Exact Exchange in Density Functional Calculations

16 3. Density Functional Theorywhere∫U H [n] = 1 2drdr ′ n(r)n(r′ )|r − r ′ |∑∫E x [n] = − 1 2f n f n ′ drdr ′ φ∗ n(r)φ n ′(r)φ ∗ n ′(r′ )φ n (r ′ )|r − r ′ |nn ′(3.20a)(3.20b)Here f α are the occupation numbers, and φ α (r) are the single particle wave functions constitutingΦ.Using (3.19), Q s is expanded asQ s [n] = 〈Ψ minn(r) | ̂T + ̂V ee |Ψ minn(r) 〉 − 〈Φmin n(r) | ̂T |Φ minn(r)(〉= 〈Φ minn(r) |̂V ee |Φ minn(r) 〉 + 〈Ψ minn(r) | ̂T + ̂V ee |Ψ minn(r) 〉 − 〈Φmin n(r) | ̂T + ̂V)ee |Φ minn(r) 〉= U H [n] + E x [n] + E c [n](3.21)where the correlation energy functional E c [n] is defined by the above expression, and accounts foreverything that is not described by the first term, i.e.E c [n] ≡ 〈Ψ minn(r) | ̂T + ̂V ee |Ψ minn(r) 〉 − 〈Φmin n(r) | ̂T + ̂V ee |Φ minn(r) 〉 (3.22)The procedure corresponds to expressing the full HK functional byF [n] = 〈Ψ minn(r) | ̂T + ̂V ee |Ψ minn(r) 〉= 〈Φ minn(r) | ̂T + ̂V ee |Φ minn(r) 〉 + E c[n]= T s [n] + U H [n] + E x [n] + E c [n](3.23)where T s is the kinetic energy of an imagined non-interacting electron gas with the same totaldensity as the real system, U H is the Hartree energy representing a mean-field approximation ofthe electron-electron interaction. The physical significance of the last two terms E x and E c is lessobvious, and will be discussed in section 3.3.This choice of model system leads to the operator ô s = ˆt = − 1 2 ∇2 , and the effective potentialv Q (r) = u H (r) + v x (r) + v c (r)where u H (r) = δU H [n]/δn(r) = ∫ dr ′ n(r ′ )/|r − r ′ | is the Hartree potential, E xc = E x + E c isthe exchange-correlation energy, and v xc (r) = δE xc [n]/δn(r) is the local exchange-correlationpotential.There is no explicit expression for the correlation energy functional and it must be approximatedby some suitable functional, and from this, its functional derivative v c (r) must be determined.As the exchange energy functional is only an implicit functional of the density, its functionalderivative is not easily determined. Typically approximations which are explicit functionals of thedensity are made for both E c and E x , such that their functional derivatives are easily found. Besidesthe difficulties of forming the functional derivative of of an implicit functional there is alsoan empirical argument for making approximations of the combined exchange-correlation. It turnsout, a posteriori, that there is a large degree of cancellation between the long range effects of E xand E c , such that local approximations for both actually performs better than treating exchangeexactly and correlation only locally.For a discussion of how to construct the functional derivative of orbital dependent functionals,see chapter 5.From (3.16) we get the Kohn-Sham equations[−12 ∇2 + u H (r) + v xc (r) + v ext (r) ] φ n = ɛ n φ n (3.24)