My PhD thesis - Condensed Matter Theory - Imperial College London

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CHAPTER 6. THE MODIFIED PERIODIC COULOMB INTERACTION IN QUASI-2D SYSTEMS similar to a conditional probability. The exchange-correlation hole for an electron at r ′ is then defined by the formula n(r|r ′ ; N) = n(r; N) + n XC (r, r ′ ; N). (6.75) In the calculations above, the external charge density was chosen to model an electron; after fixing this electron at r ′ , the change to the density of the remaining free electrons was estimated: δn ind (r|r ′ ; N) ≈ n(r|r ′ ; N + 1) − n(r; N). (6.76) Note the change from N to N + 1 in the conditional density, which arises because an electron has been added to the system. Combining these equations then gives n XC (r, r ′ ; N + 1) = n(r|r ′ ; N + 1) − n(r; N + 1) ≈ δn ind (r|r ′ ; N) + n(r; N) − n(r; N + 1). (6.77) For large N, adding an extra electron should not change the density dramatically. The following approximation may therefore be used: ( ) N + 1 n(r; N + 1) ≈ n(r; N). (6.78) N The exchange-correlation hole is then n XC (r, r ′ ; N + 1) ≈ δn ind (r|r ′ ; N) − 1 n(r; N). (6.79) N This function is plotted in figure 6.7, which shows the characteristic spreading-out of the hole near the surface. As in the idealised system, the surface hole is curtailed by the cell boundary, although the effect is weaker. This supports the idea that the remaining finite-size error is caused by the exchange-correlation hole being squashed into the simulation cell. This error would act to reduce the energy of the system, and therefore has the correct sign. 104
CHAPTER 6. THE MODIFIED PERIODIC COULOMB INTERACTION IN QUASI-2D SYSTEMS 0 -5e-05 -1e-04 -0.00015 n XC -0.0002 -0.00025 -0.0003 -0.00035 -0.0004 5 6 7 8 9 10 x middle, L = 20 middle, L = 40 middle, L = 80 surface, L = 20 surface, L = 40 surface, L = 80 Figure 6.7: The exchange-correlation hole for a realistic slab. 105
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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 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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