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Vibronic coupling (also called nonadiabatic coupling or derivative coupling) in a molecule involves the interaction between electronic and nuclear vibrational motion. The term "vibronic" originates from the combination of the terms "vibrational" and "electronic", denoting the idea that in a molecule, vibrational and electronic interactions are interrelated and influence each other. The magnitude of vibronic coupling reflects the degree of such interrelation. In theoretical chemistry, the vibronic coupling is neglected within the Born–Oppenheimer approximation. Vibronic couplings are crucial to the understanding of nonadiabatic processes, especially near points of conical intersections. 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.

Vibronic coupling describes the mixing of different electronic states as a result of small vibrations.

The evaluation of vibronic coupling often involves complex mathematical treatment.

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. This means an acceleration factor of 2N. However, the process involves intense mathematical treatment and programming. As a result, few programs have currently implemented analytic evaluation of vibronic couplings at wave function theory levels. Details about this method can be found in ref. For the implementation for SA MCSCF and MRCI in COLUMBUS, please see ref.

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. However, the derivation of the TDDFT vibronic coupling theory is not trivial, since there are no electronic wave functions in TDDFT that are available for plugging into the defining equation of the vibronic coupling. 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. The vibronic couplings between two electronic states are given by contracting their reduced transition density matrix with the geometric derivatives of the nuclear attraction operator, followed by dividing by the energy difference of the two electronic states:

This enables one to calculate the vibronic couplings at the TDDFT level, since although TDDFT does not give excited state wave functions, it does give reduced transition density matrices, not only between the ground state and an excited state, but also between two excited states. The proof of the Chernyak Mukamel formula is straightforward and involves the Hellmann Feynman theorem. While the formula provides useful accuracy for a plane wave basis (see e. g. ref. ), it converges extremely slowly with respect to the basis set if an atomic orbital basis set is used, due to the neglect of the Pulay force. Therefore, modern implementations in molecular codes typically use expressions that include the Pulay force contributions, derived from the Lagrangian formalism. (including but not limited to surface hopping and path integral molecular dynamics). 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. The singularity of vibronic coupling at conical intersections is responsible for the existence of Geometric phase, which was discovered by Longuet Higgins in this context. Geometric phase around the conical intersection involving the ground electronic state of the C6H3F3+ molecular ion is discussed on pages 385 386 of the textbook by Bunker and Jensen.

Although crucial to the understanding of nonadiabatic processes, direct evaluation of vibronic couplings has been very limited until very recently. Evaluation of vibronic couplings is often associated with severe difficulties in mathematical formulation and program implementations. 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. By comparison, vibronic couplings between the ground state and an excited state at the TDDFT level, which are easy to formulate and cheap to calculate, are more widely available. The evaluation of vibronic couplings typically requires correct description of at least two electronic states in regions where they are strongly coupled. This usually requires the use of multi reference methods such as MCSCF and MRCI, which are computationally demanding and delicate quantum chemical methods. However, there are also applications where vibronic couplings are needed but the relevant electronic states are not strongly coupled, for example when calculating slow internal conversion processes; in this case even methods like TDDFT, which fails near ground state excited state conical intersections, can give useful accuracy. Moreover, TDDFT can describe the vibronic coupling between two excited states in a qualitatively correct fashion, even if the two excited states are very close in energy and therefore strongly coupled (provided that the equation of motion (EOM) variant of the TDDFT vibronic coupling is used in place of the time dependent perturbation theory (TDPT) variant 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. Although rigorous validation of a diabatic representation requires knowledge of vibronic coupling, it is often possible to construct such diabatic representations by referencing the continuity of physical quantities such as dipole moment, charge distribution or orbital occupations. However, such construction requires detailed knowledge of a molecular system and introduces significant arbitrariness. Diabatic representations constructed with different method can yield different results and the reliability of the result relies on the discretion of the researcher.

The first discussion of the effect of vibronic coupling on molecular spectra is given in the paper by Herzberg and Teller. Although the Herzberg–Teller effect appears to be the result of either vibronic coupling or the dependence of the electronic transition moment on the nuclear coordinates, it can be shown that these two apparently different causes of the Herzberg–Teller effect in a spectrum are two manifestations of the same phenomenon (see Section 14.1.9 of the book by Bunker and Jensen