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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 Constraining to normalized orbitals
ε [n] − ∑ ε j
δφi (r ) j =1 (∫ d r φ 2 3 j ) −1 = 0 h 2∇ 2
φi + vh (r )φi + v(r )φi + XC φi = ε iφi
δ n( r )
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
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.
- Spring '12