincalculations with the HF or DFT-M062X methods. In those situations, I found that decreasing thefive Coulomb/Exchange series tolerances, and/or scaling the most diffuse lithium basis function by2 resolved the SCF instabilities.B.2 Technical Points on Basis Set ErrorsBasis set errors, which can be split into basis set incompleteness errors (BSIE) and basis set su-perposition errors (BSSE) are difficult to correct in ionic solid state calculations when employingGaussian basis sets. Formally, correcting these errors requires a very complete basis set, whichrapidly drives up computational costs for condensed phase calculations. Furthermore, many largebasis sets are optimized for molecular calculations, often fail to converge for condensed phases.At the triple-zeta level of completeness (e.g. with pob-TZVP) BSSE and BSIE may significantlyaffect results.199 A recent article by the authors of the pob-TZVP basis set discusses the issue ofBSSE in solid state calculations, proposing a revised basis set called pob-TZVP-rev2 designedto balance BSSE between different atoms.147 However, their study shows that pob-TZVP-rev2does not necessarily perform better than pob-TZVP for the calculation of lithium halide latticeparameters. Furthermore, the reported BSSE energy of the pob-TZVP-rev2 basis set associatedwith lithium, as calculated approximately via the Counterpoise method196 using hydride dimersas a stand-in for periodic solids, is larger than in pob-TZVP. As a check, I calculated some single203point energies using experimental LiX rocksalt geometries and found that total energies calculatedwith PW1PW/pob-TZVP-rev2 were very slightly elevated (0.66 – 0.92 kJmol−1) compared withPW1PW/pob-TZVP total energies for each lithium halide salt. This suggests a slightly decreasedbasis set completeness in pob-TZVP-rev2 compared to pob-TZVP for lithium halide salts.One option for correcting BSSE is Grimme’s geometrical counterpoise (gCP) method,198 whichadds an empirically parameterized exponentially decaying potential between all atoms to counter-act the artificial inward force caused by BSSE. Although gCP has great promise for improving theaccuracy of calculations with small basis sets, its unclear if gCP has utility at the triple-zeta levelor beyond. To benchmark the usefulness of gCP corrections for my study, I calculated gCP correc-tions in combination with the D3 dispersion correction scheme by re-optimizing crystal structuresusing several XC functionals. I then checked if gCP corrections improved crystal structure pre-diction, rocksalt lattice parameters, and lattice energies when compared to experiment versus D3alone.In Table B.1 I list the gCP error correction for various LiX salts in several candidate geometries,optimized at the PW1PW/pob-TZVP level of theory. Notably, the difference in gCP correctionsbetween different crystal structures of the same salt is at most 18 kJmol−1 for LiF and 9 kJmol−1for other LiX salts.LiF LiCl LiBr LiIRocksalt 94.29 12.12 43.93 33.41Wurtzite 79.88 10.06 35.65 27.40β -BeO 76.00 9.84 33.41 25.78CsCl 97.87 10.81 41.29 29.205-5 87.45 10.48 37.42 28.94NiAs 93.81 12.23 44.42 34.54Sphalerite 79.49 10.06 34.50 28.13Table B.1: BSSE energy estimates (kJmol−1) of LiX salts in their PW1PW/pob-TZVP equi-librium geometries for all candidate crystal lattices. Calculated by the gCP method withparameters optimized for GGA XC functionals and the pob-TZVP basis set.I find that the gCP method appears to overestimate BSSE corrections for LiX salts with the pob-TZVP basis set, especially in the case of LiF. Combined with D3 dispersion corrections, gCP doesnot predict the correct lowest energy crystal structure for LiF when used with several XC function-als or HF (see Fig. B.1). For example, at the PW1PW-D3(BJ)/pob-TZVP level of theory, rocksalt204is the lowest energy crystal structure of LiF by 9.56 kJmol−1. With an added gCP correction, bothwurtzite and β -BeO become lower energy crystal structures. For the remaining LiX salts, structureprediction remains unaffected. Fig. B.2 shows that gCP combined with D3 dispersion improvesthe accuracy of thermally adjusted rocksalt lattice parameters compared with DFT-D3 dispersionalone, for LiCl, LiBr, and LiI using hybrid DFT XC functionals, but does not improve accuracy oflattice parameters for LiF. Still, the most accurate lattice parameters with the pob-TZVP basis setwere those calculated from DFT calculations without gCP corrections. The addition of gCP correc-tions also systematically reduces the accuracy of rocksalt lattice energy calculations compared withexperiment for all LiX salts (see Figure B.3), causing elevated (i.e., not negative enough) theoreti-cal lattice energies. Hence, I do not recommend gCP corrections for lithium halides in conjunctionwith the pob-TZVP basis set.Figure B.1: The difference (∆EL = EL(Rocksalt)−EL(Wurtzite)) in lattice energy betweenfully minimized rocksalt and wurtzite crystal structures for a series of quantum mechan-ical calculations, with D3(BJ) dispersion and gCP corrections included. The pob-TZVPbasis set is used in all cases. Negative values favour rocksalt, in agreement with exper-iment.205Figure B.2: The relative error ((acalculated− aexperiment)/aexperiment) in calculated lattice pa-rameter a for various D3 and gCP corrected quantum mechanical calculations. Thepob-TZVP basis set is used in all cases. Overlaid hatched bars are relative errors in cal-culated rocksalt lattice parameter a where the theoretical calculation has been correctedfor thermal expansion at 298.15 K via the Quasi-Harmonic Approximation.206Figure B.3: The relative error ([EL(Theory, Rocksalt)−EL(Experiment)]/|EL(Experiment)|)in lattice energy for various quantum mechanical calculations including D3 and gCPcorrections. The pob-TZVP basis set is used in all cases for the calculation of crystalenergies, the def2-TZVPD basis set is used for the calculation of Li+, F−, Cl−, and Br−isolated ion energies, while the basis set from Ref. 86 is used for isolated I− energies.Theoretical lattice energies are those calculated from optimized rocksalt structures.B.3 Thermal Corrections to Experimental Lattice ParametersExperimental lattice parameters for the lithium halides are available to high precision from the liter-ature, given at 298 K.83 Since these are finite temperature measurements, I cannot directly comparethem with theoretical lattice parameters calculated from energy minimization of bulk crystals, asthese correspond to 0 K results without zero-point vibration. However, linear thermal expansioncoefficients α(T ) of lithium halide (and NaCl) salts at various temperatures are available in the lit-erature.306 In Fig. B.4 I plot the available experimental linear thermal expansion coefficients α(T )together with α(0) = 0 and an interpolation function (modified Akima cubic Hermite interpolationas implemented in MATLAB) used for numerical integration. I integrated the thermal expansionfunction to calculate the 0 K lattice parameter a0 bya0 = aT ′(1−∫ T ′0α(T )dT)(B.5)207Figure B.4: Experimental coefficients of thermal expansion306 for the lithium halides andNaCl are indicated by circles. The theoretical value of α(0)= 0 is also included for eachcase. The interpolated lines over the domain T = [0, 298] were numerically integratedto obtain the thermal expansion correction at 0 K.LiF LiCl LiBr LiI NaCla298 K (Ref. 83) 4.02620(5) 5.13988(4) 5.501(6) 6.012(7) 5.64009(3)a0 (this work) 4.00514 5.09608 5.447 5.958 5.59548∆a 0.02106 0.04380 0.054 0.054 0.04461Table B.2: A summary of experimental lattice parameters at 298 and 0 K for the lithiumhalides and NaCl, including uncertainties where available. All units are Angstroms.where T ′ = 298 K and aT ′ are the finite temperature lattice parameters. In Table B.2 I list themeasured 298 K lattice parameters, the calculated 0 K lattice parameters, and their difference forthe lithium halides and NaCl.B.4 Technical Points on Dispersion CorrectionsThis section gives detailed technical comparisons of dispersion interactions implemented in variousclassical and DFT models. Additionally, I derive a set of geometrical measures which quantify themost significant classes of dispersion interaction for selecting one crystal structure over another, in208pairwise models.Table B.3 lists the two-body ion-ion partitioning of dispersion energy for each salt in optimizedrocksalt crystal structures at the PW1PW-D3(BJ)/pob-TZVP, PW1PW-D4/pob-TZVP, JC-SPC/E,and TF levels of theory. Additionally, the C6 and C8 dispersion coefficients used in the classicalmodels are compared with those computed via D3 and D4 in Table B.4. Note that dispersioninteractions are damped in the D3 and D4 models, making comparison to classical models tenuousfor medium-range dispersion. Based on the results presented so far, I can now explain the interplaybetween pairwise dispersion and structure determination for the lithium halides.In the remainder of this section, I show through geometrical arguments that the second-shell,anion-anion dispersion interaction is responsible for favouring rocksalt in lithium halides. Fortwo crystal structures A and B with mth shell coordination numbers am and bm and coordinationdistances rm,a and rm,b, respectively, structure A is favoured by the mth shell dispersion whenam f (ram)< bm f (rbm), (B.6)where f (r) is the functional form of the pairwise dispersion interaction. Note that this criterionignores the possibility of many-body dispersion terms. In the case of the usual pairwise expansionof dispersion with fixed coefficients, a simple measure defined as ζnm(a,b), can be written foreach term in the expansion (n = 6,8, . . .). Structure A is favoured by the mth shell dispersion termproportional to r−n whenζnm(a,b)≡ ambm(rbmram)n> 1, (B.7)which has no explicit dependence on the dispersion coefficients.It is easy to check that the difference in dispersion energy between structure A and B (∆Edispersion =Edispersion,A−Edispersion,B, i.e. how strongly dispersion favours one structure over another) scaleswith both the size of these measures as well as the dispersion coefficients,∆Edispersion = ∑m=1am f (ram)−bm f (rbm) = ∑m=1∑k=3C2k[1−ζ(2k)m(a,b)], (B.8)The sums in Eq. (B.8) run over nearby coordination shells (m), and terms of the dispersion ex-pansion (2k) with dispersion coefficients C2k. Of course, the equilibrium values of ram and rbmthemselves depend on the full model potential, including the dispersion interaction. However, Ihave observed that the bond length is primarily determined by the repulsive and Coulombic in-209LiF LiCl LiBr LiI NaClValue % Value % Value % Value % Value %M+-X− -10.29 17 -21.77 31 -24.70 30 -28.50 27 -23.45 30M+-M+ -46.35 76 -22.69 32 -19.00 23 -15.28 14 -34.47 44X−-X− -4.20 7 -25.77 37 -38.70 47 -62.13 59 -19.86 26D3(BJ)Total -60.84 100 -70.24 100 -82.40 100 -105.90 100 -77.78 100M+-X− -4.62 30 -6.30 19 -6.31 15 -6.78 12 -5.96 22M+-M+ -3.00 20 -1.52 4 -1.15 3 -0.91 2 -1.61 6X−-X− -7.60 50 -26.05 77 -34.66 82 -49.39 87 -19.54 72D4Total -15.23 100 -33.87 100 -42.11 100 -57.07 100 -27.12 100M+-X− -22.81 68 -21.54 57 -23.71 48 -23.58 40 -24.65 69M+-M+ -0.82 2 -0.26 1 -0.18 0 -0.11 0 -1.97 6X−-X− -9.77 29 -15.84 42 -25.82 52 -34.95 60 -8.96 25JCTotal -33.40 100 -37.63 100 -49.71 100 -58.64 100 -35.59 100M+-X− -5.74 17 -3.49 6 -2.87 5 -2.24 3 -10.37 26M+-M+ -0.13 0 -0.03 0 -0.02 0 -0.01 0 -0.38 1X−-X− -27.43 82 -51.43 94 -56.66 95 -68.64 97 -28.55 73TFTotal -33.30 100 -54.94 100 -59.54 100 -70.89 100 -39.30 100Table B.3: Contributions to the dispersion energy for lithium halides and NaCl in fully min-imized rocksalt crystal structures using four different dispersion models. The energiesare further broken down into the total Cation-Anion, Cation-Cation, and Anion-Aniondispersion contributions. The D3(BJ) minimized structure is that produced by optimiza-tion at the PW1PW-D3(BJ)/pob-TZVP level of theory, ignoring 3-body dispersion terms.Similarly, the D4 minimized structure is that produced by optimization at the PW1PW-D4/pob-TZVP level of theory, ignoring 3-body dispersion terms. The JC model forSPC/E water was used here. Energies are in units of kJmol−1.teractions for the lithium halides, with a relatively small dependence on dispersion (except whendispersion is scaled far enough to completely overcome the repulsive wall, as seen for the TF modelwith large SD). Hence, the value of ζnm(a,b) for a given pair of LiX crystal structures depends pri-marily on the strength and shape of the short-range repulsive potentials of the model, combinedwith the (typically fixed) Coulombic potential.A limitation of ζnm(a,b) as expressed in Eq. (B.7) is that uniform ion-ion distances in coor-dination shells are assumed, which is often not the case. For example, the NiAs crystal structure210D3 C6 D4 C6 JC C6 TF C6 D3 C8 D4 C8 TF C8Li+ Li+ 4918.9 393.8 – 437.2 44.1 4.4 104130.2 8335.8 – 9256.1 1.8F− F− 411.3 697.8 524.3 873.2 1970.8 3343.5 1023.8Cl− Cl− 5211.7 6062.9 2716.7 6684.6 60892.5 70837.9 13429.0Br− Br− 9757.0 11085.3 6260.2 11141.0 172756.4 196275.0 25474.0I− I− 20668.4 23183.5 14172.9 22764.0 531602.2 596290.7 63835.0Li+ F− 990.1 496.7 335.0 48.2 9971.7 5002.9 36.1Li+ Cl− 4450.9 1603.4 1011.9 120.4 69999.6 25216.0 144.5Li+ Br− 6238.3 2072.7 1574.2 150.6 120775.6 40128.0 198.7Li+ I− 9306.9 3015.4 2618.0 198.7 217169.0 70362.8 319.2Table B.4: A comparison of Li+-X− dispersion coefficients for: the D3 model in the limitof large ( 1) “fractional coordination numbers”; the D4 model computed for PW1PW-D4/pob-TZVP optimized rocksalt geometries; the JC-SPC/E model; and the TF model.Units of C6 are kJmol−1 A˚6while units of C8 are kJmol−1 A˚8.metal-metal second coordination shell is arranged into two subshells: a subshell of two at shorterdistance and a subshell of six at longer distance. ζnm(a,b) from Eq. (B.7) cannot be used to assesswhether second shell dispersion favours rocksalt over NiAs since rbm are not uniform. In suchsituations a more generalized measure can be used,ζnm(a,b) =∑i amirnami∑ jbm jrnbm j> 1, (B.9)where the sums i and j run over subshells of the mth coordination shells of structures A and Brespectively. ami and bm j are the coordination numbers of the ith and jth subshells of the mth shellof structures A and B, respectively. rami and rbmi are the corresponding ion-ion distances.Tabulating values of ζnm(a,b) for rocksalt and wurtzite offers a simple, quantitative means ofchecking which dispersion terms are most important for structure determination of rocksalt versuswurtzite in the lithium halides. Table B.5 lists rocksalt/wurtzite ζnm(a,b) values for geometriesobtained at several levels of theory. These measures show that n = 6 dispersion from the Li+-X−first coordination shells (m= 1) tend to weakly favour either rocksalt or wurtzite, depending on theparticular LiX salt. Meanwhile, the first shell n = 8 term tends to favour wurtzite for all lithiumhalides. This result is in striking contrast to the m = 2 dispersion interactions, which stronglyfavour rocksalt for both n = 6 and n = 8 terms in all cases. The predictions from these dispersion211Figure B.5: The JC model for SPC/E water: dispersion scaling factor SD applied only toLi+-X− type dispersion interactions, versus the difference in lattice energy (∆EL =EL(Rocksalt)− EL(Other)) between cell-minimized rocksalt (EL(Rocksalt)) and sixother (EL(Other)) cell-optimized crystal structures with fixed ideal fractional coordi-nates.favouring measures are confirmed by Figs. B.5 to B.8, which scale dispersion interactions betweenspecific ion pairs rather than all dispersion interactions simultaneously, as presented in Chapter 3.B.5 Classical Models vs DFT at High PressureIn Figure B.9, I compare potential energy curves for ideal rocksalt and wurtzite structures as a func-tion of their lattice parameter (a stand-in for unit cell volume) for classical models and PW1PW-D3(BJ)/pob-TZVP calculations. These results show that, despite its inability to correctly predictthe lowest energy crystal structure of the lithium halides, the TF model energies match quantummechanical results much more closely than the JC model in high pressure (low a) environments.Due to the D3 “fractional coordination number” (FCN) calculation scheme,184 which predicts verylarge FCN in condensed systems, combined with the lack of suitable reference molecules of highcoordination number for lithium, D3 dispersion coefficients are static with respect to small struc-tural changes in a LiX crystal environment. However, as the LiX crystal is greatly expanded (as in212M+-X− (m = 1) X−-X− (m = 2)n = 6 n = 8 n = 6 n = 8LiFHSEsol 1.00 0.88 1.59 1.85PW1PW-D3(BJ) 1.01 0.88 1.59 1.86PW1PW-D4 1.00 0.87 1.57 1.83HF 0.98 0.86 1.56 1.80JC-SPC/E 1.15 1.06 1.82 2.22TF 0.88 0.73 1.38 1.54LiClHSEsol 0.98 0.84 1.54 1.78PW1PW-D3(BJ) 0.98 0.85 1.54 1.78PW1PW-D4 0.97 0.84 1.53 1.76HF 0.91 0.77 1.44 1.63JC-SPC/E 1.06 0.95 1.68 1.99TF 0.90 0.76 1.42 1.60LiBrHSEsol 1.02 0.90 1.61 1.89PW1PW-D3(BJ) 1.02 0.90 1.61 1.89PW1PW-D4 1.02 0.89 1.61 1.88HF 0.94 0.80 1.48 1.69JC-SPC/E 1.00 0.88 1.59 1.85TF 0.90 0.76 1.43 1.61LiIHSEsol 0.97 0.84 1.54 1.78PW1PW-D3(BJ) 0.97 0.84 1.53 1.77PW1PW-D4 0.97 0.84 1.53 1.76HF 0.90 0.76 1.42 1.60JC-SPC/E 0.94 0.81 1.49 1.71TF 0.83 0.68 1.31 1.43NaClHSEsol 1.13 1.02 1.78 2.16PW1PW-D3(BJ) 1.13 1.02 1.78 2.16PW1PW-D4 1.09 0.98 1.73 2.07HF 1.08 0.97 1.71 2.05JC-SPC/E 1.20 1.12 1.90 2.35TF 1.07 0.96 1.69 2.01Table B.5: Rocksalt vs wurtzite dispersion favouring measures, calculated from frozen-lattice optimized geometries of the given salt/theory combinations. These are definedas ζnm = ab(rbmram)nwhere a and b are the coordination numbers for the first (M+-X−)or second (X−-X−) coordination shells of rocksalt and wurtzite respectively. M is themetal, while X refers to the halide. ram and rbm are average ion-ion distances for themth shell, and n is to the power of the dispersion term. Measures greater than 1 favourrocksalt for that interaction, while values less than 1 favour wurtzite.213Figure B.6: The TF model: dispersion scaling factor SD applied only to Li+-X− type dis-persion interactions, versus the difference in lattice energy (∆EL = EL(Rocksalt)−EL(Other)) between cell-minimized rocksalt (EL(Rocksalt)) and six other (EL(Other))cell-optimized crystal structures with fixed ideal fractional coordinates.Figure B.9), the calculated FCN decreases steadily towards 1. Figure B.9 shows that D3 correctedmodels of lithium salts have a sudden non-physical dip in their potential energy curves when ex-panded too far. This occurs because the calculated FCN falls between 1 and 0 in this region, addingemphasis to the neutral lithium reference in the D3 polarizability interpolation scheme. Since neu-tral lithium has much higher polarizability than lithium in LiH, the resulting Li+-Li+ and Li+-X−dispersion coefficients rise rapidly in this region of the potential energy curves.B.6 Scalar Relativistic Effects on Heavy Lithium HalidesIn this section, I compare geometry optimized lithium halide crystals with and without applicationof the third order Douglas–Kroll–Hess (DKH) transformation.141–143 I use this approach to deter-mine the extent that scalar relativistic phenomena effect three key bulk LiBr and LiI properties: thelattice parameter a, the lattice energy EL, and the difference in energy between the octahedrally-coordinated rocksalt and tetrahedrally-coordinated wurtzite phase. In all cases, I use the pob-TZVPbasis set for in-crystal lithium. I use the Sapporo-QZP148–155 basis set for in-crystal Br− and I−anions when DKH is not applied, while the Sapporo-DKH3-QZP156 basis set is employed when214Figure B.7: The JC model for SPC/E water: dispersion scaling factor SD applied only toX−-X− type dispersion interactions, versus the difference in lattice energy (∆EL =EL(Rocksalt)− EL(Other)) between cell-minimized rocksalt (EL(Rocksalt)) and sixother (EL(Other)) cell-optimized crystal structures with fixed ideal fractional coordi-nates.third order DKH is applied. The corresponding isolated ion energies, used to calculate lattice en-ergy, employ the def2-TZVPD basis set for lithium and the diffuse-augmented Sapporo basis setequivalents for the bromine and iodine anions.Figure B.10 gives the difference in lattice energy between rocksalt and wurtzite crystals forLiBr and LiI, calculated from geometry optimized crystals at various levels of theory. For calcula-tions without dispersion corrections (i.e. PBE169 and SCAN)174 and calculations with the rVV10class of dispersion corrections,139 the relativistic corrections from third order DKH slightly favourthe rocksalt crystal structure over wurtzite. One can rationalize this result from the changing effec-tive radius ratios. Relativistic corrections tend to shrink the heavier halides (relativistic shrinking)but leave the light lithium ion unaffected. This raises the cation/anion radius ratio, favouring octa-hedral structures. Curiously, calculations employing vdW-type corrections137 (i.e. optB88-vdW170and rev-vdW-DF2)138 favour wurtzite slightly when the third order DKH transformation is applied.Figure B.11 gives the equivalent to Figure B.10 for relative errors in the rocksalt lattice energycompared with experiment. In most cases, application of the third order DKH transformation215Figure B.8: The TF model: dispersion scaling factor SD applied only to X−-X− type dis-persion interactions, versus the difference in lattice energy (∆EL = EL(Rocksalt)−EL(Other)) between cell-minimized rocksalt (EL(Rocksalt)) and six other (EL(Other))cell-optimized crystal structures with fixed ideal fractional coordinates.lowers the lattice energy slightly. This follows from the relativistic shrinking of s- and p-orbitals,leading to denser packing and stronger ionic bonds for elements with p-orbitals occupying themost outer orbitals. Figure B.12 shows this phenomenon more explicitly, where application ofthird order DKH leads to a significant shrinking of the rocksalt lattice parameter at all levels oftheory tested. As expected, the relativistic correction is more pronounced for iodine than bromine,leading to a greater correction in the case of LiI.216Figure B.9: A comparison of ideal rocksalt and wurtzite potential energy curves for threelevels of theory: PW1PW-D3(BJ)/pob-TZVP, JC-SPC/E, and TF. Curves for wurtziteare plotted as dashed lines, while rocksalt curves are solid lines. Rocksalt structureexperimental EL and a are indicated as yellow circles, while wurtzite structure exper-imental15–17 a are indicated as black dotted lines. Experimental lattice energies forlithium halides in the wurtzite crystal structure have not been measured. Note that LiFwith a wurtzite crystal structure has not been observed experimentally.217Figure B.10: The difference (∆EL = EL(Rocksalt)−EL(Wurtzite)) in lattice energy betweenfully minimized rocksalt and wurtzite crystal structures at various levels of theory.Negative values favour rocksalt, in agreement with experiment. The Sapporo-QZPand Sapporo-DKH3-QZP basis sets are used for anions in non-relativistic (solid bar)and third order DKH (hatched bar) calculations, respectively. The pob-TZVP basis setis used for lithium.218Figure B.11: The relative error ([EL(Theory, RS)−EL(Exp.)]/|EL(Exp.)|) in lattice energyat various levels of theory. The Sapporo-QZP and Sapporo-DKH3-QZP basis setsare used for anions in non-relativistic (solid bar) and third order DKH (hatched bar)calculations, respectively. The pob-TZVP basis set is used for in-crystal lithium. Theenergy of isolated ions was calculated with the def2-TZVPD basis set (Li+) or thediffuse augmented versions of the Sapporo-QZP and Sapporo-DKH3-QZP basis sets(Br− and I−).219Figure B.12: The relative error ((acalculated− aexperiment)/aexperiment) in calculated lattice pa-rameter a at various levels of theory. The Sapporo-QZP and Sapporo-DKH3-QZP ba-sis sets are used for anions in non-relativistic (solid bar) and third order DKH (hatchedbar) calculations, respectively. The pob-TZVP basis set is used for lithium.220Appendix CSupporting Materials for Chapter 4C.1 Rigid-Ion Models: Crystal Energies at (T = 0, P = 0)In this section we give the relative and absolute lattice energies of all alkali halides employing therigid-ion interaction potentials investigated in the main paper. All crystal structures are ideal withenergy minimized lattice parameters.86 Energies are plotted relative to the rocksalt crystal structure,which is the experimentally stable structure of all alkali halides under ambient conditions, exceptfor CsCl, CsBr, and CsI, which occupy the CsCl (body-centered cubic) crystal structure. In theplots, a positive relative energy indicates that the given crystal structure is energetically favouredover the rocksalt crystal structure. In the tables below each plot, we give the corresponding absolutelattice energies.221Figure C.1: Relative crystal energies of all alkali halides employing the TF12,13 interactionpotential. ERS−EStruc is the lattice energy of the given model alkali halide in a partic-ular crystal structure, relative to rocksalt. Rocksalt is the experimental crystal structureof most alkali halides under ambient conditions.222Salt Rocksalt Wurtzite β -BeO CsCl 5-5 NiAs AntiNiAs SphaleriteLiF -1043.10 -1052.12 -1047.55 -988.47 -1049.00 -1031.00 -1028.34 -1049.82LiCl -844.67 -846.96 -843.20 -795.80 -845.28 -835.71 -828.83 -845.08LiBr -795.60 -796.76 -793.12 -746.60 -795.24 -787.26 -779.08 -794.98LiI -719.94 -729.48 -726.68 -672.22 -726.89 -712.26 -704.72 -727.90NaF -918.09 -923.27 -917.88 -871.61 -921.26 -905.43 -907.30 -921.23NaCl -777.72 -762.68 -758.70 -749.15 -769.64 -769.60 -767.25 -760.94NaBr -738.27 -725.12 -721.30 -708.13 -730.88 -730.64 -727.25 -723.46NaI -684.90 -676.52 -672.94 -651.60 -679.55 -677.79 -673.05 -675.00KF -814.09 -811.79 -805.33 -771.92 -812.47 -800.34 -805.49 -809.95KCl -708.33 -690.55 -686.67 -688.20 -699.07 -700.40 -700.79 -688.96KBr -679.06 -658.78 -655.22 -662.36 -668.56 -672.02 -671.66 -657.26KI -638.64 -618.57 -615.22 -622.00 -628.06 -632.29 -630.98 -617.16RbF -780.97 -772.95 -766.11 -740.38 -776.11 -766.65 -773.02 -771.17RbCl -688.55 -665.53 -661.81 -673.89 -676.76 -680.92 -681.91 -664.00RbBr -659.38 -636.97 -633.48 -646.09 -647.82 -652.47 -652.80 -635.51RbI -624.20 -599.86 -596.66 -614.07 -611.62 -618.19 -617.79 -598.49CsF -761.56 -738.85 -732.26 -725.26 -749.18 -746.97 -754.28 -737.16CsCl -675.83 -644.23 -640.90 -670.31 -659.87 -669.01 -670.03 -642.78CsBr -648.48 -618.14 -614.98 -644.46 -633.10 -642.28 -642.86 -616.75CsI -615.86 -584.78 -581.85 -615.08 -600.08 -610.44 -610.57 -583.48Table C.1: Energy-minimized lattice energies of eight (ideal) crystal structures at (T = 0,P = 0) for all alkali halides employing the TF interaction potential. Units are kJmol−1and the lowest-energy structure for each salt is given in bold.223Figure C.2: Relative crystal energies of all alkali halides employing the JC(SPC/E) model.224Salt Rocksalt Wurtzite β -BeO CsCl 5-5 NiAs AntiNiAs SphaleriteLiF -1052.60 -1030.57 -1026.40 -986.39 -1042.15 -1042.28 -1029.42 -1028.29LiCl -870.32 -863.63 -860.38 -792.81 -866.54 -861.50 -843.52 -861.72LiBr -826.03 -826.84 -823.81 -741.37 -825.19 -817.57 -796.13 -825.00LiI -761.78 -770.25 -767.57 -674.85 -764.34 -753.91 -730.30 -768.54NaF -930.75 -899.21 -895.33 -906.43 -915.74 -922.26 -918.99 -897.21NaCl -793.14 -769.77 -766.45 -756.99 -781.89 -785.65 -779.22 -768.06NaBr -761.55 -740.67 -737.43 -718.03 -751.20 -754.25 -745.56 -739.03NaI -713.21 -696.73 -693.65 -662.05 -704.62 -706.25 -694.93 -695.18KF -833.01 -800.40 -796.80 -821.81 -817.10 -825.50 -824.71 -798.63KCl -720.74 -693.75 -690.69 -705.63 -707.63 -714.34 -712.14 -692.21KBr -695.19 -668.75 -665.73 -677.19 -682.18 -688.98 -685.92 -667.27KI -656.86 -632.26 -629.34 -633.88 -644.58 -650.90 -646.53 -630.86RbF -802.32 -769.70 -766.10 -792.21 -786.21 -794.80 -794.64 -768.00RbCl -696.88 -669.52 -666.55 -685.77 -683.51 -690.77 -689.27 -668.04RbBr -672.63 -645.60 -642.68 -659.87 -659.29 -666.73 -664.63 -644.17RbI -636.76 -611.17 -608.35 -620.39 -623.99 -631.12 -628.08 -609.82CsF -786.79 -758.96 -755.32 -771.21 -773.48 -778.53 -779.13 -757.28CsCl -689.50 -665.09 -662.14 -675.44 -677.85 -683.07 -681.88 -663.62CsBr -664.34 -640.35 -637.48 -648.99 -652.75 -658.20 -656.36 -638.93CsI -629.20 -606.58 -603.82 -610.55 -618.13 -623.39 -620.50 -605.23Table C.2: Energy-minimized lattice energies of eight (ideal) crystal structures at (T = 0,P = 0) for all alkali halides employing the JC(SPC/E) interaction potential. Units arekJmol−1 and the lowest-energy structure for each salt is given in bold.225Figure C.3: Relative crystal energies of all alkali halides employing the JC(TIP3P) interac-tion potential.226Salt Rocksalt Wurtzite β -BeO CsCl 5-5 NiAs AntiNiAs SphaleriteLiF -1093.78 -1073.80 -1069.60 -1024.32 -1084.62 -1082.96 -1069.89 -1071.43LiCl -891.53 -899.94 -896.89 -795.18 -894.23 -882.28 -857.22 -897.95LiBr -836.91 -853.10 -850.35 -738.43 -844.02 -828.17 -800.92 -851.21LiI -768.97 -790.43 -788.16 -674.36 -781.09 -760.90 -733.98 -788.68NaF -954.14 -924.01 -920.07 -928.74 -940.09 -945.21 -942.48 -921.96NaCl -811.89 -790.40 -786.95 -766.31 -801.32 -804.06 -795.26 -788.65NaBr -773.93 -755.92 -752.57 -720.95 -764.65 -766.37 -755.02 -754.25NaI -721.21 -708.61 -705.55 -663.66 -714.37 -714.05 -700.92 -707.05KF -846.00 -815.51 -811.81 -831.55 -831.40 -837.84 -837.64 -813.71KCl -730.85 -703.37 -700.18 -713.69 -717.42 -724.22 -721.74 -701.81KBr -701.73 -675.08 -671.94 -681.24 -688.51 -695.34 -691.84 -673.58KI -660.56 -636.96 -634.01 -635.41 -648.80 -654.47 -649.82 -635.54RbF -809.25 -777.91 -774.16 -796.12 -793.87 -801.04 -801.56 -776.20RbCl -701.46 -672.63 -669.53 -690.82 -687.21 -695.25 -693.79 -671.13RbBr -674.65 -646.29 -643.26 -662.34 -660.49 -668.71 -666.60 -644.86RbI -636.42 -610.67 -607.82 -620.54 -623.55 -630.76 -627.87 -609.32CsF -773.65 -742.17 -738.15 -758.27 -757.81 -764.73 -766.45 -740.53CsCl -672.21 -642.81 -639.80 -665.46 -657.52 -666.29 -665.51 -641.38CsBr -647.22 -618.18 -615.26 -640.18 -632.62 -641.64 -640.41 -616.81CsI -611.87 -585.18 -582.45 -602.38 -598.48 -606.59 -604.80 -583.89Table C.3: Energy-minimized lattice energies of eight (ideal) crystal structures at (T = 0,P = 0) for all alkali halides employing the JC(TIP3P) interaction potential. Units arekJmol−1 and the lowest-energy structure for each salt is given in bold.227Figure C.4: Relative crystal energies of all alkali halides employing the JC(TIP4PEW) model.Figure C.5: Relative crystal energies of NaCl employing the SD interaction potential.228Salt Rocksalt Wurtzite β -BeO CsCl 5-5 NiAs AntiNiAs SphaleriteLiF -1059.52 -1041.38 -1037.36 -989.72 -1051.23 -1049.01 -1035.63 -1039.07LiCl -877.65 -883.40 -880.50 -786.66 -879.51 -868.56 -845.78 -881.44LiBr -826.72 -842.64 -840.09 -730.21 -833.83 -818.08 -791.78 -840.77LiI -760.92 -783.03 -780.88 -666.47 -773.68 -752.93 -726.01 -781.30NaF -931.77 -903.04 -899.24 -905.04 -918.43 -923.10 -919.90 -901.04NaCl -800.39 -780.43 -777.16 -756.10 -790.84 -792.66 -784.39 -778.70NaBr -766.62 -750.44 -747.25 -712.94 -758.43 -759.10 -747.68 -748.78NaI -716.80 -705.80 -702.82 -657.10 -710.73 -709.65 -695.81 -704.24KF -832.09 -802.45 -798.95 -818.40 -817.97 -824.46 -823.68 -800.68KCl -724.41 -698.89 -695.83 -705.86 -712.13 -717.83 -715.19 -697.34KBr -698.51 -673.53 -670.49 -675.86 -686.25 -692.13 -688.25 -672.03KI -659.90 -637.25 -634.34 -632.05 -648.62 -653.78 -648.49 -635.83RbF -796.85 -766.31 -762.85 -785.92 -781.98 -789.47 -789.18 -764.62RbCl -695.56 -668.79 -665.83 -683.39 -682.53 -689.44 -687.77 -667.31RbBr -671.88 -645.13 -642.19 -657.56 -658.66 -665.96 -663.53 -643.70RbI -636.39 -611.37 -608.55 -618.28 -623.91 -630.71 -627.33 -610.01CsF -767.03 -736.98 -733.42 -755.01 -752.21 -759.29 -759.80 -735.35CsCl -670.81 -644.09 -641.21 -661.49 -657.73 -664.90 -663.82 -642.66CsBr -647.97 -621.16 -618.32 -637.87 -634.69 -642.32 -640.72 -619.78CsI -614.90 -589.51 -586.79 -602.22 -602.24 -609.51 -607.22 -588.20Table C.4: Energy-minimized lattice energies of eight (ideal) crystal structures at (T = 0,P = 0) for all alkali halides employing the JC(TIP4PEW) interaction potential. Units arekJmol−1 and the lowest-energy structure for each salt is given in bold.Rocksalt Wurtzite β -BeO CsCl 5-5 NiAs AntiNiAs Sphalerite-792.86 -771.83 -768.22 -739.17 -781.75 -785.18 -773.34 -770.13Table C.5: Energy-minimized lattice energies of eight (ideal) crystal structures at (T = 0,P = 0) for NaCl employing the SD54 interaction potential. Units are kJmol−1 and thelowest-energy structure is shown in bold.229Appendix DSupporting Materials for Chapter 5D.1 Systematic MP Uncertainty in the Thermodynamic LimitIn this subsection I derive and visualize the hypothetical systematic uncertainty of MP calculationsin the thermodynamic limit (N → ∞) when calculated with the algorithm described in the mainpaper. Recall that ∆Tmin sets the initial upper bound on the systematic uncertainty, but the truesystematic uncertainty of given calculation may increase beyond this if two or more indeterminatepoints are found. I can eliminate this possibility by assuming that tmax is sufficiently large suchthat a melting or freezing event will eventually occur at all temperatures, except at the exact ther-modynamic MP where a singular indeterminate point theoretically persists. By setting N → ∞,I ensure that finite size effects do not contribute to the width of the calculated MP distribution.Instead, the MP calculation can be treated as deterministic function of the input parameters. Allsimulations with temperatures set above the thermodynamic MP will melt, while all simulationswith temperatures set below the thermodynamic MP will freeze.Assume the true thermodynamic MP is Tm, the initial step size is S, the step size multiplier is γ ,the calculation stopping criteria is ∆Tmin, and current temperature undergoing simulation is T ∗. Foreach MP calculation, the temperature is initialized to a random positive number T0, which may bevery far from Tm. In the first stage of the MP algorithm, the T ∗ approaches Tm monotonically withincreasing step size governed by γ , requiring some number of steps, x, to achieve a state where botha melting event and freezing event have been simulated. At this stage, at least one freezing eventhas been simulated at a temperature less than Tm and at least one melting event has been simulatedat a temperature greater than Tm. The melting and freezing events observed that bracket Tm at thisstage, TUB and TLB respectively, form an “intermediate” interval ∆Ti = TUB(i)−TLB(i), containing230Tm, given by∆Ti = Sγx−1. (D.1)Here, the i simply indicates the ∆T interval is taken at the earliest stage of the algorithm where bothmelting and freezing events have been simulated. Next, the algorithm switches strategies, halvingthe current ∆T = TUB−TLB interval with each iteration by simulating the midpoint temperature andupdating either TUB or TLB accordingly. The ∆T midpoint check proceeds until ∆T ≤ ∆Tmin, whichoccurs in some number of iterations y yielding the final interval I, which is the final systematicuncertainty. I can be calculated from ∆Ti byI =∆Ti2y≤ ∆Tmin. (D.2)Substituting Eq. (D.1)) into Eq. (D.2) yieldsI = Sγx−12y≤ ∆Tmin. (D.3)Solving this in general requires knowledge of T0 in order to find x, the number of steps required toform ∆Ti. x can be derived by noticing that ∆Ti forms only afterx−1∑j=0Sγ j ≥ |Tm−T0| . (D.4)This equation can be solved analytically for x by employing the formula for a truncated geometricseries, yieldingx = dlogγ(1− 1− γS|Tm−T0|)e. (D.5)However, since T0 is considered a variable here, I can instead assume a positive integer distributionof x and solve Eq. (D.3) for y,y = dlog2(S∆Tmin)+(x−1) log2 (γ)e. (D.6)In general, the systematic uncertainty I of a particular MP calculation depends non-linearly on S,γ , ∆Tmin, and T0. For our results, I used S = 10 K, γ = 2, ∆Tmin = 1 K, and a random T0. In this231special case, Eq. (D.6) reduces toy = dlog2(10 K1 K)+(x−1) log2 (2)e (D.7a)y = dx−1+ log2 (10)e. (D.7b)Inserting this result back into Eq. (D.3) and simplifying yieldsI = (10 K)2x−12dx−1+log2(10)e(D.8a)I = (10 K)2x−1−dx−1+log2(10)e (D.8b)I = (10 K)2−4 (D.8c)I = 0.625 K. (D.8d)Therefore, with the calculation parameters chosen in the main paper, a hypothetical MP calculationin the thermodynamic limit yields a systematic uncertainty of 0.625 K. The distribution of suchrepeated MP calculations is uniform (for uniformly selected T0), centered at Tm, with a completedistribution width of ±0.3125 K. A visualization of the hypothetical TUB and TUB distributionswith these parameters are shown in Figure D.1, while the corresponding MP distribution is shownin Figure D.2.232Tm ! 0:625 Tm Tm + 0:625T00.20.40.60.811.21.41.61.8;(T)TLBTUBFigure D.1: A simulated probability distribution ρ(T ) of TUB and TUB in the thermodynamiclimit when S = 10 K, γ = 2, ∆Tmin = 1 K, tmax = ∞. The distribution was produced bysimulating the outcome of 108 hypothetical MP calculations with perfectly well-definedTm and randomly generated T0.Tm ! 0:3125 Tm Tm + 0:3125T00.20.40.60.811.21.41.61.8;(T)Figure D.2: A simulated probability distribution ρ(T ) of Tm in the thermodynamic limitwhen S = 10 K, γ = 2, ∆Tmin = 1 K, tmax = ∞. The distribution was produced bysimulating the outcome of 108 hypothetical MP calculations with perfectly well-definedTm and randomly generated T0.233D.2 Melting Point Distributions - Unfiltered Data1280 1282 1284 1286 1288 1290 1292 1294 1296 1298 1300Tm [K]00.10.20.30.40.50.60.70.8;(T m)N = 2000PDF: 7 = 1289:8 K, < = 1:79 KN = 11664PDF: 7 = 1289:9 K, < = 0:47 KFigure D.3: The MP distribution of repeated calculations for the NaCl/JC(SPC/E) sys-tem (rocksalt/liquid) using our automated MP calculation method. These distribu-tions includes all 100 calculations, including those with systematic error larger than∆Tmin = 1 K. To aid visualization, both distributions include a best-fit Gaussian proba-bility distribution function (PDF) envelope with mean µ and standard deviation σ givenin the legend. The initial temperatures (T0) given to the MP algorithm were randomlygenerated between 1270 – 1310 K. All calculations use ∆x∗liq = 25 %, ∆Tmin = 1 K, andtmax = 5 ns.234Appendix ESupporting Materials for Chapter 6E.1 Choice of Covariance and Acquisition FunctionsIn this section, I give some preliminary data used to guided the choice of two hyperparameters forthe Bayesian optimization calculations in Chapter 6: the acquisition function and the covariancefunction. All figures in this section involve Bayesian optimization of the same objective functionrun for 1000 iterations without further local optimization. The goal is the optimization of CLJfunction parameters towards LiBr crystal targets at T = 0, with Lennard-Jones mixing rules em-ployed. The targets are: LiBr Rocksalt lattice energy EL (target weight = 1); the rocksalt/wurtziteEL difference, as calculated using the best DFT results in Chapter 3 (target weight = 105); the exper-imental rocksalt lattice parameter a0, thermally-corrected to T = 0 (weight = 0.1); the experimentalwurtzite lattice parameters a0 and c0, thermally-corrected to T = 0 (weights = 0.05 each).E.1.1 Acquisition FunctionHere I compare the results of Bayesian optimization of the objective function described above,using eight acquisition functions. In all cases, the Mate´rn-5/2 covariance function298 with auto-matic relevance determination (ARD)301 is employed. The expected improvement plus acquisitionfunction includes a parameter called the exploration ratio,302 which allows the user to weigh theimportance of exploration vs exploitation. Several exploration ratios are tested, indicated in brack-ets. The lower confidence bound (LCB)319 has a similar parameter κ that sets the number ofstandard deviations from the mean used to define the lower confidence bound. Several values of κare tested, indicated in brackets. Since results partly depend on the initial random sample, I ran all235calculations in triplicate.Figure E.1: Traces of the best observed objective function value vs Bayesian optimizationiteration, for several acquisition functions.236Figure E.2: Comparison of the final objective function value, and crystal properties, afterBayesian optimization was applied to the objective function described above, usingseveral acquisition functions. The target properties are indicated with red boxes.From these results, it was difficult to identify the best acquisition function to employ. Mostseemed to perform similarly well, although LCB with too high a value of κ gave undesirable resultsdue to its high weight on exploration. For all further calculations,the expected improvement plus(with exploration ratio 2) and lower confidence bound (with κ = 1) were employed as described inSection 6.2.1.E.1.2 Covariance FunctionIn this section, Bayesian optimization is employed to test the quality of several covariance functionson the objective function described above. In all cases, the expected improvement plus acquisitionfunction is employed with exploration ratio 2. All covariance functions employ automatic relevancedetermination.301 At first, only a single replicate calculation for each covariance function waschosen.237Figure E.3: Traces of the best observed objective function value vs Bayesian optimizationiteration, for several covariance functions.238Figure E.4: Comparison of the final objective function value, and crystal properties, afterBayesian optimization was applied to the objective function described above, usingseveral covariance functions. The target properties are indicated with red boxes.Once again, it is quite difficult to differentiate the results with any rigor. No covariance functioncame out as the clear winner after a single replicate. Overall, the two Mate´rn covariance functionsappear to do best. These were tested for an additional four replicates as shown below.239Figure E.5: Traces of the best observed objective function value vs Bayesian optimizationiteration, for the two Mate´rn covariance functions.240Figure E.6: Comparison of the final objective function value, and crystal properties, afterBayesian optimization was applied to the objective function described above, usingreplicates of the Mate´rn covariance functions. The target properties are indicated withred boxes.The conclusion is that both Mate´rn functions appear to perform about equally as well for thisobjective function. The default Mate´rn-5/2 covariance function was selected for all productionBayesian optimization calculations discussed in Chapter 6.241E.2 Distributions of Thermal DataFigure E.7: Distributions of the results from 350 repeated thermal calculations (using differ-ent initial conditions) for a fitted LiI CBH force field, targeted to both low temperaturecrystal and high temperature thermal target data, using the methodology described inSection 6.2.2. The symbols are as described in the caption of Figure 6.7. This datareflects the approximate nondeterminism in various parts of the objective function dueto the inherent distribution of calculated property averages from molecular dynamicssimulations at finite size.242E.3 Coulomb Lennard-Jones Model: Additional DataFigure E.8: A comparison of fitted LiX forcefield results to all experimental and theoreticaldata of interest, including untargeted data. The plots and symbols are as described inthe caption of Figure 6.7. The optimization employed (T = 0, P = 0) crystal proper-ties as targets: EL(rocksalt), the rocksalt lattice parameter (as a stand-in for V ), and∆EL(wurtzite), as indicated by red boxes in the figure. Forcefield parameters are ob-tained after 1000 Bayesian optimization iterations, followed by local objective functionoptimization until convergence. In some cases, untargeted thermal data could not becalculated for a given model due to a phase change in the corresponding simulation;these results are not shown. The experimental and theoretical reference data used hereis given in Table 6.1 and 6.2, respectively.243Figure E.9: A comparison of fitted LiX forcefield results to all experimental and theoreticaldata of interest, including untargeted data. The plots and symbols are as described inthe caption of Figure 6.7. Ion absolute charge was treated as an input parameter to beoptimized, with a domain between 0.95 and 1.0. Combining rules were employed. Theoptimization employed (T = 0, P = 0) crystal properties as targets: EL(rocksalt), therocksalt lattice parameter (as a stand-in for V ), and ∆EL(wurtzite), as indicated by redboxes in the figure. Forcefield parameters are obtained after 1000 Bayesian optimizationiterations, followed by local objective function optimization until convergence. In somecases, untargeted thermal data could not be calculated for a given model due to a phasechange in the corresponding simulation; these results are not shown. The experimentaland theoretical reference data used here is given in Table 6.1 and 6.2, respectively.244Figure E.10: A comparison of fitted LiX forcefield results to all experimental and theoreticaldata of interest, including untargeted data. The plots and symbols are as described inthe caption of Figure 6.7. The optimization employed (T = 0, P = 0) crystal proper-ties as targets: EL(rocksalt), the rocksalt lattice parameter (as a stand-in for V ), and∆EL(wurtzite), as indicated by red boxes in the figure. Ion absolute charge was fixedto 1.0 and combining rules were turned off, increasing the number of optimizable pa-rameters from four to six. Forcefield parameters are obtained after 1000 Bayesianoptimization iterations, followed by local objective function optimization until con-vergence. In some cases, untargeted thermal data could not be calculated for a givenmodel due to a phase change in the corresponding simulation; these results are notshown. The experimental and theoretical reference data used here is given in Table 6.1and 6.2, respectively.245
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Patey, Grenfell
Lithium halide structural chemistry : computational analysis with machine learning, quantum chemistry, and molecular dynamics
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