Revisiting the (E plus A) ⊗ (e plus a) problems of polyatomic systems with trigonal symmetry: general expansions of their vibronic Hamiltonians

In this work, we derive general expansions in vibrational coordinates for the (E + A) circle times (e + a) vibronic Hamiltonians of molecules with one and only one C-3 axis. We first derive the expansion for the lowest C3 symmetry. Additional symmetry elements systematically eliminate terms in the expansion. We compare our expansions with the previous results for two cases, the D-3h (E' + A"(2)) circle times (e' + a"(2)) and the C3 (E + A) circle times e. The first comparison demonstrates the robustness, completeness, conciseness, and convenience of our formalism. There is a systematic discrepancy in the second comparison. We discuss the origin of the discrepancy and use a numerical example to corroborate our expansion. Our formalism covers 153 vibronic problems in 6 point groups. It also gives general expansions for the spin-orbit vibronic Hamiltonians of the p-type (E + A) circle times (e + a) problems.