My PhD thesis - Condensed Matter Theory - Imperial College London

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LIST OF FIGURES 8.12 The smoothed, cusp-corrected plasmon two-body function. The homogeneous curve is plotted for comparison. . . . . . . . . . . . . . . . 151 8.13 The one-body term χ cusp , for various values of the cut-off distance L c , calculated using equation (8.119); this equation was derived using the approximation of constant electron density within the slab. This term is designed to combat the density-changing effects of u cusp . As L c becomes large, the curves tend to a limit, because the decay of u cusp is then dominated by k c rather than L c . . . . . . . . . . . . . . . 154 8.14 The electron density profile, from LDA and VMC. The wave function used in VMC was a product of determinants made up of LDA orbitals, with no Jastrow factor; the densities are the same, to within the VMC noise. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156 8.15 The effect on the electron density of including a homogeneous twobody term of the form given in equation (8.113) in the wave function. The density becomes slightly more homogeneous, with more electrons being pushed into the vacuum regions outside the slab. . . . . . . . . 157 8.16 The electron density profile when the plasmon two-body term (u cusp + u s pl ) is included in the VMC wave function. The profile is completely different to the original curve, with almost all the electrons now at the slab edges. The change is much more pronounced than when the homogeneous (and short-ranged) u cusp alone was used. . . . . . . . . 158 8.17 The density profile when the full plasmon Jastrow factor is included in the VMC wave function. The result is very close to the original density, although not perfect. . . . . . . . . . . . . . . . . . . . . . . 159 8.18 The electron density profile for a Jastrow factor consisting of only the short-range two-body function u cusp and the corresponding one-body term χ cusp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160 8.19 Electron density profile for the bounded slab, in LDA and VMC (with no Jastrow factor). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161 12
LIST OF FIGURES 8.20 Electron density profile for the bounded slab, with a Jastrow factor containing only the short-ranged two-body term. The disturbance to the density is small; this is reflected in the VMC energy, which is significantly lower than when using no Jastrow factor. . . . . . . . . . 162 8.21 Electron density profile for the bounded slab, with the plasmon twobody term only. The long-ranged correlations alter the density drastically, as in the unbounded slab; electrons are pushed to the slab edges. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163 8.22 Electron density profile for the bounded slab, with the full plasmon Jastrow factor. The density is very close to the LDA form, and the energy is lower than when using only the short-ranged Jastrow. . . . 164 8.23 Electron density profile for the bounded slab, with the full shortranged Jastrow factor. The LDA density is almost restored, and the energy is lower than when using only u cusp , though not as low as that obtained with the full plasmon Jastrow factor. . . . . . . . . . . . . . 165 9.1 The effect of including the image tail in V XC . The simple LDA form (shown for comparison) is matched to the image potential outside the slab. The test system has density parameter r s = 2.07 and slab width s = 17.64248; the simulation cell contains 600 electrons. In addition to the exchange-correlation potential V XC , the Kohn-Sham potential V KS = V Ha + V B + V XC is also plotted. The positions of the image planes are indicated by ticks on the z-axis. . . . . . . . . . . . . . . . 168 9.2 The effect of the image tail on the highest occupied single-electron sub-band. The original orbital is shown, along with the difference between this and the modified orbitals (magnified one hundred times - note the two scales). . . . . . . . . . . . . . . . . . . . . . . . . . . 169 9.3 The electron density profile with periodic and antiperiodic boundary conditions for two different cell sizes. The infinite-cell limit is also shown (because this may be calculated in DFT). . . . . . . . . . . . 171 13
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CHAPTER 5. THE JELLIUM SLAB The jel
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CHAPTER 5. THE JELLIUM SLAB -9.125
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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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