Lecture8

# Ngs and gs thus the problem is reduced to a self

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Unformatted text preview: Using, where necessary, δ = δ n δ = φ j δ if φ j is occupied. δφ * δφ * δ n δn j j Constraining to normalized orbitals N δ ε [n] − ∑ ε j * δφi (r ) j =1 (∫ d r φ 2 3 j ) −1 = 0 h 2∇ 2 δE φi + vh (r )φi + v(r )φi + XC φi = ε iφi ⇒− 2m δ n( r ) vh(r;n) vXC(r;n) 8 If the resulting density is the same as the density, then the finial is minimized. ⇒ ngs and εgs Thus, the problem is reduced to a self-consistent field problem. h2 2 ∇ + veff (r , n) φi = ε iφi : Kohn-Sham equation − 2m N ⇒ n(r ) = ∑ φi (r ) 2 i =1 But, (ⅰ) this looks like a system of non-interacting particles in an effective potential. 9 (ⅱ) it is really a many-body theory for εgs and ngs. (ⅲ) εi → εkn is the band structure. (come back to this later.) Self-consistent Kohn-Sham equations 1. Initial guess: nσ(r) 2. Calculate effective potential: Vσeff(r) 3. Solve KS equation 4. Calculate electron density: nσ(r)=Σi fσi|Ψσi(r)|2 5. Self-Consistent? (No, go to 1; Yes, output .) 10 The Ex...
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## This document was uploaded on 03/12/2014 for the course PHYSICS 240C at UC Davis.

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