Cramer C.J. Essentials of Computational Chemistry Theories and Models

Figure 11.7 To determine α for atom k in the GB approach, the interaction of the charge q on atom k with the surrounding continuum is determined by the radial integration shown, where ρk is the cavity radius (sometimes called the Coulomb radius) of atom k and A is the area of expanding spherical shells used in the integration which depend on all other atomic radii ρk . (Note that, once the outermost shell encompasses the entire molecule, the remaining integral may be solved analytically using the Born formula, since the system is simply a sphere having charge qk .) The effective Born radius α is then determined by requiring the equality of the first and second lines on the r.h.s. of the indicated equation

2.Compute effective Born radii α for all atoms using the procedure outlined above.

3.Using those effective Born radii, compute all values of γkk .

4.Compute or arbitrarily assign the atomic partial charges.

5.Evaluate Eq. (11.20).

Insofar as both GB and PB depend parametrically on the atomic radii used to define the cavity, direct comparisons between the two methodologies must be made using identical choices. Comparisons under such conditions have been made, and the agreement between the two models has been found to be excellent for small to medium-sized molecules (see, for instance, Edinger et al. 1997; Onufriev, Bashford, and Case 2000), so for practical purposes they may be taken as essentially equivalent in terms of predictive utility, and choice of model will usually be dictated by matters of computational convenience. For larger molecules, like biopolymers, agreement is still generally good but some technical care is required to ensure that the GB protocol does not assign empty volume inside the biomolecule to be characterized erroneously by the dielectric constant of the solvent (Feig et al. 2004). Some additional issues associated with comparison of the two approaches are discussed further in Section 11.4.

A modification of GB that includes the effects of dissolved electrolytes in the formalism, i.e., an extension analogous to the Poisson–Boltzmann extension of the Poisson equation, has been proposed by Srinivasan et al. (1999). In this model, the dielectric constant is a function of the interatomic distance and the Debye –Huckel parameter (Eq. (11.7)).

to the electrostatic potential that is distributed throughout space when the Poisson equation is solved using a volume element approach. However, if the surrounding space were to be characterized by an infinite dielectric constant, i.e., if the medium were conducting, then no potential exists in the medium, and instead image charge develops on the conductor surface in contact with the solute. Such a situation considerably simplifies the necessary electrostatic equations for the calculation of polarization free energy (and also associated energy derivatives) and this approximation is made in the so-called conductor-like screening model (COSMO), first described in detail at the semiempirical SCRF level by Klamt and Schu¨urmann¨ (1993).

Of course, the response of a conductor to a solute charge distribution is ‘complete’, while that of a dielectric medium is not. So, in COSMO models, the more simply evaluated conductor-polarization free energy is scaled by a factor of 2(ε − 1)/(2ε + 1) after its computation (i.e., by the Onsager factor; in the case of the SM5C model, however, the scaling factor is (ε − 1)/ε – see Section 11.3.3).

Since its original description at the semiempirical level, COSMO has also been generalized to the ab initio and density functional levels of theory as well (Klamt et al. 1998). In addition, conductor-like modifications of the PCM formalism have also been described, and to distinguish between the conductor-like version and the original (dielectric) version, the acronyms C-PCM and D-PCM have been adopted for the two, respectively (Barone and Cossi 1998).

From a chemical perspective, dielectricand conductor-like continuum models give sufficiently similar electrostatic results that the differences in their underlying assumptions appear to have no impact. Conductor-like models seem to be slightly more computationally robust in some instances, which may make them a better choice if instability is manifest in an SCRF calculation. Some concerns were raised initially that the post facto correction for dielectric behavior might render the models appropriate only for media having reasonably high dielectric constants, but a systematic study by Dolney et al. (2000) indicated non-polar solvents to be equally amenable to treatment by a COSMO model.

Moving beyond computation of the electrostatic component of the solvation free energy, Klamt (1995) has also described using the results of COSMO calculations to model ‘real solvents’ (COSMO-RS). In this model, a molecule in solution is considered to be entirely defined by the screening charge density on its cavity surface, which is called its σ profile. That surface is then shattered into a discrete number of fragments (each carrying its own characteristic charge density σ ) and a chemical potential is defined in a statistical mechanical formalism by considering the optimal matching of all fragments with partners having charge densities of opposite sign for the collection of all fragments in the liquid. In spite of the loss of structural information associated with breaking the molecular surface into completely independent fragments, this model has proven to be particularly effective for describing the thermodynamic properties of mixtures of molecules that are not too dissimilar, for example, vapor–liquid equilibria in binary solvent systems (Spuhl and Arlt 2004). Extending this idea to charged solutes, however, has proven more challenging.

It is important to re-emphasize that the electrostatic component of the solvation free energy is not a physical observable. Thus, it is impossible to assert on any basis other

than intuition that one continuum modeling algorithm is more or less accurate than another in the computation of this quantity (Curutchet et al. 2003a). One may take as a standard for comparison numerically converged solutions of the Poisson equation, but the Poisson equation is itself a model, and not necessarily the optimal one. In order to make comparisons against experiment, it is necessary to supplement the polarization (and distortion) energies with terms corresponding to cavitation, dispersion, structural rearrangement, etc. Models that purport to compute the full free energy of solvation may then be compared one to another using experimental free energies of transfer as a common yardstick.

11.3 Continuum Models for Non-electrostatic Interactions

Just as the electrostatic component of the free energy of solvation cannot be measured, neither can the non-electrostatic components. That being said, various experimental systems may be biased so as to make one component or another likely to heavily dominate the solvation free energy. For example, the solvation free energies of charged species would be expected to be dominated by the electrostatic component, and solvation free energies for ions can be helpful in the assignment of parametric Born radii to atoms. To assess the free-energy changes associated with cavitation, dispersion, and other physical effects, different neutral model systems have been studied, and we examine some of these next.

11.3.1Specific Component Models

Noble gas atoms have no permanent electrical moments, and the lighter ones are amongst the least polarizable of chemical systems. Thus, their transfer into a solvent may be regarded as a process reasonably cleanly associated with cavitation, i.e., the introduction of the noble gas atom is like introducing a vacuum of equivalent size into the solvent. By examining solvation data for the noble gases and certain other systems, Pierotti (1976) developed a formula for the cavitation free energy, associated with a spherical cavity volume, that depends on the radius of the sphere to the first, second, and third powers. Simulation data have been used to supplement noble-gas experimental data and refine constants appearing in the Pierotti formula (Hofinger¨ and Zerbetto 2003). By viewing a non-spherical solute as a collection of atomic spheres where overlapping volumes are only accounted for once, Pierotti’s formula has been generalized to molecular cavities (Claverie 1978; Colominas et al. 1999).

Dispersion is a considerably more difficult modeling task. As first noted in Section 2.2.4, dispersion is a purely quantum mechanical effect associated with the interactions between instantaneous local moments favorably arranged owing to correlation in electronic motions. In order to compute dispersion at the QM level, it is necessary to include electron correlation between interacting fragments, which immediately sets a potentially rather high price on direct computation. More difficult still, however, is that the continuum model by construction does not include the solvent molecules in the first place.

As a result, some approaches to computing dispersion energy have involved using either experimental or theoretical data for gas-phase clusters to estimate the strength of dispersion interactions between different possible solute and solvent functional groups. However,

when the cluster interaction involves molecules with permanent electrical moments, it can be quite difficult to separate out the dispersion interaction from the overall interaction. In any case, the typical approach deriving from this work is to develop a set of atomic (or group) polarizabilities that will be used together with bulk solvent polarizabilities to estimate dispersion interactions; usually, these are combined with other methods for estimating exchange-repulsion, i.e., repulsive van der Waals effects, to come up with a complete shortrange term (see, for example, Floris, Tomasi, and Pascual-Ahuir 1991).

In practice, models that directly calculate cavitation and dispersion/repulsion tend to predict that both effects are quite large in magnitude, but with opposite sign so that there is a large degree of cancellation. This suggests the unfortunate possibility that errors in the individual models may be larger than the net result.

Other energetic components associated with the solvation process include non-electrostatic aspects of hydrogen bonding and solvent-structural rearrangements like the hydrophobic effect. Despite many years of study, the fundamental physics associated with both of these processes remains fairly controversial, and physically based models have not been applied with any regularity in the context of continuum solvation models.

11.3.2Atomic Surface Tensions

Given the somewhat ad hoc nature of most specific schemes for evaluating the non-electro- static components of the solvation free energy, a reliance on a simpler, if somewhat more empirical, scheme has become widely accepted within available continuum models. In essence, the more empirical approach assumes that the free energy associated with the non-electrostatic solvation of any atom will be characteristic for that atom (or group) and proportional to its solvent-exposed surface area. Thus, the molecular GCDS may be computed simply as

k

where k runs over atoms or groups, A is the exposed surface area, and σ is the characteristic ‘surface tension’ associated with the atom or group. Note that here the use of the term surface tension refers to the unit dimensionality of energy per area, and the atomic terms should not be confused with the surface tension of the solvent, which is a macroscopic property.

Part of the motivation behind so straightforward an approach derives from its ready application to certain simple systems, such as the solvation of alkanes in water. Figure 11.8 illustrates the remarkably good linear relationship between alkane solvation free energies and their exposed surface area. Insofar as the alkane data reflect cavitation, dispersion, and the hydrophobic effect, this seems to provide some support for the notion that these various terms, or at least their sum, can indeed be assumed to contribute in a manner proportional to solvent-accessible surface area (SASA).

It should be noted that SASA itself can be defined in many ways (see, for instance, PascualAhuir, Silla, and Tunon˜ 1994). In the simplest approach, one imagines solvent molecules to be spheres having some characteristic radius. The SASA is then generated by the center of

The surface tensions themselves in the GB/SA and MST-ST models were developed by taking collections of experimental data for the free energy of solvation in a specific solvent, removing the electrostatic component as calculated by the GB or MST model, and fitting the surface tensions to best reproduce the residual free energy given the known SASA of the solute atoms. Such a multilinear regression procedure requires a reasonably sized collection of data to be statistically robust, and limitations in data have thus restricted these models to water, carbon tetrachloride, chloroform, and octanol as solvents.

In order to be more generally applicable, the SMx models of Cramer and Truhlar address the issue of data scarcity by making the atomic surface tensions a function of quantifiable solvent properties, i.e.,

j

where j runs over the property list, is the value of a particular property in convenient units, and the quantities ηk j become the parameters needing to be fit by multilinear regression. Although this introduces multiple parameters per atom type k, it permits regression over a single data set containing solvation free energies into any solvent, so long as its required solvent properties are known. In the SM5 versions of the models, the macroscopic solvent properties include surface tension, index of refraction, hydrogen bonding acidity and basicity, and percent composition of aromatic carbon atoms and electronegative halogen atoms, and the parameterization set involves more than 2500 data in 91 different solvents (Li et al. 1999).

A separate flexibility built into the SMx models compared to most other QM continuum models augmented with surface tensions is that no assignment of atom type need be made. Instead, the SMx surface tensions are functions of local geometry, so that, for instance, a carbon-bound hydrogen atom is distinguished from an oxygen-bound hydrogen atom and assigned a different surface tension to reflect its different character. The surface tension functions are smooth and differentiable, which facilitates their use in modeling situations where an atom may change from one type to another along a reaction coordinate, for instance.

Surface-tension augmented continuum models permit the computation of full free energies of solvation and may thus be used to construct solvated potential energy surfaces in the spirit of Figure 11.1. Insofar as the solvation free energy itself and any equilibrium or kinetic quantities computed for the solvated PES are physical observables, the accuracy of the solvation models may be assessed by comparison to experimental data. We consider several such comparisons in the next section in addition to addressing certain important technical details. Prior to doing so, however, it must be mentioned that the use of atomic surface tensions has been carried to the extreme of assuming that they can account for the entire solvation free energy, i.e., the electrostatics are completely implicit and the parameters in Eq. (11.23) are fit to the full solvation free energy (recent examples include Hawkins et al. 1998, Wang et al. 2001, and Hou et al. 2002). Such models are typically designed for use with biopolymers, where there is a need for extreme efficiency and the range of atom types is rather limited. An approach that is similar in its conceptual simplicity, albeit not entirely devoid of electrostatics, is the solvation free energy density (SFED) approach of No et al. (1999) where the full free energy of solvation is computed from the accessible volume (as

opposed to surface area) of a finite shell surrounding each atom. This model, too, is primarily designed for use with biomolecular simulations, although its performance for more general small neutral solutes is perfectly acceptable.

11.4 Strengths and Weaknesses of Continuum Solvation Models

11.4.1General Performance for Solvation Free Energies

±0.1 kcal mol−1. Carefully parameterized surface-tension-augmented continuum models typically exhibit average errors over large data sets on the order of 0.5 kcal mol−1. Ionic solutes pose more difficulties experimentally, since measurement of a gas/solution equilibrium is no longer a viable methodology. However, for singly charged species, solvation free energies ranging from −40 to −110 kcal mol−1 can be obtained with accuracies of

±2–5 kcal mol−1, depending on the experimental technique. Well parameterized continuum models achieve mean absolute errors at the high end of the experimental error range, which is perhaps the best that can be expected. Reliable data for more highly charged species are extremely scarce, so no legitimate comparison can be made.

It is worth noting that the solvation free energy of the proton is a somewhat special case. Determining the solvation free energy of the proton is equivalent to determining the absolute potential of the normal hydrogen electrode (NHE), which is a tricky issue in electrochemistry (Trasatti 1986). In 1986, the International Union of Pure and Applied Chemistry (IUPAC) recommended an absolute value of 4.44 V for the NHE which corresponds to a 1 M gas phase to 1 M solution standard-state aqueous proton solvation free energy of −261.7 kcal mol−1. In the 1990s, however, Tissandier et al. (1998) used ion-cluster measurements to establish a value of −264.0 kcal mol−1 for the same standard-state process, which corresponds to an NHE potential of 4.36 V (Lewis et al. 2004). Subsequent experimental and theoretical work has been supportive of the greater accuracy of the newer value and its use can be recommended. Note that most methods for determining ionic solvation free energies experimentally rely on having a benchmark value for the proton solvation free energy, so a change in the benchmark changes all ionic solvation free energies. Thus, care should be employed in comparing tabulations of such values in the literature to ensure common standard-state conventions and proton solvation free energies.

One of the reasons that it is hard to predict accurate solvation free energies for charged species is that such predictions tend to be very sensitive to the size of the solute cavity, leading to many proposals in the literature for how to go about choosing the ‘best’ electrostatic cavity. However, insofar as the electrostatic component of the solvation free energy is not an observable, there is not much weight to these arguments. The essentially equivalent performances of surface-tension augmented models like MST-ST and SMx for full free energies of solvation, even though they use very different cavity radii in some cases and therefore determine very different electrostatic free energies of solvation (Curutchet et al. 2003a), speak to the ability of the parameterization process to mask any lack of physicality in the cavity definitions.

Table 11.3 Absolute free energies of solvation (kcal mol−1) and chloroform/water partition coefficients (log10 units) for nucleic acid bases at the SM5.4/AM1 levela

a Computational results from Giesen et al. (1997); experimental results from Cullis and Wolfenden (1981)

In the future, analysis of this problem at the SCRF level will necessarily have to focus on molecular properties other than the solvation free energy to assess the greater accuracy of one cavity compared to another. Thus, differences in the gas-phase and solvated wave functions, and their corresponding effects on such properties as NMR, IR, and UV spectral transitions, may prove useful in identifying optimal methods for handling the electrostatics.

Such differences may in principle be quite large, as already illustrated in Table 11.1. Even the solvation free energies themselves may be strongly influenced by the relaxation of the wave function in solution. In Table 11.3 are listed the SM5.4/AM1 solvation free energies of six methylated nucleic acid bases, both in chloroform and in water, computed using either the charge distribution from the gas-phase wave function or from the relaxed wave function. As discussed further in Section 11.4.2, the difference between the two may be expressed as a partition coefficient, and the two sets of partition coefficients (frozen and relaxed) are compared to experimental values. Agreement is significantly better using the relaxed solvation free energies rather than the so-called ‘no solute polarization’ solvation free energies.

Note that one implication of the importance of solute polarization is that intrinsically non-SCRF methods, like continuum solvation models associated with force fields or other fixed-charge-density representations of the solute, must somehow include the energetic effect of polarization by other means. For instance, often atomic partial charges are chosen from calculations at the HF/6-31G(d) level. This level overestimates charge separation (as judged by a consistent roughly 10% overestimation of dipole moments), but this may be regarded as a virtue, not a failing, when used with non-SCRF continuum models, because the solute polarization is ‘built-in’ through the gas-phase wave function errors. Alternatively, fixedcharge models can use cavity radii slightly smaller than those for SCRF models to offset the use of unrelaxed charges.

11.4.1.1 pKa values

Returning to ionic solvation free energies, such quantities play important roles in the computation of two common properties of interest, namely pKa values and relative redox

Figure 11.9 Free energy cycle for computation of pKa values (where n is an integer). This cycle is sometimes referred to as a Born – Haber cycle

potentials. Computation of the former is accomplished by employing the free-energy cycle depicted in Figure 11.9. Thus, gas-phase free energies of AHn and An−1 may be computed at arbitrarily high levels of theory to establish as accurately as possible the deprotonation free energy of AHn. Note that if n and/or n − 1 are negative numbers then the basis set will need to include diffuse functions in order to obtain even modest quantitative accuracy. As for the proton, its electronic energy is obviously zero, and its gas-phase free energy derives entirely from a PV enthalpy term and a translational free energy that may be computed from Eqs. (10.16) and (10.17). At 298 K in the usual 1 atm standard state the free energy of the proton is −0.00999 a.u.

To compute the deprotonation free energy in solution, we take the gas-phase free energy change, add the free energies of solvation of An−1 and H+ (see above for the latter), and subtract the free energy of solvation of AHn. However, note that most continuum solvation models compute the free energy of solvation assuming the same standard-state concentration in the gas phase as in solution. As most pKa values adopt a standard-state concentration of 1 M, we need then to compute the free energy change associated with adjusting the concentrations of all of the gas-phase species from 1 mol per 24.5 L (the concentration of an ideal gas at 1 atm pressure and 298 K) to 1 mol per 1 L. As described in Section 10.5.4, this change is RT ln(24.5) for every species. As there are two products and only one reactant in the deprotonation reaction, the net effect is to make deprotonation less favorable by 1.9 kcal mol−1 in the 1 M standard state compared to the 1 atm standard state at 298 K.

Having computed the free-energy change in solution by this protocol, we may then compute Ka as

As errors in ionic solvation free energies are often on the order of 5 kcal mol−1, and as errors in the gas-phase deprotonation free energies may be of similar magnitude even with reasonably good levels of theory, errors in predicted absolute pKa values of 5 or more pK units are not terribly unusual, which is not particularly satisfying insofar as experimental measurements can be accurate to 0.01 pK units.