SOLIDS USING THE AB INITIO ORTHOGONALIZED LINEAR COMBINATION OF
ATOMIC ORBITALS METHOD
A THESIS IN Physics
Presented to the Faculty of the University of Missouri-Kansas City in partial fulfilment of the requirements for the degree
MASTER OF SCIENCE
by JOSHUA RAY SEXTON
B. S., University of Missouri-Kansas City, 2017
Kansas City, Missouri 2020
JOSHUA RAY SEXTON
ALL RIGHTS RESERVED
ELECTRONIC STRUCTURE OF THREE CENTER TWO ELECTRON BONDS IN BORON
RICH MATERIALS: A STUDY OF MOLECULES, CRYSTALS, AND AMORPHOUS
SOLIDS USING THE AB INITIO ORTHOGONALIZED LINEAR COMBINATION OF
ATOMIC ORBITALS METHOD
Joshua Ray Sexton, Candidate for the Master of Science Degree
University of Missouri-Kansas City, 2020
It has been shown that isomers of carborane, α and β rhombohedral boron, α and β tetragonal boron, γ boron, and boron carbide tend to form icosahedral structures. If the icosahedra were formed with pure boron, it has been demonstrated that there is an insufficient number of electrons to populate all the molecular orbitals that would nominally be required for molecular cohesion. To overcome that electron deficiency, the icosahedral boron units tend to form interesting bonding structures with their neighbors in solids, and/or for molecular boron the icosahedra tends to incorporate substitutional carbon atoms.
The resultant bonding structures are often of a 3 center 2 electron nature. The visualization of these 3 center 2 electron bonds varies in the literature and this exposes a discontinuity with how multi-center bonds should be perceived. Here, I have attempted to reconcile these delocalized bonds with the common “ball-and-stick” molecular model using an extension to Mulliken Population Analysis within the Orthogonalized Linear Combination of Atomic
The Faculty listed below, appointed by the Dean of the College of Arts and Sciences, have examined a thesis titled “Electronic Structure of Three Center Two Electron Bonds in
Boron Rich Materials: A Study of Molecules, Crystals, and Amorphous Solids using the ab initio Orthogonalized Linear Combination of Atomic Orbitals Method”, presented by
Joshua Ray Sexton, candidate for the Master of Science degree, and certify that in their opinion, it is worthy of acceptance.
Paul Rulis, Ph.D., Committee Chair Department of Physics and Astronomy
Wai-Yim Ching, Ph.D. Department of Physics and Astronomy
Michelle Paquette, Ph.D. Department of Physics and Astronomy
Table of Contents
LIST OF TABLES ...... x
ACKNOWLEDGEMENTS ...... xi
1. INTRODUCTION ...... 1
Context ...... 1
Motivation ...... 6
Outline ...... 7
2. METHODS ...... 9
The OLCAO Method ...... 10
The Visualization ToolKit...... 14
Paraview ...... 16
Program ...... 17
3. RESULTS AND DISCUSSION ...... 21
Model Systems ...... 21
Carborane C2H10B12 ...... 23
α-Rhombohedral Boron ...... 24
β-Rhombohedral Boron ...... 25
γ-Boron ...... 27
α-Tetragonal Boron 51 ...... 28
β-Tetragonal Boron 190 ...... 29
Boron Carbide B11C-CBC ...... 30
Amorphous Hydrogenated Boron Carbide a-BxC:Hy ...... 31
4. CONCLUSIONS AND FUTURE WORK ...... 39
Future Work ...... 39
APPENDIX ...... 41
Created and Altered Scripts and Programs ...... 41
bond3C.F90 ...... 41
insert3cbo ...... 41
makeBOND ...... 42
Custom Paraview Filter ...... 43
REFERENCES ...... 44
VITA ...... 48
LIST OF ILLUSTRATIONS
Figure 1 Atomic structure of dodecaborane B12H12 ...... 4
Figure 2 (A)-(C) Schematic, stereographic, and broad views of α-rhombohedral boron;
(D) Bonding charge contours from  ...... 5
Figure 3 Sample VTK legacy polydata output ...... 16
Figure 4 Paraview rendering of Figure 3 ...... 17
Figure 5 Old process of bond visualization ...... 19
Figure 5 New process of visualization ...... 20
Figure 6 Traditional visualizations of ortho, para, and meta-carboranes...... 23
Figure 7 Carboranes new visualization: ortho, para, and meta ...... 24
Figure 8 Alpha boron traditional visualization ...... 24
Figure 9 Alpha boron new visualization ...... 25
Figure 10 Beta boron traditional visualization...... 25
Figure 11 Beta boron new visualization ...... 26
Figure 12 Gamma boron traditional visualization ...... 27
Figure 13 Gamma boron new visualization ...... 27
Figure 14 Alpha tetragonal boron traditional visualization ...... 28
Figure 15 Alpha tetragonal boron new visualization ...... 28
Figure 16 Beta tetragonal boron traditional visualization ...... 29
Figure 17 Beta tetragonal boron new visualization ...... 29
Figure 18 Boron Carbide traditional visualization...... 30
Figure 19 Boron Carbide new visualization ...... 30
Figure 20 a-BC:H traditional visualization ...... 31
Figure 21 a-BC:H new visualization...... 32
Figure 22 localization index of boron allotropes and a-BC:H ...... 36
Figure 23 localization index, zoomed in on the zero of energy and scaled ...... 37
Figure 24 total density of states of boron allotropes and a-BC:H ...... 38
LIST OF TABLES
Table 1- Programs and Purpose ...... 9
I would like to thank Paul Rulis, whose guidance and support helped me finish this research despite the time constraints. Fred Leibsle, who reminded me why I love physics and rekindled my passion for learning as an undergraduate. Patrick Ryan Thomas, whose intellect and work ethic were a constant inspiration to be a better scientist. My parents, who encouraged my curiosity and pushed me to be a better student and person than I thought possible. My grandmother Evelyn Sexton, who showed me that using my intellect for others would bring me more joy than squandering it. My grandfather Ernest Sexton, who kept me humble by being the example of someone whose intellect and memory will always be superior to my own. The rest of my family, without their support I doubt I could have persevered under this stress. Finally, my twin brother Jordan, whose constant positive attitude kept me from becoming too stressed, and whose overwhelming luck and grit were a constant reminder that I needed to give 120% effort in everything I do, so I never waste an opportunity to succeed.
Boron-rich materials such as macropolyhedral boranes and metallaboranes, the various allotropes of elemental boron, crystalline boron carbide, boron sub-oxide, and amorphous forms such as hydrogenated boron carbide are known for their unusual bonding patterns. Of particular interest is their tendency to form icosahedral structures in both molecular and solid-state environments. This coordination is unusual for most light elements, but because boron—a group IIIA element with three valence electrons available for bonding—is a metalloid it is often found in structures that require more delocalized bonding.
Therefore, each boron atom has two valence electrons remaining to bond to its five neighboring boron atoms. Traditionally, in a covalent bond, one electron is donated from each atom to form the bond. Such an apportionment is an inappropriate way to consider the intraicosahedral boron bonding pattern because the boron atoms cannot participate in five intraicosahedral covalent bonds each. In total the boron icosahedron has twenty-four electrons to use for intraicosahedral bonds. (That is from an initial total of thirty-six less the twelve used for bonding with the H atoms.) Using molecular orbital theory and icosahedral symmetry, Longuet-Higgins and Roberts found that thirteen molecular bonding orbitals need to be filled, requiring twenty-six electrons. Thus, a B12H12
1 molecular structure would be deficient by two electrons. In a molecular configuration, the
2- boron icosahedron in B12H12 may become negatively charged, making dianions, (B12H12) or it may contain two substitutional carbon atoms, forming B10C2H12 with para-, meta-, and ortho-carborane distributions of the carbon atoms. Either of those approaches may be used to achieve the necessary number of electrons for full population of the intraicosahedral molecular orbitals.
In an elemental or boron-rich solid-state system, such as boron allotropes and their polymorphs, the necessary twenty-six electrons are accounted for via different methods.
For the boron allotropes and crystalline boron carbide there are no H atoms with which to establish covalent bonds and so the icosahedra must bond directly with neighboring icosahedra. A convenient way to describe the electronic states of a system is to use molecular orbitals. When found in the solid state, the icosahedra acquire the two electrons needed for full occupation of the icosahedral molecular orbitals simply by not requiring every B atom to have a traditional two-center two-electron covalent bond with a B atom from a neighboring icosahedron. If we return specifically to consideration of pure B12 icosahedra as found in α-rhombohedral boron we see that the crystal contains only icosahedra positioned at the corners of a rhombohedral cell, as shown in Figures 2b, 2c, and 2d. In that case each icosahedron again requires twenty-six electrons to fill up its thirteen intraicosahedral orbitals, leaving it ten electrons for intericosahedral bonds. The polar boron atoms use six electrons for direct covalent bonds with neighboring icosahedra, leaving four electrons for the equatorial boron to use. Thus, 2/3 of an electron may be used by each equatorial boron atom to bond with neighboring icosahedra. The result is a set of three-center two-electron (3c2e) bonds between the icosahedra at the equatorial sites as
2 well as traditional two-center two-electron (2c2e) bonds at the polar sites, shown in Figure
2c. Similar, typically more complicated, arguments can be made for the other allotropes of boron.
Considering both molecular and solid-state icosahedra, the bonding patterns on the icosahedra themselves also show a mixture of covalent and delocalized metal-like bonding.
Occupation of the high-lying molecular bonding orbitals tends to shift the bonding charge from the icosahedral edges to the triangular faces, as seen in Figure 2a, which is work by
Emin . The bonding pattern is confirmed through X-ray diffraction studies  showing the location of charge accumulation and that three boron atoms share two electrons.
Other allotropes of boron have been shown to have unique 3c2e bonds as well. γ-
Boron has 3c2e bonds between its icosahedral units and its boron “dumbbell” internal structure , and the α-tetragonal, β-tetragonal, and β-rhombohedral allotropes along with boron carbide (often labeled as B4C or B11C-CBC) have been shown to have the electron deficiency indicative of these 3c2e bonds . It should be noted that while boron-rich materials were the target of this research, the method developed to study them is generalized and can be applied to any material that exhibits similar delocalized bonding, such as gallium rich compounds or metalloid rich materials.
Figure 1 Atomic structure of dodecaborane B12H12
Figure 2 (A)-(C) Schematic, stereographic, and broad views of α-rhombohedral boron;
(D) Bonding charge contours from 
Currently there is no well-established method for visualizing three-center two- electron (3c2e) bonds as one might commonly do with a traditional ball-and-stick model to visualize two-center two-electron (2c2e) bonds. Instead, most attempts to visualize 3c2e bonds are done in a somewhat roundabout manner. Figures 2a and 2d show two different kinds of 3c2e bonds both drawn in a different way. Figure 2d displays the 3c2e bond by using contour lines for the density of bonding charge, while Figure 2a displays another
3c2e bond schematically as a dotted-line triangle between vertices on three icosahedra.
References [6,10], which discuss the properties of icosahedral boron rich materials, include traditional “ball-and-stick” or wireframe models of 3c2e bonding structures. However, because the very nature of the ball-and-stick model is to represent bonds with sticks we arrive at a point of misrepresentation because the sticks are shown on the edges of the icosahedra instead of the faces where the bonds are actually located. This exposes a discontinuity or a conceptual difference in the way we currently think about 3c2e bonds.
Because the 2c2e ball-and-stick model is so ingrained in common usage, we must find a way to reconcile this conceptual difference with it. The research presented in this thesis has been pursued in order to provide a consistent method of visualization of 3c2e bonds, while also keeping with the convention of the familiar “ball-and-stick” molecular model.
The “ball-and-stick” style of molecular model is deeply entrenched, so simple changes to it may make it easier to express the concepts of multi-center and delocalized bonding.
In many ab initio electronic structure methods that use localized basis sets, normal
2c2e bonds are calculated by using Mulliken population analysis[11–14]. In the Mulliken method, the occupied states are examined to identify overlap between two atoms at each energy level. If the overlap is consistent between these two atoms across multiple energy levels then a bond is said to exist. To calculate a 3c2e bond, a more complex method must be implemented that tracks the accumulated bond between triplets of atoms. When a proper triplet is identified, the overall bond value is calculated as in the Mulliken method. If the strength of the apparent 3c2e bond exceeds a minimum threshold then it is recorded as a real three-center bond for visualization. A detail of some importance is that the position of the bond centroid is weighted by the strength of the pairs of atomic interactions, shifting it away from the geometric centroid for uneven contributions from the pairs. A more detailed description of this process will be given in Chapter 3.
Once all 3c2e bonds are calculated, a set of data files are made that contain both
3c2e and 2c2e bonds. The data is then combined and processed into a format suitable for visualization by tools such as Paraview. See the Appendix for a more thorough explanation of all of the scripts used in this research.
Chapter 2 provides an overview of the computational methods and visualization techniques used. The Orthogonalized Linear Combination of Atomic Orbitals (OLCAO) method is introduced and special attention is given to how the OLCAO method calculates bonds and bond orders, how the data are ultimately visualized, and the new method introduced with this research of extending Mulliken Analysis to calculate 3c2e bonds.
Chapter 3 demonstrates application of the method to a few specific examples of various
7 types of molecular and solid-state materials. Each material includes an icosahedral structure and is primarily boron based. Analysis of the electronic structure of the materials is used to support the visualizations. Chapter 4 contains concluding remarks and ruminations on future work that can be done to visualize multi-center bonding, or more efficient calculation of bonds and bond orders. The concluding remarks expand on the information discussed and the results obtained to give greater insight into 3c2e bonds and their properties.
Computation and then visualization of a 3c2e bond requires a combination of quantum mechanical ab initio electronic structure analysis and customizable and flexible visualization tools. There are many different methods for computing electronic structure properties, each with its own advantages and disadvantages. Among them, the density functional theory (DFT) based Orthogonalized Linear Combination of Atomic Orbitals
(OLCAO) method developed at UMKC provides a good balance between accuracy and computational efficiency, particularly for large systems such as amorphous hydrogenated boron carbide. Regarding visualization, the Visualization Tool Kit (VTK) was chosen because it is highly parallelized, open source, frequently updated, has an active user community, and has potential for further integration with the OLCAO method. VTK provides a programmatic interface to visualization tasks, and its legacy ASCII format is easy for a program to produce and easy for humans to read. Paraview was selected as the graphical user interface for VTK because it is also open source, has an active user community, and operates as a minimal overlay on top of VTK.
Table 1- Programs and Purpose
OLCAO VTK Paraview
Properties Local orbital and Free-form ability to Open source, well- DFT based, visualize large and documented, computationally diverse data types actively maintained inexpensive and data sets
Purpose Electronic structure Highly parallelized Graphical interface and bond order programmatic for VTK calculations visualization tool kit
The OLCAO Method
The OLCAO method is a program suite based on density functional theory in the local density approximation (LDA) and has been developed and built over many decades by Wai-Yim Ching and Paul Rulis at the University of Missouri – Kansas City. The method achieves a great balance between efficiency and accuracy and is reliable for systems ranging from simple molecular structures, to periodic crystalline systems, to amorphous materials. The advantages, history, and current limitations of the method are discussed in detail in the reference , so the focus here will be on the calculations that are important to this research, most notably the calculation of bonds and bond orders.
The OLCAO method expresses the solid-state electronic wavefunction, given in
Equation (1), by expanding it with a basis set of atomic orbitals (2) which are themselves expanded in a basis set of Gaussian type orbitals (GTOs) (3).
푛 ⃗⃗ ⃗⃗ 훹푛푘(푟⃗) = ∑푖,훾 퐶푖훾(푘)푏푖훾(푘, 푟⃗) (1)
1 ⃗⃗ ⃗⃗⃗⃗⃗ 푏 (푘⃗⃗, 푟⃗) = ( ) ∑ 푒푖(푘∙푅푣) 푢 (푟⃗ − 푅⃗⃗⃗⃗⃗ − 푡⃗⃗⃗⃗) (2) 푖훾 √푁 푣 푖 푣 훾
2 푁 푛−1 (−훼푗푟 ) 푢푖(푟⃗) = [∑푗=1 퐴푗 푟 푒 ] ∙ 푌푙푚(휃, 휑) (3)
The set of atomic orbitals are divided into categories that are labeled as the core orbitals, occupied valence orbitals, and a number of empty orbitals. Depending on the size
10 and properties of the system being examined, OLCAO calculations can employ minimal basis (MB), full basis (FB), or extended basis (EB). The MB contains core orbitals, all occupied valence shell orbitals, and any unoccupied orbitals of equivalent n and l quantum number as the occupied valence states. The FB contains the MB and additional empty orbitals of the next unoccupied shell. The EB contains the FB and an additional shell of higher energy unoccupied states. Usually, a FB set is more than sufficient to get accurate calculations for most basic electronic structure properties. An extended basis is used for optical properties calculations where higher energy final states are of interest. The minimal basis is commonly used for Mulliken analysis of bonds and charge transfer. The concept of a bond in OLCAO is calculated using Mulliken population analysis, which uses a
MB for the Bloch function because the Mulliken scheme prefers a localized basis. Any result from the Mulliken scheme is basis dependent so that results can only be compared to other calculations done with basis sets constructed in the same way.
The OLCAO method maintains a database of atomic basis functions based on results of past calculations, but the default basis can be changed to account for specific circumstances if necessary.
Bonds calculated in the OLCAO method, according to the Mulliken scheme, are defined by a fractional charge ρ, Equation (5), of the ith orbital of the αth atom, of a normalized state m
2 푚 1 = ∫|훹푚(푟)| 푑푟⃗ = ∑푖,훼 휌푖,훼 (4)
푚 푚∗ 푚 휌푖,훼 = ∑푗,훽 퐶푖훼 퐶푗훽 푆푖훼,푗훽 (5)
푚 where 퐶푖훼 is the eigenvector coefficient and 푆푖훼,푗훽 is the overlap matrix, j is the orbital of the βth atom. The summation in Equation (5) is over all possible paired atoms β. The overlap matrix is of the form
⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ −푖푘⋅푅휇 푆푖훼,푗훽(푘) = ⟨푏푖훼(푘, 푟⃗)|푏푗훽(푘, 푟⃗)⟩ = ∑휇 푒 ∫ 푢푖(푟⃗ − 푡⃗훼)푢푗(푟⃗ − 푅⃗⃗휇 −
where Rμ is the lattice vector and t is the position of its denoted atom in the cell. The overlap matrix describes the degree to which any basis function (occupied or not) overlaps with any other basis function. The fractional charge is used to calculate the localization index of the system
푚 2 퐿푚 = ∑푖,훼[휌푖,훼] (7)
The localization index measures the square of the fractional charge associated with each basis function summed over all basis functions for a given state m. A resulting Lm value of
1 indicates complete localization of a state onto one basis function and a value of N-1 indicates complete and uniform delocalization across all basis functions, where N is the number of electron states.
The fractional charge concept is extended to calculate the bond order (bond overlap population) between each pair of atoms. Fractional charge is computed by using the product of the wave function coefficient from a single orbital (basis function) of an atom summed against coefficients from all other orbitals (basis functions) of all atoms in the same state. A variation of the fractional charge is the effective charge, which accumulates the fractional charge for all orbitals of a given atom and further sums over all occupied states. In contrast to both the fractional and effective charge, the bond order sums the
12 product of coefficients of all orbitals (basis functions) from one atom against the orbitals from only one other atom and then sums over all occupied states. All of these calculations are accumulated over every k-point in each energy level. The bond overlap population values are >0 for a bonding orbital, <0 for an anti-bonding orbital, or 0 for a nonbonding orbital. For the case of a three-center bond the calculation becomes a bit more tricky. What has been done for this research is to introduce a novel way to use the Mulliken scheme definition of a bond order and extend it to calculate a bond between three atoms. In each state there are now three atoms and their orbitals to consider. The third atom is designated with γ and its orbitals are subscript k. A 3c2e bond is treated as a summation of 2c2e bonds from a triplet of atom pairs (ραβ, ραγ, ρβγ). Therefore, the bond order of a 3c2e calculation will have the form
휌훼훽훾 = 휌훼훽 + 휌훼훾 + 휌훽훾 ( 8) where the individual values have the forms
∗푛 푛 휌훼훽 = ∑푖 ∑푛 ∑푗(훽) 퐶푖훼 퐶푗훽푆푖훼,푗훽 ( 9)
∗푛 푛 휌훼훾 = ∑푖 ∑푛 ∑푘(훾) 퐶푖훼 퐶푘훾푆푖훼,푘훾 ( 10)
∗푛 푛 휌훽훾 = ∑푗 ∑푛 ∑푘(훾) 퐶푗훽 퐶푘훾푆푗훽,푘훾 ( 11) where the 3rd summation in each equation is over the basis functions of one specific atom and not all possible paired atoms as was the case in Equation (5) for the partial charge. The bond centroid is calculated via a combination of the atomic sites and the relative magnitudes of the component 2c2e bonds. The center point of each 2c2e bond is computed and then the weighted center point of the set of mid-points is computed and referred to as
13 the 3c2e centroid. Sometimes, due to periodic boundary conditions, it is possible for the
3c2e centroid to be located outside the periodic cell. The centroid is then shifted back into the cell. The fact that the 3c2e bond order centroid is shifted away from the geometric centroid is part of what sets this work calculation apart from past work [18,19] on 3c2e visualization. For visualization purposes, a “fake atom” is later placed at the 3c2e centroid so that traditional sticks can be drawn between it and the actual atoms of the model.
There are other methods of calculating bonds that the OLCAO method does not employ, some of which might be more accurate but also potentially more computationally expensive. Bader Charge Analysis (BCA) for example uses the Laplacian of the electron density for a system to determine critical points between atoms which indicate regions of charge separation . Perhaps using BCA, someone could partition off the center of a 3 center bond as its own volume and the bond critical points might lay between it and the atoms. I will return to discussing the Bader scheme in the ‘Future Works’ section.
The Visualization ToolKit
The Visualization Toolkit (VTK) is open source software that is designed for visualization of complicated and large three dimensional datasets, two dimensional plotting, and image processing. Development of the software began over 20 years ago , and its creators have since formed Kitware Inc. to improve and develop VTK as well as their other software packages such as CMAKE, ITK, and Paraview, in partnerships with
Sandia National Laboratories, Los Alamos National Laboratory, and others.
VTK is object-oriented software that is written in C++ but has various available language wrappers including Python. The central structure of VTK is a data pipeline ,
14 beginning with a source of information and ending as an image rendered on the screen. The source provides the initial data, which is either read in from a file or generated with a command in VTK and is then passed to filters that are specified by the user. These filters are the workhorse data modifiers in VTK, each having a unique way of altering the data.
A custom filter was produced as a part of this research. Before any filtered data can be visualized it must first be mapped with a mapper onto a so-called actor. Once an actor is generated, its visible properties such as color, opacity, and reflectance can be adjusted or modified. Finally, a renderer is chosen and the actor is displayed in a rendering window.
A specific example of a legacy VTK ASCII input file is shown in Figure 3. This type of source file is what the OLCAO generates when completing the 3CBO calculations.
At the time of writing this paper no other molecular models have been done using this type of file. The merit in using the VTK legacy format for this research, is its flexibility and generalizability when constructing a model and when applying glyphs. The raw data file depicted contains a header with data specifications, a set of points, lines, vertices on the points, and a scalar based color table applied to the points. VTK reads this file and then waits on user commands to either apply filters, such as glyphs, to specified parts of the data, or to use the read in data as is; this is now an object. At this step, the user selects a mapper to transform the data, including any metadata obtained from applied filters, into an actor. The actor is now ready to be rendered with another user command. The use of
Paraview streamlines this process.
Figure 3 Sample VTK legacy polydata output
Paraview is an open-source, scalable, visualization application and data analysis tool. It began as a joint effort between Kitware Inc. and Los Alamos National Laboratory in the year 2000, and its first edition was released in 2002 . Visualizations can be built quickly and analyzed with quantitative or qualitative techniques. Paraview can be run on desktop/laptop computers for smaller data sets or HPC servers/supercomputers for large data sets. The architecture of Paraview is extensible, so that if a user wants to make a custom application, plugin, data filter, or Python script, he or she is able to do so with ease
The user interface of Paraview displays the toolbars, rendering window, pipeline browser, and object inspector. Paraview objects include filters, representations and views, and methods for finding and selecting data. Files are read into Paraview and can be passed through filters before rendering or rendered immediately. The visualization and data
16 analysis capabilities of Paraview are built directly on top of VTK software and the user interface allows data pipelines of varying complexity to be constructed easily. Paraview comes with a built-in Python interface, so that anyone who is familiar with VTK and
Python may easily control their rendered scenes or create a custom filter from a combination of existing filters and altered parameters. Paraview provides its own renderer and automatically chooses the mapper for the user based on the initial data file and filters selected. The research here is unique in that, while Paraview has been used for molecular modeling for several years now, no research has used VTK legacy format as a source for that visualization. Figure 4 shows what the data file presented in Figure 3 is rendered as, after a spherical glyph filter is applied to the points and rendered in Paraview.
Figure 4 Paraview rendering of Figure 3
Calculating the set of 3c2e bond order values for a particular model begins with construction of a skeleton input file as described in the OLCAO reference text.
Skeleton files can be produced manually or through one of the many file format converters that comes with the OLCAO program suite. Then the “makeinput” command is executed with appropriate options and sub-options for the system under consideration. For example, the number of k-points must be selected and the atomic types might need to be assigned.
The makeinput script produces several data files in a new “inputs” directory in addition to a file for submitting a default set of OLCAO calculations to a compute queue. One of the data files produced, the “olcao.dat” file, contains a flag near the end of the file in the
BOND_INPUT_DATA section that is used to tell the OLCAO program to compute 3c2e bond orders (3CBOs) when an otherwise normal bond order calculation is submitted. When the OLCAO program for bond order calculation is then run, it outputs raw data files containing information on regular 2c2e bond orders and 3CBOs. Because the infrastructure for visualizing two-center bond order (2CBO) data is already prevalent in OLCAO and many other third-party programs, the main post-processing goal is to generate data files that look like 2CBO data file but which contain 3CBO data. This is accomplished with a script called “inser3cbo”. The two raw data files (for 2CBO and 3CBO) are used along with the structure.dat file created during the makeinput step to produce a new raw data file and a new structure.dat file. Both new files contain important information for 2CBO and
3CBO visualization, but it is all expressed in the form of a set of traditional 2CBOs.
Because of that, the resultant files can be easily processed with the “makeBOND” script that was modified to produce a legacy VTK file for Paraview. Once in Paraview a customized filter that combined glyphs and tubes could be used to modify the data.
Spherical glyphs are applied to each atomic site representing the atoms, and tube glyphs can be applied to the lines representing bonds between atoms. The glyphs and tubes can be colored and scaled based on specified scalars in the VTK legacy file, which were also computed as part of the “makeBOND” script. The “fake atom” centroids of the 3c2e bonds are smoothly integrated into the visualization as simply a small sphere with tubes drawn from the sites of the participating atoms to the “fake atom”.
A schematic illustration of the flow of information and commands in the OLCAO suite and afterward inside Paraview is shown in Figure 5.
Figure 5 Old process of bond visualization
Figure 6 New process of visualization
RESULTS AND DISCUSSION
The materials selected for this research range from simple boron-rich molecular structures, crystalline solid allotropes of boron, and an amorphous boron-rich material. The molecular structures are three dicarborane isomers, the crystalline solids are five allotropes of boron and boron carbide, and the amorphous material is amorphous hydrogenated boron carbide. Each of the materials is primarily composed of one or more icosahedra of boron.
These icosahedra contain the majority of the desired 3c2e bonds (intraicosahedral), though a few of the allotropes have 3c2e bonds between the icosahedral clusters (intericosahedral).
The carboranes are each composed of one icosahedron of ten boron and two carbon, surrounded by twelve hydrogen atoms. The allotropes contain multiple icosahedra, either bonded together by single boron atoms in each icosahedron, conjoined on an icosahedral face, or a combination of both, all on their respective lattices.
The configurations of the dicarboranes all have very simple structures. Each has a single icosahedron of ten boron atoms and two carbon atoms with each one of those sites connected to a hydrogen. The 3c2e bond centroids are located on the faces of the icosahedron, and the position of the carbon atoms changes the weighted positions of a few of the bond centroids.
α-rhombohedral boron, as evidenced from the calculations done by Fujimori et al, experiences 2c2e bonds between its polar boron atoms on the icosahedra, and 3c2e bonds between the equatorial boron atoms on them[8–10,23,24]. β-rhombohedral boron contains, at the center of its cell, two three-fused icosahedral units (face-sharing), which
21 display their own unique 3c2e bonding pattern[1,9,10,24] based on the work done by van
Setten et al. The γ-boron used for this research has “dumbbells” of two boron atoms connecting its icosahedra, and 3c2e bonds connecting to those dumbbells from the icosahedra[8,9,27]. α-tetragonal boron has a relatively unknown structure so the phase selected for this research has 51 atoms in its unit cell, as described by Wang et al. β- tetragonal boron is another allotrope of boron with a complex structure. Having been found to contain 190 atoms in its unit cell due to low occupancy of four of its atom sites, this model contains eight icosahedra, four icosahedral clusters, and ten non-icosahedral boron atoms. Boron carbide consists of an icosahedron on the corner of a rhombohedral lattice just like α-rhombohedral boron, but it also contains a carbon–boron–carbon (CBC) chain that runs through the center of the unit cell, as described by Aryal et al. Note that a rhombohedral cell can be transformed into a hexagonal cell and that this operation was done for this research. While its general shape is similar to α-rhombohedral boron, boron carbide lacks the equatorial 3c2e bonds between icosahedra that the α-boron allotrope possesses because of the CBC chain, and so all of its 3c2e bonds are intraicosahedral.
The amorphous material used in this research, using the model generated by
Belhadj-Larbi et al, cannot be described using the traditional language of crystal symmetry. Due to not being crystalline, the system required a very large number of atoms to produce a pseudo non-periodic structure. That is, the model of the amorphous material is defined within a cubic periodic cell, but it was made to be sufficiently large that the periodic replicas of atoms from neighboring cells do not influence atoms in the central cell. Because of the complexity, the exact 3c2e bonding of this completely amorphous material is unknown; 3c2e bonds may form in structures, in interconnected spaces, or the
22 bonding may be satisfied in different ways. Each bond in these systems was given a maximum allowable length, to prevent showing unphysical bonds between boron atoms on opposite sides of their respective icosahedra.
What follows is a series of figures (Figures 6-21) that illustrate the differences between visualization with only traditional ball-and-stick models and visualization when
3c2e bonds are explicitly taken into account. Standard 2c2e covalent bonds and 3c2e bonds are shown using the same type of stick. The main difference is that the 2c2e covalent bonds always have atomic sites for both vertices but the 3c2e bonds have atomic sites for three vertices and one central non-atomic site as a final vertex. Following the figures obtained as results of the research, there will be an in-depth discussion about the impact of the number and relative positions of 3c2e bonds in a system.
Figure 7 Traditional visualizations of ortho, para, and meta-carboranes.
Figure 8 Carboranes new visualization: ortho, para, and meta
Figure 9 Alpha boron traditional visualization
Figure 10 Alpha boron new visualization
Figure 11 Beta boron traditional visualization
Figure 12 Beta boron new visualization
Figure 13 Gamma boron traditional visualization
Figure 14 Gamma boron new visualization
α-Tetragonal Boron 51
Figure 15 Alpha tetragonal boron traditional visualization
Figure 16 Alpha tetragonal boron new visualization
β-Tetragonal Boron 190
Figure 17 Beta tetragonal boron traditional visualization
Figure 18 Beta tetragonal boron new visualization
Boron Carbide B11C-CBC
Figure 19 Boron Carbide traditional visualization
Figure 20 Boron Carbide new visualization
Amorphous Hydrogenated Boron Carbide a-BxC:Hy
Figure 21 a-BC:H traditional visualization
Figure 22 a-BC:H new visualization
The simpler boron allotropes, α-rhombohedral boron and γ-boron, distinctly show how bonding between the icosahedra differs compared to the conventional representation, and are consistent with the bonding charge density calculated by Haussermann and
Mikhaylushkin. α-tetragonal boron 51 showed the expected intraicosahedral 3c2e bonds, and two intericosahedral 3c2e bonds, but at the time of writing this no research was found to compare the bonding charge density of this phase with. The bonding to the CBC chain in the selected example of boron carbide was unaltered after the inclusion of three center bonds. This calculation is consistent with the model described by Emin. The more complex allotropes of boron, such as both types of β boron, are a bit more difficult to describe structurally. β-rhombohedral boron may be described in terms of its “icosahedral clusters” which here refer to its face-sharing icosahedra. β-rhombohedral boron showed unique 3c2e bonds its face-sharing icosahedral clusters, and the central boron atom was connected by a 3c2e bond to one of those clusters. This type of unique bonding of its central units was described by Jemmis and Balakrishnarajan. The amount and close proximity of the 3c2e bond centroids in the icosahedral clusters of this material may indicate that, for structures formed of fused icosahedra, the bonds exhibit more metallic than covalent behavior. Because these 3c2e bonds act more metallic than they do covalent, the density of them may explain why some allotropes of boron are semiconductors and others are metals.
The α-rhombohedral and γ allotropes, having simpler structures with adequate space between icosahedra or icosahedral clusters, would then be semiconductors. The more complex allotropes, like the α and β tetragonal phases, having more densely compacted icosahedra or icosahedral clusters and thus shorter 2c2e bonds and numerous inter- and intra- icosahedral 3c2e bonds, would behave more like metals. β-rhombohedral boron is
33 harder to analyze by looking at these bonds because it has icosahedral clusters with densely compacted 3c2e bonds, but these clusters are not densely compacted to each other.
Recent research has revealed that β-rhombohedral boron is a semiconductor, and that the other materials in this research behave as described above . Specifically, for the case of amorphous hydrogenated boron carbide, its exceptionally complex nature and mostly unknown structure makes interpreting its visualization difficult. The amorphous structure means that the amount of 3c2e bonds could wildly vary in their density and spacing. The best estimate would be that it is a semiconductor, which is supported by previous work . To support the findings from the visualizations of 3c2e bonding, the localization index (LI) (Equation 7) was calculated for each system along with the total density of states (TDOS) (Figures 22, 23, and 24). The LI for energies between -20 and 20 eV are plotted in Figure 22 while Figure 23 shows a zoom-in on the energy range of -5 to
5 eV that also affects a scaling of the data so that smaller crystals can be directly compared with larger crystals. Normally, as described in the Methods chapter, the LI quantifies the degree of localization of a particular electronic state with 1 indicating complete localization and N-1 indicating complete delocalization where N is the number of electronic states.
Because small systems with fewer states will have a numerical value for complete delocalization that is larger than the complete delocalization value for large systems Figure
23 shows modified LI data that were scaled by ratios of the number of states so that the smallest LI value possible is the same across all systems. Another point of clarification regarding the figures is that the highest occupied state in the TDOS plot (Figure 24) is fixed at a reference energy of 0 eV. Each point in the LI plots represents an average across multiple k-points, hence the highest energy occupied state in the LI is not exactly at 0 eV
34 as it would be for the TDOS. The vertical scale for the LI is the same for all systems while for the PDOS it is different for each system.
The TDOS shows clean gaps for α-rhombohedral boron and γ-boron, a gap for β- rhombohedral boron with one mid-gap defect state, and no gap for α- and β-tetragonal boron. For α-tetragonal boron the metallic TDOS is obvious while for β-tetragonal boron there appear to be a large number of defect-like states that fill the energy range in what would otherwise be a gap. Although the TDOS of a-BC:H appears to show metallic behavior, it is most likely that those near-gap states are a result of a similar crowding of defect-like states as seen in β-tetragonal boron.
Figures 23 clearly shows increased localization near the top of the occupied states for α- and β-tetragonal boron, γ-boron, and β-rhombohedral. But it also shows generally delocalized behavior for all valence electrons in α-rhombohedral while the absolute localization values for all the other systems are much larger, but a factor of ten compared to a-BC:H and by factors of at least five for the others. The interpretation indicates that the boron icosahedral systems tend to show an interesting degree of localized metallicity. That is, we see metal-like bonding that is confined to progressively more localized regions as the systems become more complicated.
Setting a maximum bond length for this system may not be the correct method here.
The close proximity of a few of the bond centroids may indicate the presence of higher multicenter bonding, maybe with 4, 5, or even 6 centers [34,35] that have a greater influence such that the material behaves more metallic.
Figure 23 localization index of boron allotropes and a-BC:H
Figure 24 localization index, zoomed in on the zero of energy and scaled
Figure 25 total density of states of boron allotropes and a-BC:H
CONCLUSIONS AND FUTURE WORK
Boron rich materials, mainly those that contain icosahedral structures, display several types of 3c2e bonding. Intraicosahedral 3c2e bonds are prevalent among all boron icosahedra, and depending on how they bond to each other, intericosahedral bonds can vary between 3c2e and traditional 2c2e bonds. The side-by-side view of the carborane isomers clearly show that, for the 3c2e cases, the position of the carbon atoms shifts the location of the bond centroid. 3c2e bonds, being delocalized, can be treated as more of a metallic bond than a covalent bond. Consequently, if a material has an interconnected network of 3c2e bonds through the periodic boundary conditions, that channel would be more conductive than significant numbers of 2c2e bonds. This determination is more difficult with amorphous boron-rich materials because the structure is mostly unknown. An extension to the Mulliken scheme for calculating the existence, strength, and position of 3CBOs for the
OLCAO method is demonstrated in this research. The method is accurate for all tested systems and it successfully updates traditional ball-and-stick molecular models to include
3c2e bonds that are positionally weighted according to atom pair contributions.
Since Mulliken Analysis is basis set dependent, work should be done to determine if utilizing a different basis set in the calculation of the 3c2e bond centroids would lead to the same centroids from another basis set. Since Mulliken Analysis is employed in a multitude of ab initio methods, this developed extension to it can be utilized in those software packages as well. Another thing than can be done is trying to achieve the same
39 results through another scheme, such as Bader Charge Analysis (BCA) or Adaptive Natural
Density Partitioning (AdNDP). This type of method, while much more computationally expensive, might yield even more accurate results. Or, using AdNDP, work could be done to visualize three center bonds as isosurfaces instead of ball-and-stick additions[34–41].
This could extend into visualizing even higher multi center bonds as isosurfaces in the
OLCAO method in the future. The production of the legacy VTK file itself could also be minorly improved. Visualization of bond strength could be added to a “CELL_DATA” section of the VTK file to color bonds, and the lattice information could be printed to a
VTK file. This work could also be viewed as the first step at integrating VTK and Paraview into the OLCAO method, so that future specialized scripts could produce VTK files for manipulation in Paraview.
Created and Altered Scripts and Programs bond3C.F90
This is the core program within the OLCAO program suite that calculates the three center bonds of each system. This is a tricky calculation, so the finer details of how it works will not be included here and are instead embedded directly as comments in the source code. The program starts by storing each atomic site, type, and number of valence states. Then the program creates a list of expected two-center bonded atoms. The program computes the minimum distance between each possible pair of atoms. For each pair that is sufficiently close to form a likely bond the program further computes the distance to any other third atom. Once all the distances in a triplet have been identified, the program then filters out all of the possible bonds that are not populated. Then, it calls a subroutine that calculates the 3CBO for the current triplet, as well as the 2CBO for each pair in that triplet.
When all the bond information has been accumulated for each state, k-point, and spin orientation, the program applies a final rejection criteria. The 2CBO of each pair in a triplet must exceed a value of 0.1, otherwise the triplet could be viewed as a set of two center bonds instead of a single three center bond. Lastly, the centroid of each 3CBO is weighted by the strength of the interaction between the atoms, and the information on each 3CBO is written to a raw data file to be used in the script “insert3cbo.”
This script takes the data from the raw data file containing the three center bonds produced by the bond3C.F90 program and combines it with the raw data file containing
41 the regular bonds and the structure file containing the atomic positions and lattice parameters of the system, both produced from the regular bond program. This produces two new files: a data file and a structure file, both which now contain the information about both two center and three center bonds. First the script reads in all the information from the two center bond data file. When this data is stored, it opens and begins reading the three center data file. The script removes from the two center bond data any atom involved in a three center bond. New bonds to a fake atom located at the weighted bond centroid are added from each atom involved in a three center bond. These new bonds and fake atoms are added to the list of bond order data. The revised bond order data is then written to a new raw data file. Finally, the script opens and reads the regular structure file, adds the
3CBO information, and writes it to a new structure file. These two new files will be used in the “makeBOND” script, to store the data in a way that can be visualized in Paraview.
This script takes the files produced by “insert3cbo” and can produce a number of things with them. For the purposes of this research, it is used to output the structure and data of a system into the legacy VTK format which can then be viewed in Paraview. It begins by reading and storing the information from the specified data file, control file, and structure file. Since this script was originally written to process the information and write it in a format to be used by a different visualization tool called OpenDX, some subroutines were repurposed for this research because VTK and OpenDX files are structured somewhat similarly. For this subroutine, first the script obtains the positions of periodic replicated atoms that are outside the central cell, prints the lattice info to its own file, then creates a
42 list of atoms and bonds for the model. Locating periodic atoms outside the central cell is important so that bonds can be drawn that extend through the edges of the cell. The list of atoms, bonds, and color information is then printed to a legacy VTK file, and the script ends. The produced file can be loaded and viewed in Paraview.
Custom Paraview Filter
This filter was constructed using a special function of Paraview that allows the user to create a brand-new filter by combining functions and features of several existing elements using a graphical interface. It combines the Paraview “glyph” filter, which applies glyphs to points specified as vertices, and the “tube” filter, which applies cylinders over lines. With the “create custom filter” option in the Paraview menu, these two filters are selected from the pipeline, when already being used, and then certain qualities of each are selected to be manipulatable. They are then combined and given a user specified name.
This allows for near automatic scaling of spherical glyphs with respect to atomic radii, and easily selectable color schemes for both the glyphs and the tubes. However, the radius of the tube has a set default that must be manually changed.
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Joshua Ray Sexton was born on March 16, 1992 in Kansas City, Missouri. He received education from local public schools and graduated from Winnetonka High School in 2010. He graduated with a Bachelor of Science in Physics from the University of
Missouri – Kansas City with Department of Physics and Astronomy Honors in 2017.
Mr. Sexton began the Masters of Physics program at the University of Missouri –
Kansas City in 2017.