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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
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 )
+ 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]
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 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.
- Spring '12