By Holtje, Hans-Dieter
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Additional resources for A Chemists Guide to Density Functional Theory
The electrons’ spins are parallel, the last term in equation (2-12) will , σ1 = σ 2 r r not vanish but yields <σ(si) | σ(si)> = 1 (σ = α, β). Hence, ρ HF ( H , H ) does not reduce 2 r r to the simple, uncorrelated product of individual probabilities. Rather, for r1 = r2 , the third HF r r term exactly cancels the first two and we indeed arrive at ρ 2 ( N , N ) = 0 . Thus, we rederived the conclusions from the end of the preceding chapter that the correlation due to the antisymmetry of the wave function is covered by the HF scheme – after all no surprise since a Slater determinant is antisymmetric in the coordinates of any two electrons.
How does the Coulomb hole look like in our H2 laboratory molecule? Of course, this is a particularly simple case, because there are no parallel spin contributions and the Coulomb hole refers only to the interaction of electrons of antiparallel spins. If one of the two electrons is found at, say, the left proton, the probability to find the other one at the right nucleus will be higher and vice versa. The larger the distance between the two H atoms the more pronounced this effect will be. Thus, also the Coulomb hole will be delocalized with a negative part at the nucleus where the reference electron sits and a positive part, i.
However, billiard balls are a pretty bad model for electrons. First of all, as discussed above, electrons are fermions and therefore have an antisymmetric wave function. Second, they are charged particles and interact through the Coulomb repulsion; they try to stay away from each other as much as possible. Both of these properties heavily influence the pair density and we will now enter an in-depth discussion of these effects. Let us begin with an exposition of the consequences of the antisymmetry of the wave function.
A Chemists Guide to Density Functional Theory by Holtje, Hans-Dieter