EXCHANGE ENERGY AND POTENTIAL USING THE LAPLACIAN OF THE DENSITY (master thesis)
Abstract
The challenge of density functional theory is the useful approximation of the exchange - correlation energy. This energy can be approximated with the local electron density and the gradient of the density. Many different generalized gradient approximations (GGA) have been made recently and there is controversy over the best overall functional. Recent Monte Carlo simulations give evidence that the Laplacian of the density might be a better starting place than the gradient to correct the local density approximation. It also gives a better representation of the exchange potential at the nuclear cusp and of bonding between atoms. We have tested several Laplacian based GGA models for exchange for small atoms. We use known constraints on the exchange energy used in current GGA’s. In many models unphysical oscillations occur in the potential, and understanding and eliminating them is part of the focus of this research. Our results suggest that smaller values for the short and long range constraints in the literature give more physically reasonable results in the Laplacian models. We also find that mixing s 2 and q seems to give a better result than only using one or the other.
Contents
1 Introduction 1
1.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Summary of Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2 Background 4
2.1 Thomas - Fermi Theory . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.2 The Basis For Density Functional Theory . . . . . . . . . . . . . . . . 5
2.3 Implementation: The Kohn - Sham Equation . . . . . . . . . . . . . . 6
2.4 Exchange-Correlation Energy . . . . . . . . . . . . . . . . . . . . . . 9
2.5 The Local Density Approximation . . . . . . . . . . . . . . . . . . . . 10
2.6 The Generalized Gradient Approximation . . . . . . . . . . . . . . . 12
2.6.1 The PBE Functional and Friends . . . . . . . . . . . . . . . . 14
2.7 The Meta-GGA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.8 Using The Laplacian . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.8.1 The Gradient Expansion . . . . . . . . . . . . . . . . . . . . . 21
2.8.2 Prior Work with the Laplacian . . . . . . . . . . . . . . . . . 22
3 Methodology 27 3.1 Atomic Densities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.2 Noble Gases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
3.3 Numerical Grid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3.4 Numerical Method and Code . . . . . . . . . . . . . . . . . . . . . . . 32
4 Results 36 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
4.2 Initial Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
4.2.1 Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
4.2.2 Simple Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
4.3 Mixing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
4.3.1 Motivation for Mixing . . . . . . . . . . . . . . . . . . . . . . 46
4.3.2 Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
4.3.3 Energy and Potential with Optimized Mixing . . . . . . . . . 52
4.4 Advanced forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
5 Discussion 57
5.1 Research Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
5.2 Final Thoughts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
Bibliography 67
To download the thesis click on the link below:
https://cardinalscholar.bsu.edu/bitstream/handle/123456789/195955/WagnerC_2012-2_BODY.pdf;jsessionid=D9E4847BCDC8B2F15EFDF77170719C1E?sequence=1
The challenge of density functional theory is the useful approximation of the exchange - correlation energy. This energy can be approximated with the local electron density and the gradient of the density. Many different generalized gradient approximations (GGA) have been made recently and there is controversy over the best overall functional. Recent Monte Carlo simulations give evidence that the Laplacian of the density might be a better starting place than the gradient to correct the local density approximation. It also gives a better representation of the exchange potential at the nuclear cusp and of bonding between atoms. We have tested several Laplacian based GGA models for exchange for small atoms. We use known constraints on the exchange energy used in current GGA’s. In many models unphysical oscillations occur in the potential, and understanding and eliminating them is part of the focus of this research. Our results suggest that smaller values for the short and long range constraints in the literature give more physically reasonable results in the Laplacian models. We also find that mixing s 2 and q seems to give a better result than only using one or the other.
Contents
1 Introduction 1
1.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Summary of Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2 Background 4
2.1 Thomas - Fermi Theory . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.2 The Basis For Density Functional Theory . . . . . . . . . . . . . . . . 5
2.3 Implementation: The Kohn - Sham Equation . . . . . . . . . . . . . . 6
2.4 Exchange-Correlation Energy . . . . . . . . . . . . . . . . . . . . . . 9
2.5 The Local Density Approximation . . . . . . . . . . . . . . . . . . . . 10
2.6 The Generalized Gradient Approximation . . . . . . . . . . . . . . . 12
2.6.1 The PBE Functional and Friends . . . . . . . . . . . . . . . . 14
2.7 The Meta-GGA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.8 Using The Laplacian . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.8.1 The Gradient Expansion . . . . . . . . . . . . . . . . . . . . . 21
2.8.2 Prior Work with the Laplacian . . . . . . . . . . . . . . . . . 22
3 Methodology 27 3.1 Atomic Densities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.2 Noble Gases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
3.3 Numerical Grid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3.4 Numerical Method and Code . . . . . . . . . . . . . . . . . . . . . . . 32
4 Results 36 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
4.2 Initial Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
4.2.1 Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
4.2.2 Simple Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
4.3 Mixing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
4.3.1 Motivation for Mixing . . . . . . . . . . . . . . . . . . . . . . 46
4.3.2 Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
4.3.3 Energy and Potential with Optimized Mixing . . . . . . . . . 52
4.4 Advanced forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
5 Discussion 57
5.1 Research Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
5.2 Final Thoughts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
Bibliography 67
To download the thesis click on the link below:
https://cardinalscholar.bsu.edu/bitstream/handle/123456789/195955/WagnerC_2012-2_BODY.pdf;jsessionid=D9E4847BCDC8B2F15EFDF77170719C1E?sequence=1
No comments