General formalism of vibronic Hamiltonians for tetrahedral and octahedral systems: problems that involve A-type states and a-type vibrations

In this work, we derive expansion formulas up to arbitrary order in vibrational coordinates for the tetrahedral and octahedral vibronic Hamiltonians that involve A-type states and a-type vibrations. The root-branch approach and modularized approach enable us to derive vibronic Hamiltonians including up to two vibrational modes for 5 problems in T symmetry and 92 problems in Td symmetry within one paper. These formulas can be easily adapted to problems of Th, O, and Oh symmetries. Finishing this work, we have derived general vibronic Hamiltonians for all unimodal and bimodal Jahn-Teller and pseudo-Jahn-Teller problems of cubic group systems. These bimodal formulas can be extended to cover problems that involve more than two modes.