Exact Exchange in Density Functional Calculations

5.3 Approximations to the Optimized Potential Method 33If the energy functional also depend on the KS eigenvalues, there would be an additional termδɛ i ∂E= ∑ i |φ i(r)| 2 ∂E/∂ɛ i on the right hand side of (5.12). The part∑iδv s(r) ∂ɛ i∫dr ′ G i (r, r ′ ) ( v(r ′ ) − ˆv NL) φ i (r ′ )is sometimes denoted the orbital shift ψ i (r) [64]. It is the first order shift of the orbitals with thelocal potential, towards those with the non-local potential. In terms of the orbital shifts, the OEPequation is ∑ i f iψ ∗ i (r)φ i(r) + c.c. = 0.The OEP equation (5.12) is quite difficult to solve numerically. First of all it is an integralequation, and secondly the summation in the Green’s function (5.10) run over all states, occupied aswell as unoccupied. The denominator of the Green’s function will obviously increase monotonically,so highly excited states becomes less important, but in practice a large number of unoccupiedstates have to be included to obtain convergence. The difficulties of solving the integral equationis illustrated by the fact that it has so far only been solved directly for spherically averaged isolatedatoms.5.3 Approximations to the Optimized Potential MethodIn the Krieger-Li-Iafrate (KLI) approximation [57], the energy difference in the denominator ofthe Green’s function is approximated by a constants ɛ l − ɛ k = ∆¯ɛ. Using the closure relation∑i φ i(r)φ i (r ′ ) = δ(r−r ′ ) the Green’s function thus becomes G i (r, r ′ ) ≈ (δ(r−r ′ )−φ i (r)φ ∗ i (r′ ))/∆¯ɛ.When this is done, the actual value of the constant disappears, and the resulting equations arev(r) = 1 ∑f i φ ∗ i (r) ( 〈i|ˆv − ˆv NL |i〉 + ˆv NL) φ i (r) + c.c.2n(r)i= 1 ∑ [f i φ∗n(r)i (r)ˆv NL φ i (r) + |φ i (r)| 2 〈i|ˆv − ˆv NL |i〉 ] (5.14)iwhere∫〈i|ˆv − ˆv NL |i〉 =drφ ∗ i (r) ( v(r) − ˆv NL) φ i (r) (5.15)The approximation of a constant energy denominator in G i (r, r ′ ) might seem crude, but it turnsout that the exact same equation can be derived using a mean-field approximation (the onlyapproximation done is neglecting a term which when averaged over the density is zero).For the exact exchange potential, the KLI equation can be written asv x (r) = vxSla (r) + ∑ i|φ i (r)| 2f i 〈i|ˆv x − ˆv x NL |i〉 (5.16)n(r)A different approximation of the OEP equation for exact exchange can be reached, by assumingthat the orbitals of HF and exact exchange only DFT are identical. This approximation leads tothe equationv x (r) = vxSla (r) + ∑ ijφ ∗ if i f (r)φ j(r)j 〈j|ˆv x − ˆv x NL |i〉 (5.17)n(r)which is known as the localized Hartree-Fock (LHF) method [51].The equations (5.16) and (5.17) only define the potential to within a constant. The usualasymptotic behavior v x (r → ∞) = 0 is recovered if one neglects the HOMO term in the summationof (5.16) or the HOMO-HOMO term of (5.17).Since v(r) appears on both sides of the KLI equation (5.14), the potential has to be determinedby a self-consistent approach in general. In the case of exact exchange one can derive a set oflinear equations determining the constants 〈i|ˆv|i〉, thus making it possible to solve (5.16) in oneevaluation.