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 stro ngly 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 condi tional pair density for both spins will approach the total density. Corresp ondingly, the Laplacian of the conditional pair density, whose local concen trations indicate the positions where the density of the remaining electron s are most likely to be found for a fixed position of a reference pair, app roaches 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 ref erence electrons, exhibits a homeomorphism with the Laplacian of the electr on density. This homeomorphism approaches an isomorphic mapping of one fiel d onto the other, as the reference electron pair becomes increasingly local ized 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 condensa tion, regions with greater than average probabilities of occupation by a si ngle pair of electrons, as has been previously surmized on empirical ground s. This paper establishes a mapping of the essential aspects of electron pa iring, determined in six-dimensional space, onto the three-dimensional spac e of the electron density. The properties of the conditional pair density e nable 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 o f the electron localization domains predicted by the Laplacian of the condi tional pair density differ in important aspects from those predicted by ELF , the electron localization function. (C) 1999 American Institute of Physic s. [S0021-9606(99)30743-1].