SHARING OF ELECTRONS IN MOLECULES

An index which gives a quantitative measure of the degree of sharing o f an electron between two points in space in systems containing many e lectrons is introduced. This sharing index, denoted by I(zeta;zeta') i s defined as the absolute value squared of the matrix element of a sha ring amplitude, [zeta;zeta'], which in turn is the square root of the first-order density matrix. These quantities are invariant under trans formations of the orbitals in terms of which the wavefunction is typic ally expressed and are independent of the basis set provided it is suf ficiently complete. The sharing amplitude has many of the characterist ics of a wavefunction. By integration of the sharing index over volume s assigned to atoms, indices which measure the degree of sharing of an electron between atoms in molecules are found. Bond indices (numbers) are shown to be twice the value of the atomic sharing indices. On the basis of this, prototype double and triple bonds have bond indices of 2 and 3. That the sharing index automatically accounts for the interf erence and/or the localization of the electron is illustrated by the v alues of the bond indices for He-2+ and for He-2, calculated using eit her delocalized orbitals or using localized orbitals. One consequence of interference is that the contributions of antibonding orbitals to t he sharing indices tend to cancel the contributions of bonding orbital s. To further illustrate the present definition, the values of the bon d indices are found for the first excited 1SIGMA(g)+ state of H2 and f or the pi-electrons in benzene and in 1,3-butadiene in the Huckel appr oximation. The present bond indices for the pi-electrons differ from t he covalent bond indices recently defined by Cioslowski and Mixon. In the case of benzene, the procedure of these authors leads to an infini ty of sets of equivalent localized orbitals, each set giving different values for the covalent bond indices and each set breaking the symmet ry of the benzene wavefunction. In contrast, the bond indices arising from the sharing indices retain the underlying symmetry of the wavefun ction. The covalent bond indices for 1,3-butadiene, a case in which th ere is no broken symmetry, all differ from those obtained from the sha ring indices. The relation between bond indices and the bond orders of Coulson, in the Huckel approximation, is found to be similar to that between the sharing index and the sharing amplitude. The addition of c orrelation to a simple molecular orbital wavefunction for the ground 1 SIGMA(g)+ state of H2 is shown to decrease the interatomic sharing at the equilibrium internuclear distance. At large internuclear separatio ns the addition of correlation results in the expected value of zero f or the interatomic sharing, in contrast to the nonzero value for a sin gle determinant wavefunction. A volume-point sharing index is defined by integrating the point-point sharing index over but one index. A sim plified description of the electronic structure of benzene, including sigma-electron contributions, demonstrates how this volume-point shari ng index can be used to discuss in quantitative detail the geometry of the sharing of electrons which are associated with an atom. This part icular sharing index therefore gives a microscopic picture of the shap e of the valence of an atom in a molecule. Again using a simplified de scription of benzene, we show that the two point sharing amplitude its elf gives a clear indication of the degree to which electrons are loca lized at various points in a molecule, e.g., in the regions traditiona l associated with sigma- or pi-bonds, in the core region surrounding a nucleus, etc. These amplitudes can then be interpreted in terms of su ch familiar concepts as s-p hybridization, localized orbitals, delocal ized orbitals of which an example is the pi-orbital contribution in be nzene, and so on. Unlike orbitals, however, the sharing amplitudes and sharing indices have the virtue that they depend only on the complete many electron wavefunction. They therefore describe, at the one elect ron level, the electronic structure of a many electron system in a fas hion which is invariant to orbital transformations. The present indice s are quite general and are not limited in applications to electrons i n molecules.