By Holtje, Hans-Dieter
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Additional resources for A Chemists Guide to Density Functional Theory
2 The Coulomb Hole From equations (2-17) and (2-21) it is obvious that the Coulomb hole must be normalized to zero, i. e. the integral over all space contains no charge: r r r ∫ h C ( r1; r2 ) dr2 = 0. (2-25) This makes good physical sense since for electrons of unlike spin the probability of finding an electron of spin σ anywhere in space is of course the total number of electrons of this spin, i. , Nσ. This result is independent of the positions of electrons with spin σ’ ≠ σ. Also, there is no need for a self-interaction correction.
This interaction depends on the distance between two electrons weighted by the probability that this distance will occur. , where we have integrated over the spin coordinates) which contains just this information Eee = Ψ N N 1 ∑∑ r i j > i ij Ψ = r r 1 ρ2 (r1 , r2 ) r r dr1dr2 . 2 ∫ ∫ r12 (2-18) r r r r r r r Using ρ 2 ( r1 , r2 ) = ρ( r1 )ρ( r2 ) + ρ( r1 )h XC ( r1; r2 ) (cf. 3 Fermi and Coulomb Holes E ee 1 = 2 r r ρ( r1 )ρ( r2 ) r r 1 ∫ ∫ r12 dr1dr2 + 2 r r r ρ( r1 )h XC ( r1; r2 ) r r d r1dr2 .
It applies equally well to neutral fermions and – also this is very important to keep in mind – does not hold if the two electrons have different spin. This effect is known as exchange or Fermi correlation. As we will show below, this kind of correlation is included in the Hartree-Fock approach due to the antisymmetry of a Slater determinant and therefore has nothing to do with the correlation energy E HF C discussed in the previous chapter. Next, let us explore the consequences of the charge of the electrons on the pair density.