Ground State Structural Searches for Boron Atomic Clusters Using Density Functional Theory
By John Kabaa
()
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In this project we used a computational scheme to predict the best atomic arrangements of boron clusters. A completely unbiased search mechanism was implemented to determine optimal boron clusters. An approximate model was first used to locate minimum energy conformations followed by a more accurate first principles calculation to get the global minimum. Our results were then validated by comparisons to those reported in literature. The objective was to perform a consistent search of cluster sizes ranging from sizes n = 2-14, 16, 18 and 20. This was done to not only study where boron makes its transition from flat to three-dimensional clusters, but also to determine patterns in their evolution with size. The motivation behind our work and the long-term goal involved exploring the viability of the existence of large cage-like boron clusters, like B80, which have been proposed in literature as being extremely stable.
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Ground State Structural Searches for Boron Atomic Clusters Using Density Functional Theory - John Kabaa
GROUND STATE STRUCTURAL SEARCHES FOR BORON ATOMIC CLUSTERS
USING DENSITY FUNCTIONAL THEORY
John Kabaa Kamau
A thesis submitted in partial fulfillment of
the requirements for the degree of
Master of Science
Department of Physics
Central Michigan University
Mount Pleasant, Michigan
July, 2008
ACKNOWLEDGEMENTS
I am grateful for God’s provision, grace and guidance during the entire course. I thank all of my family members and my friends both near and far for their continuing support and encouragement.
I would like to thank Dr. Alan Jackson, my advisor, who spent many hours of his time reviewing this document, and made many contributions to the final product. I acknowledge Dr. Veronica Barone who assisted me and provided direction when this project began. I thank the other members of the ad hoc Thesis Requirements Committee, Dr. Stanley Hirschi and Dr. Juan Peralta, for their careful examination of this thesis and for their valuable discussions.
I also thank and appreciate all of the Physics Department faculty members especially those who have taught me. I thank the entire staff that have provided the support and equipment I needed to complete my thesis. Many thanks, to all of my colleagues in the department for their assistance and sharing this enjoyable endeavor. I remain indebted to Dr. Li Ma, C. C. Dharmawardhana, Diego Franco Ocampo and Eric McDonald for many helpful suggestions throughout the course of this work. Finally, I wish to acknowledge the support of Central Michigan University in producing this work.
iii
ABSTRACT
GROUND STATE STRUCTURAL SEARCHES FOR BORON ATOMIC CLUSTERS
USING DENSITY FUNCTIONAL THEORY
by John Kabaa Kamau
Elemental boron has a fascinating chemical versatility that is unique among the elements of the periodic table. The study of the evolution of boron atomic clusters and their chemical and physical properties is of fundamental interest to researchers interested in nanotechnology.
One of the most difficult problems in the study of boron clusters is finding their ground state cluster structures. The atomic arrangements in clusters are generally very different from those in corresponding bulk materials, such that chemical intuition cannot be trusted to generate optimal structures. An unbiased search method was used to search for these stable boron structures. It took advantage of the relative speed of the density functional-based tight binding (DFTB) method to identify low-lying local structures and then used the more accurate density functional theory (DFT) to find the ground state structures.
The lowest energy boron clusters for cluster sizes N = 2-14, 16, 18 and 20 and their respective binding energies per atom were investigated. Planar, three-dimensional (3D) cage-like and tubular ring-shaped structures were found. Preliminary results, for B80, at the DFTB level suggest that cages are not the most stable cluster structures.
iv
TABLE OF CONTENTS
LIST OF TABLES……………………………………………………………………….vii
LIST OF FIGURES……………………………………………………………………..viii
CHAPTER
I.
ITRODUCTIO……………………………………………………………….1
1.1 Overview of Atomic Clusters………………………………………...1
1.2 Boron Clusters……………………………………………………......2
1.3 Previous Studies of Boron Clusters…………………………………..4
1.4 Problem Statement …………………………………………………...7
II.
THEORETICAL METHODS FOR ELECTROIC STRUCTURE
CALCULATIOS…..….……………….…………………………..….……………8
2.1 Overview……..…………..…………………………………………...8
2.2 Density Functional Theory (DFT) …………………………………...8
2.3 The NRLMOL Code …………………………………………………9
2.4 Density Functional Based Tight Binding (DFTB) Method ………...10
2.5 Comparison of DFT and DFTB Methods...………………………....11
III.
METHODOLOGY…………………………..…………………………………13
3.1 The Structure Prediction Problem...………………………………...13
3.2 Energy Minimization Methods………………………………….…..14
3.3 The BIG BANG Algorithm………...…………….............................16
3.4 Energy Correlations for DFT vs. DFTB and DFT vs. DFT1….........18
IV.
GROUD STATE STRUCTURAL SEARCHES FOR BORO
CLUSTERS…...………………………………………………………………....24
4.1 Creating Different Geometrical Volumes….………………….…….24
4.2 Tests to Find Optimal Parameters…………………………………...26
4.3 DFT vs. DFT1 Correlation for B12…………….……………………28
4.4 Potential Energy Comparison for DFT and DFTB.......……………..30
4.5 Searching Different Sizes……….………………………………….....31
V.
RESULTS……………………………….………………………………………33
5.1 The Lowest Energy Structure and Isomers for Bn Clusters (n = 2-14, 16, 18 and 20)…..…………….…...……...………….………………...33
v
5.2 DFTB Results for B80 ……………….……...………....………..…...41
5.3 Binding Energy of the Clusters…………………………………...…43
VI.
COCLUSIOS …..…………………………………………………………...45
6.1 Summary of Findings………...……………………………………...45
6.2 Recommendations for Future Research…...…………………….......46
APPENDICES………………………………………………………………………...…47
REFERENCES……………………………………………………………………….…...53
vi
LIST OF TABLES
TABLE
PAGE
1.
The first column indicates the total number of random initial configurations formed, followed by the lowest DFTB local minimum that emerged, the number of times the same minimum was found and the last column shows the total minima from each run…………………………………………….………...26
2.
Optimal input parameters that were used in Bn searches (n =