9768
J. Phys. Chem. 1995,99, 9768-9783
Sharing of Electrons in Molecules: Basin-Basin and Basin-Point Sharing Indices for Some Simple Molecules Robert L. Fulton* and Stacey T. Mixon Department of Chemistry, The Florida State University, Tallahassee, Florida 32306-3006 Received: February 3, 1995; In Final Form: March 30, 1995@
The behavior of an electron in a many-electron system has been shown to be described by the point-point sharing index. Volume-volume and volume-point sharing indices, given by the integration of the pointpoint sharing index over the appropriate volumes, are quantitative measures of the sharing of an electron between two volumes and between a volume and a point. When the volumes are associated with the atomic basins of Bader, basin-basin and basin-point sharing indices are obtained. These give measures of the sharing of an electron between atoms in a molecule and between an atom in a molecule and a point. In this paper bond indices are defined to be twice the value of the basin-basin sharing indices. Basin-basin sharing indices are determined for the sequence of molecules Hz, LiH, C h , C2H6, C2H4, C2H2, and P4 at both the Hartree-Fock and MP2 levels of approximation. The evaluations give (1) the trends of the indices in this sequence and (2) the effect correlation has on the bond indices. The addition of correlation to the HartreeFock wave functions describing the ground electronic states tends to decrease the interbasin sharing indices, in most cases by 10-22%, with the lower percentage being for single hydrogen-hydrogen, carbon-hydrogen, and carbon-carbon bonds, while the higher percentages are found for multiple bonds in the hydrocarbons and for the phosphorus-phosphorus bond in P4. At the MP2 level of approximation, the nearest neighbor bond indices are ordered according to the commonly assigned, but qualitatively determined, bond indices. For example, the interbasin bond indices of the covalent carbon-carbon and carbon-hydrogen single bonds tend to be in the range 0.85-0.90, and the bond index for LiH is 0.20, in line with the ionic character of the molecule, while the bond indices associated with the double and triple bonds in the unsaturated hydrocarbons are 1.55 and 2.22. The separation of the double and triple bonds into n-and a-orbital contributions is discussed, and it is shown that correlation is important in determining the n-contribution to the sharing indices. The self-sharing indices and the valences are suggested as quantitative measures of the localization of an electron in a many-electron system. In order to show finer details of bonding which are not apparent from the basinbasin sharing indices, basin-point sharing indices are evaluated for various basins in the same sequence of molecules. The structures of these volume-point sharing indices are shown to vary from the simple (in H2) to the complex (in P4). In spite of the inclusion of correlation in the approximations to the ground state wave functions, these sharing structures are interpretable in terms familiar from the concepts of hybrid orbitals, such as simple 1s structures, hybrid 2s2p structures, and (in the case of P4) more complicated hybrid 3s3p structures. The addition of correlation to Hartree-Fock approximations to the ground state wave functions of H2 and to the n-contributions of ethylene is shown to make these volume-point sharing indices more compact. Unlike traditional visualizations of orbitals and hybrids, the sharing structures are independent of the basis set used provided only that the basis sets are sufficiently complete.
I. Introduction A quantitative measure of the sharing of an electron between the two points 5 and 5' is given by the point-point sharing index l(
