Interactions between Solvated Electrons
Interactions between Solvated Electrons. 1. Electron-Electron, Electron-Solvent, Solvent-Solvent Interactions in Ammonia. Valence Bond Approximation Paul 9. Schettler, Jr.,* and Gerard Lepoutre Department of Chemistry, Juniata College, Huntingdon, Pennsylvania 16652 (Received July 23, 1975)
The binary interaction between two electrons solvated in ammonia is considered by treating the electrons in the valence bond approximation with dispersion forces added. Solvent-solvent and solvent-electron interactions vary as a function of distance between the solvated electrons. These changes are considered within the continuum approximation along with the electronic interaction. A major result of this study is that the ground (singlet state) is separated from the lowest triplet state by kT at distances of separation as large as 10.5 A. The theoretical treatment is general in that it does not need to be restricted to the valence bond approach or to two electron problems.
Introduction The general properties of metal-ammonia solutions are well known and have been amply documented elsewhere.1-6 Basically the solutions show strong deviations from “ideality” as measured by many thermodynamic and nonequilibrium methods but surprisingly no deviation from ideality as measured by others. Although the concept of a solvated electron has been successful in explaining experimental results extrapolated to infinite dilution, the attempts to explain the interactions of solvated electrons in terms of chemical species (e.g., e22- (solvated), dimers) has not enjoyed a similar success. The most explicit statement of the problems involved has been presented by Dye7 who points out that existing (and generally accepted) data are paradoxical when interpretation is attempted in terms of models involving only the formation of (various) chemical species as a description of interelectronic interaction. As a specific example the solutions, initially paramagnetic due to all electrons being unpaired, become primarily diamagnetic at concentration as low as 0.05 M (0.1 MPM). This has been widely interpreted in terms of the formation of some sort of paired electron chemical species whereas data from optical and infrared spectroscopy would seem to mitigate against the formation of any such species. It should be noted that the problem is common to all models of solvated electron-electron interaction which seek explanation in terms of the appearance of any species (e.g., e22-) as a chemical entity. Thus the dilemma raises a number of important theoretical questions. Dye’s dilemma would apply at concentrations of 0.005 M and above as this concentration is the lower limit of paramagnetic susceptibility data which demonstrate the significant presence of spin pairing.8 This is well below the concentration range of most recent theoretical attention which has focused on the nonmetal-metal transition and formation of metallic clusters8 as explanations of properties above 0.4M (1MPM). Whereas critical fluctuations formation may be important at high concentrations, the results of studying binary interactions should help to explicate properties at low concentration. Work has already been done for two important cases of trap separation, R = m corresponding to the energy of two single solvated e1ectronsl0-l2 and R = 0 corresponding to
two electrons in the same trap.13-15 In this previous work the presupposition of spherical symmetry allows consideration of the effects of cavity formation and a partial treatment of the discrete nature of the solvent as well. This has not been attempted here for the new problem at hand. Theoretical Treatments of bonding in the valence bond approximation are plentiful and need not be discussed here.16J7 HOWever for the case of solvated electrons two important modifications are necessary. First the diffuse nature of the trapping potential (as contrasted to the point-point l / r interaction for electrons with a nucleus) necessitates the replacement of algebraic operators by integral operators in the Hamiltonian since the solvent orientation at all points contributes to the potential of an electron. Second, as bonding arises from changes in the electronic wave function as a function of R , these changes will be reflected in changes in solvent orientation. This consideration is one of utmost importance as solvent and electronic wave function have a “bootstrapping” interrelationship. Thus the effect of perturbations on the electronic wave function is intensified by the inclusion of solvent changes. Attention here is focused on these necessary modifications to the ordinary valence bond treatment. The total Hamiltonian will contain terms dealing with the electron-trap interaction and the trap-trap interaction and these terms will be considered now. This problem has two components: first, the electrostatic potential created by the trap and, second, the nonelectrostatic work necessary to create that potential. Electrostatic Terms. If the charge density of an electron is associated with the electric displacement, D, Poissons equation gives
+
where is a normalized electronic wave function. Following standard treatmentl8Jg the trapping effect of the solvent continuum can be described by a polarization vector, PD, associated with that part of the total polarization which is associated with a long time constant. 4TPD =
