The Kohn-Sham Equations • Applying the variational principle to ...

Computational ChemistryThe Kohn-Sham Equations• Applying the variational principle to determine the orbitals, withthe usual orthonormality constraints, leads to the Fock-likeKohn-Sham equationsVorlesung TC2ˆfψ i (x 1 ) = ǫ i ψ i (x 1 )ˆf = − 1 2 ∆2 + v ne + v j + v xcwhere the first three terms are the kinetic energy and Coulombterms as in Hartree-Fock.The exchange-correlation potential is defined viav xc = ∂E xc(ρ)∂ρ• Since the density depends on the orbitals, the KS equationsmust be solved iteratively until self-consistency is reached.

Computational ChemistryDensity functionalsVorlesung TC2• Local density approximation (LDA)– Treat density locally as a uniform electron gas:∫= −C x ρ 4/3 (r)drE LDAx– Additional terms for correlation energy of uniform electrongas (Vosko, Wilk, Nusair (VWN) functional, Can. J. Phys. 58(1980) 1200)• Generalized Gradient approximation (GGA)– Include terms that depend on the gradient of the density.– Exchange functionals: Perdew-Wang (PW86),Becke (B88), Perdew-Wang (PW91)– Correlation functionals: Lee, Yang, Parr (LYP),Perdew (P86), Perdew-Wang (PW91), Becke (B95)