A formalism for expansions of all Jahn-Teller and pseudo-Jahn-Teller Hamiltonian operators in all axial symmetries is presented. The formalism provides Hamiltonian expansions up to arbitrarily high order and including an arbitrary number of vibra?tional modes, which are of arbitrary types. It consists of three equations and two tables. The formalism is user-friendly since it can be used without understanding its derivation. An example of E00 3 ⊗ e 0 1 Jahn-Teller interaction of cycloheptatrienyl cation is used to demonstrate the correctness of the formalism. A Python program is devel?oped to automate the generation of Hamiltonian expansions for all axial Jahn-Teller and pseodo-Jahn-Teller problems, , and interface the expansions to quantum dynamics simulation program. This is the first unified Hamiltonian formalism for axial Jahn?Teller and pseudo-Jahn-Teller problems. And it is the only one. File list (2) download file view on ChemRxiv paper.pdf (3.24 MiB) download file view on ChemRxiv SI.pdf (491.06 KiB)

Spin–orbit coupling and vibronic coupling are both closely related to orbital degeneracy of electronic states. Both types of coupling play significant roles in determining properties of heavy element compounds and shall be treated on the same footing. In this work, we derive a unified one-electron Hamiltonian formalism for spin–orbit and vibronic interactions for systems in all axial symmetries. The one-electron formalism is usually adequate as the spin–orbit interaction can often be approximated as a one-electron interaction. For the first time, the formalism covers spin–orbit and vibronic couplings in all axial symmetries from C1 to D∞h, arbitrary types of vibrational modes in those symmetries, and an arbitrary number of those modes and gives Hamiltonian expansions up to an arbitrary order.