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.

Bond paths linking two bonded hydrogen atoms that bear identical or similar charges are found between the ortho-hydrogen atoms in planar biphenyl, between the hydrogen atoms bonded to the C1-C4 carbon atoms in phenanthrene and other angular polybenzenoids, and between the methyl hydrogen atoms in the cyclobutadiene, tetrahedrane and indacene molecules corseted with tertiary-tetra-butyl groups. It is shown that each such H-H interaction, rather than denoting the presence of "nonbonded steric repulsions", makes a stabilizing contribution of up to 10 kcal mol(-1) to the energy of the molecule in which it occurs. The quantum theory of atoms in molecules-the physics of an open system-demonstrates that while the approach of two bonded hydrogen atoms to a separation less than the sum of their van der Waals radii does result in an increase in the repulsive contributions to their energies, these changes are dominated by an increase in the magnitude of the attractive interaction of the protons with the electron density distribution, and the net result is a stabilizing change in the energy. The surface virial that determines the contribution to the total energy decrease resulting from the formation of the H-H interatomic surface is shown to account for the resulting stability. It is pointed out that H-H interactions must be ubiquitous, their stabilization energies contributing to the sublimation energies of hydrocarbon molecular crystals, as well as solid hydrogen. H-H bonding is shown to be distinct from "dihydrogen bonding", a form of hydrogen bonding with a hydridic hydrogen in the role of the base atom.

A combined experimental and theoretical study on the bond characterization of Cr-L (L ) O, N, C) multiple bonds is applied to a series of Cr-complexes: [(CO) 4 (Cl)Cr (I) ; and [(TPPOMe)Cr (IV) O] 4, where [TPPOMe ) (5,10,15,20-p-methoxyphenyl)porphyrin)]. Compounds 1 and 2 were investigated by accurate single-crystal X-ray diffraction. Detailed descriptions of Cr-C carbyne , Cr-N nitrido , Cr-N imido , and Cr-O oxo bonds will be given based on the natural bonding orbital (NBO) analyses and Fermi hole function. The bonding feature of all these multiple bonds is essentially a triple bond character consisting of one σ and two π bonds. The σ character of the Cr-L multiple bonds is a highly polarized one with the electron density strongly polarized toward L (O, N, C) ligands. The π character of the Cr-L multiple bonds depends on the nature of L ligand, i.e., with electron density polarized toward the Cr and O center for Cr-C carbyne and Cr-O oxo bond, respectively, whereas electron density is roughly equally distributed at both Cr and N for the Cr-N nitrido and Cr-N imido bonds. Bond characterizations are also shown in terms of Laplacian of electron density where the inner valence shell charge concentration (i-VSCC) is embedded. The isovalue surface of zero Laplacian of electron density reveals the shape of such i-VSCC at each chromium atom. Shapes around the Cr atom are a pressed disklike for 1 and 3 and inverted square pyramid for 2 and 4. The topological properties associated with the bond critical point (BCP) of Cr-L multiple bonds in these compounds indicate a strong covalent bond character. The order of the binding interaction is Cr-N nitrido > Cr-O oxo > Cr-N imido > Cr-C carbyne . The combined study of experiment and theory on 1 and 2 demonstrates good agreement between experiment and theory.

Eighty years have elapsed since Lewis introduced the concept of an electron pair into chemistry where it has continued to play a dominant rôle to this day. The pairing of electrons is a consequence of the Pauli exclusion principle and is the result of the localization of one electron of each spin to a given region of space. It is the purpose of this paper to demonstrate that all manifestations of the spatial localization of an electron of a given spin are a result of corresponding localizations of its Fermi hole. The density of the Fermi hole determines how the charge of a given electron is spread out in the space occupied by a second same-spin electron, thereby excluding an amount of same-spin density equivalent to one electronic charge. The Fermi hole is an electron's doppelgängerit goes where the electron goes and vice versa:  if the hole is localized, so is the electron. The topologies of two fields have been shown to provide information about the spatial localization of electronic charge:  the negative of the Laplacian of the electron density, referred to here as L(r), and the electron localization function ELF or η(r). The measure provided by L(r) is empirical. It is based upon the remarkable correspondence exhibited by its topology with the number and arrangement of the localized electron domains assumed in the VSEPR model of molecular geometry. η(r) is based upon the local behavior of the same-spin probability, and it is shown that the picture of electron localization that its topology provides is a consequence of a corresponding localization of the Fermi hole density. This paper provides a complete determination and comparison of the topologies of L(r) and η(r) for molecules covering a wide spectrum of atomic interactions. The structures of the two fields are summarized and compared in terms of the characteristic polyhedra that their critical points define for a central atom interacting with a set of ligands. In general, the two fields are found to be homeomorphic in terms of the number and arrangement of electron localization domains that they define. The complementary information provided by the similarities in and differences between these two fields extends our understanding of the origin of electron pairing and its physical consequences.