Electronic Correlation - Mathematical Viewpoint

Mathematical Viewpoint

For two independent electrons a and b,

where ρ(r_a,r_b) represents the joint electronic density, or the probability density of finding electron a at r_a and electron b at r_b. Within this notation, ρ(r_a,r_b) dr_a dr_b represents the probability of finding the two electrons in their respective volume elements dr_a and dr_b.

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").