11.2 Chemical Solvent Models

Ab initio quantum chemistry makes possible the study of gas-phase molecular properties from first principles. In liquid solution, however, these properties may change significantly, especially in polar solvents. Although it is possible to model solvation effects by including explicit solvent molecules in the quantum-chemical calculation (e.g. a super-molecular cluster calculation, averaged over different configurations of the molecules in the first solvation shell), such calculations are very computationally demanding. Furthermore, cluster calculations typically do not afford accurate solvation energies, owing to the importance of long-range electrostatic interactions. Accurate prediction of solvation free energies is, however, crucial for modeling of chemical reactions and ligand/receptor interactions in solution.

Q-Chem contains several different implicit solvent models, which differ greatly in their level of sophistication. These are generally known as self-consistent reaction field (SCRF) models, because the continuum solvent establishes a “reaction field” (additional terms in the solute Hamiltonian) that depends upon the solute electron density, and must therefore be updated self-consistently during the iterative convergence of the wave function. The simplest and oldest of these models that is available in Q-Chem is the Kirkwood-Onsager model,^{477, 712, 478} in which the solute molecule is placed inside of a spherical cavity and its electrostatic potential is represented in terms of a single-center multipole expansion. More sophisticated models, which use a molecule-shaped cavity and the full molecular electrostatic potential, include the conductor-like screening model⁴⁸⁴ (COSMO) and the closely related conductor-like PCM (C-PCM),^{974, 42, 186} along with the “surface and simulation of volume polarization for electrostatics” [SS(V)PE] model.¹⁶³ The latter is also known as the “integral equation formalism” (IEF-PCM).^{122, 123}

The C-PCM and IEF-PCM/SS(V)PE are examples of what are called “apparent surface charge” SCRF models, although the term polarizable continuum models (PCMs), as popularized by Tomasi and coworkers,⁹⁶³ is now used almost universally to refer to this class of solvation models. Q-Chem employs a Switching/Gaussian or “SWIG” implementation of these PCMs.^{531, 532, 533, 376, 529} This approach resolves a long-standing—though little-publicized—problem with standard PCMs, namely, that the boundary-element methods used to discretize the solute/continuum interface may lead to discontinuities in the potential energy surface for the solute molecule. These discontinuities inhibit convergence of geometry optimizations, introduce serious artifacts in vibrational frequency calculations, and make ab initio molecular dynamics calculations virtually impossible.^{531, 532} In contrast, Q-Chem’s SWIG PCMs afford potential energy surfaces that are rigorously continuous and smooth. Unlike earlier attempts to obtain smooth PCMs, the SWIG approach largely preserves the properties of the underlying integral-equation solvent models, so that solvation energies and molecular surface areas are hardly affected by the smoothing procedure.

Other solvent models available in Q-Chem include the “Langevin dipoles” model;^{260, 261} as well as versions 8 and 12 of the SM$x$ models, and the SMD model, developed at the University of Minnesota.^{635, 633, 632} SM8 and SM12 are based upon the generalized Born method for electrostatics, augmented with atomic surface tensions intended to capture non-electrostatic effects (cavitation, dispersion, exchange repulsion, and changes in solvent structure). Empirical corrections of this sort are also available for the PCMs mentioned above, but within SM8 and SM12 these parameters have been optimized to reproduce experimental solvation energies. SMD (where the “D” is for “density") combines IEF-PCM with the non-electrostatic corrections, but because the electrostatics is based on the density rather than atomic point charges, it is supported for arbitrary basis sets whereas SM8 and SM12 are not.

Table 11.1 summarizes the implicit solvent models that are available in Q-Chem. Solvent models are invoked via the SOLVENT_METHOD keyword, as shown below. Additional details about each particular solvent model can be found in the sections that follow. In general, these methods are available for any SCF level of electronic structure theory, though in the case of SM8 only certain basis sets are supported. Post-Hartree–Fock calculations can be performed by first running an SCF + PCM job, in which case the correlated wave function will employ MOs and Hartree-Fock energy levels that are polarized by the solvent.

Table 11.2 summarizes the analytical energy gradient and Hessian available with implicit solvent models. For unsupported methods, finite difference methods may be used for performing geometry optimizations and frequency calculations.

Note:  The job-control format for specifying implicit solvent models changed significantly starting in Q-Chem version 4.2.1. This change was made in an attempt to simply and unify the input notation for a large number of different models.

Before going into detail about each of these models, a few potential points of confusion warrant mention, with regards to nomenclature. First, “PCM” refers to a family of models that includes C-PCM and SS(V)PE/IEF-PCM (the latter two being completely equivalent¹²³). One or the other of these models can be selected by additional job control variables in a $pcm input section, as described in Section 11.2.2. COSMO is very similar to C-PCM but includes a correction for that part of the solute’s electron density that penetrates beyond the cavity (the so-called “outlying charge”).^{482, 36} This is discussed in Section 11.2.7.

Two implementations of the SS(V)PE model are also available. The PCM implementation (which is requested by setting SOLVENT_METHOD = PCM) uses a solute cavity constructed from atom-centered spheres, as with most other PCMs. On the other hand, setting SOLVENT_METHOD = ISOSVP requests an SS(V)PE calculation in which the solute cavity is defined by an isocontour of the solute’s own electron density, as advocated by Chipman.^{163, 164, 161} This is an appealing, one-parameter cavity construction, although it is unclear that this construction alone is superior in its accuracy to carefully-parameterized atomic radii,⁴¹ at least not without additional, non-electrostatic terms included,^{770, 771, 772, 773} which are available in Q-Chem’s implementation of the isodensity version of SS(V)PE (Section 11.2.6). Moreover, analytic energy gradients are not available for the isodensity cavity construction, whereas they are available when the cavity is constructed from atom-centered spheres. One additional subtlety, which is discussed in detail in Ref. 533, is the fact that the PCM implementation of the equation for the SS(V)PE surface charges [Eq. (11.2)] uses an asymmetric $𝐊$ matrix. In contrast, Chipman’s isodensity implementation uses a symmetrized $𝐊$ matrix. Although the symmetrized version is somewhat more computationally efficient when the number of surface charges is large, the asymmetric version is better justified, theoretically.⁵³³ (This admittedly technical point is clarified in Section 11.2.2 and in particular in Table 11.3.)

Regarding the accuracy of these models for solvation free energies ($\mathrm{\Delta }{G}_{298}$), SM8 achieves sub-kcal/mol accuracy for neutral molecules, based on comparison to a large database of experimental values, although average errors for ions are more like 4 kcal/mol.¹⁸⁸ To achieve comparable accuracy with IEF-PCM/SS(V)PE, non-electrostatic terms must be included.^{483, 770, 772} The SM12 model does not improve upon SM8 in any statistical sense,⁶³³ but does lift one important restriction on the level of electronic structure that can be combined with these models. Specifically, the Generalized Born model used in SM8 is based on a variant of Mulliken-style atomic charges, and is therefore parameterized only for a few small basis sets, e.g., 6-31G*. SM12, on the other hand, uses a variety of charge schemes that are stable with respect to basis-set expansion, and can therefore be combined with any level of electronic structure theory for the solute. Like IEF-PCM, the SMD model is also applicable to any basis sets, and its accuracy is comparable to SM8 and SM12.⁶³² Quantitative fluid-phase thermodynamics can also be obtained using Klamt’s COSMO-RS approach,^{485, 480} where RS stands for “real solvent”. The COSMO-RS approach is not included in Q-Chem and requires the COSMOtherm program, which is licensed separately through COSMOlogic,^COSMOlogic but Q-Chem can write the input files that are need by COSMOtherm.

The following sections provide more details regarding theory and job control for the various implicit solvent models that are available in Q-Chem. In addition, recent review articles are available for PCM methods,⁹⁶³ SM$x$,¹⁸⁸ and COSMO.⁴⁸⁶ Formal relationships between various PCMs have been discussed in Refs. 164, 533.