My PhD thesis - Condensed Matter Theory - Imperial College London

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Chapter 3 Quantum Monte Carlo methods The term ‘Quantum Monte Carlo’ embraces several different techniques for solving the quantum many-body problem, all of which employ random sampling. This chapter will (briefly) describe the theory behind two of these techniques: variational and fixed-node diffusion QMC. 1 Both are zero-temperature methods, and both are routinely used in the calculation of properties of real solids. An important aspect of the application of this theory — the general form of the trial wave function — will also be described. The aim of this chapter is not to present all the technical details associated with implementing a modern QMC algorithm; rather, the fundamental ideas which underlie the technique will be described, along with some of the more important aspects of the implementation. All the QMC calculations contained in this <strong>thesis</strong> were carried out using the CASINO program developed in Cambridge [61]. 1 These versions of QMC are discussed in detail in the review by Foulkes and coworkers [24], and, along with various others, in the book by Hammond et al. [30]. 30
CHAPTER 3. QUANTUM MONTE CARLO METHODS 3.1 Random sampling To calculate expectation values of observables in quantum mechanics, we need to evaluate expressions of the form: ∫ Ψ ∗ 〈Ô〉 = (X)ÔΨ(X) dX ∫ Ψ∗ (X)Ψ(X) dX , (3.1) where Ô is a Hermitian operator and X is a vector representing all the electron coordinates. For large systems, the dimension of the integrals is correspondingly large. Computation of such integrals by conventional quadrature methods becomes impractical, because the error in these calculations scales poorly in high dimensions. For example, the error in a Simpson’s Rule calculation scales as M −4/d , where M is the number of grid points and d is the dimensionality of the integral. As d increases, improving the accuracy of the calculation becomes prohibitively expensive. In contrast, the Monte Carlo method of integration generates a statistical error in the value of the integral which scales as M −1/2 , independent of the dimension. While this is worse than grid-based techniques when d is small, it is far superior for large d. One way of evaluating the integral ∫ I = g(r) dr (3.2) Ω is to sample a set of M random vectors {r i } from a distribution which is uniform over the region of integration Ω. Each vector is assigned a score g(r i ) and the integral is estimated as I ≈ Ω M M∑ g(r i ). (3.3) i=1 This is the Monte Carlo method of integration, and in this form, it is not very efficient. It can be much improved by using importance sampling. The integral is first rewritten as ∫ I = f(r)P (r) dr, (3.4) 31
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CHAPTER 5. THE JELLIUM SLAB At the
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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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