Polaron effects and electron correlations in two-electron systems: Arbitrary value of electron-phonon interaction

Abstract

The energy of two-electron systems [exchange-coupled pairs of paramagnetic centers (${D}^{\ensuremath{-}}$ centers) and bipolarons] is calculated for various distances between paramagnetic centers with regard to polaron effects for arbitrary coupling of electrons with a phonon field. Interaction of electrons with a phonon field is found by the Buymistrov-Pekar method. The calculations are made with a wave function (WF) in the form of expansion in Gaussians. Both the electronic correlations (direct dependence of the WF of a system on the interelectronic distance) and the permutation symmetry of the two-electron WF are taken into account. The lowest singlet $^{1}\ensuremath{\Sigma}_{g}^{+}$ and triplet $^{3}\ensuremath{\Sigma}_{u}^{+}$ terms are considered. Effects of electronic correlations are exemplified by the dependence of the energy and spatial distribution of the bipolaron WF on the distance between the centers of polaron polarization wells. A bipolaron corresponding to a two-center configuration is energetically unstable. The only minimum on the curve for the energy dependence of two polarons on the distance between the centers of their polarization wells corresponds to a one-center bipolaron configuration. For AgBr and AgCl we present the energies of the lowest singlet and triplet states of ${F}_{2}$ centers ($^{1}\ensuremath{\Sigma}_{g}$ and $^{3}\ensuremath{\Sigma}_{u}$ terms) and those for $1{\ensuremath{\sigma}}_{g}$ and $1{\ensuremath{\sigma}}_{u}$ terms of ${F}_{2}^{+}$ centers as a function of the distance between them (with a graph of various contributions into these energies). Control calculations performed for a hydrogen molecule with the use of a variational function suggested in the work yield the energies of the singlet and triplet states equal to $\ensuremath{-}1.17416$ and $\ensuremath{-}0.78315\phantom{\rule{0.3em}{0ex}}\mathrm{a.u.}$ respectively, the equilibrium internuclear distance corresponds to ${R}_{m}=1.4011\phantom{\rule{0.3em}{0ex}}\mathrm{a.u.}$ The contribution of phonons into the exchange interaction between paramagnetic centers has antiferromagnetic character. The exchange interaction caused by phonons is comparable in the order of magnitude with Coulomb exchange.