Lecture8

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Unformatted text preview: icular 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 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 = <Ψg.s.|H|Ψg.s.> = T[n] + U[n] + V[n] = ε[n] : ground state energy ⇓ K.E. ⇓ P.E. ⇓ interaction with external potential 5 Also, the density that minimizes ε[n] for a fixed number of electrons is the ground state density: δ δ n(r ) [ε [ n ] − µ N [ n ] ] |n g . s . = 0 δε [n] = µ = co n s t . δ n(r ) δT δU + + v(r ) − µ = 0 δ n(r ) δ n(r ) 6 Kohn-Sham procedure Separate ε[n] in a different way ε[n] = To[n] + UH[n] +∫d3r vn(r) + Exc[n] Here, UH[n]=1/2∬d rd r′ n(r)vc(r-r′)n(r′) : Hartree energy, 3 3 e2 vc (r ) ≡ r T0[n]= K.E. of non-interacting system with density n, i.e., with N n(r ) = ∑ φi (r ) , 2 i =1 N h 2∇ 2 h2 3 T0 [n] ∑ ∫ d r φ − φi =∑ ∫ d r − ∇φi i =1 i =1 2m 2m N 3 * i 7 2 Now, minimize ε[n] w.r.t. ortials,...
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This document was uploaded on 03/12/2014 for the course PHYSICS 240C at UC Davis.

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