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Towards A New Exchange-Correlation Density Functional For More Accurate Band Gap Predictions

Towards A New Exchange-Correlation Density Functional For More Accurate Band Gap Predictions

ABSTRACT

Density-Functional Theory (DFT) offers a simplification to electronic structure problems by using the electron density instead of the wave-function. Unlike the wave function which is a function of 3N variables (excluding spin) for an N-electron system, the density depends only on three variables, irrespective of the number of electrons in the system. While DFT, in principle, gives an accurate description of ground state properties, practical applications of DFT are based on approximations to the so-called exchange-correlation (xc) potential. The exchange-correlation potential describes the effects of the Pauli exclusion principle and the electron-electron Coulomb repulsion beyond a purely electrostatic interaction of the electrons. A common description of exchange-correlation functional is the so-called local density approximation (LDA) which locally substitutes the exchange-correlation energy density of an inhomogeneous system by that of an electron gas evaluated at that local density. While many ground state properties (such as lattice constants and bulk moduli) are well described in the LDA, the band gap is underestimated by as
much as 50% in LDA compared to experiments.

In this thesis, we focus on the development of an exchange-correlation functional with adjustable parameters which can give more accurate band gap energies. This functional is based on the xc potential derived in 1988 from a tight-binding approximation by Hanke and Sham (HS)[1]. Our contribution consists in expressing the HS potential in terms of the electron density and its gradient. This new expression for the xc functional was parameterized for the Si and Ge bulk systems and found to reduce the error in the LDA band gap prediction by an average of 22:3% for the
systems (ZnO, MgO, ZnS, LiF, FeO and GaAs) that we tested it on.

TABLE OF CONTENTS

Abstract i
Acknowledgments ii
Dedication iv
Contents vi
1 Introduction 1
1.1 Successes and Failures of DFT . . . . . . . . . . . . . . . . . . . . . . 2
1.2 The Band Gap Problem . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.3 Atomic Units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2 Theoretical Background 6
2.1 Electronic Structure Problem . . . . . . . . . . . . . . . . . . . . . . 6
2.2 Wave-Function Based Methods . . . . . . . . . . . . . . . . . . . . . 9
2.2.1 The Hartree-Fock Formalism . . . . . . . . . . . . . . . . . . . 9
2.2.2 Correlated Methods Beyond Hartree-Fock . . . . . . . . . . . 13
2.3 Density-Functional Theory . . . . . . . . . . . . . . . . . . . . . . . . 13
2.3.1 The Thomas-Fermi (TF) Model . . . . . . . . . . . . . . . . . 14
2.3.2 The Hohenberg-Kohn (HK) Theorem . . . . . . . . . . . . . . 15
2.4 The Kohn-Sham (KS) Scheme . . . . . . . . . . . . . . . . . . . . . . 17
2.5 Interpretation of Kohn-Sham energies . . . . . . . . . . . . . . . . . . 20
2.6 Failure of DFT for Band Gap Energies . . . . . . . . . . . . . . . . . 20
Contents vi
2.7 Past Work on Correcting DFT Band Gap Energies . . . . . . . . . . 22
3 Functional Development 25
3.1 Exchange-Correlation Functionals . . . . . . . . . . . . . . . . . . . . 25
3.1.1 The Local-Density Approximation (LDA) . . . . . . . . . . . . 26
3.1.2 The Generalized-Gradient Approximation (GGA) . . . . . . . 27
3.1.3 Meta-GGA (mGGA) . . . . . . . . . . . . . . . . . . . . . . . 28
3.1.4 Hybrid Schemes . . . . . . . . . . . . . . . . . . . . . . . . . . 29
3.2 A New Density Functional . . . . . . . . . . . . . . . . . . . . . . . . 30
3.2.1 X-Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
3.2.2 The Corresponding xc Energy Exc . . . . . . . . . . . . . . . . 32
4 Methods 35
4.1 Basis Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
4.1.1 Plane Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
4.1.2 Pseudo-potentials . . . . . . . . . . . . . . . . . . . . . . . . . 37
4.2 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
5 Results, Discussion and Conclusion 42
5.1 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
5.2 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
5.3 Conclusion and Perspective . . . . . . . . . . . . . . . . . . . . . . . 48
A Hanke-Sham (HS) xc Potential 49
A.1 Green’s Function and Self-Energy Operator . . . . . . . . . . . . . . 49
A.2 A Tight-Binding (TB) Model for the Self-Energy (xc) . . . . . . . . 51
A.3 A vxc for Insulators and Semiconductors . . . . . . . . . . . . . . . . 54
Bibliography 57

CHAPTER ONE

Introduction

Density-Functional Theory (DFT) is one of the most popular and successful Quantum Mechanics (QM) approach for large systems. It is a widely used methods for “ab initio” calculations of the structure of atoms, molecules, crystals, surfaces and their interactions. It is nowadays routinely applied for calculating e.g., the binding energy of molecules in chemistry and the band structure of solids in physics. First application relevant for fields traditionally considered more distant from quantum mechanics, such as biology and mineralogy are beginning to appear. Superconductivity, atoms in the focus of strong laser pulses, relativistic effects in heavy elements and in atomic nuclei, classical liquids, and magnetic properties of alloys have been studied with DFT [2].

The Hamiltonian for a real material consisting of N electrons and M nuclei is given by
H =
XM
=1
~2
2M
r2
􀀀
XN
i=1
~2
2me
r2i
+
X
i<j
e2
40jri 􀀀 rj j
􀀀
X
i;
Ze2
40jri 􀀀 Rj
+
X
<
ZZe2
40jR 􀀀 Rj
(1.1)
Here R, Z, and M represent the position, nuclear charge, and mass of the-th nucleus and ri is the position of electron i. The variables m and e are the mass and charge (magnitude) of the electron. If we invoke the Born-Oppenheimer approximation in which nuclei masses are set to infinity, we arrive at the ‘electronic’


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