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Unformatted text preview: changeCor 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 HellmanFeynman 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): noninteracting system λ=1: ε1 = Ts[n] + ∫d3r v1 n(r) + UH[n] + EXC[n]
(By definition, EXC contains TTs and UUH. EXC can be decomposed as
EX+EC, where EX is due to Pauli principles and EC is due to correlations.)
13 Then,
• ε1ε0= ∫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.
 Spring '12
 WarrenE.Pickett
 Physics

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