Chem. Rev. 1999, 99, 2161−2200 2161 Implicit Solvation Models: Equilibria, Structure, Spectra, and Dynamics Christopher J. Cramer* and Donald G. Truhlar* Department of Chemistry and Supercomputer Institute, University of Minnesota, Minneapolis, Minnesota 55455-0431 Received January 4, 1999 (Revised Manuscript Received June 4, 1999) Contents 1. Introduction 2161 2. Previous Reviews 2162 3. Elements of Continuum Solvation Theory 2163 4. Models for Equilibrium Solvation 2165 4.1. Widely Used Models 2165 4.2. Recent Methodological Extensions 2168 4.3. Electron Correlation 2169 4.4. Direct Reaction Field 2169 4.5. Individual First-Solvation-Shell Effects 2170 4.6. Universal Solvation Models 2171 4.7. Explicit Solvent in the First Solvation Shell 2172 4.8. Effect of Solute Volume 2174 5. Dipole Moments and Charge and Spin 2174 Chris Cramer received his A.B. degree summa cum laude in Chemistry Distributions and Mathematics from Washington University (St. Louis) in 1983 and his 6. Solvent Effects on Equilibria 2176 Ph.D. degree in Chemistry from the University of Illinois in 1988. Following four years of national service, he joined the faculty of the University of 7. Solvent Effects on Spectra 2177 Minnesota, where he is now an Associate Professor of Chemistry, 7.1. Electronic Spectra 2177 Chemical Physics, and Scientific Computation. His research interests 7.2. Solvent Effects on Vibrational Spectra 2183 revolve around the development and application of methods for modeling 8. Dynamics 2184 the structure and reactivity of, well, everything. 8.1. Equilibrium Solvation 2184 8.1.1. Theory 2184 8.1.2. SES and ESP Applications 2187 8.2. Nonequilibrium Solvation 2188 8.3. Nonhomogeneous Media 2190 9. Conclusion 2190 10. Acknowledgments 2191 11. References 2191 1. Introduction The present review is concerned with continuum and other implicit models of solvation effects. We will concentrate on the elements required to make such Don Truhlar received his B.A. degree in Chemistry summa cum laude models successful and on the factors that limit their from St. Mary’s College of Minnesota in 1965 and his Ph.D. degree in accuracy. We will discuss explicit solvent models only Chemistry from Caltech in 1970. In 1969, he joined the faculty of the to the extent that they provide a complementary University of Minnesota, where he is now Institute of Technology picture or a context, since physical models of solva- Distinguished Professor of Chemistry, Chemical Physics, and Scientific tion, e.g., nonequilibrium solvent coordinates, can Computation and Director of the University of Minnesota Supercomputing often be achieved using either a continuum treatment Institute for Digital Technology and Advanced Computations. His research fields are theoretical chemical dynamics, molecular structure, modeling, of solvent or one that recognizes individual solvent and scientific computation. molecules. There are two principal advantages of continuum 1-octanol. Observable structural and dynamical prop- models. The first is a reduction in the system’s erties of a solute must be averaged over these degrees number of degrees of freedom. If we treat 200 of freedom, typically by Monte Carlo or molecular molecules of a solvent explicitly, this adds 1800 dynamics techniques. If, however, we can treat the degrees of freedom for water, 9000 degrees of freedom solvent as a continuous medium bathing the solute, for ethyl ether, and 16 200 degrees of freedom for the averaging becomes implicit in the properties 10.1021/cr960149m CCC: $35.00 © 1999 American Chemical Society Published on Web 07/30/1999 2162 Chemical Reviews, 1999, Vol. 99, No. 8 Cramer and Truhlar attributed to the bath. Although approximate models continuum solvation models, a comparison of alter- are seldom perfect, we may be able to make the errors native methods, and a summary of selected applica- small enough that our attention can be focused tions. elsewhere, e.g., on a better quantitative treatment Rivail and Rinaldi5 reviewed this field in a book of the solute or the dynamics, rather than on further chapter entitled “Liquid State Quantum Chemistry: improving the description of solvent effects. Computational Applications of the Polarizable Con- The second advantage is appreciated when one tinuum Models”. This review pays particular atten- realizes that complementary methods based on ex- tion not only to the nature of the solute-solvent plicit solvent are also imperfect. Only when such interface (e.g., ideal vs nonideal cavities, size of the models are considerably refined do they model the cavities, etc.), but also to the means by which the solvent electric polarization as well as a continuum solute charge distribution is represented. The Nancy model based on the experimental dielectric constant. group has carefully examined the degree to which the Thus, the second advantage of continuum solvent continuous solute charge distribution can be realisti- models is that they provide a very accurate way to cally represented as a truncated multipolar expan- treat the strong, long-range electrostatic forces that sion at one or more positions within the solute cavity. dominate many solvation phenomena. There is a This is an alternative approach to the solution of third advantage in practice. Most explicit-solvent Poisson’s equation in terms of apparent surface simulations still neglect solute electronic polarization charges. They also review the use of analytic gradi- due to cost (a notable exception being an extensive ents for geometry optimization to study solvent body of work of Gao and co-workers1,2), whereas there influences on molecular geometries and reaction is over 20 years worth of experience with polarizable dynamics. solutes using continuum theory. At about the same time, the present authors reviewed the field in two book chapters,6,7 both 2. Previous Reviews entitled “Continuum Solvation Models”. The former review was particularly complete on the generalized- Before proceeding to discuss implicit models of Born-plus-atomic-surface-tensions approach and con- solvation, we note that the present review is but one tained extensive comparisons of the results of differ- in a steady stream of reviews of this subject in recent ent approaches applied to the same systems. The years. This high review activity stems, of course, from latter review provided a fairly complete recapitula- a large number of original papers on the subject and tion of SCRF theory and the derivation of the a concomitant steep rate of research progress. For appropriate one-electron operator for inclusion of example, the 1994 review of continuum solvation solvent effects in molecular orbital theory within the theories and calculations by Tomasi and Persico3 has context of linear response theory. A systematic clas- 838 references, with over 300 of them from 1988 or sification scheme for different continuum models was later. Furthermore, the subject continues to be in an proposed, and a complete enumeration of extant exponential growth stage with no end in sight. Thus, models to that point was attempted. In addition, the not only will our review of the literature be selective, subject of dynamical processes within the context of even our review of reviews (i.e., the following para- continuum solvation was discussed. Finally, a thor- graphs) will be selective. We note the following ough review of continuum calculations of solvent reviews that particularly complement the one pre- effects on tautomeric equilibria, particularly those of sented here (with a brief description of each). heterocycles, was presented. The 1994 review of Tomasi and Persico,3 entitled Richards8 has provided the most recent review of “Molecular Interactions in Solution: An Overview of continuum solvation models as part of a larger work Methods Based on Continuous Distributions of the focusing on the application of semiempirical quantum Solvent”, provides a particularly complete discussion mechanical methods to biological systems. Detailed of the historical development of various continuum derivations and descriptions of popular models are approximations for computing the electrostatic com- provided. We also mention the 1994 review of Rashin ponent of the solvation free energy. It describes how and Bukatin9 which discussed additional thermody- early research on idealized systems having analytic namic quantities beyond the free energy of solvation solutions to Poisson’s equation led to more general where continuum models of hydration can be com- approaches for realistic cavities and, moreover, pro- pared to experiment. vides a thorough analysis of the relative utility of Five other reviews are mentioned because of their different numerical recipes for solving Poisson’s discussion of solvation modeling in general, including equation recast as a surface integral over apparent both continuum and discrete methods. In particular, surface charges. Finally, this review provides com- the reviews of Tapia10 in 1992, Warshel11 and Smith parisons of different approaches for including the and Pettitt12 in 1994, Orozco et al.13 in 1996, and solvation free energy in the one-electron Fock or Hummer et al.14 in 1998 provide useful comparisons Kohn-Sham operator needed for various quantum of the underlying assumptions of continuum vs chemical calculations, i.e., so-called self-consistent- discrete models (including lattice dipole models) and reaction-field (SCRF) computations. A less detailed of their relative performance for different classes of but still edifying summary of accounting for nonelec- problems. The Warshel and Orozco-Luque groups trostatic components of the solvation free energy is have been particularly active in comparing different also provided. A 1997 review by Tomasi et al.4 theoretical models against one another in a variety presents an updated analysis of quantum mechanical of systems; for the general practitioner, such com- Implicit Solvation Models Chemical Reviews, 1999, Vol. 99, No. 8 2163 parisons can be very helpful in analyzing how first radii, is probably the single most important contribu- to attack a specific problem. tor to the negative impression some researchers Finally the reader is referred to two recent mono- formed of continuum models.