My PhD thesis - Condensed Matter Theory - Imperial College London

More documents

Recommendations

Info

CHAPTER 10. CONCLUSIONS the electronic ground-state wave function. A new approach to this connection has been presented, based on physically intuitive ideas. The resulting prescription for the long-wavelength correlations in the wave function agrees with previous work; the value of the formalism presented here is that it is now much easier to incorporate prior knowledge about plasmon normal modes in any given system. This idea has been tested on the jellium slab system, with some success: the plasmon-predicted Jastrow factor reduces the energy in VMC. However, the strong long-wavelength correlations disrupt the electron density significantly, and although the predicted one-electron term almost restores the original (and presumably correct) density, this restoration is not perfect. In contrast, the short-ranged correlation term developed here has a much weaker effect on the density, and the corresponding analytic one-electron term is able to restore the original density almost exactly. One important conclusion is that it appears to be more important to have the correct density than to include all the long-wavelength correlations, and this makes the short-ranged Jastrow factor more attractive. Presumably, the best Jastrow factor would use both long- and short-ranged correlations, and take the predicted one-electron term as a starting point; adding some variational freedom to this term would then allow the correct density to be restored while maintaining all the correlations. However, the aim here was to minimise the need for optimisation, since this procedure is awkward for the jellium slab; therefore only the short-ranged correlation term (and the corresponding density-restoring function) were used in the calculations of the final chapter. The density-restoring term corresponding to the short-ranged correlation function was derived analytically, using a relationship which has been postulated by other authors and which is automatically satisfied by the plasmon Jastrow factor. This suggests that the relationship may hold generally in inhomogeneous systems, and can provide a way to make the optimisation process much more efficient: a starting-point for the one-electron term is provided analytically, depending on the two-body term. Two other techniques were shown to have little impact: alternative k-point sam- 182
CHAPTER 10. CONCLUSIONS pling and orbitals derived from a potential which includes the correct image-like asymptotic behaviour when an electron leaves the slab (in contrast to LDA orbitals). The fact that the finite-size error is not affected by alternative k-point sampling lends support to the ‘hole-squashing’ argument put forward to explain this error. The final calculation of the jellium surface energy draws on all the advances made here. While this is only a VMC result, it suggests that earlier extended-system QMC simulations were flawed; the new result is in line with density-functional theory and finite-system QMC calculations. The earlier calculations did not take careful account of the Coulomb finite-size effect, and this has been shown to lead to a significant error. However, the most important error comes from comparing the results of slab and bulk calculations; it appears that bulk wave functions are significantly better than those available for slabs, which leads to an overestimate of the surface energy. This is an important consideration for any future surface energy calculations; the error is magnified in jellium, because the surface energy itself is so small. 183
Page 1 and 2:
Improving the accuracy of surface s
Page 3 and 4:
Acknowledgements I am greatly indeb
Page 5 and 6:
CONTENTS 3.3.4 Importance-sampled d
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 23 and 24:
Page 25 and 26:
Page 27 and 28:
Page 29 and 30:
Page 31 and 32:
CHAPTER 3. QUANTUM MONTE CARLO METH
Page 33 and 34:
Page 35 and 36:
Page 37 and 38:
Page 39 and 40:
Page 41 and 42:
Page 43 and 44:
Page 45 and 46:
Page 47 and 48:
Page 49 and 50:
Page 51 and 52:
Page 53 and 54:
Page 55 and 56:
Page 57 and 58:
Chapter 4 Errors in QMC simulations
Page 59 and 60:
CHAPTER 4. ERRORS IN QMC SIMULATION
Page 61 and 62:
Page 63 and 64:
Page 65 and 66:
Page 67 and 68:
Page 69 and 70:
Page 71 and 72:
CHAPTER 5. THE JELLIUM SLAB The jel
Page 73 and 74:
CHAPTER 5. THE JELLIUM SLAB 0.08 0.
Page 75 and 76:
CHAPTER 5. THE JELLIUM SLAB The DMC
Page 77 and 78:
CHAPTER 5. THE JELLIUM SLAB -9.125
Page 79 and 80:
CHAPTER 5. THE JELLIUM SLAB -0.3 Su
Page 81 and 82:
CHAPTER 5. THE JELLIUM SLAB 0.14 be
Page 83 and 84:
CHAPTER 5. THE JELLIUM SLAB At the
Page 85 and 86:
CHAPTER 6. THE MODIFIED PERIODIC CO
Page 87 and 88:
Page 89 and 90:
Page 91 and 92:
Page 93 and 94:
Page 95 and 96:
Page 97 and 98:
Page 99 and 100:
Page 101 and 102:
Page 103 and 104:
Page 105 and 106:
Page 107 and 108:
CHAPTER 7. THE ELECTRONIC GROUND-ST
Page 109 and 110:
Page 111 and 112:
Page 113 and 114:
Page 115 and 116:
Page 117 and 118:
Page 119 and 120:
Page 121 and 122:
CHAPTER 8. APPLYING THE PLASMON NOR
Page 123 and 124:
Page 125 and 126:
Page 127 and 128:
Page 129 and 130:
Page 131 and 132: CHAPTER 8. APPLYING THE PLASMON NOR
Page 167 and 168: CHAPTER 9. ENERGY A NEW CALCULATION
Page 181: Chapter 10 Conclusions The original
Page 185 and 186: APPENDIX A. THE QUASI-2D EWALD SUM
Page 193 and 194: APPENDIX B. THE CUSP CONDITIONS may
Page 195 and 196: APPENDIX B. THE CUSP CONDITIONS wit
Page 197 and 198: APPENDIX C. INTEGRATING THE CUSP FU
Page 199 and 200: APPENDIX C. INTEGRATING THE CUSP FU
Page 201 and 202: APPENDIX D. RECONSTRUCTING A PROBAB
Page 207 and 208: Bibliography [1] P. H. Acioli and D
Page 209 and 210: BIBLIOGRAPHY [19] A. G. Eguiluz and
Page 211 and 212: BIBLIOGRAPHY [39] P. R. C. Kent, R.
Page 213 and 214: BIBLIOGRAPHY [59] H. J. Monkhorst a
Page 215: BIBLIOGRAPHY [80] C. J. Umrigar, K.
show all

My PhD thesis - Condensed Matter Theory - Imperial College London

Create successful ePaper yourself

Delete template?

Save as template?