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 $T_d$ symmetry within one paper. These formulas can be easily adapted to problems of $T_h,O$, and $O_h$ 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.