On-Top Pair Density as a Measure of Left-Right Correlation in Bond Breaking

To better understand left-right electron correlation and the effect of bond breaking on the on-top pair density, analytic expressions for the total density, the on-top pair density, and the ratio of the on-top pair density to the square of the total density were derived for H₂ for both a restricted Hartree-Fock wave function and a complete active space self-consistent-field wave function with two electrons in two active orbitals. These quantities are compared for the two wave functions for various points in space around the molecule as functions of internuclear distance. At some points in space, in the CASSCF(2,2) wave function, electron correlation, perhaps counterintuitively, increases the probability that two electrons are at the same point in space. At the Coulson-Fischer point, the on-top pair density for the complete active space wave function starts to rapidly approach zero, and this can be taken as an indicator of bond breaking.