Numerical Frequency Calculation:

The calculation of harmonic frequencies raises the problem of non-equilibrium solvation in the cosmo framework, because the molecular vibrations are on a time scale that do not allow a re-orientation of the solvent molecules. Therefore, the total response of the continuum is split into a fast contribution, described by the electronic polarization, and a slow term related to the orientational relaxation. As can be shown [126] the dielectric energy for the disturbed state can be written as

$${\frac{{1}}{{2}}} \varepsilon \bf q \bf P^{0}_{} \bf\Phi \bf P^{0}_{} {\frac{{1}}{{2}}} \bf q \bf P^{\Delta}_{} \bf\Phi \bf P^{\Delta}_{} \varepsilon \bf q \bf P^{0}_{} \bf\Phi \bf P^{\Delta}_{}$$

where $\bf P^{\Delta}_{}$ denotes the density difference between the distorted state and the initial state with density $\bf P^{0}_{}$. The interaction is composed of three contributions: the initial state dielectric energy, the interaction of the potential difference with the initial state charges, and the electronic screening energy that results from the density difference. The energy expression can be used to derive the correspondent gradients, which can be applied in a numerical frequency calculation. Because the cosmo cavity changes for every distorted geometry the initial state potential has to be mapped onto the new cavity in every step. The mapped potential of a segment of the new cavity is calculated from the distance-weighted potentials of all segments of the old cavity that fulfill a certain distance criterion. The mapped initial state screening charges are re-calculated from the new potential.