Equation of Motion for the Solvent Polarization Apparent Charges in the Polarizable Continuum Model: Application to Real-Time TDDFT

When a solute charge density is evolving in time, e.g., due to an external perturbation, the solvent reaction field also becomes time-dependent, in a nontrivial way due to the delayed response of the solvent polarization rooted in its frequency-dependent dielectric constant, In polarizable continuum models, the time-dependent reaction field is represented by time-dependent apparent surface charges. Here, we derive general expressions for such charges. In particular, for all the main flavors of PCM, including IEF-PCM, we show how the frequency-dependent dielectric function terms can be singled-out in diagonal matrices, most convenient for Fourier transforming. For spherical cavities such formulation highlights the relation with multipolar solvation models and, when applied to the related context of metal nanoparticles, discloses a direct connection with multipolar plasmons. Using the Debye dielectric function, we derive a simple equation of motion for the apparent charges, free from system history. Such an equation has been coupled to real time time-dependent density functional theory (RT-TDDFT), to simulate the time evolution of the solute density rigorously accounting for the delayed solvent reaction field. The presented method seamlessly encompasses previous nonequilibrium approaches limited to an instantaneous-solute potential change (e.g., a sudden electronic excitation), does not require additional assumptions besides the basic PCM's, and is not limited to iterative inversion procedures. Numerical examples are given, showing the importance of accounting for the delayed solvent-response effects.