General formalism of vibronic Hamiltonians for tetrahedral and octahedral systems: problems that involve T, E states and t, e vibrations

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 Td symmetries involving up to two vibrational modes. The formulas can be easily generalized to problems of Th, O, and Oh 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.