My PhD thesis - Condensed Matter Theory - Imperial College London

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LIST OF FIGURES 6.2 The total energy per electron as a function of system size, obtained using fixed-node DMC. . . . . . . . . . . . . . . . . . . . . . . . . . . 91 6.3 The time taken to calculate the electron-electron interaction during an 8000-move VMC simulation, comparing the quasi-2D versions of the MPC and Ewald interactions. . . . . . . . . . . . . . . . . . . . . 92 6.4 The charge disturbance created by an electron-like test charge at the surface (z 0 = 0) and in the bulk (z 0 = 10). The plots show the value of δn ind (r) at the points (x, 0, z 0 ) (labelled x) and (0, 0, z 0 + z) (labelled z). In the bulk, the x and z curves are identical. . . . . . . . 99 6.5 The effect of cell size on the shape of the hole. Systems of size L = 20, L = 40 and L = 80 are compared at a distance of 10 from the centre of the imposed charge; this corresponds to the edge of the smallest cell, where any difference should be most clearly visible. The curves for the bulk hole at each of the three sizes are indistinguishable on this scale. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 6.6 The Thomas-Fermi hole for different cell sizes in a more realistic model of the slab. The induced electron density δn ind is plotted at the points (x, 0, z 0 ), where z 0 corresponds to the slab centre (labelled ‘middle’) or the conventional slab edge (labelled ‘surface’). . . . . . . 102 6.7 The exchange-correlation hole for a realistic slab. . . . . . . . . . . . 105 8.1 The metal slab. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 8.2 The surface charge density. . . . . . . . . . . . . . . . . . . . . . . . . 125 8.3 The dispersion relation for a slab with s = 200 and density parameter r s = 2.0 (in Hartree atomic units). This density corresponds to a plasma frequency ω p = 0.6124. The crosses represent the results of applying a numerical root-finding algorithm to equation (8.39); the smooth lines correspond to the small-k approximation given in equation (8.42). In the limit of large k, ω tends to ω p / √ 2. . . . . . . 128 10
LIST OF FIGURES 8.4 The x- and z-components of the longitudinal electric field associated with the surface plasmon oscillations. . . . . . . . . . . . . . . . . . . 131 8.5 The dispersion relation for surface plasmons in the electrostatic theory. The slab width is 20 au. For metallic densities, the Fermi wave vector is of order 1, whereas k p ∼ 0.01. The plasmon frequency very rapidly reaches the large-k limit of ω p / √ 2; this is the result obtained for a semi-infinite system, and indicates that the coupling between surface plasmon modes on the two sides of the slab is weaker than that obtained when using the full dynamical theory. . . . . . . . . . . 136 8.6 The plasmon contribution to the two-body function in the Jastrow factor. In this and subsequent graphs, one electron is fixed (here at z ′ = 2.0, inside the slab); the plot shows the dependency of u ∞ pl on the parallel separation ∆r ‖ and the z-coordinate of the other electron. 143 8.7 The fixed electron is at z ′ = 1.0: inside the slab but close to the surface.144 8.8 The fixed electron is now just outside the slab at z ′ = −1.0; the bulk plasmons no longer contribute to u ∞ pl . . . . . . . . . . . . . . . . . . . 145 8.9 The fixed electron is now further outside the slab (z ′ = −2.0), and the strength of the correlation is reduced. . . . . . . . . . . . . . . . . 146 8.10 Removing the cusps from u pl at the slab boundaries. In each plot, one electron is fixed while the other is scanned along a line in the z-direction; the x- and y-coordinates are chosen to be the same, so that ∆r ‖ = 0. The smoothed and unsmoothed curves are almost indistinguishable when the electrons are far from the slab edges; the most pronounced difference appears when one electron is just inside the slab edge (the black curves). This and the following plots were calculated for a cell containing 600 electrons, with s = 17.64248 and r s = 2.07 (in Hartree atomic units). . . . . . . . . . . . . . . . . . . 149 8.11 Removing the cusps from χ bulk at the slab boundaries; χ s bulk is compared with χ bulk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150 11
Page 1 and 2: Improving the accuracy of surface s
Page 3 and 4: Acknowledgements I am greatly indeb
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CHAPTER 4. ERRORS IN QMC SIMULATION
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CHAPTER 5. THE JELLIUM SLAB The jel
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CHAPTER 5. THE JELLIUM SLAB 0.08 0.
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CHAPTER 5. THE JELLIUM SLAB The DMC
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CHAPTER 5. THE JELLIUM SLAB -9.125
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CHAPTER 5. THE JELLIUM SLAB At the
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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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