My PhD thesis - Condensed Matter Theory - Imperial College London

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CHAPTER 6. THE MODIFIED PERIODIC COULOMB INTERACTION IN QUASI-2D SYSTEMS 0.00015 1e-04 5e-05 δn ind 0 -5e-05 -0.0001 -0.00015 -0.0002 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 6.6: 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’). the 3D lattice and that the Θ-functions now also become periodic. The transform of δ ˜ρ ext in the z-direction appearing in equation (6.68) is then ∫ w e ik′ zz ′ δ ˜ρ ext (k ‖ , z ′ ) dz ′ = − 1 (k 2 0 L 2 e−σ2 ‖ +k′2 z )/2+ik zz ′ 0 + 1 ) zs/2( sin(k ′ z s/2) L 2 eik′ δ k zs/2 ′ k‖ ,0. (6.69) Combining this with equations (6.30), (6.31), (6.34) and (6.68) gives the induced charge density: δn ind (r) = 4 ( ) 1/3 3n(z) ∑ e ∑ [ −ik·r ũ n (k z ) ∑ ũ ∗ L 2 π λ k n n − k n(k z) ′ ‖ 2 k z ′ ( ) ] − e ik′ zs/2 sin(k ′ z s/2) δ k zs/2 ′ k‖ ,0 . e ik′ z z 0−σ 2 (k 2 ‖ +k′2 z )/2 (6.70) This is plotted in figure 6.6. On comparing figures 6.5 and 6.6, one major difference stands out immediately: far away from the imposed charge, the induced electron 102
CHAPTER 6. THE MODIFIED PERIODIC COULOMB INTERACTION IN QUASI-2D SYSTEMS density tends to zero in the idealised surface system, but to some non-zero positive value in the more realistic slab model. This is because the net change in the electron density is forced to be zero in the slab system (which is finite); since the electron density is reduced in the vicinity of the charge disturbance, it must be increased elsewhere. This constraint does not apply to the idealised surface system (which is infinite). cell: To see this, consider integrating the original differential equation (6.29) over the ∫ cell ∫ ∫ ∇ 2 δφ tot (r) d 3 r − 4π δn ind (r) d 3 r = −4π δρ ext (r) d 3 r. (6.71) cell cell The external charge density was chosen to be neutral overall, giving ∫ δρ ext (r) d 3 r = 0. (6.72) cell When the potential δφ tot is forced to be periodic, the first integral in equation (6.71) also gives zero. This is evident on substitution of the Fourier series representation: ∫ ∫ ∇ 2 δφ tot (r) d 3 r = ∇ ∑ 2 δ ˜φ tot (k)e −ik·r d 3 r cell cell k = − ∑ ∫ k 2 δ ˜φ tot (k) e −ik·r d 3 r (6.73) k cell = 0. Thus, for a finite system with periodic boundary conditions, the net change to the electron density is zero: ∫ cell δn ind (r) d 3 r = 0. (6.74) This effect becomes smaller as the system size increases, as can be seen in figure 6.6; in an infinite system like the idealised surface, it disappears. Since the aim is to investigate the exchange-correlation hole, it is helpful to establish a link between this entity and δn ind . For this, a new notation is required. The electron density in a system of N electrons at the point r is labelled n(r; N). When one electron is fixed at the position r ′ , the density at r is n(r|r ′ ; N); this is 103
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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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