My PhD thesis - Condensed Matter Theory - Imperial College London

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Chapter 6 The modified periodic Coulomb interaction in quasi-2D systems Finite-size errors related to the implementation of the Coulomb interaction in periodic systems were described in chapter 4. In systems with 3D periodicity, it has been shown [25] that using the Ewald interaction introduces an error which decays as the reciprocal of the number of electrons in the system. An effective cure for this finite-size error has been found: the model periodic Coulomb (MPC) interaction [84], which has the additional benefit of being significantly less computationally expensive than the Ewald sum. In this chapter, the theory of the MPC interaction will be outlined, and the extension of the interaction to quasi-2D systems will be investigated. 6.1 The MPC interaction The analysis of the Coulomb finite-size error in section 4.1.2 is not specific to systems with 3D periodicity. There, the problem was shown to be caused by the use of the Ewald interaction in the exchange-correlation energy: UXC EW = 1 ∫∫ n(r)n XC (r, r ′ )[v E (r − r ′ ) − ξ] dr dr ′ . (6.1) 2 cell 84
CHAPTER 6. THE MODIFIED PERIODIC COULOMB INTERACTION IN QUASI-2D SYSTEMS In fact, since the exchange-correlation hole is contained entirely within the simulation cell and converges to the shape it would have in an infinite system rapidly as the system size increases, the correct interaction should have the usual Coulomb 1/|r − r ′ | form, with the proviso that the vector (r − r ′ ) is first reduced into the Wigner-Seitz cell of the simulation lattice. This reduction of (r − r ′ ) is known as the minimum image convention, and ensures that the interaction remains periodic (as it must). The correct interaction will be written f(r − r ′ ), where f(r − r ′ ) = 1 |(r − r ′ ) MI | . (6.2) The solution to the problem is therefore simply to replace [v E (r − r ′ ) − ξ] with f(r − r ′ ) in equation (6.1): U MPC XC = 1 2 ∫∫ The total electron-electron interaction energy is then cell n(r)n XC (r, r ′ )f(r − r ′ ) dr dr ′ . (6.3) E MPC e−e = U Ha + UXC MPC = 1 ∫∫ n(r, r ′ )f(r − r ′ ) dr dr ′ 2 cell + 1 2 ∫∫ n(r)n(r ′ )[v E (r − r ′ ) − f(r − r ′ )] dr dr ′ . (6.4) cell The ground-state wave function Ψ associated with this expression for the electronelectron energy must minimise the total energy E[Ψ] = 〈Ψ| ˆT + ˆV ext |Ψ〉 + E MPC e−e [Ψ], (6.5) where ˆT and ˆV ext are operators for the kinetic energy and the external potential. At the same time, normalisation of Ψ must be preserved; this yields the variational principle δ ( E[Ψ] − λ〈Ψ|Ψ〉 ) = 0 (6.6) 85
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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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