[3][4] The direct calculation of vibronic couplings used to be uncommon due to difficulties associated with its evaluation, but has recently gained popularity due to increased interest in the quantitative prediction of internal conversion rates, as well as the development of cheap but rigorous ways to analytically calculate the vibronic couplings, especially at the TDDFT level.
Each component of the vibronic coupling vector can be calculated with numerical differentiation methods using wave functions at displaced geometries.
Evaluating derivative couplings with analytic gradient methods has the advantage of high accuracy and very low cost, usually much cheaper than one single point calculation.
As a result, few programs have currently implemented analytic evaluation of vibronic couplings at wave function theory levels.
[10] The computational cost of evaluating the vibronic coupling using (multireference) wave function theory has led to the idea of evaluating them at the TDDFT level, which indirectly describes the excited states of a system without describing its excited state wave functions.
[5] In 2000, Chernyak and Mukamel[11] showed that in the complete basis set (CBS) limit, knowledge of the reduced transition density matrix between a pair of states (both at the unperturbed geometry) suffices to determine the vibronic couplings between them.
Therefore, modern implementations in molecular codes typically use expressions that include the Pulay force contributions, derived from the Lagrangian formalism.
This happens in the neighbourhood of an avoided crossing of potential energy surfaces corresponding to distinct electronic states of the same spin symmetry.
At the vicinity of conical intersections, where the potential energy surfaces of the same spin symmetry cross, the magnitude of vibronic coupling approaches infinity.
The large magnitude of vibronic coupling near avoided crossings and conical intersections allows wave functions to propagate from one adiabatic potential energy surface to another, giving rise to nonadiabatic phenomena such as radiationless decay.
When the potential energy surfaces of both the initial and the final electronic state are approximated by multidimensional harmonic oscillators, one can compute the internal conversion rate by evaluating the vibration correlation function, which is much cheaper than nonadiabatic molecular dynamics and is free from random noise; this gives a fast method to compute the rates of relatively slow internal conversion processes, for which nonadiabatic molecular dynamics methods are not affordable.
[15] The singularity of vibronic coupling at conical intersections is responsible for the existence of Geometric phase, which was discovered by Longuet-Higgins[16] in this context.
As a result, the algorithms to evaluate vibronic couplings at wave function theory levels, or between two excited states, are not yet implemented in many quantum chemistry program suites.
[18] When even an approximate calculation is unrealistic, the magnitude of vibronic coupling is often introduced as an empirical parameter determined by reproducing experimental data.
Alternatively, one can avoid explicit use of derivative couplings by switch from the adiabatic to the diabatic representation of the potential energy surfaces.