Vibronic coupling

Vibronic coupling (also called nonadiabatic coupling or derivative coupling) in a molecule involves the interaction between electronic and nuclear vibrational motion.^[1][2] 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.^[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.^[5][6][7]

Definition

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

${\displaystyle \mathbf {f} _{k'k}\equiv \langle \,\chi _{k'}(\mathbf {r} ;\mathbf {R} )\,|\,{\hat {\nabla }}_{\mathbf {R} }\chi _{k}(\mathbf {r} ;\mathbf {R} )\rangle _{(\mathbf {r} )}}$

Evaluation

The evaluation of vibronic coupling often involves 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.^[8]

First order accuracy can be achieved with forward difference formula:

${\displaystyle (\mathbf {f} _{k'k})_{l}\approx {\frac {1}{d}}\left[\gamma ^{k'k}(\mathbf {R} |\mathbf {R} +d\mathbf {e} _{l})-\gamma ^{k'k}(\mathbf {R} |\mathbf {R} )\right]}$

Second order accuracy can be achieved with central difference formula:

${\displaystyle (\mathbf {f} _{k'k})_{l}\approx {\frac {1}{2d}}\left[\gamma ^{k'k}(\mathbf {R} |\mathbf {R} +d\mathbf {e} _{l})-\gamma ^{k'k}(\mathbf {R} |\mathbf {R} -d\mathbf {e} _{l})\right]}$

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.

${\displaystyle \gamma ^{k'k}(\mathbf {R} _{1}|\mathbf {R} _{2})=\langle \chi _{k'}(\mathbf {r} ;\mathbf {R} _{1})\,|\,\chi _{k}(\mathbf {r} ;\mathbf {R} _{2})\rangle _{(\mathbf {r} )}}$

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 methods, 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

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.[9] For the implementation for SA-MCSCF and MRCI in COLUMBUS, please see ref.^[10]

TDDFT-based methods

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.^[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. 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:

${\displaystyle (\mathbf {f} _{k'k})_{l}={\frac {1}{E_{k}-E_{k'}}}\sum _{pq}\langle \psi _{p}|{\frac {\partial }{\partial \mathbf {e} _{l}}}{\hat {V}}_{\rm {ne}}|\psi _{q}\rangle (\gamma ^{k'k}(\mathbf {R} |\mathbf {R} ))_{pq}}$

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

Vibronic coupling is large in the case of two

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. Therefore, one of the most important applications of vibronic couplings is the quantitative calculation of internal conversion rates, through e.g. nonadiabatic molecular dynamics^[13] (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.^[14]

The singularity of vibronic coupling at conical intersections is responsible for the existence of

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

Alternatively, one can avoid explicit use of derivative couplings by switch from the

The first discussion of the effect of vibronic coupling on molecular spectra is given in the paper by Herzberg and Teller.[19] 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^[16]). Calculations of the lower

• Jahn–Teller effect – Mechanism of spontaneous symmetry breaking
• Born–Huang approximation
• Born–Oppenheimer approximation – The notion that the motion of atomic nuclei and electrons can be separated
• Conical intersection – the location of a discrete degeneracy between two electronic statesPages displaying wikidata descriptions as a fallback

References