Ground State Structural Searches for Boron Atomic Clusters Using Density Functional Theory

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 =