My PhD thesis - Condensed Matter Theory - Imperial College London

More documents

Recommendations

Info

CHAPTER 8. APPLYING THE PLASMON NORMAL MODE THEORY TO SLAB SYSTEMS 50 40 smoothed unsmoothed 30 χ bulk (z) 20 10 0 0 s z Figure 8.11: Removing the cusps from χ bulk at the slab boundaries; χ s bulk is compared with χ bulk. taking this precaution would risk the introduction of yet more unwanted cusps in the wave function. A factor of e −r2 ij /L2 c is included for this purpose. The cut-off distance L c should be larger than kc −1 , to avoid interfering with the short-range behaviour, but significantly less than the Wigner-Seitz radius of the cell. The final form of the cusp function is ( ) ( ) me e 2 1 1 u cusp (x i , x j ) = e −kcr ij−r 4πɛ 0 2 ij 2 /L2 c . (8.113) 2k c 1 + δ σi σ j The full two-body function is plotted in figure 8.12, along with the usual homogeneous form (equation (3.79)) for comparison. The purpose of the two-body terms is to incorporate correlations into the wave function: principally to keep electrons apart. However, a secondary effect (in nonhomogeneous systems) is to alter the electron density. In the case of the slab, using only a u term forces electrons away from the centre, towards the slab edges; cluding correlations) are treated in the minimum-image scheme described in chapter 6. 150
CHAPTER 8. APPLYING THE PLASMON NORMAL MODE THEORY TO SLAB SYSTEMS 1.5 u hom 1 u →→ → upl → → u hom (∆r ll = 0, z, z’= s/2) (∆r ll = 0, z, z’= s/2) u pl → (∆r ll = 0, z, z’= -2.0) u pl → (∆r ll = 0, z, z’= 2.0) u pl → (∆r ll = 0, z, z’= s/2) (∆r ll = 0, z, z’= s/2) 0.5 0 -2 0 2 s z Figure 8.12: The smoothed, cusp-corrected plasmon two-body function. The homogeneous curve is plotted for comparison. the density spreads out more. This effect is undesirable, because the initial twodeterminant wave function usually gives an accurate estimate of the density. The χ function serves to correct this problem, restoring the original density. The plasmon χ function should be expected to correct the density changes created by u pl . It should not be expected to account for the additional effect of using u cusp ; thus, it is necessary to make a final modification to the wave function. Returning to equation (8.87), it is evident that the plasmon one-body function has the form ∫ χ bulk (r) = u bulk (r, r ′ )¯n(z ′ ) d 3 r ′ . (8.114) V To within a factor of (2N −1)/2N, this is in agreement with the findings of Malatesta et al. [56], which are based on a plausibility argument. The difference arises because in that work, the terms with i = j in the double sums used to define u are neglected; in the explicit plasmon formulation used here and in the work of Gaudoin et al. [28], it is more natural to include these terms. 151
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: CHAPTER 6. THE MODIFIED PERIODIC CO
Page 107 and 108: CHAPTER 7. THE ELECTRONIC GROUND-ST
Page 121 and 122: CHAPTER 8. APPLYING THE PLASMON NOR
Page 149: CHAPTER 8. APPLYING THE PLASMON NOR
Page 167 and 168: CHAPTER 9. ENERGY A NEW CALCULATION
Page 181 and 182: Chapter 10 Conclusions The original
Page 183 and 184: CHAPTER 10. CONCLUSIONS pling and o
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 203 and 204:
Page 205 and 206:
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?