By Harrison D.M.

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The equations which define Exc [ρ], starting with Eq. 13) for E[ρ], are hopelessly complicated. The utility of the Kohn-Sham formulation is that the exchange-correlation terms in Eg , Eq. 30), are small, so an approximation is acceptable. 31) where X(ρ(r)) is a function of ρ(r), not a functional. 28), the exchange-correlation potential Vxc (r) also becomes a function of the local density ρ(r), Vxc (r) = d[ρ(r)X(ρ(r))] . 32) Kohn and Sham (1965) pointed out that the local density approximation is correct for a uniform electron gas, and in fact they limited their original derivation to the local density approximation.

It will then be very useful to split the electronic excitation Hamiltonian into a part evaluated at the reference structure, and the remaining part which sees the motion of the nuclei. Positions of the nuclei in the reference structure are denoted RK , K = 1, · · ·, N . The potential of an electron at r, due to the nuclei in configuration {rK }, has been denoted V(r), and is more completely specified as V(r; {rK }). Let us write V σ (r) to denote evaluation at the nuclear reference structure, so that V σ (r) = V(r; {RK }).

17) . 18) λ where δVλλ = ψλσ |δVg |ψλσ November 6, 2002 15:30 WorldScientific/ws-b8-5x6-0 TOTAL HAMILTONIAN WallaceBook 29 Notice once again that both δVλλ and δEλ depend parametrically on the nuclear positions, and they also have an implicit dependence on the reference structure itself. 8) can now be written σ H= ˙ HN + HEX + δHEX . 19) This is a valuable result. First, since the nuclear positions do not appear as σ σ variables in HEX , then HN and HEX commute, which means there is no interaction between the motion of the nuclei and the excitation of reference σ structure electrons.

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