The mapping of the conditional pair density onto the electron density

This paper shows that the Fermi hole of a reference electron can be so strongly localized to a given region of space, as to cause the conditional pair density for same-spin electrons to approach the one-electron spin density outside the region of localization and for a closed-shell system, the conditional pair density for both spins will approach the total density. Correspondingly, the Laplacian of the conditional pair density, whose local concentrations indicate the positions where the density of the remaining electrons are most likely to be found for a fixed position of a reference pair, approaches the Laplacian of the density. The Laplacian of the conditional pair density generated by a sampling of pair space by an alpha,beta pair of reference electrons, exhibits a homeomorphism with the Laplacian of the electron density. This homeomorphism approaches an isomorphic mapping of one field onto the other, as the reference electron pair becomes increasingly localized to a given region of space. Thus the local charge concentrations (CCs) displayed by the Laplacian of the electron density, the local maxima in L(r)=- del(2)rho(r), signify the presence of regions of partial pair condensation, regions with greater than average probabilities of occupation by a single pair of electrons, as has been previously surmized on empirical grounds. This paper establishes a mapping of the essential aspects of electron pairing, determined in six-dimensional space, onto the three-dimensional space of the electron density. The properties of the conditional pair density enable one to determine which CCs of L(r) are coupled and represent the same localized pair of electrons. It is found that the pattern and properties of the electron localization domains predicted by the Laplacian of the conditional pair density differ in important aspects from those predicted by ELF, the electron localization function. (C) 1999 American Institute of Physics. [S0021-9606(99)30743-1].