SHARING OF ELECTRONS IN MOLECULES

被引:102
作者
FULTON, RL
机构
[1] Department of Chemistry, Florida State University, Tallahassee
关键词
D O I
10.1021/j100131a021
中图分类号
O64 [物理化学(理论化学)、化学物理学];
学科分类号
070304 ; 081704 ;
摘要
An index which gives a quantitative measure of the degree of sharing of an electron between two points in space in systems containing many electrons is introduced. This sharing index, denoted by I(zeta;zeta') is defined as the absolute value squared of the matrix element of a sharing amplitude, [zeta;zeta'], which in turn is the square root of the first-order density matrix. These quantities are invariant under transformations of the orbitals in terms of which the wavefunction is typically expressed and are independent of the basis set provided it is sufficiently complete. The sharing amplitude has many of the characteristics of a wavefunction. By integration of the sharing index over volumes 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 interference and/or the localization of the electron is illustrated by the values of the bond indices for He-2+ and for He-2, calculated using either delocalized orbitals or using localized orbitals. One consequence of interference is that the contributions of antibonding orbitals to the sharing indices tend to cancel the contributions of bonding orbitals. To further illustrate the present definition, the values of the bond indices are found for the first excited 1SIGMA(g)+ state of H2 and for the pi-electrons in benzene and in 1,3-butadiene in the Huckel approximation. The present bond indices for the pi-electrons differ from the covalent bond indices recently defined by Cioslowski and Mixon. In the case of benzene, the procedure of these authors leads to an infinity of sets of equivalent localized orbitals, each set giving different values for the covalent bond indices and each set breaking the symmetry of the benzene wavefunction. In contrast, the bond indices arising from the sharing indices retain the underlying symmetry of the wavefunction. The covalent bond indices for 1,3-butadiene, a case in which there is no broken symmetry, all differ from those obtained from the sharing 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 correlation to a simple molecular orbital wavefunction for the ground 1SIGMA(g)+ state of H2 is shown to decrease the interatomic sharing at the equilibrium internuclear distance. At large internuclear separations the addition of correlation results in the expected value of zero for the interatomic sharing, in contrast to the nonzero value for a single determinant wavefunction. A volume-point sharing index is defined by integrating the point-point sharing index over but one index. A simplified description of the electronic structure of benzene, including sigma-electron contributions, demonstrates how this volume-point sharing index can be used to discuss in quantitative detail the geometry of the sharing of electrons which are associated with an atom. This particular sharing index therefore gives a microscopic picture of the shape of the valence of an atom in a molecule. Again using a simplified description of benzene, we show that the two point sharing amplitude itself gives a clear indication of the degree to which electrons are localized at various points in a molecule, e.g., in the regions traditional associated with sigma- or pi-bonds, in the core region surrounding a nucleus, etc. These amplitudes can then be interpreted in terms of such familiar concepts as s-p hybridization, localized orbitals, delocalized orbitals of which an example is the pi-orbital contribution in benzene, 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 electron level, the electronic structure of a many electron system in a fashion which is invariant to orbital transformations. The present indices are quite general and are not limited in applications to electrons in molecules.
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页码:7516 / 7529
页数:14
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