7.13.10 State-Specific PCM with RAS-SF

The polarizable continuum model (PCM) is an efficient way to incorporate dielectric boundary conditions into a ground-state quantum chemistry calculation as described in Section 11.2.3, although some additional choices exist regarding how to model electronic excited states within this formalism. ⁵⁰⁵ Herbert J. M.
Wiley Interdiscip. Rev.: Comput. Mol. Sci.
(2021), 11, pp. e1519. Link For vertical transition energies computed using RAS-SF wave functions, both equilibrium and non-equilibrium versions of a state-specific PCM solvation correction are available. ³² Alam B. et al.
J. Chem. Phys.
(2022), 156, pp. 194110. Link Ground-state polarization is described using the solvent’s static dielectric constant, and in the nonequilibrium solvation approach that polarization is modified upon vertical excitation using the solvent’s optical dielectric constant.

The equilibrium formulation solves the state-specific Schrödinger equation

where ${\widehat{R}}_k$ is the reaction field operator for state $k$. Since ${\widehat{R}}_k$ depends on $|{\mathrm{\Psi }}_k⟩$, this equation must be solved iteratively. The fully relaxed (equilibrium) excitation free energy is

(It is a free energy because the continuum description of the solvent implicitly includes averaging over solvent degrees of freedom.) We use $G$ to represent a free energy and $W$ to represent polarization work, with subscripts to indicate the electronic state in question.

The nonequilibrium version of the solvation model uses a perturbative framework to solve the Schrödinger equation. This is described in more detail in Section 11.2.3.3. Briefly, this approach is based on a state-specific Hamiltonian that is partitioned according to

The zeroth-order Hamiltonian (${\widehat{H}}_0$) includes the ground-state, equilibrium reaction-field operator ${\widehat{R}}_0\equiv R_0^{\text{s+f}}={\widehat{R}}_0^{\text{s}}+{\widehat{R}}_0^{\text{f}}$. The solvent polarization is divided into “slow” (nuclear) and “fast” (electronic) components, but for equilibrium solvation both components are included. The perturbation (indicated by perturbation parameter $\lambda$ corrects the fast polarization upon excitation, $|{\mathrm{\Psi }}_0⟩\to |{\mathrm{\Psi }}_k⟩$. This nonequilibrium correction is described via first-order perturbation theory. The result is a perturbative, state-specific (ptSS) correction to the excitation energy:

For further details see Ref.  32 Alam B. et al.
J. Chem. Phys.
(2022), 156, pp. 194110. Link .