My PhD thesis - Condensed Matter Theory - Imperial College London

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CHAPTER 2. MECHANICS THE SIMPLIFICATION OF MANY-ELECTRON QUANTUM and now contains all the difficult parts of E. T s [n] is the kinetic energy of a noninteracting system of density n, and is therefore easily calculable: T s [n] = − 1 2 N∑ ∫ i=1 ψ i (r)∇ 2 ψ i (r) dr. (2.38) The ψ i are the occupied single-electron orbitals of the non-interacting system. This system has been defined to have density n, giving N∑ ∑ |ψ i (r)| 2 = n(r). (2.39) i=1 σ E[n] must be minimised subject to the constraint ∫ n(r) dr = N: This gives where ( ∫ δ E[n] − µ δn(r) ∫ v s (r) = v ext (r) + with the exchange-correlation potential µ = v s (r) + δT s[n] δn(r) ) n(r) dr = 0. (2.40) (2.41) n(r ′ ) |r − r ′ | dr′ + v XC (r) (2.42) v XC (r) = δE XC[n] δn(r) . (2.43) This is exactly equivalent to the result which would be obtained for a non-interacting system with the external potential v s . The required density may therefore be calculated by solving the one-electron equations ( − 1 ) 2 ∇2 + v s (r) ψ i (r) = ɛ i ψ i (r) (2.44) and using equation (2.39). Because v s itself depends on n, equations (2.42), (2.39), and (2.44) must be solved self-consistently. Having obtained n, the energy E[n] may be calculated immediately, as long as E XC is known. This brief outline of the theory has not addressed the important question of establishing the existence of a non-interacting system whose density is exactly equal 28
CHAPTER 2. MECHANICS THE SIMPLIFICATION OF MANY-ELECTRON QUANTUM to that of the interacting system being studied. This and other issues too detailed to belong here are described in the books by Dreizler and Gross [18] and Parr and Yang [64]. Unfortunately, E XC is not known, although various approximations and parameterisations exist. One possibility is to apply the local density approximation (LDA), in which the exchange-correlation energy functional is taken to be ∫ EXC LDA [n] = n(r)ɛ XC (n(r)) dr (2.45) where ɛ XC (n) is the exchange-correlation energy per electron of a uniform electron gas with density n. This approximation would be expected to give reasonable results for slowly-varying densities; however, it also works for a surprising number of strongly-inhomogeneous systems. The next logical step is to allow ɛ XC to depend on ∇n as well as n: this is the generalised gradient approximation (GGA) [50, 8, 67]. Higher derivatives may also be included, giving the meta-GGA [68]. In fact, even ɛ XC (n) is not known analytically. The parameterised versions being used today are based on theoretical predictions combined with results from quantum Monte Carlo calculations [70]. 29
Page 1 and 2: Improving the accuracy of surface s
Page 3 and 4: Acknowledgements I am greatly indeb
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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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CHAPTER 5. THE JELLIUM SLAB At the
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CHAPTER 6. THE MODIFIED PERIODIC 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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