I are approximation to the single particle

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Unformatted text preview: Green’s function φ j (r)φ * (r′) j GDFT (r, r′; ε ) = ∑ j ε − ε j + iδ • Then, − 1 π εF Im ∫ d ε G DFT (r , r ′; ε ) εF 2 = − Im −iπ ∑ ∫ d ε δ (ε − ε i ) φi (r ) π i 1 r =r′ 1 1 Q = P − iπδ ( x), x = 0+ x + iδ x 17 or, − 1 π εF Im ∫ d ε G DFT (r , r ′; ε ) r =r′ = ∑ ε i <ε F 2 φi (r ) = n(r ) From many-body theory, the exact single particle Green’s function satisfies − 1 π εF Im ∫ d ε G (r , r ′; ε ) r = r ′ = n(r ) almost from its definiton ⇒ Thus, GDFT contains a great deal of G! • Equations the Green’s functions satisfy: (complex energy z) Full: { } − z − ∇ 2 + v(r ) + vH (r ) G ( r , r ′; z ) + ∫ dr ′′ ∑ (r , r ′′; z )G (r ′′, r ′; z ) = δ (r − r ′) r 18 DFT: { } 2 − z − ∇ r + v(r ) + vH (r ) G DFT ( r , r ′; z ) + v XC (r )G DFT (r , r ′; z ) = δ (r − r ′) Thus, vXC can be viewed as a local (r=r′), real, static approximation to self-energy Σ(r,r′;z). εi are approximation to the single particle excitations, a good mean field approximation to excitation energies. Both Σ(r,r′;z) and vXC give rise to the same density. 19 DFT& the Many Body Problem The Hamiltonian can be written H = T +V +UH = T + V +Vhxc + (UH – Vhxc) = H0 + ∆V where, of course, Vhxc could be anything. However, if we choose Vhxc = ∫ d 3 r [ vH ( r ) + v XC ( r ) ] n( r ) N = ∑ [ vH ( ri ) + v XC ( ri ) ] , where vH = i =1 20 δUH δ n( r ) then h 2...
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This document was uploaded on 03/12/2014 for the course PHYSICS 240C at UC Davis.

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