Lecture8

# E to vv1v1 and full coulomb repulsion denote h t v

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Unformatted text preview: change-Cor relation Energy Functional “Adiabatic Connection”: Coupling constant integration Introduce λ in vC ,λ e2 =λ r − r′ , scaling of Coulomb repulsion. In DFT, functionals depend only on n and e2, ħ, m. With λ, DFT holds again, but functionals depends on λ. Thus, for a given n, there exists an external potential vλ that gives n as the ground state density. Then, let n(r) be the ground state density corresponding to λ=1, 11 i.e., to v≡vλ=1≡v1 and full Coulomb repulsion. Denote Hλ = T + Vλ + λU. Here, Vλ≡∫d3r vλ(r) n(r), where at each λ, vλ is that potential which gives n, i.e., n(r) is fixed as λ varies between 0 and 1. Also, HλΨλ = ελΨλ 12 Hellman-Feynman theorem gives ∂H λ ∂Vλ = +U ∂λ ∂λ and 1 Also, λ=0: ∫ dλ 0 dελ ∂H λ ψλ = ψλ dλ ∂λ . ∂ε λ = ε1 − ε 0 . ∂λ ε0 = T[n] + ∫d3r v0 n(r) = Ts[n] + ∫d3r v0 n(r): non-interacting system λ=1: ε1 = Ts[n] + ∫d3r v1 n(r) + UH[n] + EXC[n] (By definition, EXC contains T-Ts and U-UH. EXC can be decomposed as EX+EC, where EX is due to Pauli principles and EC is due to correlations.) 13 Then, • ε1-ε0= ∫d3r v1 n(r) - ∫d3r v0 n(r) + UH[n]...
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## This document was uploaded on 03/12/2014 for the course PHYSICS 240C at UC Davis.

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