This is a good question. In order to go beyond Born-Oppenheimer approximation, one has to first make a choice of electronic basis - diabatic or adiabatic - in order to define the vibronic coupling (vibrational-electronic coupling). The adiabatic choice seems to be more practical since it is a well-defined basis (adiabatic states are eigenstates of the electronic Hamiltonian) and most quantum chemistry program make use of them anyway. So the "only" thing remains to compute is the non-adibatic coupling vectors (NADVEC) between different electronic states. These calculations are now possible with programs such as MOLPRO, COLUMBUS, CFOUR and others. Adiabatic basis is often used in conjunction with "on-the-fly" molecular dynamics methods such as the popular Surface Hopping scheme.
On the other hand, most vibronic problems are actually studied using diabatic basis, and for good reason. While diabatic states are not unique (and NOT eigenstates of electronic Hamiltonian), they preserve the electronic character with the change in nuclear position and are therefore "smoothly varying". Not only this has better mathematical properties (potential energy surface is always differentiable, unlike in adiabatic basis, where PES may have "cusps"), but also it has more "chemical" meaning. Think of diabatic states as the valence bond structures (like resonance structure of benzene!).
A large body of work has been done to model vibronic problems in diabatic basis. The most popular approach is the "vibronic model Hamiltonian" proposed by Koppel-Domcke-Cederbaum (KDC) in 1984 . It remains a classic and is an excellent place to start for all things vibronic. Even today the linear variant of vibronic models (Linear Vibronic Model or commonly known as LVC) is used widely to model many different kind of processes. The parameters of the vibronic models can be obtained in many different ways, and the process of extracting these parameters is known as "diabatization". Since diabatic states are not unique, many diabatization procedures have been proposed in the literature. However, this depends on the problem and is often also a matter of taste, and does not take away the power of vibronic model Hamiltonian ansatz!
PS: Vibronic model Hamiltonian is also referred to as KDC Hamiltonian for obvious reasons!
- H. Köuppel, W. Domcke, L. S. Cederbaum, Multimode Molecular Dynamics Beyond the Born‐Oppenheimer Approximation, Advances in Chemical Physics, Volume 57, 1984. https://doi.org/10.1002/9780470142813.ch2