7.3.4 TDDFT + PCM for Excitation Energies and Excited-State Properties

As described in Section 11.2.3, polarizable continuum models (PCMs) are a simple means of including solvation effects in quantum chemistry calculations, at the level of a dielectric continuum description.⁴⁶¹ The “conductor-like PCM” (C-PCM) is one such model, which can be combined with TDDFT to include the effects on solvent on electronic spectra. The TDDFT/C-PCM combination can also be used to perform excited-state geometry optimizations and vibrational frequency calculations in solution.⁷⁰⁴

When PCMs are used in the context of vertical excitation energy calculations, the solvent around the vertically excited solute is out of equilibrium because the (implicit) solvent molecules cannot reorient in response to a vertical excitation process.⁴⁶¹ Whereas the solvent’s static or zero-frequency dielectric constant (${\epsilon }_0$) describes all of the solvent response mechanisms (electronic, vibrational, and orientational), only the electronic part of the response is appropriate to include in a vertical excitation energy calculation. Polarization of the electrons (only) is encoded in the solvent’s optical or infinite-frequency dielectric constant (${\epsilon }_{\mathrm{\infty }}$), which is equal to the square of the solvent’s index of refraction (${\epsilon }_{\mathrm{\infty }}=n_{\text{refr}}^2$). The difference between these two values can be stark, e.g., ${\epsilon }_0=78$ versus ${\epsilon }_{\mathrm{\infty }}=1.8$ for water. For geometry optimizations in solution, however, it is probably appropriate to use the numerical value of the static dielectric constant for ${\epsilon }_{\mathrm{\infty }}$, on the assumption that the solvent molecules have time to reorient in response to changes in the solute’s geometry, even in an electronically excited state.

The sample job provided below computes the a PCM contribution to the TDDFT linear-response matrix using a dielectric constant ${\epsilon }_{\mathrm{\infty }}$ (specified using the keyword OpticalDielectric in the $pcm input section), whereas the value ${\epsilon }_0$) (specified using Dielectric in the $pcm section) is used to polarize the ground-state MOs. This corresponds to “full linear response theory” (LR-PCM), as described in Ref. 704. A perturbative approximation to full LR-PCM is also available, along with state-specific corrections that are somewhat more theoretically rigorous,⁴⁶¹ and have also been implemented for TDDFT.^{776, 1261} Those approaches are described in Section 11.2.3.3 following a thorough introduction to the PCM approach. See Ref. 461 for a general overview of the theory of nonequilibrium dielectric continuum methods. Additional PCM job control options are discussed in Section 11.2.