My PhD thesis - Condensed Matter Theory - Imperial College London

More documents

Recommendations

Info

CHAPTER 9. ENERGY A NEW CALCULATION OF THE JELLIUM SURFACE 5 4 VMC fitted quadratic 3 2 Energy (mHa) 1 0 -1 -2 -3 -4 1.4 1.6 1.8 2 2.2 2.4 2.6 α Figure 9.5: Optimising the parameter α. The VMC energies are shown along with the quadratic function fitted to them. These results are for a slab width of 11.7783 with a cell containing 588 electrons; the minimum is at α = 2.26. the difference has the correct sign to be associated with the additional Coulomb finite-size error arising from the periodicity in the z-direction. More computationally-intensive DMC simulations were also carried out for some of the smaller systems: the results are listed in table 9.5. 9.3.2 Analysis The best estimate of the jellium surface energy is obtained using figure 9.6. The graph illustrates that there is a significant finite-size effect: comparison with the DMC energies and the results of chapter 6 shows that this is due to the Jastrow factor. The Coulomb finite-size correction has a negative sign and is smaller in magnitude. The optimised Jastrow factor becomes significantly better for larger system sizes; the axes of the graph have been chosen to demonstrate that the error scales approx- 176
CHAPTER 9. ENERGY A NEW CALCULATION OF THE JELLIUM SURFACE -0.5 -1 -1.5 -2 ε slab (mHa) -2.5 -3 -3.5 -4 s = 11.7783 -4.5 s = 15.1317 s = 18.4851 -5 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 s/N Figure 9.6: The slab energy per electron, plotted against s/N, where s is the slab width and N is the number of electrons. The abscissa is proportional to 1/L 2 , and was chosen to demonstrate the form of the finite-size errors. imately as 1/L 2 . The reason that the Jastrow factor performs less well in small systems is almost certainly the parameter L c . This parameter, which keeps the wave function free of unwanted cusps, depends on the cell size, and is not allowed to be optimised. It modifies the long-range behaviour of the two-body term; it does not appear to affect the electron density adversely (by rendering the derived one-body term less efficient). Most importantly, this finite-size error is independent of the slab width: the evidence for this is the fact that the different curves in figure 9.6 remain equallyspaced. This allows an accurate calculation of the surface energy to be made. First, the curves s = 11.7783 and s = 15.1317 are interpolated on the abscissae of the third curve, s = 18.4851. Working with the largest possible system size (and thus effectively eliminating the Coulomb finite-size error), the slab energies per electron at this point are (in mHa) -3.713 ± 0.063, -4.060 ± 0.051 and -4.593 177
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:
CHAPTER 8. APPLYING THE PLASMON NOR
Page 125 and 126: CHAPTER 8. APPLYING THE PLASMON NOR
Page 167 and 168: CHAPTER 9. ENERGY A NEW CALCULATION
Page 175: 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 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

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?