11 Molecules in Complex Environments: Solvent Models, QM/MM and QM/EFP Features, Density Embedding

11.2 Chemical Solvent Models

(June 30, 2021)

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 multipolar expansion method, also known as a Kirkwood-Onsager model, 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. (This should not be confused with the Onsager model, in which a dipole approximation is used, and to avoid confusion the multipolar expansion method is perhaps preferable.) More sophisticated models, which use a molecule-shaped cavity and the full molecular electrostatic potential, include the conductor-like PCM (C-PCM), 61 Barone V., Cossi M.
J. Phys. Chem. A
(1998), 102, pp. 1995.
Link
, 226 Cossi M. et al.
J. Comput. Chem.
(2003), 24, pp. 669.
Link
the conductor-like screening model (COSMO), the “surface and simulation of volume polarization for electrostatics” [SS(V)PE] model, 199 Chipman D. M.
J. Chem. Phys.
(2000), 112, pp. 5558.
Link
and the closely-related “integral equation formalism” (IEF-PCM). 152 Cancès E., Mennucci B., Tomasi J.
J. Chem. Phys.
(1997), 107, pp. 3032.
Link
, 153 Cancès E., Mennucci B.
J. Chem. Phys.
(2001), 114, pp. 4744.
Link
For an overview of all of these methods and their interconnections, see the review by Herbert.

The C-PCM, IEF-PCM, and SS(V)PE are examples of what are called “apparent surface charge” SCRF models, and the term polarizable continuum models (PCMs), as popularized by Tomasi and coworkers, 1093 Tomasi J., Mennucci B., Cammi R.
Chem. Rev.
(2005), 106, pp. 2999.
Link
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,606, 607, 609, 446, 604 which 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. 606 Lange A. W., Herbert J. M.
J. Phys. Chem. Lett.
(2010), 1, pp. 556.
Link
, 607 Lange A. W., Herbert J. M.
J. Chem. Phys.
(2010), 133, pp. 244111.
Link
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; 316 Florián J., Warshel A.
J. Phys. Chem. B
(1997), 101, pp. 5583.
Link
, 317 Florián J., Warshel A.
J. Phys. Chem. B
(1999), 103, pp. 10282.
Link
as well as versions 8 and 12 of the SMx models, and the SMD model, developed at the University of Minnesota. 733 Marenich A. V. et al.
J. Chem. Theory Comput.
(2007), 3, pp. 2011.
Link
, 731 Marenich A. V., Cramer C. J., Truhlar D. G.
J. Chem. Theory Comput.
(2013), 9, pp. 609.
Link
, 730 Marenich A. V., Cramer C. J., Truhlar D. G.
J. Phys. Chem. B
(2009), 113, pp. 6378.
Link
SM8 and SM12 are based upon the generalized Born method for electrostatics, augmented with atomic surface tensions intended to capture nonelectrostatic effects (cavitation, dispersion, exchange repulsion, and changes in solvent structure). Empirical corrections of this sort are also available for the PCMs mentioned above, although these have not always been particularly successful. In contrast, the nonelectrostatic interactions in SM8 and SM12 are carefully parameterized to reproduce experimental free energies of solvation. The SMD model (in which the “D” is for “density”) combines IEF-PCM with similar nonelectrostatic corrections. Statistically speaking, SMD is not any more or less accurate than other SMx models, but has the advantage that its electrostatic interactions are density-based and this model can be used in arbitrary basis sets. The SM8 model uses Mulliken atomic charges and thus can only be used in certain small basis sets for which it was parameterized, while the SM12 model replaces these with Hirshfeld-based atomic charges that are more robust with respect to the choice of basis set.

Model Cavity Non- Basis
Construction Discretization Electrostatic Sets
Terms? Supported
Kirkwood-Onsager spherical point charges no all
Langevin Dipoles atomic spheres dipoles in no all
(user-definable) 3-d space
C-PCM atomic spheres point charges or user- all
(user-definable) smooth Gaussians specified
SS(V)PE/ atomic spheres point charges or user- all
IEF-PCM (user-definable) smooth Gaussians specified
COSMO predefined point charges none all
atomic spheres
Isodensity SS(V)PE isodensity contour point charges none all
CMIRS isodensity contour point charges automatic all
SM8 predefined automatic 6-31G*
atomic spheres N/Aa 6-31+G*
6-31+G**
SM12 predefined N/Aa automatic all
atomic spheres
SMD predefined point charges automatic all
atomic spheres
aGeneralized Born electrostatic model; does not require cavity construction.
Table 11.1: Summary of implicit solvation models available in Q-Chem, indicating how the solute cavity is constructed and discretized, whether non-electrostatic terms are (or can be) included, which basis sets are available for use with each model, and whether analytic first and second derivatives are available for optimizations and frequency calculations.

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, which the caveat that only certain basis sets are supported for SM8. 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. This represents a “zeroth-order” inclusion of the solvent effects at the correlated level of theory, but is perfectly adequate for many applications as higher-order corrections are usually small. Table 11.2 also summarizes the availability of analytical energy gradients for implicit solvent models. (Finite-difference gradients and Hessians are requested automatically for calculations where the requisite analytic derivatives are not available.)

Note:  The 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.

Energy Derivatives C-PCM SS(V)PE & CMIRS COSMO SM8 SM12 SMD
  IEF-PCM
SCF energy gradient yes yes no yes yes no yes
SCF energy Hessian yes no no yes no no no
CIS/TDDFT energy gradient yes no — unsupported —
CIS/TDDFT energy Hessian yes no — unsupported —
MP2 & double-hybrid energy — unsupported —
derivatives
Coupled cluster methods — unsupported —
Table 11.2: Summary of analytic energy gradient and Hessian available with implicit solvent models.

SOLVENT_METHOD
       Sets the preferred solvent method.
TYPE:
       STRING
DEFAULT:
       0
OPTIONS:
       0 Do not use a solvation model. ONSAGER Use the Kirkwood-Onsager model (Section 11.2.1). PCM Use an apparent surface charge, polarizable continuum model (Section 11.2.2). ISOSVP Use the isodensity implementation of the SS(V)PE model (Section 11.2.5). COSMO Use COSMO (Section 11.2.7). SM8 Use version 8 of the Cramer-Truhlar SMx model (Section 11.2.8.1). SM12 Use version 12 of the SMx model (Section 11.2.8.2). SMD Use SMD (Section 11.2.8.3). CHEM_SOL Use the Langevin Dipoles model (Section 11.2.9).
RECOMMENDATION:
       Consult the literature, e.g., Ref. Herbert:2021b. PCM is a collective name for a family of models and additional input options may be required in this case, in order to fully specify the model. (See Section 11.2.2.) Several versions of SM12 are available as well, as discussed in Section 11.2.8.2.

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, SS(V)PE, and IEF-PCM. The latter two models are formally equivalent at the level of integral equations, 153 Cancès E., Mennucci B.
J. Chem. Phys.
(2001), 114, pp. 4744.
Link
but exhibit some some differences in their numerical implementation. 609 Lange A. W., Herbert J. M.
Chem. Phys. Lett.
(2011), 509, pp. 77.
Link
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”), 554 Klamt A., Jonas V.
J. Chem. Phys.
(1996), 105, pp. 9972.
Link
although later work cast doubt on the theoretical justification for this and other ad hoc charge renormalization procedures. 198 Chipman D. M.
J. Chem. Phys.
(1999), 110, pp. 8012.
Link
[The IEF-PCM and SS(V)PE methods already contain an implicit correction for outlying charge, as does the C-PCM method that is derived as an approximation to these models, 198 Chipman D. M.
J. Chem. Phys.
(1999), 110, pp. 8012.
Link
although this was not recognized at the time that COSMO was formulated. See Ref. Herbert:2021b for a historical discussion of these developments.] In any case, COSMO is described 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 in conjunction with appropriate job-control variables in the $pcm input section) uses a solute cavity constructed from atom-centered spheres, in keeping with 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. 196 Chipman D. M., Dupuis M.
Theor. Chem. Acc.
(2002), 107, pp. 90.
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, 192 Chen F., Chipman D. M.
J. Chem. Phys.
(2003), 119, pp. 10289.
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, 201 Chipman D. M.
J. Chem. Phys.
(2006), 124, pp. 224111.
Link
This is an appealing, one-parameter cavity construction that avoid many of the problems with cusps in “van der Waals” cavity surfaces that are constructed from atom-centered spheres, and the isodensity implementation of SS(V)PE forms the basis of a physics-based continuum solvation model called CMIRS that is described in Section 11.2.6. 879 Pomogaeva A., Chipman D. M.
J. Chem. Theory Comput.
(2011), 7, pp. 3952.
Link
, 880 Pomogaeva A., Chipman D. M.
J. Phys. Chem. A
(2013), 117, pp. 5812.
Link
, 881 Pomogaeva A., Chipman D. M.
J. Chem. Theory Comput.
(2014), 10, pp. 211.
Link
, 882 Pomogaeva A., Chipman D. M.
J. Phys. Chem. A
(2015), 119, pp. 5173.
Link
, 1224 You Z.-Q., Herbert J. M.
J. Chem. Theory Comput.
(2016), 12, pp. 4338.
Link
CMIRS is competitive in accuracy (for solvation free energies) with the best-available SMx solvation models, despite using far fewer fitting parameters. However, analytic energy gradients are not available for the isodensity cavity construction and therefore not available for CMIRS, whereas these gradients are available when the cavity surface is constructed from atom-centered spheres.

Regarding the accuracy of these models for solvation free energies (ΔG298), the SMx models generally achieve 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. 228 Cramer C. J., Truhlar D. G.
Acc. Chem. Res.
(2008), 41, pp. 760.
Link
(Note that the SM12 and SMD models generally do not improve too much upon SM8 in any statistical sense, 730 Marenich A. V., Cramer C. J., Truhlar D. G.
J. Phys. Chem. B
(2009), 113, pp. 6378.
Link
, 731 Marenich A. V., Cramer C. J., Truhlar D. G.
J. Chem. Theory Comput.
(2013), 9, pp. 609.
Link
but do extend these models to arbitrary basis sets whereas SM8 is limited to a few small basis sets.) To achieve accuracy comparable to the SMx models within the PCM class of solvent models, nonelectrostatic terms must be added. Among the various PCMs described above, the only one that constitutes a “black box” model for solvation energies is CMIRS, which slightly outperforms the SMx models in a statistical sense, 1224 You Z.-Q., Herbert J. M.
J. Chem. Theory Comput.
(2016), 12, pp. 4338.
Link
although it is only available for a few solvents.

The following sections provide more details regarding theory and job control for the various implicit solvent models that are available in Q-Chem. Ref. Herbert:2021b contains both formal comparisons amongst these models as well as a side-by-side comparison of the accuracy of solvation free energies.