My PhD thesis - Condensed Matter Theory - Imperial College London

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CHAPTER 8. APPLYING THE PLASMON NORMAL MODE THEORY TO SLAB SYSTEMS Several standard integrals involving Bessel functions have been used. In the limit s → ∞, the surface term simplifies to u s→∞ surf (r, r ′ ) = √ 2e 2 ɛ 0 ω p L 2 ∑ k ‖ 1 k ‖ cos k ‖ · ∆r ‖ e −k ‖(|z|+|z ′ |) . (8.98) When the limit L → ∞ is also taken, the prescription of equation (8.95) for converting the summation to an integral gives u ∞ surf(r, r ′ ) = = = = √ 2e 2 ∫ ∞ ∫ π k dk dθ e−k ‖(|z|+|z ′ |) cos k 4π 2 ‖ · ∆r ‖ ɛ 0 ω p 0 0 k √ 2e 2 ∫ ∞ ∫ π dk e −k ‖(|z|+|z ′ |) dθ cos ( k 4ɛ 0 π 2 ‖ ∆r ‖ cos(θ − φ) ) ω p 0 0 √ 2e 2 ∫ ∞ ( ) dk e −k ‖(|z|+|z ′ |) πJ 4π 2 0 k‖ ∆r ‖ ɛ 0 ω p 0 √ 2e 2 √ 4πɛ 0 ω p (|z| + |z′ |) 2 + (∆r ‖ ) . (8.99) 2 The full plasmon two-body function in this limit is therefore [ ( ) u ∞ pl (r, r ′ e 2 ) = Θ(z)Θ(z ′ 1 ) 4πɛ 0 ω p |r − r ′ | − 1 √ (z + z′ ) 2 + (∆r ‖ ) 2 √ ] 2 + √ . (|z| + |z′ |) 2 + (∆r ‖ ) 2 (8.100) This function is plotted in figures 8.6, 8.7, 8.8 and 8.9. The contribution from the bulk plasmons is only relevant when both electrons are inside the metal; when the electrons are deep inside, u ∞ pl tends to the expected homogeneous electron gas form. The correlation is boosted for electrons closer to the boundary. The singularities present in equation (8.100) are a feature of the approximations which have been made; they do not appear in the original expressions relevant to a finite slab and cell. In any case, the plasmon theory is not expected to predict electron-electron correlation accurately at short range; the cusp conditions described in appendix B and discussed in the next section give more information about this regime. 142
CHAPTER 8. APPLYING THE PLASMON NORMAL MODE THEORY TO SLAB SYSTEMS 1 0.8 0.6 0.4 0.2 0 0 1 2 3 4 ∆r ll 5 6 7 8 -4 -3 -2 -1 0 1 z 2 3 4 Figure 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. 8.3 Creating a realistic Jastrow factor Equation (8.89) and (8.93) are the final results of applying the plasmon theory to the jellium slab system, and they are obtained from a consideration of the plasmon degrees of freedom only. The full electronic Hamiltonian (equation (2.9)) can be separated into long-range plasmonic and short-range terms: Ĥ = Ĥpl + Ĥsr. (8.101) The plasmon Hamiltonian Ĥpl is the one described in chapter 7 and appearing in different forms in equations (7.11), (7.16), (7.19) and (7.47). The long-range part of 143
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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 4 Errors in QMC simulations
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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 -0.3 Su
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CHAPTER 5. THE JELLIUM SLAB 0.14 be
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CHAPTER 5. THE JELLIUM SLAB At the
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CHAPTER 6. THE MODIFIED PERIODIC CO
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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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