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Lecture8 - Kwan-Woo Lee HK electron density contains in...

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Kwan-Woo Lee HK: “electron density contains in principle all the information contained in a many-electron wave function.” 1
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Hohenberg-Kohn Theorem 1 . V=V[n] V is a functional of ground state density n(r). Proof) Suppose for two systems with the same # of particles, and V (r) V(r) + const. (but that n (r)=n(r) .) • H| Ψ > = (H 0 + V)| Ψ > = ε | Ψ > • H | Ψ′ >=(H 0 + V )| Ψ′ >= ε′ | Ψ′ >, where | Ψ′ > | Ψ >. 2
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Then, ε = < Ψ |H| Ψ > < < Ψ′ |H| Ψ′ > ( Ψ′ is not the g.s. wave function of H.) RHS = < Ψ′ |H 0 + V + (V-V )| Ψ′ > = ε′ + < Ψ′ |V - V | Ψ′ > = ε′ + d 3 r n(r)[v(r) – v (r)] So, ε = ε′ + d 3 r n(r)[v(r) – v (r)] Exchange (primed unprimed) ε′ = ε + d 3 r n(r)[v (r) – v(r)] 3
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Adding two equations, ε′ + ε < ε + ε′ Contradiction n (r) n(r) ! Therefore, two different potentials cannot give the same ground state density. I.e., given n(r), Some particular v(r) , or v is a functional of n : V=V[n] Of course, also n is a finial of v , so v n is 1-to-1. 4
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2. In principle, any property of the system is a finial of n! (Each is its own functional.) In particular, the total energy of the system; ε = < Ψ g.s.|H| Ψ g.s.> = T[n] + U[n] + V[n] = ε [n] : ground state energy K.E. P.E. interaction with external potential 5
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Also, the density that minimizes ε [n] for a fixed number of electrons is the ground state density : . . [ [ ] [ ]] | 0 ( ) g s n n N n n r δ ε µ δ = [ ] . ( ) ( ) 0 ( ) ( ) n const n r T U v r n r n r δε µ δ δ δ µ δ δ = = + + = 6
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