Dynamic Jahn–Teller Effect in Trigonally Distorted Cubic Systems

Abstract

The Jahn–Teller problem in trigonally distorted cubic systems is discussed. Particular attention is devoted to the effects of a trigonal distortion on cubic systems in which a triply degenerate $\mathrm{(T)}$ electronic state interacts with a doubly degenerate $\mathrm{(e)}$ vibrational mode. The $T$ state is split into singly and doubly degenerate ($A$ and $E$ ) states and both “genuine” and “pseudo” Jahn–Teller interactions are present. It is shown how the Jahn–Teller problem can be formulated such that the relation of the trigonal and cubic problems is apparent. This leads to the definition of “isotropic” and “anisotropic” Jahn–Teller parameters, the latter arising from the trigonal distortion and vanishing in the cubic limit. When the trigonal distortion is small, it can be treated by perturbation theory. Explicit calculations are made of the trigonal splitting of the ground triplet state of the cubic vibronic system for several specific cases. It is shown that, to first order, this depends on both the $\mathrm{A–E}$ splitting in the trigonal geometry and the “anisotropic” first‐order Jahn–Teller interaction. The inclusion of second‐order Jahn–Teller terms and other electronic perturbations, such as spin–orbit coupling, is also discussed. The results are employed to complete the analysis of the zero‐phonon levels of the ${}^3{T}_{\text{2g}}$ state of V³⁺ / Al₂O₃. The different quenching of the first‐order electronic trigonal field and spin–orbit splittings, previously observed, is attributed to the “anisotropy” in the Jahn–Teller interaction. Values for both the “isotropic” and “anisotropic” first‐order Jahn–Teller parameters are derived.