My PhD thesis - Condensed Matter Theory - Imperial College London

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CHAPTER 8. APPLYING THE PLASMON NORMAL MODE THEORY TO SLAB SYSTEMS 1 0.8 0.6 0.4 0.2 0 0 1 2 3 4 ∆r ll 5 6 7 8 -4 -3 -2 -1 0 1 z 2 3 4 Figure 8.7: The fixed electron is at z ′ = 1.0: inside the slab but close to the surface. the Coulomb interaction is subsumed into Ĥpl; the remaining screened interaction stays in Ĥsr. The Schrödinger equation has been effectively separated; the groundstate eigenfunction is then a product of the form Ψ pr = Ψ pl Ψ sr . (8.102) The basis for the separation is the physical observation that plasmons exist. The short-range wave function does not contain information about the longrange correlations between electrons; however, it should contain the short-range correlations. The conventional trial wave functions used in QMC have Ψ sr = D ↑ D ↓ , (8.103) where the determinants are made up of orbitals obtained from a mean-field calculation. This wave function does not include the necessary information about short- 144
CHAPTER 8. APPLYING THE PLASMON NORMAL MODE THEORY TO SLAB SYSTEMS 1 0.8 0.6 0.4 0.2 0 0 1 2 3 4 ∆r ll 5 6 7 8 -4 -3 -2 -1 0 1 z 2 3 4 Figure 8.8: The fixed electron is now just outside the slab at z ′ = −1.0; the bulk plasmons no longer contribute to u ∞ pl . range correlation between electrons; this must therefore be introduced artificially. The behaviour of the wave function when two electrons are in close proximity is analysed in appendix B; the divergence in the potential energy in this limit must be compensated by a cancelling divergence in the kinetic energy, which can only arise as a consequence of a cusp (gradient discontinuity) in the wave function. The cusp conditions, originally due to Kato [37], specify only the gradient with respect to the separation of the two electrons in the limit of this separation tending to zero; they do not determine the value of the wave function. In appendix B, it is shown that a wave function with the desired short-range behaviour is [ Ψ = exp − 1 2 ∑ i≠j ] u cusp (r i , r j ) Ψ smooth , (8.104) 145
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Improving the accuracy of surface s
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Acknowledgements I am greatly indeb
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CONTENTS 3.3.4 Importance-sampled d
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CONTENTS 10 Conclusions 181 A The q
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LIST OF FIGURES 5.3 The LDA energy
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LIST OF FIGURES 8.4 The x- and z-co
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LIST OF FIGURES 8.20 Electron densi
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List of Tables 8.1 The function F k
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Chapter 1 Introduction In recent de
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CHAPTER 1. INTRODUCTION A successfu
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CHAPTER 2. MECHANICS THE SIMPLIFICA
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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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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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