Exact Exchange in Density Functional Calculations

5.5 Conclusion 355.5 ConclusionTo include exact exchange self-consistently in the SCF, one must in some way have access tothe exchange potential. In the hybrid HF-KS schemes of section 3.2.1, the exchange potential issimply the non-local Fock potential of Hartree-Fock theory, this however redefines the nature ofthe correlation functional, making established approximations less usable. In Kohn-Sham theory,the exchange-correlation potential must be a local multiplicative potential. Constructing a localpotential from an orbital dependent functional is a quite complicated, i.e. computationally timeconsuming, process and approximations are needed to make the approach feasible in practice. Constructionof the local exchange potential can be achieved at several levels, climbing the followingladder of accuracy (and complexity):• Slater potential: v x (r) = vxSla (r)• KLI: v x (r) = v Slax• LHF: v x (r) = v Slax(r) + ∑ i f i|φ i (r)| 2 〈i|ˆv x − ˆv NL |i〉/n(r)(r) + ∑ ij f if j φ ∗ i (r)φ j(r)〈j|ˆv x − ˆv xNL |i〉/n(r))φi (r ′ ) + c.c. = 0• OEP: ∑ i f i∫dr ′ φ ∗ i (r)G i(r, r ′ ) ( v x (r ′ ) − ˆv NLxwhere the OEP level can also be reached by an iterative scheme from one of the lower rungs ofthe ladder [55].x