Vibronic coupling

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.

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

The evaluation of vibronic coupling often involve complex mathematical treatment.

The form of vibronic coupling is essentially 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, ${\displaystyle \mathbf {e} _{l}}$ is a unit vector along direction ${\displaystyle l}$. ${\displaystyle \gamma ^{k'k}}$ is the transition density between the two electronic states.

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