Analytical derivative couplings within the framework of time-dependent density functional theory coupled with conductor-like polarizable continuum model: Formalism, implementation, and applications.

The nonadiabatic phenomena, which are characterized by a strong coupling between electronic and nuclear motions, are ubiquitous. The nonadiabatic effect of the studied system can be significantly affected by the surrounding environment, such as solvents, in which such nonadiabatic process takes place. It is essential to develop the theoretical models to simulate these processes while accurately modeling the solvent environment. The time-dependent density functional theory (TDDFT) is currently the most efficient approach to describe the electronic structures and dynamics of complex systems, while the polarizable continuum model (PCM) represents one of the most successful examples among continuum solvation models. Here, we formulate the first-order derivative couplings (DCs) between the ground and excited states as well as between two excited states by utilizing time-independent equation of motion formalism within the framework of both linear response and spin flip formulations of TDDFT/CPCM (the conductor-like PCM), and implement the analytical DCs into the Q-CHEM electronic structure software package. The analytic implementation is validated by the comparison of the analytical and finite-difference results, and reproducing geometric phase effect in the protonated formaldimine test case. Taking 4-(N,N-dimethylamino)benzonitrile and uracil in the gas phase and solution as an example, we demonstrate that the solvent effect is essential not only for the excitation energies of the low-lying excited-states but also for the DCs between these states. Finally, we calculate the internal conversion rate of benzophenone in a solvent with DC being used. The current implementation of analytical DCs together with the existing analytical gradient and Hessian of TDDFT/PCM excited states allows one to study the nonadiabatic effects of relatively large systems in solutions with low computational cost.