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UBC Theses and Dissertations

Computational studies on functional materials Chen, Yuzhe 2016

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 Computational Studies on Functional Materials  by  Yuzhe Chen  Honours B.Sc., The University of British Columbia, 2013  A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF   MASTER OF SCIENCE  in  The Faculty of Graduate and Postdoctoral Studies  (Chemistry)    THE UNIVERSITY OF BRITISH COLUMBIA (Vancouver)  August 2016  ©Yuzhe Chen, 2016    ii  Abstract  Carbon nanotubes (CNTs) are of great significance due to their almost unlimited potential applications. Transition metal doped CNTs in particular can be developed into small molecule sensors or CNT based catalysis and theoretical studies are important methods to predict their structures and properties. The Project I of this thesis focused on the development of a model that is specifically used for the study of transition metal doped single wall CNT (SWCNT) systems at a reduced computational cost for neutrally charged systems. Longer SWCNT segments of about 20 layers of carbon and terminated by hydrogen or nitrogen atoms with single transition metals doped in exo and endo modes as well as the transition state of the exo-endo transformation were utilized to benchmark our new model. Numerical results show that the new model with a shorter length of SWCNT segment is able to accurately represent geometrical and electronic structures of longer ones based on conventional termination schemes. Computational results further indicate that the transition barrier heights between the exo and endo modes of various metal-doped systems are ranging from 0.1 to 5 eV. Further research work include the study of other elements or metal clusters as dopants, the ionized version of our doped SWCNT systems, as well as to test with other functional methods.  Finding the weakest bond within a molecule is crucial for understanding the mechanisms behind chemical reactions and the bond dissociation energy (BDE) is used frequently for locating the weakest bond. However, it requires one to perform tedious calculations of BDEs for all bonds within a molecule. To save time and computational resources, we made attempts to design reasonable yet simple structural indicators to identify weak chemical bonds in Project II. Based on the commonly available structural property indicators for bond strength, such as bond length, the Mulliken interatomic electron number, and the Wiberg bond order, we have created several new bond-strength indicators that can be directly used to efficiently identify almost all weak bonds. In the future, different chemical systems will be analyzed by the new indicators to test their reliabilities.     iii  Preface  All projects of this thesis were done under the supervision of Prof. Alexander Yan Wang, with the computational facilities provided by the Department of Chemistry at the University of British Columbia and the WestGrid of Compute Canada. While the first project was completed solely by the author of this thesis, the second project was a collaboration between Wang Group and Gui-Xiang Wang, a visiting professor from Nanjing University of Science and Technology in P.R.China, whose contribution was to carry out the BDE calculations for given molecules of interest. The author of this thesis was involved in the second project by providing Bader Analysis of the given chemical systems.                       iv  Table of Contents                                                                                                     Abstract ................................................... ii Preface   ................................................... iii Table of Contents  ............................................ iv List of Tables  ............................................... v List of Figures   .............................................. vi Acknowledgements  ........................................... viii   Project I, Section 1 Introduction  .................................. 1 Project I, Section 2 Computational details   ........................... 7 Project I, Section 3 Results and Discussion  ........................... 12 Project I, Section 4 Conclusions   .................................. 39   Project II, Section 1 Introduction   ................................. 41 Project II, Section 2 Computational details    .......................... 48 Project II, Section 3 Results and Discussion   .......................... 48 Project II, Section 4 Conclusions  .................................. 62   Closing Remarks  ............................................. 64   References .................................................. 65       v  List of Tables                                                                                                                       Project I:  Table I.3.1 Table of Energetic Data and CPU Time for                     Benchmark Systems  .................................... 13 Table I.3.2 Structural Data for Benchmark Systems  ....................... 15 Table I.3.3 Energetic Data for All Dopants ............................. 16 Table I.3.4 Average Metal Carbon Bond Distances of Three States  ............. 24 Table I.3.5 Various Atomic Radii for Elements of Interest   .................. 28 Table I.3.6 Natural Charges for Metal Centers in Doped Systems  .............. 34 Table I.3.7 Electronegativity Values of Each Dopant and                     Comparison with Carbon ................................. 35    Project II:  Table II.1.1 Bond length and Bond Strength of Selected Common Bonds ......... 42 Table II.3.1 Bond strength indicators for selected bonds                       of TNT, TNM, and PNT ................................. 49 Table II.3.2 Bond strength indicators for selected bonds                       of TNCr, CDNAPY, and PNT  ............................. 50 Table II.3.3 Bond strength indicators for selected bonds                       of AMNA, AMNFMC, and NMDACB  ....................... 51 Table II.3.4 Average Bond Distances and Bond Energies                       of Common Bonds   .................................... 53 Table II.3.5 MIEN Values of Various Basis Sets for TNT  ................... 58       vi  List of Figures                                    Project I:  Figure I.1.1 Illustration of CNTs .................................... 1 Figure I.1.2 Naming and Chirality of SWCNTs  .......................... 2 Figure I.1.3 Diagram of Reaction Coordinates and Doping Modes .............. 3 Figure I.1.4 Depiction of Existing Schemes for CNT Termination .............. 4 Figure I.1.5 Structures of the Edges of Hydrogen and Nitrogen Termination  ....... 5 Figure I.1.6 Illustration for the Scheme of H and N Doped Edges  .............. 6 Figure I.1.7 Demonstration of Our Nitrogen Terminated “EAR” Model  .......... 7 Figure I.2.1 Illustration of Our Nitrogen Terminated Model and                       Conventional Hydrogen Terminated Model  .................... 8 Figure I.2.2 Notations and Definitions Used  ............................ 9 Figure I.3.1 3D Surface Plot for Energetic Data .......................... 17 Figure I.3.2 Plot of the Energy Differences Between Exo                       and Endo Modes for all Dopant Systems ...................... 19 Figure I.3.3 Hg Doped SWCNT in Endo mode  .......................... 20 Figure I.3.4 Plot of the Exo and Endo Barrier Heights for                       All Dopant Systems  ................................... 21 Figure I.3.5 Illustration of the Transition State Geometry for Cu Doped System  .... 22 Figure I.3.6 Plot of the Average Bond Distances Between                       Dopants and Adjacent Carbon against Barrier Height   ............. 23 Figure I.3.7 Plot of Average Metal Carbon Bond Distances                       of Each Systems Against Respective Barrier Heights .............. 25 Figure I.3.8 Plot of Various Types of Atomic Radii  ....................... 27 Figure I.3.9 Depiction of Relationship Between Metal                       Carbon Bond Distances and Metallic Radii ..................... 31 Figure I.3.10 Representation of Metallic Radii   .......................... 32 Figure I.3.11 Illustration of Natural Charges and                         Electronegativities of All Dopants .......................... 36 Figure I.3.12 Plot of Natural Charges of Metal Centers Against                         Respective Differences in Electronegativity Relative to Carbon ...... 37 Figure I.3.13 3D Surface Plot for Natural Charges of all Dopants .............. 38 vii  List of Figures                                    Project II:  Figure II.1.1 The Illustration of Electron Density of A                        Homo-diatomic Chemical Entity ........................... 44 Figure II.1.2 Profile of the Electron Density Distribution of H2 ................ 45 Figure II.1.3 Structures and Names of Selected Molecules ................... 47 Figure II.3.1 Plot of Bond Energy per unit Bond Length                        Against Inverse of Average Bond Length ..................... 53   Figure II.3.2 Comparison of Various Bond Strength                        Indicators for Breaking Single Bonds ........................ 54                     viii  Acknowledgements  I would like to express my great appreciation to Prof. Yan Alexander Wang for providing me with opportunities to join his research group. I would also like to offer my gratitude to the whole Wang group, whose members inspired me to work in this field. Special thanks are owed to my parents, who have supported me throughout my years of undergraduate education, both morally and financially, as well as to my friends for providing emotional support.   For Project I, special thanks to UBC and Natural Sciences and Engineering Research Council (NSERC) of Canada for providing financial support for the research. For Project II in particular, special thanks to visiting Professor Gui-Xiang Wang (Nanjing University of Science and Technology) for carrying out the BDE calculations for given molecules of interest. Once again, special thanks to UBC and NSERC for providing the financial support.        1 Project I. Non-junction Single-Walled Carbon Nanotubes Doped with Transition Metal Atoms  I.1 Introduction  Ever since the reintroduction of carbon nanotubes (CNTs) to the scientific community by Iijima [1.1], [1.2] in early 1990s, carbon nanotubes have become the hot spot of discussion owing to their intriguing physical and chemical properties. Such properties, including mechanical strength, chirality dependant electric conductivity, thermal conductivity and chemical inertness, were originated from their unique pure carbon structures that the carbon atoms were connected to each other via sp2 bonds, opening up the gates to the development of new structural and functional materials [1.3]. The applications of CNTs can be found in molecular electronics [1.4], nano motors [1.5] and sensors [1.6]. Among them, one of the most encouraging one is the first CNT based computer processors made by groups from Stanford University in 2013 [1.7], as it is not just theoretical studies but an actual experiment to prove that CNTs can be built into real logic circuits.     One way to classify CNTs is by their number of layers. For CNTs with only one layer, they are called single wall carbon nanotubes (SWCNTs) [1.2], [1.8], while the multi-layered ones are called multiwall carbon nanotubes (MWCNTs) [1.1], [1.9], as Figure I.1.1 illustrated. There are two types of MWCNTs, the most common one is the coaxial, Russian Doll, as Figure I.1.1 (b) depicted, and the other is the Parchment, more like the rolled up of single sheet of paper.            Figure I.1.1: Illustration of CNTs. (a) SWCNT [1.8] and MWCNT (b) [1.9]. Note that only the Russian Doll (coaxial) type is shown for MWCNT. (a) (b) 2 Additionally, SWCNTs can be named by an index. When constructing the model of a SWCNT, one can imagine wrapping a one-atom-thick sheet of graphite, or graphene, into a cylinder. This segment of SWCNT can be represented by a pair of indices, (n,m), in which the integers, n and m, denoted the number of unit vectors along two directions (a1, a2) in the crystal lattice of graphene as shown in Figure I.1.2. The determination of indices (n,m) is crucial because the properties of SWCNTs, especially electronic properties, depend strongly upon it. With varying values of m and n, the conductivity of SWCNT can range from metallic to semiconductor. Special cases for indexes are when n = m or m = 0, the SWCNTs are called armchair and zigzag respectively, otherwise they are all chiral. Exceptions exist in terms of electronic conductivity for smaller diameter CNTs, in which the curvature effect plays a role in altering this property [1.10]. For MWCNTs, they can be considered as a few co-axial SWCNTs with different diameters inside each other.                     Figure I.1.2: Naming and Chirality of SWCNTs [1.8]. This figure also shows how to construct SWCNT models via wrapping of graphene sheets and the three possible outcomes, which are armchair, zigzag and chiral. The index (n,m) is determined by the way of warping, in which n, m are the number of hexagons along two directions,  1a  and 2a . 3 Computer simulations have an advantage over the experimental studies as they can provide insights on the subject of interest in the case when actual experiments are very difficult to conduct or limited by current technology or funding. Hence computational modeling can be essential when designing new materials. Previous computational studies done in our group suggested that transition metal doped SWCNTs have the potential to be small gas detectors as the electronic structures of such materials are highly influenced by the presence of gases [1.11]. Upon binding of the targeted molecules onto the doped transition metal, the SWCNT system will undergo a change in its conductance and electronic structures, and by monitoring the changes, one can easily detect the presence of these molecules. When doping the transition metals onto SWCNT, the relative spaces of dopant will be divided by the nanotube wall into interior and exterior, in which the dopants experience either concave or context surfaces that results in different physical and chemical properties. The model in which the dopant is located at the outside (convex tube surface) of the SWCNT is called exo-doping (Figure I.1.3), while the inside (concave tube surface) version of the model is endo-doping (Figure I.1.3). According to Y.K. Chen et al., the system would generally prefer exo-                  Transition State ΔE ~ 2eV Endo Exo Endo-Barrier Height Exo Barrier Height  Figure I.1.3: Diagram of Reaction Coordinates and Doping Modes [1.12]. The picture on the upper right refers to the exo doping mode, while the one below refers to the endo one. Exo- (Endo-) Barrier Height refers to the energy difference between ground state exo- (or endo-) doped CNT and their respective transition state. Reaction Progress Energy 4 doping over endo ones due to the lowering of overall energies, and for dopants from 4th row transition metals, the energy difference between two doping modes is about 2eV [1.12]. Another study which utilizes carbene (CH2), imine (NH), and atomic oxygen as dopants on both sides of SWCNTs indicates that such exo-endo difference is due to the lack of σ bonding interactions between the dopant and adjacent carbon atoms of the CNT wall when the dopants are located inside the nanotube [1.13]. For the dopants doped outside of the tube, they can have strong σ bonding interactions with adjacent carbon, yet the dopants inside SWCNT will only interact with the adjacent carbon atoms via a partial sp3-hybridized state. This study further suggests that, such poor σ bonding interactions are results from the concave tube surface geometry of the interior of SWCNT, so when doping inside the SWCNT, the dopants were held mostly by the π interactions with the carbon atoms on SWCNT side wall. Now, identifying the transition state between the two doping modes would be the next sensible step, so that the magnitudes of the barrier heights are fully appreciated. Such study would be essential for the applications of these transition metal doped SWCNTs, especially when there is transition between exo and endo mode involved, or when we want to know the maximum energy we can input to the system without causing the transition. Currently, there are two major methods of constructing SWCNT models [1.14], the first one is infinite length with periodic boundary conditions (PBC), and the second is finite length with C or H termination, as well as the open-ended (no termination) ones, as illustrated in Figure I.1.4 below.              (a) (b) (c)                     Figure I.1.4: Depiction of Existing Schemes for CNT Termination [1.14]. Types of finite length ending: (a) with H termination, (b) with C termination, and (c) with no termination. 5 In the case of C termination model, the proper carbon “caps” (cut portion of fullerenes) would have to be placed on both ends of the CNT (Figure I.1.4, (b)), which increases the computational cost as it enlarges the size of the system under study, and the fitting of carbon cabs onto the CNT edges may not be easy as the position and size of the cab can be difficult to determine. On the other hand, the H termination model has more H atoms, as well as an extra carbon-atom layer on both end of CNT (Figure I.1.4, (a)), added into the system, which increases computational cost. Moreover, the terminating H atoms have a tendency to enlarge the opening of both ends of the CNT when we were performing the calculations. The third one is the open ended model (Figure I.1.4, (c)) with no termination left the openings chemically active, which might interference with the site of interest in the middle of the tube due to reactions with other chemical entities. Generally, the use of the open ended model was limited to circumstances in which there are needs in chemical modifications or reactions at the openings. There also exist models which involve combinations of different methods mentioned above.  In this project, we employed a different terminating model, which is the N termination, and it was built simply by replacing the terminal CH atom pairs with the isoelectronic N atoms. The terminating N atoms also participated in the aromatic system, behaved similar to the original carbon atoms in CH pair, yet the convergence of the systems were improved, and computational costs were reduced. A representation showing how the CH pairs can be replaced by the nitrogen atom of the two terminating methods is portrayed in Figure I.1.5 below.              Figure I.1.5: Structures of the Edges of Hydrogen and Nitrogen Termination. (a) H termination, (b) N termination, and the red circles indicate the location of CH replacement by N. Note: the grey balls are C atoms, the blue balls are N atoms, and the smaller white balls are H atoms.  (a) (b) 6                    From Xu et al. [1.15], whose group done a comprehensive study regarding the nitrogen edge-doped effect of SWCNTs, in which they found that replacing CH pairs by N atoms one by one will result in a gradual decrease in the diameter of the opening. As illustrated in Figure I.1.6 above, when completely doped with N atoms, the N terminated opening of the tested (5,0) super short SWCNT will have a diameter of 3.84 Å, which is about 6% smaller than the H terminated ones, whose edge has a diameter of 4.08 Å. Their study also indicated that N-terminated models are more stable than H-terminated ones. Our practices in these computational tests have shown that N termination did help the system to converge easier and faster in most situations, which is likely due to the above edge effects of terminating with nitrogen atoms. In this project, we adopted the N terminated model since it actually improve the efficiency. To obtain a balance between the accuracy and efficiency for locating the transition states, we further modified our model by adding an extra pair of “ears” on both ends of the tube (5,0) super short SWCNT with H termination on both edges Figure I.1.6: Illustration for the Scheme of H and N Doped Edges [1.15]. A segment of (5,0) SWCNT terminated with hydrogen on both edges was being used. The lower row shows the graduate substitution of CH pairs by N atoms. The picture on the upper right provides a visualization of diameters comparisons between fully N doped and fully H doped edges, which illustrates that a fully N doped opening would have a smaller diameter than the H doped counter parts. Illustration of the diameters of N (DU) and H (DL) termination 3.84 Å  4.08 Å  7 to have more buffering between the doping site and the edges. This ensures that the edge would have smaller influence to the doping site, as Figure I.1.7 shown, and thus named the “ear” model. In this way, we left two more C atoms on the edges that were closest to the dopant atom, so the number of carbon-atom layers between the doping site and the edge is increased along the tubular direction.                I.2 Computational Details All computational tasks were performed by Gaussian 09 package [1.16]. DFT calculations were done via exchange-correlation density functional of PBEPBE, by Perdew, Burke, and Ernzohoff [1.17~1.18]. On the other hand, the electronic wave functions were expanded in user defined basis sets, in which all non-metal elements, such as carbon and nitrogen, were calculated using 6-31G basis set [1.19], while all transition metals appearing in this project were calculated using the pseudo-potential LanL2DZ basis function [1.20~1.23]. Moreover, all models calculated were neutral and non-charged. A section of (5,5) unbranched SWCNT with two “ears” extended around the doping site was used for all transition state computations. Initially, a 13-carbon-atom layer (5,5) SWCNT consists of pure carbons was obtained from the online nanotube generator developed by Figure I.1.7: Demonstration of Our Nitrogen Terminated “EAR” Model. Note: the grey balls are C atoms, the blue balls are N atoms, and the red ball represents the dopant. A further discussion of this model will be provided latter, in the Computational Details session.   8 University of Delaware, the tubegen [1.24]. Then the original tube segment was cut to left with two “EARs” (two more carbon atoms at each end) at both ends, and all exterior carbon atoms replaced by nitrogen. Next, a central carbon atom was replaced by a generic transition metal X either above or below the position of the initial carbon site, to produce symmetrical models with two doping modes (exo or endo), and a molecular formula of C93N10X, as shown in Figure I.2.1. The resulting models were first optimized using different spin multiplicities to produce a ground                           9 core C layers Terminating layers Terminating layers 10.0 Å  2.5 Å  6.6 Å  Figure I.2.1: Illustration of Our Nitrogen Terminated Model and Conventional Hydrogen Terminated Model. (a) Our N terminated, “EAR” model, (b) a segment of the H terminated, conventional straight cut model. Note: the grey balls are C atoms, the blue balls are N atoms, the smaller white balls are H atoms, and the red ball represents the dopant. The red circle in (a) demonstrates one layer of carbon atoms. For our model, the core carbon (CC) layers have a length of about 10 Å, and each terminating layers have a length of about 2.5 Å. The diameter of the opening in our model is 6.6 Å, in oppose to 7.0 Å in the conventional H terminated straight cut model, while both of the models were based on the same (5,5) SWCNTs.  One layer of C atoms 7.0 Å Terminating layers Core C layers (a) (b) 9 state structure, which were confirmed with vibrational analysis. Symmetry was enabled in order to accelerate the calculation as all exo and endo modes are belonging to Cs point group. An initial guess of transition state was made by placing the dopant on the side wall of SWCNT, together with the structures of endo and exo modes, transition state calculations were done for most of the transition metals on periodic table. The resulting transition state structures were also confirmed with vibrational analysis.  An illustration of some terminologies and definitions, as well as some geometrical measurements, of our “EAR” model have been depicted in Figure I.2.1. This illustration also provided a comparison of the diameters of the opening between our model and the conventional straight cut, H terminated model. As Figure I.2.1 shows, even thought our model and the conventional H terminated model are both based on the (5,5) armchair type SWCNTs, our modified N terminated model has an opening diameter of 6.6 Å, which is about 5% smaller than the 7.0 Å of the opening diameter of the H terminated ones. For any finite length SWCNT model, its carbon atom layers can be divided as terminating layer or core carbon layers, whereas a layer of carbon atoms is defined as a ring of carbon atoms perpendicular to the direction of the tube, as the red circle in Figure I.2.1 (a) demonstrates. Moreover, the total length of our model is about 15 Å, in which the core carbon atom layers consists of around 10.0 Å, while the two terminating layers each spans roughly 2.5 Å. Further details of the doping site, as well as other notations used,              Figure I.2.2: Notations and Definitions Used. The red ball marked with X is the dopant metal, while grey balls are carbon atoms, among which the atoms adjacent to X were marked with 1, 2 and 3 are referring to as C1, C2 and C3.  X C3 C2 C1 10 were included in Figure I.2.2. Carbons at positions 1, 2 and 3 are referring to as C1, C2 and C3. X–C1, X–C2 and X–C3 are the bond distances between Metal X and C1, C2 and C3. The bond angles and dihedral angles, were defined as the following:          In order to obtain a reasonable initial guess for the transition state, a series of trials were performed. We first select a trial element from the 4th row, as elements from this row are smaller in size comparing to latter rows, and we hoped a smaller atom would undergo the transition between endo and exo modes easily, which would make the identification of transition state more promising. We have chosen Ni to start with because it has a relatively small atomic radius within the row, so it should be an easy case to do exo-endo transition. After a valid transition state has been obtained, we switched to Sc to repeat the same transition state search. Note that, this time we used the geometry of transition state for Ni system obtained previously as the new initial guess for Sc system. Since Sc has the largest radius for 4th row transition metals, its system should be the hardest case to do comparing to other systems with 4th row transition metal dopants. After the valid transition state of Sc is acquired, its geometry will be used as the initial guess for the rest of the 4th row dopant systems, as well as going down the Sc group. The idea behind such procedure is that, within 4th row dopants, if the most difficult case (Sc) works, then the rest of the 4th row transition metal doped systems should be easy replications of the most difficult one. Moreover, the geometry for Sc system should be good initial guess for the Y and La since they are from the same group. Sc was used in the benchmark systems to compare other SWCNT models because its systems are of moderate difficulty when considering the trends of atomic sizes in rows and groups of periodic table. Prior to adopting our “EAR” model to all other intended transition metal elements, we have done some benchmarking tests against the periodic boundary conditions and the existing Bond angle 1: C1–X–C2 Bond angle 2: C1–X–C3 Bond angle 3: C2–X–C3 Dihedral angle 1: X–C1–C2–C3 Dihedral angle 2: X–C2–C3–C1 Dihedral angle 3: X–C3–C1–C2 11 finite length straight cut CH and N termination. One-dimension periodic boundary conditions were utilized for the infinite model, in which the primitive cell contains 20 carbon layers of (5,5) SWCNT, or 199 carbon atoms plus 1 dopant atom Sc. In addition to the PBC models, straight cut 21-corecarbon-atom (21CC) layer (5,5) SWCNT models doped with Sc, and terminated with N and H, respectively, were used as the benchmark for our new models. Structural information such as bond length, bond angle and dihedral angle were obtained and compared against each other, so as to the energy differences between exo, endo, and transition state. For the purpose of performance comparison, the job CPU time was also listed for each system. To find the very first transition state for Ni, we first used an initial guess in which Ni was placed at the wall of nanotube, while froze all the atoms except the dopant. The purpose is to significantly reduce the computational cost and improve the chances of convergence. After we got a converged result, we used the output geometry as the initial guess to perform a new round of calculation, yet this time we freed up the three adjacent carbons around the dopant, allowing four flexible atoms in total, including the dopant, while all other atoms remained frozen. After this computation was terminated normally, we repeated the above step for another round using the newest output as initial guess, but now we set all atoms to be flexible to do this round of calculation. Finally the last round of calculation in which all atoms are free should yield the correct transition state, which can be check by doing a vibrational analysis. If the latest attempt of freeing all atoms cannot yield any valid transition state geometries, then we have to go back to previous step yet only free up a few carbon atoms adjacent to the doping site, and repeat the above finding process.  While searching for the transition state, if we restrict the symmetry of transition state to Cs, which is the point group of the ground state of both exo and endo mode, the task would result in a failure, or even if it converged at the end, the resulting transition state would not pass the vibrational analysis by producing two or more negative frequencies. The reason being that, during the transition between exo and endo modes, the system may not retaining Cs symmetry at all times, or may have different symmetries at the transition state. Hence, in this project, all transition state calculations were done with the symmetry disabled completely, while it was enabled for the geometry optimization of exo and endo modes. To display and compare all the data obtained from the calculations at once, the 3D surface plots, which were done using OriginPro 9.0 [1.25], were necessary in addition to the 2D plots. 12 I.3 Results and Discussion We have chosen the (5,5) SWCNT to be the system of interest because previous studies [1.12] have shown that it is the most optimum system for hetero-atom doping. The benchmark test results between existing models and ours are presented in Table I.3.1 and Table I.3.2. As we also want to see how PBC, H terminated and N terminated models differ from each other, the PBC and 21CC H terminated model were also included. Since H termination and N termination models are different in energy, we should use the N terminated model to compare ours against.  From the energetic perspective, as Table I.3.1 (a) shows, when comparing the 20CC PBC with 21CC N and H terminated models, the latter two yielded exo-endo energy differences of 2.4 eV and 2.3 eV, while the PBC ones produced a value of 2.7 eV. Note that, only the exo-endo difference is available for PBC calculations, since the transition state calculation is almost impossible for PBC models. When only look at the exo-endo energy difference, the 21CC N terminated model appear to be more desirable because it can yield results closer to the PBC ones, which is supposed to be more accurate. However, when imposing the 1D periodic boundary condition, not only the segments of the SWCNT, but also the doping sites and dopants were being copied along the tube directions. This essentially created a multi-doping nanotube system, in which the distance between each dopant is not infinite, and it is very different from our intended single doping model. Such multi-doping PBC models introduce interactions between each doping sites, and the spin multiplicity might also be affected as the system is now open shell. Therefore, the PBC model is not the most accurate one for benchmarking as expected, we still have to go with the very long tube models.  When comparing between different N terminated models, the straight cut 9CC N terminated model failed to produce an exo-endo energy difference that is in good agreement with the longer 21CC ones, so as to its endo-barrier height, although its exo-barrier height is in excellent agreement. On the other hand, our modified 9CC N terminated model produced results that are more or less in good agreement with the exo-endo energy difference, as well as the exo- and endo- barrier heights.  Speaking of the time spent on each computational task, as Table I.3.1 (b) shown, the PBC ones took the longest job CPU time, followed by H terminated models. While the 21CC N terminated systems ranked at the middle for all three types of tasks, its transition state search task took less than half of the time comparing to the 21CC H terminated ones. Our EAR model 13 ranked 4th among all models for all tasks, as the 9CC straight cut N terminated model took strikingly lowest time. The reason for such small amount of CPU time is because the input files for 9CC ones were obtained by removing the two “ears” from the already converged EAR models. Considering the computational costs, the N terminated models should be better than the H terminated ones. This intriguing behaviour might be owing to the fact that the N edge opening models are more stable than the H edge ones, as suggested by studies done by Xu et al. [1.15], and this stabilization from N edge effect can contribute to better convergence. Thus, a computationally cost efficient way of doing termination of same core carbon length should use the N edges.      CNT Systems Exo (hr) Endo (hr) TS (hr) 20CC PBC 215 739 N/A 21CC N-Term 72 129 392 21CC H-Term 138 148 817 9CC  N-Term 11 19 86 9CC  EAR N-Term 24 115 206      CNT Systems Exo−Endo (eV) Exo-Barrier (eV) Endo-Barrier (eV) 20CC PBC 2.773 N/A N/A 21CC N-Term 2.405 3.970 1.565 21CC H-Term 2.260 3.755 1.494 9CC  N-Term 2.615 3.967 1.352 9CC  EAR N-Term 2.442 3.934 1.492 Table I.3.1: Table of Energetic Data and CPU Time for Benchmark Systems. (a) Energetic data of each system. (b) Job CPU time of each system. The energy differences are all in eVs and they are all absolute values, while the CPU times are rounded in hours (hr). Note that the N/A here means Not Available and CC means core-carbon layers. (b) (a) 14 From the geometrical perspective, as Table I.3.2 demonstrates, all the benchmarking systems, even the H terminated ones, produced very similar results in terms of bond lengths, bond angles and dihedral angles. In this case, it appears that the geometry around the doping site is not influenced much by varying termination methods. Hence, we can conclude that our 9CC EAR model can produce results that are in excellent agreement to its longer version with a reduced job CPU time. So we are safe to employ our EAR model further to explore the transition states for other transition metal dopings.   For the purpose of a quick overview of all available energetic data for transition state calculations shown in Table I.3.3 in a clear and efficient way after they were collected, 3D plots were very useful. To serve this purpose, a 3D surface plot in which all energetic data entries were being presented and compared at the same time was created. The three different perspectives of this plot were showed, including Figure I.3.1 (a), the top view, Figure I.3.1 (b), the exo barrier view, and Figure I.3.1 (c), the endo barrier view. Since we only care about energy differences, when we were creating the plot, the energies of all exo modes were set to 0 eV, and the energies of transition states or endo modes were shifted and measured with respect to this new reference point in eVs. Utilizing OriginPro 9.0, we plotted a 3D surface graph in which the groups were listed as the X-axis, while the states (exo, TS, and endo) as the Y-axis, and the energies being the Z-axis. In the plot, a colour scale for the energy in eVs was used instead of the contours for the clarity of the plot. Please note that, the elements were bundled together and labelled by their group numbers on the periodic table, which means, the elements were sorted by going down a group going from Group 3 to Group 12, for example, Group Number 3 (or IIIB) family have members Sc, Y and La being listed in sequence.  As can be seen from the 3D surface plot in Figure I.3.1, it has an interesting behaviour that is similar to the periodic trends, in which the transition state barrier height peaked when the dopant is at the beginning and gradually decrease as the dopant moving to the right side of periodic table, and the barrier heights were also increasing going down the group. However, after the barrier heights reached minimum at the middle of a row, when approaching to the right end of a transition metal period from left, the barrier height usually observe some sudden raises.  As the 3D surface plot provided a comparison of all data at the same time, it has a drawback due to the fact that we zeroed all the exo mode energies and shifted other energies created such plot. The consequence of the above is, we can visualize and compare the exo-barrier  15     Exo X-C1 (Å) X-C2 (Å) X-C3 (Å) Bond angle 1 C1–X–C2 Bond angle 2 C1–X–C3 Bond angle 3 C2–X–C3 Dihedral 1 X–C1–C2–C3 Dihedral 2 X–C2–C3–C1 Dihedral 3 X–C3–C1–C2 20 CC PBC 2.081 2.081 2.131 80.34° 84.69° 84.69° 57.80° 59.85° 59.84° 21CC N-Term 2.083 2.083 2.125 80.54° 84.20° 84.20° 58.17° 59.89° 59.89° 21CC H-Term 2.081 2.081 2.122 80.77° 84.58° 84.58° 57.88° 59.73° 59.73° 9CC  N-Term 2.070 2.070 2.131 80.54° 84.14° 84.14° 58.20° 59.90° 59.90° 9CC EAR N-Term 2.086 2.086 2.129 80.31° 84.42° 84.42° 57.91° 59.95° 59.95° Endo X-C1 (Å) X-C2 (Å) X-C3 (Å) Bond angle 1 C1–X–C2 Bond angle 2 C1–X–C3 Bond angle 3 C2–X–C3 Dihedral 1 X–C1–C2–C3 Dihedral 2 X–C2–C3–C1 Dihedral 3 X–C3–C1–C2 20 CC PBC 2.023 2.024 2.075 84.74° 90.24° 90.21° -54.03° -56.79° -56.79° 21CC N-Term 2.023 2.023 2.109 85.74° 87.68° 87.68° -57.02° -56.64° -56.64° 21CC H-Term 2.023 2.023 2.106 85.91° 88.05° 88.05° -56.70° -56.51° -56.51° 9CC  N-Term 2.023 2.023 2.102 86.64° 88.42° 88.42° -56.46° -56.13° -56.13° 9CC EAR N-Term 2.024 2.024 2.131 85.63° 86.82° 86.82° -58.10° -56.70° -56.70° Transition States X-C1 (Å) X-C2 (Å) X-C3 (Å) Bond angle 1 C1–X–C2 Bond angle 2 C1–X–C3 Bond angle 3 C2–X–C3 Dihedral 1 X–C1–C2–C3 Dihedral 2 X–C2–C3–C1 Dihedral 3 X–C3–C1–C2 20 CC PBC N/A N/A N/A N/A N/A N/A N/A N/A N/A 21CC N-Term 2.005 1.951 1.981 98.37° 117.2° 142.5° -6.786° -13.98° -8.455° 21CC H-Term 2.007 1.952 1.983 98.49° 116.7° 143.1° -6.311° -13.18° -7.798° 9CC  N-Term 2.009 1.952 1.986 100.9° 114.2° 141.8° -8.655° -17.12° -10.07° 9CC EAR N-Term 2.009 1.953 1.982 98.19° 116.0° 143.0° -8.053° -16.91° -9.894° Table I.3.2: Structural Data for Benchmark Systems. Note that the N/A here means Not Available and CC means core-carbon layers.  16      Row Element Index Exo−Endo (eV) Exo-Barrier (eV) Endo-Barrier (eV) 4th Row 21Sc 1 2.44 3.93 1.49 22Ti 2 1.95 3.40 1.45 23V 3 2.14 3.03 0.89 24Cr 4 2.34 2.93 0.59 25Mn 5 2.07 2.71 0.63 26Fe 6 2.13 2.89 0.77 27Co 7 2.01 2.61 0.60 28Ni 8 1.86 2.36 0.49 29Cu 9 1.61 1.80 0.19 30Zn 10 2.46 3.11 0.66 5th Row 39Y 16 2.30 5.59 3.28 40Zr 17 2.48 4.86 2.38 41Nb 18 2.54 4.45 1.91 42Mo 19 3.19 4.08 0.90 43Tc 20 3.41 3.94 0.53 44Ru 21 3.49 4.09 0.59 45Rh 22 3.21 4.03 0.82 46Pd 23 2.78 3.14 0.36 47Ag 24 0.71 3.00 2.29 48Cd 25 0.16 4.56 4.39 6th Row 57La 31 2.28 7.82 5.53 72Hf 32 2.49 4.81 2.32 73Ta 33 2.47 4.21 1.74 74W 34 3.21 3.88 0.67 75Re 35 3.52 3.87 0.35 76Os 36 3.84 3.95 0.11 77Ir 37 3.56 3.88 0.32 78Pt 38 3.13 3.16 0.02 79Au 39 1.49 2.97 1.48 80Hg 40 0.49 6.81 6.32 Table I.3.3: Energetic Data for All Dopants. The indexes assigned to each element are for the purpose of 2D plots, so that the dopants from the same row will be bundled together. The energy differences as well as the barrier heights are all in eVs and they are all absolute values. 17                                (a) (b) 18                     height as well as the exo-endo energy gap easily, yet the endo-barrier height would be hard to tell. Therefore, this 3D plot can only serve as a crude overview, and the 2D plots are still essential to investigate further detailed relations. When doing the 2D plots, we assigned each dopant element an index to bundle them by their rows, as shown in the Table I.3.3. For example, Index 1 to 10 represents 4th row transition metals, Sc to Zn, while Index 31 to 40 refers to 6th row transition metals, La to Hg, excluding other elements in Lanthanide series. The indexes were designed so that there were blanks left between different rows as separations between different rows.  First of all, we created a 2D plot for the exo-endo energy differences with respect to the assigned indexes of elements. By examining this plot, Figure I.3.2, the energy difference plot, an interesting trend can be observed. While both of the 5th and 6th row dopant systems have a peak Figure I.3.1: 3D Surface Plot for Energetic Data. The three different perspective diagrams were shown: (a) top view, (b) exo view, (c) endo view. A colour scale was used for the energy in eVs instead of the contours for the clarity of the plot, in which the red color indicated higher values in energy, while blue color indicated lower values. Also note that the lines are interpolations to show that the exo, TS, and endo data are from the same element.   (c) 19 for the energy difference at the middle of their series, displaying an inverted V-shape, the 4th row dopant systems have peaks at the two ends of its series, with a local maxima at the middle, making it a W-shape. Moreover, the Ag, Au, Cd and Hg doped systems exhibit a very low energy gap between the two doping modes, which means there will be almost no preference on doping inside or outside, which indicates that there might be little difference in terms of bonding between their exo and endo modes.               By taking a close look at the endo mode of these systems, we can find the dopant will be sitting at the center of the CNT model we tested, as illustrated in Figure I.3.3, is an example using Hg as dopant. This means their dopants would not prefer forming bonding interactions with the carbons on CNT walls when being doped inside. Since the energy gap is very small, the energy of exo mode is almost the same as endo mode, we can tell that even when doping outside, these dopants will form very weak interactions with the CNT walls. For Ag and Au, they are chemically inert and generally not reacting with other chemical entities, so it makes sense if they do not readily interact with the carbons on CNT; while for Cd and Hg, in our neutral, non-charged models, their ns orbitals and (n−1) d-orbitals were filled, in which n stands for their row number, so that they do not have reactivities comparable to other d-block elements [1.26]. Thus, this results in very poor bonding interactions between metal and carbon for Cd and Hg dopants. 0.00.51.01.52.02.53.03.54.00 5 10 15 20 25 30 35 40Energy (eV)IndexFigure I.3.2: Plot of the Energy Differences Between Exo and Endo Modes for all Dopant Systems. The indexes of dopants were assigned so that they can be bundle by their rows. For detailed assignment of indexes, please refer to Table I.3.3.  20                Another issue associated with the latter elements of Cu and Zn family is the bond distances. When examining the endo modes of Ag, Au, Cd, and Hg, on (5,5) SWCNT, the observation is that the dopant will stay exactly at the center of the tube. At extreme cases, if the bond distances between the dopants and carbon atoms exceeds the radius of the tube, they will be forced to stay at the middle of the tube due to the small size of (5,5) SWCNT. To investigate this issue, we have performed additional geometry optimizations for above systems using models based on (8,8) SWCNTs, in which the only difference comes from the diameter of the CNT. After measurements done, we found that the diameter of our (5,5) SWCNT model is around 6.6 Å, while for (8,8) ones is about 10.8 Å, and the average distances between dopant and adjacent carbons are from 3.5 to 4.5 Å in (8,8) SWCNT models, which means the dopant would stay at some location close to the doping site but not exactly at the middle if they were doped onto a (8,8) SWCNT.  We have also tried placing the dopant exactly at the middle of a (8,8) model, yet that initial guess failed to converge or system will return to the previously mentioned location. The (5,5) model in which dopants stay at the middle will produce an average distance of 3.3 Å between dopant and doping site, which is smaller than the distance when testing with (8,8) CNT models.  Hence, we can conclude that the previous (5,5) model is not big enough to study the Figure I.3.3: Hg Doped SWCNT in Endo mode. (a) Hg doped (5,5) SWCNT and (b) Hg doped (8,8) SWCNT. Inside the smaller diameter tube (a), the doped Hg was located almost at the center, while inside the bigger diameter ones (b), the doped Hg was located at 4.5 Å away from the doping site. (a) (b) 6.6 Å  10.8 Å 21 endo doping of above four elements and not suitable to describe the endo doping of these dopants. In the near future, we will redo all calculations for our systems with the (8,8) models to better describe the endo mode of that four dopant and for the consistency. Meanwhile, it will be a good idea to perform DFT calculations using several different functionals to compare against the results. When speaking of the barrier heights, from Figure I.3.4, there is a clear trend in the magnitudes of the for transition metal dopants in the same period: they were peaked at the beginning and the end of the period, but are much smaller in the middle. This means the dopants from the middle of periodic table will be able to switch between the inside and outside of the nanotube walls easily in comparison to those from the two ends. This is understandable since the transition metals from the two ends possess certain characters which hinders their abilities to migrate across the wall. While the elements on right side within a row are smaller in size, they are less capable of forming more chemical interactions with other chemical species due to their electronic configurations. Apparently, during an exo-endo transition, the transition metal must first have a good size so that when it is at the wall (at transition state), it can squeeze through and does not displace the adjacent carbon atoms too far away from their original locations. Otherwise it would create a relatively large disturbance to the system, especially to the three adjacent               Figure I.3.4: Plot of the Exo and Endo Barrier Heights for All Dopant Systems. The blue solid circles are exo systems, while the yellow diamonds are endo systems. For detailed assignment of indexes, please refer to Table I.3.3. 0.01.02.03.04.05.06.07.08.00 5 10 15 20 25 30 35 40Energy (eV)IndexExo BarrierHeightEndo BarrierHeight22 carbon atoms at the doping site, and this certainly rise the energy of transition state. Figure I.3.5 depicts the geometry of transition state for the Cu doped system as an example of the transition state in general, whose barrier heights were among the smallest ones for all metals tested. From this depiction, we can see that even the dopant with one of the smallest barrier heights still have a transition state geometry that have obvious disruption to adjacent carbon atoms. Comparing to Figure I.2.2, the carbon C3 is the most displaced atom, as it can be seen easily that it is out of the tubular wall. In addition, the three carbon atoms and the dopant generally form an absolute dihedral angle of less than 20o, while for the exo or endo models, this value is generally above 50o. This is a clear indication that the doping site will be disturbed by the dopant at the transition state. Moreover, the transition state must be stabilized by forming more interactions with the surrounding carbons, yet the transition metals from the rear ends are less capable of forming bonding with carbons, which means they will experience more difficulties during such transition.  As to the group trends, we found that the first-row transition metal dopants usually have the lowest barrier heights, and the barrier heights will increase moving down a group. This is cause by the increasing size of the atoms moving down a group. Because the sizes of the atoms are bigger for latter rows, it becomes significantly more difficult for the dopant atom to come near the wall of the SWCNT to reach the transition state.                C1 C2 C3 X This carbon atom, marked as C3, is clearly out of the tubular wall of CNT system. The three adjacent carbon atoms generally form a dihedral angel of about 10o with the dopant, making them almost co-planar. Figure I.3.5: Illustration of the Transition State Geometry for Cu Doped System. The red ball marked with X is the dopant Cu, while grey balls are carbon atoms, among which the atoms adjacent to X were marked with 1, 2 and 3 are referring to as C1, C2 and C3.   23 To further investigate the size effect on barrier heights, the average bond distances between dopant and the three adjacent carbon atoms of endo, exo and transition state were collected for each of our systems, as summarized in Table I.3.4. In the event that there is no bonding, which can be found if the bond length is much longer than the normal transition metal carbon bonds, it will be marked as N.B. beside its value. Also note that, in the case of endo doping for Zn, Y and La, only two bonds were formed so their average bond length was only between the two. After taking the average for bond distances of endo and exo modes, a plot was made for average bond distances against endo and exo barrier heights respectively, as shown in Figure I.3.6, from which one can observe a weak correlation between average bond distances and corresponding barrier heights.  The relationship demonstrated in Figure I.3.6 is not highly linear since the plot from it contains all the bond distances data for every system, including the non-bonding ones. The purpose of that figure is to provide a rough picture of the relation for all systems tested, and a more accurate plot excluding the non-bonding systems is provided in the Figure I.3.7. As demonstrated in Figure I.3.7, most of the transition metal dopants tested have a linear relation between their bond lengths of exo or endo mode and the respective exo or endo barrier heights. For transition metal dopants from latter rows (5th and 6th), the R-square values are higher than 0.9 for both exo and endo cases, yet for 4th row doped systems the linearity is relatively lower,                y = 4.7872x - 5.8392R² = 0.6215y = 2.8246x - 4.5076R² = 0.66780.02.04.06.08.01.6 2.0 2.4 2.8 3.2 3.6Barrier Height (eV)Bond Distance (Å)Exo RelationsEndoRelationsFigure I.3.6: Plot of the Average Bond Distances Between Dopants and Adjacent Carbon against Barrier Height. This is done for exo and endo modes respectively.  24  Row Element Index Exo Average (Å) Endo Average (Å) T.S. Average(Å) 4th Row 21Sc 1 2.10 2.06 1.98 22Ti 2 1.96 1.92 1.88 23V 3 1.90 1.84 1.82 24Cr 4 1.90 1.83 1.81 25Mn 5 1.85 1.81 1.81 26Fe 6 1.81 1.74 1.75 27Co 7 1.82 1.76 1.78 28Ni 8 1.85 1.81 1.76 29Cu 9 1.93 2.16 (N.B.) 1.86 30Zn 10 2.05 2.11 (N.B.) 1.93 5th Row 39Y 16 2.25 2.22 2.09 40Zr 17 2.10 2.07 2.00 41Nb 18 2.01 1.99 1.94 42Mo 19 1.98 1.94 1.88 43Tc 20 1.94 1.89 1.84 44Ru 21 1.92 1.87 1.85 45Rh 22 1.94 1.89 1.87 46Pd 23 1.98 1.97 1.90 47Ag 24 2.13 2.65 (N.B.) 1.97 48Cd 25 2.29 (N.B.) 3.47 (N.B.) 2.04 6th Row 57La 31 2.42 2.41 2.21 72Hf 32 2.07 2.03 1.99 73Ta 33 1.99 1.97 1.94 74W 34 1.97 1.93 1.89 75Re 35 1.93 1.91 1.85 76Os 36 1.92 1.88 1.85 77Ir 37 1.93 1.88 1.88 78Pt 38 1.97 1.90 1.91 79Au 39 2.05 2.71 (N.B.) 2.00 80Hg 40 2.85 (N.B.) 3.59 (N.B.) 2.08     Table I.3.4: Average Metal Carbon Bond Distances of Three States. The N.B. means no bonding observed in the final converged geometry.  25 y = 5.0422x - 5.7545R² = 0.9709y = 8.3006x - 14.965R² = 0.95350.01.53.04.56.01.8 1.9 2.0 2.1 2.2 2.3Energy (eV)Bond Distance (Å)                                 y = 4.6486x - 5.8361R² = 0.8293y = 3.2939x - 5.221R² = 0.70790.01.02.03.04.01.7 1.8 1.9 2.0 2.1 2.2Energy (eV)Bond Distance (Å)Zn (a) y = 8.4503x - 12.669R² = 0.9392y = 10.3712x - 19.2422R² = 0.96400.02.04.06.08.01.8 1.9 2.0 2.1 2.2 2.3 2.4 2.5Energy (eV)Bond Distance (Å)Figure I.3.7: Plot of Average Metal Carbon Bond Distances of Each Systems Against Respective Barrier Heights. Solid circles are exo systems, while diamonds are endo ones, and (a) 4th row dopants, (b) 5th row dopants, (c) 6th row dopants. The outstanding points that were excluded from the linear plots were either marked as tilted crosses (exo) in blue, or vertical (endo) crosses in red, otherwise they were excluded from this figure because of their large value off the scale. Cu Cu Zn Cd Ag Pd Pd Au (b) (c) Ni Pt 26 having R-square of around 0.8 and 0.7 for exo and endo systems respectively. From the general significance of atomic radii described by Slater [1.27], when two atoms come together to form covalent bonds, there will be an overlap of their outer shell orbitals. This overlapping would be at a maximum when the two atoms are separated by a distance that is approximately the sum of their atomic radii. Hence, the trends above make sense: since the bigger the dopant, the further away the dopants are from the doping sites, and the more energies would be required to push them to the other side of the nanotube, so the more significant the sizes of dopants are for the transition between exo and endo. The exceptions to the above linear relations occur mostly on dopants from Cu (Group 11) and Zn (Group 12) family, in which there are poor bonding interactions between metal and carbons.  However, there is one case that the truly exceptions occur, which is the Ni (Group 10) family doped system. Even though their metal carbon bonds have lengths fall into the range for usual metal carbon bond in both exo and endo mode, their barrier heights, both exo-barrier and endo-barrier, are much lower than other dopants from same period, as summarized in Table I.3.3 and Table I.3.4. This is most obvious for Pd, as illustrated in the plot for row 5, the points for Pd in both exo and endo mode were remove from the linear fitting and plotted as outstanding points since they will severely affect the linearity. For Ni and Pt, their behaviours were better, so they were still plotted in the linear fitting, yet the data points corresponding to their exo modes still shown strong deviation from the trend line. Such abnormal behaviour is worth further investigation regarding the bonding details of group 10 element doped systems in the future. One possible reason is, when go through an exo − endo transition, in which the dopants and the three adjacent carbon atoms will be co-planar, the group 10 transition metals will be able to form more interactions with the adjacent carbons comparing to others, which provides more stabilizations for the transition and results in much lower barrier heights than expected.  Another interesting trend observed from Figure I.3.7 is that the bond length and barrier height relationship in 4th row dopant system is not that highly linear in comparison to latter rows. This is most likely due to the fact that 4th row elements do not have atomic radii as big as the latter rows, so that the bond distances are not a significant factor to impact the exo – endo transition. On the other hand, the 5th and 6th row dopants are large enough such that during the exo – endo transition, their sizes become a real challenge when going through the wall of carbon nanotube.   27 0.60.81.01.21.41.61.82.02.20 5 10 15 20 25 30 35 40Radii (Å)IndexTheoreticalEmpiricalMetallicCovalentIonicFrom the previously discussed periodic trends crossing rows and descending groups, as well as the relationship of average bond distance against barrier height, we can conclude that the size of a dopant does have an impact on its barrier height. Therefore, it is crucial to have a review regarding the atomic size of dopants in our systems. There are different ways of defining the atomic radii in various situations since the atoms do not have a definite size, rather, when being far away from the nucleus, the electron density only approaches, but not equal to zero. Thus, the review on estimation of atomic sizes under different circumstances is essential. First of all, we should have a look on different types of atomic radii for transition metals obtained from previous researches, which are the theoretical radii by Clementi et. al. [1.28~1.29], the empirical radii by Slater again [1.27], the metallic radii by Pauling [1.30], the recent summary of covalent radii by Cordero et. al. [1.31], as well as the ionic radii by Shannon and Prewitt [1.32~1.33], all listed in Table I.3.5.  As seen in Figure I.3.8, except for the trend for theoretical radii, which is decreasing all the way from left to right of a period, all other radii has a trend that roughly fits that of endo and exo barrier heights illustrated in Figure I.3.4. When within the same period, the systems with dopants from the beginning of transition metal series have higher endo and exo barrier, then the barrier heights of both will decrease moving from left to right of the period. Since the heights increase again once reach the last few elements, the systems with element from the middle of a               Figure I.3.8: Plot of Various Types of Atomic Radii. The theoretical radii for La (index 31) is 6.22 Å, which is not shown on this figure. For detailed assignment of indexes, please refer to Table I.3.3.  28   Atomic Centers Radii (Å) Element Atomic # Index Theoretical Empirical Metallic Covalent(sig) Ionic Sc 21 1 1.84 1.60 1.62 1.70 0.885 Ti 22 2 1.76 1.40 1.47 1.60 0.810 V 23 3 1.71 1.35 1.34 1.53 0.780 Cr 24 4 1.66 1.40 1.28 1.39 0.755 Mn 25 5 1.61 1.40 1.27 1.39 0.720 Fe 26 6 1.56 1.40 1.26 1.32 0.690 Co 27 7 1.52 1.35 1.25 1.26 0.685 Ni 28 8 1.49 1.35 1.24 1.24 0.700 Cu 29 9 1.45 1.35 1.28 1.32 0.680 Zn 30 10 1.42 1.35 1.34 1.22 0.880 Y 39 16 2.12 1.80 1.80 1.90 1.040 Zr 40 17 2.06 1.55 1.60 1.75 0.860 Nb 41 18 1.98 1.45 1.46 1.64 0.860 Mo 42 19 1.90 1.45 1.39 1.54 0.830 Tc 43 20 1.83 1.35 1.36 1.47 0.785 Ru 44 21 1.78 1.30 1.34 1.46 0.820 Rh 45 22 1.73 1.35 1.34 1.42 0.805 Pd 46 23 1.69 1.40 1.37 1.39 0.900 Ag 47 24 1.65 1.60 1.44 1.45 0.890 Cd 48 25 1.61 1.55 1.51 1.44 1.090 La 57 31 6.22 1.95 1.87 2.07 1.117 Hf 72 32 2.08 1.55 1.59 1.75 0.850 Ta 73 33 2.00 1.45 1.46 1.70 0.860 W 74 34 1.93 1.35 1.39 1.62 0.800 Re 75 35 1.88 1.35 1.37 1.51 0.770 Os 76 36 1.85 1.30 1.35 1.44 0.770 Ir 77 37 1.80 1.35 1.36 1.41 0.820 Pt 78 38 1.77 1.35 1.39 1.36 0.940 Au 79 39 1.74 1.35 1.44 1.36 0.990 Hg 80 40 1.71 1.50 1.51 1.32 1.160       Table I.3.5: Various Atomic Radii for Elements of Interest. 29 row usually have the lowest barrier heights. This is very similar in the trends for atomic radii, in which the sizes of atoms gradually decrease moving from left to right, yet after passing the minimum located at the middle elements, the sizes start bouncing back.   It is also worth a while to discuss why the theoretical radii produce such a trend. The way that theoretical radii was obtained is to calculate the location where the maximum charge density occurs in the outermost shell of the atom, which means this method considers the size of atom when it is alone and the shapes of its orbitals were not modified by bonding with other atoms. Therefore, the theoretical radii for tested metals produce a periodic trend as expected. In the case of La, its theoretical radius is not show in Figure I.3.8, since it is too large in value and not suitable for the clear displaying on the trends of data. The reason for this extremely large value is due to the diffusiveness of the 4f orbital, which happens to be the outer most shell for La, and states by Clementi et al., tremendous trouble was involved when obtaining the results related to 4f orbitals. Thus, for La, its theoretical radius would not be accurate. Another way to estimate the size of atoms via a somewhat theoretical perspective is by Bader’s atoms in molecules theory. In Bader’s theory [1.34], the shape of an atom would be the space enclosed within a zero flux surface of the electron density around the nucleus, which may provide an insight to the atomic size. However, in most of molecules, except for noble gases, atoms do not exist on their own, rather, they form bonding interactions with each other and do not necessarily have spherical shapes which is not easy to deduce a radius from. Therefore, the radii obtained from pure calculations, such as the theoretical radii, as well as the size deduced from the Bader’s atoms in molecules, would not be able to produce accurate results that can match with experimental measurements. On the contrary, the radii predicted from methods in which experimental data were involved can produce trends that are more consistent to the barrier height results. The empirical radii was developed from Bragg’s idea of treating the atoms in crystals as hard spheres, and measuring atomic distances across several bonds as the sums of radii [1.35]. All these measured distances were from the early X-ray crystallization results. Slater’s role in empirical radii was to apply Bragg’s idea by careful comparison of bond distances in over thousands of bond types in ionic, metallic, and covalent crystals and molecules, so that a relatively complete data of empirical radii across the periodic table was made available. The drawback of above method is 30 that the atoms are not behaving as hard spheres all the time, and the averaging nature of this method does not provide predictions under special circumstances where exceptions occur.   A more recent and elaborate set of data developed from the empirically measured radii was published by Cordero, et al [1.31]. In their paper, they defined their covalent radii as the bond distance from the atom of interest to N, C and O, whose radii were determined first. The bond lengths were obtained from the Cambridge Structural Database, in which more than thousands of experimentally measured X-ray crystallization data entries were collected. For elements in which their data were not readily available, they did interpolation or extrapolations from neighboring elements or comparing with the general trends. Moreover, they searched for data of specific elements that represent its typical oxidation state and coordination numbers. Owing to the statistical natural of this collected covalent data, there is a trade off in specificity for generality, as they admitted themselves. For this reason, the atomic radii for tested dopants obtained from this relatively new data set of covalent radii yielded a trend that agrees with our barrier height to some extent, but not perfectly well.   Now we have finished the discussion of covalent radii, for the completeness of discussion, it is essential to touch the ionic radii a bit. These ionic radii were derived from experimental interatomic distances and linear relationship between ionic volume and unit cell volume. Because the data used were exclusively from compounds whose bonds between atomic centers were highly ionic, such as metal halide, metal oxide or other chalcogenides salts, the ionic radii will be a great predictor when studying ionic compounds, yet not very suitable in our systems, in which the metal carbon bonds have more covalent content.  The metallic radii proposed originally by Pauling is the most suitable one for our systems. For the purpose of comparison, a plot in which the average metal carbon bond distances for endo, exo, transition states and the metallic radii was made, followed by a linear fitting involving only the bond lengths from transition state structures and the metallic radii. From Figure I.3.9, the trends for metallic radii roughly match with all three sets of average bond distances, while the fitting shows that the metallic radii match especially well for the bond lengths of transition states, as seen on the high linearity of the plot for average bond lengths of transition state against the metallic radii. Hence, we can conclude that the average metal carbon bond distance of each system is highly correlated to their respective dopants’ metallic radii.   31 To fully appreciate this intriguing observation, one will have to take a careful look into the ideas behind metallic radii. From his paper back to early decades of the twentieth century,  Pauling suggested that the bonds between different metal atoms within metal crystals are more similar to the covalent bonds due to the equal sharing of electrons in between. These covalent bonds can somehow resonate themselves across different interatomic positions, which contributes to the conductivity of electricity and heat in metals. In this sense, the modern day textbook definition of metallic radii by Shriver and Atkins [1.25], is given as, half of the distance between the two adjacent metal nuclei in the metallic lattice as the radii of metal atoms. Such radii will be a dependent of the chemical environment of the atoms of interest, for example, the coordination numbers. Therefore, the data published for metallic radii were already corrected as if they all have 12 coordination numbers and forming single bonds.                    Figure I.3.9: Depiction of Relationship Between Metal Carbon Bond Distances and Metallic Radii. (a) Plot of Average Metal Carbon Bond Distances of all Systems Along with the Metallic Radii of Each Dopant. (b) Plot of Average Metal Carbon Bond Distances for transition states against the Metallic Radii of Each Dopant. For detailed assignment of indexes, please refer to Table I.3.3.  1.01.52.02.53.03.54.00 5 10 15 20 25 30 35 40AverageBond Distance (Å)IndexExoEndoTSMetallic Radiiy = 0.6292x + 1.0169R² = 0.82911.501.752.002.252.501.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9Average Bond Distances (Å)Metallic Radii (Å)(a) (b) 32 Based on the ideas above, the metal atoms behave like hard spheres when they were packed in the simple, homo-atomic substance form. In our systems, all tested metal dopants have very similar chemical environments, in which the metal atoms formed single bonds to three adjacent carbons in almost every system, as long as there is good bonding interaction between them, so the difference would only come from the nature of dopant atoms. That is, in our systems, the trends of metal carbon bond distances are more resembling to the trends of the dopant’s size in their metallic form. In this way, the metallic radii are highly predictive to the trends in bond distances in our systems, particularly for transition states, in which the metal dopants were at the point of being squeezed into the wall of nanotubes.   As a summary of the atomic radii review, we can conclude that the metallic radii, which can be considered as some sort of covalent radii between metal atoms of the same kind, is the most suitable one when studying metal dopants on carbon nanotubes. What follows next would be the general covalent radii, which is a statistical average of metals bonding with C, N, and O. The ionic radii will not be good for describing our systems since they concern about chemical species with highly ionic bonds. The size of atoms deduced from theoretical methods are generally not good because the atomic radii are greatly system dependent.  From the trends of barrier heights and energy gaps we have discussed before, we can gain some insights on the applications which involves the doping of transition metals on CNTs. For applications that require frequent transitions between the exo and endo doping modes, those transition metals in the middle of the period or earlier rows should be used. On the other hand, for applications that require rigid dopant conformation either inside or outside of the SWCNT, the transitions between the exo and endo doping modes can be suppressed by employing the transition metals from the two ends of a period, or from latter rows. Specifically, Group-12 metals from the Zn family are very difficult to converge computationally, mainly because of their filled (n-1) d orbitals, in which n stands for their period (row) number. Another issue associated with applying the model is related to the ionization energy of this model and its stability upon the process of ionizing. When doing the exo-endo transition, an external energy will be supplied to the system for it to overcome corresponding barrier heights. Figure I.3.10: Representation of Metallic Radii (p23, Atkins) [1.24]. 33 However, if the barrier height is too high, higher than the ionization energy of the system, we might not get the results we want after supplying the required energy, since ionization will take place, yet the stability and behaviour of the ionized model are still unknown to us. Due to time constraints of this project, the ionized structures and energetics of our model were not being invested, but will be done in the near future.   In terms of the Natural Charges obtained from NBO analysis, as listed in Table I.3.6, the dopants from the two ends of the periodic table have much higher charges than the middle ones, regardless their doping modes, as seen in Figure I.3.11 (a). This is interesting because the transition metals from the two ends are less likely to form chemical bonding with other atoms, yet they have higher partial charges. One can see that for endo modes of Cd and Hg, they are essentially neutral, meaning they do not have bonding interactions with adjacent carbon atoms. This is an exception and it is consistent with the geometry we obtained for their endo modes, in which the metal dopant stayed relatively far away from the doping site, and not likely to have chemical interactions with the nearby carbon, as discussed before.  The trend observed for the natural charges can also be related to the electronegativity of elements, and a table containing electronegativities of all tested elements along with carbon was prepared as Table I.3.7. For the dopants from the middle of the periodic table, their electronegativities are similar to that of carbon, which makes them neutral in terms of natural charges. Below the plot of natural charges, the difference in electronegativity of each dopant with respect to carbon in Pauling scale [1.35] was plotted in Figure I.3.11 (b) for better visualization of the trends. Obviously, the natural charges of all metal centers demonstrated some sort of relationship to their difference in electronegativity (in Pauling scale) comparing to carbon, especially for metals from 5th and 6th row. For the completeness of discussion, a series of linear fittings were done to check how much they are related to each other.  As the Figure I.3.12 illustrated, for 5th row and 6th row metal doped systems, their differences in electronegativity (in Pauling scale) relative to carbon are highly correlated, with R-square values of about 0.87 and 0.7 for row 5 and 6 systems respectively. On the other hand, such linearity was not observed for the 4th row systems, as the R-square values of three states ranging from around 0.2 to 0.5. These trends are intriguing since normally the larger the difference in electronegativity relative to carbon would result in a higher value of NPA charges in the metals between metal carbon bonds. The discrepancies for the 4th row doped systems  34  Row Element Index Exo Endo T.S. 4th Row 21Sc 1 1.412 1.107 1.336 22Ti 2 1.022 0.619 1.046 23V 3 0.671 0.296 0.518 24Cr 4 0.579 0.309 0.446 25Mn 5 0.445 0.383 0.537 26Fe 6 0.262 0.105 0.211 27Co 7 0.359 0.222 0.273 28Ni 8 0.519 0.409 0.374 29Cu 9 0.726 0.825 0.666 30Zn 10 1.178 0.572 1.205 5th Row 39Y 16 1.721 1.715 1.751 40Zr 17 1.502 1.313 1.629 41Nb 18 1.067 0.914 0.975 42Mo 19 0.754 0.623 0.575 43Tc 20 0.527 0.442 0.282 44Ru 21 0.327 0.269 0.232 45Rh 22 0.403 0.312 0.200 46Pd 23 0.584 0.561 0.360 47Ag 24 0.734 0.594 0.729 48Cd 25 1.162 0.010 1.326 6th Row 57La 31 1.720 1.891 1.752 72Hf 32 1.656 1.502 1.893 73Ta 33 1.284 1.189 1.175 74W 34 0.939 0.861 0.829 75Re 35 0.673 0.541 0.411 76Os 36 0.521 0.470 0.330 77Ir 37 0.515 0.397 0.370 78Pt 38 0.670 0.484 0.482 79Au 39 0.743 0.004 0.792 80Hg 40 0.650 -0.092 1.294     Table I.3.6: Natural Charges for Metal Centers in Doped Systems. 35      Row Element Index Value Difference to Carbon (2.55) 4th Row 21Sc 1 1.36 1.19 22Ti 2 1.54 1.01 23V 3 1.63 0.92 24Cr 4 1.66 0.89 25Mn 5 1.55 1.00 26Fe 6 1.83 0.72 27Co 7 1.88 0.67 28Ni 8 1.91 0.64 29Cu 9 1.90 0.65 30Zn 10 1.65 0.90 5th Row 39Y 16 1.22 1.33 40Zr 17 1.33 1.22 41Nb 18 1.60 0.95 42Mo 19 2.16 0.39 43Tc 20 1.90 0.65 44Ru 21 2.20 0.35 45Rh 22 2.28 0.27 46Pd 23 2.20 0.35 47Ag 24 1.93 0.62 48Cd 25 1.69 0.86 6th Row 57La 31 1.10 1.45 72Hf 32 1.30 1.25 73Ta 33 1.50 1.05 74W 34 2.36 0.19 75Re 35 1.90 0.65 76Os 36 2.20 0.35 77Ir 37 2.20 0.35 78Pt 38 2.28 0.27 79Au 39 2.54 0.01 80Hg 40 2.00 0.55 Table I.3.7: Electronegativity Values of Each Dopant and Comparison with Carbon [1.36]. The electronegativity values in this table are all in Pauling scale and the differences are in absolute values.  36 0.000.250.500.751.001.251.500 5 10 15 20 25 30 35 40Pauling ScaleIndex                                -0.250.000.250.500.751.001.251.501.752.000 5 10 15 20 25 30 35 40NPA ChargesIndexExoEndoT.S.Figure I.3.11: Illustration of Natural Charges and Electronegativities of All Dopants. (a) Plot Natural Charges for all Metal Centers in Three Doping Systems, and (b) Plot of Differences in Electronegativities of each Dopant Metal With Respect to Carbon in Pauling Scale.  (a) (b) 37 y = 1.3824x - 0.4702R² = 0.464y = 0.8079x - 0.2093R² = 0.2418y = 1.5352x - 0.6576R² = 0.50770.000.501.001.500.60 0.70 0.80 0.90 1.00 1.10 1.20NPA ChargesDifference in Electronegativity                                 y = 1.1752x + 0.0565R² = 0.8942y = 1.1348x - 0.0238R² = 0.869y = 1.4502x - 0.2079R² = 0.87490.000.501.001.502.000.20 0.40 0.60 0.80 1.00 1.20 1.40NPA ChargesDifference in Electronegativityy = 0.8101x + 0.4413R² = 0.7398y = 1.1097x + 0.0454R² = 0.7091y = 0.9287x + 0.3645R² = 0.6103-0.500.000.501.001.502.000.00 0.20 0.40 0.60 0.80 1.00 1.20 1.40 1.60NPA ChargesDifference in Electronegativity(b) (c) Figure I.3.12 Plot of Natural Charges of Metal Centers Against Respective Differences in Electronegativity Relative to Carbon. The electronegativity values are in Pauling scale, the blue filled solid circles are exo modes, while yellow filled diamonds are endo modes, and green tilted crosses are transition states. The lines of best fit are: black dotted lines for exo systems, red dotted lines for endo systems, and green dotted lines for transition states. The transition metal dopants are from (a) 4th row, (b) 5th row, (c) 6th row. (a) 38 suggest that either the NPA charge analysis or the electronegativity in Pauling scale, or both, need improvement. For comparison of all natural charges data at the same time, a 3D plot similar to that of energetic data as in Figure I.3.1 was presented, in which the elements were again sorted and bundle into groups that were listed as the X-axis, while the states (exo, TS, and endo) were listed as the Y-axis, yet this time the natural charges being the Z-axis, presented as colour scale as well. Another intriguing observation comes from the vibrational analysis, which is, upon examining the vibrational movies of the transition state in a very long tube model, for the moving of transition metal dopant across the CNT walls, the adjacent carbon atoms to the dopant will have to delocalize themselves so that there will be enough space for the dopant to go through. This delocalization will cause some distortions near the doping site, in which about three layers of carbon atoms around the site would be affected. The carbon atoms connected directed to the dopant will delocalize the most, and they are the most activated carbon atoms.                   Figure I.3.13: 3D Surface Plot for Natural Charges of all Dopants. A colour scale was used to demonstrate the magnitude of natural charges for the clarity of the plot, in which the red color indicated higher values in charge, while blue color indicated lower values. Also note that the lines are interpolations to show that the exo, TS, and endo data are from the same element.    39 Therefore, we suggest that when designing CNT models with transition states of a dopant to go across the wall, there should be at least three layers of unrestricted carbon atoms surrounding the dopant to ensure the minimum interference from the edges, and this is the central idea of how we came up with our EAR model.  The NBO charge analysis, as well as the barrier height against bond distance relationship, certainly involved the studies of the natural of chemical bonds. Therefore, in the second chapter of this thesis, we have included our attempts of developing a new method which locates the bond with the smallest bond dissociation energy within a given molecule. In that portion, a bond strength indicator will be introduced, which we hope for a part of the future work, beyond its original purpose of finding weakest bonds, it can also be used to compare the carbon metal bond strength between good dopant (transition metals from the middle parts of a period) systems and bad dopant (such as Cu and Zn family) systems. This will enable us to develop numerical scheme to describe how strong the bonding interactions are between carbon atoms and dopants for these systems.  I.4 Conclusion In this project, we have successfully developed and tested EAR model for computations of straight CNTs with transition metal dopings. At the same time, transition states and barrier heights were located and calculated for EAR models doped with the first three rows of transition metals, including Lanthanum. The transition for dopants to go across the wall will affect about three layers of carbon atoms and displaces the three adjacent carbon atoms so that the dopant and these three atoms are almost co-planar, with a dihedral angle of less than 20o. The N-termination models, both straight cut and EAR type, works well in practice. In addition, the EAR model can significantly reduce the computational cost. The barrier height for transition between the exo- and endo-doping modes is higher at the two ends of a period or descending down a group.   In the near future, more elements and various sizes of nanotube models will be investigated and compared against existing results. Moreover, the structural and energetics of the ionized models will be further studies using ideas from the bond strength indicator chapter. Additionally, we would like to benchmark against the one of the newest functionals developed by the Truhlar group, which is the MN12-SX functional [1.37], along with other well know ones 40 such as B3lyp [1.38~1.40]. The purpose of this is to check whether the energy gaps and barrier heights will be stable under different functionals.            41  Project II. Bond Strength Indicator  II.1. Introduction In the domain of Physical Sciences, the study of Chemistry in particular concerns about the transformation between matters, or what we refer to as chemical transformations. During a chemical transformation, there are old bonds being broken and new bonds being form, in which the weakest bonds within the reactants play crucial roles in initiating the reaction. To quantitatively assess and evaluate the weak bonds, bond dissociation energy (BDE) is of importance because it is the measurement of strength within a chemical bond. By definition [2.1], it is the standard (under 1 bar of pressure) enthalpy change when a bond is cleaved by hemolysis at 0 K. For example, the BDE between two atoms within a generic molecule A – B can be represented as:                                                        BDE = (EA + EB) – EAB     To compute the BDE accurately, we need to perform vibrational analysis to obtain the zero-point corrections and to estimate the basis set superposition errors (BSSE), which emerges from inconsistency of the basis set used before and after the reaction. However, the drawback is quite obvious, the process itself is very time consuming and tedious, as all bonds need to be investigated individually and thoroughly.  Over the course of the development of Sciences, there are established short cuts for overcoming the above obstacle, which are the indicators to help identifying the weak bonds. Among them, there are bond length (R) [2.2], bond order (within MO theory) [2.2], Mulliken interatomic electron number (MIEN) [2.3], Wiberg bond order (WBO) [2.4~2.5], and the electron density ρ(Rc) at Bader’s bond critical point within Atoms in Molecule (AIM) theory [2.6]. All of these are quick and easy to obtain, yet they ae not sensitive enough to differentiate bonds within a molecule, especially between same kinds of bonds, or with similar bond lengths and bond order. ⋅+⋅ →− BABA       breaking  bond42   Firstly, among the all indicators mentioned above, the bond length (R) is generally a very good indicator for the bond strength [2.2], since it can be measured experimentally using x-ray crystallography, and usually inversely related to BDE. However, exceptions do exist, for example, the bond between carbon and fluoride, as shown in Table II.1.1, listed along with carbon hydrogen bond and other carbon halide bonds for comparison. Even though C−F bond is longer than C−H bond, it has higher BDE and thus stronger bonds, yet if comparing carbon halide bonds exclusively, they all follow the general relationship between bond lengths and BDE nicely. This example also tells us, the bond length indicator may not be valid when comparing different types of bonds.    Bond Bond Length (pm) Bond Dissociation Energy (kJ/mol) C−H 109 414 C−F 142 464 C−Cl 178 355 C−Br 193 309 C−I 214 228     Speaking of the bond order, which is defined by the MO theory, it can only tell the difference between different bond orders, yet it cannot do comparisons within the same bond order. Hence, it will not be so useful if the objective is to find out the weakest bonds among the bonds with the same bond order. The next indicator, MIEN, stands for Mulliken Interatomic Electron Number, which can be obtained via Mulliken Population Analysis. According to Mulliken [2.3], after SCF is done, the i-th MO can be expressed as:                                                              ,Atom AO,MO∑∑∈=A AAAiiCψµµµφTable II.1.1: Bond Length and Bond Strength of Selected Common Bonds [2.7]. 43  in which the electron density for the i-th MO is:                                                The number of electrons occupying each MO is:                       Therefore, the total number of electrons in the entire molecule is given by      The density matrix element is:                                   The overlap matrix element is:                               For the total number of electrons in the entire molecule, it can be partitioned as:       The first term represents the number of electrons locating at the atomic centers, while the second term represents the interatomic electron numbers (MIEN) between atomic pairs, which can be used as one of the bond strength indicators.  ∑ ∑∑∑∑∑∑∑∈ ∈∈∈===Atom Atom AO AO*,*,Atom AO,Atom AO*,2MO     )(A B A BBABiAiiB BBBiA AAAiiiiiCCfCCfψfµ ννµνµνννµµµφφφφρ∑ ∑ ∑ ∑ ∫∫∈ ∈==Atom  Atom AO AO*,*,         A B A BBABiAiiiidCCfdnµ ννµνµτφφτρ∑ ∑ ∑ ∑∑ ∑ ∑ ∑ ∑ ∫∑∫∑∈ ∈∈ ∈====Atom  Atom AO AOAtom  Atom AO AO MO*,*,MOMO             A B A BABABA B A B iBABiAiiiiiiSDdCCfdτρnNµ νµνµνµ ννµνµτφφ∑=MO,*,iBiAiiAB CCfDνµµντφφνµµνdSBAAB∫=*444 3444 21444 344 21MIENA AB AABBABA A AAAAAA B A BABABMOiiMOiiSDSDSDnN∑ ∑∑∑∑∑∑∑ ∑ ∑∑∑∫∑≠ ∈ ∈∈ ∈∈ ∈+====Atom Atom AO AOcentered   atomicAtom AO AOAtom  Atom AO AO2  µµννµνµ νµνµνµ νµνµνφ44  Done in a similar way to MIEN, Wiberg Bond Order (WBO) was introduced by Wiberg and Mayer [2.4~2.5] to describe the bond order between a pair of atoms A and B. Start with the Mulliken Population Analysis, after obtaining the density elements, WBO can be expressed as:     In his own paper back in 1981 [2.6], Bader proposed several parameters to indicate the relative bond strength using the idea associated with bond critical points (BCPs). These bond critical points were located at the interatomic surface between a pair of atoms A and B, at which the electron density reaches minimum in one dimension, yet reaches maximum in the other two. For a generic homo-diatomic chemical species, its electron density would appear as what the Figure II.1.1 demonstrates, in which the two atomic centers are located at the bases of the tall “density chimney”. The chimney shape is caused by the fact that electron density at the atomic center would approach to infinitely high. The contour map on the right side of the same figure further illustrates this. The BCP of this chemical species would be at the middle between the two nuclei. Take H2 as an example, as shown in Figure II.1.2, the BCP of H2 is located right at the middle point between the two H atoms, as expected for homo-diatomic species. As Bader originally suggested, the charge density at this bond critical point, ρ(Rc), can be used to indicate the Bader bond order and thus indicate the strength of the bonds.             BAA BABABDSDSνµµ νµν)()(    WBOAO AO⋅= ∑ ∑∈ ∈Figure II.1.1: The Illustration of Electron Density of A Homo-diatomic Chemical Entity. It also demonstrated as a contour map on the right [2.8]. 45                  On the other hand, the Bader bond energy is an integral of the charge densities over the associated interatomic surface between A–B atomic pair; however, there is a dimensionless proportionality pre-factor that is not universal, i.e., system dependent. Moreover, Bader’s method can only be used to compare the same kind of bonds between the same pair of atoms within very similar local chemical environment and might not be able to indicate the strengths of different kinds of bonds even within the same molecule.  In simple words, none of the existing indicators can work well on its own, as will be further illustrated in the discussion section of this chapter, and our purpose in this project is to recombine some of the existing indicators to produce a more reliable one. We utilized the following parameters that have mentioned previously, to develop our own Bond Strength Indicator (BSI) K, which are the Bond length (R), the Mulliken interatomic electron number (MIEN), and the Wiberg bond order (WBO). In addition, M = MIEN/R, W = WBO/R, and K = W · M. Although W or M by itself separately is not a very good and consistent indicator, their product, K, is a well-balanced and reliable one. Over the course of developing this indicator, we also found interesting behaviours for MIEN, which leads us to do a survey on this quantity using different basis sets.  Figure II.1.2: Profile of the Electron Density Distribution of H2. This distribution reflects electron density along the Interatomic Axis The interatomic separation is 1.4 Å. Charge density in atomic units [2.8]. 46  We carried out calculations on nine selected compounds that serve the potentials of being high-energy density explosives as shown in Figure II.1.3. The reason for choosing these explosives being that they are very unstable and will easily undergo hemolytic radical bond cleavage reactions. Such cleavage is called the pyrolysis, which is crucial in the initiation of explosion process. Hence, the weakest bond within a molecule of interest would be the trigger bond, and its identification is certainly essential in the study of the safety, reliability, and explosion mechanism of the explosives. The selection of these compound contained various functional groups and geometric shapes, so that we can exam different types of single bonds under diverse chemical environment.  47     TNT 2,4,6-trinitrotoluene TNM 1,3,5-trimethyl-2,4,6-trinirobenzene PNT Pentanitrotoluene    TNCr 1-methyl-3-hydroxy-6-amino-2,4,6-trinitrobenzene CDNAPY 2-chloro-3,5-dinitro-6-aminopyridine PAM 1-methoxy-2,4,6-trinitrobenzene    AMNA N-amino-methyl-nitramine NMDACB 1-nitro-3-methyl-1,3-diazacyclobutane AMNFMC 1-azide-methyl-N-nitro-N-fluoro-methyl-carbamate   Figure II.1.3: Structures and Names of Selected Molecules. 48  II.2. Computational details In this project, all computational works were done within Gaussian 09 program package [2.9], with the DFT-B3lyp method [2.10~2.11], while the electronic wave functions of the nine selected molecules were expanded using 6-31G* [2.12~2.13]. The survey of the signs of MIEN for TNT was done using a wide range of basis sets, including the Pope’s [2.12], Dunning’s [2.14] and Steven’s [2.15] basis sets. The selected molecules were first optimized to obtain their ground state geometries, tested with vibrational analysis, then the MIEN and WBO were obtained from Mulliken Population Analysis [2.3] and Natural Bond Orbital [2.16] calculations. On the other hand, the Bader Analysis was done using AIMAll (Version 14.04.17) package [2.6].  II.3. Results and Discussion  The existing BSIs, such as WBO, MIEN, R and ρ(Rc), as well as BDE were listed along with our proposed indicators W, M and K, for tested molecules in Table II.3.1, Table II.3.2, and Table II.3.3. First of all, let us exam the bond length (R) as the bond strength indicators within TNT and its derivatives, in which there is a six-membered aromatic ring at the center, connected with a few nitro groups, as well as some other various functional groups. These include TNT, TNM, PNT, TNCr, CDNAPY, and PAM. In these systems, the carbon nitrogen bonds between the aromatic ring and nitro groups are the weakest bonds, yet they might not be the longest bonds. For example, in TNT, the C1−N14 and C5−N8 bonds both having the smallest BDE, 237.75 kJ/mol, and bond lengths of 1.481 Å, satisfy the general trend between bond length and bond strength relation comparing to the third C N bond, C3−N11, which has BDE of 265.01 kJ/mol and bond length of 1.4749 Å. However, when comparing other types of single bonds within this molecule, discrepancies would occur.  The C6−C7 bond of TNT, being the longest bond of this molecule, have a bond length of 1.5089 Å, while its BDE value is 403.65 kJ/mol, almost twice of that for C1−N14 and C5−N8. An ideal fit for bond length and strength relationship would be the case of PAM, in which the longest bonds being C2−N7 and C4−N9, have a value of 1.4769 Å , and both having the lowest BDE, 217.54 kJ/mol. We admitted that our K indicator may not be able to identify the weakest bonds at all times, yet, if we round it up to two decimal places to get the K’, it can always isolate the weaker bonds. From these weaker bonds, we can either perform accurate BDE calculations or just use Bader analysis to find out the truly weakest bond.   49   TNT WBO MIEN R (Å) BDE (kJ/mol) W M K K’ ρ(Rc) C1–N14 0.9138 0.1504 1.4810 237.75 0.6170 0.1015 0.0627 0.06 0.2621 C3–N11 0.9214 0.1468 1.4749 265.01 0.6247 0.0995 0.0622 0.06 0.2653 C5–N8 0.9138 0.1504 1.4810 237.75 0.6170 0.1015 0.0627 0.06 0.2621 C6–C7 1.0375 0.3819 1.5089 403.65 0.6876 0.2531 0.1740 0.17 0.2502 C7–H19 0.9062 0.3559 1.0895 360.77 0.8318 0.3267 0.2717 0.27 0.2758 C7–H20 0.9062 0.3559 1.0895 360.77 0.8318 0.3267 0.2717 0.27 0.2758 C7–H21 0.8851 0.3611 1.0944 361.26 0.8087 0.3299 0.2668 0.27 0.2699           TNM WBO MIEN R (Å) BDE (kJ/mol) W M K K’ ρ(Rc) C1–N16 0.8880 0.1532 1.4807 232.21 0.5997 0.1035 0.0621 0.06 0.2614 C3–N12 0.8984 0.1458 1.4778 233.19 0.6079 0.0987 0.0600 0.06 0.2625 C5–N8 0.9009 0.1447 1.4772 232.24 0.6099 0.0980 0.0598 0.06 0.2629 C2–C15 1.0278 0.3611 1.5106 404.02 0.6804 0.2390 0.1626 0.16 0.2498 C4–C11 1.0286 0.3762 1.5113 403.3 0.6806 0.2489 0.1694 0.17 0.2495 C6–C7 1.0282 0.3709 1.5108 403.41 0.6806 0.2455 0.1671 0.17 0.2499 C7–H19 0.8910 0.3614 1.0945 365.27 0.8140 0.3302 0.2688 0.27 0.2705 C15–H26 0.9085 0.3518 1.0893 366.21 0.8340 0.3230 0.2694 0.27 0.2764           PNT WBO MIEN R (Å) BDE (kJ/mol) W M K K’ ρ(Rc) C2–N19 0.8881 0.1484 1.4828 208.11 0.5989 0.1000 0.0599 0.06 0.2659 C3–N16 0.8904 0.1284 1.4837 200.62 0.6001 0.0865 0.0519 0.05 0.2671 C4–N13 0.8890 0.1313 1.4839 197.65 0.5991 0.0885 0.0530 0.05 0.2671 C5–N10 0.8912 0.1263 1.4838 200.63 0.6006 0.0851 0.0511 0.05 0.2669 C6–N7 0.8856 0.1533 1.4847 208.4 0.5965 0.1033 0.0616 0.06 0.2650 C1–C22 1.0284 0.3673 1.5097 419.52 0.6812 0.2433 0.1657 0.17 0.2493 C22–H23 0.8878 0.3600 1.0946 365.39 0.8111 0.3289 0.2668 0.27 0.2704 C22–H24 0.9039 0.3674 1.0935 365.21 0.8266 0.3360 0.2778 0.28 0.2714 C22–H25 0.9033 0.3526 1.0880 365.00 0.8303 0.3240 0.2690 0.27 0.2778      Table II.3.1: Bond Strength Indicator Data for Selected Bonds of TNT, TNM, and PNT. Here, W = WBO/R, M = MIEN/R, K=M×W, and K’ is K rounded into two decimal places. The entries with yellow background refers to the bond with smallest Bond Dissociation Energy (BDE), while the numbers in red for K (or K’) and ρ(Rc) indicates the occurrence of smallest value that is not in consistency with BDE.  50  TNCr WBO MIEN R (Å) BDE (kJ/mol) W M K K’ ρ(Rc) C2–N8 0.8908 0.1555 1.4793 235.36 0.6022 0.1051 0.0633 0.06 0.2623 C4–N10 1.0545 0.1707 1.4307 294.13 0.7371 0.1193 0.0880 0.09 0.2779 C6–N12 0.9805 0.1574 1.4548 250.81 0.6740 0.1082 0.0729 0.07 0.2694 C5–N11 1.3579 0.3829 1.3367 434.3 1.0159 0.2865 0.2910 0.29 0.3413 C1–C7 1.0321 0.3656 1.5111 386.34 0.6830 0.2419 0.1652 0.17 0.2497 C3–O9 1.1728 0.3734 1.3220 453.39 0.8871 0.2824 0.2505 0.25 0.3223 C7–H19 0.8892 0.3588 1.0942 367.69 0.8126 0.3279 0.2664 0.27 0.2699 C7–H20 0.9054 0.3655 1.0903 367.31 0.8304 0.3352 0.2784 0.28 0.2742 C7–H21 0.9044 0.3461 1.0894 367.12 0.8302 0.3177 0.2637 0.26 0.2770 O9–H22 0.5986 0.2117 0.9967 391.96 0.6006 0.2124 0.1276 0.13 0.3106 N11–H23 0.7385 0.2746 1.0112 424.61 0.7303 0.2716 0.1983 0.20 0.3264 N11–H24 0.7360 0.2695 1.0134 405.53 0.7263 0.2659 0.1931 0.19 0.3245           CDNAPY WBO MIEN R (Å) BDE (kJ/mol) W M K K’ ρ(Rc) C2–N8 0.9397 0.1335 1.4657 248.56 0.6411 0.0911 0.0584 0.06 0.2695 C4–N9 0.9729 0.1237 1.4520 278.67 0.6701 0.0852 0.0571 0.06 0.2730 C5–N10 1.3124 0.3491 1.3386 455.55 0.9804 0.2608 0.2557 0.26 0.3415 C1–Cl7 1.0973 0.3046 1.7380 325.22 0.6314 0.1752 0.1106 0.11 0.2057 C3–H15 0.8624 0.3366 1.0824 470.41 0.7967 0.3110 0.2478 0.25 0.2841 N10–H16 0.7516 0.2808 1.0114 435.63 0.7431 0.2777 0.2064 0.21 0.3255 N10–H17 0.7875 0.2942 1.0102 434.71 0.7796 0.2913 0.2271 0.23 0.3284           PAM WBO MIEN R (Å) BDE (kJ/mol) W M K K’ ρ(Rc) C2–N7 0.9191 0.1475 1.4769 217.54 0.6223 0.0999 0.0621 0.06 1.1958 C4–N9 0.9191 0.1475 1.4769 217.54 0.6223 0.0999 0.0621 0.06 1.1958 C6–N10 0.9241 0.1501 1.4736 262.69 0.6271 0.1019 0.0651 0.06 1.2627 C3–O8 1.0529 0.3040 1.3398 354.45 0.7859 0.2269 0.1785 0.18 1.3156 O8–C17 0.8424 0.2252 1.4534 263.82 0.5796 0.1550 0.0924 0.09 1.0756 C17–H20 0.9267 0.3626 1.0899 398.18 0.8503 0.3327 0.3030 0.28 0.9198 C17–H21 0.9267 0.3626 1.0899 398.18 0.8503 0.3327 0.2832 0.28 0.9198 C17–H22 0.9305 0.3856 1.0896 398.18 0.8540 0.3539 0.2847 0.30 0.9158    Table II.3.2: Bond Strength Indicator Data for Selected Bonds of TNCr, CDNAPY, and PAM. Here, W = WBO/R, M = MIEN/R, K=M×W, and K’ is K rounded into two decimal places. The entries with yellow background refers to the bond with smallest Bond Dissociation Energy (BDE), while the numbers in red for K (or K’) and ρ(Rc) indicates the occurrence of smallest value that is not in consistency with BDE.   51  AMNA WBO MIEN R (Å) BDE (kJ/mol) W M K K’ ρ(Rc) N2–N4 0.9807 0.1603 1.4221 107.87 0.6896 0.1127 0.0777 0.08 0.3241 N2–N3 1.0575 0.2451 1.3965 248.42 0.7572 0.1755 0.1329 0.13 0.3306 C1–N2 0.9571 0.2703 1.4632 233.11 0.6541 0.1847 0.1208 0.12 0.2575 C1–H7 0.9131 0.3737 1.0884 376.14 0.8390 0.3433 0.2880 0.29 0.2807 C1–H8 0.924 0.3729 1.0989 376.41 0.8409 0.3394 0.2854 0.29 0.2718 C1–H9 0.9141 0.3713 1.0895 376.59 0.8390 0.3408 0.2860 0.29 0.2789 N3–H10 0.8192 0.3215 1.0177 315.18 0.8050 0.3159 0.2543 0.25 0.3333 N3–H11 0.8231 0.3109 1.0237 314.42 0.8040 0.3037 0.2442 0.24 0.3274           AMNFMC WBO MIEN R (Å) BDE (kJ/mol) W M K K’ ρ(Rc) N7=N8 1.4789 0.2938 1.2439 141.55 1.1889 0.2362 0.2808 0.28 0.4313 N5–N11 0.9167 0.1381 1.4494 124.16 0.6325 0.0953 0.0603 0.06 0.3031 C1–O3 0.9845 0.2689 1.3609 326.62 0.7234 0.1976 0.1429 0.14 0.3014 C1–N5 0.9998 0.1857 1.4139 320.4 0.7071 0.1314 0.0929 0.09 0.2963 O3–C4 0.8441 0.1903 1.4432 264.75 0.5849 0.1319 0.0771 0.08 0.2393 C4–N7 0.9896 0.2696 1.4541 267.37 0.6805 0.1854 0.1262 0.13 0.2683 C4–H15 0.9075 0.3719 1.0925 358.88 0.8307 0.3404 0.2828 0.28 0.2853 N5–C6 0.9583 0.2199 1.4436 330.94 0.6638 0.1523 0.1011 0.10 0.2701 C6–F10 0.8574 0.2802 1.3782 418.79 0.6221 0.2033 0.1265 0.13 0.2465           NMDACB WBO MIEN R (Å) BDE (kJ/mol) W M K K’ ρ(Rc) C1–N2 0.9456 0.2486 1.4792 196.21 0.6393 0.1680 0.1068 0.11 0.2590 C1–N3 0.9685 0.3084 1.4786 170.88 0.6550 0.2086 0.1361 0.14 0.2704 C1–H10 0.9004 0.3552 1.0983 367.45 0.8198 0.3234 0.2679 0.27 0.2774 N2–N6 0.9756 0.1878 1.3886 157.49 0.7026 0.1352 0.1007 0.09 0.3469 N3–C5 1.0026 0.3014 1.4545 294.3 0.6893 0.2072 0.1412 0.14 0.2704 C5–H14 0.9212 0.3525 1.1061 378.11 0.8329 0.3187 0.2672 0.27 0.2666        Table II.3.3: Bond Strength Indicator Data for Selected Bonds of AMNA, AMNFMC, and NMDACB. Here, W = WBO/R, M = MIEN/R, K=M×W, and K’ is K rounded into two decimal places. The entries with yellow background refers to the bond with smallest Bond Dissociation Energy (BDE), while the numbers in red for K (or K’) and ρ(Rc) indicates the occurrence of smallest value that is not in consistency with BDE. 52   The above observed violations of the general trends can also be found in the other three remaining molecules, in which we can conclude that the general relationship between bond length and bond strength will certainly hold in the case of similar chemical environment (i.e. between same type of bonds), as briefly mentioned in the introductory section at Table II.1.1. For the completeness of discussion, we further extra average bond length and average bond energy data from the same textbook [2.7], to investigate a more general trend.  Since the general trend suggested that longer the bond, the weaker the bond, we manipulated the data after the extraction to reflect this trend. We proposed that the bond length would be inversely proportional to the bond energy per unit bond length. Then, we have: dRcRBE+= 1, in which c and d are arbitrary constants, and RBEwill be plotted as y axis, whereas R1 as x axis.  As the plot demonstrated in Figure II.3.1, which is the graphical representation of the Table II.3.4, the average bond distances and energies from common bonds, the square of the inverse of average bond length and the division of average bond energy by bond length are highly correlated, whose linear fitting has an R-square value of 0.9745. The only two outstanding points that were not included in the linear fit were N−N and H−F bonds. This proves that the assumption of BE per unit bond length and R being inversely correlated to each other is valid. Hence, in our finalized effective BSI formulation, there will be terms involving bond lengths at the denominator part.  A series of graphical representations for all indicators listed in Table II.3.1, Table II.3.2, and Table II.3.3 are created and illustrated in Figure II.3.2, from which we can compare their relative performances. In terms of other existing indicators such as WBO, MIEN and ρ(Rc), they cannot effectively indicate the weakest bond as well, just similar to the case of bond lengths R. However, we found that when dividing the WBO and MIEN by bond length R respectively, they produce indicators W and M, which have slightly improved performances. Moreover, when we multiply W and M, we obtain our effective BSI, K, that can pin point the weakest bonds within a molecule. Hence, our proposed BSI has the expression: 2RMIENWBOK⋅= .   53                   Bond Average Bond Distance (R) (pm) Average Bond Energy (BE) (kJ/mol) 1/R (pm−1) BE/R  ( kJ∙mol−1∙ pm−1) H – H 74.14 436 0.0135 5.88 F – H 91.70 565 0.0109 6.16 O – H 97.00 464 0.0103 4.78 N – H 100.00 389 0.0100 3.89 C – H 110.00 414 0.0091 3.76 Cl – H 127.40 431 0.0078 3.38 S – H 132.00 368 0.0076 2.79 Br – H 141.40 364 0.0071 2.57 C – O 143.00 360 0.0070 2.52 N – N 145.00 163 0.0069 1.12 C – N 147.00 305 0.0068 2.07 C – C 154.00 347 0.0065 2.25 I – H 160.00 297 0.0063 1.86 C – Cl 178.00 339 0.0056 1.90 Cl – Cl 199.00 243 0.0050 1.22 Br – Br 228.00 193 0.0044 0.85 I – I 266.00 151 0.0038 0.57 y = 568.64x - 1.5101R² = 0.97450.01.02.03.04.05.06.07.02.00E-03 4.00E-03 6.00E-03 8.00E-03 1.00E-02 1.20E-02 1.40E-02BE/R (kJ mol−1 pm−1)1/R (pm−1)H−F N−N dRcRBE+=21Figure II.3.1: Plot of Bond Energy per unit Bond Length Against Inverse of Average Bond Length. This can only provide a general guide: even the same type of bonds can become very different due to local chemical environments, and the cases of H−F and N−N were excluded due to their uncommon bond length and strength relationship. Table II.3.4: Average Bond Distances and Bond Energies of Common Bonds [2.7]. 54  As Figure II.3.2 illustrated, the existing bond strength indicators, even the most accurate one from Atoms in Molecule theory [2.6], the electron density at bond critical point ρ(Rc), failed to produce a trend that is similar to that of BDE. The reason being that, as previously mentioned, the ρ(Rc) is only effective when comparing between identical type of bonds within molecules with similar chemical environment. On the other hand, our proposed indicator K, exhibited a trend that closely resemble to that of BDE, so that the ranking of bonds by K would produce an order which can be used to determine the weakest bonds. Most of the time, our K indicator can narrow down the selection of weakest bonds to one.  However, under special circumstances, such as same type of bonds within very similar chemical environment, it can only yield a collection of weak bonds with almost the same K values that contain the real weakest bond. Whenever the above case occurs, we can use the electron density at bond critical point ρ(Rc) to identify the weakest bond amongst the weaker bonds picked out by K, since the conditions of identical type of bonds and similar chemical environment now satisfied for the use of ρ(Rc). Alternatively, one can use the conventional way of calculating BDE to find the weakest bonds from the weak bonds selected by K. Either way, the work load of identifying the weakest bond will be certainly reduced due to the narrow down of choices for the weakest bonds by employing K.      (a) Breaking Bonds 55        Breaking Bonds Breaking Bonds Breaking Bonds Breaking Bonds (b) 56         Figure II.3.2: Comparison of Various Bond Strength Indicators for Breaking Single Bonds. (a) TNCr, (b) TNT, TNM, PNT and CDNAPY, (c) PAM, AMNA, NMDACB, and AMNFMC. Breaking Bonds Breaking Bonds Breaking Bonds Breaking Bonds (c) 57   In the process of developing K, there is another intriguing discovery. While we were determining the MIEN, we found out that using different basis sets, this value will differ from each other, and the most intriguing thing is, depending on the basis sets used, the MIEN values would become negative. From the definition of MIEN, it is a measurement of the number of electrons (or electron density) between atomic centers, and it can be obtained by summing up individual overlapping integrals from density matrix between generic atomic pairs A−B, which is: ∑ ∑∑∑≠ ∈ ∈=Atom Atom AO AO A AB AABBABSDMIENµµννµν.  In this sense, we expect the MIEN values to be always positive. As we did find negative MIEN for some basis sets, it is worth a while to exam the values of MIEN across different basis sets that are currently available. For this evaluation, we have chosen TNT to be the molecule of interest because it is a representative molecule of its kind, and employed a large range of different basis sets to determine MIEN for selected bonds. We have summarized the results in Table II.3.5 below, in which the occurrences of negative MIEN were marked as red and filled with yellow in background.  By surveying Table II.3.5, it can be conclusive that negative MIEN occurred when there is no polarization functions being added, or when there is diffuse functions being added, in which case even if there is polarization added together, the resulting MIEN would still be negative.  For example, the 6-31G basis set produced negative MIEN for all the C−N bonds tested, yet the 6-31G* and the 6-31G** yielded positive values for these bonds. However, with diffuse functions added, such as 6-31+G and 6-31++G, the MIEN values for these bonds become negative again. If we suggest that polarization functions would increase the values of MIEN in general, and diffuse function would act in oppose to this effect. In addition, the effect of diffuse function would out weight that of polarization, as the combination of polarization and diffuse functions, such as 6-31+G* or 6-31+G**, still produce negative MIEN.  The trends for MIEN behaviour makes sense if we take a careful look on the theories behind polarization and diffuse functions. According to Levine [2.17], the addition of polarization functions would alter the shapes of the original atomic orbitals, allowing the electrons to have more rooms to account for bonding interactions, such that the electron density would be shifted away from the atomic center, towards the inter-atomic bonding region in the molecule of interest. In this way, the electron densities (or numbers of electron) in between 58         Basis Set Bond MIEN  Basis Set Bond MIEN 3-21G C1–N14 0.01583  4-31G** C1–N14 0.19651 C3–N11 −0.00125  C3–N11 0.19797 C5–N8 0.01582  C5–N8 0.19651 C6–C7 0.29206  C6–C7 0.39290 C7–H19 0.34545  C7–H19 0.37284 C7–H20 0.36248  C7–H20 0.37645 C7–H21 0.34545  C7–H21 0.37285 3-21+G C1–N14 3.43251  6-31G C1–N14 −0.06080 C3–N11 −5.60807  C3–N11 −0.11784 C5–N8 3.43164  C5–N8 −0.06081 C6–C7 67.14607  C6–C7 0.34230 C7–H19 0.34096  C7–H19 0.36003 C7–H20 0.30988  C7–H20 0.36590 C7–H21 0.34100  C7–H21 0.36004 4-31G C1–N14 −0.00897  6-31G* C1–N14 0.15031 C3–N11 −0.05270  C3–N11 0.14683 C5–N8 −0.00898  C5–N8 0.15031 C6–C7 0.34916  C6–C7 0.38238 C7–H19 0.36391  C7–H19 0.35582 C7–H20 0.36936  C7–H20 0.36114 C7–H21 0.36391  C7–H21 0.35582 4-31G* C1–N14 0.19723  6-31G**  C1–N14 0.14943 C3–N11 0.20011  C3–N11 0.14456 C5–N8 0.19723  C5–N8 0.14943 C6–C7 0.39095  C6–C7 0.38800 C7–H19 0.36118  C7–H19 0.36730 C7–H20 0.36572  C7–H20 0.37129 C7–H21 0.36118  C7–H21 0.36730 Table II.3.5: MIEN Values of Various Basis Sets for TNT (Part 1). The data entries in red numbers and yellow background indicated the occurrence of negative MIEN value. 59   Basis Set Bond MIEN  Basis Set Bond MIEN 6-31+G C1–N14 −1.72257  6-31++G* C1–N14 −0.75564 C3–N11 −2.34520  C3–N11 −1.43247 C5–N8 −1.72277  C5–N8 −0.75581 C6–C7 0.11076  C6–C7 2.26025 C7–H19 0.44751  C7–H19 0.41653 C7–H20 0.37787  C7–H20 0.36154 C7–H21 0.44752  C7–H21 0.41653 6-31++G C1–N14 −2.31570  6-31++G** C1–N14 −0.71132 C3–N11 −1.92030  C3–N11 −1.40684 C5–N8 −2.31579  C5–N8 −0.71153 C6–C7 3.95723  C6–C7 2.05747 C7–H19 0.41522  C7–H19 0.46441 C7–H20 0.43654  C7–H20 0.41724 C7–H21 0.41517  C7–H21 0.46441 6-31+G* C1–N14 −0.67140  6-311G C1–N14 −0.07191 C3–N11 −1.53432  C3–N11 −0.10414 C5–N8 −0.67160  C5–N8 −0.07192 C6–C7 0.72109  C6–C7 0.21386 C7–H19 0.42121  C7–H19 0.37873 C7–H20 0.34562  C7–H20 0.37187 C7–H21 0.42123  C7–H21 0.37873 6-31+G** C1–N14 −0.60680  6-311G* C1–N14 0.14235 C3–N11 −1.60700  C3–N11 0.17667 C5–N8 −0.60703  C5–N8 0.14235 C6–C7 0.07339  C6–C7 0.30745 C7–H19 0.45461  C7–H19 0.37813 C7–H20 0.39246  C7–H20 0.37835 C7–H21 0.45463  C7–H21 0.37813        Table II.3.5: MIEN Values of Various Basis Sets for TNT (Part 2). The data entries in red numbers and yellow background indicated the occurrence of negative MIEN value. 60   Basis Set Bond MIEN  Basis Set Bond MIEN 6-311G** C1–N14 0.14024  6-311+G** C1–N14 −1.12152 C3–N11 0.16809  C3–N11 −1.62221 C5–N8 0.14024  C5–N8 −1.12172 C6–C7 0.35725  C6–C7 −0.04397 C7–H19 0.39699  C7–H19 0.35834 C7–H20 0.39461  C7–H20 0.34943 C7–H21 0.39699  C7–H21 0.35835 6-311+G C1–N14 −2.79096  6-311++G* C1–N14 −0.95607 C3–N11 −3.35910  C3–N11 −1.85648 C5–N8 −2.79117  C5–N8 −0.95632 C6–C7 0.85960  C6–C7 1.35452 C7–H19 0.31683  C7–H19 0.24534 C7–H20 0.30877  C7–H20 0.33228 C7–H21 0.31684  C7–H21 0.24531 6-311++G C1–N14 −2.34045  6-311++G** C1–N14 −0.94279 C3–N11 −3.37558  C3–N11 −1.57422 C5–N8 −2.34066  C5–N8 −0.94303 C6–C7 3.16244  C6–C7 1.96397 C7–H19 0.23047  C7–H19 0.31886 C7–H20 0.41464  C7–H20 0.40112 C7–H21 0.23038  C7–H21 0.31882 6-311+G* C1–N14 −1.11310  CEP-4G C1–N14 0.32276 C3–N11 −1.84265  C3–N11 0.32185 C5–N8 −1.11330  C5–N8 0.32276 C6–C7 −0.29004  C6–C7 0.37153 C7–H19 0.33234  C7–H19 0.39918 C7–H20 0.29308  C7–H20 0.38872 C7–H21 0.33236  C7–H21 0.39918      Table II.3.5: MIEN Values of Various Basis Sets for TNT (Part 3). The data entries in red numbers and yellow background indicated the occurrence of negative MIEN value. 61    Basis Set Bond MIEN  Basis Set Bond MIEN CEP-4G* C1–N14 0.27812  CEP-121G* C1–N14 0.11145 C3–N11 0.27491  C3–N11 0.14590 C5–N8 0.27813  C5–N8 0.11145 C6–C7 0.35329  C6–C7 0.13799 C7–H19 0.39556  C7–H19 0.35974 C7–H20 0.38370  C7–H20 0.37436 C7–H21 0.39556  C7–H21 0.35973 CEP-31G C1–N14 −0.31203  cc-pVDZ C1–N14 0.16369 C3–N11 −0.51513  C3–N11 0.20324 C5–N8 −0.31207  C5–N8 0.16369 C6–C7 0.14070  C6–C7 0.47665 C7–H19 0.32197  C7–H19 0.36057 C7–H20 0.30719  C7–H20 0.36376 C7–H21 0.32197  C7–H21 0.36057 CEP-31G* C1–N14 0.07893  aug-cc-pVDZ C1–N14 −0.73036 C3–N11 0.00589  C3–N11 −0.94467 C5–N8 0.07891  C5–N8 −0.73048 C6–C7 0.17254  C6–C7 −0.06991 C7–H19 0.31443  C7–H19 0.03261 C7–H20 0.30517  C7–H20 0.17447 C7–H21 0.31443  C7–H21 0.03261 CEP-121G C1–N14 −0.15023  cc-pVTZ C1–N14 0.23749 C3–N11 −0.27595  C3–N11 0.33543 C5–N8 −0.15024  C5–N8 0.23750 C6–C7 0.20277  C6–C7 0.36698 C7–H19 0.36149  C7–H19 0.39958 C7–H20 0.35629  C7–H20 0.38608 C7–H21 0.36148  C7–H21 0.39958 aug-cc-pVTZ C1–N14 −0.60816  cc-pVQZ C1–N14 0.28738 C3–N11 −0.38491  C3–N11 0.38494 C5–N8 −0.60818  C5–N8 0.28738 C6–C7 −0.18733  C6–C7 0.47007 C7–H19 0.45090  C7–H19 0.38871 C7–H20 0.47185  C7–H20 0.39439 C7–H21 0.45087  C7–H21 0.38872 Table II.3.5: MIEN Values of Various Basis Sets for TNT (Part 4). The data entries in red numbers and yellow background indicated the occurrence of negative MIEN value. 62  generic bonding pairs A−B will be greater than the ones without polarization. Meanwhile, basis sets with added polarization will be able to describe the bonding characters more accurately as well. This also explains why the computing results from basis sets with no polarization or diffuse function are generally not as accurate, and in our case, produced negative MIEN.   On the other hand, the diffuse functions with very small orbital exponents, are being added to the basis sets, so that the maximum of electron density will be attained at relatively large distance to the nucleus. One consequence of this is the reduction of the calculated electron densities in the inter-nuclei space, or the smaller MIEN. The addition of diffuse functions is absolutely essential for describing anions, compounds with lone pairs, or hydrogen bonded complexes, in which there involves weakly bonded systems, whose electron densities are still significant when being away from the nucleus and thus must be accounted for. Yet in our project, none of the molecules selected fall into the conditions of using diffuse functions, hence the results from any basis sets with diffuse functions will not be accurate. Now there is no doubt why basis sets with diffuse functions produce negative MIEN for TNT molecule, and we should avoid using them for the computation of similar chemical systems. Moreover, we can utilize the sign of MIEN values as an indicator to check whether the basis sets used are suitable for the system of interest. The presence of negative MIEN values certainly indicates the poor choice of basis set, and one should definitively considering the revision of selected basis sets.     Throughout this survey on the sign of MIEN obtained using different basis sets on TNT molecule, we should emphasize the importance of good basis sets selection. For molecules similar to TNT or its kind, the use of polarization function is necessary to deliver a reliable result. When considering the computational cost and time efficiency, the use of 6-31G(*) would be adequate, and that is why we choose to employ this basis set to complete the computation of all bond strength indicators for all other compounds of interest.     II.4. Conclusion In this project, we have successfully developed a very effective bond strength indicator for the tested chemical systems. When given any molecular geometry, we hope we can use our BSI to single out a few weaker bonds, from which we can carry out more accurate BDE calculations or Bader analysis on these bonds to pick out the weakest bond among them if 63  necessary. For the future works, we would like to investigate different systems to test the reliability of our BSI and to further improve it.                              64  Closing Remarks  After tremendous efforts of computations, we have successfully developed models for quick transition state calculations of exo-endo transitions of substitutionally doped transition metal SWCNT systems, as well as effective bond strength indicators that can help identifying the weak bonds of a molecule. Since both the models and the indicators can really speed up the computational time, these efforts were worth. Specifically for Project I, we found out that using N terminated models would have faster and easier convergence than the H terminated ones. In addition, we obtained the transition state geometries and energies for exo-endo transitions of 4th, 5th, 6th (including La but excluding other lanthanides) row transition metal doped SWCNT segments, from which we deduced the respective barrier heights. From these data, we observed that dopants from the middle of a row or earlier row would have lower barrier heights, and thus are ideal for applications involving frequent exo-endo transitions of doped SWCNTs. Moreover, the geometry of the transition state demonstrates that the dopant and its three adjacent carbon atoms were almost co-planar, with a dihedral angle of less than 20o. On the other hand, for Project II, we have developed a very effective indicator, K, for bond strength. Although this indicator may not always pinpoint the weakest bond, when rounding up its value to two significant digits, it can at lease isolate the weaker bonds, from which we can further pick out the weakest bond via dedicated BDE calculations or Bader’s AIM analysis.  Speaking of the future work, structural and energetic results of transition states obtained in Project I will be tested with other functionals such as B3lyp or MN12SX. 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