My PhD thesis - Condensed Matter Theory - Imperial College London

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CHAPTER 4. ERRORS IN QMC SIMULATIONS To understand the origin of the finite-size error, it is helpful to separate E EW e−e into Hartree and exchange-correlation terms. For this analysis, the two-electron density is required: ∫ n(r, r ′ ) = cell|Ψ(X)| ∑ 2 δ(r − r i )δ(r ′ − r j ) dX. (4.20) i≠j With this definition, the electron-electron energy becomes Ee−e EW = 1 ∫∫ n(r, r ′ )v E (r − r ′ ) dr dr ′ + 1 Nξ. (4.21) 2 2 cell If the electrons were completely uncorrelated, then the two-electron density n(r, r ′ ) would simply be the product of the one-electron densities n(r) and n(r ′ ). However, the electrons are not uncorrelated; the relationship between one- and two-electron densities defines the exchange-correlation hole: n(r, r ′ ) = n(r)n(r ′ ) + n(r)n XC (r, r ′ ). (4.22) The exchange-correlation hole describes the way that electrons avoid each other; the reduction in the electron density at r when one electron is fixed at r ′ is described by n XC (r, r ′ ). Integrating equation (4.22) with respect to r ′ reveals the important property ∫ cell n XC (r, r ′ ) dr ′ = −1. (4.23) This is a manifestation of the fact that the hole consists of the absence of a single electron from the overall density. Applying this to equation (4.21) gives Ee−e EW = 1 ∫∫ n(r)n(r ′ )v E (r − r ′ ) dr dr ′ + 1 ∫∫ 2 2 cell = U Ha + U EW XC . cell n(r)n XC (r, r ′ )[v E (r − r ′ ) − ξ] dr dr ′ (4.24) The first term, the Hartree energy, is the classical self-interaction energy per simulation cell of a static periodic charge density n(r) (compare equation (2.11)). The second term is the interaction energy of the electron with the exchange-correlation hole; this dynamical correction appears because the electron motions are correlated. 64
CHAPTER 4. ERRORS IN QMC SIMULATIONS The Ewald interaction v E is the correct interaction to use for the Hartree energy, 6 but not for the exchange-correlation energy. The reason for this is that the entire exchange-correlation hole (a total charge deficit of one electron) is contained within the simulation cell; there should be no images of the exchange-correlation hole. The Ewald interaction, which includes the effects of image charges, is therefore inappropriate for the calculation of U XC . 4.2 Fixed-node errors After many time steps, the walkers in a conventional DMC simulation are distributed according to the fixed-node density Ψ T Ψ FN 0 rather than the desired density Ψ T Ψ 0 . Any fixed-node estimate of the ground-state energy must be variational: E FN 0 ≥ E 0 . The equality holds only when the nodes are exact; while this may be achievable for a one-electron system, it is almost impossible in many-electron calculations. An indication of the difficulty in correctly guessing the nodal surface is the high dimensionality of that surface: (Nd − 1), for a system of N electrons moving in d dimensions. It is not possible to deduce the nodal surface from the condition that the wave function be zero when two electrons coincide: this defines a surface of only (N − 1)d dimensions. The fixed-node approximation is uncontrolled; the size of the error it introduces cannot be calculated analytically. It is not surprising that a great deal of time and effort has been devoted to overcoming this problem, with limited success. The release-node algorithm of Ceperley and Alder [12] uses separate populations of positive and negative walkers which are allowed to cross the nodes of the trial wave function. The problem with this method is that both walker populations grow geometrically in time, leading to exponentially-increasing statistical fluctuations; the method becomes a race to obtain convergence to the ground state before the 6 Although the Hartree energy defined by equation (4.24) depends on the value of ξ, the total energy does not, because of the corresponding terms in the ion-ion energy. The exchange-correlation energy defined by the same equation does not depend on ξ. 65
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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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CHAPTER 7. THE ELECTRONIC GROUND-ST
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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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