My PhD thesis - Condensed Matter Theory - Imperial College London

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APPENDIX A. THE QUASI-2D EWALD SUM Here, equation (A.27) has again been used to convert the reciprocal space sum to one in real space. To recover the original sum, we integrate from 0 to β: S(β) − S(0) = The two remaining integrals are ∫ β 0 = √ 2 ∑ π 2 √ β ∑ √ (1 − R 2 β)e −R2β dβ π R R≠0 R ∫ √ β 0 2t(1 − R 2 t 2 )e −R2 t 2 dt = √ 4 ∑ (I 2 − R 4 I 4 ) + 4 ∫ √ β √ t 2 dt. π π 0 (A.32) and I 4 = ∫ √ β 0 t 4 e −R2 t 2 dt = − β√ β 2R 2 e−R2β + 3 2R 2 I 2 (A.33) R≠0 ∫ √ β I 2 = t 2 e −R2 t 2 dt 0 √ √ β π ( = − 2R 2 e−R2β + 4R erf R √ ) β . 3 (A.34) Substituting these results into equation (A.32), and using the fact that S(0) = 0, S(β) = √ 4 ∑ [( √ β π 4R − β√ ) √ β π ( e −R2β − 2 2 8R erf R √ ) ] β + 4β√ β 3 3 √ π = 4β√ β 3 √ π − 1 2 ∑ R≠0 1 ( R erf R √ ) [ ] β + O e −L2 β . 3 (A.35) Returning to the original summation, ∑ ( ) πk σk A erfc = 2 k≠0 = 4 3σ 3√ π − 1 ∑ 2 R≠0 4 3σ 3√ π − C [ L + O e −L2 /σ 2] 3 ( ) 1 R R erf + O [e −L2 /σ 2] 3 σ (A.36) where the constant C is given by C = 1 2 ∞∑ ′ (m 2 + n 2 ) −3/2 (A.37) m,n=−∞ 190
APPENDIX A. THE QUASI-2D EWALD SUM and the prime here indicates that the term with m = n = 0 should be excluded from the sum. It may be shown [10] that this sum reduces to 4β(3/2)ζ(3/2) (A.38) where β and ζ are the Dirichlet beta and Riemann zeta functions respectively, giving C = 4.516810842. Combining equations (A.36) and (A.29) gives the final result: v E (r) − ξ = 1 r − C L 3 (z2 − r2 ‖ 2 ) + O [ r 4] [ + O e −L2 /σ 2] . (A.39) 191
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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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Page 215: BIBLIOGRAPHY [80] C. J. Umrigar, K.
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My PhD thesis - Condensed Matter Theory - Imperial College London

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