My PhD thesis - Condensed Matter Theory - Imperial College London

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CHAPTER 6. THE MODIFIED PERIODIC COULOMB INTERACTION IN QUASI-2D SYSTEMS 0.035 -0.214 -0.2145 0.034 -0.215 -0.2155 VMC energy 0.033 0.032 -0.216 -0.2165 -0.217 Exchange energy 0.031 Exchange VMC (Ewald) VMC (MPC) -0.2175 -0.218 -0.2185 0.03 0 500 1000 1500 2000 Number of electrons in simulation cell -0.219 Figure 6.1: The total VMC energy per electron as a function of system size, comparing the Ewald and MPC interactions. The exchange energy given by equation (6.25) is also plotted (note the different vertical scale). Energies, as usual, are in Ha. cusp conditions (see section 3.4.1 and appendix B). Note that the geometry is such that χ(z i ) is symmetric about the centre of the slab. The value of the parameter L c is chosen to ensure that u(r ij ) decays to zero before r ij approaches the size of the simulation cell; this avoids introducing unwanted gradient discontinuities into the wave function. The single-electron orbitals which make up the determinants D ↑ and D ↓ are taken from LDA calculations. Both VMC and DMC simulations were performed; the results are shown in figures 6.1 and 6.2. Because of the difficulty of optimising the variational parameters in the Jastrow factor, the VMC simulations were carried out with no Jastrow factor in order to avoid introducing a bias caused by optimisation of varying quality. For 90
CHAPTER 6. THE MODIFIED PERIODIC COULOMB INTERACTION IN QUASI-2D SYSTEMS -0.0075 -0.008 Energy per electron -0.0085 -0.009 -0.0095 -0.01 Ewald MPC -0.0105 0 200 400 600 800 1000 Number of electrons in simulation cell Figure 6.2: The total energy per electron as a function of system size, obtained using fixed-node DMC. this reason, the exchange energy 2 is also plotted in figure 6.1: ∑ ∫∫ [ vE (r − r ′ ) − ξ ] φ ∗ n(r)φ n (r ′ )φ n ′(r)φ ∗ n ′(r′ ) dr dr ′ (6.25) E X = − ∑ n n ′ cell where φ n is the nth single-electron LDA orbital. Since the Jastrow factor cannot alter the location of the nodes, and therefore does not affect the fixed-node DMC energy, it was included in the DMC trial wave functions to improve efficiency. A correction of the form (E LDA ∞ − EN LDA ) has been applied to all the results to account for the independent-particle finite-size effect (see section 4.1.1). It is evident from figures 6.1 and 6.2 that the results obtained using the MPC and Ewald interactions are in good agreement, both displaying a fairly slow convergence with respect to the system size. This is unexpected, and contrasts with the 3D case, where the MPC interaction improves the rate of convergence significantly; the 2 This quantity is not the true exchange energy, which requires optimised Hartree-Fock orbitals; rather, it is a hybrid of the LDA and Hartree-Fock theory. 91
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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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