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Hydrogen atom transfer reactions and the effects of non-redox active metal cations van Santen, Jeffrey A. 2018

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Hydrogen Atom Transfer Reactionsand the Effects of Non-Redox ActiveMetal CationsbyJeffrey A. van SantenB.Sc. Hons. Chemistry, The University of British Columbia, 2015A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFMASTER OF SCIENCEinTHE COLLEGE OF GRADUATE STUDIES(Chemistry)THE UNIVERSITY OF BRITISH COLUMBIA(Okanagan)January 2018© Jeffrey A. van Santen, 2018The following individuals certify that they have read, and recommend to theCollege of Graduate Studies for acceptance, a thesis/dissertation entitled:Hydrogen Atom Transfer Reactions and the Effects of Non-RedoxMetal Cationssubmitted by Jeffrey A. van Santen in partial fulfilment of the requirements ofthe degree of Master of ScienceGino DiLabio, I. K. Barber School of Arts & SciencesSupervisorStephen McNeil, I. K. Barber School of Arts & SciencesSupervisory Committee MemberFrederic Menard, I. K. Barber School of Arts & SciencesSupervisory Committee MemberMichael Deyholos, I. K. Barber School of Arts & SciencesUniversity ExamineriiAbstractHydrogen atom transfer (HAT) reactions are a fundamental step in many bi-ological processes, but can initiate the free-radical induced oxidation of cellularcomponents. Although HAT reactions appear fundamentally elementary, there aremany poorly understood factors that influence HAT. In this thesis, three aspects ofHAT reactivity are investigated using quantum chemical techniques.First, the importance of pre-reaction complex formation in considering the ki-netics of HAT reactions were investigated. For a set of nearly-thermoneutral HATreactions involving oxygen-centred radicals, the relationship between pre-reactioncomplex non-covalent binding energies and Arrhenius pre-exponential factors (A-factors) was investigated. It is demonstrated that for HAT reactions that take placethrough similar mechanisms, there is a strong correlation between pre-reaction com-plex binding energies and A-factors. This suggests that non-covalent interactionsmay directly affect the kinetics of certain HAT reactions.Next, the relationship between bond dissociation energies (BDEs) and reactionrates for abstraction of a hydrogen from a C H bond by the CumO radical areinvestigated in the context of the Bell-Evans-Polanyi (BEP) principle. The appli-cability of the BEP principle is examined by exploring a hypothesis: If the BEPprinciple is a valid linear free-energy relationship, there should exist two linear re-lationships for BDE against the logarithm of HAT rate constant, one for incipientradicals that are allylic or benzylic, and one for alkyl radicals. It is demonstratedthat there is a reasonably strong correlation for allylic/benzylic C H bonds, but notiiiAbstractfor alkyl ones. The BEP principle should not be used for quantitative prediction,but remains useful as a conceptual framework.Finally, the effect of non-redox active metal cations on HAT reactions involv-ing small models for proteins and oxygen-centred radicals is studied. Previous ex-perimental evidence demonstrated that Lewis acid-base interactions between metalcations and substrates can inhibit HAT reactions, and that the cations may serveas a form of chemo-protection in biological systems. The results herein demonstratethat metal-substrate interactions can deactivate certain CH bonds. Metal-radicalinteractions may promote HAT reactions. On the basis of these limited results,non-redox active metal cations might not act as natural chemo-protective agents.ivTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiiList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xList of Schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiiiList of Symbols and Abbreviations . . . . . . . . . . . . . . . . . . . . xivAcknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xviChapter 1: Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Details of HAT reactions . . . . . . . . . . . . . . . . . . . . . . . . . 41.3 Research goals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10Chapter 2: Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.1 The quantum mechanical approach . . . . . . . . . . . . . . . . . . . 172.1.1 Spin and Spatial Orbitals . . . . . . . . . . . . . . . . . . . . 182.1.2 Slater determinants . . . . . . . . . . . . . . . . . . . . . . . 202.1.3 The Hartree-Fock approximation . . . . . . . . . . . . . . . . 212.1.4 Basis sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242.1.5 Post-Hartree-Fock methods . . . . . . . . . . . . . . . . . . . 292.1.6 The complete basis set limit . . . . . . . . . . . . . . . . . . . 332.1.7 Composite quantum chemistry methods . . . . . . . . . . . . 342.1.8 Density-functional theory . . . . . . . . . . . . . . . . . . . . 362.2 Applying theory to chemical problems . . . . . . . . . . . . . . . . . 432.2.1 Geometry optimization . . . . . . . . . . . . . . . . . . . . . 432.2.2 Molecular vibrations . . . . . . . . . . . . . . . . . . . . . . . 442.2.3 Thermochemistry . . . . . . . . . . . . . . . . . . . . . . . . . 462.2.4 Modelling solvent . . . . . . . . . . . . . . . . . . . . . . . . . 49vTABLE OF CONTENTS2.2.5 Rate constants and transition state theory . . . . . . . . . . . 50Chapter 3: The Relationship Between Arrhenius Pre-factors withNon-Covalent Binding . . . . . . . . . . . . . . . . . . . . 603.1 Prologue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 603.2 Computational methods and details . . . . . . . . . . . . . . . . . . 633.3 Results and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . 653.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78Chapter 4: Interrogation of the Bell-Evans-Polanyi Principle: In-vestigation of the Bond Dissociation Enthalpies Cor-related with Hydrogen Atom Transfer Rate Constants 804.1 Prologue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 804.2 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 864.2.1 Quantum chemical composite procedures . . . . . . . . . . . 874.2.2 Transition state calculations . . . . . . . . . . . . . . . . . . . 904.3 Comparison of composite method for the prediction of BDEs . . . . 914.4 Analysis of the Bell-Evans-Polanyi principle . . . . . . . . . . . . . . 1014.5 Transition state analysis . . . . . . . . . . . . . . . . . . . . . . . . . 1054.6 Is the Bell-Evans-Polanyi principle valid? . . . . . . . . . . . . . . . 110Chapter 5: Do non-redox active metal cations have the potentialsto behave as chemo-protective agents? The Effects onMetal Cations on HAT Reaction Barrier Heights . . . 1155.1 Prologue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1155.2 Computational methods and details . . . . . . . . . . . . . . . . . . 1285.3 Exploring the nature of metal cation substrate interactions . . . . . 1305.4 HAT reactions involving non-redox active metals . . . . . . . . . . . 1415.4.1 DMA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1415.4.2 DIA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1485.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151Chapter 6: Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . 154References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159Appendices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182Appendix A: Chapter 3 Additional Data . . . . . . . . . . . . . . . . . 183Appendix B: Chapter 4 Additional Data . . . . . . . . . . . . . . . . . 184Appendix C: Chapter 5 Additional Data . . . . . . . . . . . . . . . . . 188viTABLE OF CONTENTSC.1 Benchmarking DFT based methods for the binding of alkali and al-kaline earth metals to organic substrates and oxygen-centred radicals 188C.1.1 Metal cation basis set convergence . . . . . . . . . . . . . . . 190C.1.2 High level results and evaluation of various density-functionaltheory based methods . . . . . . . . . . . . . . . . . . . . . . 195C.2 HAT reactions involving non-redox active metals . . . . . . . . . . . 199C.2.1 DMA + HO . . . . . . . . . . . . . . . . . . . . . . . . . . . 199C.2.2 Strong hydrogen bond accepting substates . . . . . . . . . . . 201viiList of TablesTable 3.1 Table of results for (nearly) thermoneutral reactions studied. 61Table 4.1 Bond dissociation enthalpies of the species used to investigatethe accuracy of composite methods. . . . . . . . . . . . . . . . 92Table 4.2 Reaction barrier heights for reactions of substrates with CumOcalculated in the gas phase at 298 K. . . . . . . . . . . . . . . 106Table 5.1 Ionic concentrations inside a mammalian heart cell and in theblood plasma. . . . . . . . . . . . . . . . . . . . . . . . . . . . 116Table 5.2 Summary of rate constants for reactions of CumO with var-ious organic substrates in the presence of alkali and alkalineearth metal salts. . . . . . . . . . . . . . . . . . . . . . . . . . 118Table 5.3 Bond dissociation enthalpies of DMA, DMSO, MeCN, andDIA with and without metal cations. . . . . . . . . . . . . . . 134Table 5.4 Calculated free energy (enthalpy) barrier for direct HAT fromdifferent C H bonds in DMA by CumO and BnO , with andwithout NaCl. . . . . . . . . . . . . . . . . . . . . . . . . . . . 142Table 5.5 Calculated free energy (enthalpy) for direct HAT from differ-ent C H bonds in DIA by CumO , with and without NaCl. . 151Table B.1 Summary of experimental rate constants and literature bonddissociation enthalpies. . . . . . . . . . . . . . . . . . . . . . . 186Table C.1 Total energy of alkali and alkaline earth-metal cations. . . . . 195Table C.2 Benchmark gas-phase binding energies of alkali and alkalineearth-metals with small organic substrates and radicals. . . . 196Table C.3 Evaluation of DFT-based methods for alkali and alkaline metalbinding to organic substrates and radicals. . . . . . . . . . . . 196Table C.4 Comparison of single point and relaxed binding energies foralkali and alkaline metal binding with DFT-based methods. . 198Table C.5 Calculated free energy (enthalpy) barrier for direct HAT fromdifferent C H bonds in DMA by HO , with and without NaCl. 200viiiLIST OF TABLESTable C.6 Calculated free energy (enthalpy) barrier for HAT betweenDMSO and CumO , and conventional HAT and radical H-atom donation with BnO , with and without NaCl. . . . . . . 202Table C.7 Calculated free energy (enthalpy) barrier for HAT betweenHMPA and TBPO with CumO with and without NaCl. . . . 207ixList of FiguresFigure 1.1 A typical reaction coordinate diagram. . . . . . . . . . . . . 11Figure 2.1 Schematic representation of a quantum mechanical compositemethod. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35Figure 2.2 A reaction coordinate diagram for a generic reaction. . . . . 52Figure 2.3 Quantum mechanical tunnelling occurs when a particle pen-etrates a reaction barrier, rather than surmounting it. . . . . 57Figure 3.1 Plot of logarithm of A-factor against binding energy. . . . . 66Figure 3.2 Three-dimensional structures of pre-reaction complexes 2 (TEMPO-H and 4-oxo-TEMPO) and 3 (di-t-butyl-hydroxylamine anddi-t-butyl-nitroxyl). . . . . . . . . . . . . . . . . . . . . . . . 68Figure 3.3 Three-dimensional structure of pre-reaction complex 4 be-tween 2,4,6-tri-t-butylphenol and 4-t-butylphenoxyl. . . . . . 70Figure 3.4 Three-dimensional structures of pre-reaction complexes 8 (t-butylperoxyl and phenol) and 9 (t-butylperoxyl and 2-naphthol). 72Figure 3.5 Three-dimensional structure of pre-reaction complex 10 be-tween t-butylperoxyl and α-tetralin peroxide. . . . . . . . . . 73Figure 3.6 Three-dimensional structure of pre-reaction complex 6 be-tween N,N -diphenylhydroxylamine and N,N -diphenylnitroxyl. 75Figure 3.7 Three-dimensional structure of pre-reaction complex 7 be-tween 2-naphthol and phenoxyl. . . . . . . . . . . . . . . . . 76Figure 3.8 Three-dimensional structures of pre-reaction complexes 1 (2,4,6-tri-t-butylphenoxl and 2,4,6-tri-t-butylphenoxyl) and 5 (2,4,6-tri-t-butylphenol and t-butylperoxyl). . . . . . . . . . . . . . 77Figure 3.9 Reaction coordinate qualitatively illustrating the proposedmechanism for HAT in complexes 1 and 5. . . . . . . . . . . 77Figure 4.1 Energy profiles for a series of related exothermic reactionsillustrating the Bell-Evans-Polanyi principle. . . . . . . . . . 81Figure 4.2 Bell-Evans-Polanyi plot of experimental rate constants againstliterature BDEs. . . . . . . . . . . . . . . . . . . . . . . . . . 85Figure 4.3 Summary of deviations of BDEs from literature for compositequantum chemical methods. . . . . . . . . . . . . . . . . . . 98xLIST OF FIGURESFigure 4.4 One-to-one plot of BDEs from literature and as calculated bythe W1BD composite method. . . . . . . . . . . . . . . . . . 99Figure 4.5 One-to-one plot comparing BDEs calculated by ROCBS-QB3to literature and W1BD BDEs. . . . . . . . . . . . . . . . . . 100Figure 4.6 Bell-Evans-Polanyi plot of experimental rate constants (nor-malized for the number of equivalent hydrogen atoms) forHAT between CumO and substrates against BDEs calcu-lated using the ROCBS-QB3 method. . . . . . . . . . . . . . 102Figure 4.7 Further breakdown of Bell-Evans-Polanyi plot of experimen-tal rate constants (normalized for the number of equivalenthydrogen atoms) for HAT between CumO and alkyl sub-strates against BDEs calculated using the ROCBS-QB3 method.104Figure 4.8 Structures of TS for HAT between CumO and toluene withSOMO and HOMO. . . . . . . . . . . . . . . . . . . . . . . . 107Figure 4.9 Bell-Evans-Polanyi plot of calculated enthalpic barriers forHAT between CumO and substrates against BDEs calculatedusing the ROCBS-QB3 method. . . . . . . . . . . . . . . . . 111Figure 5.1 Plot of observed rate constant against concentration of DMFand DMA for reaction with CumO at 298 K in the presenceof 0.2 M Mg(ClO4)2. . . . . . . . . . . . . . . . . . . . . . . . 121Figure 5.2 Plot of observed rate constant against concentration of DMAfor reaction with CumO at 298 K in the presence of 0.2 MNaClO4 and Mg(ClO4)2. . . . . . . . . . . . . . . . . . . . . 123Figure 5.3 Plot of observed rate constant against concentration of DIAfor reaction with CumO at 298 K in the presence of 0.2 MMg(ClO4)2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124Figure 5.4 Potential energy surface of binding energy between DMA andsodium cation and sodium chloride. . . . . . . . . . . . . . . 132Figure 5.5 Potential energy surface of binding energy between DMA andmagnesium cation and magnesium chloride. . . . . . . . . . . 133Figure 5.6 Structures of the DMA-NaCl complex and associated radicalcomplexes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136Figure 5.7 The resonance forms of DMA. . . . . . . . . . . . . . . . . . 138Figure 5.8 Structures of the DIA-NaCl complex and radical complex. . 139Figure 5.9 TS structures of HAT reaction between DMA and CumOincluding and excluding NaCl. . . . . . . . . . . . . . . . . . 145Figure 5.10 TS structures of HAT reaction between DMA and BnO in-cluding NaCl. . . . . . . . . . . . . . . . . . . . . . . . . . . 148Figure 5.11 TS structures for HAT reaction between DIA and CumOincluding NaCl. . . . . . . . . . . . . . . . . . . . . . . . . . 152xiLIST OF FIGURESFigure A.1 Molecular orbitals of hydrogen peroxide-peroxyl self-exchangereaction TS complex, demonstrating a PCET mechanism. . . 183Figure B.1 One-to-one plots of composite methods compared to litera-ture and W1BD. . . . . . . . . . . . . . . . . . . . . . . . . . 184Figure B.2 One-to-one plots comparing experimental and calculated rateconstants for HAT reactions between CumO and various or-ganic substrates. . . . . . . . . . . . . . . . . . . . . . . . . . 187Figure C.1 Basis set convergence for alkali and alkaline earth-metal cations.192Figure C.2 Basis set convergence for sodium and magnesium ions withcore-correlation basis sets. . . . . . . . . . . . . . . . . . . . 193Figure C.3 Explicitly correlated basis set convergence for alkali and al-kaline earth-metal cations. . . . . . . . . . . . . . . . . . . . 194Figure C.4 TS structures of HAT reaction between DMSO and CumO ,and the conventional HAT and radical H-atom donation re-actions with BnO excluding and including NaCl. . . . . . . 203Figure C.5 Reaction profiles for HAT between HMPA with BnO andTBPO with BnO . . . . . . . . . . . . . . . . . . . . . . . . . 206Figure C.6 TS structures of HAT reaction between HMPA and TBPOwith CumO and BnO including NaCl. . . . . . . . . . . . . 208xiiList of Schemes1.1 Common reactions involved in protein oxidation. . . . . . . . . . . . 31.2 Self-exchange reactions of the benzyl-toluene couple through directHAT, and the phenoxyl-phenol couple through PCET. . . . . . . . . 61.3 Possible cisoid (stacked) and transoid TS structures for the benzyl-toluene couple, the peroxyl-toluene couple, and the iminoxyl-oximecouple. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.4 Hyperconjugative overlap in tetrahydrofuran and the effect of non-redox active metal cations on the transition state complex. . . . . . 154.1 Unimolecular decay of the cumyloxyl radical. . . . . . . . . . . . . . 874.2 Locally-dense basis set partitioning used in the calculation of BDEs. 894.3 A generic HAT transition state structures and possible resonance forms.1145.1 Chemical structures of the species studied herein. . . . . . . . . . . . 127C.1 Initial proposed benchmark set of substrates/radicals and metal cations.189C.2 Revised benchmark set of small substrates and cations. . . . . . . . . 191C.3 The HAT reactions of DMSO with CumO and BnO . . . . . . . . . 202xiiiList of Symbols andAbbreviationsA-factor Arrhenius pre-exponential factorAO atomic orbitalsATP adenosine triphosphateBDE bond dissociation enthalpyBEP Bell-Evans-PolanyiBnO benzyloxyl radicalBSIP basis set incompleteness potentialCHD 1,4-cyclohexadieneCumO cumyloxyl radicalCBS complete basis setDABCO diazobicyclo[2.2.2]octaneDFT density-functional theoryDIA N,N -diisobutylacetamideDMA N,N -dimethylacetamideDMF N,N -dimethylformamideDMSO dimethyl sulfoxideEa activation energyGGA generalized-gradient approximationGTO Gaussian-type orbitalsH Hamiltonian operatorHAT hydrogen atom transferHF Hartree-FockHMPA hexamethylphosphoramideHOMO highest-occupied molecular orbitalIP ionization potentialk rate constantK equilibrium constantLDA local-density approximationLCAO linear combination of atomic orbitalsLFER linear free energy relationshipLFP laser flash photolysisLP lone pairxivList of Symbols and AbbreviationsLUMO lowest occupied molecular orbitalMeCN acetonitrileMO molecular orbitalMP Møller-PlessetNBO natural bonding orbitalNCI non-covalent interactionNPA natural population analysisPCET proton coupled electron transferPES potential energy surfaceQM quantum mechanicsRDF radial distribution functionROS reactive oxygen speciesSOMO singly occupied molecular orbitalsSTO Slater-type orbitalsTBPO tributylphosphineTEA triethylamineTHF tetrahydrofuranTS transition stateTST transition state theoryXC exchange-correlationZPE zero-point vibrational energy∆G Gibbs free energy of reaction∆G‡ Gibbs free energy barrier of reaction∆H enthalpy of reaction∆H‡ enthalpic reaction barrier of reaction∆S entropic change of reaction∇2 Laplacian operatorxvAcknowledgementsI would like to first express my deepest gratitude to my supervisor, Prof. GinoDiLabio, for his guidance throughout the course of my degree. His depth of knowl-edge and continuing enthusiasm were always there to support and inspire me. I amtruly indebted to him and cannot say thank you enough. I would also like to thankall the past and current members of the DiLabio group, without whom the compu-tational research path would have been exceedingly isolated. A special thank youto Dr. Alberto Otero de la Roza for all his help with many research and technicalmatters.Thank you to Dr. Steve McNeil, Dr. Fred Menard, Dr. Paul Shipley, Dr. JimBailey, and Dr. Sandra Mecklenburg for their advice, knowledge, and dedication toteaching. Your collective efforts as instructors, mentors, and advisors have helpedtremendously.I would also like to thank my family for their endless support, encouragement,and love: Michael, Mama, Papa, Nicola, Rob, and Arlene. Words cannot expresshow much you all mean to me.And finally I would like to thank Grace. Your never ending support and lovehave been fundamental to my success.xviChapter 1Introduction1.1 BackgroundRadicals are chemical species that tend to be highly reactive due to the presenceof one or more unpaired electrons. Living systems depend on radical processesas part of normal metabolism1 but biomaterials, such as proteins, are susceptibleto radical-induced damage. Radical-induced oxidation of biomaterials has beenimplicated in a number of degenerative disease states, including cancer, Alzheimer’s,Parkinson’s, and multiple sclerosis.2–5In biological systems, radicals are derived from many sources. Exogenous sourcesinclude solar radiation and air pollutants, while endogenous sources include in vivotransition metal-ion redox processes, such as in the electron transport chain involvedin cellular respiration.6 Some processes in the electron transport chain may transferan electron to molecular oxygen, forming the superoxide anion (O –2 ). Superoxide isnot a strong oxidant on its own, however it may become protonated to form the morereactive hydroperoxyl radical,7 or disproportionate spontaneously or catalyticallythrough metalloenzymes such as superoxide dismutase, leading to the formationof highly reactive oxygen-centred radicals. Oxygen-centred radicals derive fromreactions of O2 with redox-active metals.1Oxygen-centred radicals, known as reactive oxygen species (ROSs) in biology,are particularly important and common due to the previously described nature ofaerobic respiration. The ROSs that are of primary concern are the highly reac-11.1. Backgroundtive hydroxyl radicals (HO ), alkoxyl radicals (RO ), superoxide (HOO /O –2 ), andperoxyl radicals (ROO ).1 The oxidation of proteins by ROSs occurs through a rad-ical chain mechanism that has been studied experimentally in detail.8,9 This chainreaction occurs when an ROS initiates a radical chain reaction through hydrogenatom transfer (HAT), single electron transfer, or addition reactions with proteinsubstrates, leading to rapid propagation and formation of new radicals. HAT is anextremely important reaction in the context of oxidative damage. The focus of mywork is the development of an understanding of the fundamental chemistry involvedin protein oxidation through studying small model systems.Proteins are the most abundant biomaterial in most mammalian biological sys-tems,10 thus understanding their degradation is essential to understanding degener-ative disease. Because proteins are composed of as many as 20 natural amino acidside-chains, as well as the common peptide backbone, there are a large number ofpossible reactions. Some of the reactions involved in protein oxidation are shownin Scheme 1.1.Initial abstraction (Reaction a) often occurs at the α-carbon position (α CH),forming a carbon-centred radical (α C ) that is partially delocalized in the pi-systemof the neighbouring amide and carbonyl groups. Studies have indicated that thestability of α C is determined by stereo-electronic effects related to the planarityof the amide group.11 As such, steric bulk in the side-chains, as well as local proteinstructure (helix, sheet, etc.) can constrain radical geometries. For example, themost stable α-carbon radicals occur at glycine residues in antiparallel β-sheets,whereas other bulkier residues and secondary structures lead to loss of captodativestabilization.12 Amino acid side-chains are also susceptible to oxidation. Those side-chains containing sulfur,13 as well as tyrosine (which has a fairly weak phenolic O-Hbond of about 89 kcal mol−1),14 are particularly susceptible to oxidation.21.1. BackgroundHN CHROCHN CROCROHa)nO2HN CROCOOHN CROCHN CROCOH+c)HN CROCe)HN CROCO HN CRC NOO+NH2 CHNC+OROf)R OInitiationRadical Chain ReactionHN CHROCHN CROCOOHb)TerminationHN CROCOOd)R3CROorA H R3CHROHorAg)Radical Mediated Protein OxidationHN CR'OCNHCRCONHC R'COScheme 1.1: Common reactions involved in protein oxidation. The reactions areas follows: a) initiation of radical chain through abstraction by an oxygen-centredradical to generate an α-carbon-centred radical, b) radical addition of molecularoxygen, c) propagation of the radical chain reaction generating another α-carbonradical and a peroxide. d) Radical mediated protein oxidation proceeds throughmultiple steps involving oxygen centred radicals and molecular oxygens and resultsin the generation of a reduced amide (alcohol). Termination of the radical chainreaction can occur in several ways, including: e) cross-linking of two carbon-centredradicals, f) fragmentation of an oxygen-centred radical intermediate, or g) HATwith an antioxidant (AH).31.2. Details of HAT reactionsPropagation of the radical chain reaction occurs through various processes. Inthe presence of molecular oxygen, rapid addition occurs at the newly formed α C(Reaction b), generating a peroxyl radical, which can carry forward through furtherHAT reactions (Reaction c).15 The mechanism involved in the radical mediatedoxidation of proteins has been studied experimentally using techniques involvingionizing radiation.16,17 The course of this process is dependent on the availabilityof either singlet oxygen (1O2), or superoxide (O–2 ) or the protonated form, peroxylradical ( OOH). A detailed analysis of this process is outside the scope of this thesis,but ultimately, these reactions lead to the generation of a hydroxyl-amide (Reactiond).The radical chain reaction can be terminated through several mechanisms, in-cluding protein-protein cross-linking (Reaction e), or protein fragmentation (Reac-tion f). Reactions with antioxidants (A H, Reaction g) also terminate the chainreaction by removing the radical from the protein system. The sum total of allthese processes contribute to the accumulation of oxidized proteins that are associ-ated with many degenerative diseases.18 HAT reactions, which are important stepsin the initiation, propagation, and termination reactions of protein oxidation, areinvestigated through small molecular models in this thesis.1.2 Details of HAT reactionsDeveloping an understanding of protein and other biomolecular oxidation re-quires an understanding of the deceptively simple HAT reactions involved. FormalHAT reactions are a fundamental radical chemical transformation that have beenstudied for more than a century.19,20 From an experimental perspective, HAT re-actions that involve oxygen-centred radicals and non-radical organic substrates arereasonably well characterized: the effects of different solvents are well understood.2141.2. Details of HAT reactionsHowever, the main challenge faced by many experiments is elucidating the mech-anistic details of a reaction. This is a problem that can be examined by quantumchemistry, which is the approach that I shall take. Background on the theory usedin this thesis is given in Chapter 2.In order to investigate HAT reactions, we need to consider the mechanism in de-tail. For a simple HAT reaction, there exist several possible mechanisms by whichthis transformation can occur. The two most common concerted mechanisms aredirect HAT and proton-coupled electron transfer (PCET). At the basic level, directHAT involves the transfer of an electron and proton through the same set of ac-ceptor/donor orbitals, while PCET involves the transfer of an electron and protonthrough different sets of orbitals. In practise, this distinction is poorly described,and is still an active topic of discussion in the literature.22–32The prototypical example demonstrating the difference between direct HAT toPCET comes from the computational work of Mayer et al.,23 that describes the self-exchange reactions of benzyl-toluene and phenoxyl-phenol, shown in Scheme 1.2.These complexes are oriented so that the aromatic rings are anti relative to oneanother. In this geometry, the benzyl-toluene pair undergoes direct HAT, withthe 2p− pi orbital of the benzylic carbon radical oriented at the benzylic hydrogenon toluene. This is described as direct HAT, as the orbital containing the radicaloverlaps with the C H σ∗ anti-bonding orbital, and thus the transfer of the H atomoccurs through the same set of orbitals (see Scheme 1.2 A).For the phenoxyl-phenol pair (see Scheme 1.2 B), a fairly strongly hydrogenbonded pre-reaction complex is first formed with a predicted gas-phase bindingenthalpy of -8.1 kcal mol−1. As a result of this strong interaction, the TS structureis such that the oxygen 2p-orbital of the phenoxyl radical that nominally containsthe unpaired e− is perpendicular to the hydrogen bond. Therefore, in order to51.2. Details of HAT reactionsCHHCHHHCHHCHHHA.OOHOOHB.Scheme 1.2: Self-exchange reactions of A. the benzyl-toluene couple through directHAT, and B. the phenoxyl-phenol couple through PCET.undergo direct HAT, the hydrogen bond between the phenol OH and phenoxyl Olone-pair (LP) must break, and a new, weaker hydrogen bond with the nominally O-centred radical must form. Alternatively, the hydrogen bonded pre-reaction complexgeometry allows the orbital containing the radical to overlap with the 2p LP of thephenol moiety, and the conjugated aromatic pi-systems in the TS complex. Thisoverlap results in a TS complex with a singly occupied molecular orbital (SOMO)that is of pi-symmetry and highly delocalized. Accordingly, the proton is transferredthrough the hydrogen bond and the electron is transferred through the pi-system.This reaction has an enthalpic barrier height (∆H‡) of 5.0 kcal mol−1 relative to thehydrogen bonded complex, so that the barrier is 3.1 kcal mol−1 below the separatedreactants.The work by Mayer et al. 23 suggests that hydrogen bonding is a necessary,but not sufficient, condition for PCET to occur. This then implies that PCETis not possible between molecules that do not possess hydrogen bonding moieties,61.2. Details of HAT reactionssuch as carbon atoms. Work by other authors has shown this to be untrue.26,33 Inparticular, DiLabio and Johnson 26 demonstrated that Mayer et al. 23 neglected theimportant contributions of pi−pi interactions and LP-pi interactions. Computationalstudies revealed the existence of a TS structure for the benzyl-toluene couple thatis 3.7 kcal mol−1 lower in energy than previously reported. This structure hasthe aromatic rings in an optimal “parallel-displaced” or “stacked” conformation,as observed in the benzene-benzene non-covalently bound dimer.34 Analysis of theTS structure highest occupied molecular orbital (HOMO) reveals bonding characterbetween the two pi-systems, while the SOMO shows anti-bonding character betweenthe pi-systems, as well as both C H bonds. Thus, there exists a net partial bondinginteraction between the two pi-systems, opening up an additional electronic channelfor electron transfer to occur, while the overlap between the C H σ∗ orbital and thesingly occupied C p-orbital allows for proton transfer. DiLabio and Johnson alsosuggested that the phenol-phenoxyl couple also prefers a pi-stacked TS structure,and compared this to a structural analogue, a naturally occurring tyrosyl-tyrosinecouple in the RNR enzyme, in which the HAT reaction proceeds through a PCETmechanism. Other authors have confirmed the existence of a pi-stacked TS structurefor the phenol-phenoxyl couple.31,32,35Hammes-Schiffer has argued that molecular orbital based analysis is insufficientto describe the distinction between PCET and HAT mechanisms, as it is not alwaysconclusive.31,36 Instead, Hammes-Schiffer has proposed the use of quantitative diag-nostics that measure the non-adiabatic effects of a reaction, that is, the breakdownof the Born-Oppenheimer approximation.31,35 Under these criteria, electronicallynon-adiabatic and adiabatic proton-transfer occur for PCET and HAT mechanisms,respectively. This was demonstrated using the non-stacked TS structures for theself-exchange reactions of the phenol-phenoxyl and benzyl-toluene couples. To com-71.2. Details of HAT reactionsplicate the matter further, work by Inagaki et al. 37 and Mun˜oz-Rugeles et al. 32 hasdemonstrated that in the favoured stacked TS structures for the above self-exchangereactions, there are no non-adiabatic effects. This can be interpreted in two dif-ferent ways: Firstly, pi-pi-stacking changes the mechanism of the phenol-phenoxylself-exchange reaction from PCET to HAT, as suggested by Inagaki et al. 37 Alter-natively, Mun˜oz-Rugeles et al. 32 demonstrated that pi-pi-stacking orbital interactionscan “turn off” non-adiabatic effects, while the electron and proton are transferredthrough different sets of acceptor and donor orbitals, thus fitting within the def-inition of a PCET mechanism. While this highlights the active discussion of theHAT/PCET field, in this work I shall utilize the latter interpretation, which treatsMO overlap as a qualifier for the PCET mechanism.HAT can also occur through a PCET mechanism for species that have LP-pior LP-LP overlap in the TS complex.26,38 DiLabio and Johnson 26 showed that theformal HAT reaction between phenol and the t-butyl peroxyl radical exhibits orbitaloverlap between the non-radical O-lone pair of t-butyl peroxyl and the aromatic pi-system of phenol in the lowest energy cisoid TS complex. As with the benzyl-toluenecouple, there is both a bonding interaction in the HOMO of the TS structure, andan anti-bonding interaction in the SOMO of the TS structure. Therefore, thereexists a net partial bonding interaction that allows electron transfer through theLP-pi interaction and proton transfer through the hydrogen bond. DiLabio andIngold 38 also showed that iminoxyl-oxime self-exchange reactions occur through afive-centred PCET TS complex. The lowest energy transition states for iminoxyl-oxime couples are cisoid, such that the LPs of the nitrogen centres overlap, openinga channel for electron transfer, while proton transfer occurs between the two oxygencentres. Examples of cisoid and transoid TS structures are shown in Scheme 1.3.Bearing in mind there is not an obvious way to explore the related differences81.2. Details of HAT reactionsScheme 1.3: Possible cisoid (stacked) and transoid TS structures for A. the benzyl-toluene couple, B. the peroxyl-toluene couple, and C. the iminoxyl-oxime couple.91.3. Research goalsin mechanism experimentally, computational examination of formal HAT reactionsenables analysis of the mechanism of these reactions. Herein, I shall consider theexistence of these mechanisms on a continuum: Reactions where an unambiguouspi-pi stacked PCET TS structure is formed (for example the PhO + PhOH self-exchange reaction) have the most PCET character, and therefore have the lowestreaction barriers; while reactions where an unambiguous direct HAT TS structureis formed (for example the CH3 + CH4 self-exchange reaction) have the most HATcharacter, and therefore have the highest reaction barriers. In the middle are PCETreactions that may occur through either LP-pi (more PCET character) or LP-LPinteractions (less PCET character). The use of this analysis provides importantinsight into the electronic behaviour of these reactions. In this vein, the investigationof the physico-chemical nature of HAT reactions shall be the central theme of thisthesis.1.3 Research goalsConsider for a moment the potential energy surface (PES) for an arbitrary chem-ical reaction. Theoretical methods can be used to generate a full PES However, thisquickly becomes computationally unfeasible as the number of atoms in a systemincreases. Typically this problem can be simplified by examining only the relevantdegrees of freedom. Often, the two most important coordinates can be isolated,giving a three-dimensional PES. Furthermore, in chemistry we often simplify thisproblem to two dimensions using the so-called intrinsic reaction coordinate, whichis the lowest energy cross section of a higher dimension PES. This yields a reactioncoordinate diagram, as is illustrated below in Figure 1.1.In a typical reaction coordinate diagram, the reactants begin to interact and forma pre-reaction complex. Given sufficient energy, the reaction will proceed over the101.3. Research goalsFigure 1.1: A typical reaction coordinate diagram.top of the energy barrier through a TS complex. After the chemical transformationis completed, a post-reaction complex is formed and then the products separate.This is a somewhat simplified description, as it only broadly describes a chemicaltransformation. In particular, the roles of substrate-radical and substrate-radical-medium interactions along the reaction coordinate are not fully described. This isin fact a key point, as a thorough understanding of these interactions continues tobe lacking in the literature.Consequently, recent work from our group, in collaboration with our experi-mental colleagues at the University of Rome Tor Vergata, has focused on the im-portance of substrate-radical interactions in determining the kinetics of HAT re-actions. Specifically, it has been shown that the three-dimensional structures ofoxygen-centred radicals, as well as the organic substrates, impact the nature of theinteractions involved in HAT reaction pathways.39 In our work, we utilize primarilythe benzyloxyl (BnO ) and cumyloxyl (CumO ) radicals, which serve as proxies forbiological oxygen-centred radicals. This is primarily due to the fact that reactionsinvolving BnO and CumO are relatively long lived in solution, and can be mon-111.3. Research goalsitored using time-resolved laser flash photolysis (LFP) techniques. These radicalsare somewhat different than biologically relevant radicals such as HO , and as aresult, the reactivity trends pertaining to the substrates can be somewhat maskedby the properties of the radical, such as steric bulk,40 or non-covalent binding.41Nonetheless, through a careful combination of theoretical and experimental tech-niques, reactions involving BnO and CumO with a variety of organic substrateshave been used to develop a great deal of insight with respect to the role of structurein both the radicals and substrates, and resulting intermolecular interactions.With respect to the work in this thesis, in Chapter 3 the importance of the left-hand side of Figure 1.1 shall be examined by studying how the pre-reaction compleximpacts HAT reactions. There has been limited investigation of the importance ofpre-reaction complex formation for HAT reaction.42 This is problematic, as oxygen-centred radicals can hydrogen bond with substrates as both acceptors and donors.43These hydrogen bonding interactions, in addition to the other non-covalent inter-actions between the radical and substrate, lead to the formation of a pre-reactioncomplex. Accordingly, the formation of a pre-reaction complex is a fundamentalstep in the model systems that have been used to study HAT.The specific aim of Chapter 3 is to investigate the effects of non-covalent bindingin the pre-reaction complex, with respect to the well-known, but phenomenological,Arrhenius equation. As of yet, there is no framework that relates the non-covalentlybound pre-reaction complex to kinetic results. I ask the simple question: Does thereexist a direct correlation between the Arrhenius pre-factor and the non-covalentbinding that occurs in the pre-reaction complex formed for HAT reactions? To ad-dress this question, I examine the non-covalent binding in the pre-reaction complexin a series of related HAT reactions. Arrhenius parameters for the systems of inter-est in this work were previously tabulated,38 and consist of thermoneutral or nearly121.3. Research goalsthermoneutral reactions involving the formation and destruction of oxygen-centredradicals. These reactions are related to the phenol-phenoxyl self-exchange reaction,where a relatively strong pre-reaction complex is expected.Then in Chapter 4, the right-hand side of Figure 1.1 is considered, where theeffects of bond dissociation enthalpies (BDEs) on HAT rate constants are examined.BDEs are central to the understanding of reactions with respect to thermodynamics.In addition to this, there exists a tremendous amount of literature in which BDEs arelinked to chemical reactivity, especially for HAT reactions.19,25,44–46 There is a linearfree energy relationship (LFER) called the Bell-Evans-Polanyi (BEP) principle,47,48which states that the difference in activation energy (Ea) for two related reactionsis proportional to the differences in reaction enthalpy (∆H):Ea = E0 + α∆H (1.1)where E0 is the activation energy of a reference reaction, and α, a constant thatcharacterizes the position of the TS along the reaction coordinate. This relation-ship has been generally used to compare larger families of reactions. Despite thewidespread use of the BEP principle, the validity of this relationship is not welldescribed.I probe the generality of the BEP principle for a series of HAT reactions fromC H bonds, with the aim to determine how generally it may be applied. Thisis achieved by relating accurate, theoretically determined C H BDEs for speciesthat undergo abstraction at the appropriate C H position, to the experimentallydetermined HAT rate constants. HAT reaction rate constants depends on manyfactors. However, by using rate constants determined under specific conditions (LFPwith CumO in acetonitrile at 298K), the differences in reactivity depend mainly onthe differences in chemical properties of the substrates of interest. Therefore, if the131.3. Research goalsBEP relation is valid, there should exist two relationships for C H bonds: one inwhich the incipient radical is delocalized into a pi-system (benzylic-allylic), and theother in which the remaining alkyl radicals are largely localized.Finally, recent experimental results show that non-redox active metal cations,which are found ubiquitously in biological systems, have an inhibitory effect on HATreactions involving oxygen-centred radicals. This has been demonstrated experi-mentally for substrates that undergo abstraction from sites adjacent to heteroatoms(e.g. amines, amides, and ethers). Under various stoichiometric ratios, these metalcations have effects ranging from full inhibition to partial deactivation of HAT reac-tivity.49–51 This effect has been attributed partially to the effects of hyperconjugativeoverlap. Take for example tetrahydrofuran (THF), shown in Scheme 1.4. In the ab-sence of other species, there exists C H bond weakening hyperconjugative overlap ofelectron density from one of the oxygen LPs and the adjacent C H σ∗ anti-bondingorbitals. The interaction of a metal cation with the oxygen LPs removes electrondensity from the C H σ∗, thus increasing the C H bond strength. As a result, thereactivity of this bond is decreased, as observed from the experimentally-measured3.2-fold decrease in the rate constant for HAT with CumO in acetonitrile from 5.8×106 M−1s−1 to 1.8 ×106 M−1s−1 in the presence of 1.0 M Mg(ClO4)2.49The nature of the interactions between non-redox active metal cations and or-ganic substrates is poorly understood. This problem is explored in Chapter 5, withthe aim to understand the fundamental physico-chemical properties that lead to theobserved trends in reactivity. The experimentally observed effects have led us to hy-pothesize that the presence of non-redox active metal cations has a chemo-protectiveeffect against the radical-induced oxidation of biomaterials such as proteins.In using theory to study HAT reactions, I hope to contribute to a better un-derstanding of the fundamental properties that govern these reactions, and thus141.3. Research goalsOHHHHσ∗A.OHHHHCumOMn+B.Scheme 1.4: A. Hyperconjugative overlap in tetrahydrofuran. B. The non-redoxactive metal cation accepts electron density from the heteroatom lone pair, reduc-ing overlap with the C H σ∗ anti-bonding orbital, and increasing the C H bondstrength, thus destabilizing the TS complex.151.3. Research goalsdevelop insights into the many important biological processes in which HAT takesplace.16Chapter 2Theory2.1 The quantum mechanical approachThe fundamental properties governing all of chemistry are dictated by the quan-tum mechanical wave functions, Ψ. Therefore, in quantum chemistry we seek solu-tions to the non-relativistic time-independent Schro¨dinger equationH |Ψ〉 = E |Ψ〉 (2.1)where H is the Hamiltonian operator for a system of nuclei and electrons, and Ψis the wave function, defined as the set of eigenvectors with energy eigenvalues E.52For a system with N electrons and M nuclei, the full Hamiltonian in atomic unitsisH =−N∑i=112∇2i −M∑A=112MA∇2A −M∑i=1M∑A=1ZAriA+N∑i=1N∑j>i1rij+M∑A=1M∑B>AZAZBRAB(2.2)In this equation, ZA is the atomic number of nucleus A with a mass MA divided bythe mass of an electron. The Laplacian operators ∇2i and ∇2A represent differentia-tion with respect to the coordinates of the ith electron and Ath nucleus. The firstand second terms are the kinetic energies of the electrons and nuclei, respectively.172.1. The quantum mechanical approachThe third term represents the Coulomb attraction between electrons and nuclei withdistance riA. The fourth and fifth terms represent repulsion between two electronswith distance rij , and between two nuclei with distance RAB, respectively.Nuclei move slowly relative to electrons, due to their much greater mass. Thisis the central pillar of the Born-Oppenheimer approximation that is nearly alwaysapplied in molecular electronic structure calculations. The application of this ap-proximation allows for the simplification of Equation 2.2: using a separation ofelectronic and nuclear variables, the second term for nuclear kinetic energy is solvedseparately. Also, the last term of nuclear repulsion is constant, and thus is generallyignored. This leaves us with the electronic HamiltonianHelec = −N∑i=112∇2i −M∑i=1M∑A=1ZAriA+N∑i=1N∑j>i1rij(2.3)Unfortunately, it is only possible to exactly solve the Schro¨dinger equation for thefull electronic Hamiltonian Helec in the simplest of cases: when there is only oneelectron (H, H +2 , He+, Li2+, etc). Note that since we will always work within theBorn-Oppenheimer approximation, the subscript elec is usually dropped. In order toproceed to systems with multiple electrons, we must make further approximations.2.1.1 Spin and Spatial OrbitalsWe will refer to the wave function of a single particle as an orbital. Naturallythen, as we will deal with electrons in molecules, we shall refer to their wave functionsas molecular orbitals (MOs). To fully describe electrons we must consider a spatialand spin component to the overall wave function. A spatial orbital ψi(r), is afunction of the position vector r, and describes the distribution of an electron in allspace. It is usually assumed that spatial MOs form an orthonormal set such that182.1. The quantum mechanical approach〈ψi(r)|ψj(r)〉 =∫drψ∗i (r)ψj(r) = δij (2.4)where the left-hand side is standard Dirac bra-ket notation representing the sameintegral in the middle. The right-hand side of Equation 2.4 is the standard Kroneckerdelta.The spin of an electron is represented by two orthonormal functions α(ω) andβ(ω), or spin-up and spin-down. If a wave function describes both the spatialdistribution and spin of an electron it is a spin orbital, χi(r), where x representsboth the spatial distribution and spin coordination of an electron (x = {r, ω}).Since ψi(r) and α(ω)/β(ω) are orthonormal, so too is χi(x)〈χi(x)|χj(x)〉 = δij (2.5)The first steps towards describing an N electron wave function come from thework in the late 1920s by Hartree. The early Hartree method took an approach inwhich the wave function of N non-interacting electrons (ΨHP ) is described by theproduct of N spin orbitals, known as a Hartree product :ΨHP (x1,x2, . . . ,xN ) = χi(x1)χj(x2) . . . χk(xN ) (2.6)In such a system the Hamiltonian has the form of a sum of N independent operatorsH =N∑i=1hˆ(i) (2.7)where hˆ(i) ishˆ(i) = −12∇2i + V (ri) (2.8)192.1. The quantum mechanical approachsuch that the first term describes an electron’s kinetic energy, and the second termdescribes potential felt by a single electron. If we consider the case that ignoreselectron-electron repulsion, then V describes only the nuclear-electron attraction.Alternatively, the electron-electron repulsion may be included as an average poten-tial.Solutions to the Schro¨dinger equation for this system of non-interacting electronsare facile to obtain as each h(i) depends only on the variables of χi(xi), so thatH∣∣ΨHP 〉 = E ∣∣ΨHP 〉 (2.9)gives an eigenvalue energy solution E that is the sum of N spin orbital energies εiE = ε1 + ε2 + · · ·+ εN (2.10)While this theory does allow one to calculate energies for an N electron system,it has a basic deficiency: the antisymmetry principle of wave functions is not obeyed.The antisymmetry principle states that the electronic wave function must changesign (be antisymmetric) with respect to the exchange of spacial and spin coordinateof any two electrons. Hartree accounted for this by nominally applying the Pauliexclusion principle. However, this description is still incomplete in the sense that itdoes not describe the statistical nature of quantum particles.2.1.2 Slater determinantsIn order to satisfy the antisymmetry principle, a linear combination of Hartreeproducts can be taken. Although the method was first utilized independently byHeisenberg53 and Dirac,54 this method is called a Slater determinant after Slater.55For an N electron system, a Slater determinant is written as202.1. The quantum mechanical approachΨ(x1, . . . ,xN) =1√N !∣∣∣∣∣∣∣∣∣∣∣∣∣χi(x1) χj(x1) · · · χk(x1)χi(x2) χj(x2) · · · χk(x2)....... . ....χi(xN ) χj(xN ) · · · χk(xN )∣∣∣∣∣∣∣∣∣∣∣∣∣(2.11)where 1/√(N !) is a normalization factor. This simple mathematical trick ensuresantisymmetry since the interchange of two electrons requires the exchange of tworows in the determinant, which changes the sign. Normally the short-hand form,which implicitly includes the normalization factor and assumes the ordering of elec-trons is x1,x2, . . . ,xN , is written as only the diagonal elements of the determinant:Ψ(x1,x2, . . . ,xN ) = |χiχj . . . χk〉 (2.12)Slater determinants are completely dependent on the spin orbitals from which itis formed, to within a sign. Therefore, Slater determinants also form an orthonormalset. Additionally, the introduction of antisymmetry into the Hartree product incor-porates so-called exchange correlation. This means that the motion of two electronswith parallel spin are correlated because each spin-orbital can only contain one elec-tron. However, since the motion of electrons with opposite spin are not correlated,a single determinant wave function is said to be uncorrelated.2.1.3 The Hartree-Fock approximationNow that we have a method for describing many-electron wave functions, wecan consider the computation of molecular properties. The cornerstone of quantumchemistry is the Hartree-Fock method (HF), otherwise known as the self-consistentfield method. The main principle of the HF method is to approximate electron-electron interactions with an average potential. We begin with a single Slater de-212.1. The quantum mechanical approachterminant for an N electron system in the ground state:|Ψ0〉 = |χ1χ2 . . . χN 〉 (2.13)By applying the variational method to the Schro¨dinger equation, we hope to findthe lowest possible ground state energy, E0. One applies the variational principle bychoosing a trial wave function (φ) dependent on some number of parameters. Theseparameters are optimized so that that expectation value of the energy is minimized:E0 ≤ 〈φ|H |φ〉 (2.14)The trial wave function minimizes E0 only when φ = Ψ0, the ground state wavefunction.Within the Hartree-Fock approximation, we approximate the full electronic Hamil-tonian H with a related operator Hˆ0:Hˆ0 =N∑i=1fˆ(i) (2.15)where fˆ(i) is the Fock operator of the i− th electron, defined asfˆ(i) = −12∇2i +M∑A=1ZAriA+ vHF (i) (2.16)The first two terms are the non-interacting one electron Hamiltonian, hˆ(i). Thethird term, vHF (i), is the average potential experienced by each electron in thepresence of other electrons.With these approximations, the quantum problem is now reduced to solving theeigenvalue Hartree-Fock equation of the form222.1. The quantum mechanical approachfˆ(i)χ(xi) = iχ(xi) (2.17)Solving Equation 2.17 directly is computationally very challenging, as there areinfinite possible solutions. However in 1951, Roothaan 56 demonstrated that theproblem can be simplified by expanding each spin orbital into a linear combinationof a known finite number K basis functions:χi =K∑µ=1Cµ,iφµ (2.18)where Cµ,i is a weighting coefficient and φµ is a basis function. As K approaches∞, the set {φµ} becomes more complete and the energy approaches the so-calledHartree-Fock limit, or the exact energy in the Hartree-Fock approximation. One is,however, always limited to a finite number of basis functions, leaving deficiencies inthe desired wave function Ψ0.The expansion of spin orbitals into a basis allows Equation 2.17 to be written interms of the Roothaan matrix equationFC = SCε (2.19)where F =∑l,m 〈χl| fˆ(i) |χm〉 is the Fock matrix, S =∑l,m = 〈χl|χm〉 is theorbital overlap matrix. C is the orbital coefficient matrix, and ε is the diagonalmatrix of orbital energies εi, which are generally the desire solutions. By performinga transformation of basis to an orthonormal basis, the overlap matrix S becomesthe identity matrix 1, and simplifies the problem. Thus, utilizing Equation 2.19reduces the problem to the of diagonalization F. Unfortunately, this must be doneiteratively, as F depends on its own solution, hence the name self-consistent fieldmethod.232.1. The quantum mechanical approach2.1.4 Basis setsChoosing optimal basis functions can help significantly in terms of determiningthe ground state wave function Ψ0. Quantum chemists rely on the choice of basissets, defined as the vector space in which an ab initio problem is defined. Basissets usually refer to the set of one particle functions, which are used to form MOsin a linear combination of atomic orbitals (LCAO-MO) like approach. For a sys-tem with N electrons, the LCAO-MO approach gives N/2 occupied orbitals in theground state. The remaining basis functions in a set are combined to give virtual(unoccupied) orbitals.Early basis sets were composed of Slater-type orbitals (STOs), due to their re-semblance to the atom orbitals (AOs) of the hydrogen atom. These are functions ofthe formφSTOi (ζ, n, a, b, c, x, y, z) = Nrn−1e−ζrxaybzc (2.20)where N is a normalization constant, ζ is a constant related to the effective nuclearcharge of the nucleus, r is the distance of the electron from the atomic nucleus, nis a natural number that plays the role of the principal quantum number, and x,y, and z are Cartesian coordinates. The angular component xaybzc describes theshape of the function, such that if a+b+c=0 φSTOi is of s-type; if a+b+c = 1, φSTOiis of p-type, and so forth. Although STOs approximate the long and short rangebehaviour of atomic orbitals correctly, performing integration with these functionsis computationally very demanding, due primarily to the complexity of the integralsinvolved describing in electron-electron interactions.In order to make the integration over orbitals computationally efficient, modernbasis sets are almost exclusively composed of Gaussian-type orbitals (GTOs), which242.1. The quantum mechanical approachcan generally be represented asφGTOi (α, a, b, c, x, y, z) = Ne−αr2xaybzc (2.21)where N is a normalization constant, α is the orbital exponent coefficient, x, y, and zare Cartesian coordinates, r is the radius (r2 = x2+y2+z2), and the angular portionis described the same as in an STO. It takes a linear combination of several GTOsto represent the same function as an STO. These linear combinations of GTOS areknown as contracted GTOs (CGTO) with n GTOs combined asφCTGOi (α, a, b, c, x, y, z) = Nn∑i=1cie−αr2xaybzc (2.22)where ci is referred to as the contraction coefficient that describes the weighting ofeach GTO. Although it requires more GTOs than STOs to accurately describe theatomic orbitals, the integrals can be computed 4–5 times faster, and thus calculationsinvolving GTOs are much more efficient.57Basis set nomenclatureStandard basis sets are composed of basis functions used to represent atomicorbitals where each basis function is a CGTO composed of several GTOs. A minimalbasis set is one in which each AO is represented by a single basis function. To moreaccurately represent AOs, more basis functions should be used, although basis setsize needs to be balanced with computational cost. Larger basis sets are referredto by their cardinal number, the number of basis functions that represent each AO.When two basis functions are used to represent each AO, this is called a double-zetabasis set. If three basis functions represent each AO, this is called a triple-zeta basisset. Generalized, a basis set is N -zeta in size when N basis functions are used per252.1. The quantum mechanical approachAO.A split-valence basis set is one in which a single basis function is used to representeach core AO, while more basis functions are used to represent the valence AOs.Constructing basis sets in this way can help reduce the computational cost whilestill accurately representing the electrons that are most important to chemistry.Additional basis functions are often added to basis sets in order to correctlydescribe molecular properties. Polarization functions are basis functions that areone or more angular momentum channels greater than the natural electronic config-uration of an atom. For example, a single p-type basis function can be added to theminimal basis of a hydrogen atom. Polarization functions are essential to accuratelydescribe chemical bonding, as the presence of other atoms distorts the spherical sym-metry of a single atom’s AOs.58 Diffuse functions are basis functions that extendfurther into space, typically by the inclusion of a very shallow Gaussian function(small ζ exponent). Diffuse functions are necessary to accurately describe anions,very electronegative atoms, and large systems in which non-covalent interactions(NCIs) are important.Commonly used basis setsA large number of basis sets currently exist in the literature.59 While not allbasis sets are created equally, we shall briefly consider four of the most commonlyused basis sets used in quantum chemistry.Pople-style basis setsPerhaps the most utilized basis sets in chemistry are those arising from the groupof Pople.60–62 These basis sets were defined by minimizing the HF energy. Theearliest of these basis sets are the minimal STO-NG basis sets, where N describesthe number of GTOs that go into each contraction.262.1. The quantum mechanical approachThe practise of using minimal basis sets has diminished significantly as tech-nology has advanced, thus these basis sets are largely considered out of date. It ismore common to utilize the split-valence basis sets, denoted as n− ijG or n− ijkGfor double and triple zeta split-valence basis sets, respectively. In this system ofnotation, n represents the number of GTOs that describe the core AOs, and i, j, kdescribe the number of GTOs for contractions in the valence AOs. Polarizationfunctions are denoted either with asterisks or with the specific shell and number offunctions that are being added. Diffuse functions are denoted with either a singleor double “+”, indicating diffuse s and p-type functions for non-hydrogen atoms,and the addition of diffuse s-type functions for hydrogen, respectively. For example,the 6–31+G(d,p) ≡ 6–31+G** double-zeta basis set for C and H is one that has: 6GTOs per core AO, 3 GTOs for the first valence set of AOs, and 1 GTO for the sec-ond, along with s and p diffuse functions of the heavy atoms, a single d polarizationfunction of heavy atoms, and a single p polarization function of hydrogen atoms.Correlation consistent basis setsPost-Hartree-Fock methods (vide infra) are commonly used in quantum chem-istry. In 1989, Dunning63–65 identified that the use of basis sets optimized forHF were inappropriate for post-HF methods. The basis sets that came from Dun-ning and co-workers, which are referred to as “correlation consistent” basis setsare commonly used in, but not limited to, state of the art wave function calcula-tions. These basis sets are said to be correlation consistent as they treat electroncorrelation (vide infra) in a manner that systematically approached the completebasis set limit. Correlation consistent basis sets are denoted as “cc-VNZ”, whereN=2(D),3(T),4(Q),5,6,. . . is the cardinal number of the basis set. These are largesets containing polarization functions by default and can be additionally augmentedwith diffuse functions, denoted by the prefix “aug.” A variant of these basis sets272.1. The quantum mechanical approachthat are used for including core-correlation are denoted by “cc-pCVNZ”.66 A com-monly used basis set is aug-cc-pVTZ, which is a triple-zeta basis set with implicitpolarization functions and specified diffuse functions on all atoms.Polarization consistent basis setsThe polarization consistent basis sets have been developed by Jensen and cowork-ers.67–70 The polarization consistent basis sets have been developed to systematicallycomplete basis set limit in density-functional theory calculations through the use ofhigher order polarization functions. The notation adopted is “pc-X”, where X is thecardinal number of the basis set minus one (i.e. X = N -1). Polarization functionsare included by default in these basis sets, and additional diffuse functions can bespecified with the same “aug” prefix used for the correlation consistent basis sets.Ahlrich basis setsThe last basis sets we will mention are those developed by Ahlrich and cowork-ers.71,72 These are the “Def2” basis sets, named as such because they are the secondgeneration of default basis set in the Turbomole quantum chemistry package.73 Ad-ditionally, these basis sets have been developed so that consistent errors are obtainedfor nearly every element on the periodic table: a unique trait among modern basissets. The nomenclature for these basis sets is fairly straightforward where either SVis used for split valence, or NZ is used for cardinal number. Addition of polarizationand diffuse functions is specified with a P and D, respectively. For example, Def2-SVP is the basis set of split-valence double-zeta quality with polarization functions;Def2-TZVP is the triple-zeta basis set with polarization functions; Def2-QZVPD isthe quadruple-zeta basis set with polarization and diffuse functions.282.1. The quantum mechanical approach2.1.5 Post-Hartree-Fock methodsThe HF method gives an approximation to the ground state wave function ofa molecule for a reasonable computational cost (scaling with N4 number of basisfunction). There is however, a lack of the complete description of dynamical elec-tron correlation,74 and thus significant deviations from experimental results can beobserved. Dynamical electron correlation is a measure of how much one electron’smovement is affected by the motion of other electrons. As described previously, theHF method includes correlation through the average electron field potential term,however this field is in general, not static, thus correlation must be treated directlyin order to obtain accurate results. The majority of methods take the HF wavefunction Ψ0 as the starting point. The correlation energy is defined as the differencebetween the exact energy and the HF energy:Ecorr = Eexact − E0 (2.23)Ecorr is the difference between the full non-relativistic energy from the Schro¨dingerequation, Eexact, and a reference ground state energy E0, usually the HF energy.We shall briefly consider two important methods for accounting for electroncorrelation and obtaining Ecorr: Møller-Plesset perturbation theory, and the relatedconfiguration interaction and coupled cluster theories.Møller-Plesset perturbation theoryMøller-Plesset (MP) perturbation theory is a special case of Rayleigh-Schro¨dingerperturbation theory in which the Hamiltonian for a system can be approximated byHˆ = Hˆ0 + λVˆ (2.24)292.1. The quantum mechanical approachwhere Hˆ0 is an unperturbed Hamiltonian, Vˆ is a small perturbation, and λ is anarbitrary parameter that controls the size of the perturbation. The perturbed wavefunction and energy are expressed as a power series in λ:Ψ = limm→∞m∑i=0λiΨ(i) (2.25)E = limm→∞m∑i=0λiE(i) (2.26)The MP method applies perturbations to HF by defining a shifted Fock operatorHˆ0 and correlation potential Vˆ asHˆ0 = Fˆ + 〈φ0| (Hˆ − Fˆ ) |φ0〉 (2.27)Vˆ = Hˆ − Hˆ0 (2.28)where φ0 is the ground state Slater determinant of the Fock operator.Within this formulation, the zeroth-order energy is the expectation of Hˆ, whichgives the HF energy. The first-order energy isEMP1 = 〈φ0| Vˆ |φ0〉 = 0 (2.29)by Brillouin’s Theorem of singly excited determinants. Thus, the first useful correc-tion occurs at the second order of perturbation, which is known as MP2. Additionalorders of perturbation are referred to as MP3, MP4, etc. The MP2 method has beenpopular in quantum chemistry because it scales with N5 number of basis functionsand is a significant improvement on the treatment of electron correlation comparedto HF. One may expect higher order of perturbation theory to more accurately de-302.1. The quantum mechanical approachscribe a system. Practically however, the expansions used in MPN theory do notin all cases converge smoothly to a limit with higher order of perturbation.75 As aresult, for molecular properties calculated with MP3 or higher are not guaranteedto give more accurate results than MP2.Configuration interaction and coupled cluster theoryThe solutions to the HF method give a single determinant wave function thatonly describes the ground state electronic configuration. Configuration interaction(CI) is a post-HF method that describes a linear combination of Slater determinantsto more accurately represent a system’s wave function. The additional Slater de-terminants represent excited electronic configurations and can be singly excited (S),doubly excited (D), and so forth. This is represented as follows:|Ψ〉 = (1 +N∑j=1Cj) |φ0〉 (2.30)where Cj are operators that describes the j-th excitations of electrons. If all possibleexcitations are included in the CI equation, this is referred to as full CI (FCI).Extending FCI to an infinite basis set gives the exact solution to the Schro¨dingerequation.Coupled cluster (CC) theory76 is a similar approach to CI, but uses the so-calledexponential ansatz|Ψ〉 = eTˆ |φ0〉 (2.31)where Tˆ is the cluster operator, defined by n-electron excitation operators Tˆn:Tˆ = Tˆ1 + Tˆ2 + Tˆ3 + . . . (2.32)312.1. The quantum mechanical approachWithin the exponential ansatz, eTˆ is usually truncated and expanded in a Taylorseries. For example, truncation at the Tˆ2 excitation operator gives|Ψ〉 = eTˆ1+Tˆ2 |φ0〉= (1 + Tˆ1 + Tˆ2 +12!Tˆ 21 + Tˆ1Tˆ2 +12!Tˆ 22 + . . . ) |φ0〉(2.33)Considering both CI and CC with single and double excitation (CISD and CCSD),the wave functions will include similar excitations, however inclusion of cross terms(Tˆ1Tˆ2) in CCSD implicitly includes higher excitation levels. Additionally, the useof the exponential operator makes the CC formulation size consistent, which is thelargest short coming of the CI method. Size consistency refers to the additivity ofenergies for an ensemble of molecules. That is, for a pair of molecules A and B,their energies must follow the relationEAB(r →∞) = EA + EB (2.34)Size consistency is a necessary requirement of a theoretical treatment to treat sys-tems of molecules accurately. It is for this reason that CC has superseded CI as thedominant highly correlated method in quantum chemistry.The inclusion of higher order excitations become less important with increasingorders of excitation; however, the inclusion of triples is often found to be necessaryfor the accurate description of electron correlation (i.e. CCSDT). The computationof triples is prohibitively expensive in all but the simplest of systems, thus approxi-mations based on perturbation theory are often used instead. The most commonlyused perturbative triples method is CCSD(T), where the parenthesis indicate theuse of perturbative arguments. Note also that traditionally, the use of CCSD(T)implies excitation of only the valence electrons, unless otherwise stated.322.1. The quantum mechanical approachCCSD(T) is commonly referred to as the gold standard in quantum chemistryand is often used to obtain benchmark quality results for thermochemistry andNCIs.77 However, CCSD(T) scales with N7 number of basis functions, and is thussignificantly more computationally expensive than HF or MP2, restricting its appli-cation to small systems of molecules. Quadratic configuration interaction (QCI) isclosely related to CC, except that is uses quadratic operators in place of exponentialones. QCISD(T) and CCSD(T) gives very similar results.782.1.6 The complete basis set limitComplete basis set extrapolationIn accordance with the variational principle, the energy obtained by a partic-ular method will always be greater than or equal to the exact energy. The ex-act energy can only be achieved with an infinite basis set, a value known as thecomplete basis set (CBS) limit.79 Since this is computationally infeasible, specificapproaches have been developed to approximate the CBS limit. Specifically, molec-ular properties calculated using the HF and post-HF methods have been shown toasymptotically approach the CBS limit in a smooth manner when appropriate basissets are used. Therefore, to obtain results estimating a molecular property at theCBS limit (Y (∞)), properties can be fit to three-parameter80,81 or two-parameterfunctions:82,83Y (x) = Y (∞) +Ae−x/B (2.35)Y (x) = Y (∞) +A/x3 (2.36)where the molecular property as a function of basis set cardinal number Y (x) is332.1. The quantum mechanical approachfit using parameters A and B. Typically calculations of this nature are performedusing the correlation consistent basis sets (cc-pVNZ), however there is evidencethat for DFT based calculations, the polarization consistent basis sets (pc-X) morerapidly approach the CBS limit for some molecular properties.84 The true goldstandard in quantum chemistry is referred to as CCSD(T)/CBS, which typicallymeans CCSD(T) with complete basis set extrapolation with aug-cc-pVNZ basissets, where N=D, T, Q, or 5. Although extrapolation is useful for approximatinghighly accurate results, there is an inherent amount of uncertainty associated withthe final fitted results, which may be unclear from the nomenclature.Explicit correlation methodsA new technique that is gaining popularity among post-HF methods is the in-clusion of so called explicit correlation.85,86 The introduction of additional functionsdependent on inter-electronic distance coordinates allows for explicit correlation ofelectrons.87 As a result the dynamical correlation of electrons is treated more ac-curately with reduced basis sets, therefore accurate results can be achieved at areduced computational cost. Basis set extrapolation can also be performed on ex-plicitly correlated results: this is quickly becoming the standard approach.882.1.7 Composite quantum chemistry methodsIn order to calculate thermochemical and kinetic properties that are within asub-kcal mol−1 range of experiment, multistep ab initio procedures that are re-ferred to as composite methods have been developed.89 These procedures work byincluding important energy terms that contribute to molecular properties. Gener-ally, composite methods make use of a combination of low correlation methods withlarge basis sets and high correlation methods with small basis sets, as is illustrated342.1. The quantum mechanical approachin Figure 2.1. Some of the relevant energy terms include: core-valence, relativis-tic, spin-orbital, Born-Oppenheimer, and zero-point vibrational energy corrections.There exist many composite methods, each of which makes use of a variety of quan-tum mechanical (QM) methods and different basis set extrapolation techniques inorder to best approximate energy terms that are relevant, with the ultimate goalof achieving the exact energy of a system. In this work, I have made use of sev-eral composite methods including: the G4 and G4(MP2) methods,90,91 CBS-QB3and CBS-APNO methods,92–94 and the W1BD method.95 A description of thesemethods will be provided in Chapter 4.Basis SetCorrelationMethod 1Method 2Composite MethodFigure 2.1: Schematic representation of a quantum mechanical composite method.The exact energy can only be achieved at the limits of an infinite basis set and com-plete correlation. Using a combination of Method 1 (low correlation/large basis set)and Method 2 (high correlation/small basis set), the composite method approachesthe exact energy.352.1. The quantum mechanical approach2.1.8 Density-functional theoryDensity-functional theory (DFT) is the most popular quantum chemical methodapplied to date. It relies on the two Hohenberg-Kohn theorems, the first of whichstates that there exists a unique electron density ρ that defines the properties ofa many-electron system. The second theorem defines an energy functional of theelectron density and demonstrates that the correct ground state electron densityminimizes the energy functional through the variational theorem.96,97 These theo-rems alone do not provide the solutions to the Schro¨dinger equation.It wasn’t until the formulation of Kohn-Sham DFT98 that the theory begangaining ground as a useful quantum theory. Kohn-Sham DFT scales formally withN3 number of electrons74 which is better than HF by a factor of N . In addition,DFT is a complete theory like FCI; however, there is no straightforward way todetermine the correct functionals of the electron density as the exact form of thefunctionals is unknown. Nonetheless, the drive for the development of the correctdensity-functional has been one of the main endeavours in quantum chemistry inthe last two decades.The framework behind conventional DFT is built into the description of the fullenergy functional E:E[ρ] = Tni[ρ] + Vne[ρ] + Vee[ρ] + ∆T [ρ] + ∆Vee[ρ] (2.37)where Tni is the kinetic energy of non-interacting electrons, Vne is the potentialof nuclear-electron interactions, and Vee is the classical electron-electron repulsion.The last two terms are collectively referred to as the exchange-correlation (XC)functionals, where ∆T is the dynamical correlation term, and ∆Vee is the non-classical correction to electron-electron repulsion. All the functionals, except the362.1. The quantum mechanical approachXC functionals have an exact form. It is therefore the XC functionals in whichthere is currently empiricism.The ultimate goal in describing XC functionals is to find the “correct” XC func-tional that gives the exact energy of a system from the electron denisty. At thispoint, this must be done using approximations, for which there are several degreesof complexity. These approaches follow a hierarchical scheme, commonly referred toas the “Jacob’s ladder” of DFT.99 The first rung represents the simplest approxima-tion that is known as local-density approximations (LDAs), which approximate theexchange-correlation density at a given point by the electron density at that samepoint. The form of these functionals is:ELDAXC =∫ρ(r)εXC(ρ(r))dr (2.38)where εXC(ρ(r)) is the exchange-correlation energy per particle (energy density) ofa uniform electron gas of density ρ(r). This approximation is overly simple andapplies only when the electron density if constant at all points, and are thus notgenerally applied in chemical problems. Nonetheless, LDA based approaches arecommonly employed in solid state physics.The second rung on the ladder corresponds to generalized-gradient approxima-tion (GGA) based functionals, which are still amongst some of the most populardensity-functionals. GGAs depend on both the electron density and the gradient ofthe electron density at a point:EGGAXC =∫ρ(r)εGGAXC (ρ(r),∇ρ(r))dr (2.39)where, ε(ρ(r),∇ρ(r))dr is the energy density associated with a given GGA. GGAfunctionals provide a substantial improvement over LDAs, and most are constructed372.1. The quantum mechanical approachso that they correct the LDA energy:εGGAXC (ρ(r),∇ρ(r)) = εLDAXC (ρ(r))dr + ∆εXC( ∇ρ(r)ρ4/3(r))(2.40)A step above GGAs on the third rung of the ladder are meta-GGAs, whichdepend on the electron density, as well as the first derivative of electron density ata point, and the kinetic-energy density, τ(r), defined as:τ(r) =occupied∑i12|∇χi(r)|2 (2.41)where χi(r) are the self-consistently determined Kohn-Sham orbitals. Meta-GGAsimprove upon the accuracy of GGAs at a comparable cost.74The XC functionals described up to this point (LDAs, GGAs, meta-GGAs) de-pend only on the electron density and derivatives of the electron density. The fourthand fifth rungs of the ladder improve upon the prior functionals by inclusion of termsdependent on additional properties. While these approaches improve upon the ac-curacy of these functionals, they comes with an increase in computational cost. Onthe fourth rung sit functionals that depend to some percentage on the HF exact ex-change. When the ratio of HF exchange is fixed, these functionals are termed hybridfunctionals. Alternatively, functionals are said to be range-separated corrected if adifferent amount of exact-exchange to describe long and short-range behaviours. Inthe cases of hybrid and range-separated functionals, the added computational costcomes from the calculation of the HF exact exchange (N4).Alternatively, one can describe the fourth rung functionals as the dependingupon the properties of the occupied molecular orbitals. The fifth rung then, is saidto depend on the properties of unoccupied molecular orbitals. These functionals aretypically referred to as double-hybrids, and incorporated correlation energy from382.1. The quantum mechanical approacha post-HF method, typically MP2.100 Double-hybrid DFT methods are once againmore accurate than the lower rung methods, however, calculating the MP2 correla-tion energy is considerably more computationally demanding than traditional DFTapproaches. Therefore, double-hybrid DFT methods have not gained popularity inthe literature.There are many published XC functionals. Fortunately, there is a fairly stan-dard system of nomenclature, such that density functionals are described as exchangefunctional -correlation functional. The most commonly used density functional is theB3-LYP, which uses the 3-parameter hybrid exchange functional of Becke,101 andthe correlation functional of Lee, Yang, and Parr.102 There are also standalone func-tionals that have built in exchange and correlation functionals. A common exampleof these are the Minnesota family of functionals from the Truhlar group.103,104Aside from the problem of choosing density-functionals, solving DFT is compu-tationally very similar to the HF method. Within Kohn-Sham (KS) DFT, we definea fictitious system of non-interacting electrons with the same electron density asthe real system. This is achieved by the use of a Hamiltonian in which there is aneffective local potential, Vs(r):Hˆs = −12N∑i∇2i |N∑iVS(ri) (2.42)The ground state wave function of this non-interacting Hamiltonian is representedby a single Slater determinant with spin orbitals (χ), completely analogous to theHF problem. These spin orbitals, referred to as Kohn-Sham orbitals are determinedbyhˆKSi χi = εiχi (2.43)392.1. The quantum mechanical approachwhere the one-electron Kohn-Sham operator hˆKS is defined ashˆKSi = −12∇2 + Vs(r) (2.44)It is crucial to realize that this procedure does not give us the exact energy of asystem, but rather is used to determine an electron density that represents our realsystem. The connection between this fictitious system comes from the chose of theeffective potential such that the density of our real system is a result of summingover the squared moduli of the KS orbitals:ρ(r) =∑i|χi|2 (2.45)Once again in analogy to HF theory, one applies the variational principle to min-imize the total energy functional in Equation 2.37 with respect to χ. The effectivepotential that variationally minimizes the energy is given by105Vs(r) =δJ [ρ]ρ(r)+δEXC [ρ]δρ(r)+M∑AZAriA=∫ρ(r2)r12+ VXC +M∑AZAriA(2.46)where the first term describes the Coulombic potential between two electrons, thelast term is the potential between the electron and each nucleus. The middle termis once again the unknown XC potential. The electron density obtained from thefictitious system of non-interacting particles is finally used in Equation 2.37 to findthe total energy of the system. Since Vs(r) depends on the electron density, theseequation must be solvent iteratively, as with HF theory. Note however, that if theexact form of EXC [ρ] was known, this method would give the exact ground state402.1. The quantum mechanical approachelectron density of the system, and thus the exact energy.Challenges for density-functional theory methodsPure DFT has low computational cost and potentially good accuracy, henceits popularity as a quantum chemical treatment. However, there are several prob-lems that common DFT methods experience that lead to erroneous results in manycases.106 It is well established that traditional DFT methods completely fail atdescribing non-covalent interactions.107,108 This shortcoming leads to poor descrip-tions of chemistry beyond equilibrium geometries, including transition states. For-tunately, there are several methods that can correct for this problem, commonlythrough the addition of an energy correction term Edisp to the DFT energy EDFT ,asEtot = EDFT + Edisp (2.47)It is common to employ the empirical D3 pair-wise correction of Grimme,109 pairedwith of the Becke-Johnson damping function,110 denoted as D3(BJ). This correctionworks by calculating the dispersion interactions between all pairs of atoms A and Bseparated by distance RAB, with the following equation:Edisp =∑A>BCAB6R6ABf6(RAB) + s8CAB8R8ABf8(RAB) (2.48)where C6 and C8 are dispersion coefficients, s8 is an empirically determined scalingparameter, and fn are the damping functions that limit the range of dispersioncorrection, avoiding near singularities at small RAB. Another approach to correctingfor dispersion is to add parameters directly to the functional, as is the case in theMinnesota functionals.103,103 Both of these empirical corrections have the benefit412.1. The quantum mechanical approachof adding negligible computational time, but must be parametrized for each DFTmethod with which they are employed.Another striking issue with DFT is the unphysical ability of an electron to inter-act with itself, termed self-interaction error. This is most obvious in what is knownas delocalization error, which is a result of many-electrons interacting with them-selves, or many-electron self-interaction error. In HF theory, self-interaction erroris exactly cancelled, thus DFT methods that have a high portion of HF exchangein their formulation are able to account for this issue. Consider for a moment aone electron system: there should be exactly zero electron correlation. In terms ofthe energy functionals shown in Equation 2.37, the electronic repulsion term Vee[ρ]should cancel exactly with the XC term (Vee[ρ] = −EXC [ρ]).74 Unfortunately, allpure DFT methods fail to reproduce this expected behaviour. An obvious mani-festation of delocalization error is the incorrect treatment of charge-transfer in in-tramolecular interactions,111,112 as well as in transition state complexes. Even forthe simplest HAT reaction H2 + H H + H2, the calculated barrier height is un-derestimated by 8–9 kcal mol−1 using a GGA functional.113 Charge-transfer occurswhen a fraction of an electron is transferred between molecular entities. Specifically,charge-transfer is mistreated at longer ranges, thus either high percentage exact ex-change hybrid functionals, or range-separated functionals are suggested for systemsin which charge-transfer may occur.As is the case for most experimental methods, identifying the correct theoreticalmethods requires the careful consideration of the problem at hand. Choosing a DFTbased method requires calibration. However, once a method has been tested andis known to provide reasonably accurate results, DFT methods have the ability tohelp understand chemical problem with relatively low computational costs.422.2. Applying theory to chemical problems2.2 Applying theory to chemical problems2.2.1 Geometry optimizationAll QM methods depend parametrically on the geometry of a molecular system.That is the electronic energy of a system depends on the positions of the nuclei.While the wave functions can describe any arbitrary geometry, we are typicallyonly interested in certain geometries of a molecule. These geometries of interestare normally stationary states along the PES of a system, that is, points wherethe gradient of energy with respect to nuclear coordinates is zero. Therefore, weperform geometry optimization calculations to determine these points.Molecular systems have complex PESs. For a non-linear molecule, the nuclearPES has 3N -6 dimensions, where N is the number of nuclei present.114 In geometryoptimization, we seek the local minima (reactants, products, or intermediates) andlocal maxima (TS complexes). Consider only local minima for a moment. Oftencomplex molecules have more than one possible conformation, and each conforma-tion represents a different local minimum along the PES. It is therefore importantto ensure the correct conformation, typically the lowest energy structure (globalminimum), is used when approaching chemical problems.In order to efficiently perform geometry optimization, numerical analysis tech-niques are employed. All geometry optimization methods follow the same generalframework.115 First, energy and necessary derivatives are computed from an initialgeometry. Second, the geometry is modified to step towards the nearest stationarystate. And lastly, some test is performed to determine if the new geometry is nearenough to the stationary state along the PES. The most efficient method to do thisis the Newton method, in which the energy is expanded in a Taylor series (truncatedat the second order point) about the current point, x0:432.2. Applying theory to chemical problemsE(x) = E0 + g0∆x +12∆xH0∆x (2.49)where E0, g0, and H0 are the energy, gradient (Jacobian), and second derivative(Hessian) at point x0, and ∆x = xi − x0. The aim of the Newton method is tominimize the gradient of the Taylor expansion, g(x), such thatg(x) = g0 + H0∆x (2.50)Solving for ∆x gives the so-called Newton step that leads to minimization:∆x = −H−10 g0 (2.51)The analytic computation of the Hessian is very expensive, especially for largesystems. Therefore, to simply the problem at the beginning of geometry optimiza-tion, the Hessian matrix is approximated and updated at each step in the optimiza-tion, using clever algorithms.115 This is called the quasi-Newton method, and is thedefault optimization routine in the Gaussian116 quantum chemistry package, as wellas many other quantum chemistry packages.Some additional caution must be taken in optimizing molecular structures. Nor-mal algorithms that optimize structures stop when the gradient of energy is suffi-ciently close to zero; however, often PES can be flat or very shallow in regions andstructures that are not fully optimized can be obtained. To avoid this, geometriesare always subject to molecular vibration analysis.2.2.2 Molecular vibrationsThe computation of molecular vibrations can be performed simply given a setof molecular coordinates.117 Assuming a non-linear molecule, we start with 3N -6442.2. Applying theory to chemical problemsinternal coordinates that are non-coupled (orthogonal). We then apply the harmonicapproximation, in which we assume each normal mode follows Hooke’s LawF = kx (2.52)where F is the force, k is the force constant, and x is the displacement along onenormal mode’s coordinates. This approximation assumes the PES along the normalmode is parabolic, which in general is not true, but is a good approximation nearthe minima. Deviations from this approximation are known as anharmonicity. Inpractise, however, at normal temperatures (∼ 298K) the harmonic approximation issufficient to describe molecular vibrations as displacements are assumed to be small.Typically to obtain molecular frequencies, one computes the mass-weighted Hes-sian matrix elements FijFij =1√mimjHij (2.53)where the partial derivatives of internal coordinates xi of the potential energy Uare taken for 3N atoms with mass m. One then seeks to diagonalize this 3N × 3Nmatrix to obtain eigenvalues λi, which describe the force constant of each normalmode. The harmonic frequencies νi are then obtained byνi =√λi2pi(2.54)and the lowest 6 modes are then discarded to account for 3N -6 translations androtations. These lowest energy modes generally correspond the internal rotations,and thus must be discarded to correctly obtain thermochemical corrections.From these frequencies, the zero-point vibrational energy (ZPE, EZPE) is calcu-lated:452.2. Applying theory to chemical problemsEZPE =3N−6∑i=1hνi2(2.55)The ZPE is an important quantum correction to the classical potential, giving theelectronic potential energyU = Eelec + EZPE (2.56)where Eelec is the QM electronic energy.If a normal mode describes a non-minimum along the PES, the energy gradientwill be negative (imaginary) instead of positive. Only energy maxima or saddle-points (TS structures) should have a single imaginary mode. Therefore, if a non-TSmolecular structure calculation yields one or more imaginary modes, the geometryoptimization has yielded a structure that is not at minimum on the PES. In thissituation additional steps must be taken to find a corrected structure.2.2.3 ThermochemistryUp until this point we have been viewing molecules from a microscopic per-spective; however, this is not useful for describing properties of bulk systems. For-tunately, fundamental statistical thermodynamics can be used to approximatelydescribe a system in bulk.118,119 We approximate our system as an ensemble ofnon-interacting particles: using the Ideal Gas approximation. Within statisticalthermodynamics, the fundamental starting point is the partition function Q,120from which all thermodynamic properties can be calculated. For our ensemble, themolecular partition function isQ =∑Jeεj/kBT (2.57)462.2. Applying theory to chemical problemswhere a Boltzmann distribution of j energy states ε is taken at temperature T , andkB is the Boltzmann constant. All calculations herein are defined under conditionsof temperature T = 298.15 K and pressure P = 1 atm, and concentration = 1 M.Normally, the molecular partition function is decomposed into contributions fromtranslational, vibrational, rotational, and electronic motion:Q = qtransqvibqrotqelec (2.58)The equation describing the translational partition function qtrans isqtrans =(2pimkBTh2)3/2 kBTP(2.59)where m is the mass of the molecule, h is Planck’s constants.The vibrational partition function qvib depends on the contributions of each of Kvibrational modes. Only the 3N−6 (or 3N−5 for linear molecules) real vibrationalmodes of a molecule are considered, and imaginary frequencies are ignored. There-fore, for molecules that posses an imaginary frequency this thermodynamic analysisis invalid. TS complexes do posses a single imaginary frequency that is ignored, asit is assumed to not contribute to the overall vibrational partition function as noformal bond is said to be formed in the acceptor-donor system. Each vibrationalmode has a characteristic vibrational electronic temperature, Θν,K = hν/kB, andthe partition function isqvib =∏Ke−Θν,K/2T1− e−Θν,K/T (2.60)The rotational partition function depends on the geometry of a system. For asingle atom qrot=1. For a linear molecule, the rotational partition function is472.2. Applying theory to chemical problemsqrot =1σr(TΘr)(2.61)where σr is the symmetry number for rotation that depends on the molecular sym-metry, and Θr = h2/8pi2IkB. I is the moment of inertia. Finally, for a non-linearpolyatomic molecule, the rotational partition function isqrot =√piσr(T 3/2√Θr,xΘr,yΘr,z)(2.62)where Θr,x,Θr,y, and Θr,z describe contributions of the moment of inertia in eachof the x, y, and z-planes.Finally, we make an important assumption that electronic contributions are as-sumed to exist in only the ground state, as excited states are generally safely assumedto be much larger than kBT in energy. The full electronic partition function isqelec =∑i=0ωie−i/kBT (2.63)where ω is the degeneracy of an energy level with energy . Applying our assumption,and by setting the ground state energy 0 = 0, our problem simplifies dramatically,such that qelec = ω0, which is simply the spin multiplicity of the molecule.We now have all the information needed to calculate the thermodynamic quanti-ties we are interested in. In chemistry we are concerned with the Gibbs free energyG, which is defined by the entropy S and enthalpy H asG = H − TS (2.64)From each of the partition functions, the entropy of a system with N moles, Stot =Strans + Svib + Srot + Selec, is calculated using the relation482.2. Applying theory to chemical problemsS = NkB +NkB ln(QN)+NkBT(∂ lnQ∂T)V(2.65)Similarly, the internal energy of a system, Eint,tot = Eint,trans +Eint,vib +Eint,rot +Eint,elec, is given by the relationEint = NkBT2(∂ lnQ∂T)V(2.66)Finally, the enthalpy is obtained fromHtot = Eint,tot + kBT (2.67)Using very simple statistical thermodynamic arguments, the properties of a bulksystem are easily computed. It is important to emphasize that these results are forparticles in the gas phase, thus additional steps must be taken if one desires tocompare results to experiments performed in solvent.2.2.4 Modelling solventIt is in principle possible to include solvent molecules explicitly in QM calcula-tions: this is in practise, extremely cost prohibitive. In order to approximate theimportant contributions of solvation, so-called implicit continuum solvent modelsare generally employed.74,121 Mathematically, one describes this asHˆtot(rm) = Hˆmol(rm) + Vˆmol+sol(rm) (2.68)where a perturbation Vˆ mol+sol dependent only on the coordinates of the solute (rm;thus implicit) is applied to the Hamiltonian of the solute. The perturbation term iscomposed of interaction operators that contribute to the net free energy:492.2. Applying theory to chemical problemsGsolv = Gcavity +Gelectrostatic +Gdispersion +Grepulsion +Gsolv kinetic (2.69)where the total solvation free energy Gsolv contains terms from: the formation ofa solvation cavity Gcavity, the electrostatic interactions between solvent and soluteGelectrostatic, the dispersion interactions between solvent and solute Gdispersion, theQM exchange repulsion between solvent and solute Grepulsion, and the movement ofsolvent molecules Gsolv kinetic.A widely used model for solvation comes from the Truhlar group, and is knownas SMD.122 The main parameter in implicit solvent models is the solvent dielectricconstant (ε) with contributions from surface tension and the solvent-solute interface.SMD also includes terms that depend on the electron density of the solute. Whilemany other implicit solvent models require the use of the same QM method asthey were parametrized,123 SMD is a universal model that was parametrized usingseveral QM methods. Therefore, it does not require the use of a specific QM methodand can be applied broadly in both single point energy and geometry optimizationcalculations.2.2.5 Rate constants and transition state theoryIn the discussion of chemical kinetics, the rate (r) of a reaction involving twospeciesaA + bB cC + dD (2.70)is determined by the rate law, which can generally be described as502.2. Applying theory to chemical problemsr =1cdCdt=1ddDdt= k[A]α[B]β (2.71)where k is the rate constant, t is time, A,B,C, and D are chemical species withstoichiometric coefficients a, b, c, and d, α and β are the order of the reaction withrespect to reactants A and B, and k is the rate constant. Computational chemistryis in general, not useful for determining rate laws: this must be done experimentally.Where computational studies can be useful, is in determining reaction mechanisms,and how the reaction barrier height can be altered. In doing so, we focus entirelyon k.Most chemists are intimately familiar with the phenomenological ArrheniusequationkArr = Ae−Ea/RT (2.72)where A is a constant, R is the gas constant, and Ea is the activation energy, whichis an experimental measure of the reaction barrier height. This equation dates backto the 1880s, when Arrhenius noticed that the reactions depended more heavily ontemperature than was intuitive, and thus introduced the A constant, known oftenas the Arrhenius pre-factor.124 The Arrhenius pre-factor is an empirical measure ofthe frequency of collisions which have the correct orientation for a reaction to occur.From the perspective of theory, Equation 2.72 has little meaning as the parametersare empirical. Thus, to study rate constants theoretically we must turn to transitionstate theory.512.2. Applying theory to chemical problemsTransition state theoryThe study of transition state theory (TST) originates in the 1930s, and wasdeveloped primarily by Eyring.124,125 In TST we focus on the TS complex, which isdefined as a transient species that exists at the top of the energy barrier of a reaction.If we consider the same reaction in Equation 2.70, and set all the coefficients to 1,then TST states the reaction proceeds in two steps, the first of which includes aquasi-equilibrium between the reactants and TS complexA + B AB‡ C + D (2.73)with an equilibrium constant (K‡c ) expressionK‡c =[AB‡]/c◦[A]/c◦[B]/c◦(2.74)where c◦ is the standard-state concentration (normally taken to be 1 M).Figure 2.2: A reaction coordinate diagram for the reaction of Equation 2.73. TheTS complex is defined to exist in the small region δ on top the reaction barrier.In TST, we define the TS complex to exist within a small region of width δ ontop of the reaction barrier (Figure 2.2). From the second step of the reaction in522.2. Applying theory to chemical problemsEquation 2.73, we can define a reaction rate dependent on the concentration [AB‡]and vc, a factor that defines the frequency with which the complexes proceed overthe barrier:r = vc[AB‡] (2.75)From Equations 2.70 and 2.73, we now have two equivalent expressions for thereaction rate, which allows us to derive the followingr = k[A][B] = vc[AB‡] (2.76)and solving Equation 2.74 for [AB‡] results inr = vc[A][B]K‡cc◦(2.77)ork =vcK‡cc◦(2.78)We must now invoke statistical thermodynamics to make sense of Equation 2.78.We can rewrite the equilibrium expression K‡c in terms of partition functions of eachmolecular species:K‡c =[AB‡]/c◦[A]/c◦[B]/c◦=(q‡/V )c◦(qA/V )(qb/V )(2.79)where V is the volume, and qA, qB, and q‡ are the partition functions of A, B, andAB‡, respectively.Since we have defined the reaction to be occurring with one degree of freedom,the translational partition function qtrans can be defined as532.2. Applying theory to chemical problemsqtrans =√2pim‡kBThδ (2.80)where m‡ is the mass of the TS complex. The partion function of the TS complexcan be written as the product q‡ = qtransq‡int, where the second term accounts forall remaining degrees of freedom of the TS complex. We can use this and rewriteEquations 2.79 and 2.78 asK‡c =√2pim‡kBThδ(q‡int/V )c◦(qA/V )(qb/V )(2.81)andk = vc√2pim‡kBThc◦δ(q‡int/V )c◦(qA/V )(qb/V )(2.82)We are now left with the two terms vc and δ that are ill-defined. However, theproduct of these two terms is the average speed at which the TS complex crossesthe barrier, 〈uTS〉 = vcδ. A Maxwell-Boltzmann distribution is used to calculatethe value of 〈uTS〉:〈uTS〉 =(m‡2pikBT)1/2 ∫ ∞0ue−m‡u2/2kBTdu =(m‡2pikBTm‡)1/2(2.83)Substituting Equation 2.83 into Equation 2.82 for vcδ yieldsk =√kBThc◦(q‡int/V )c◦(qA/V )(qb/V )=kBThc◦K‡c (2.84)We now define the standard Gibbs free energy of activation (∆‡G◦) to be thechange in Gibbs free energy in going from reactants to TS. The thermodynamicalexpression is542.2. Applying theory to chemical problems∆‡G◦ = −RT lnK‡c (2.85)which can be substituted into Equation 2.84k =kBThc◦e−∆‡G◦/RT (2.86)The standard Gibbs free energy of activation can be expressed in terms of en-thalpy and entropy as∆‡G◦ = ∆‡H◦ − T∆‡S◦ (2.87)which upon substitution gives the equationk =kBThc◦e∆‡S◦/Re−∆‡H◦/RT (2.88)At this point, we can draw a direct comparison to the Arrhenius equation (Equa-tion 2.72) by expressing Ea in terms of ∆‡H◦ and A in terms of ∆‡S◦. We mustdifferentiate the natural logarithm of Equation 2.84, as well as Equation 2.72 (as-suming that A is independent of temperature):d ln kdT=1T+d lnK‡cdT(2.89)d ln kArrdT=EaRT 2(2.90)Next, we use the fact that d lnKc/dT = ∆U◦/RT 2 for an ideal gas, then Equation2.89 becomes552.2. Applying theory to chemical problemsd ln kdT=1T+∆‡U◦RT 2(2.91)Additionally, ∆‡H◦ = ∆‡U◦+RT∆‡n (∆‡n = 1), so Equation 2.91 can be rewrittenasd ln kdT=∆‡H◦ + 2RTRT 2(2.92)Therefore, by comparison of Equation 2.92 and 2.90, we getEa = ∆‡H◦ + 2RT (2.93)which then converts Equation 2.88 into the formk =e2kBThc◦e∆‡S◦/Re−Ea/RT (2.94)Therefore, a natural statistical thermodynamical picture of the Arrhenius equa-tion arises from TST, and the Arrhenius pre-factor A can be expressed asA =e2kBThc◦e∆‡S◦/R (2.95)In practise, we use the form of Equation 2.86 to compute the rate constantof a reaction, which we shall denote as kTST . The conventional TST makes anassumption that the reaction coordinate is static along the lowest energy pathway.This can be corrected by the use of variational transition state theory.126 We shallnot consider variational TST in this work, as with careful application, conventionalTST does a remarkably good job at accounting for the magnitude and temperaturedependence of a wide range of reactions.125 Additionally, if one makes correctionsfor QM tunnelling, conventional TST can easily give a more complete description of562.2. Applying theory to chemical problemsthe rate constant.Quantum mechanical tunnellingAtoms are quantum mechanical particles, and are thus subject to the probabilis-tic behaviours observed at the microscopic level. QM tunnelling refers to the abilityof particles to penetrate the reaction barrier, rather than surmounting it classically(Figure 2.3). While all reactions are subject to QM tunnelling, we will show thatdue to the low mass of the hydrogen atom, QM tunnelling can play a significant rolein HAT reactions.Figure 2.3: Quantum mechanical tunnelling occurs when a particle penetrates areaction barrier, rather than surmounting it.In order to determine the effects of scattering, one must find transmission co-efficients (κ) by solving the Schro¨dinger equation.52 This is done by approximat-ing the reaction barrier with an analytical potential, thus simplifying the problemmathematically. The earliest model potentials were introduced by Bell, who useda parabolic function to approximate the reaction barrier.127 To obtain κ, and thusthe observed rate constant (kobs), the following equations were used:572.2. Applying theory to chemical problemskobs = κAe−Ea/RT (2.96)κ =eαβ − α(βe−α − αe−β)(2.97)α = Ea/RT (2.98)β =2api2(2mEa)1/2h(2.99)where the Arrhenius equation was used to estimate the rate constant, m is the massof the tunnelling particle, and 2a is the width of the barrier. Since the equation isdependent on the mass of the particle, tunnelling occurs more often when lighterparticles are involved. As a consequence, tunnelling is more common in HAT reac-tions than other atom transfer reactions. Also, the height and width of the barrierare important factors in determining the contributions to tunnelling: reactions withsmall barriers have low tunnelling contributions; narrow barriers result in highertunnelling contributions.The Bell model is a poor representation of an actual reaction barrier. One thatis a much better approximation is the Eckart potential.128 The form of this potentialisV = − Ay1− y −By1− y2 (2.100)y = −e2pix/L (2.101)where x is the variable along the reaction coordinate and L is called the characteristiclength. If A = 0 the potential becomes a symmetric function, further simplifyingthe problem; however, most reactions do not have a symmetric potential. A, B and582.2. Applying theory to chemical problemsL are related to the change in barrier height in the forward and reverse direction,∆V1 and ∆V2, respectively:A = ∆V1 −∆V2 (2.102)B = ((∆V1)1/2 + (∆V2)1/2)2 (2.103)L2pi= (− 2F ∗)1/2[1(∆V1)1/2+1(∆V2)1/2]−1 (2.104)where F ∗ = d2V/dx2 evaluated at the maximum of the potential. In this formu-lation, V is a placeholder energy. Note that if a reaction is endoergic, tunnellingdoes not occur. Alternatively, one says tunnelling only occurs in exoergic or energy-neutral reactions.The solutions to the Schro¨dinger equation for the Eckart potential are analytical,thus the transmission coefficient κ can easily be computed using standard numericaltechniques. These tunnelling corrections will be applied, where applicable askcalc = κkTST = κkBThc◦e−∆‡G◦ (2.105)59Chapter 3The Relationship BetweenArrhenius Pre-factors withNon-Covalent Binding3.1 PrologueDiLabio and Ingold38 previously investigated the formal HAT reaction of theiminoxyl/oxime self-exchange reaction. In that paper, they compiled a table of pa-rameters from the phenomenological Arrhenius equation for a series of interestingreactions, which appear here in Table 3.1.42,129–135 These are thermoneutral hy-drogen atom self-exchange reactions involving oxygen-centered radicals, and othernearly thermoneutral reactions involving the destruction and formation of oxygen-centered radicals, reactions 3.1 and 3.2, respectively:pi RO + ROH ROH + pi RO ∆H = 0 (3.1)pi R′O + ROH R′OH + pi RO ∆H ≈ 0 (3.2)Although it is well known that reactions of this nature involve remarkably lowactivation energies (Ea),136–139 the Arrhenius pre-exponential factors (A), or as theyshall be referred to herein, A-factors, as a well as rate constants, span a wide range(summarized in Table 3.1): The measured A-factors range from 103.5–108.3 M−1s−1and the rate constants range from 10–10×107 M−1s−1. In the past, this range has603.1. PrologueTable 3.1: Table of results for (nearly) thermoneutral reactions studied. Units for∆H, Ea, and calculated pre-reaction complex binding energy (BE) are kcal mol−1,logA are log M−1s−1, and k are M−1s−1. References to the original literature areincluded with the Complex ID number. †Calculated binding energies involve struc-tures that could not be fully optimized and contain one or more small imaginaryfrequencies. Adapted with permission from Reference 38. Copyright (2005) Ameri-can Chemical Society.ID RO /R′O ROH ∆H log A Ea k BE142 Ot-But-But-Bu OHt-But-But-Bu 0.0 3.7 1.2 3.3×102 -10.82129 N OO N OH -2.0 3.8 3.8 10 -14.8342 N Ot-But-BuN OHt-But-Bu0.0 5.1 3.5 3.3×102 -10.1†4130,131 Ot-But-But-Bu OHt-Bu 4.2 5.5 4.8 93 -10.0†5132 t-BuOO OHt-But-But-Bu -7.0 4.2 0.5 7×103 -6.5642 Ph2NO Ph2NOH 0.0 >7 - >107 -13.6†7133 OOH-2.2 8.3 2.3 4×106 -8.68134 t-BuOO OH 0.3 7.2 5.2 3×103 -5.5†9134 t-BuOOOH-1.9 6.4 2.6 3×104 -5.6†10135 t-BuOOOOH1.4 6.0 4.5 7×102 -8.0†been attributed to steric shielding around the oxygen atoms, resulting in a largerentropic barriers.38 Importantly, it was noted that the degree of steric shielding onthe oxygen atom appears to play an important role in the order of the A-factor;613.1. Prologuesystems with greater bulk have lower A-factors, while non-shielded systems havelarger A-factors.Stereo-electronic effects are known to play an important role in HAT, and havebeen studied extensively.40,140–146 Although the abstraction of a specific hydrogenatom may be more thermodynamically favourable than others on a given substrate,if it is not accessible due to steric constraints, abstraction will not occur at thissite. Otherwise, additional steric bulk can lead to significant reductions in reactiv-ity, through destabilization of the TS complex, or by forcing additional processesinvolving conformational changes in order to reach the appropriate TS structure.For example, in reactions of tertiary acetamides with CumO ,146 where abstractionoccurs mainly from C H bonds α to the nitrogen atom, a two-fold decrease in therate constant (normalized for the number of equivalent hydrogen atoms) is observedin going from N,N -dimethylacetamide to N,N -diisobutylacetamide (kH = 2.0×105and 7.8× 104 M−1s−1, respectively). The decrease in rate constant is attributed tothe steric clash between the methyl groups of CumO and the isobutyl groups ofN,N -diisobutylacetamide.As all of the reactions in Table 3.1 are nearly thermoneutral, thermochemicaleffects on the rates of reaction are expected to be minimal. Therefore, the largedegree of variance in their rate constants (k) is somewhat surprising. These reactionsare closely related to the self-exchange reaction between phenol and phenoxyl,23 inwhich a strong molecule-radical pre-reaction complex akin to those listed in Table 3.1is formed, ca. 10 kcal mol−1 below the separated reactants. It is therefore expectedthat most, if not all, of the systems in Table 3.1 should exhibit a similar molecule-radical complex; granted, the strength of the interaction will vary because of stericrepulsion. As such, it is plausible that the strength of this interaction may directlyinfluence the rate of formal hydrogen atom transfer.623.2. Computational methods and detailsCurrently, there has been no comprehensive investigation of the relationshipbetween the pre-reaction complex and the kinetics of a reaction. On the basis ofthe reaction data in Table 3.1, we ask the question: Do A-factors have a correlationwith non-covalent binding energies of the pre-reaction complex? This is a reasonablequestion as non-covalent binding and steric hinderance represent a loss of degreesof freedom and therefore entropy,i which ultimately determines the A-factor magni-tude. If the answer to the question is yes, then non-covalent binding may be usefulas a diagnostic for the “looseness” or “tightness” of a TS complex, in addition toproviding an important link between theory and experiment.3.2 Computational methods and detailsDensity-functional theory (DFT) calculations were carried out using the Gaussian-09 software package.116 Care was taken to obtain minimum energy structures throughdetailed conformational analysis. For this, the BLYP density-functional102,147 wasutilized, paired with the empirical D3 dispersion correction109 with the recom-mended Becke-Johnson damping functions,110 as well as our groups’ own basis setincompletion potentials (BSIPs),148 and minimal MINIs basis sets.149 The use ofminimal basis sets corrected for basis set incompleteness allows DFT-based meth-ods to be used efficiently in performing a large number of calculations. Minimumenergy conformers of the monomers (substrates and radicals) were first obtained bymanual manipulation of the necessary dihedral bond angles, followed by geometryoptimization and vibrational analysis.The lowest energy radicals and substrates were combined to generate the appro-priate pre-reaction complexes. These pre-reaction complexes were subject to confor-iRecall from Equation 2.95 that the A-factor can be related to TST such that the primary variableis entropy (∆‡S◦).633.2. Computational methods and detailsmational analysis using the same BLYP-D3(BJ)-BSIP/MINIs method. Geometrieswere initially manipulated by hand. It became apparent that manual manipulationresulted in an unsatisfactory exploration of the conformational space. To solve this,all the necessary dihedral angles were scanned systematically using a combinationof scripts.150 All manipulated geometries were subject to optimization. For eachcomplex, the top 5–10 complex geometries were subject to further optimization us-ing a higher level of theory (BLYP-D3(BJ)-BSIP/pc-1) to obtain the final minimumenergy pre-reaction complex structures. Due to the free rotation of groups such ast-butyl and methyl, some of the optimized pre-reaction complex structures containsmall imaginary frequencies, and thus do not represent proper stationary states.Several measures were taken to resolve this; no resolution was obtained in manycases. Regardless, the complexes adequately represent the pre-reaction complexand differences in “true” binding energies can likely be ignored.To obtain accurate pre-reaction complex binding energies, the substrates andcomplexes were subject to single-point energy calculations using the LC-ωPBE long-range corrected density functional151,152 with D3(BJ) dispersion corrections andpc-2 basis sets with f -type basis functions removed (pc-2-spd).153 This method wasselected on the recommendation of work by Johnson et al. 153 , which demonstratedthe accuracy of this method for the calculation of NCIs. On the basis of the reportedmean absolute error in Reference 153 for the S66 benchmark set of sixty-six differentnon-covalently interacting dimers,154 the calculated binding energies reported hereinfrom the LC-ωPBE-D3(BJ)/pc-2-spd level of theory carry an estimated 0.2 kcalmol−1 margin of error.643.3. Results and discussion3.3 Results and discussionThe theoretically determined electronic binding energies calculated for the lowestenergy pre-reaction complex of each system are listed in Table 3.1. The logarithmof A-factor against binding energy was plotted, as shown in Figure 3.1. The overallcorrelation is quite poor (R2 = 0.33), however much of the data is grouped abouta single, well correlated line (R2 = 0.95). The intercept of the fitted line thatcorresponds to zero binding energy is 8.63, a value that is in line with what has beencited as the expected A-factor for HAT reactions, viz. 108.5±0.5 M−1s−1.155 Theseresults suggest that the observed correlation is genuine, that is, NCIs may have animpact on A-factors. I shall demonstrate that the data that do not correlate arereasonable outliers. In fact, using simple rationale I shall demonstrate that differentregimes of steric bulk results in different mechanistic processes leading to the TScomplex.In order to understand the deviations from the expected linear trend of A-factoragainst pre-reaction complex binding energies, it is important to consider the specificreaction mechanisms taking place. Recall from Chapter 1 that we are focussed ontwo important possible reaction mechanisms, namely direct HAT and PCET.For direct HAT to occur, the SOMO of the radical must overlap with the O-Hσ∗ anti-bonding orbital. In the case of hydrogen abstraction from a phenolic com-pound, this may require the rotation of the hydrogen atom donating hydroxyl groupout of the plane. The rotation of a phenolic hydroxyl group has an energy barrierthat follows a cos2 θ relationship.156 As a reference point, the rotational barrier ofphenol157 is 3.1 kcal mol−1, thus sterically hindered phenols may have a greaterrotational barrier. For a PCET mechanism to occur, there are two possible geome-tries: The nominally singly-occupied O 2p-orbital of the radical can overlap with thecorresponding oxygen LP 2p-orbital, as seen in the work of Mayer et al. 23 . Alterna-653.3. Results and discussion16 14 12 10 8 6 4Binding Energy (kcal mol 1)345678log 10 A23489101567R2 = 0.949 y = 0.328 x + 8.626Figure 3.1: Plot of logarithm of A-factor against binding energy. Only the blackpoints were included in the line fitting (slope = 0.328 kcal mol−1, intercept = 8.626kcal mol−1, and R2 = 0.949). Red points with open faced markers indicate outliers,vide infra. The inclusion of complexes 1, 5, and 7 result in an R2=0.334. Complex6 is always omitted from line fitting as the experimental A-factor is approximate.663.3. Results and discussiontively, a LP-pi, LP-LP, or pi-pi bonding overlap between the radical and substrate canoccur, as seen in the work of DiLabio and Ingold 38 , and DiLabio and Johnson 26 .As described in Chapter 1, there remains no clear physical criteria to distinguishdirect HAT from PCET, a topic of active discussion in the literature.22–32 Onepossible solution is consider the existence of these mechanisms on a continuum;the rate constant (and thus A-factor) for formal HAT (kHAT ) can be describedas a combination of the rate constants direct HAT (kdirect) and PCET (kPCET )mechanisms, i.e.:kHAT = kdirect + kPCET (3.3)Before elaborating on Equation 3.3, we must first discuss the role of the pre-reaction complexes in formal HAT reactions. As a radical and substrate approachsufficient proximity for a reaction to take place, NCIs lead to the formation of aweakly-bound complex. If this complex has the appropriate geometry for a hydrogentransfer to occur, it is considered a pre-reaction complex, otherwise it is consideredan initial encounter complex. An initial encounter complex must pass over an addi-tional energy barrier to reach the appropriate pre-reaction complex. With respect tothe species in Table 3.1, the complexes formed involve various degrees of pi-pi, LP-pi, and LP-LP interactions, which contribute to the weakly attractive, dispersioninteractions. Furthermore, these same orbital interactions in the TS complex canlead to the formation of an additional electronic channel, enabling contributions ofa PCET mechanism to the overall mechanism.26,38 Returning now to Equation 3.3,the different types of orbital interactions may control the contributions of kPCET tothe overall rate constants. Our hypothesis, which cannot be tested until a quantita-tive diagnostic for PCET vs HAT is developed, is that pi-pi interactions contributeto a PCET dominated mechanism (kHAT ≈ kPCET ). On the other hand, LP-LP673.3. Results and discussioninteractions are weaker and thus PCET of this nature contributes only weakly tothe overall mechanism.While the data herein explore only the geometries of the pre-reaction complexesinvolved in the hydrogen transfer reactions, the presumptive TS structures willhave similar structures and more importantly, orbital interactions. Therefore, byconsidering the similarities between pre-reaction and TS complexes, it is possible torationalize the deviations from the observed trendline in Figure 3.1.A. B.Figure 3.2: Three-dimensional structures of A complex 2 (TEMPO-H and 4-oxo-TEMPO) and B complex 3 (di-t-butyl-hydroxylamine and di-t-butyl-nitroxyl).Bond distances are shown in units of A˚. The elements are coloured as white forcarbon, light blue for hydrogen, red for oxygen, and blue for nitrogen.I shall begin by examining the points that fall on the expected line, complexes2–4 and 8–10. The examination of all these pre-reaction complexes reveals that anadditional rearrangement that has a moderate energetic barrier is required in order683.3. Results and discussionfor the hydrogen transfer to proceed. Complexes 2 and 3 are shown in Figure 3.2, andare very similar in structure. Both are hydroxylamine-nitroxyl couples with similardegrees of steric bulk adjacent to the reacting centres. The t-butyl groups of 3, andthe methyl groups of 2 (the NO-ON dihedral angles are 65◦ and 68◦, respectively)prevent the overlap of the N LP orbitals in of the NO-H-ON frameworks to allow forPCET. Thus, while the presumptive TS structure may have some degree of LP-LPorbital interaction, the overall mechanism is dominated by direct HAT.In the most stable stacked conformation, complex 4, as seen in Figure 3.3, stericclash of the para-position t-butyl groups obstructs pi-pi overlap between the aro-matic rings. It is likely that this steric clash does not allow any significant orbitalinteraction, suggesting that the reaction is dominated by direct HAT. In order toreact via direct HAT, the hydroxyl group must rotate farther out of the aromaticplane, or the bulky para-position t-butyl groups must come into close proximity.Alternatively, an open conformation for complex 4 is possible, which lies ca. 2 kcalmol−1 higher in energy than the stacked complex, a result that is also consistentwith the observed trend-line. From the open conformation, PCET is still not pos-sible due to the steric bulk of the ortho-position t-butyl groups of the radical, thusthis reaction likely also proceeds through a direct HAT mechanism. For complex 4,steric clash prevents pi-pi orbital intactions, therefore the reaction proceeds througha direct HAT dominated mechanism.Complexes 8 and 9 are similar systems, in which t BuOO reacts with un-hindered phenolic substrates. As seen by the structures in Figure 3.4, the boundcomplexes are somewhat dissimilar. The hydroxyl group of complex 8 is rotated outof the plane 24◦, while in complex 9 the hydroxyl group lies entirely in the plane. Itis likely that the larger aromatic system of 2-naphthol results in a larger OH rota-tional barrier. Therefore, the most favourable conformation for this complex places693.3. Results and discussionA. B.Figure 3.3: Three-dimensional structure of pre-reaction complex 4 between 2,4,6-tri-t-butylphenol and 4-t-butylphenoxyl. A demonstrates the hydrogen bond distancesin units of A˚, and the out-of-plane rotation by 35.2◦ of the phenolic hydroxyl group.B demonstrates the steric clash (highlighted by red box) between the para-position t-butyl groups. The elements are coloured as white for carbon, light blue for hydrogen,and red for oxygen.703.3. Results and discussionthe phenolic hydroxyl group entirely in the plane. In complex 9, there is also a weakhydrogen bond between the C H bond in the ortho-position and the non-radicalO-centre of t BuOO , contributing further to the stabilization of the planar confor-mation. Complex 8 was previously studied by DiLabio and Johnson 26 , where it wasdemonstrated that a partial bonding interaction exists between the peroxyl LP andphenolic pi-system in the TS complex. However, this interaction is likely weak andthus contributes weakly to the overall rate constant. That is, kHAT is dominatedby kdirect. Also, although the pre-reaction complexes are somewhat dissimilar, theconformational changes necessary to reach the TS complex, similar to that reportedin reference 26, are likely not dramatically different in terms of energetic barriers.Any small differences result in noise in the observed trend.Complex 10, shown in Figure 3.5 is unique in that it is the only reaction betweena peroxide and a peroxyl radical. Therefore, this system represents the best casescenario for LP-LP overlap to occur. The self-exchange reaction between HOOand HOOH can be considered the simplest reference for the reaction of α-tetralinperoxide with t-butylperoxyl. To the best of my knowledge, the mechanism of thehydroperoxyl-hydrogen peroxide couple has not been characterized as either PCETor direct HAT previously in the literature, although the TS structure has been pre-viously reported.158 Using this structure, calculations reveal a LP-LP interactionleading to partial bonding in the TS, i.e. a PCET mechanism. (See Appendix A, Fig-ure A.1). The hydroperoxyl-hydrogen peroxide couple contains a H O O H dihe-dral angle of 90◦, so that the two non-reacting hydrogen atoms oriented 180◦ awayfrom one another. Orienting substituents directly away from one another is likelythe most stable TS structure for all peroxyl-peroxide formal HAT reactions.The orientation of t-butylperoxyl and α-tetralin peroxide at exactly 180◦ awayfrom one another is unfavourable in complex 10, due to steric clash. Nonetheless,713.3. Results and discussionA. B.Figure 3.4: Three-dimensional structures of pre-reaction A complex 8 (t-butylperoxyl and phenol) and B complex 9 (t-butylperoxyl and 2-naphthol). Bonddistances are shown in units of A˚. Complex 8 has an out of plane rotation of thephenolic hydroxyl group of 24.1◦. The elements are coloured as white for carbon,light blue for hydrogen, and red for oxygen.723.3. Results and discussionthere may still be some LP-LP overlap contributing to a weak kPCET contributionto kHAT . On the basis of our hypothesis, LP-LP interactions do not allow forPCET to contribute strongly to the overall mechanism. This will require additionalinvestigation. For complex 10, kHAT is dominated by kdirect.A.B.Figure 3.5: Three-dimensional structure of pre-reaction complex 10 between t-butylperoxyl and α-tetralin peroxide. A demonstrates the hydrogen bond distancesin units of A˚. B demonstrates the likely steric clash preventing strong LP-LP over-lap. The elements are coloured as white for carbon, light blue for hydrogen, and redfor oxygen.Once again, complexes 2–4 and 8–10 follow the observed trend. In all cases,these complexes may have some PCET contribution to kHAT through either LP-pior LP-LP orbital overlap. Interpretation of Figure 3.1 in this manner allows for twopossible explanations. The simplest is that all these complexes proceed through amechanism in which kHAT ≈ kdirect (kPCET  kdirect). In this case, the A-factor is adirect reflection of kHAT and this the pre-reaction complex binding energy correlates733.3. Results and discussionwell with the A-factor.Alternatively, there may be increasing contributions of PCET leading to anincrease in the A-factor. This effect can be rationalized on the basis of a strongerinteraction in the case of LP-pi overlap, as compared to LP-LP overlap. Withinthis framework, complexes 2, 3, and 4 may have little or no overlap due to stericclashing, and complexes 8 and 9 have a higher A-factor than complex 10 due toLP-pi vs. LP-LP overlap. Further work is necessary to discern this effect.Consider next the points that sit above the trendline, complexes 6 and 7, shownin Figure 3.6 and Figure 3.7. The A-factor for complex 6 is approximate and thusdoes not get factored into the line fitting. In both cases, the non-covalently boundcomplexes are in a slipped-parallel pi-stacked conformation, allowing for pi-pi orbitaloverlap. Complex 7 in particular is very similar to the phenol-phenoxyl couple,except with 2-naphthol instead of phenol. In both cases, the pi-stacked pre-reactioncomplex is very close to the presumptive TS structure. Therefore, it is possible toinfer that both of these reactions take place through mechanism in which kPCETis dominant. The key difference from the points that fall on the trendline is thatkHAT ≈ kPCET .Lastly, consider the points that fall below the trendline, complexes 1 and 5. Inboth cases, a high degree of steric repulsion likely does not allow for a PCET mech-anism through orbital overlap. It is important to study the “encounter complex”that represents the first pre-reaction complex, i.e. prior to any reorganization, asthis will be the complex that affects the A-factor with regards to simple collisiontheory. Complex 1 is the self-exchange reaction between the very bulky 2,4,6-tri-t-butylphenol/2,4,6-tri-t-butylphenoxyl couple, as seen in Figure 3.8 A. As a resultof steric shielding around the reaction centres, the encounter complex is stacked tomaximize dispersion interactions, but does not have a hydrogen bond. Therefore,743.3. Results and discussionA. B.Figure 3.6: Three-dimensional structures of pre-reaction complex 6 between N,N -diphenylhydroxylamine and N,N -diphenylnitroxyl. A demonstrates the hydrogenbonding interaction while B demonstrates the pi-pi interaction. Distances in unit ofA˚ and angles are shown in degrees. The elements are coloured as white for carbon,light blue for hydrogen, blue for nitrogen, and red for oxygen.753.3. Results and discussionA. B.Figure 3.7: Three-dimensional structures of pre-reaction complex 7 between 2-naphthol and phenoxyl. A demonstrates the hydrogen bonding interaction whileB demonstrates the pi-pi interaction. Distances in unit of A˚ and angles are shownin degrees. The elements are coloured as white for carbon, light blue for hydrogen,and red for oxygen.an additional rearrangement is required in order to get to the presumptive TS struc-ture. That is, there must be a higher-energy hydrogen-bonded pre-reaction complexthat leads to the direct HAT mechanism. Note however, that there is a barrier torotation of the hydroxyl group to 90◦ out of the plane for direct HAT to occur.Complex 5 is the 2,4,6-tri-t-butylphenol/t-butylperoxyl reaction couple. The en-counter pre-reaction complex also does not contain a hydrogen bond. As with com-plex 1, an encounter complex without a hydrogen bond must form first. However,in complex 5 there is less steric clashing. As a result the formation of a hydrogenbond is favourable and the “true” pre-reaction complex is about 0.7 kcal mol−1 morestable than the encounter complex. In contrast, for complex 1 the true pre-reactioncomplex is about 0.6 kcal mol−1 less stable than the encounter complex. Note alsothat there is a barrier to rotationii of the hydroxyl group that can be estimated asabout 4.1 kcal mol−1. This is illustrated schematically in Figure 3.9.iiCalculated as the difference in energy between the in-plane and out-of-plane structures of 2,4,6-763.3. Results and discussionA. B.Figure 3.8: Three-dimensional structures of pre-reaction A complex 1 (2,4,6-tri-t-butylphenoxl and 2,4,6-tri-t-butylphenoxyl) and B complex 5 (2,4,6-tri-t-butylphenol and t-butylperoxyl). Distances in unit of A˚ and angles are shown indegrees. The elements are coloured as white for carbon, light blue for hydrogen,and red for oxygen.Figure 3.9: Reaction coordinate qualitatively illustrating the proposed mechanismfor HAT in complexes 1 and 5. React. = reactants, EC = encounter complex,PRC-1/5. = true pre-reaction complex 1/5, TS1 = transition state associated without-of-plane rotation of the OH group, TS2 = (presumptive) TS associated withHAT, Post. = post-reaction complex, Prod. = products.773.4. SummaryFor both complex 1 and 5, steric clashing prevents significant pi-pi overlap or LP-pi overlap. Therefore, the reactions likely proceed through a direct HAT dominatedmechanism (kHAT ≈ kdirect). One might then expect these data to fall on thetrendline, however the formation of an encounter complex that does not lead directlyto HAT results in a different overall process from the other complexes. As a result ofthe necessary initial process complex 1 and 5 have lower A-factors as fewer collisionsare likely to lead to successful formal HAT.3.4 SummaryIn this investigation, a series of thermoneutral or nearly thermoneutral HATreactions were considered. I have plotted the logarithm of experimentally deter-mined A-factors against the theoretically determined electronic binding energies.These results demonstrate that the A-factors for (nearly) thermoneutral HAT re-actions correlate to some extent with the pre-reaction complex binding energies,given that the reactions proceed through similar mechanisms and energetically sim-ilar pathways. The results herein can be sorted into three bins by considering thecontributions of kdirect and kPCET to the overall transformation, kHAT :1. Complexes that have weak kPCET contributions due to either LP-LP or LP-piorbital overlap, and are therefore dominated by kdirect. This is the case forthe data that fall on the trendline.2. Complexes in which kPCET is the dominant contribution to kHAT , as is thecase for complexes 6 and 7.3. Complexes in which the encounter complex does not lead directly the HATTS complex, as was the case for complexes 1 and 5.tri-t-butylphenol at the LC-ωPBE-D3/6-311+G(2d,2p) level of theory.783.4. SummaryThese results indicate that different regimes of electronic and steric interactionslead to different chemical processes in seemingly similar reactions. As a results,non-covalent binding can be used as a metric for kinetics parameters, however, itcannot describe in full the entropic factors that contribute to the A-factor. Onemust first consider the mechanistic details in which formal HAT occurs.Additional work is necessary to extend these results. In particular, the mainquestion that remains is whether pi-pi PCET is “better” than other forms of orbitaloverlap. To answer this a larger sample of data points, and a diagnostic for PCETmust be used. Regardless, the results herein represent a novel attempt to linktheory and experiment. Given that obtaining the full PES for large molecules iscurrently computationally impractical, these results serve as a seed for developinga fundamental understanding of complex formal HAT reactions.79Chapter 4Interrogation of theBell-Evans-Polanyi Principle:Investigation of the BondDissociation EnthalpiesCorrelated with HydrogenAtom Transfer Rate Constants4.1 PrologueThe Bell-Evans-Polanyi (BEP) principle is a conceptual framework that states,for two closely related reactions, the difference in activation energy is proportionalto the difference in their enthalpies of reaction.47,48,159 This is commonly expressedas the linear free energy relationship (LFER): Ea = E0 + α∆H (Equation 1.1).Initially, the BEP principle was used as a simple model to explain the Brønstedcatalysis law, which states that the stronger an acid is, the faster the catalyzedreaction will proceed.160 A key assumption associated with the BEP principle is thatthe position of the TS along the reaction coordinate is the same for all reactions. Therelationship is described schematically in Figure 4.1: the more stable the product,the lower the reaction barrier.A modern utilization of the BEP principle is to estimate rate constants of relatedreactions. The main purpose of LFERs is to apply understanding of known systemsto new systems in order to develop novel chemical insight. For example, much804.1. PrologueFigure 4.1: Energy profiles for a series of related exothermic reactions illustratingthe Bell-Evans-Polanyi principle.of our groups’ work focuses on studying simple protein models. By thoroughlyinvestigating small systems with ab initio approaches, it is possible to extrapolatethe fundamental concepts to large-scale systems. Furthermore, if one can establishthat there exists a LFER between activation energy and bond strength for a specificmodel, then the difference in bond dissociation enthalpy (BDE) can be used toestimate HAT reaction rates in a large-scale protein system.The application of the BEP principle in HAT reactions, utilizes the relation-ship between the logarithm of the rate constant (kH) and the BDEs: log(kH) =α∆H+ constant. For HAT reactions involving abstraction by CumO , the enthalpyof reaction (∆H) is directly related to the strength of the breaking bond: ∆H =BDE(C H) − BDE(CumO H). If the relationship holds for a series of related HATreactions, then BDEs should correlate with the activation energy (Equation 1.1). Itwould then stand to reason that an increase in bond strength represents a desta-bilization in the TS complex, and thus a decrease in reaction rate. This concept814.1. Prologueis also important for the work in Chapter 5, where the interaction of non-redoxactive metal cations results in an increase in effective bond strength, and decreasein rate constant. It is also important to note that if the BEP principle breaks downfor reactions that appear related, then additional physico-chemical factors, such asnon-covalent binding (viz. Chapter 3), or stereo-electronics may be influencing thereaction barrier.An interesting application of the BEP principle is the work of Pratt et al. 161 ,in which the free radical oxidation of unsaturated lipids was examined. They stud-ied the correlation of theoretically determined allylic or benzylic C H and C OObond strengths with experimentally-measured HAT rate constants and O2 additionrate constants, respectively. BEP plots (log k vs. BDE) for a large range of polyun-saturated fatty acid models show good correlation (R2 = 0.92) for C OO bondsexamined, and reasonable correlation (R2 = 0.82) for C H bonds. This demon-strates that factors which affect BDEs also affect reaction barrier height, in linewith the BEP principle. Additionally, these results provide the ability to predictrate constants for HAT and oxygen addition reactions related to unsaturated lipidmodels, by means of calculating BDEs. Another area of research in which the BEPprinciple is often applied is heterogenous catalysis.162There is a significant gap in the literature on the BEP principle: there are nocriteria for how broadly the BEP principle can be utilized. In fact, the theoreticalvalidity of the BEP relationship has come into question, and a call has been madeto theoreticians for a detailed analysis of the BEP principle.163 In this work, I ex-plore the relationship between BDEs and log(kH). In order to achieve this, I havestudied HAT reactions involving the abstraction of C H bonds by CumO underthe same experimental conditions, for which many rate constants have been pub-lished.140,141,143,144,164–167 Additional unpublished rate constants have been provided824.1. Prologueby our experimental colleagues in Rome. The substrates of interest are diverse innature and include branched and cyclic alkanes, linear and cyclic amines and ethers,and substrates with allylic or benzylic C H bonds. The BDEs for these substratesrange from ∼76–100 kcal mol−1. The experimentally determined rate constants arelisted in Appendix B, Table B.1.The above studies, as well as many others, have used CumO and the closelyrelated t-butoxyl radical (t BuO ) as models for reactive oxygen-centred radicals instudying oxidative damage of biological macromolecules,168–170 as well as in studyingthe mechanism and efficiency of antioxidants.171–175 Using these radicals to studybiomolecular oxidation has an important caveat: The fundamental chemistry ofthese radicals is less well understood than is often assumed.40,41,176The BDE of CumO H is 106.7 kcal mol−1, a value that is larger than all theC H bonds studied herein.177 Therefore, HAT reactions involving CumO and theorganic substrates of interest are all exothermic on the order of 5–32 kcal mol−1.Hammond’s postulate178 states that the transition states for these HAT reactionshould most closely resemble the reactants in structure (i.e. an early TS), and theBEP α values should all be less than 0.5.179 If the BEP principle holds as a LFER,the substrates should be considered as if the BDEs were controlled by substituenteffects. For example, if one considers methane as the reference C H bond model,the BDE of toluene should reflect the effect of replacing one hydrogen with a phenylgroup. This is also the basis for schemes that utilize group additivity to predictmolecular heats of formation.155Considering this group additivity-like approach, our colleague in Rome, MassimoBietti, hypothesized that there should exist two general BEP relations for C Hbond: one in which the incipient radical is delocalized into a pi-system (benzylic orallylic), and one in which the remaining alkyl radicals are largely localized. Plotting834.1. Prologuethe experimental rate constants against literature BDEs (Figure 4.2) there appearsto be evidence for the two BEP relations.There is a considerable amount of scatter in Figure 4.2, thus possible outlierswere excluded from the initial linear-regression analysis. The scatter may be dueto differences in experimental procedures in the measurement of BDEs, which weredetermined using a large number of different experimental techniques. A great dealof data exists in the literature, much of which has conveniently been compiled inthe de facto reference for BDEs: the Handbook of Bond Dissociation Energies inOrganic Compounds.180 However, caution must be taken with experimentally de-termined BDEs, as not all experimental methods give reliable data. For example,BDEs from the Bordwell181 thermochemical cycle are possibly unreliable.144,182 Thiswas demonstrated for the BDE of dimethyl sulfoxide (DMSO), for which the experi-mentally determined BDE is about 8 kcal mol−1 lower than the best computationalestimate.144 Therefore, quantum chemistry is a useful tool for studying BDEs, asit is relatively simple to compute BDEs. The BDE for an arbitrary X H bond isgiven by:∆H(BDE) = H(X ) +H(H )−H(X H) (4.1)where ∆H(BDE) is the BDE, and the right-hand terms are the enthalpies of theradical product, the hydrogen atom, and the substrate, respectively. By computingthe most accurate BDEs possible, we are able to discern if the BEP principle holdsfor C H bond hydrogen abstraction by CumO .Many DFT-based methods have been shown to give reasonably reliable relativeBDEs, that is, the difference in BDE from a reference molecule (often CH4).183–185However, highly correlated wave function based methods are required to predictchemically accurate (sub-kcal mol−1) BDEs. For this purpose, we shall use com-844.1. Prologue1 1,4-diazobicyclo[2.2.2]octane 2 2,2-dimethylbutane3 2,2-dimethylbutane 4 Benzaldehyde5 Diethyl ether 6 Dimethyl sulfoxide7 Dioxane 8 Hexamethylphosphoramide9 Morpholine 10 Piperazine11 Piperidine 12 Pyrroldiine13 Tetrahydrofuran 14 Triethylamine15 1,4-cyclohexadiene 16 9,10-dihydroanthracene17 Cumene 18 Diphenylmethane19 Ethylbenzen 20 Toluene21 Adamantane (2◦) 22 Adamantane (3◦)23 Cycloheptane 24 Cyclohexane25 Cyclooctane 26 Cyclopentane27 Acetone 28 Acetonitrile29 Benzyl alcohol 30 Dibenzyl etherFigure 4.2: Bell-Evans-Polanyi plot of experimental rate constants (normalized forthe number of equivalent hydrogen atoms) for HAT between CumO and substratesagainst weakest literature C H BDEs of the substrates. BDEs for dimethyl sulfoxideand hexamethylphosphoramide are from Ref. 144, while all other BDEs are fromRef. 180. Points 21–26 and 29–30 were excluded from linear regression as possibleoutliers.854.2. Methodsposite quantum chemical procedures. Unfortunately, due to the computational costof some of these procedures, calculations are often limited to small molecules. Forinstance, Chan and Radom 184 recently published a diverse set of high-level BDEsfor small molecules with at most 5 heavy (non-hydrogen) atoms. There is a lackof literature that compares the ability of common composite methods to predictaccurate C H BDEs for relatively large molecules. Therefore, another aim of thework is to determine that composite procedure can be used to calculate accurateBDEs for relatively large molecules.4.2 MethodsExperimental rate constants were either provided from unpublished results fromour colleagues in Rome, or come from literature sources.140,141,143,144,164–167 All rateconstants come from laser flash photolysis (LFP) experiments of CumO with thesubstrates of interest. Nitrogen or argon saturated acetonitrile solvent and ambientconditions (298 K and 1 atm) were used in all cases. For those results that areunpublished, CumO is generated by laser pulses at either 266 nm or 355 nm insolutions of excess dicumyl peroxide. Many of the literature results are also fromthe Bietti group, where the same procedure is used. Other results may have smallvariations in experimental details; all results are well time-resolved.Observed rate constants (kobs) are generally obtained from 2–8 averaged trials,which are reproducible to within 5%. Transient absorption decay traces of CumOmonitored at 485 nm are used to determine kobs. The observed rate constant isplotted against concentration of substrate to provide the bimolecular HAT rateconstant (kH) as the slope (kobs = k0 + kH [substrate]). The CumO radical decaysunimolecularly through the β-scission of a methyl group, giving acetophenone and amethyl radical, as shown in Scheme 4.1. The unimolecular decay rate constant186,187864.2. Methodsfor CumO (k0) in acetonitrile is on the order of 6.3 ×105 s−1 at 298 K.OOCH3k0Scheme 4.1: Unimolecular decay of the cumyloxyl radical.All quantum chemical calculations were performed using the Gaussian 09 soft-ware package.116 Several composite quantum chemical methods that are imple-mented in Gaussian 09 were used in this work: W1BD, CBS-QB3 and the restrictedopen-shell variant ROCBS-QB3, CBS-APNO, G4 and the MP2 variant G4(MP2).An approach using ROCCSD(T) with locally-dense basis sets188,189 (LDBS) was alsoutilized in order to approximate W1BD results. Each of these methods is brieflydescribed below.4.2.1 Quantum chemical composite proceduresW1BDThe highest-accuracy method used is W1BD, which employs seven different cal-culations to obtain highly-correlated electronic energies, as well as thermochem-ically corrected quantities. This method is very computationally expensive, andthus cannot be applied to the larger species of interest in this work. Geometriesand thermochemical corrections come from DFT-based B3LYP calculations withnearly complete cc-pVTZ+d basis sets. A frequency scaling factor of 0.985 is usedto obtain thermochemical corrections. The electronic energy comes from severaladditive corrections involving the Brueckner Doubles95 (BD) variation of coupledcluster and large basis sets extrapolated to the complete basis set limit. Correctionsfor core-electron correlation and relativistic contributions are computed using an874.2. Methodsuncontracted variant of the cc-pVTZ+2df basis sets, known as MTsmall.190LDBS approachLocally-dense basis sets have been used in the past to calculated BDEs for rel-atively large molecules.188,189,191 The principle behind LDBS is to use large basissets to treat the atomic centres that are directly involved in the chemistry takingplace, and use progressively smaller basis sets for “remote” portions of the molecule,thus taking advantage of error cancellation. We chose a method that best approx-imates W1BD results for a small subset of molecules. The scheme utilized hereininvolves geometry optimization and scaled frequency calculations from DFT-basedB3LYP/cc-pVTZ+d, as used in the W1BD procedure. Single-point energy calcula-tions are then performed using ROCCSD(T) and an LDBS partitioning scheme wedenote as pc-3/3/2/1, demonstrated in Scheme 4.2, using the polarization consistentbasis sets.67884.2. MethodsHHigh-LevelMedium-LevelLow-LevelScheme 4.2: Locally-dense basis set partitioning used in the calculation of BDEs.The scheme is referred to as pc-3/3/2/1, where for the shown cumene molecule,the centre of C H cleavage and the immediately adjacent groups are treated withhigh-level pc-3 basis sets. The next groups are treated with medium-level pc-2 basissets, and all other atoms/groups are treated with low-level pc-1 basis sets.CBS methodsThe Complete Basis Set (CBS) methods of Petersson and colleagues92–94,192 arewidely used because of the relatively low computational cost (compared to othercomposite procedures), and well-established accuracy.193,194 CBS-QB392,93 utilizesDFT-based B3LYP optimization and scaled frequencies (factor = 0.990) with modi-fied triple-zeta Pople style basis sets. Electronic energies are obtained by extrapolat-ing medium basis set MP2 single point energy calculations to the complete basis setlimit, along with corrections for electron correlation from MP4(SQD) and CCSD(T).Small empirical corrections are added to achieve more accurate results compared tothe parametrization sets.195 ROCBS-QB3 is a similar procedure to CBS-QB3, ex-cept spin-restricted wave functions are used in place of unrestricted wave functions.894.2. MethodsThis is done to eliminate spin contamination, and the use of a restricted open-shelldefinition has been shown to produce more accurate BDEs.183 The (RO)CBS-QB3methods have been implemented for the first, second, and third periods of elements.Gn methodsThe Gaussian−n (Gn) series of methods originates from the Pople group,196 andG4 is the fourth generation. G4 utilizes moderately large basis sets and extrapo-lation techniques with CCSD(T) calculations to obtain highly correlated electronicenergies. G4(MP2) uses MP2 in place of CCSD(T) and is thus less computationallyexpensive, but also gives a less complete description of electron correlation. Bothmethods use the B3LYP/6-31(2df,p) level of theory for optimization and frequencycalculations with a frequency scaling factor of 0.9854. G4 results have been de-scribed as generally on par with CBS-QB3 results,193,194 but calculations are morecomputationally expensive.4.2.2 Transition state calculationsCalculations were performed to identify the lowest energy TS complex of severalreactions between CumO and organic substrates. In all cases cisoid and transoidconformations were explored. All optimization calculations were performed at theB3LYP-D3(BJ)/6-31+G∗ level of theory, followed by single-point energy calcula-tions at the B3LYP-D3(BJ)/6-311+G(2d,2p) level of theory with the SMD con-tinuum solvent model122 to approximate acetonitrile solvent effects. Transitionstates were visualized using the Chemcraft program197 to confirm a single imag-inary frequency connecting reactants to products. In some cases, a small secondaryimaginary frequency was observed, indicating a TS complex that is not fully op-timized. Necessary steps were taken to re-optimize the structures and eliminate904.3. Comparison of composite method for the prediction of BDEsthe small imaginary frequencies, but this was not always successful. Nonetheless, Iam confident the structures reported herein sufficiently represent the true TS com-plex geometries and relative energies. Results from structures that are not fullyoptimized are indicated appropriately as such.4.3 Comparison of composite method for theprediction of BDEsIn order to determine the best method for BEP principle analysis, and to inves-tigate which is the most efficient yet accurate composite method, the BDEs of 50species have been calculated. This set of species contains a wide variety of chemicalfunctionalities with BDEs ranging from 75–113 kcal mol−1, thus this set may bedescribed as a comprehensive test of these methods for C H BDEs. Given thatW1BD is the most accurate method used, these results have been used for compar-ison to other composite methods. However, due to computational cost restrictions,BDEs for only 33 out of the 50 species studied could be calculated using W1BD;hard disk capacity was insufficient for large systems. Therefore, literature BDEsfrom Luo 180 for the 49 species that have literature values in the set are also usedfor comparison. The literature and calculated BDEs are listed in Table 4.1.914.3. Comparison of composite method for the prediction of BDEsTable 4.1: Bond dissociation enthalpies of the species used to investigate the accu-racy of composite methods. Structures show an explicit C H bond for that whichis cleaved. All values are in kcal mol−1. Statistics are listed at the bottom of thetable.Molecule Structure Lit. W1BD ROCBS-QB3 G41,3-pentadieneH83.0 82.9 81.7 81.61,4-cyclohexadieneH76.0 76.3 75.0 75.21,4-diazabicyclo[2.2.2]-octaneNN H93.4 98.8 96.71,4-pentadieneH76.6 76.2 75.0 75.12,2-dimethylbutaneH98.0 99.3 99.3 97.52,3-dimethylbutane H 95.4 97.8 97.8 96.22-methylbutaneH95.8 97.3 97.1 95.99,10-dihydroanthraceneH76.3 78.1AcetaldehydeOH94.3 95.9 95.7 94.9924.3. Comparison of composite method for the prediction of BDEsAcetoneOH 96.0 96.9 96.7 95.4Acetonitrile NH97.0 96.9 96.6 96.3Adamantane (2◦)H98.4 100.4 97.8Adamantane (3◦)H96.2 99.9BenzaldehydeOH88.7 91.4 89.3Benzene H 112.9 113.1 113.0Benzyl AlcoholOHH79.0 83.2 83.4Cumene H 83.2 86.9 86.9CycloheptaneH94.0 95.8 93.9CyclohexaneH99.5 99.2 99.3 97.5CyclooctaneH94.4 92.4 90.2934.3. Comparison of composite method for the prediction of BDEsCyclopentaneH95.6 96.3 96.3 95.6Cyclopropane H 106.3 109.0 109.2 108.2Dibenzyl etherOH85.8 82.7Diethyl etherOH93.0 95.3 95.5 93.8DimethylamineHN H94.2 92.6 92.8 92.0Dimethylsulfoxide SOH 94.0 102.0 102.3 100.9DioxaneOOH96.5 97.3 97.6 95.7DiphenylmethaneH84.5 82.8EthaneH100.5 101.3 101.5 100.7EthylbenzeneH85.4 87.6 87.6EthyleneH110.9 110.8 110.9 109.9FluoreneH82.0 81.9FormaldehydeOH88.0 88.6 88.9 88.2944.3. Comparison of composite method for the prediction of BDEsHexamethyl-phosphoramidePONNNH93.9IndeneH83.0 80.1 79.0Methane H3C H 105.0 105.0 105.2 104.5Methanol HOH96.1 96.4 96.8 96.0Methylamine H2NH93.9 93.1 93.3 92.7MorpholineNHOH92.0 93.3 91.8N,N-dimethylacetamideONH91.4 99.6 99.5 97.6PiperazineNHHNH93.0 93.4 93.5 91.9PiperidineNHH89.5 92.1 92.2 90.7PropaneH100.9 101.6 101.8 100.7PyrrolidineNHH89.0 90.8 90.7 89.5954.3. Comparison of composite method for the prediction of BDEsTetrahydro-2H-pyran OH96.0 96.3 96.5 94.7TetrahydrofuranOH92.1 93.7 93.8 92.2TolueneH89.7 90.5 89.7 89.8Trichloromethane ClClClH 93.8 93.5 93.7 92.4TriethylamineNH90.7 91.2 89.4TrifluormethaneFFFH106.4 107.2 107.4 105.8Statistics Lit. W1BD ROCBS-QB3 G4Number of BDEs (N) 49 33 50 43Relative to Literature ValuesMAE 0.82 1.64 1.21Max. Error 1.59 3.15 4.19Min. Error -8.22 -8.25 -6.86Relative to W1BD ValuesMAE (W1BD) 0.18 0.70Max. Error 1.26 2.05Min. Error -0.35 0.37964.3. Comparison of composite method for the prediction of BDEsMean absolute error (MAE) is used to assess the quality of computational meth-ods, where errors are calculated with respect to benchmark values for a given dataset.198 The MAE is calculated asMAE =1N∑|Eref − Ecalc| (4.2)where, for a set of N reference values, the MAE is the average of the mean differ-ences of the reference energy (Eref ) and the calculated value (Ecalc). The MAE withrespect to W1BD and literature shall be reported herein as “MAEW1BD (MAELit.)”.An additional semi-quantitative metric that I used to evaluate the accuracy of com-posite procedures to reproduce experimental results is a bar chart that summarizesthe number of deviations from literature within given error ranges. This bar chart isreported in Figure 4.3. Note that calculations for some species with some methodsfailed to converge, thus number of BDEs out of 49 are also shown in Figure 4.3.Also, an alternative method that I shall utilize for reporting these data is throughthe use of one-to-one plots, in which BDEs from two methods are directly compared.An ideal plot should have a slope = 1 and y-intercept = 0.Comparing W1BD results to literature, the MAE is 0.82 kcal mol−1, and themajority of the data match to within 1–2 kcal mol−1 of literature. Thus, W1BDis largely consistent with the literature values. Additionally, the one-to-one plotscomparing W1BD to literature in Figure 4.4 show reasonable agreement with slopeof 0.98 and a y-intercept of 2.35. There are two notable outliers: DMSOiii andN,N -dimethylacetamide, for which experiment underestimates the BDEs by -8.0and -8.2 kcal mol−1, respectively. DMSO and N,N -dimethylacetamide are consis-tently outliers amongst all composite methods, suggesting the literature BDEs areincorrect.iiiThe experimental BDE for dimethyl sulfoxide was previously identified as being inaccurate by974.3. Comparison of composite method for the prediction of BDEsFigure 4.3: Summary of deviations of BDEs from literature or composite quantumchemical methods. Errors are units of kcal mol−1 and are relative to Ref. 180. aIncludes BDEs for 33/49 substrates. b Includes BDEs for 40/49 substrates.984.3. Comparison of composite method for the prediction of BDEsFigure 4.4: One-to-one plot of BDEs from literature180 and as calculated by theW1BD composite method. The red dashed-line represents the least squares line ofbest fit, while black line represents a perfect one-to-one correlation.The method that gives the best combined agreement with W1BD and literatureis ROCBS-QB3 with an MAE from W1BD (literature) = 0.18 (1.64) kcal mol−1. Itis also apparent, from the one-to-one plots in Figure 4.5, that ROCBS-QB3 matcheswell with literature and experiment. In comparison, CBS-QB3 has an MAE = 0.32(1.88) kcal mol−1, while CBS-APNO has an MAE = 0.20 (1.40) kcal mol−1. TheLDBS approach also performs well with an MAE = 0.22 (1.60) kcal mol−1. The G4method deviates from the W1BD reference by about 0.5 kcal mol−1 more, however,it appears to give reasonable agreement with experimental results (MAE = 0.70(1.21) mol). The use of the MP2 variant of G4 gives somewhat questionable results,with an MAE of 0.88 (1.60) kcal mol−1, as well as a large outlier of 6.2 kcal mol−1Salamone et al.144994.3. Comparison of composite method for the prediction of BDEsthat is not present in the other data from composite methods. One-to-one plots ofall other methods are presented in Appendix B.Figure 4.5: One-to-one plot comparing calculated BDEs calculated by the ROCBS-QB3 to reference literature180 and W1BD BDE values, respectively. The red dashed-line represents the least squares line of best fit, while black line represents a perfectone-to-one correlation.In summary, ROCBS-QB3 performs best for the calculation of C H BDEs whileG4(MP2) performs worst. Given these data, and considering the relative compu-tational cost, I recommend the ROCBS-QB3 for the calculation of accurate BDEs,particularly for large molecules for which more expensive computational methods arenot possible. Importantly, we can now confidently continue investigating the BEPrelationships using reliably calculated BDE data from the ROCBS-QB3 method.Furthermore, these results can be extended to even larger systems as the ROCBS-QB3 approach is one of the least computationally-expensive composite methods.For example, calculations on the cyclohexane molecule take about 20 minutes usingROCBS-QB3 on SGI Altix compute nodes with 6 processors and 8 GB RAM, whileG4 takes approximately 27 times longer, the LDBS approach takes about 500 timeslonger, and W1BD takes about 1100 times longer.1004.4. Analysis of the Bell-Evans-Polanyi principle4.4 Analysis of the Bell-Evans-Polanyi principleWe turn now to the application of accurate BDEs to the BEP principle. Exper-imental HAT rate constants have been collected for 32 reactions involving CumOand organic substrates. The BEP plot of the logarithm of rate constants dividedby the number of equivalent H atoms (i.e. normalized) against BDEs is shownin Figure 4.6.As with the experimental results in Figure 4.2, there clearly exists two distinctregions in Figure 4.6. This is congruent with the initial hypothesis that there shouldexist two linear relations: one for allylic/benzylic C H bonds, and another for alkylC H bonds. However, there remains a considerable amount of scatter in the data,thus correlation of the expected BEP relations is poor. For the allylic/benzylic seriesof C H BDEs, for which bond scissions results in a radical that is delocalized, thecoefficient of determination is R2 = 0.89. This result is consistent with work ofPratt et al.,161 which obtained an R2 = 0.82 from the BEP plot for the abstractionof C H bonds from models for unsaturated fatty acids. Most of the rate constantsused in the work of Pratt et al. are for the abstraction of C H by peroxyl radicals,which were obtained through an experimental method that gives estimated HATrate constants with large associated errors. Thus, they suggested that the degreeof scatter is associated with experimental errors. The same however cannot be saidfor the rate constants associated with this work. Therefore, there may be additionalphysico-chemical factors at play.The alkyl C H BDEs show very weak correlation with CumO HAT rate con-stants, with an R2 = 0.63. One possibility is that applying the BEP principle tosuch a large grouping of substrates is inappropriate. Thus, I have re-plotted thisdata in Figure 4.7, breaking the data into several smaller chemical groupings: cyclicalkanes, other alkanes (branched and adamantane), hydrogen bond donating (H-1014.4. Analysis of the Bell-Evans-Polanyi principle1 Acetone 2 Acetonitrile3 Cyclopentane 4 Cyclohexane5 Cycloheptane 6 Cyclooctane7 2,2-dimethylbutane 8 2,3-dimethylbutane9 Adamantane (2◦) 10 Adamantane (3◦)11 Diethyl amine 12 Piperazine13 Piperidine 14 Pyrrolidine15 Morpholine 16 Propylamine17 Triethylamine 18 1,4-diazobicyclo[2.2.2]octane19 Tetrahydrofuran 20 Dioxane21 Dimethyl sulfoxide 22 Benzaldehyde23 Hexamethylphosphoramide 24 Diethyl ether25 1,4-cyclohexadiene 26 Toluene27 Benzyl alcohol 28 Ethylbenzene29 Cumene 30 Diphenylmethane31 Dibenzyl ether 32 9,10-dihydroanthraceneFigure 4.6: Bell-Evans-Polanyi plot of experimental rate constants (normalized forthe number of equivalent hydrogen atoms) for HAT between CumO and substratesagainst BDEs calculated using the ROCBS-QB3 method. Acetone and acetonitrileare note included in fitting as the experimental rate constants are approximate.1024.4. Analysis of the Bell-Evans-Polanyi principleBond Donors) species, and other C H bonds with heteroatom neighbours (Het.Neighbours). Doing so appears to reveal one reasonably well-correlated trend forC H bonds with heteroatom neighbours (R2 = 0.84). There are two data pointsfor the tertiary amines (triethylamine and 1,4-diazobicyclo[2.2.2]octane) that do notfit well with the expected trend, however it is unclear why they do not fit into theother points in the heteroatomic neighbours trend. Excluding points for the tertiaryamines results in an R2 = 0.92.The cyclic alkanes are somewhat poorly correlated (R2 = 0.73). Within the“other alkanes” grouping there are two branched alkanes, 2,3-dimethylbutane and2,2-dimethylbutane, as well as the secondary and tertiary C H positions of adaman-tane. There may be a separate correlation for each of these, but there are too fewdata points to make this assertion. Extremely poor correlation is observed for boththe hydrogen bond donating species (R2 = 0.02). This is likely due to the formationof a hydrogen bonded pre-reaction complex that does not allow for HAT to occurwithout some subsequent rearrangement (See Chapter 3). In general, there areno evident reasons on the basis of group-additivity based arguments that explainthe poor correlations observed. Thus, the lack of simple relationships is perhapsevidence against the validity of the BEP principle. However, before making anyconclusions, we must consider if there are any explanations that arise from examin-ing the transition state structures.1034.4. Analysis of the Bell-Evans-Polanyi principle3 Cyclopentane 4 Cyclohexane5 Cycloheptane 6 Cyclooctane7 2,2-dimethylbutane 8 2,3-dimethylbutane9 Adamantane (2◦) 10 Adamantane (3◦)11 Diethyl amine 12 Piperazine13 Piperidine 14 Pyrrolidine15 Morpholine 16 Propylamine17 Triethylamine 18 1,4-diazobicyclo[2.2.2]octane19 Tetrahydrofuran 20 Dioxane21 Dimethyl sulfoxide 22 Benzaldehyde23 Hexamethylphosphoramide 24 Diethyl etherFigure 4.7: Further breakdown of Bell-Evans-Polanyi plot of experimental rate con-stants (normalized for the number of equivalent hydrogen atoms) for HAT betweenCumO and substrates.1044.5. Transition state analysis4.5 Transition state analysisIn order to determine if there are any reasons for the breakdown of the BEPprinciple, I have calculated TS structures for 20 of the reactions at the B3LYP-D3(BJ)/6-311+G(2d,2p)//B3LYP-D3(BJ)/6-31+G∗ level of theory. Note that theinclusion of the SMD continuum solvent model decreases the agreement in calculatedrate constant with experiment, and so gas-phase results are reported herein (SeeAppendix B, Figure B.2). The experimental and calculated HAT reaction rateconstants (kH) agree reasonably well (within 2 orders of magnitude) and are listedin Table 4.2, along with the free-energy barrier heights (∆G‡) and the decompositioninto enthalpic and entropic terms: ∆G‡ = ∆H‡ − T∆S‡.First, consider some general features associated with the TS complexes listedin Table 4.2. One factor that may lead to deviations from the BEP principle isthe possibility for different HAT reaction mechanisms, i.e. direct HAT or PCET.Consider first the reaction of toluene with CumO . As this reaction is similar tothe self-exchange reaction of the benzyl-toluene couple as described by DiLabioand Johnson,26 one might expect the reaction to proceed via PCET. The lowest-energy TS complex has a partially pi-stacked conformation with the rings orientedapproximately 40◦ relative to one another. Examination of the SOMO and HOMOreveals no pi-pi partial bonding interaction, as can be seen in Figure 4.8. The electrondensity of the SOMO is largely localized on the toluene portion of the complex.This is likely due to the additional non-conjugated carbon centre of CumO , whichprevents an additional electron channel for PCET to occur. Therefore, this reactiontakes place through direct HAT, as has been previously described.140 This behaviouris specific to the CumO radical, thus all the reactions likely also take place througha direct HAT dominant mechanism, and this should not factor into the deviationsin the observed BEP principle relationships.1054.5. Transition state analysisTable 4.2: Reaction barrier heights for reactions of substrates with CumO calcu-lated in the gas phase at 298 K at the B3LYP-D3(BJ)/6-311+G(2d,2p)//B3LYP-D3(BJ)/6-31+G∗ level of theory. Rate constants are in units of M−1s−1, while allother values are in units of kcal mol−1. ID numbers match those in Figure 4.6. †TSstructure could not be fully optimized and contains a small additional imaginaryfrequency.ID Substrate kH(expt.) kH(calc.) ∆G‡ ∆H‡ −T∆S‡Excluded1 Acetone < 1×104 2.8×102 14.1 0.1 14.02 Acetonitrile < 1×104 2.9×102 14.1 2.6 11.5Cyclic Alkanes3 Cyclopentane† 9.5×105 5.5×104 11.0 -2.0 13.04 Cyclohexane 1.1×106 5.8×104 10.9 -1.1 12.16 Cyclooctane† 3.0×106 2.7×104 11.4 -2.7 14.1Other Alkanes7 2,2-dimethylbutane 9.5×104 1.1×106 9.2 -1.2 10.48 2,3-dimethylbutane 5.6×105 2.5×106 8.7 -3.3 12.09 Adamantane (2◦) 6.9×106 2.0×105 10.2 -1.8 12.010 Adamantane (3◦) 6.9×106 1.1×107 7.9 -3.9 11.8H-Bond Donor11 Diethylamine 1.1×108 8.2×109 3.9 -9.1 13.0Heteroatom Neighbours18 1,4-diazobicyclo[2.2.2]octane 9.6×106 5.1×107 6.9 -5.3 12.320 Dioxane 8.2×105 1.2×107 7.8 -4.0 11.821 Dimethyl sulfoxide 1.8×104 2.7×104 11.4 -0.5 11.922 Benzaldehyde† 1.2×107 2.5×107 7.4 -5.6 12.923 Hexamethylphosphoramide 1.9×107 3.9×107 7.1 -7.3 14.424 Diethyl ether 2.6×106 5.9×107 6.8 -4.6 11.4Allylic/Benzylic25 1,4-cyclohexadiene 6.6×107 1.1×108 6.5 -6.5 13.026 Toluene 1.8×105 2.0×105 10.2 -3.5 13.729 Cumene 5.6×105 1.9×107 7.5 -5.8 13.330 Diphenylmethane† 8.7×105 1.8×106 8.9 -5.6 14.532 9,10-dihydroanthracene† 5.0×107 2.2×108 6.1 -8.5 14.61064.5. Transition state analysisA.B.Figure 4.8: Structures of TS for HAT between CumO and toluene with A. SOMOand B. HOMO. The elements are coloured as grey for carbon, white for hydrogen,and red for oxygen. MOs are shown with an isovalue of 0.02 e−/A˚3.1074.5. Transition state analysisFor all of the TS structures of the reactions in Table 4.2, a conformation thatmaximizes non-covalent interactions while minimizing steric repulsion is adopted. Inthe cases of acetone, acetonitrile, hexamethylphosphoramide (HMPA), and DMSO,a very weak hydrogen bonding interaction is formed between the X O (or C Nfor acetonitrile) moieties and the C H of the methyl of CumO . In all but twocases, this involves a cisoid (partially-stacked) complex so that dispersion interac-tions are maximized. The two outliers are benzaldehyde and cyclooctane. In orderfor benzaldehyde to adopt a cisoid TS structure, there are two possibilities. First, aT-shaped conformation could be adopted, rather than a slipped-parallel pi-stackedconformation. On the basis of the benzene-benzene non-covalently bound dimer,34the T-shaped conformation is very slightly favourable compared to the pi-stackedconformation by circa 0.1 kcal mol−1. However, forming the T-shaped TS structurewould require a rotation of nearly 90◦ of the C(CH3)2O of CumO , which has a pre-dicted energetic costiv of 4.2 kcal mol−1, and so this conformation is unlikely. Onthe other hand, the C(CH3)2O of CumO could rotate to accommodate a partiallyslipped-parallel pi-stacking conformation in the TS complex. Note that I was unableto obtain TS structures for either of the described possible cisoid conformations forthe benzaldehyde-CumO TS complex, as geometry optimization calculations didnot converge. For cyclooctane, the difference in free energy between the cisoid andtransoid TS structures is 1.8 kcal mol−1Both structures contain a secondary smallimaginary frequency, and thus do not represent true TS structures. The reason forthe transoid TS structure being more stable is somewhat unclear, but it is possiblethat the non-optimal nature of the TS structures is the cause. Furthermore, it ispossible that the cyclooctane molecule undergoes a conformational change in form-ing the TS complex that was not accounted for in these calculations. Cyclooctane1084.5. Transition state analysishas many conformations that are close in relative energy.199TS complex structures and mechanism aside, there is one striking feature in thereaction barriers calculated for HAT reactions involving CumO : all the reactionsstudied are entropy-controlled at 298 K. This means that the free-energy barrier,and thus rate constant, is controlled by the entropic contributions, rather thanthe enthalpic contributions, i.e., −T∆S‡ > ∆H‡. From the results in Table 4.2,it can be said that −T∆S‡ >> ∆H‡ for hydrogen abstraction by CumO . Oneinterpretation of these results is that CumO is so highly reactive that HAT isgoverned by trajectory, orientation, and degrees of freedom, factors that are normallyassociated with the A-factor in Arrhenius theory. In fact, in many cases, ∆H‡ iscalculated to be negative with respect to the separated reactants. This implies thata pre-reaction complex is formed, which, as was demonstrated in Chapter 3, canhave significance on HAT reactivity with respect to the magnitude of the A-factor.Pre-reaction complex structures were not calculated in this work. In some cases thesystems have been studied in combined experimental and theoretical work. Someexamples of previously calculated CumO + substrate pre-reaction complex bindingenthalpies are: HMPAv ∆H ≈ -6 kcal mol−1, DMSO ∆H ≈ -5 kcal mol−1, and1,4-diazobicyclo[2.2.2]octane (DABCO)41 ∆H ≈ -0.1 kcal mol−1. Note that thecalculated enthalpic barrier herein is -5.3 kcal mol−1 for DABCO, a result thatcan be ascribed to differences in computational methods. In Ref. 41, no dispersioncorrection was used, thus accounting for a less stable TS complex and pre-reactioncomplex. The calculated difference in enthalpy from pre-reaction complex to TScomplex for DABCO was found to be only 1.0 kcal mol−1.The fact that hydrogen abstraction by CumO is entropy-controlled is perhapsunsurprising, given the work of Finn et al. 40 , which demonstrated that HAT re-ivCalculated at the B3LYP-D3(BJ)/6-311+G(2d,2p) level of theory.vDMSO and HMPA were studied in Ref. 144 at the B3LYP-DCP200/6-31+G(2d,2p) level of theory1094.6. Is the Bell-Evans-Polanyi principle valid?actions involving various organic substrates and the closely related radical t BuOare also entropy-controlled at room temperature. Furthermore, it has been shownthat CumO and t BuO display very similar hydrogen atom abstraction reactiv-ities.140,201–203 It is surprising then that these radicals remain so often applied asproxies for reactive oxygen species in kinetic studies. Future work should use ex-treme caution in applying CumO and t BuO as chemical probes, as has been notedin the past.40,41,140 Note also that it is uncommon to encounter entropy-controlledreactions in organic chemistry, and they are often associated with non-Arrheniuskinetic behaviour. Other examples of reported entropy-controlled reactions includethe addition of a carbene across a multiple bond,204,205 and radical-radical recom-bination reactions.206Classical physical organic chemical literature can explain why the reactions thatare entropy-controlled do not follow “normal” LFERs.207 Blackadder and Hinshel-wood 208 defined three classifications for different types of LFERs, the first of whichapplies to the BEP principle: A series of reactions with constant entropy are con-trolled by enthalpy changes that are based on electronic effects that do not affect theform of the TS. Therefore, reaction rates that involve non-isoentropic TS complexformation will not correlate with bond strengths, as is observed herein. It seemsprudent at this point to suggest that expecting reactions to be isoentropic withrespect to transition state formation is an over-simplification, especially given thenumber of factors that contribute to entropy in solvent phase chemistry.4.6 Is the Bell-Evans-Polanyi principle valid?The question still remains whether the BEP principle is valid or not. Recallfrom Equation 2.93 that Ea is related directly to ∆H‡. Thus, if the BEP principlestill holds for HAT reactions between CumO and organic substrates, then the cal-1104.6. Is the Bell-Evans-Polanyi principle valid?culated values of ∆H‡ should be a function of C H BDE. These data are plottedin Figure 4.9.1 Acetone 2 Acetonitrile3 Cyclopentane 4 Cyclohexane6 Cyclooctane 7 2,2-dimethylbutane8 2,3-dimethylbutane 9 Adamantane (2◦)10 Adamantane (3◦) 11 Diethyl amine18 1,4-diazobicyclo[2.2.2]octane 20 Dioxane21 Dimethyl sulfoxide 22 Benzaldehyde23 Hexamethylphosphoramide 24 Diethyl ether25 1,4-cyclohexadiene 26 Toluene29 Cumene 30 Diphenylmethane32 9,10-dihydroanthraceneFigure 4.9: Bell-Evans-Polanyi plot of calculated enthalpic barriers for HAT betweenCumO and substrates against BDEs calculated using the ROCBS-QB3 method.Perhaps unsurprisingly at this point, there is once again a great deal of scatterin the data. The cyclic alkanes fit into a linear relationship that is very well corre-lated (R2 = 0.98). However, all other chemical groupings show very poor correla-1114.6. Is the Bell-Evans-Polanyi principle valid?tion. Therefore, the correlation seen for the cycloalkanes is an adventitious exampleof the BEP principle showing a linear relation between ∆H‡ and BDE. Even thesubstrates with allylic/benzylic C H bonds show only weak correlation in a BEPrelation (R2 = 0.63), although the experimental results show a reasonable correla-tion between log(kH/n) and calculated BDEs. Therefore, the experimental resultsare likely serendipitous, especially considering the reactions are entropy-controlledand non-isoentropic.Further analysis of the allylic/benzylic relation demonstrates a clear breakdownin the BEP principle. If one begins with toluene with a BDE of 89.7 kcal mol−1and ∆H‡ of -3.5 kcal mol−1, then the addition of two methyl substituents formscumene, with a BDE of 86.9 kcal mol−1 and ∆H‡ of -5.8 kcal mol−1, indicatingthe relative stabilization of the TS by substituents. However, if one adds anotherphenyl group instead of two methyl groups, diphenylmethane is obtained, which hasa BDE of 82.8 kcal mol−1. This indicates that phenyl is a better radical stabilizinggroup, however ∆H‡ is -5.6 kcal mol−1, which is slightly higher than that of cumene.The difference can be partially attributed then to differences in progress along thereaction coordinate. Evidence of this difference is the spin density localized on theO-centre of CumO in the TS complex, which should go to zero as the reactantsmove to products. The O spin densities are 0.477 e−, 0.528 e−, and 0.518 e−for toluene, cumene, and diphenylmethane, respectively. Therefore, the progressalong the reaction coordinate is furthest for toluene, and progressively less far fordiphenylmethane and cumene. Note, that the O spin densities for cyclopentane,cyclohexane, and cyclooctane are 0.455 e−, 0.452 e−, and 0.458 e−, respectively.Therefore the progress along the reaction coordinate for the cycloalkanes is roughlythe same, which may explain the R2 of nearly 1.Such contradictory data makes it very difficult to draw any conclusions. Instead,1124.6. Is the Bell-Evans-Polanyi principle valid?I shall make some suggestions as to why the BEP principle is an incomplete theo-retical construct for describing the HAT reaction of CumO with organic substrates:1. HAT reactions between CumO and these organic substrates may be decidedlyexothermic, resulting in reactions with no enthalpic barrier associated withthe breaking of a C H bonds and the formation of an O H bond. This issupported by the fact that the calculated enthalpic barriers are all very low oreven negative. Therefore, any remaining nominal activation energy is a resultof stereo-electronic interactions between CumO and the substrate. The highreactivity of CumO also suggests that abstraction from the weakest bond ina substrate will not always occur. The site of abstraction will most likely bedetermined by the orientation of the substrate upon collision. This is likely anadditional reason why log kH/n does not correlate with the calculated C HBDEs.2. Polar effects have been shown to be extremely important in the stability ofthe TS complex.209 The species involved in HAT reactions are often neu-tral radicals, thus the influence of charge transfer in the TS complex canhave important implications. Consider the TS of a generic HAT reactionin Scheme 4.3, there are four obvious resonance forms. Oxygen-centred rad-icals are electrophilic in nature, thus the importance of the third resonancestructure increases. The BEP principle does not account for polarity in the TScomplex, as these effects are not captured by the BDE of the substrate, thus∆H‡ does not correlate well with BDE. This issue was addressed by Robertsand Steel 210 , who suggested an extension of the BEP principle to includesimple empirical parameters that capture the polar effects in the transitionstate.1134.6. Is the Bell-Evans-Polanyi principle valid?[O H C]‡[O H C]‡ [O HC ]‡ [O:–H C+]‡ [O+H C:–]‡Scheme 4.3: A generic HAT transition state structures and possible resonance forms.3. The BEP principle is an over-simplification that does not capture nearlyenough of the physics associated with the deceptively complex hydrogen ab-straction reactivity of CumO (or t BuO ). Therefore, I suggest that theBEP principle should not be used as a tool for predicting activation energiesor rate constants. One method that has been popularized by Mayer is theuse of Marcus cross-relations.29 This predictive method has also been used toexplain reactions that have negative enthalpic barriers.129 An alternative ap-proach is that of Zavitsas, that predicts activation energies based on so-called“triplet repulsion”vi and radical delocalization.211,212 It is clear from the anal-ysis herein that the BEP principle is valid only as a conceptual framework,rather than a broadly applied linear relationship.viZavitsas uses the term “triplet repulsion” to describe repulsion between the parallel spins of thehydrogen atom acceptor and donor atoms (↑↓↑ or ↓↑↓) in the TS complex.114Chapter 5Do non-redox active metalcations have the potentials tobehave as chemo-protectiveagents? The Effects on MetalCations on HAT ReactionBarrier Heights5.1 PrologueMetal cations are ubiquitous in biological systems and play an important rolein biological function. As such, there is a great deal of interest in studying met-als in biological systems. Proteins in particular are often associated with metals,and in the worldwide Protein Data Bank,213,214 over one-third of crystal structurescontain metals. Redox active metals, such as copper and iron, act as co-factors inmetalloenzymes for important catalytic processes.215Non-redox active metal cations are equally as important in biological function asredox active metals, where they are essential to protein structure and function, alongwith cellular and neuronal signalling.216 Sodium and calcium ions are most abun-dant extracellularly, while potassium and magnesium are dominant inside of cells.Specific ionic concentrations vary dramatically depending on physiological condi-tions; estimates for equilibrium concentrations in both mammalian heart cells217and blood plasma218 are listed in Table 5.1. As sodium and magnesium are the1155.1. Prologuemost abundant alkali and alkaline earth metals found in biologically relevant sys-tems, they are of prime interest for my investigation.Table 5.1: Ionic concentrations inside a mammalian heart cell and in the bloodplasma. Concentrations are in units of mM. Values are rounded to one significantfigure. Data are from Ref. 217 and 218.Ion Conc. Mammalian Cells Blood PlasmaNa+ 10 100–200Mg2+ 10 1K+ 100 4Ca2+ 0.1 2Extensive crystallographic surveys indicate that metals bind predominantly tooxygen centres in proteins.219–221 Divalent metals are most often found bound di-rectly to proteins. Calcium binds anywhere from 4 to 6 binding sites in proteincrystal structures, while magnesium binds only 1 or 2. Monovalent metals, on theother hand, are often heavily solvated and so they appear in solvent cavities of pro-teins, although sodium or potassium are sometimes found bound directly to carbonylor carboxylate oxygen centres.213A great deal of research has focussed on Ca2+ in the context of reactive oxygen-centred radical production.222 Specifically, Ca2+ ions are important in the mito-chondria, where, depending on physiological conditions and concentrations, theycan act as inhibitors or promoters of free-radical production in the electron trans-port chain.223 One explanation is that Ca2+ induce conformational changes of theproteins involved in the electron transport chain that are responsible for radicalgeneration.224 Mitochondrial free-radicals, when present in moderate amounts, canact as cell signalling molecules to activate pro-growth responses.225 However, “dys-functional” mitochondria can produce excess radicals leading to oxidative damagethat has been linked to degenerative diseases.Given the significant importance alkali and alkaline earth metals play in biolog-1165.1. Prologueical systems, their impact on protein oxidation must be considered. However, untilrecently, kinetic studies of protein oxidation have not investigated the mechanisticrole of non-redox active metals. In a series of three papers,49–51 Bietti and colleaguesshowed that alkali and alkaline earth metals have an inhibitory effect on HAT re-actions involving CumO and organic substrates. Some of the experimental rateconstants from these papers are summarized in Table 5.2. All rate constants wereobtained by time-resolved LFP in nitrogen or argon-saturated acetonitrile (MeCN)at 298 K, as was previously described in Section 4.2. The experimental results havebeen rationalized on the basis of Lewis acidic metal cation interactions with Lewisbasic substrates.For hydrocarbons, cyclic ethers, and tertiary amines, rate constants for hydrogenabstraction by CumO in the presence of excess concentrations of lithium and mag-nesium salts were measured.49 In the presence of LiClO4 and Mg(ClO4)2, the rate ofabstraction by CumO from 1,4-cyclohexadiene (CHD) increases very slightly. SinceCHD has no Lewis basic centres, the increase in HAT rate constant was explainedon the basis of metal cation interactions with CumO , very slightly increasing thehydrogen abstraction ability by withdrawing electron density from the aromaticring. Metal cations were also shown to increase the unimolecular decay of CumOby β-scission (See Section 4.2). The largest kinetic effect was observed with Li-ClO4 with kβ = 1.8×106 s−1, which is a roughly 3-fold increase as compared tothe rate in MeCN at 298 K (kβ = 6.3×105 s−1).187 This effect is significantly lessthan the observed kinetic solvent effect on CumO β-scission measured in H2O or2,2,2-trifluoroethanol (kβ = 1.0×107 and 6.1×106 s−1, respectively).226,227 There-fore, the kinetic effects of these alkali and alkaline metal salts interacting via Lewisacid-base interactions with the oxygen-centre of CumO are less than the effects ofhydrogen-bonding by solvents.1175.1. PrologueTable 5.2: Summary of rate constants for reactions of CumO with various organicsubstrates in the presence of alkali and alkaline earth metal salts.Substrate Conditions kH (M−1s−1) kH(MeCN)/kH(Mn+)1,4-cyclohexadiene 6.7×107(CHD) LiClO4 1.0 M 7.5×107 0.89Mg(ClO4)2 1.0 M 7.0×107 0.96tetrahydrofuran 5.7×106(THF) LiClO4 1.0 M 2.9×106 1.7LiOTf 1.0 M 2.8×106 2.0Mg(ClO4)2 1.0 M 1.8×106 3.2triethylamine 2.0×108(TEA) LiClO4 1.0 M 9.4×107 2.1Mg(ClO4)2 0.005 M <1×106 >200N,N -dimethylformamide 1.2×106(DMF) LiClO4 0.5 M kH1 = 8.9×105 1.3kH2 = 1.5×106 0.80NaClO4 0.2 M kH1 = 9.6×105 1.3kH2 = 1.4×106 0.86Mg(ClO4)2 0.2 M kH1 = 5.8×105 2.1kH2 = 1.1×106 1.1Ca(ClO4)2 0.2 M kH1 = 1.0×106 0.83N,N -dimethylacetamide 1.2×106(DMA) LiClO4 0.2 M kH1 = 8.5×105 1.4kH2 = 1.5×106 0.8NaClO4 0.2 M kH1 = 1.1×106 1.1kH2 = 1.3×106 0.92Mg(ClO4)2 0.2 M kH1 = 4.7×105 2.6kH2 = 2.4×105 5.0kH3 = 1.1×106 1.1Ca(ClO4)2 0.2 M kH1 = 1.2×106 1.01185.1. PrologueNext, the HAT rate constants for abstraction from tetrahydrofuran (THF) de-crease in the presence of non-redox active metal salts. Both LiClO4 and LiOTfdecrease kH by a factor of about 2, indicating the nature of the counter-anionplays a negligible role in the Lewis acid-base interactions between metal cationsand substrates. The addition of Mg(ClO4)2 has a greater effect on HAT reactiv-ity, decreasing kH by a factor of 3. The magnesium ion is a stronger Lewis acidthan lithium,228 supporting the notion of Lewis acid-base interactions between theoxygen lone-pair and the metal cations. The decrease in kH has been partially at-tributed to the reduction in electron density in the C H σ∗ anti-bonding orbitalwhich is normally present due to hyperconjugative overlap with the neighbouringoxygen lone-pair (See Scheme 1.4), as a consequence of the metal cation withdrawingelectron density from the oxygen lone-pair.A 2-fold decrease in kH for the tertiary amine, triethylamine (TEA), is observedupon the addition of LiClO4, for which an analogous orbital interaction explanationis also appropriate. Interestingly, the addition of 1.0 M Mg(ClO4)2 was reportedto immediately form a precipitate. This precipitate was identified as the formationof a strong TEA-Mg2+ Lewis acid-base adduct. This observation is once againconsistent with the stronger Lewis acidity of Mg2+ as compared to Li+, and alsothe significantly greater Lewis basicity of TEA vs THF.49,229 It was also pointedout that MeCN will competitively bind with metal cations, but it is a weaker Lewisbase than both THF and TEA. Measurements of kH for HAT between CumO andTEA in the presence of 0.005 M Mg(ClO4)2 were successful only up until [TEA] =9.6 mM, at which point a precipitate began to form. Nonetheless, an upper limit tothe hydrogen abstraction rate constant was estimated as kH < 1×106 M−1s−1, orat least a 200 fold decrease relative to no metal salt. Very similar results for bulkiertertiary amines were also obtained. Thus, the addition of strong Lewis acids in the1195.1. Prologuepresence of Lewis basic sites on hydrogen atom donors can deactivate C H bonds.Next, we turn to the more relevant models for the work of this thesis, the tertiaryamides N,N -dimethylformamide (DMF) and N,N -dimethylacetamide (DMA). Aswith THF, normal hyperconjugative overlap between the conjugated amide pi-systemand the adjacent C H σ∗ anti-bonding orbitals weakens the C H bonds. There-fore, metal binding to the amide oxygen-centre should result in a decrease in thisorbital interaction, strengthen the C H bonds, and decrease HAT reactivity. Intheir study, Salamone et al. 50 measured CumO abstraction rate constants fromDMF and DMA in the presence of stoichiometric equivalents of LiClO4, LiOTf,NaClO4, Mg(ClO4)2, and Ca(ClO4)2 (in contrast to the excess used in Reference49). Figure 5.1a,b shows the plots of kobs against [substrate] for the reactions ofCumO with DMF and DMA in MeCN containing 0.2 M Mg(ClO4)2, respectively.For both DMF and DMA, there are three distinct regions in the plots: weak C Hbond activation for [amide]/[Mg2+]≤ 2, followed by strong C H bond deactivationfor 2<[amide]/[Mg2+]≤4, and no deactivation for [amide]/[Mg2+]<4.The addition of both LiClO4 and LiOTf decrease to a similar extent the rateconstants for abstraction from DMF and DMA by CumO . However, in contrastto Mg(ClO4)2, the lithium salts strongly deactivate C H bonds for 2 equivalents,followed by weak deactivation for another 2 equivalents, and no deactivation for[amide]/[Li+]<4. Salamone et al. were not able to give a clear cut explanation,but suggest that the different patterns are a result of differences in charge density,which is greater for Mg2+ than Li+, as well as different coordination geometriesof the two ions. A coordination number of 4 is most common for Li+, while anoctahedral geometry with the coordination of 6 ligands is almost always observedfor Mg2+.230,231 As a result, interactions of the ions with solvent and counter-anionswere suggested to be more important for Mg2+ than Li+.1205.1. PrologueFigure 5.1: a) Plot of observed rate constant against concentration of DMF forreaction with CumO at 298 K in the presence of 0.2 M Mg(ClO4)2. 0–0.4 M [DMF]range (black circles), kH1 = 5.8×105 M−1s−1; 0.8–2.2 M [DMF] range (white circles),kH2 = 1.3×106 M−1s−1. b) Plot of observed rate constant against concentration ofDMA for reaction with CumO at 298 K in the presence of 0.2 M Mg(ClO4)2. 0–0.4M [DMA] range (black circles), kH1 = 4.7×105 M−1s−1; 0.4–0.8 M [DMA] range(grey circles), kH2 = 2.4×105 M−1s−1; 0.8–2.2 M [DMA] range (white circles), kH3= 1.1×106 M−1s−1. Reprinted with permission from Reference 50. Copyright 2015American Chemical Society.1215.1. PrologueNaClO4 and Ca(ClO4)2 influence HAT between CumO and DMA to differentextents than both LiClO4 and Mg(ClO4)2. Figure 5.2a,b shows the plots of kobsagainst [substrate] for the reactions of CumO with DMA in MeCN containing 0.2 MNaClO4 and Mg(ClO4)2, respectively. For NaClO4, an almost negligible deactivationof C H bonds is observed for up to 4 equivalents of DMA. This was explained onthe basis of the weaker Lewis acidity of Na+ as compared to Li+. With regardsto Ca(ClO4)2, binding to DMA fully deactivates C H bond abstraction up to 4equivalents of DMA. The first region of Figure 5.2b ([DMA] = 0–0.2 M, blackcircles) represents the decrease in kβ of CumO as Ca2+ preferentially binds toDMA over CumO . Interestingly, for both DMF and DMA, the same experimentsin dimethyl sulfoxide (DMSO) solvent show no inhibition of HAT reactivity by metalcations. This was rationalized on the basis of the stronger Lewis basicity of DMSOas compared to both MeCN and the amides, thus the metals preferentially bind thesolvent rather than amide substrate.Finally, Salamone et al. 51 examined the effects of substrate structure on HATreaction between CumO and sterically bulky tertiary alkanamides in the presence ofalkali and alkaline earth metal ions. For N,N -dialkylacetamides, the steric bulk ofthe N -alkyl groups was previously characterized.146 Steric repulsion between CumOand the N -alkyl groups can decreases the HAT rate constant, as evident by the 3-fold decrease in kH in going from DMA to N,N -diisobutylacetamide (DIA; 1.2×106and 3.1×105 M−1s−1, respectively). For reactions of CumO with DIA addition of0.2 M LiClO4 or Ca(ClO4)2 to results in the same trends in C H bond deactivationobserved for DMA. This indicates that the influence of metal cation-substrate bind-ing is not significantly influenced by the steric bulk of N -alkyl groups. The sameis true for the addition of 0.2 M Mg(ClO4)2 to abstraction from DIA by CumO ,as shown in Figure 5.3. Once again, a slight decrease in reactivity is observed for1225.1. PrologueFigure 5.2: a) Plot of observed rate constant against concentration of DMA forreaction with CumO at 298 K in the presence of 0.2 M NaClO4. 0–0.8 M [DMA]range (black circles), kH1 = 9.6×105 M−1s−1; 0.8–1.4 M [DMA] range (white circles),kH2 = 1.4×106 M−1s−1. b) Plot of observed rate constant against concentration ofDMA for reaction with CumO at 298 K in the presence of 0.2 M Ca(ClO4)2. 0.8–1.7M [DMA] range (white circles), kH1 = 1.2×106 M−1s−1. Adapted with permissionfrom Reference 50. Copyright 2015 American Chemical Society.the first 2 equivalents of DIA, followed by strong C H bond deactivation for anadditional two equivalents, and no deactivation beyond that. No additional insightwas provided by Salamone et al. as to the reason for this reactivity. The plausibleexplanation provided was once again that Mg2+ has a high charge density. Theseresults show that Lewis acid-base interactions between alkali or alkaline earth metalcations can greatly depress hydrogen abstraction by alkyoxyl radicals.With these results in mind, I am interested in the possibility that alkali andalkaline earth metal cations found in biological system can protect C H bonds inproteins from HAT to reactive oxygen-centred radicals. However, the experimentalresults do not answer some of the key physico-chemical determinants that may makethis possible. Specifically, I have composed several important research questions thatremain unclear from the experimental results.The first question I have is one of methodology: Can DFT-based methods can ac-1235.1. PrologueFigure 5.3: Plot of observed rate constant against concentration of DIA for reactionwith CumO at 298 K in the presence of 0.2 M Mg(ClO4)2. 0–0.4 M [DIA] range, kH1= 3.6×105 M−1s−1; 0.8–1.4 M [DIA] range, kH2 = 2.9×105 M−1s−1. Reprinted fromTetrahedron, 72, Salamone et al., Hydrogen atom transfer from tertiary alkanamidesto the cumyloxyl radical. The role of substrate structure on alkali and alkaline earthmetal ion induced CH bond deactivation, 7757–7763, 2016, with permission fromElsevier.1245.1. Prologuecurately treat alkali/alkaline metal cation binding to organic substrates or radicals?There exists limited ab initio data describing these interactions.232–235 Therefore,I have conducted a benchmark quality study involving Li+, Na+, Mg2+, K+, andCa2+. To the best of my knowledge, this represents the first systematic benchmarkstudy of these metal cations with both organic substrates and radicals.Secondly, the nature of the binding of metal ions to substrates is still poorlydescribed, especially given the odd stoichiometric effects observed for Mg(ClO4)2with alkylamides. Specifically, I wish to address the range of these interactions, andhow much the metals effect the C H being broken. To address this I have utilizedboth Na+ and Mg2+ in my calculations. These metal ions were chosen to capturethe large differences in Lewis acidity and ion size associated with these third-periodions, and because they are two of the most biologically relevant metal ions.Thirdly, I address the effect that metal ions have on the HAT barrier heights. Ex-periments demonstrate that under certain conditions, the presence of metal ions candecrease HAT reactivity. If metal ions do effectively increase C H bond strengths,this will be a contributing factor to the free energy barrier as per the BEP princi-ple47,48 (see Chapter 4). There will likely be additional factors such as polarizationin the TS complex, or other effects of possible charge transfer from the substrateto metal ions (or vice versa). To investigate this, I have primarily studied HATreactions involving DMA and oxygen-centred radicals. Given there are only exper-imental data for CumO , this is the primary subject. However, I was interested instructural differences of the oxygen-centred radical, thus I have utilized BnO aswell, which differs significantly in that it has the ability to form strong pre-reactioncomplexes with hydrogen bond accepting substrates.144,166 I also investigated theeffect metal cations have on the abstraction from DMA by the more biologicallyrelevant hydroxyl radical (See Appendix C). I have also performed calculations with1255.1. Prologuethe bulkier DIA substrate and CumO to verify whether or not steric bulk does havean influence on the ability of a metal cation to affect HAT reactions.Finally, given that reactions of DMA with CumO in the presence of metal saltsshow no deactivation, I was interested in studying the reactivity of alkoxyl radicalswith strong Lewis bases. Strong Lewis basic compounds are important as they areoften used as solvents in physical organic chemical experiments. Furthermore, strongLewis acids are common in biological systems. Specifically, phosphates represent animportant functionality as part of the DNA backbone, as well being important inadenosine triphosphate (ATP), the so-called “energy currency.” Sulfur containingamino acids are also susceptible to oxidation into sulfoxides and disulfoxides.236Therefore, understanding the HAT reactivity of strong Lewis basic compounds withoxygen-centred radicals also contributes to the understanding of oxidative stress inbiological systems.HAT reactions involving alkoxyl radicals and strong Lewis bases have been previ-ously studied,144,237 and can possess interesting and unusual chemical reactivity. Forinstance, we recently showed that for the HAT reaction between BnO and DMSO,BnO acts as a hydrogen atom donor rather the acceptor.237 In light of this, I ex-amined the effect of metal cations on the expected HAT reactivity between CumOand DMSO, as well as the radical H-atom donation reactivity between BnO andDMSO. I also performed calculations to determine the effects of metal cations andif there exists reverse reactivity for two other strong Lewis basic substrates: hexam-ethylphosphoramide (HMPA) and tributylphosphine oxide (TBPO). These resultsare reported in Appendix C. The chemical structures of all the species studies hereinare shown Scheme 5.1.1265.1. PrologueO OH HNODMANODIAOHOSOPONNNDMSO HMPAOHPOTBPOBnO CumOScheme 5.1: Chemical structures of the species studied herein.1275.2. Computational methods and details5.2 Computational methods and detailsAll quantum mechanical calculations were performed using either the Gaussian09 software package,116 or the TURBOMOLE software package.73 Detailed bench-mark studies of metal cation-substrate interactions were carried out, the full dataand discussion of which is presented in Appendix C. Calculations for the bench-mark quality data of metal binding to substrates were first optimized at the LC-ωPBE-D3(BJ)/6-31+G(2d,2p) level of theory,109,110,151,152 and later re-optimizedwith larger 6-311+G(3df,3pd) basis sets. Single-point energy calculations were thencarried out using the coupled cluster methodology with single, double and pertur-bative triples with full core correlation, CCSD(T,Full), and various basis sets. Finalbenchmark quality binding energies have been calculated using the F12∗ explicitlycorrelated method with Def2-QZVPPD primary basis sets and Def2-QZVPP aux-iliary basis sets required for the resolution-of-the-identity (RI) approximation asimplemented in TURBOMOLE. The RI approximation is used to reduce the com-putational cost associated with calculating MO integrals.238 A total of 31 differentDFT-based methods with nearly complete 6-311+G(3df,3pd) and moderate sized 6-31+G(2d,2p) basis sets were tested both by single-point energy calculations on theLC-ωPBE-D3(BJ)/6-311+G(3df,3pd) optimized reference structures. Geometry op-timization calculations starting from the reference structures were also performedfor three of the best performing DFT-based method, in order to verify their abilityto capture the minimum energy bound structures. The final DFT-based methodselected from this benchmark work is M05-2X.103To test the effects of metal cations on HAT barrier heights, calculations were firstperformed for the reactions not involving metal cations. Geometry optimizationswere performed at the M05-2X/6-31+G∗∗ level of theory. Transition state (TS)structures were obtained by first freezing the abstraction donor-hydrogen-acceptor1285.2. Computational methods and detailsbond lengths with multiple initial orientations. The frozen bonds were then relaxedto obtain the final TS structures, which were then used to identify the appropriatepre- and post-reaction complexes. All structures were subjected to harmonic vibra-tional frequency calculations, which were visualized using the Chemcraft program197to verify minima (or saddle-points with a single imaginary frequency connectingreactants to products for TS structures). Single-point energy calculations were per-formed at the M05-2X/6-311+G(2d,2p) level of theory. The effects of MeCN solventwere estimated by inclusion of the SMD122 continuum solvent model in single-pointenergy calculations.The inclusion of metal cations into the TS structures proved to be techni-cally challenging. It was my expectation that I could simply include metal cationsand necessary counter-anions into the minimum energy complex structures and re-optimize; this was not the case. TS structures were once again obtained by con-strained optimization with the inclusion of the metal cation and counter-anion andfreezing the abstraction donor-hydrogen-acceptor bond lengths, providing a guessTS structure. However, in most cases the force constants (which are necessary fora TS optimization calculation) from the guess TS structure were not representa-tive of the true TS structure, thus force constants were recalculated for every stepalong the optimization, using the “CalcAll” keyword in Gaussian. This is a verycomputationally expensive procedure. Even using this method, many TS structuresincluding metal cations failed to converge. Therefore, guess TS structures that con-tain a single imaginary frequency connecting reactants to products are used in place.This technique provides genuine TS structures, although they may not be the TSassociated with the minimum energy reaction pathway. Nonetheless, the guess TSstructure can be used to provide an estimate of the reaction barrier height that areverified with calculations that were successful. Where available, final TS structures1295.3. Exploring the nature of metal cation substrate interactionswere used to identify the appropriate pre- and post-reaction complexes.Natural bond order (NBO) and natural population analysis (NPA) were utilizedin order to investigate the electronic structures involved in the HAT reactions and theeffects of metal cation binding.239–241 Version 3.1 of the NBO software package,242as implemented in the Gaussian 09 package was used in all cases.116 NBO analysisprovides a means for estimating the physical effects of chemically intuitive orbitalinteractions while NPA charges are a means for calculating the occupancies andcharges of atomic centres.243,2445.3 Exploring the nature of metal cation substrateinteractionsThe first step to understanding the effect non-redox active metal have on HATreaction barrier heights is investigating the nature of the binding interaction. Fig-ure 5.4a,b show the potential energy surfaces (PESs) of the binding of sodium ion,and sodium chloride to DMA, respectively. There are three surfaces in each plot rep-resenting the same potential energy surface in the gas-phase (black circles), and inan SMD122 continuum solvent field of MeCN (grey squares) or water (white circles).For both sodium ion and sodium chloride, the gas-phase PES demonstrates strongbinding with respect to a solvated system. This is indicated by a much deeper wellthan both solvents and a long-range interaction that does not tail off within 6 A˚due to the lack of screening. This underscores the importance of including solventeffects in studying the effects of metal cations. Interestingly, for the effects of watercompared to MeCN solvent appears to be quite small. In both cases the differencebetween the minimum of the water PES is about 2.5 kcal mol−1. Furthermore, theminimum of the PES well in all cases falls at about the same distance (2.1–2.2 A˚) in1305.3. Exploring the nature of metal cation substrate interactionsthe gas-phase and in both solvents. The small differences in binding interactions in-dicate that the effects as measured in MeCN may also apply to the more biologicallyrelevant aqueous system.For both the ion and the salt of sodium in water and MeCN, the binding inter-action only approaches zero slowly. Even at 6 A˚, the predicted binding energy ofDMA in water with Na+ and NaCl are 1.1 and 0.7 kcal mol−1, respectively. This isan indication that M05-2X-SMD does not properly capture long-range interactionsof DMA with sodium. This may be a result of not fully capturing the screeningeffect of solvent. Nonetheless, at a distance of 5 A˚, which corresponds well with thesize of the first solvation shell of the sodium ion,245 the interaction between DMAand sodium is nearly zero. This result is consistent with literature that studies theHofmeister series, where it has been shown that biologically relevant cations areonly able to influence their immediate solvation shell.246,247 Furthermore, Heydaet al. 248 utilized molecular dynamics simulations of N -methylacetamide in aqueoussolutions of NaCl, NaBr, KCl, and KBr to obtained radial distribution functions(RDFs). The RDFs are in agreement with the calculated PESs in that the mostprobable distance to find Na+ from the amidic oxygen-centre is at about 2–2.5 A˚separation. Heyda et al. also showed that Na+ binds more strongly with the amidethan K+, and that the nature of the halide counteranion does not contribute sig-nificantly to the overall interaction. This is consistent with the results of Salamoneet al., which showed that the nature of the counteranion plays a negligible role tothe effect on rate constants.49 The calculated binding energies of NaCl and NaClO4to DMA are very similar in magnitude (-16.0 and -18.1 kcal mol−1, respectively).From these results it is possible to draw an important conclusion: The use of Cl– inthe calculations may reasonably reflect the trends observed by Salamone et al. withClO –4 and OTf–.1315.3. Exploring the nature of metal cation substrate interactionsa. b.Figure 5.4: Potential energy surface of binding energy between DMA and a) sodiumcation and b) sodium chloride as a function of O-Na interaction distance (A˚). Theblack line and points represent gas-phase results, the grey squares and line is incontinuum MeCN solvent, and white circles and dashed line is in continuum watersolvent. Calculated as a rigid scan from the M05-2X/6-31+G∗∗ minimized complexstructure at the M05-2X/6-311+G(2d,2p) level of theory with the SMD solventmodel.Figure 5.5a,b show the PESs of the binding of magnesium ion and magnesiumdichloride to DMA, respectively. As is the case for Na+, the gas-phase PES of Mg2+is very strongly bound. In fact, the binding energy does not asymptotically approachzero. At an O–Mg distance of 12 A˚, there is calculated binding interaction of -48.5kcal mol−1, which is actually greater than at 6 A˚ by about 5 kcal mol−1. This isa result of the the ionization potentials (IP) of the metals with respect to that ofDMA. The experimental IP249–251 of DMA is 8.8–9.2 eV, the first IP of Na is 5.1eV, and the second IP of Mg is 15.0 eV (calculated with M05-2X/6-311+G(2d,2p)= 8.9, 5.0, and 14.9 eV, respectively). In a solvated system, magnesium ions existinvariably in the +2 oxidation state. However, due to nature of the IPs, DFT-basedmethods prefer to transfer charge, resulting in a non-convergent binding interactionbetween DMA and Mg2+, but not for Na+.1325.3. Exploring the nature of metal cation substrate interactionsa. b.Figure 5.5: Potential energy surface of binding energy between DMA and a) magne-sium cation and b) magnesium chloride as a function of O-Mg interaction distance(A˚). The black line and points represent gas-phase results, the grey squares and lineis in continuum MeCN solvent, and white circles and dashed line is in continuumwater solvent. Calculated as a rigid scan from the M05-2X/6-31+G∗∗ minimizedcomplex structure at the M05-2X/6-311+G(2d,2p) level of theory with the SMDsolvent model.The inclusion of MeCN and water solvent appear at first glance to alleviatethis problem. Note however that both PESs cross over zero binding at about 3 A˚,indicating there is still charge transfer to some extent. Additionally, for magnesiumchloride in the gas-phase, there also appears to be charge transfer occurring, asevident by the PES crossing zero just below 4 A˚. On the other hand, the inclusionof either water or MeCN solvent with MgCl2 give reasonable PESs with a bindinginteraction of about -25 kcal mol−1 for both water and MeCN, and a tailing off atabout 3 A˚, or slightly larger than the size of the first hydration shell of Mg2+ (ca. 2A˚).252 Therefore, studying the effects of magnesium on HAT reaction barriers maybe possible using MgCl2. Given the difficulties with Mg in general, I elected to focusonly on NaCl for future considerations.Next, I performed calculations to determine if the interaction of metal cations1335.3. Exploring the nature of metal cation substrate interactionssystematically increase the bond strengths of C H bonds in the models of interestin this work. This effect is hypothesized to occur due to a decrease in electrondensity in the C H σ∗ orbital, which is normally present due to hyperconjugativeoverlap. The BDEs for several substrates in the presence of Na+ and NaCl arelisted in Table 5.3. The ROCBS-QB3 BDEs for each of the substrates is includedto demonstrate that the relative order of BDEs for substrates with multiple C Hbonds is reasonable. The BDE for an arbitrary metal substrate complex (M X H)can be calculated as:BDE = E(M X ) + E(H )− E(M X H) (5.1)Table 5.3: Bond dissociation enthalpies of DMA, DMSO, MeCN, and DIA with Na+,NaCl, and without metal cations (Bare) calculated at the M05-2X-SMD(MeCN)/6-311+G(2d,2p)//M05-2X/6-31+G∗∗ level of theory. ROCBS-QB3-SMD(MeCN)BDEs without metals are included for reference. All values are in kcal mol−1.ROCBS-QB3 M05-2XSubstrate Bare Bare Na+ NaClDMA (acetyl) 101.0 98.5 97.8 98.4DMA (cis) 94.8 92.2 93.2 94.0DMA (trans) 94.3 91.6 92.8 92.6DMSO 104.3 103.4 104.4 103.7MeCN 99.0 97.4 98.3 98.1DIA (acetyl) 99.7 97.8 97.5 97.7DIA (α-cis) 97.0 95.7 96.5 95.3DIA (β-cis) 98.3 96.4 94.8 93.1DIA (α-trans) 94.6 93.0 93.8 94.0DIA (β-trans) 97.3 95.3 95.5 95.5For DMA, the N -methyl groups cis and trans relative to the carbonyl are theweakest, and therefore most thermodynamically favourable for abstraction. Sala-mone et al. 166 showed that both BnO and CumO prefer to abstract from theN -methyl groups of DMA, rather than the acetyl group. The complexation of Na+increases the cis BDE by 1.0 kcal mol−1 and the trans BDE by 1.2 kcal mol−1. The1345.3. Exploring the nature of metal cation substrate interactionscomplexation of NaCl to DMA increases the cis BDE by 1.8 kcal mol−1 and thetrans BDE by 1.0 kcal mol−1. One would expect the greater positive charge of thesodium cation (as compared to the sodium moiety of NaCl) to have a greater effecton the BDEs. However, these calculations indicate this may not always be the case.Specifically, the cis BDE is 0.8 kcal mol−1 weaker for the DMA NaCl complex ascompared to the DMA Na+ complex. These results, which are inconsistent withthe hypothesis, can be explained by additional analysis of both the parent and radi-cal complexes. The structures of the DMA-NaCl complex and three possible radicalcomplexes are shown in Figure 5.6a-d.NBO analysis can be used that give a qualitative description based on perturba-tion theory of the energy contribution of specific NBO interactions.244 Using NBOanalysis on DMA, the estimated effect of hyperconjugative overlap reveals a weaken-ing of the N -methyl C H bonds by approximately 5.6 and 5.9 kcal mol−1 for the cisand trans positions, respectively. Upon complexation of Na+, the bond weakeningeffects become 3.7 and 4.1 kcal mol−1 for the cis and trans positions, respectively.Therefore, from an NBO perspective, the cis and trans N -methyl bond strengthscan be described as strengthening upon complexation of Na+. Furthermore, thepredicted effect of hyperconjugative overlap in the DMA NaCl complex are 5.4 and5.5 kcal mol−1 for the cis and trans positions, respectively. Again, from an NBOperspective, the complexation of NaCl can be said to strengthen the N -methyl C Hbonds in DMA, but not as much as Na+. NBO analysis of the parent complexesthus supports the hypothesis.However, BDEs can be affected by both parent and radical stability. The hy-pothesis considers only the stability of the parent molecule, where complexationof a metal cation stabilizes the parent by withdrawing electron density from ananti-bonding orbital. In the case of complexation of NaCl, secondary interactions1355.3. Exploring the nature of metal cation substrate interactions(a) DMA-NaCl (b) Acetyl radical(c) Cis radical (d) Trans radicalFigure 5.6: Structures of a the DMA-NaCl complex, b the DMA-NaCl acetyl radicalcomplex, c the DMA-NaCl cis radical complex, and d the DMA-NaCl trans radicalcomplex. Key interaction distances are shown in units of A˚. Element colour key:white is carbon, light blue is hydrogen, red is oxygen, blue is nitrogen, purple issodium, and green is chlorine.1365.3. Exploring the nature of metal cation substrate interactionsbetween the Cl moiety of NaCl and the radical can result in the destabilization ofthe radical complex and a further increase in the predicted C H BDE. This is thecase for the cis-DMA-NaCl radical complex (Figure 5.6c). The Cl moiety of NaClis attracted to the partial positive charge of the acetyl hydrogen of DMA, which isgreater than the partial charge on the cis N -methyl hydrogens (0.25 e− vs. 0.21e−). This leads to the donation of electron density from Cl to the C H σ∗ orbital.NBO estimates this orbital interaction to amount to about 1 kcal mol−1 of bondweakening. Therefore, the hypothesis is incomplete in that it does not account forradical complex stability. Furthermore, with respect to TS complexes, additionalinteractions may further complicate the picture.Interestingly, the predicted BDE for the acetyl position of DMA decreases, ratherthan increasing as hypothesized, with both Na+ and NaCl. This can be attributedto the change in electronic structure associated with the complex. Consider the twopossible resonance forms of DMA shown in Figure 5.7. Glendening and Hrabal253utilized natural resonance theory to estimate that the right-hand resonance struc-ture in the closely related formamide contributes about 30% to the overall resonancehybrid. On the other hand, NBO analysis predicts a bond order in DMA of 1.5 be-tween the C and O and the C and N. Nonetheless, the complexation of DMA to Na+slightly increases the contribution of the zwitterionic form, resulting in a decreasein electron density at the carbonyl carbon. This is evidenced by the increase inNPA charge at the carbonyl carbon from +0.72 in DMA to +0.74 in the DMA-Na+complex. The partially positively-charged carbon centre inductively withdraws elec-tron density, stabilizing the acetyl radical. This increases the pi-bonding characterbetween the two carbon centres, and decreases the effective BDE.The observed inductive effects also manifests in the decrease in the carbonyl-acetyl C-C bond lengths in the acetyl radical, which decrease from 1.457 A˚ to 1.4431375.3. Exploring the nature of metal cation substrate interactionsA˚ upon complexation of Na+. Complexation of NaCl results in a bond length of1.451 A˚. These results are consistent with the ordering of the calculated acetyl BDEs(DMA > DMA NaCl > DMA Na+). On the other hand, the amidic nitrogenbecomes net more positive upon complexation of either Na+ or NaCl, but still hasan NPA charge that is negative. As a results, there is no inductive stabilizationeffect for the N -methyl radical complexes.ONONFigure 5.7: The resonance forms of DMA.For MeCN and DMSO the complexation of either Na+ or NaCl results in anincrease in C H BDE. In MeCN, the electron density in the C H σ∗ anti-bondingorbital decreases as a result of the interaction between Na and the nitrogen centre.DMSO has a non-Lewis electronic structure, making it difficult to analyze orbital in-teractions of valence-bond orbitals. Nonetheless, there is normally hyperconjugativeoverlap between the sulfur centre and the C H σ∗ anti-bonding orbitals, which wasconfirmed by NBO analysis. The electron density in the C H σ∗ orbital decreasesas a result of the interaction of Na+ with the oxygen-centre of DMSO.Upon complexation of NaCl, the BDEs of the more sterically bulky amide sub-strate DIA follow the same trend that is observed as for DMA: The acetyl C HBDE decreases due to inductive stabilization, while the C H bonds α to the amidicnitrogen centre increase as a result of decreased C H σ∗ occupancy. Alkoxyl radi-cals are not expected to abstract from C H BDEs of the bonds β to the nitrogencentre in DIA or other longer chain N -alkyl amides, as the incipient radical is not1385.3. Exploring the nature of metal cation substrate interactionsstabilized by the amidic pi-system. However, due to steric repulsion, the α-radicalsof DIA cannot lie directly plane of allowing conjugation with the pi-system. As suchthe α-C H BDEs of DIA are greater than those of DMA by 2-3 kcal mol−1, and arecloser to the β-C H BDEs than perhaps expected. The effects of sodium bindingto the amidic oxygen are almost nil for the β-trans C H bond of DIA, but thereis a significant decrease in the β-cis C H BDE. Figure 5.8a,b shows the structuresof the DIA-NaCl complex and the β-cis radical complex, where it can seen thatthe metal cation interacts with both the oxygen-centre and the carbon-centred rad-ical. This interaction stabilizes the radical complex and thus decreases the effectiveBDE; This interaction is likely not possible in the TS structure and is not expectedto contribute to a reduction of the barrier height.(a) DIA-NaCl (b) β-cis radicalFigure 5.8: Structures of a the DMA-NaCl complex, b the DIA-NaCl β-cis radicalcomplex. Key interaction distances are shown in units of A˚. Element colour key:white is carbon, light blue is hydrogen, red is oxygen, blue is nitrogen, purple issodium, and green is chlorine.1395.3. Exploring the nature of metal cation substrate interactionsAll of these results together confirm that specific metal-substrate interactionscan increase the effective BDEs of abstractable C H bonds by decreasing C H σ∗occupancy. However, this is complicated by the possibility for secondary interac-tions between the metal or counter-anion with the formed radicals. Furthermore,additional factors such as induction can alter the effects. It appears that for C Hbonds α to atoms with LPs that hyperconjugatively overlap with the C H σ∗ anti-bonding orbital, the complexation of non-redox active metals increases the C Hbond strength. However, if the C H bond is adjacent to an electron-poor centre,such as the carbon of a carbonyl, metal complexation actually decreases the bondstrength slightly by stabilizing the carbon-centred radical. In the context of HATreaction barrier heights, increasing the effective C H bond strength should decreasethe reaction-rate slightly by destabilizing the TS complex. However, there are otherimportant factors to consider such as how the metal effects the dipole moment inthe TS complex. Furthermore, it is important to note that experiments showed thatNaClO4 did not significantly affect the HAT rate constants for reactions betweenCumO and organic substrates. Therefore, the effects observed herein likely are anexaggeration of what is truly occurring in situ. Nonetheless, these theoretical cal-culations may be useful in developing an understanding of the subtle nature of theeffects of non-redox active metal cations on HAT reactions in general.With regards to the implications these results have on protein systems, sinceabstraction occurs predominantly from an α-C H bonds, it is likely that the natureof the amino acid, and the three dimensional structure of the protein will havesignificant importance. As the geometry of the peptide backbone becomes morestrained by steric interactions, the α hydrogen will become more difficult to abstract,as might be exemplified by the higher C H BDEs in DIA as compared to DMA.In the absence of secondary interactions, alkali or alkaline earth-metals are able to1405.4. HAT reactions involving non-redox active metalsbind to a given carbonyl site on the surface of a protein, they may exert a chemo-protective effect by increasing the BDE of an adjacent C H bond. However, thecomplexity of biological systems makes the absence of secondary interaction unlikely.Therefore, the possibility for chemo-protection to occur is remote.5.4 HAT reactions involving non-redox active metals5.4.1 DMAHydrogen abstraction reactions involving the oxygen-centred radicals BnO andCumO and DMA in MeCN have been previously investigated experimentally andtheoretically.166 The HAT reaction between DMA and BnO was determined tooccur predominantly through a direct HAT mechanism from the N -methyl groupcis relative to the carbonyl, and is kinetically limited by the formation of a strongpre-reaction complex between the relatively acid α-C-H of BnO and the amidicoxygen centre. On the other hand, CumO cannot form a strong hydrogen-bondinginteraction, and thus forms a non-specific dispersion-bound pre-reaction complexes.Abstraction by CumO still takes place from one of the N -methyl groups, but therate constant is 2 orders of magnitude less than for BnO . Recall that the inclusion ofmetal salts in reactions of DMA with CumO were previously investigated.50 On thebasis of the exaggerated effects observed in the changes in BDEs and the technicaldifficulties associated with these studies, the goal of this work is not to reproduceexperimental results, but rather, develop insights into the possible changes that canoccur as a result of metal salt addition to HAT reactions.Herein, I have calculated the reaction barrier heights for all three abstractablepositions of DMA for HAT reactions involving CumO and BnO , both with andwithout NaCl. These data are summarized in Table 5.4. Perhaps alarmingly, the1415.4. HAT reactions involving non-redox active metalsfree energy barriers calculated at the M05-2X-SMD(MeCN)/6-311+G(2d,2p)//M05-2X/6-31+G∗∗ are systematically higher than those previously calculated by about8.5 kcal mol−1.166 The previously calculated results were in reasonable agreementwith experimental results. The reason for this discrepancy is unclear, given thatM05-2X has previously been used successfully to calculate accurate HAT rate con-stants.254 Furthermore, the optimized minimum energy structures from both meth-ods do not differ significantly, with the exception of slightly shorter abstracting C Hpartial bonds and slightly elongated O H partial bonds in the TS structures ex-cluding NaCl (ca. 0.03 to 0.05 A˚). However, the relative ranking and differences inenergies for the reaction barrier heights for the different C H bonds are consistentwith previous results. Therefore, although these results cannot be used to predictrate constants, they are useful for studying the change in barrier height due to thecomplexation of NaCl.Table 5.4: Calculated free energy (enthalpy) barrier (∆G(H)‡, kcal mol−1) for directHAT from different C H bonds in DMA by CumO and BnO , with and withoutNaCl. The change in barrier height (∆∆G(H)‡) is calculated relative to the sameabstraction site without the inclusion of NaCl. All barrier heights are relative toseparated reactants (or complexed DMA-NaCl) and were calculated at the M05-2X-SMD(MeCN)/6-311+G(2d,2p)//M05-2X/6-31+G∗∗ level of theory.Reaction Abstraction Site ∆G(H)‡ ∆∆G(H)‡DMA + CumO trans 17.3(3.4)cis 17.5(3.8)acetyl 21.6(7.5)DMA-NaCl + CumO trans 20.3(3.7) 3.0(0.3)cis 18.4(1.2) 0.9(-2.6)acetyl 21.0(4.3) -0.6(-3.2)DMA + BnO trans 16.5(3.7)cis 17.5(3.6)acetyl 20.8(7.8)DMA-NaCl + BnO trans 18.6(1.7) 2.1(-2.0)cis 17.8(4.7) 0.3(1.1)acetyl 22.0(4.7) 1.2(-3.1)1425.4. HAT reactions involving non-redox active metalsFocussing first on the barrier heights for HAT between DMA and CumO , theresults of complexation with NaCl is variable. For each of the acetyl, cis, and transC H bond positions of DMA, there are three distinct effects upon complexation ofNaCl. For the trans position, both the free energy and enthalpic barriers increase, forthe cis position the free energy barrier increases and the enthalpic barrier decreases,and for the acetyl position both the free energy and enthalpic barriers decrease. Thereasons for this can be understood by examining the TS structures, which are shownin Figure 5.9a-f.First, note for all the TS structures in Figure 5.9, the complexation of NaClresults in a shortening of the O H bond that is being formed as a result of the HATreaction. This indicates that the TS structure shifts towards the product side alongthe reaction coordinate as a results of interactions with NaCl. By Hammond’s pos-tulate,178 this indicates a more endothermic reaction, giving evidence for increasedreaction barrier heights.For the TS structure representing abstraction from the trans position (relativeto the carbonyl) C H bond of DMA by CumO (Figure 5.9a), there is a calculated0.3 kcal mol−1 increase in ∆H‡, which is somewhat less than the predicted increasein BDE of 1.0 kcal mol−1. This result is consistent with the BEP principle, asthe change is ∆H‡ is necessarily less than or equal to the change in ∆H due tothe constant α in Equation 1.1. This difference can possibly be ascribed to theeffect of charge transfer: NPA indicates a 0.04 e− transfer from DMA to NaCl.As a result, there is less electron density available for hyperconjugative donationto the C H σ∗ orbital, and thus there is a lesser effect upon ∆H‡. Furthermore,TSs for HAT between C H bond and oxygen-centred radicals are characterizedby a degree of charge separation.209 NPA indicates that in the trans position TSstructure excluding NaCl the charge transfer from DMA to CumO is 0.24 e−, but1435.4. HAT reactions involving non-redox active metals(a) Trans DMA + CumO (b) Trans DMA-NaCl + CumO(c) Cis DMA + CumO (d) Cis DMA-NaCl + CumOFigure 5.9: Continued on following page.1445.4. HAT reactions involving non-redox active metals(e) Acetyl DMA + CumO (f) Acetyl DMA-NaCl + CumOFigure 5.9: TS structures of HAT reaction between DMA and CumO includingNaCl for different C H bonds: a trans, b trans with NaCl, c cis, d cis with NaCl,e acetyl, and f acetyl with NaCl. Key interatomic distances are shown in units ofA˚. Element colour key: white is carbon, light blue is hydrogen, red is oxygen, blueis nitrogen, purple is sodium, green is chlorine, and peach is a dummy atom in thecentre of an aromatic ring.the charge transfer increases with the inclusion of NaCl to 0.26 e−. This increasedcharge separation results in a lower enthalpic barrier than expected solely on thebasis of the increase in C H BDE. While orbital analysis does not indicate anyPCET type orbital interactions, charge separation between DMA and CumO inthe TS structure may be considered as a partial ionization of the hydrogen atom.Therefore, by increasing charge separation in the TS structure, it becomes easierto abstract the hydrogen atom as there is an increase in the proton-transfer likecharacter of the hydrogen atom. The increase in ∆G‡ is 3.0 kcal mol−1, thereforethe complexation of metal cations increases the entropic penalty in forming the TSstructure.In the abstraction at the cis position C H bond of DMA by CumO , there is acalculated 2.6 kcal mol−1 decrease in ∆H‡ and an increase of 0.9 kcal mol−1 in ∆G‡.The decrease in enthalpic barrier is inconsistent with the predicted increase in BDE1455.4. HAT reactions involving non-redox active metalsof 1.8 kcal mol−1 upon complexation with NaCl. The TS structure in Figure 5.9cshows a possible long range interaction between Na and the aromatic ring of CumOthat draws electron density and increases the reactivity. Additionally, NPA predictsa 0.07 e− charge transfer between DMA and Na. The combination of these twofactors stabilizes the TS and decreases ∆H‡. The entropic penalty associated withcomplexation of NaCl results in an increase in the free energy barrier.Abstraction by CumO from the acetyl C H bond of DMA was previously de-scribed as being a minor reaction pathway.166 In light of the reduction in BDE atthe acetyl position of amides, it may be reasonable to expect the reaction barrierto decrease. This indeed appears to be the net effect of complexation of NaCl toDMA, as ∆H‡ decreases by 3.1 kcal mol−1 and ∆G‡ decreases by 0.6 kcal mol−1.Figure 5.9f shows that in the TS structure, Na interacts with the aromatic systemof CumO , which also stabilizes the TS and decreases the barrier, but not enoughto make it the lowest barrier.For HAT between DMA and BnO , the interaction between NaCl and BnO , arestronger as compared to CumO , as indicated by to the shorter distance betweenNa and the centre of the aromatic ring, as shown in Figure 5.10a-f. Note that, asin reactions with CumO , the O H partial bond in the TS structures are shorterupon complexation with NaCl, indicating greater product-like character in the TScomplex. This shorter distance is likely possible due to the easier rotation of DMArelative to BnO , as compared to CumO . As a result, the enthalpic barriers for theabstraction from DMA by BnO decrease upon complexation of NaCl to both theacetyl and trans position C H bonds. For the cis position C H bond of DMA, theenthalpic barrier increases.This can be explained on the basis of the presence of an interaction betweenCl and the α-C H bond of BnO . While Na withdraws electron density from the1465.4. HAT reactions involving non-redox active metalsaromatic system of BnO , Cl is able to donate electron density back to BnO , coun-teracting the effect of the interaction of Na. Charge analysis confirms this, suchthat the NPA charge on Cl in the cis position abstraction TS complex is -0.88 e−,as compared to -0.91 e− in abstraction from the trans position. As a result, the en-thalpic barrier increases as predicted on the basis of the increase of the cis positionC H BDE of DMA.(a) Trans DMA + BnO (b) Trans DMA-NaCl + BnO(c) Cis DMA+ BnO (d) Cis DMA-NaCl + BnOFigure 5.10: Continued on following page.For both CumO and BnO , the presence of possible secondary interactions be-1475.4. HAT reactions involving non-redox active metals(e) Acetyl DMA + BnO (f) Acetyl DMA-NaCl + BnOFigure 5.10: TS structures of HAT reaction between DMA and BnO includingNaCl for different C H bonds: a trans, b trans with NaCl, c cis, d cis with NaCl,e acetyl, and f acetyl with NaCl. Key interatomic distances are shown in units ofA˚. Element colour key: white is carbon, light blue is hydrogen, red is oxygen, blueis nitrogen, purple is sodium, green is chlorine, and peach is a dummy atom in thecentre of an aromatic ring.tween NaCl and the radicals obfuscates the results. Therefore, in order to determineif non-redox active metal cations may act as chemo-protective agents in biologicalsystems, I have performed calculation involving the more biologically relevant HOradical. Although there is no literature value for kH of the HAT reaction betweenDMA and HO , I have used the Snelgrove-Ingold equation255 to estimate the rateconstant as 1.5×1010 M−1s−1, which is two and five orders of magnitude greaterthan BnO and CumO , respectively. Unfortunately, I was unsuccessful in perform-ing full optimization calculations in the presence of the metal salt. Therefore, I havelisted these preliminary results and the analysis thereof in Appendix C.5.4.2 DIANext, to study the effect steric bulk has on the HAT reactions between amidesand oxygen-centred radical, I have performed a study of HAT between DIA and1485.4. HAT reactions involving non-redox active metalsCumO . The HAT reaction between DIA and CumO was previously studied bySalamone et al. 146 , but only the α-N-alkyl positions were studied theoretically.Since the BDEs for α- and β-N-alkyl C H positions are relatively close in energy,I calculated the reaction barriers for these positions as well. The calculated freeenergy (enthalpic) barriers excluding and including NaCl are listed in Table 5.5.Interestingly the predicted reaction barriers for abstraction of the β-positions of DIAare lower than the α-positions. By applying the Boltzmann distribution about 69%of abstractions by CumO from DIA will take place from either the cis- or trans-βpositions of DIA. Therefore, abstraction from bulky amides by bulky oxygen-centredradicals are likely controlled by steric considerations. Note also that many of theTS optimizations were not successful with the inclusion of NaCl and in which case“guess” TS structures have been used to estimate the barrier heights. The TSstructures for the HAT reaction between DIA and CumO including NaCl are shownin Figure 5.11.As was observed in the barrier height calculations involving NaCl with DMAwith BnO and CumO , the results vary depending on the presence or absence ofsecondary interactions of Na with CumO . For the β-cis and acetyl position of DIA(Figure 5.11d-e), ∆H‡ decreases by 2.3 and 0.2 kcal mol−1, respectively, as a resultof relatively long range interactions of Na with the aromatic system of CumO . Forabstraction from the α-cis position of DIA (Figure 5.11b), Na is able to interactwith both the amidic oxygen lone pair, and a lone pair on the oxygen of CumO ,resulting in a significant decrease in ∆H‡ by 7.8 kcal mol−1.For the α-trans position there is no interaction between Na and CumO , yetthe complexation of NaCl to DIA results in a 2.5 kcal mol−1 decrease in ∆G‡.Comparing the TS structures including and excluding NaCl, there is very littledifference. Therefore, it is likely that the “guess” TS structure including NaCl does1495.4. HAT reactions involving non-redox active metals(a) α−trans DIA-NaCl +CumO(b) α−cis DIA-NaCl + CumO(c) β−trans DIA-NaCl +CumO(d) β−cis DIA-NaCl + CumOFigure 5.11: Continued on following page.1505.5. SummaryTable 5.5: Calculated free energy (enthalpy) (∆G(H)‡, kcal mol−1) for direct HATfrom different C H bonds in DIA by CumO , with and without NaCl. The changein barrier height (∆∆G(H)‡) is calculated relative to the same abstraction sitewithout the inclusion of NaCl. All barrier heights are relative to separated reac-tants (or complexed DIA-NaCl) and were calculated at the M05-2X-SMD(MeCN)/6-311+G(2d,2p)//M05-2X/6-31+G∗∗ level of theory. ∗Indicates estimated barrierbased on “guess” TS structure.Reaction Abstraction Site ∆G(H)‡ ∆∆G(H)‡DIA + CumO α-trans 19.5(6.2)α-cis 19.1(5.4)β-trans 18.6(6.0)β-cis 18.4(6.5)acetyl 19.1(7.4)DIA-NaCl + CumO α-trans∗ 17.0(3.9) -2.5(-2.3)α-cis∗ 12.7(-2.4) -6.4(-7.8)β-trans 19.8(6.8) 0.7(1.4)β-trans∗ 19.6(6.9) 0.5(1.5)β-cis∗ 16.8(4.2) -1.6(-2.3)acetyl∗ 17.8(3.8) 0.8(-0.1)acetyl 17.9(3.7) 0.9(-0.2)not appropriately represent the “true” TS structure for this particular abstractionsite. Note that this may also be the case in other systems.Abstraction by CumO from the β-trans position of DIA does not allow for aninteraction between both the amidic oxygen and CumO . However, there shouldbe no effect from decreased electron density in the C H σ∗ orbital. Consequently,∆H‡ increases by 1.4 kcal mol−1 as a result of the effect of 0.04 e− charge transferfrom DIA to Na in the TS structure.5.5 SummaryIn this investigation, the effects of non-redox active metal cations upon thebarrier heights of HAT reactions between small models for proteins with oxygen-centred radicals were studied. I began by examining the effects of the complexation1515.5. Summary(e) Acetyl DIA-NaCl + CumOFigure 5.11: TS structures for HAT reactions between DIA and CumO includingNaCl for different C H bonds: a α-trans, b α-cis, c β-trans, d β-cis, and e acetyl.Key interatomic distances are shown in units of A˚. Element colour key: white iscarbon, light blue is hydrogen, red is oxygen, blue is nitrogen, purple is sodium,green is chlorine, and peach is a dummy atom in the centre of an aromatic ring.of NaCl upon the C H bond strengths of the substrates of interest in this work. Thisin particular is central to the hypothesis of this work: complexation of a non-redoxactive metal to a heteroatom of a substrate reduces the electron density donated toadjacent C H σ∗ orbitals, increasing the C H bond strength. By the BEP principle(See Chapter 4), the HAT reaction barrier height increases as a function of the C HBDE.Specific metal-substrate interactions can increase the effective C H BDEs inthe parent molecule, but this is complicated by the possibility for metal-radicalinteractions following bond cleavage. Metal-radical interaction can either increaseor decrease C H BDEs depending on the nature of the orbital interactions of theproduct radical and metal cation. The aforementioned hypothesis does not take intoaccount the effects of metal-radical interaction.1525.5. SummaryNext, I studied the effects of metal complexation on the HAT reaction barrierheights. I calculated the barrier heights for two small models for proteins, DMAand DIA, including and excluding NaCl, with two different oxygen-centred radical,BnO and CumO . In the absence of any secondary interactions in the TS complex,the enthalpic HAT reaction barrier increases as a results of metal complexation.However, in the event of interactions between either the sodium or chlorine moietieswith the oxygen-centred radical or extensions of the substrate in the TS complex,the effect of complexation can increase or decrease the barrier height, depending onthe specific interaction. This further motivates the idea that the central hypothesisis incomplete in describing the effects of non-redox active metals on HAT reactionbarrier heights.The results obtained herein do not align with those from experiment. Recallthat previous experimental results for the HAT reaction between DMA and CumOdemonstrated clear inhibition upon the addition of alkali and alkaline earth-metalsalts. There are several possible explanations for this discrepancy: The modelsused herein may not properly capture the behaviour of bulk systems with respect tosolvation or stoichiometry (the binding of multiple amide substrates to a single metalcation). Another possible explanation is the use of Cl– as a counteranion herein,whereas ClO –4 and OTf– were utilized in experiment; the nature of the counteranionmay play a larger roll than has been previously demonstrated.The sum of these results suggests that non-redox active metal cations may notalways deactivate C H bonds in complex chemical environments. Therefore, on thebasis of these limited results, it seems unlikely that a natural chemo-protective effectis observed due to the presence of alkali and alkaline earth-metal cations in proteinor other biological systems. Additional work should be carried out to corroboratethese results.153Chapter 6ConclusionsHAT reactions are amongst the simplest radical chemical transformations. Thiscan be deceiving, as there are many poorly understood factors that influence HAT.It is important to develop a full understanding of HAT reactions as they are afundamental step in many biochemical processes. The radical-induced oxidation ofbiomaterials is often trigged by HAT reactions, and has been implicated in a numberof degenerative disease states. In this thesis, three aspects of HAT reactivity wereexplored using quantum chemical techniques.First, the role of non-covalent binding in the pre-reaction complex of HAT re-actions was investigated. In particular, the relationship between the calculatedpre-reaction complex binding energies and experimentally determined Arrheniuspre-exponential factors (A-factors) was examined for a series of thermoneutral ornearly thermoneutral reactions involving the formation and destruction of oxygen-centred radicals. The interpretation of the results of this investigation relies on anassumption: the mechanism for formal HAT, which is normally described as eitherdirect HAT of PCET, exists on a continuum that is described by Equation 3.3.It was demonstrated that there may be a correlation between A-factors and pre-reaction complex binding energies for (nearly) thermoneutral HAT reactions giventhat the reactions follow similar reaction mechanisms. This is an important caveat,as it is clear that binding energies do not serve as a diagnostic for fully describing allthe entropic contributions to A-factors. Specifically, for a set of ten self-exchange andpseudo-self-exchange reaction, six out of ten of the reactions studied share similar154Chapter 6. Conclusionsreaction mechanisms, and there is a strong correlation (R2 = 0.949) of pre-reactioncomplex binding energies with A-factors. In order to apply this analysis to furthersystems, future work should aim at determining a quantitative diagnostic for thecontributions of PCET and direct HAT to the overall hydrogen transfer reactionmechanism.Next, the validity of the Bell-Evans-Polanyi (BEP) principle was investigated byanalyzing a series of HAT reactions in which a hydrogen atom is abstracted from acarbon centre by the CumO radical. A hypothesis on the basis of group-additivitystates that if the BEP principle is valid, there should exist two linear relationshipsfor C H bonds, namely, one in which the incipient radical is delocalized into a pi-system (benzylic or allylic), and the other in which the remaining alkyl radicals arelargely localized. Detailed analysis of experimentally determined log(kH/n) plottedagainst theoretically determined C H BDEs demonstrated that there was reason-able correlation (R2 = 0.889) in the case of allylic/benzylic substrates, however thecorrelation was not strong for all other alkyl substrates (R2 = 0.641). Breakingthe larger group of alkyl substrates into smaller chemical groups suggested thatspecific groups of alkyl containing species (e.g. cyclic alkanes and alkyl group withheteroatomic neighbours) may demonstrate linear relationships, however more dataare needed to support this.To explore this further, calculations were performed to determine the structuresof relevant transition state complexes and reaction barrier heights. It was demon-strated that HAT reactions involving CumO likely proceed through a mechanismthat is dominated by direct HAT. As a result differences in mechanistic details maybe ruled as a factor contributing to the observed poor correlation. Decompositionof the free energy barrier heights revealed that HAT reactions involving CumO areentropy-controlled (−T∆S‡ >> ∆H‡), and non-isoentropic. It is established in the155Chapter 6. Conclusionsliterature that entropy-controlled processes do not follow LFERs consistently.207Therefore, while these results do not invalidate the BEP principle as a LFER, itis apparent that the model is an over-simplification of the complexity associatedwith HAT reactions involving CumO . This is likely to be the case for many HATreactions. As a result, the BEP principle should not be used to as a quantitativeprediction tool, but should remain a conceptual framework to qualitatively describechanges in reaction rates.Finally, recent experimental evidence demonstrated that non-redox active metalcations can have an inhibitory effect on HAT reactions involving oxygen-centred rad-icals. As these metals are found ubiquitously in biological systems, it was suggestedthat this may be a form of chemo-protection against radical induced oxidation ofbiomaterials, in particular proteins. It was previously reported that the mechanismfor this inhibition relies on C H bond deactivation, which may be the result of ametal cation binding to a substrate. Herein, I sought to determine the exact mech-anism by which this occurs. It was previously hypothesized that the metal boundto the substrate via an electron rich centre, such as the oxygen of a carbonyl. Then,electron density is withdrawn from a neighbouring C H σ∗ orbital, which is nor-mally populated through hyperconjugation. Withdrawing electron density from theC H σ∗ orbital strengthens the bond, and thus the HAT reaction barrier heightmay increase, as per the BEP principle.Two amides, DMA and DIA, were used as small models for protein systems.The calculated C H BDEs of these substrates both with and without metal cationsdemonstrate that specific metal-substrate interactions can increase the effectiveC H BDEs in the parent molecule. The calculated N -methyl C H BDEs of DMAincrease by an average of 1.4 kcal mol−1 upon complexation of NaCl. This is howevercomplicated by the fact that the product radical can also interact with the metal156Chapter 6. Conclusionscation, which can either increase or decrease the BDE. For example, the acetyl C HBDE of DMA decreases by 0.1 kcal mol−1 upon complexation of NaCl. As a resultof metal-radical interaction, BDEs may not properly reflect the effect metal cationshave on HAT barrier heights.The hypothesis that C H bond strengths are increased by interactions of themetal cation with the substrate does not account for interactions of that metal withspecies other than the substrate. This became abundantly obvious in calculatingthe reaction barrier heights including NaCl of DMA and DIA with the radicalsCumO and BnO . Specifically, the predicted changes in barrier heights as a resultof binding of NaCl to the substrate were found to both increase and decrease. For theHAT reaction of DMA and CumO including NaCl, the predicted reaction barrierheight for abstraction from an N -methyl hydrogen trans to the carbonyl increasesas there are no interactions of NaCl with CumO . However, for abstraction from anN -methyl cis to the carbonyl decrease as a result of a charge transfer interactionbetween NaCl and CumO . Analysis of the interactions of NaCl with both theradical and extensions of the substrate demonstrated that secondary interactioncan either stabilize or destabilize the TS complex.The aforementioned hypothesis does not account for secondary interactions, how-ever it was based on experimental results which demonstrate that alkali and alkalineearth metal salts do inhibit HAT reactions of amides with CumO . The results ob-tained herein do not support the hypothesis. An explanation for this discrepancyremains unclear, however I have speculated that this may be a result of problemswith the model chosen. Specifically, the experimental data suggests that stoichio-metric ratios of substrates and metal cations are somehow related to the inhibitoryeffect. No multiple coordinating stoichiometries were considered herein.Furthermore, the complexity of protein systems increases the likelihood that157Chapter 6. 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Phys. 2005, 123, 174101.[279] Kabsch, W. A solution for the best rotation to relate two sets of vectors. ActaCrystallogr. 1976, A32, 922–923.[280] Calculated using the ROCBS-QB3 method.180References[281] Salamone, M.; DiLabio, G. A.; Bietti, M. Hydrogen Atom Abstraction Reac-tions from Tertiary Amines by Benzyloxyl and Cumyloxyl Radicals: Influenceof Structure on the Rate-Determining Formation of a Hydrogen-Bonded Pre-reaction Complex. J. Org. Chem. 2011, 76, 6264–6270.181Appendices182Appendix AChapter 3 Additional DataFigure A.1: Molecular orbitals of hydrogen peroxide-peroxyl self-exchange reactionTS complex, demonstrating a PCET mechanism. Left is the HOMO-1 and rightis the SOMO. Together they demonstrate a lone pair-lone pair net half bondinginteractions, consistent with PCET. MOs are shown with an isovalue of 0.02 e−/A˚.183Appendix BChapter 4 Additional DataBelow are one-to-one plots comparing literature data to data calculated hereinusing theoretical approaches.Figure B.1: One-to-one plots of composite methods compared to literature andW1BD.184Appendix B. Chapter 4 Additional DataFigure B.1: Continued: One-to-one plots of composite methods compared to litera-ture and W1BD.Figure B.1: Continued: One-to-one plots of composite methods compared to litera-ture and W1BD.185Appendix B. Chapter 4 Additional DataTable B.1: Summary of experimental rate constants (M−1s−1) and literature180bond dissociation enthalpies (BDEs, kcal mol−1).Molecule kH Normalized kH BDE1,4-cyclohexadiene 6.65±0.02 ×107 1.66 ×107 76.01,4-diazabicyclo-[2.2.2]octane 9.6±1.4 ×106 8.0 ×105 93.42,2-dimethylbutane 9.5±0.3 ×104 4.8 ×104 98.02,3-dimethylbutane 5.6±0.2 ×105 2.8 ×105 95.49,10-dihydroanthracene 5.04±0.01 ×107 1.26 ×107 76.3Acetone < 1×104 2 ×103 96.0Acetonitrile < 1×104 3 ×103 97.0Adamantane (2◦) 6.90 ×106 5.75 ×105 98.4Adamantane (3◦) 6.90 ×106 1.73 ×106 96.2Benzaldehyde 1.20 ×107 1.20 ×107 88.7Benzyl alcohol 2.97 ×106 1.49 ×106 79.0Cumene 5.6±0.3 ×105 5.6 ×105 83.2Cycloheptane 2.20±0.02 ×106 1.57 ×105 94.0Cyclohexane 1.1±0.1 ×106 9.2 ×104 99.5Cyclooctane 2.98±0.02 ×106 1.86 ×105 94.4Cyclopentane 9.54±0.08 ×105 9.54 ×104 95.6Dibenzyl ether 5.60 ×106 1.40 ×106 85.8Diethyl ether 2.6 ×106 6.5 ×105 93.0Dimethyl sulfoxide 1.8 ×104 6.0 ×103 94.0Diethylamine 1.10 ×108 2.75 ×107 88.6Dioxane 8.2 ×105 1.0 ×105 96.5186Appendix B. Chapter 4 Additional DataDiphenylmethane 8.71±0.03 ×105 4.36 ×105 84.5Ethylbenzene 7.9±0.1 ×105 4.0 ×105 85.4Hexamethylphorsphoramide 1.87 ×107 1.04 ×106Morpholine 5.00 ×107 1.25 ×107 92.0Piperazine 2.26±0.01 ×108 2.84 ×107 93.0Piperidine 1.07±0.01 ×108 2.68 ×107 89.5Pyrrolidine 1.24±0.05 ×108 3.10 ×107 89.0Tetrahydrofuran 5.8±0.1 ×106 1.5 ×106 92.1Toluene 1.85±0.08 ×105 6.17 ×104 89.7Triethylamine 2.1 ×108 3.5 ×107 90.7Figure B.2: One-to-one plots comparing experimental and A gas-phase calculatedand B solvent-phase rate constants for HAT reactions between CumO and variousorganic substrates.187Appendix CChapter 5 Additional DataC.1 Benchmarking DFT based methods for thebinding of alkali and alkaline earth metals toorganic substrates and oxygen-centred radicalsIn order to be confident of the results of quantum mechanical mechanistic stud-ies, the method of choice must be calibrated. While DFT-based methods have beenwidely applied to these studied, few studies have previously investigated alkali andalkaline earth-metal cation binding to organic substrates.232–235 Most importantly,benchmark quality data for a wide variety of metals binding to biologically rele-vant substrates and oxygen-centred radicals does not exist to calibrate DFT-basedmethods. Therefore, I performed a benchmark study which incorporated all thebiologically relevant alkali and alkaline earth-metal cations, models for dipeptidesincluding amino acid side-chains, oxygen-centred radicals, and solvents which areutilized in the experimental mechanistic studies involved in probing these systems.Unfortunately, due to computational restrictions (vide infra), benchmark qualitycalculations on the originally proposed benchmark set were not possible. Full de-tails of the originally proposed benchmark set are shown in Scheme C.1.Benchmark quality binding energies are generally calculated using the “goldstandard” approach, CCSD(T)/CBS, where correlation consistent basis sets257,258(cc-pVX Z, X =T,Q,5) developed by Dunning and151,152 co-workers are used for com-plete basis set extrapolation. For the alkali and alkaline earth-metals, Iron et al. 259188C.1. Benchmarking DFT based methodsR1=H, Me, CH2OH, CH(CH3)2, CH2COOH, CH2SH, CH2CONH2M=Na+, Li+, Mg2+, K+, Ca2+SOPONNNNOH HOOHOOHH2NOHNR1 HNC NHOScheme C.1: Initial proposed benchmark set of substrates/radicals and metalcations. Note this set consists of all combinations of substrates and metal cation,thus there are 75 complexes in the set. Conformational analysis using the Hyperchempackage256 to identify the lowest energy conformers of all the substrates was com-pleted using the AM1 semi-empirical approach. Geometry optimizations were thenperformed without metal cations at the LC-ωPBE-D3(BJ)/6-31+G(2d,2p) level oftheory. Several binding sites were investigated and optimized at the same levelof theory. Benchmark quality structures have been optimized at the LC-ωPBE-D3(BJ)/6-311+G(3df,3pd) level of theory. I am awaiting computational resourcesto performed CCSD(T)-F12∗/Def2-QZVPPD calculations.189C.1. Benchmarking DFT based methodsdemonstrated that additional d-type basis functions are necessary to obtain reason-able results. It is also necessary to include core-correlation of at least the first coreshell in alkali and alkaline earth metals, thus it would be appropriate to use corevalence basis sets such as cc-pCVXZ.66 Iron et al. 259 also developed core-valencebasis sets for the alkali and alkaline earth-metals, however I was not able to obtainthese basis sets until very recently.viiThese basis sets should be considered for futurebenchmarking work. Given these difficulties, I originally chose the augmented ver-sion of the polarization consistent basis sets of Jensen and co-workers67–70 (aug-pc-N, N =2,3,4), which have been shown to converge to the CBS limit systematically84and are available for all the elements of interest.While performing CCSD(T)/CBS calculations, I observed that the metal cations(and neutral metal atoms), did not converge smoothly to the complete basis setlimit. As a consequence, complete basis set extrapolation is not feasible. In lightof this problem, I decided to re-evaluate the size scope of the benchmark set beingused. In order to facilitate future DFT-based work and probe the issue of basis setconvergence of alkali and alkaline earth metals, a benchmark set of small substrateswas proposed. This new set is shown in Scheme C.2. The new, small benchmark setwas selected to include important functional groups and radicals found in biologicalsystems, and one of the most common solvents used in physical organic experiments,acetonitrile.C.1.1 Metal cation basis set convergenceIn order to perform complete basis set (CBS) extrapolation, the total energy of amolecule/atom should converge smoothly to the CBS limit.79 However, CCSD(T,Full)/aug-pcN (N=1,2,3,4) calculations for alkali and alkaline earth-metals convergence ofviiSee http://theochem.weizmann.ac.il/web/papers/group12.html for the CVNZ basis sets for Li,Be, Na, Mg, K, Ca.190C.1. Benchmarking DFT based methodsONH2HOOHNOHO OH H2O NH3Mn+ = Li+, Na+, Mg2+, K+, Ca2+Scheme C.2: Revised benchmark set of small substrates and cations. Note thisset consists of all combinations of substrates and metal cations, i.e., there are 35complexes in the set.poorly to the CBS limit (See Figure C.1). Examining the energy of each ion relativeto the smallest basis set, for Li+ the value appears to converge reasonably, howeverthis is because there are only 2 electrons in this ion. For all of Na+, Mg2+, K+,and Ca2+, there appears to be no convergence to the CBS limit as no asymptote isreached. This is problematic as it means that CBS extrapolation would result in asignificant degree of uncertainty in the estimated CBS limit total energy.The poor convergence was thought to be a result of poorly suited basis setsto full core-correlation. However, the same CCSD(T,Full) calculations using thecore-correlation (cc-pCVNZ) basis sets also show unsatisfactory convergence forNa+ and Mg2+ (See Figure C.2). Therefore, I was tasked with finding a methodwhich would give results which best approximate alkali and alkaline metal bindingat the complete basis set limit. I decided to use an explicitly correlated CCSD(T)191C.1. Benchmarking DFT based methodsFigure C.1: Basis set convergence of CCSD(T,Full)/aug-pc-N (N=1,2,3,4) for alkaliand alkaline earth-metal cations. The relative energy of each basis set relative tothe aug-pc-1 for each metal. The cardinal number of the aug-pc-N basis sets isX = N + 1.192C.1. Benchmarking DFT based methodstreatment known as “F12∗” to more rapidly approach the CBS limit.87 I testedboth the core-correlation consistent basis set developed for used with explicitlycorrelation (cc-pCVXZ-F12),260 and the Ahlrich basis sets (Def2-SVP,-TZVPPD,and -QZVPPD).261 Both these basis sets combined with the CCSD(T,Full)-F12∗methodology gave satisfactory convergence to the CBS limit for the sodium andmagnesium ion (See Figure C.3) for the convergence of all metal ions calculatedwith the CCSD(T,Full-F12∗/Def2-QZVPPD method). Given that Def2-QZVPPDis available for almost every atom on the periodic table, and the observed conver-gence to the CBS limit, this basis set was selected for benchmark quality bindingenergies.Figure C.2: Basis set convergence of CCSD(T,Full)/cc-pVCXZ (X=3,4,5) forsodium and magnesium ions. The relative energy of each basis set relative to thecc-pVCDZ for each metal. The cardinal number of the basis sets is X.To the best of my knowledge, there is no precedent for extrapolating the Ahlrich193C.1. Benchmarking DFT based methodsFigure C.3: Explicitly correlated CCSD(T,Full)-F12∗/Def2-X(X=SVP,TZVPPD,QZVPPD) basis set convergence for alkali and alkalineearth-metal cations. The relative energy of each basis set relative to the Def2-SVPfor each metal. The cardinal number of the basis sets is X.194C.1. Benchmarking DFT based methodsbasis sets, thus the final benchmark energies are at the CCSD(T)-F12∗/Def2-QZVPPDlevel of theory, without extrapolation. The convergence of the total energies of thecations can be estimated as the sum of the experimental ionization energies of theions. These results are listed in Table C.1. The calculated values are too high(i.e., not at the CBS limit), and deviate from experiment from 0.16 and 0.66 AU(4.4–18 eV). Deviations of this magnitude are rather significant, and are likely dueto the increasing contribution of “relativistic effects” with increasing atomic num-ber. Additionally, there may be cumulative experimental error, as the experimentalionization energies range from 4–5500 eV. Relativistic effects were not consideredherein, thus, the calculated binding energies herein are likely the best availableapproximation to the non-relativistic gas-phase metal-substrate binding energy.Table C.1: Total energy of alkali and alkaline earth-metal cations from experimen-tal ionization energies251 (Expt.) and calculated (Calc.) at the CCSD(T,Full)-F12∗/Def2-QZVPPD level of theory. All values are in units of AU.Ion Expt. Calc.Li+ -7.47798 -7.27983Na+ -162.43089 -162.24203Mg2+ -200.32523 -199.49171K+ -601.93332 -601.77381Ca2+ -680.19158 -679.53065C.1.2 High level results and evaluation of variousdensity-functional theory based methodsTable C.2 lists the benchmark binding energy values calculated at the CCSD(T,Full)-F12∗/Def2-QZVPPD//LC-ωPBE-D3(BJ)/6-311+G(3df,3pd) level of theory. Somegeneral trends are that alkaline earth-metals bind more strongly than alkali earth-metals. Also, the order of binding follows the Lewis acidity of the metal ions: Mg2+> Ca2+ > Li+ > Na+ > K+. The metals all appear to bind most strongly to the195C.1. Benchmarking DFT based methodsamidic oxygen-centre, reflecting the higher Lewis basicity. The metals also bindweakest to the oxygen-centred radicals, with greater binding to HOO as comparedto HO .Table C.2: Benchmark gas-phase binding energies of alkali and alkaline earth-metals with small organic substrates and radicals. Values are calculatedat the CCSD(T,Full)-F12∗/Def2-QZVPPD//LC-ωPBE-D3(BJ)/6-311+G(3df,3pd)level of theory. All values are in kcal mol−1.Li+ Na+ Mg2+ K+ Ca2+H2O -34.7 -24.4 -82.0 -17.8 -56.8NH3 -39.9 -28.2 -98.1 -19.8 -65.3MeCN -44.4 -33.0 -113.1 -24.9 -80.7Formamide -50.7 -36.9 -128.2 -28.5 -96.1Formic acid -38.4 -27.0 -101.9 -20.0 -72.6HO -21.3 -16.8 -57.0 -12.4 -40.7HOO -27.1 -19.1 -72.2 -13.9 -49.0Next, 31 DFT-based methods combined with a moderate basis set (6-31+G(2d,2p))and a large basis set (6-311+G(3df,3pd)) were tested for their ability to estimate thebinding energy between metal cations and substrates. The mean absolute/signederrors (MAE/MSE) and maximum and minimum errors for each method are listedin Table C.3.Table C.3: Evaluation of DFT-based methods for alkali and alkaline metal bindingto organic substrates and radicals. All values are in kcal mol−1. Negative valuesindicate under-binding.Method MAE/MSE Max./Min MAE/MSE Max./Min.6-311+G(3df,3pd) 6-31+G(2d,2p)B3101LYP102 1.49/1.35 5.12/-0.57 1.59/-0.17 4.67/-7.28B3P86262 0.94/0.47 3.87/-0.96 1.36/-1.08 1.99/-7.38B3PW91263 0.95/-0.14 2.74/-1.64 1.89/-1.70 1.47/-8.76BH+H264LYP 1.89/1.84 5.29/-0.59 1.93/0.63 4.65/-5.64196C.1. Benchmarking DFT based methodsB147LYP 1.60/1.07 5.56/-1.51 1.88/-0.75 5.30/-8.80BMK265 0.90/-0.70 1.13/-2.40 1.98/-1.93 0.86/-8.75BP86 1.63/-0.25 4.61/-3.21 2.27/-2.14 1.55/-9.38CAM-B3LYP266 2.40/2.40 6.25/ 0.21 1.98/1.04 5.82/-5.25LC-ωPBE151,152 0.78/0.58 2.95/-0.73 1.34/-0.74 2.19/-8.00M05-2X103 1.11/1.11 3.21/ 0.15 1.24/-0.17 2.55/-5.75M06267 1.05/-0.62 2.36/-4.83 1.83/-1.63 1.76/-9.03M06-2X267 1.13/1.13 3.68/ 0.11 1.26/-0.07 3.00/-6.63M06L268 1.52/-1.14 2.64/-6.94 2.55/-2.48 1.21/-11.2MOHLYP269 2.30/-2.02 1.52/-5.40 4.04/-3.96 0.89/-15.2PBE0270,271 1.22/1.18 4.19/-0.31 1.25/-0.34 3.30/-7.25PBE272 1.70/1.46 6.09/-0.87 1.58/-0.46 4.68/-8.15TPSS273 1.38/0.95 4.88/-1.12 1.60/-0.98 2.91/-8.12B97274D3109 1.50/0.47 5.94/-2.19 1.69/-1.37 2.67/-8.41ωB97275 0.61/0.23 2.13/-1.72 1.41/-0.97 1.78/-7.57ωB97XD276 1.12/-0.94 1.02/-4.52 2.24/-2.21 0.65/-8.64HSEH1PBE277,278 1.30/1.29 4.28/-0.16 1.23/-0.23 3.45/-6.95B3LYP-D3(BJ)109,110 2.86/2.86 7.50/ 0.34 1.92/1.34 7.05/-4.33BLYP-D3(BJ) 2.89/2.88 8.40/-0.14 1.83/1.06 8.14/-5.30B3PW91-D3(BJ) 1.47/1.40 5.81/-0.44 1.02/-0.14 3.88/-5.69BMK-D3(BJ) 1.03/0.80 4.05/-1.06 1.02/-0.43 2.03/-5.49BP86-D3(BJ) 1.77/1.29 7.78/-1.05 1.26/-0.60 3.96/-6.19CAM-B3LYP-D3(BJ) 3.19/3.19 7.50/ 0.78 2.27/1.82 7.07/-3.54LC-ωPDE-D3(BJ) 1.47/1.46 4.07/-0.06 1.33/0.14 3.30/-6.15197C.1. Benchmarking DFT based methodsPBE0-D3(BJ) 1.92/1.92 5.37/ 0.20 1.33/0.40 4.47/-5.74PBE-D3(BJ) 2.24/2.23 7.57/-0.22 1.44/0.30 5.90/-6.67TPSS-D3(BJ) 2.03/1.99 6.93/-0.30 1.25/0.06 4.56/-6.03Given the magnitude of the gas-phase binding energies, the overall agreementbetween the benchmark values and the DFT-based method values with both moder-ate and large basis sets is very good. Interestingly, the application of the empiricalD3(BJ) dispersion correction systematically decreases agreement with benchmarkvalues as it increases over-binding. Also, going from moderate to large basis setssystematically increases the predicted binding energies, as indicated by an increasein MSE across the board. I chose three of the best performing methods (BMK-D3(BJ), TPSS-D3(BJ), and M05-2X) and performed geometry optimizations withmoderate basis sets on the reference structures to determine if the choice of methodwould significantly impact the minimum energy bound structure. These results arelisted in Table C.4.Table C.4: Comparison of single point (SP on benchmark structure) and relaxed(optimized with method) binding energies for alkali and alkaline metal binding withDFT-based methods and 6-31+G(2d,2p) basis sets. Mean absolute error (MAE)values are in kcal mol−1 and average root mean squared deviation (RMSD)(Rootmean square deviation was calculated using the Kabsch algorithm279 as implementedin the rmsd package available on GitHub (Calculate RMSD for two XYZ structures,GitHub, http://github.com/charnley/rmsd, accessed Nov. 18, 2016)) of geometryare in A˚.Method MAE(SP) MAE(Relaxed) Average RMSDBMK-D3(BJ) 1.02 1.24 0.012M05-2X 1.24 1.17 0.020TPSS-D3(BJ) 1.25 1.21 0.026For all the three methods tested, the average of the root mean square devia-tions (RMSD) from reference structures are very small (0.012–0.026 A˚). For BMK-198C.2. HAT reactions involving non-redox active metalsD3(BJ), re-optimization of the structures results in a slight increase in MAE, whilefor TPSS-D3(BJ) and M05-2X, the opposite is true. As a whole, it seems that DFT-based methods are capable of predicting gas-phase binding energies for alkali andalkaline metal ions with organic substrates and radicals. M05-2X appears to be oneof the best performing DFT-based methods. Furthermore, M05-2X is recommendedby the QM-ORSA254 method, which outlines “best principles” for calculating ac-curate HAT rate constants in solution. And finally, M05-2X was previously usedby our group in the study of the HAT reaction between DMSO and BnO ,237 thusI have selected this method for further study of the effects of alkali and alkalineearth metal cations on the barrier heights of HAT reactions. Note also that M05-2Xis a hybrid density-functional with 56% HF-exchange, and thus should not suffersignificantly from delocalization error.C.2 HAT reactions involving non-redox active metalsC.2.1 DMA + HOI was unsuccessful in performing full optimization calculations in the presenceof the metal salt. However, the “guess” TS structures obtained by freezing thehydrogen atom acceptor donor bonds (as described in the Chapter 5 section) hereinrepresent approximated transition state structures and therefore provide an estimateof the effects of metal salts in a more biologically relevant model. The calculatedreaction barriers are presented in Table C.5.For the direct HAT reaction between OH and DMA abstraction occurs primarilyfrom the cis position C H bond of DMA. The calculated free energy (enthalpic)barrier is 7.9(-2.7) kcal mol−1, which gives a calculated rate constant of 1.0 ×107M−1s−1, or three orders of magnitude lower than the predicted rate constants of199C.2. HAT reactions involving non-redox active metalsTable C.5: Calculated free energy (enthalpy) barrier (∆G(H)‡, kcal mol−1) for di-rect HAT from different C H bonds in DMA by HO , with and without NaCl.The change in barrier height (∆∆G(H)‡) is calculated relative to the same ab-straction site without the inclusion of NaCl. All barrier heights are relative toseparated reactants (or complexed DMA-NaCl) and were calculated at the M05-2X-SMD(MeCN)/6-311+G(2d,2p)//M05-2X/6-31+G∗∗ level of theory. ∗Indicatesestimated barrier based on “guess” TS structure.Reaction Abstraction Site ∆G(H)‡ ∆∆G(H)‡DMA + HO trans 8.9(0.0)cis 7.9(-2.7)acetyl 9.9(0.5)DMA-NaCl + HO trans∗ 13.1(1.3) 4.2(1.3)cis∗ 12.9(0.2) 5.0(2.9)acetyl∗ 16.6(2.0) 7.7(1.5)1.5 ×1010 M−1s−1. This result in unsurprising given the poor agreement of thecalculated values with experiment of HAT reaction between DMA and BnO andCumO . For abstraction by HO , the complexation of NaCl to DMA increases theestimated free energy reaction barriers across the board by 4.2–7.7 kcal mol−1. Thissuggests that if Na+ interacts closely with DMA in vivo, then non-redox activemetals may have a chemo-protective effect against hydrogen abstraction by HO .For the acetyl position C H bond of DMA, the enthalpic barrier increases by 1.5kcal mol−1, even though the calculated BDE decreases slightly upon complexationof NaCl. This can once again be explained by the effects of charge transfer in theTS complex. For the TS structure representing direct HAT between HO the acetylposition of DMA-NaCl, there is a calculated charge transfer from DMA to Na+ of0.02 e−, which results in a decrease in charge separation between HO and DMAfrom 0.26 e− without NaCl to 0.25 e− with NaCl. Although this effect is small, itappears to significantly effect the ability of HO to abstract a hydrogen atom fromDMA. The same argument applies to both the cis and trans C H bond positionsof DMA, for which the enthalpic barrier increases by more than the calculated200C.2. HAT reactions involving non-redox active metalschange in BDE. For the significantly more reactive HO radical, the metal is lesslikely to interact with the oxygen centre, thus the effects observed depend only oninteractions with DMA. As a result, the reaction barrier increases in all cases uponcomplexation of NaCl.C.2.2 Strong hydrogen bond accepting substatesIn the investigation of the effects of metal cations on HAT reactions betweenDMA and CumO Salamone et al. demonstrated that in DMSO solvent, HATreactivity is not significantly affected by the addition of metal salts.50 This can beexplained on the basis of the stronger Lewis basicity of DMSO as as compared toDMA, resulting in the competitive binding of Mn+ to DMSO over DMA. The rateconstant for HAT between DMSO and CumO in DMSO solvent is 1.8 ×104 M−1s−1while that for DMSO and CumO is 1.2 ×106 M−1s−1 in MeCN solvent at 298 K,therefore only small changes in kobs are expected for HAT between CumO andDMA in DMSO solvent with metal salts. I am interested in how metal salts affectthe HAT reactivity of DMSO and other related strong hydrogen bond acceptingsubstrates. We previously showed that DMSO and BnO react via a radical H-atomdonation reaction, where BnO acts counterintuitively as the hydrogen atom donorrather than acceptor. The net reaction yields benzaldehyde, dimethyl sulfide, andHO as the DMSO-H radical decomposes in a concerted manner following the radicaldonation of the hydrogen atom by the radical.An important driving force in this reaction is the formation of benzaldehyde bycleavage of an acidic α-C H bond. This is not possible in CumO therefore CumOdoes not react through radical H-atom donation. I have performed computationalstudies to determined if metal cations could affect this reactivity and if the related(HMPA and TBPO) substrates displayed similar reactivity.201C.2. HAT reactions involving non-redox active metalsSOO SOHO+ +SOHHO SOHHHO+ +SOHHO S+ +OH+OH123Scheme C.3: The HAT reactions of DMSO with 1 CumO , 2 the radical H-atomdonation reaction with BnO , and 3 the conventional HAT reaction with BnO .Table C.6: Calculated free energy (enthalpy) barrier (∆G(H)‡, kcal mol−1) for HATbetween DMSO and CumO , and conventional HAT and radical H-atom donationwith BnO , with and without NaCl. The change in barrier height (∆∆G(H)‡) iscalculated relative to the same reaction without the inclusion of NaCl. All barrierheights are relative to separated reactants (or complexed DMSO-NaCl) and werecalculated at the M05-2X-SMD(MeCN)/6-311+G(2d,2p)//M05-2X/6-31+G∗∗ levelof theory.Reaction ∆G(H)‡ ∆∆G(H)‡1 20.9(8.9)1 + NaCl 22.4(5.2) 1.5(-3.7)2 10.4(-2.2)2 + NaCl 13.4(-3.4) 2.0(-1.2)3 22.7(10.4)3 + NaCl 22.1(5.2) -0.6(-5.2)202C.2. HAT reactions involving non-redox active metals(a) DMSO + CumO (b) DMSO-NaCl + CumO(c) Radical H-atom DonationDMSO + BnO(d) Radical H-atom DonationDMSO-NaCl + BnO(e) HAT DMSO + BnO (f) HAT DMSO-NaCl + BnOFigure C.4: TS structures of HAT reaction between DMSO and CumO , and theconventional HAT and radical H-atom donation reactions reaction with BnO ex-cluding and including NaCl. Key interatomic distances are shown in units of A˚.Element colour key: white is carbon, light blue is hydrogen, red is oxygen, yellow issulfur, purple is sodium, green is chlorine, and peach is a dummy atom in the centreof an aromatic ring.203C.2. HAT reactions involving non-redox active metalsFocussing first of the reactions of DMSO with CumO and BnO in the presenceof NaCl. The reactions studied are shown in Scheme C.3 and the calculated freeenergy (enthalpic) barriers are listed in Table C.6. Also, the TS structures forthe reactions in Scheme C.3 are shown in Figure C.4a-f. Firstly, for HAT betweenDMSO and CumO in the presence of NaCl (Reaction 1, Figure C.4b), the freeenergy barrier increases by 1.5 kcal mol−1 however the enthalpic barrier decreasesby 3.7 kcal mol−1. This can be explained in a similar manner to the reactions ofDMA: Na+ interacts both with the oxygen of DMSO and the aromatic system ofCumO , resulting in a stabilization of the TS and a decrease in enthalpic barrier inspite of a predicted increase in C H BDE. This binding however results in a tighterTS structure, thus the entropic cost increases and so does the ∆G‡.Next, the reaction of DMSO with BnO was previously characterized144 as oc-curring through the rate determining formation of a strong pre-reaction complex,however, we more recently showed the reaction likely takes place through a reac-tion in which BnO acts as a hydrogen atom donor, rather than acceptor.237 Forthis radical H-atom donation reaction between DMSO and BnO in the presence ofNaCl (Reaction 2, Figure C.4d), the free energy barrier increases as a result of NaClcomplexation. This is appears to be an entropic effect as a result of binding, as∆H‡ decreases by 1.2 kcal mol−1 due to the interaction of Na with DMSO and thearomatic ring of BnO . In this TS structure excluding NaCl there is some chargeseparation, however, DMSO (the acceptor) has a partial positive charge and BnO(the donor) has a partial negative charge. This is contrary to typical HAT reactionsbetween oxygen-centred radicals and C H bonds where in the TS structure, thedonor typically has a partial positive charge and the acceptor has a partial positivecharge. Therefore, the significant reduction in charge separation from 0.25 e− to0.11 e− appears to play a lesser role. This may be due to the unusual electronic204C.2. HAT reactions involving non-redox active metalsstructure of sulfoxide compounds, which cannot easily be defined using simple Lewisstructures. The radical H-atom donation reaction is likely driven by the concertedcleavage of the S O bond of DMSO-H.While the conventional HAT reaction between DMSO and BnO is unlikely tooccur as the barrier is ca. 12 kcal mol−1 higher than the radical H-atom donationreaction, it is still interesting to consider the effect that NaCl has on the reactionbarrier height. Figure C.4f shows that in the TS structure for the HAT reactionbetween DMSO and BnO in the presence of NaCl, there is a relatively close inter-action with both the oxygen of DMSO and the aromatic ring of BnO . As a result,the free energy (enthalpic) barrier decreases by 0.6(5.2) kcal mol−1. NPA estimatesa relatively large (as compared to the DMA HAT reactions) 0.10 e− charge transferfrom DMSO to Na, resulting in a decrease in charge separation in the TS structurefrom 0.27 e− to 0.21 e−.Turning now to the reactions of BnO with the substrates HMPA and TBPO,which are closely related phosphine oxide compounds that are commonly used asorganic solvents. These two substrates are also closely related to DMSO in thatthey are strong hydrogen bond accepting substrates, and therefore may undergo aradical H-atom donation reaction in which BnO donates an acidic α-C H hydrogenatom, rather than accepting in the conventional manner. The reaction coordinatediagrams from HMPA with BnO and TBPO with BnO are shown in Figure C.5aand b, respectively. In the possible radical H-atom donation reactions for bothsubstrates the free energy barrier is lower than the conventional HAT barrier by 6kcal mol−1 for HMPA and 10 kcal mol−1 for TBPO. However, in both cases theprocess is energetically uphill, and so is unlikely to occur. For both the radicalproducts HMPA-H and TBPO-H, the cleavage of the P O bond for the subsequentloss of HO in analogy to the reaction of DMSO results in an additional 20 kcal mol−1205C.2. HAT reactions involving non-redox active metalsenergy cost owing to the significantly greater strength of P O bonds as compared toS O bonds (The BDEs are 86 kcal mol−1 in DMSO vs. 147 kcal mol−1 in HMPA).280Therefore, radical H-atom donation reactivity of BnO to DMSO is likely partiallydriven by the cleavage of the S O bond in DMSO.(a) HMPA + BnO(b) TBPO + BnOFigure C.5: Reaction profiles for HAT between a HMPA with BnO b and TBPOwith BnO . Relative free energies in kcal mol−1 are shown for the HAT (black) andradical H-atom donation (red) reactions.As BnO does not donate a hydrogen atom to HMPA or TBPO, I did not seek206C.2. HAT reactions involving non-redox active metalsthe determine the effects of NaCl on these reaction barriers. Note also, that unlikeDMSO, NBO analysis reveals no strong hyperconjugative overlap between the P Oorbitals and the abstractable C H bonds of TBPO or HMPA. From an orbitalenergetic standpoint, this is expected, given the overall higher energy of the P Obond, as compared to the S O bond. Therefore, the effects of metal complexationto the phosphine oxygen (where Na will bind) should not significantly affect theC H bond strengths or enthalpic barrier heights. The HAT reaction barrier heightsof NaCl with with HMPA and TBPO for the HAT reaction with CumO are listedin Table C.7. The TS structures for direct HAT reactions between HMPA or TBPOwith CumO and BnO including NaCl are shown in Figure C.6.Table C.7: Calculated free energy (enthalpy) barrier (∆G(H)‡, kcal mol−1) for HATbetween HMPA and TBPO with CumO with and without NaCl. The change inbarrier height (∆∆G(H)‡) is calculated relative to the same reaction without theinclusion of NaCl. All barrier heights are relative to separated reactants (or com-plexed HMPA/TBPO-NaCl) and were calculated at the M05-2X-SMD(MeCN)/6-311+G(2d,2p)//M05-2X/6-31+G∗∗ level of theory. ∗Indicates estimated barrierbased on “guess” TS structure.Reaction ∆G(H)‡ ∆∆G(H)‡HMPA + CumO 17.4(3.8)HMPA-NaCl + CumO 12.8(-1.0)∗ -4.6(-4.8)HMPA + BnO 16.0(2.6)HMPA-NaCl + BnO 14.5(1.6)∗ -1.5(-1.0)TBPO + CumO 20.1(6.8)TBPO-NaCl + CumO 21.6(6.0) 1.5(-0.8)TBPO + BnO 19.0(6.9)TBPO-NaCl + BnO 19.1(4.8) 0.1(-2.1)For HMPA with CumO and BnO in the presence of NaCl, the HAT reactionbarrier decreases in both cases. For the reaction with CumO (Figure C.6a), thiscan be explained on the basis of the interaction between Na with both the phosphineoxygen and the aromatic system of CumO . However for the reaction with BnO(Figure C.6b), there is not such interaction with the aromatic system of BnO . In207C.2. HAT reactions involving non-redox active metals(a) HMPA-NaCl + CumO (b) HMPA-NaCl + BnO(c) TBPO-NaCl + CumO (d) TBPO-NaCl + BnOFigure C.6: TS structures of HAT reaction between HMPA and TBPO with CumOand BnO including NaCl. Key interatomic distances are shown in units of A˚.Element colour key: white is carbon, light blue is hydrogen, red is oxygen, orangeis phosphorous, purple is sodium, green is chlorine, and peach is a dummy atom inthe centre of an aromatic ring.208C.2. HAT reactions involving non-redox active metalsthis case, the decrease in reaction barrier can be ascribed to the lack of rotation of theCH2O moiety of BnO . Previously computational investigations144,281 demonstratea typical rotation of the CH2O moiety of BnO of about 30 to 45◦, while the TS inFigure C.6b has an angle of about 5◦. In the HAT reaction between HMPA and BnOexcluding NaCl, there is a rotation of this moiety by ca. 30◦. The complexationof NaCl breaks the hydrogen bond between the α-C H bond of BnO with theoxygen of HMPA, and forms a now interaction between Cl– and the α-C H bond ofBnO . This new interaction stabilizes the TS and allows the oxygen-centre of BnOto remain in the plain of the aromatic system.Finally, for the reactions of TBPO with CumO and BnO there are long rangeinteractions of Na with the aromatic systems resulting in a decrease in ∆H‡. Thefree energy barrier for CumO is not significantly affected, while it increases byabout 1.5 kcal mol−1 for BnO . This is likely due to the longer range interaction ofNa with CumO as compared to BnO (4.0 A˚ vs. 3.6 A˚, respectively).209

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