My PhD thesis - Condensed Matter Theory - Imperial College London

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Contents Table of Contents 4 List of Figures 8 List of Tables 15 1 Introduction 17 2 The simplification of many-electron quantum mechanics 20 2.1 First steps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 2.2 The single-electron approach . . . . . . . . . . . . . . . . . . . . . . . 22 2.3 Hartree-Fock theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 2.4 Density-functional theory . . . . . . . . . . . . . . . . . . . . . . . . . 25 3 Quantum Monte Carlo methods 30 3.1 Random sampling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 3.2 Variational Monte Carlo . . . . . . . . . . . . . . . . . . . . . . . . . 33 3.2.1 Importance sampling in VMC . . . . . . . . . . . . . . . . . . 34 3.2.2 The Metropolis algorithm . . . . . . . . . . . . . . . . . . . . 34 3.2.3 Advantages and disadvantages of VMC . . . . . . . . . . . . . 36 3.3 Diffusion Monte Carlo . . . . . . . . . . . . . . . . . . . . . . . . . . 36 3.3.1 Simulating classical diffusion . . . . . . . . . . . . . . . . . . . 37 3.3.2 Application to quantum mechanics . . . . . . . . . . . . . . . 38 3.3.3 The ground-state wave function . . . . . . . . . . . . . . . . . 40 4
CONTENTS 3.3.4 Importance-sampled diffusion Monte Carlo . . . . . . . . . . . 41 3.3.5 Fermions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 3.3.6 Estimators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 3.4 Trial wave functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 3.4.1 The Jastrow factor . . . . . . . . . . . . . . . . . . . . . . . . 53 3.4.2 Optimisation . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 4 Errors in QMC simulations 57 4.1 Finite-size errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 4.1.1 The independent-particle finite-size effect . . . . . . . . . . . . 58 4.1.2 Coulomb finite-size errors . . . . . . . . . . . . . . . . . . . . 61 4.2 Fixed-node errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 4.3 Statistical noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 4.4 Surface calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 4.4.1 System geometry . . . . . . . . . . . . . . . . . . . . . . . . . 67 4.4.2 The surface energy . . . . . . . . . . . . . . . . . . . . . . . . 69 5 The jellium slab 70 5.1 Defining the system . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 5.1.1 Constraining the electrons . . . . . . . . . . . . . . . . . . . . 71 5.2 The surface energy of jellium . . . . . . . . . . . . . . . . . . . . . . . 72 5.3 Preliminary investigations . . . . . . . . . . . . . . . . . . . . . . . . 76 5.4 The jellium slab in QMC . . . . . . . . . . . . . . . . . . . . . . . . . 80 6 The modified periodic Coulomb interaction in quasi-2D systems 84 6.1 The MPC interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 6.2 Quasi-2D systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 6.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 6.4 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 6.5 The exchange-correlation hole . . . . . . . . . . . . . . . . . . . . . . 94 5
Page 1 and 2: Improving the accuracy of surface s
Page 3: Acknowledgements I am greatly indeb
Page 7 and 8: CONTENTS 10 Conclusions 181 A The q
Page 9 and 10: LIST OF FIGURES 5.3 The LDA energy
Page 11 and 12: LIST OF FIGURES 8.4 The x- and z-co
Page 13 and 14: LIST OF FIGURES 8.20 Electron densi
Page 15 and 16: List of Tables 8.1 The function F k
Page 17 and 18: Chapter 1 Introduction In recent de
Page 19 and 20: CHAPTER 1. INTRODUCTION A successfu
Page 21 and 22: CHAPTER 2. MECHANICS THE SIMPLIFICA
Page 31 and 32: CHAPTER 3. QUANTUM MONTE CARLO METH
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CHAPTER 3. QUANTUM MONTE CARLO METH
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Chapter 4 Errors in QMC simulations
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CHAPTER 4. ERRORS IN QMC SIMULATION
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CHAPTER 5. THE JELLIUM SLAB The jel
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CHAPTER 5. THE JELLIUM SLAB 0.08 0.
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CHAPTER 5. THE JELLIUM SLAB The DMC
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CHAPTER 5. THE JELLIUM SLAB -9.125
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CHAPTER 5. THE JELLIUM SLAB -0.3 Su
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CHAPTER 5. THE JELLIUM SLAB 0.14 be
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CHAPTER 5. THE JELLIUM SLAB At the
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CHAPTER 6. THE MODIFIED PERIODIC CO
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CHAPTER 7. THE ELECTRONIC GROUND-ST
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CHAPTER 8. APPLYING THE PLASMON NOR
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CHAPTER 9. ENERGY A NEW CALCULATION
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Chapter 10 Conclusions The original
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CHAPTER 10. CONCLUSIONS pling and o
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APPENDIX A. THE QUASI-2D EWALD SUM
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APPENDIX B. THE CUSP CONDITIONS may
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APPENDIX B. THE CUSP CONDITIONS wit
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APPENDIX C. INTEGRATING THE CUSP FU
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APPENDIX C. INTEGRATING THE CUSP FU
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APPENDIX D. RECONSTRUCTING A PROBAB
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Bibliography [1] P. H. Acioli and D
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BIBLIOGRAPHY [19] A. G. Eguiluz and
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BIBLIOGRAPHY [39] P. R. C. Kent, R.
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BIBLIOGRAPHY [59] H. J. Monkhorst a
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BIBLIOGRAPHY [80] C. J. Umrigar, K.
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My PhD thesis - Condensed Matter Theory - Imperial College London

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