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 is not common due to difficulties associated with its evaluation. Definition Vibronic coupling describes the mixing of different electronic states as a result of small vibrations. Evaluation The evaluation of vibronic coupling often involve complex mathematical treatment. Numerical gradients The form of vibronic coupling is essentially the derivative of the wave function. Each component of the vibronic coupling vector can be calculated with numerical differentiation methods using wave functions at displaced geometries. This is the procedure used in MOLPRO. First order accuracy can be achieved with forward difference formula: Second order accuracy can be achieved with central difference formula: Here, is a unit vector along direction . is the transition density between the two electronic states. Evaluation of electronic wave functions for both electronic states are required at N displacement geometries for first order accuracy and 2*N displacements to achieve second order accuracy, where N is the number of nuclear degrees of freedom. This can be extremely computationally demanding for large molecules. As with other numerical differentiation method, the evaluation of nonadiabatic coupling vector with this method is numerically unstable, limiting the accuracy of the result. Moreover, the calculation of the two transition densities in the numerator are not straightforward. The wave functions of both electronic states are expanded with Slater determinants or configuration state functions (CSF). The contribution from the change of CSF basis is too demanding to evaluate using numerical method, and is usually ignored by employing an approximate diabatic CSF basis. This will also cause further inaccuracy of the calculated coupling vector, although this error is usually tolerable. Analytic gradient methods 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