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First Principles Investigation Into the Atom in Jellium Model System

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First Principles Investigation Into the Atom in Jellium Model System First Principles Investigation into the Atom in Jellium Model System Andrew Ian Duff H. H. Wills Physics Laboratory University of Bristol A thesis submitted to the University of Bristol in accordance with the requirements of the degree of Ph.D. in the Faculty of Science Department of Physics March 2007 Word Count: 34, 000 Abstract The system of an atom immersed in jellium is solved using density functional theory (DFT), in both the local density (LDA) and self-interaction correction (SIC) approxima- tions, Hartree-Fock (HF) and variational quantum Monte Carlo (VQMC). The main aim of the thesis is to establish the quality of the LDA, SIC and HF approximations by com- paring the results obtained using these methods with the VQMC results, which we regard as a benchmark. The second aim of the thesis is to establish the suitability of an atom in jellium as a building block for constructing a theory of the full periodic solid. A hydrogen atom immersed in a finite jellium sphere is solved using the methods listed above. The immersion energy is plotted against the positive background density of the jellium, and from this curve we see that DFT over-binds the electrons as compared to VQMC. This is consistent with the general over-binding one tends to see in DFT calculations. Also, for low values of the positive background density, the SIC immersion energy gets closer to the VQMC immersion energy than does the LDA immersion energy. This is consistent with the fact that the electrons to which the SIC is applied are becoming more localised at these low background densities and therefore the SIC theory is expected to out-perform the LDA here. DFT is used within the framework of the effective medium theory (EMT) to calculate Wigner-Seitz radii for solids made up of atoms up to and including the 4d transition metals. The EMT uses, as a building block, calculations of the constituent atom of the solid immersed in infinite jellium. The calculated Wigner-Seitz radii are found to reproduce the same trends observed in the experimental Wigner-Seitz radii as a function of atomic number. To my Family Acknowledgments Thanks to my supervisor James Annett and also to Balazs Gy¨orffy. Authors Declaration I declare that the work in this thesis was carried out in accordance with the regulations of the University of Bristol. The work is original except where indicated by special reference in the text and no part of the thesis has been submitted for any other degree. Any views expressed in the thesis are those of the author and in no way represent those of the University of Bristol. The thesis has not been presented to any other University for examination either in the United Kingdom or overseas. SIGNED: ......................................... DATE: .......................... Contents 1 Introduction 1 2 Solving the Many-Electron Schr¨odinger equation 17 2.1 The Many-Electron Problem . 17 2.1.1 Single-Electron Theories . 18 2.1.2 The Variational Principle . 18 2.2 Hartree Fock Theory . 19 2.3 Density Functional Theory . 21 2.3.1 Minimising the Energy Functional . 21 2.3.2 The Kohn-Sham Equations . 25 2.3.3 Self-Consistent Solutions . 28 2.3.4 The Exchange-Correlation Energy and Potential . 29 2.3.5 Self-Interaction Correction . 32 2.4 Variational Quantum Monte Carlo . 34 2.4.1 A Variational Theory . 34 2.4.2 The Monte Carlo Technique . 35 2.4.3 The Variational Quantum Monte Carlo Method . 36 2.4.4 Metropolis Algorithm . 37 2.4.5 Equilibration and Serial Correlation . 39 2.4.6 The Choice of the Trial Wavefunction . 40 2.4.7 Updating the Slater Determinants . 41 2.4.8 Calculating the Local Energy . 42 2.4.9 Cusp Conditions . 44 2.4.10 Correlated Sampling . 46 2.4.11 Blocking Analysis to Calculate Error on Mean . 48 xi xii CONTENTS 2.4.12 Calculating the Probability Density . 50 2.4.13 HF Calculations . 50 3 An Atom in Infinite Jellium Solved using DFT 53 3.1 Solving the Schr¨odinger Equation . 53 3.1.1 The Radial Schr¨odinger Equation . 53 3.1.2 The Electron Density . 55 3.1.3 Potential Mixing . 56 3.1.4 Criterion for Convergence . 57 3.1.5 Simplifying the Coulomb Potential for the Case of Spherical Symmetry 58 3.2 Scattering States . 61 3.2.1 Introduction . 61 3.2.2 Boundary Conditions on Scattering States . 61 3.2.3 Matching to the Boundary Condition . 62 3.2.4 Normalisation of Scattering States . 64 3.2.5 Calculating the Scattering State Density . 65 3.2.6 Friedel Oscillations . 66 3.2.7 Friedel Sum Rule . 68 3.2.8 Properties of the Phase-Shift . 70 3.3 Numerical Algorithm for Solving the Radial Schr¨odinger Equation . 73 3.3.1 Radial Schr¨odinger Equation Solutions in the Limits r → 0 and r → ∞ 73 3.3.2 The Runge-Kutta Algorithm . 74 3.3.3 Bound State Calculation . 75 3.3.4 Scattered State Calculation . 78 3.4 The Immersion Energy . 78 3.4.1 Derivation of Immersion Energy . 78 3.4.2 Finite Radius Corrections . 82 3.4.3 Numerical Parameters and Error Analysis . 83 3.4.4 Results . 87 3.5 The Effective Medium Theory . 90 3.5.1 Background Theory . 90 3.5.2 Results . 96 3.6 Cerium Solved using the LDA and SIC . 97 CONTENTS xiii 3.6.1 Introduction . 97 3.6.2 Cerium . 99 3.6.3 Spin-polarised LDA for Cerium . 100 3.6.4 Imposing Orthogonality when Applying SIC . 102 3.6.5 SIC-LDA for Cerium . 106 3.6.6 Magnetic Solution of Cerium . 108 4 Hydrogen Immersed in a Finite Jellium Sphere 111 4.1 Hydrogen in Finite Jellium Spheres using the LDA . 111 4.1.1 Energy of An Atom in a Finite Jellium Sphere . 112 4.1.2 Filling of Orbitals . 114 4.1.3 Applying SIC to a Hydrogen Atom in a Finite Jellium Sphere . 115 4.1.4 Results . 117 4.2 Hydrogen in Finite Jellium Spheres using VQMC . 126 4.2.1 The Choice of the Trial Wavefunction . 126 4.2.2 Calculating the Local Energy . 129 4.2.3 Results . 131 5 Conclusions 145 A Local Kinetic Energy Calculation for Atom in Jellium 149 xiv CONTENTS List of Tables 4.1 Total energies of hydrogen in 10-electron jellium spheres . 132 4.2 Total energies of 10-electron jellium spheres . 135 4.3 Immersion energies of hydrogen in 10-electron jellium spheres . 136 xv xvi LIST OF TABLES List of Figures 1.1 The probability of finding two electrons a separation |r| apart from one another for parallel and anti-parallel spins. The system is an electron gas solved using Hartree-Fock theory, and shows how the correlation between electrons due to exchange is captured by the theory (see the reduced prob- ability of two same spin electrons being close to one another) but the cor- relation due to the Coulomb interaction is not (no reduced probability in the different spin case) [1]. 3 1.2 Wigner-Seitz radii for transition metal elements as calculated by Moruzzi et al [2] using LDA for a full periodic solid (circles) and the experimental values (crosses). 6 1.3 Bulk moduli for transition metal elements as calculated by Moruzzi et al [2] using the LDA for a full periodic solid (circles) and the experimental values (crosses). 7 1.4 Band-gaps predicted by the LDA (triangles) are too small by up to 3eV compared to the experimental values (diamonds and circles) [3]. Squares are the GW approximation. 8 1.5 The model used. The full crystal is approximated as a positive ion of charge Z surrounded by the smeared out effective charge of all the surrounding ions. The assumption of spherical symmetry is made in the final step, which is consistent with our omission of details regarding the shape of the unit cell. Ω is the atomic volume, rWS is the Wigner-Seitz radius, nbs is the number of bound states per atom and nval is the number of valence electrons per atom. Charges are in units of the electron charge, e. 11 xvii xviii LIST OF FIGURES 1.6 The background density,n ¯i, in a given cell i is made up of the sum of the density tails of all the other atoms, averaged over cell i. This picture applies to the EAM and the EMT. Figure taken from a paper by Yxklinten et al [4]. 14 2.1 Re-blocking analysis for hydrogen immersed in a 10-electron jellium sphere −3 of density 0.03aB . The error on the mean levels off at just under 0.0031eV and therefore this is the error we quote on the total energy. 49 3.1 Phase-shifts (top panel) and the corresponding density of states (lower panel) for a cerium atom embedded in jellium of rs = 5.3 . 71 3.2 The l = 0 phase-shifts for a hydrogen atom immersed in infinite jellium of −3 −3 background densities 0.01aB (top panel) and 0.05aB (bottom panel). 72 3.3 The quantity dUoutwards(r = rmatch)/dr − dUinwards(r = rmatch)/dr (as described in the main text) for l = 0 is plotted as a function of energy. The system is a Technetium atom immersed in jellium of background density −3 0.03aB , and is non-magnetic, so the curve applies for both spin-up and spin- down electrons The l = 0 bound state energies are at the points where the curve crosses the x-axis, I.e.
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