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Lecture8

# d 3r nr xc r n g g r r n 15 e xc n

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Unformatted text preview: + EXC[n] Now, other term is ∂ε ∂Vλ dλ λ = ∫ dλ ψ λ +U ψλ ∫ ∂λ 0 ∂λ 0 1 • ε1-ε0 = 1 3 dvλ 1 3 3 ′n(r )vC (r − r ′)n(r ′) ψ λ n(r ) + ψ λ ∫∫ d rd r = ∫ dλ ∫ d r dλ 2 0 1 1 1 = ∫ d r {v1 ( r ) − v0 ( r )} n ( r ) + ∫ d λ ∫∫ d 3 r d 3 r ′vC ( r − r ′) ψ λ n ( r ) n ( r ′) ψ λ uuuuuuuuuuuuuuuuuuu u ru 20 3 ≡ n(r)n(r′)gλ(r,r′) gλ(r,r′) is pair correlation function. 14 Putting two equations together, 1 1 E XC [n] = ∫∫ d 3rd 3r ′v(r − r ′)n(r )n(r ′) ∫ d λ gλ (r , r ′) − U H [n] 2 0 e2 = 2 ∫∫ d rd r ′n(r ) 3 3 g (r , r ′) − 1 n(r ′) r − r′ 1 (Since g ( r , r ′) ≡ ∫ d λ g λ ( r , r ′) : Coupling-Constant averaged pair 0 correlation function.) ≡ ∫ d 3r n(r )ε XC (r ; n), {g = g (r, r′; n)} 15 ∴ E XC [n] ≡ ∫ d 3 r n(r )ε XC (r ; n) g (r , r ′) − 1 where ε XC (r ; n) ≡ ∫ d r ′ n(r ′) r − r′ 3 In the local density approximation (LDA), ε XC (r, n) → ε (n(r)) : homogenous system h XC Simple! Surprisingly good! 16 Meaning of eigen values εi, etc. KS equation: {-∇2 +v +vH+vXC}Φi=εiΦi Corresponding...
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