Spatial degeneracy of electronic states closely connects
spin–orbit
coupling and vibronic coupling, which together determine properties
of materials, especially heavy element compounds. Accurate description
of those materials entails accurate mathematical formulas for spin–orbit
vibronic Hamiltonians. For the first time ever, we in this work derive
the Hamiltonian formalism to describe all spin–orbit Jahn–Teller
and pseudo-Jahn–Teller vibronic problems in all axial symmetries.
The conventional one-electron approximation of spin–orbit coupling,
which was the foundation of all previous studies in this field, is
not involved in the present work. Actually, the present formalism
is applicable to all time-reversal symmetric hermitian Hamiltonian
that has a Rank-1 dependence on the spin operator, without any restriction
on the type and the number of term symbols and vibrational modes.