A Symmetry Adapted Approach to the Dynamic Jahn-Teller Problem

Abstract

In this article we present a symmetry-adapted approach aimed to the accurate solution of the dynamic Jahn-Teller (JT) problem. The algorithm for the solution of the eigen-problem takes full advantage of the point symmetry arguments. The system under consideration is supposed to consist of a set of electronic levels $${\Gamma }_{1},{\Gamma }_{2}\ldots {\Gamma }_{n}$$ labeled by the irreducible representations (irreps) of the actual point group, mixed by the active JT and pseudo JT vibrational modes $${\Gamma }_{1},{\Gamma }_{2}\ldots {\Gamma }_{f}$$ (vibrational irreps). The bosonic creation operators b +(Γγ) are transformed as components γ of the vibrational irrep Γ. The first excited vibrational states are obtained by the application of the operators $${b}^{+}(\Gamma \gamma )$$ to the vacuum: $${b}^{+}(\Gamma \gamma )\vert n = 0,{A}_{1}\rangle = \vert n = 1,\Gamma \gamma \rangle$$ and therefore they belong to the symmetry Γγ. Then the operators b +(Γγ) act on the set $$\vert n = 1,\Gamma \gamma \rangle$$ with the subsequent Clebsch-Gordan coupling of the resulting irreps. In this way one obtains the basis set $$\vert n = 2,{\Gamma }^{{\prime}}{\gamma }^{{\prime}}\rangle$$ with $${\Gamma }^{{\prime}}\in \Gamma \otimes \Gamma $$ . In general, the Gram-Schmidt orthogonalization is required at each step of the procedure. Finally, the generated vibrational bases are coupled to the electronic ones to get the symmetry adapted basis in which the full matrix of the JT Hamiltonian is blocked according to the irreps of the point group. The approach is realized as a computer program that generates the blocks and evaluates all required characteristics of the JT systems. The approach is illustrated by the simulation of the vibronic charge transfer (intervalence) optical bands in trimeric mixed valence clusters.