Mathematical Viewpoint
For two independent electrons a and b,
where ρ(ra,rb) represents the joint electronic density, or the probability density of finding electron a at ra and electron b at rb. Within this notation, ρ(ra,rb) dra drb represents the probability of finding the two electrons in their respective volume elements dra and drb.
If these two electrons are correlated, then the probability of finding electron a at a certain position in space depends on the position of electron b, and vice versa. In other words, the product of their independent density functions does not adequately describe the real situation. At small distances, the uncorrelated pair density is too large; at large distances, the uncorrelated pair density is too small (i.e. the electrons tend to "avoid each other").
Read more about this topic: Electronic Correlation
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