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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
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
© 2020
JOSHUA RAY SEXTON
ALL RIGHTS RESERVED
iii
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
ABSTRACT
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
Orbitals method.
iv
v
APPROVAL PAGE
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.
Supervisory Committee
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
vi
Table of Contents
ABSTRACT................................................................................................................................................ iii
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
vii
Boron Carbide B11C-CBC ......................................................................................... 30
Amorphous Hydrogenated Boron Carbide a-BxC:Hy ................................................ 31
4. CONCLUSIONS AND FUTURE WORK ....................................................................................... 39
Conclusion..................................................................................................................... 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
viii
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 [6] ........................................................................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
ix
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
x
LIST OF TABLES
Table 1- Programs and Purpose .......................................................................................... 9
xi
ACKNOWLEDGEMENTS
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.
1
CHAPTER 1
INTRODUCTION
Context
Boron-rich materials such as macropolyhedral boranes and metallaboranes[1], the
various allotropes of elemental boron, crystalline boron carbide[2], boron sub-oxide[3],
and amorphous forms such as hydrogenated boron carbide[4] 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.
In an icosahedral molecular structure, such as B12H12 borane, each boron atom in
the icosahedron is covalently bonded to an external hydrogen atom, as shown in Figure 1.
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.[5] Thus, a B12H12
2
molecular structure would be deficient by two electrons. In a molecular configuration, the
boron icosahedron in B12H12 may become negatively charged, making dianions, (B12H12)2-
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
3
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[1].
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 [6]. The bonding pattern is confirmed through X-ray diffraction studies [7] 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 [8], 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 [9]. 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.
4
Figure 1 Atomic structure of dodecaborane B12H12
5
Figure 2 (A)-(C) Schematic, stereographic, and broad views of α-rhombohedral boron;
(D) Bonding charge contours from [6]
6
Motivation
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.
7
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[15]. See the Appendix for a more thorough
explanation of all of the scripts used in this research.
Outline
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
8
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.
9
CHAPTER 2
METHODS
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)[16] 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
DFT based,
computationally
inexpensive
Free-form ability to
visualize large and
diverse data types
and data sets
Open source, well-
documented,
actively maintained
10
Purpose Electronic structure
and bond order
calculations
Highly parallelized
programmatic
visualization tool kit
Graphical interface
for VTK
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 [17], 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).
𝛹𝑛𝑘(𝑟) = ∑ 𝐶𝑖𝛾𝑛 (��)𝑏𝑖𝛾(��, 𝑟)𝑖,𝛾 (1)
𝑏𝑖𝛾(��, 𝑟) = (1
√𝑁)∑ 𝑒𝑖(��∙𝑅𝑣 )𝑣 𝑢𝑖(𝑟 − 𝑅𝑣
− 𝑡𝛾 ) (2)
𝑢𝑖(𝑟) = [∑ 𝐴𝑗𝑁𝑗=1 𝑟𝑛−1𝑒(−𝛼𝑗𝑟
2)] ∙ 𝑌𝑙𝑚(𝜃, 𝜑) (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
11
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[17], 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
1 = ∫|𝛹𝑚(𝑟)|2𝑑𝑟 =∑ 𝜌𝑖,𝛼
𝑚𝑖,𝛼 (4)
𝜌𝑖,𝛼𝑚 = ∑ 𝐶𝑖𝛼
𝑚∗
𝑗,𝛽 𝐶𝑗𝛽𝑚𝑆𝑖𝛼,𝑗𝛽 (5)
12
where 𝐶𝑖𝛼𝑚 is the eigenvector coefficient and 𝑆𝑖𝛼,𝑗𝛽 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
𝑆𝑖𝛼,𝑗𝛽(��) = ⟨𝑏𝑖𝛼(��, 𝑟)|𝑏𝑗𝛽(��, 𝑟)⟩ = ∑ 𝑒−𝑖��⋅𝑅𝜇 ∫𝑢𝑖(𝑟 − 𝑡𝛼)𝑢𝑗(𝑟 − ��𝜇 −𝜇
𝑡𝛽)𝑑𝑟 (6)
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
𝐿𝑚 = ∑ [𝜌𝑖,𝛼𝑚 ]
2𝑖,𝛼 (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
13
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
𝜌𝛼𝛽𝛾 = 𝜌𝛼𝛽 + 𝜌𝛼𝛾 + 𝜌𝛽𝛾 ( 8)
where the individual values have the forms
𝜌𝛼𝛽 = ∑ ∑ ∑ 𝐶𝑖𝛼∗𝑛𝐶𝑗𝛽
𝑛 𝑆𝑖𝛼,𝑗𝛽𝑗(𝛽)𝑛𝑖 ( 9)
𝜌𝛼𝛾 = ∑ ∑ ∑ 𝐶𝑖𝛼∗𝑛𝐶𝑘𝛾
𝑛 𝑆𝑖𝛼,𝑘𝛾𝑘(𝛾)𝑛𝑖 ( 10)
𝜌𝛽𝛾 = ∑ ∑ ∑ 𝐶𝑗𝛽∗𝑛𝐶𝑘𝛾
𝑛 𝑆𝑗𝛽,𝑘𝛾𝑘(𝛾)𝑛𝑗 ( 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
14
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 [20]. 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 [16],
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 [21],
15
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.
16
Figure 3 Sample VTK legacy polydata output
Paraview
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 [22]. 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
[15].
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
17
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
Program
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[17].
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.
18
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”.
19
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
20
Figure 6 New process of visualization
21
CHAPTER 3
RESULTS AND DISCUSSION
Model Systems
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[23], 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
22
display their own unique 3c2e bonding pattern[1,9,10,24] based on the work done by van
Setten et al[25]. The γ-boron used for this research[26] 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[28] 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[9]. β-
tetragonal boron is another allotrope of boron with a complex structure. Having been found
to contain 190 atoms in its unit cell[29] due to low occupancy of four of its atom sites, this
model contains eight icosahedra, four icosahedral clusters, and ten non-icosahedral boron
atoms[9]. 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[30]. 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[4], cannot be described using the traditional language of crystal
symmetry[31]. 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
23
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.
Carborane C2H10B12
Figure 7 Traditional visualizations of ortho, para, and meta-carboranes.
24
Figure 8 Carboranes new visualization: ortho, para, and meta
α-Rhombohedral Boron
Figure 9 Alpha boron traditional visualization
25
Figure 10 Alpha boron new visualization
β-Rhombohedral Boron
Figure 11 Beta boron traditional visualization
26
Figure 12 Beta boron new visualization
27
γ-Boron
Figure 13 Gamma boron traditional visualization
Figure 14 Gamma boron new visualization
28
α-Tetragonal Boron 51
Figure 15 Alpha tetragonal boron traditional visualization
Figure 16 Alpha tetragonal boron new visualization
29
β-Tetragonal Boron 190
Figure 17 Beta tetragonal boron traditional visualization
Figure 18 Beta tetragonal boron new visualization
30
Boron Carbide B11C-CBC
Figure 19 Boron Carbide traditional visualization
Figure 20 Boron Carbide new visualization
31
Amorphous Hydrogenated Boron Carbide a-BxC:Hy
Figure 21 a-BC:H traditional visualization
32
Figure 22 a-BC:H new visualization
33
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[32]. 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[6]. 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[24]. 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
34
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 [9]. 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 [33]. 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
35
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.
36
Figure 23 localization index of boron allotropes and a-BC:H
37
Figure 24 localization index, zoomed in on the zero of energy and scaled
38
Figure 25 total density of states of boron allotropes and a-BC:H
39
CHAPTER 4
CONCLUSIONS AND FUTURE WORK
Conclusion
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.
Future Work
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
40
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.
41
APPENDIX
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[42]. 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.”
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
42
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.
makeBOND
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
43
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.
44
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48
VITA
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.
