12.2.2 Polarizable Continuum Models

The PCM literature has a long history⁹²³ and there are several different models in widespread use; connections between these models have not always been appreciated.^{174, 137, 175, 525} Chipman^{174, 175} has shown how various PCMs can be formulated within a common theoretical framework; see Ref. 373 for a pedagogical introduction. The PCM takes the form of a set of linear equations,

$$\mathrm{𝐊𝐪}=\mathrm{𝐑𝐯},$$

(12.2)

in which the induced charges $q_i$ at the cavity surface discretization points (organized into a vector $𝐪$ in Eq. (12.2)) are computed from the values $v_i$ of the solute’s electrostatic potential at those same discretization points. The form of the matrices $𝐊$ and $𝐑$ depends upon the particular PCM in question. These matrices are given in Table 12.3 for the PCMs that are available in Q-Chem.

The oldest PCM is the so-called D-PCM model of Tomasi and coworkers,⁶³⁹ but unlike the models listed in Table 12.3, D-PCM requires explicit evaluation of the electric field normal to the cavity surface, This is undesirable, as evaluation of the electric field is both more expensive and more prone to numerical problems as compared to evaluation of the electrostatic potential. Moreover, the dependence on the electric field can be formally eliminated at the level of the integral equation whose discretized form is given in Eq. (12.2).¹⁷⁴ As such, D-PCM is essentially obsolete, and the PCMs available in Q-Chem require only the evaluation of the electrostatic potential, not the electric field.

The simplest PCM that continues to enjoy widespread use is the Conductor-Like Screening Model (COSMO) introduced by Klamt and Schüürmann.⁴⁷⁷ Truong and Stefanovich⁹³⁴ later implemented the same model with a slightly different dielectric scaling factor ($f_{\epsilon }$ in Table 12.3), and called this modification GCOSMO. The latter was implemented within the PCM formalism by Barone and Cossi et al.,^{57, 195} who called the model C-PCM (for “conductor-like” PCM). In each case, the dielectric screening factor has the form

$$f_{\epsilon }=\frac{\epsilon -1}{\epsilon +x},$$

(12.3)

where Klamt and Schüürmann proposed $x=1/2$ but $x=0$ was used in GCOSMO and C-PCM. The latter value is the correct choice for a single charge in a spherical cavity (i.e., the Born ion model), although Klamt and coworkers suggest that $x=1/2$ is a better compromise, given that the Kirkwood-Onsager analytical result is $x=\mathrm{\ell }/(\mathrm{\ell }+1)$ for an $\mathrm{\ell }$th-order multipole centered in a spherical cavity.^{477, 51} The distinction is irrelevant in high-dielectric solvents; the $x=0$ and $x=1/2$ values of $f_{\epsilon }$ differ by only 0.6% for water at 25${}^{\circ }$C, for example. Truong⁹³⁴ argues that $x=0$ does a better job of preserving Gauss’ Law in low-dielectric solvents, but more accurate solvation energies (at least for neutral molecules, as compared to experiment) are sometimes obtained using $x=1/2$ (Ref. 57). This result is likely highly sensitive to cavity construction, and in any case, both versions are available in Q-Chem.

Whereas the original COSMO model introduced by Klamt and Schüürmann⁴⁷⁷ corresponds to Eq. (12.2) with $𝐊$ and $𝐑$ as defined in Table 12.3, Klamt and coworkers later introduced a correction for outlying charge that goes beyond Eq. (12.2).^{475, 51} Klamt now consistently refers to this updated model as “COSMO”,⁴⁷⁹ and we shall adopt this nomenclature as well. COSMO, with the outlying charge correction, is available in Q-Chem and is described in Section 12.2.7. In contrast, C-PCM consists entirely of Eq. (12.2) with matrices $𝐊$ and $𝐑$ as defined in Table 12.3, although it is possible to modify the dielectric screening factor to use the $x=1/2$ value (as in COSMO) rather than the $x=0$ value. Additional non-electrostatic terms can be added at the user’s discretion, as discussed below, but there is no explicit outlying charge correction in C-PCM. These and other fine-tuning details for PCM jobs are controllable via the $pcm input section that is described in Section 12.2.3.

As compared to C-PCM, a more sophisticated treatment of continuum electrostatic interactions is afforded by the “surface and simulation of volume polarization for electrostatics” [SS(V)PE] approach.¹⁷⁴ Formally speaking, this model provides an exact treatment of the surface polarization (i.e., the surface charge induced by the solute charge that is contained within the solute cavity, which induces a surface polarization owing to the discontinuous change in dielectric constant across the cavity boundary) but also an approximate treatment of the volume polarization (arising from the aforementioned outlying charge). The “SS(V)PE” terminology is Chipman’s notation,¹⁷⁴ but this model is formally equivalent, at the level of integral equations, to the “integral equation formalism” (IEF-PCM) that was developed originally by Cancès et al..^{138, 924} Some difference do arise when the integral equations are discretized to form finite-dimensional matrix equations,⁵²⁵ and it should be noted from Table 12.3 that SS(V)PE uses a symmetrized form of the $𝐊$ matrix as compared to IEF-PCM. The asymmetric IEF-PCM is the recommended approach,⁵²⁵ although only the symmetrized version is available in the isodensity implementation of SS(V)PE that is discussed in Section 12.2.5. As with the obsolete D-PCM approach, the original version of IEF-PCM explicitly required evaluation of the normal electric field at the cavity surface, but it was later shown that this dependence could be eliminated to afford the version described in Table 12.3.^{174, 137} This version requires only the electrostatic potential, and is thus preferred, and it is this version that we designate as IEF-PCM. The C-PCM model becomes equivalent to SS(V)PE in the limit $\epsilon \to \mathrm{\infty }$,^{174, 525} which means that C-PCM must somehow include an implicit correction for volume polarization, even if this was not by design.⁴⁷⁵ For $\epsilon \gtrsim 50$, numerical calculations reveal that there is essentially no difference between SS(V)PE and C-PCM results.⁵²⁵ Since C-PCM is less computationally involved as compared to SS(V)PE, it is the PCM of choice in high-dielectric solvents. The computational savings relative to SS(V)PE may be particularly significant for large QM/MM/PCM jobs. For a more detailed discussion of the history of these models, see the lengthy and comprehensive review by Tomasi et al..⁹²³ For a briefer discussion of the connections between these models, see Refs. 175, 525, 373.

Construction of the cavity surface is a crucial aspect of PCMs, as computed properties are quite sensitive to the details of the cavity construction. Most cavity constructions are based on a union of atom-centered spheres (see Fig. 12.1), but there are yet several different constructions whose nomenclature is occasionally confused in the literature. Simplest and most common is the van der Waals (vdW) surface consisting of a union of atom-centered spheres. Traditionally,^{107, 925} and by default in Q-Chem, the atomic radii are taken to be 1.2 times larger than vdW radii extracted from crystallographic data, originally by Bondi (and thus sometimes called “Bondi radii”).¹⁰⁸ This 20% augmentation is intended to mimic the fact that solvent molecules cannot approach all the way to the vdW radius of the solute atoms, though it’s not altogether clear that this is an optimal value. (The default scaling factor in Q-Chem is 1.2 but can be modified by the user.) An alternative to scaling the atomic radii is to add a certain fixed increment to each, representing the approximate size of a solvent molecule (e.g., 1.4 Å for water) and leading to what is known as the solvent accessible surface (SAS). From another point of view, the SAS represents the surface defined by the center of a spherical solvent molecule as it rolls over the vdW surface, as suggested in Fig. 12.1. Both the vdW surface and the SAS possess cusps where the atomic spheres intersect, although these become less pronounced as the atomic radii are scaled or augmented. These cusps are eliminated in what is known as the solvent-accessible surface (SES), sometimes called the Connolly surface or the “molecular surface". The SES uses the surface of the probe sphere at points where it is simultaneously tangent to two or more atomic spheres to define elements of a “re-entrant surface” that smoothly connects the atomic (or “contact”) surface.⁵²¹

Having chosen a model for the cavity surface, this surface is discretized using atom-centered Lebedev grids^{542, 543, 540} of the same sort that are used to perform the numerical integrations in DFT. (Discretization of the re-entrant facets of the SES is somewhat more complicated but similar in spirit.⁵²¹) Surface charges $q_i$ are located at these grid points and the Lebedev quadrature weights can be used to define the surface area associated with each discretization point.⁵²³

A long-standing (though not well-publicized) problem with the aforementioned discretization procedure is that it fails to afford continuous potential energy surfaces as the solute atoms are displaced, because certain surface grid points may emerge from, or disappear within, the solute cavity, as the atomic spheres that define the cavity are moved. This undesirable behavior can inhibit convergence of geometry optimizations and, in certain cases, lead to very large errors in vibrational frequency calculations.⁵²³ It is also a fundamental hindrance to molecular dynamics calculations.⁵²⁴ Building upon earlier work by York and Karplus,¹⁰³⁵ Lange and Herbert^{523, 524, 521} developed a general scheme for implementing apparent surface charge PCMs in a manner that affords smooth potential energy surfaces, even for ab initio molecular dynamics simulations involving bond breaking.^{524, 373} Notably, this approach is faithful to the properties of the underlying integral equation theory on which the PCMs are based, in the sense that the smoothing procedure does not significantly perturb solvation energies or cavity surface areas.^{523, 524} The smooth discretization procedure combines a switching function with Gaussian blurring of the cavity surface charge density, and is thus known as the “Switching/Gaussian” (SWIG) implementation of the PCM.

Both single-point energies and analytic energy gradients are available for SWIG PCMs, when the solute is described using molecular mechanics or an SCF (Hartree-Fock or DFT) electronic structure model, except that for the SES cavity model only single-point energies are available. Analytic Hessians are available for the C-PCM model only. (As usual, vibrational frequencies for other models will be computed, if requested, by finite difference of analytic energy gradients.) Single-point energy calculations using correlated wave functions can be performed in conjunction with these solvent models, in which case the correlated wave function calculation will use Hartree-Fock molecular orbitals that are polarized in the presence of the continuum dielectric solvent (i.e., there is no post-Hartree–Fock PCM correction).

Researchers who use these PCMs are asked to cite Refs. 524, 525, which provide the details of Q-Chem’s implementation, and Ref. 521 if the SES is used. (We point the reader in particular to Ref. 524, which provides an assessment of the discretization errors that can be anticipated using various PCMs and Lebedev grids; default grid values in Q-Chem were established based on these tests.) When publishing results based on PCM calculations, it is essential to specify both the precise model that is used (see Table 12.3) as well as how the cavity was constructed.

For example, “Bondi radii multiplied by 1.2”, which is the Q-Chem default, except for hydrogen, where the factor is reduced to 1.1,⁸⁰³ as per usual. Radii for main-group elements that were not provided by Bondi are taken from Ref. 604. Absent details such as these, PCM calculations will be difficult to reproduce in other electronic structure programs.

In vertical excitation or ionization, the solute undergoes a sudden change in its charge distribution. Various microscopic motions of the solvent have characteristic times to reach certain polarization response, and fast part of the solvent response (electrons) can follow such a dynamic process while the remaining degrees of freedom (nuclei) remain unchanged as in the initial state. Such splitting of the solvent response gives rise to nonequilibrium solvation. In the literature, two different approaches have been developed for describing nonequilibrium solvent effects: the linear response (LR) approach^{134, 193} and the state-specific (SS) approach.^{925, 135, 192, 409} Both are implemented in Q-Chem,¹⁰³⁹,at the SCF level for vertical ionization and at the corresponding level (CIS, TDDFT or ADC, see Section 7.9.7) for vertical excitation. A brief introduction to these methods is given below, and users of the nonequilibrium PCM features are asked to cite Refs. 1039 and 635. State-specific solvent-field equilibration for long-lived excited states to compute e.g. emission energies is implemented for the ADC-suite of methods as described in section 7.9.7. Users of this equilibrium-solvation PCM please cite and be referred to Ref. 634.

The LR approach considers the solvation effects as a coupling between a pair of transitions, one for solute and the other for solvent. The transition frequencies when the interaction between the solute and solvent is turned on may be determined by considering such an interaction as a perturbation. In the framework of TDDFT, the solvent/solute interaction is given by³⁹⁸

(12.4)

where $\chi$ is the charge density response function of the solvent and ${\rho }^{\mathrm{tr}}(𝐫)$ is the solute’s transition density. This term accounts for a dynamical correction to the transition energy so that it is related to the response of the solvent to the charge density of the solute oscillating at the solute transition frequency ($\omega$). Within a PCM, only classical Coulomb interactions are taken into account, and Eq. (12.4) becomes

(12.5)

where $𝒬$ is PCM solvent response operator for a generic dielectric constant, $\epsilon$. The integral of $𝒬$ and the potential of the density ${\rho }^{\mathrm{tr}}$ gives the surface charge density for the solvent polarization.

The state-specific (SS) approach takes into account the capability of a part of the solvent degrees of freedom to respond instantaneously to changes in the solute wave function upon excitation. Such an effect is not accounted for in the LR approach. In SS, a generic solvated-solute excited state ${\mathrm{\Psi }}_i$ is obtained as a solution of a nonlinear Schrödinger equation

$$\left({\widehat{H}}^{\mathrm{vac}}+{\widehat{V}}_0^{\mathrm{slow}}+{\widehat{V}}_i^{\mathrm{fast}}\right)|{\mathrm{\Psi }}_i⟩=E_i^{\mathrm{SS}}|{\mathrm{\Psi }}_i⟩$$

(12.6)

that depends upon the solute’s charge distribution. Here ${\widehat{H}}^{\mathrm{vac}}$ is the usual Hamiltonian for the solute in vacuum and the reaction field operator ${\widehat{V}}_i$ generates the electrostatic potential of the apparent surface charge density (Section 12.2.2.1), corresponding to slow and fast polarization response. The solute is polarized self-consistently with respect to the solvent’s reaction field. In case of vertical ionization rather than excitation, both the ionized and non-ionized states can be treated within a ground-state formalism. For vertical excitations, self-consistent SS models have been developed for various excited-state methods,^{409, 615} including both CIS and TDDFT.

In a linear dielectric medium, the solvent polarization is governed by the electric susceptibility, $\chi =[\epsilon (\omega )-1]/4\pi$, where $\epsilon (\omega )$ is the frequency-dependent permittivity. In case of very fast vertical transitions, the dielectric response is ruled by the optical dielectric constant, ${\epsilon }_{\mathrm{opt}}=n^2$, where $n$ is the solvent’s index of refraction. In both LR and SS, the fast part of the solvent’s degrees of freedom is in equilibrium with the solute density change. Within PCM, the fast solvent polarization charges for the SS excited state $i$ can be obtained by solving the following equation:¹⁹²

$${𝐊}_{{\epsilon }_{\mathrm{opt}}}{𝐪}_i^{\mathrm{fast},\mathrm{SS}}={𝐑}_{{\epsilon }_{\mathrm{opt}}}\left[{𝐯}_i+𝐯({𝐪}_0^{\mathrm{slow}})\right].$$

(12.7)

Here ${𝐪}^{\mathrm{fast},\mathrm{SS}}$ is the discretized fast surface charge. The dielectric constants in the matrices $𝐊$ and $𝐑$ (Section 12.2.2.1) are replaced with the optical dielectric constant, and ${𝐯}_i$ is the potential of the solute’s excited state density, ${\rho }_i$. The quantity $𝐯({𝐪}_0^{\mathrm{slow}})$ is the potential of the slow part of the apparent surface charges in the ground state, which are given by

$${𝐪}_0^{\mathrm{slow}}=\left(\frac{\epsilon -{\epsilon }_{\mathrm{opt}}}{\epsilon -1}\right){𝐪}_0.$$

(12.8)

For LR-PCM, the solvent polarization is subjected to the first-order changes to the electron density (TDDFT linear density response), and thus Eq. (12.7) becomes

$${𝐊}_{{\epsilon }_{\mathrm{opt}}}{𝐪}_i^{\mathrm{fast},\mathrm{LR}}={𝐑}_{{\epsilon }_{\mathrm{opt}}}𝐯({\rho }_i^{\mathrm{tr}}).$$

(12.9)

The LR approach for CIS/TDDFT excitations and the self-consistent SS method (using the ground-state SCF) for vertical ionizations are available in Q-Chem. The self-consistent SS method for vertical excitations is not available, because this method is problematic in the vicinity of (near-) degeneracies between excited states, such as in the vicinity of a conical intersection. The fundamental problem in the SS approach is that each wave function ${\mathrm{\Psi }}_i$ is an eigenfunction of a different Hamiltonian, since Eq. (12.6) depend upon the specific state of interest. To avoid the ordering and the non-orthogonality problems, we compute the vertical excitation energy using a first-order, perturbative approximation to the SS approach,^{133, 140} in what we have termed the “ptSS” method.⁶³⁵ The zeroth-order excited-state wave function can be calculated using various excited-state methods (currently available for CIS and TDDFT in Q-Chem) with solvent-relaxed molecular orbitals obtained from a ground-state PCM calculation. As mentioned previously, LR and SS describe different solvent relaxation features in nonequilibrium solvation. In the perturbation scheme, we can calculate the LR contribution using the zeroth-order transition density, in what we have called the "ptLR" approach. The combination of ptSS and ptLR yields quantitatively good solvatochromatic shifts in combination with TDDFT but not with the correlated variants of ADC, for which the pure ptSS approach was shown to be superior.^{1039, 635}

The LR and SS approaches can also be used in the study of photon emission processes.⁴¹⁰ An emission process can be treated as a vertical excitation at a stationary point on the excited-state potential surface. The basic requirement therefore is to prepare the solvent-relaxed geometry for the excited-state of interest. TDDFT/C-PCM analytic gradients and Hessian are available.

Section 7.3.5 for computational details regarding excited-state geometry optimization with PCM. An emission process is slightly more complicated than the absorption case. Two scenarios are discussed in literature, depending on the lifetime of an excited state in question. In the limiting case of ultra-fast excited state decay, when only fast solvent degrees of freedom are expected to be equilibrated with the excited-state density. In this limit, the emission energy can be computed exactly in the same way as the vertical excitation energy. In this case, excited state geometry optimization should be performed in the nonequilibrium limit. The other limit is that of long-lived excited state, e.g., strongly fluorescent species and phosphorescence. In the long-lived case, excited state geometry optimization should be performed with the solvent equilibrium limit. Thus, the excited state should be computed using an equilibrium LR or SS approach, and the ground state is calculated using nonequilibrium self-consistent SS approach. The latter approach is implemented for the ADC-based methods as described in Section 7.9.7.