My PhD thesis - Condensed Matter Theory - Imperial College London

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CHAPTER 4. ERRORS IN QMC SIMULATIONS neously an eigenfunction of Ĥ and of the set of all such translations: ĤΨ = EΨ (4.2) ˆT jR Ψ = T R Ψ for all R, j. (4.3) The eigenvalue T R does not depend on j because of the antisymmetry of the wave function. Two consecutive translations correspond to another single translation; it follows that T R T R ′ = T (R+R ′ ). (4.4) The eigenvalues therefore have the exponential form T R = e ik·R , (4.5) where at this stage the components of k are arbitrary complex numbers. 3 The unitarity of the translation operator means that the eigenvalues must have modulus 1, which in turn implies that k is in fact real. The result is just Bloch’s theorem: ˆT jR Ψ = e ik·R Ψ. (4.6) Thus, when the Hamiltonian operator is periodic in real space, the wave function must have the form ( Ψ({r i }) = exp ik · N∑ r i )U({r i }) (4.7) where ˆT jR U = U. The wave vector k may be reduced into the first Brillouin zone i=1 of the simulation cell: any remaining factors of ( exp iK · N∑ r i ), (4.8) where K is a simulation cell reciprocal lattice vector, are periodic and may be included in U. 3 More detail can be found in the book by Ashcroft and Mermin [4]; this section follows their proof of Bloch’s theorem. i=1 60
CHAPTER 4. ERRORS IN QMC SIMULATIONS There is an infinite number of permissible k-vectors in the first Brillouin zone; imposing strictly periodic boundary conditions on the wave function, rather than on Ĥ, corresponds to taking k = 0. However, there is no a priori reason why this particular k-point should be better than any other. In a non-interacting system, the single-electron wave functions are plane waves, and the infinite-system limit is obtained by integrating over all k- points. In an interacting system, this may also be the ideal approach [55]. The aim is always to choose the k-point(s) which best reproduce(s) the infinite-system result; several studies have investigated ways to achieve this goal in the context of QMC simulations [74, 39]. The use of alternative k-points is also discussed in chapter 9. Another way to correct this kind of finite-size error in QMC results is to apply a correction of the form (E DFT ∞ − E DFT N ), where EDFT N and E DFT ∞ are the DFT results for finite and infinite cells respectively. This is a valid procedure because finite-cell DFT calculations suffer from the same k-point sampling errors, and it is usually easier to extrapolate the results of DFT calculations to the infinite-cell limit, thus obtaining E∞ DFT . However, DFT calculations do not suffer from the second type of finite-size error: that related to the Coulomb interaction. 4.1.2 Coulomb finite-size errors A requirement of any QMC simulation is the ability to determine the Coulomb energy of a given configuration. In a finite system, this is trivial; in a finite system which is designed to model an infinite system, it is not. Referring to figure 4.1, each ‘real’ electron in the simulation cell must interact with all the others; there must also be some interaction between these electrons and the ‘imaginary’ electrons outside the simulation cell. The conventional approach to this problem is to solve Poisson’s equation for the charges in the cell with periodic boundary conditions. 4 The result is known as the 4 Note that the simulation cell as a whole must have zero net charge; all the electron charges must be cancelled by an equivalent amount of positive charge (provided by ions in real materials, 61
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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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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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