General Formalism of Vibronic Hamiltonians for Tetrahedral and Octahedral Systems: Problems That Involve <i>T</i>, <i>E</i> States and <i>t</i>, <i>e</i> Vibrations

Abstract

We derive expansion formulas up to arbitrary order in vibrational coordinates for the tetrahedral and octahedral vibronic Hamiltonians that involve T and E states, and t and e vibrations. These states feature both Jahn-Teller (JT) and pseudo-Jahn-Teller (pJT) effects, and the vibrations are the most JT and pJT active. We first derive the formulas for 92 problems of T and T_d symmetries involving up to two vibrational modes. The formulas can be easily generalized to problems of T_h, O, and O_h symmetries, as well as problems involving more than two vibrational modes. They can also be adapted to describe spin-orbit vibronic Hamiltonians of tetrahedral p-type problems. Overall, this work makes crucial preparations for future studies on vibronic coupling problems of tetrahedral and octahedral systems. Most importantly, a new, simple, modularized approach to construct vibronic Hamiltonians for a set of related problems, instead of particular problems one by one, is presented.