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The electron pair density, in conjunction with the definition of an atom in a molecule, enables one to determine the average number of electron pairs that are localized to each atom and the number that are formed between any given pair of atoms. Thus, it is through the pair density that the Lewis model of electronic structure finds physical expression. The pairing of electrons is a consequence of the Pauli principle whose effect is made manifest through the creation of the Fermi hole. The density describing the spatial distribution of the Fermi hole for an electron of given spin determines how the density of that electron is spread out in space, excluding an equivalent amount of same-spin density. The averaging of the Fermi density over single atoms or pairs of atoms determines the corresponding contributions to the total Fermi correlation. It is these terms that yield the localization and delocalization indices that determine the intra- and interatomic distribution of electron pairs that enables one to compare the pairing predicted by theory with that of a Lewis structure. The agreement is best at the Hartree−Fock level, where the Fermi hole is the sole source of correlation between the electrons. The introduction of the remaining correlation, the Coulomb correlation, disrupts the sharing of electron pairs between the atoms, and its effect is therefore, most pronounced for shared interactions. For example, Coulomb correlation reduces the number of shared pairs in N2 from the Hartree−Fock value of three to just above two. In ionic systems, the electrons are strongly localized within each atomic basin and the effect of Coulomb correlation on the atomic pairing is minimal, approaching zero over each of the atomic basins, as it does for the total molecule.

This article describes an algorithm for the calculation of the average properties of an atom in a molecule. The atom is defined within the topological theory of molecular structure, a theory which defines atoms, bonds, structure, and structural stability in terms of the topological properties of a system's charge distribution. The average properties of the atom so defined are uniquely determined by quantum mechanics. Results for a number of hydrocarbon molecules, obtained by the program PROAIM (properties of atoms in molecules) which implements this algorithm, are given. In general, this program enables one to calculate the average energy of an atom in a molecule to an accuracy of ±1 kcal/mol.