Hamiltonian formalism of spin-orbit Jahn-Teller and pseudo-Jahn-Teller problems in trigonal and tetragonal symmetries

A formalism for expansions of all bimodal spin-orbit Jahn-Teller and pseudo-Jahn-Teller Hamiltonian operators in trigonal and tetragonal symmetries is presented. With the formalism, we can easily obtain expansion formulas of the Hamiltonian matrix elements in symmetry-adapted vibrational coordinates up to arbitrary order. The formalism is presented as a set of generic matrices and lookup tables, which are convenient to use even without understanding the derivation of the formalism. Three examples are used to demonstrate the correctness, completeness, and conciseness of the formalism. One of the examples is also used to demonstrate how to obtain expansion formulas in more than two vibrational modes by using the bimodal formalism. This work lays a foundation for deriving a unified formalism for spin-orbit and non-spin-orbit (pseudo-)Jahn-Teller Hamiltonians in general axial symmetries.