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

One approach for reducing the errors associated with the prediction of pKa values is to employ an isodesmic reaction. To illustrate with a specific example, it may be very hard to correctly predict the free energy change for the aqueous reaction

However, if the theoretical target is instead the free energy change for the isodesmic equation

(11.27)

one may well expect this to be computed far more accurately, since errors in levels of theory should largely cancel from left to right. Provided experimental data are available for the unsubstituted case

(11.28)

+ H +

then the free energy change for Eq. (11.26) may be estimated from the difference between the computed value for Eq. (11.27) and the experimental value for Eq. (11.28). Chen and MacKerell (2000) have provided a more detailed demonstration of the utility of this approach for a series of substituted pyridines using a variety of different levels of theory for the gas phase and computed solvation free energies.

An alternative approach for improving predicted pKa values has been suggested by Klicic et al. (2002), who developed functional-group-specific linear regression corrections for pKa values computed from a particular DFT SCRF PB formalism. Correction of the raw computed pKas increases the model’s accuracy to about 0.5 pK units for those acidic functional groups well represented in their parameterization set.

11.4.1.2 Redox potentials

Oxidation and reduction potentials in solution are also computed via reference to particular thermodynamic cycles as illustrated in Figure 11.10. In this case, however, the

Figure 11.10 Thermodynamic cycles for one-electron oxidation (left) and reduction (right) potentials in solution

thermodynamic cycles perforce involve open-shell species and free electrons. Note that the oxidation and reduction cycles in the gas phase correspond conceptually to ionization potentials and electron affinities, respectively, except that IPs and EAs are enthalpies, not free energies, so thermal and entropic terms must be included therein. For the free electron, like the free proton, the electronic energy is zero, but the sum of the PV and translational terms leads to a total gas-phase free energy at 298 K and 1 atm of −0.00001 a.u. (it is a coincidence that for this standard state the free energy associated with the translational entropy almost exactly cancels the enthalpy).

Another key feature of redox thermodynamic cycles is that the free energy change in solution is still defined to involve a gas-phase electron, that is, the solvation free energy of the electron is happily not an issue. And, once again, redox potentials in solution typically assume 1 M standard states for all species (but not always; in this chapter’s case study, for instance, all redox potentials were measured and computed for chloride ion concentrations buffered to 0.001 M). So, free energy changes associated with concentration adjustments must also be properly taken into account.

Once the free energy change in solution has been computed, the absolute redox potential Eo may be computed as

where n is the number of electrons transferred and F is the Faraday constant equal to 23.061 kcal mol−1 V−1. Note that while Figure 11.10 presents thermodynamic cycles for one-electron processes, analogous cycles involving multiple electrons are readily constructed and may sometimes be more amenable to experimental determination.

In practice, experimental redox potentials are reported relative to a standard electrode. If the standard is the NHE, one subtracts 4.36 V from the absolute reduction potential (the ‘cost’ of the free electron) or adds 4.36 V to the absolute oxidation potential (the ‘return’ from the removed electron) in order to determine the relative potential. Adjustment to other standard electrodes is straightforward, since their potentials relative to the NHE are well known.

With respect to accuracy, it is again important to employ basis sets including diffuse functions when anions are present as either reactants or products. With large well balanced basis sets, B3LYP for gas-phase energetics, and a PB SCRF solvation model, Baik and

Friesner (2002) have reported average errors of about 150 mV for various organometallic species in different organic solutions. As already discussed for pKas, still better accuracy in redox potentials can often be achieved through the use of isodesmic equations or functional- group-specific correction schemes (see, for example, Winget et al. 2000).

11.4.1.3 Supermolecular solutes

Some final technical points merit attention. In SCRF models that use the full electronic distribution as part of the representation of the density, a problem arises in that the wave function has non-zero amplitude in the space outside the cavity. Thus, the construction of the cavity truncates the charge distribution, so that, for instance, neutral molecules have a small net positive charge inside the cavity. To return to integral charge values, the charge inside the cavity must somehow be renormalized. There are many different approaches to rectifying this problem; early methods tended to introduce considerable instability into the solvation computation, although more modern approaches seem reasonably robust (see, for example, Curutchet et al. 2004). Methods that do not suffer from the charge-penetration problem include all those that represent the density as either a singleor multi-center multipole or monopole expansion (this then includes GB methods). In addition, approaches have been developed that specifically handle, as a separate physical component, the polarization energy associated with penetration of charge into the solvent, and these models too seem to be well balanced (Chipman 2002).

The charge-penetration problem is in some sense related to a specific drawback of current continuum models, namely, that they have no mechanism to account for possible charge transfer between the solute and the surrounding solvent. It is not yet clear to what extent such solute/solvent charge transfer is important.

Of course, the simplest way to account for charge transfer would be to ‘materialize’ one or more solvent molecules around the solute and to treat the resulting cluster as a supermolecule embedded in the continuum. Pliego and Riveros (2002) and Fu et al. (2004) have recently suggested that such an approach provides a more robust protocol for the computation of accurate pKa values, for instance. However, while this model has conceptual merits, it can introduce significant computational overhead. First, the supermolecule is obviously bigger than the solute, and depending on the level of theory employed the difference in computational time for a single SCRF calculation may be large. Second, clusters tend to generate fairly complex PESs, with many minima, and any attempt to compute free energy must sample over all of the minima in a statistically correct fashion. Since part of the motivation for using a continuum model is to avoid the sampling issues associated with explicit models, the representation of specific solvent molecules is usually not undertaken in the absence of compelling need.

One case, however, where materialization of a specific solvent molecule out of the continuum is indeed critical is when that solvent molecule loses its ‘solvent’ character. For instance, a water molecule tightly bound as both a hydrogen-bond donor and acceptor in a chain involving two solute functional groups clearly should be regarded as a unique fragment in what is fundamentally a two-piece supermolecule. Unfortunately, it is not always

[X]gas

Figure 11.11 Thermodynamic relationship between partitioning free energies and free energies of solvation. Knowledge of any two free energies permits prediction of the third since any cycle around the free-energy triangle must sum to zero

Table 11.4 Chloroform/water partition coefficients (log10 units) for nucleic acid bases at different SCRF levels

a See Table 11.3.

b Orozco, Colominas, and Luque 1996.

c Young and Hillier 1993; Young, Hillier, and Gould 1994.

certain instances, this has proven relatively straightforward. One example is the extension of the PCM model to include handling liquid crystals as solvents. In the case of a liquid crystal, the ordering of the solvent gives rise to a dielectric tensor as opposed to a single uniform dielectric constant. In order to extend the continuum model, an absolute reference frame is chosen and the x, y, and z components of the PCM equations are solved separately using the appropriate dielectric constant values; in addition, non-isotropic effects on cavitation energies have been considered (Mennucci, Cossi, and Tomasi 1996).

Another interesting case is a supercritical fluid. Near their critical points, supercritical fluids can exhibit very large changes in density (and density-related properties) in response to very small changes in conditions. By including possible density changes and their effects into the

SCRF equations governed according to the experimentally measured solvent compressibility, Luo and Tucker (1995) have been able to model these effects efficiently.

While both of the above examples are gas/condensed-phase solvation phenomena, there are also many interesting cases of partitioning phenomena where one phase is non-isotropic. Perhaps the most common is the case where one phase is a pure solid and the other a liquid solvent, in which case the partitioning phenomenon corresponds to solubility. Within the context of the free energy cycle of Figure 11.11, if we consider A to be the solid phase for solute X, the free-energy change from the solid to the gas corresponds to sublimation. To compute the free energy of sublimation rigorously, one must know the crystal packing energy. However, even when the unit cell geometry of the crystal is known (which it often is not ), it is by no means trivial to compute the free energy of interaction of one monomer with the full crystal. A very rough estimate can be had by assuming that organic nonelectrolytes have crystal packing energies similar to the solvation energies that the solute would have in a solvent ‘similar’ to itself. Thus, for instance, a highly non-polar hydrocarbon would be assumed to have a crystal packing energy equal to its solvation free energy in n-hexadecane. In essence, this treats solid/liquid partitioning as just a typical liquid/liquid partitioning problem (see, for instance, Reinwald and Zimmermann 1998 and Thompson, Cramer, and Truhlar 2003). While this approach can work well for non-polar solutes, it is less secure when more complicated functionality is present. In such instances, modern work typically includes some combination of solvation free-energy estimates combined with statistical analysis over data sets of molecules having similar functionality for which solubilities have been measured in order to make predictions (generating a so-called quantitative structure –property relationship (QSPR); see, for example, Lipinski et al. 1997).

Liquid/liquid partition constants within pharmaceutical chemistry have been of primary interest because of their correlation with liquid/membrane partitioning behavior. A sufficiently fluid membrane may, in some sense, be regarded as a solvent. With such an outlook, the partitioning phenomenon may again be regarded as a liquid/liquid example, amenable to treatment with standard continuum techniques. Of course, accurate continuum solvation models typically rely on the availability of solvation free energies or bulk solvent properties in order to develop useful parameterizations, and such data may be sparse or non-existent for membranes. Some success, however, has been demonstrated for predicting such data either by intuitive or statistical analysis (see, for example, Chambers et al. 1999).

Indeed, the utility of the continuum approach for modeling non-homogeneous phases has even been extended to the modeling of soil. The partitioning behavior of organic compounds between aqueous phases and soil is an important factor affecting the persistence of organic contaminants in the environment. Thus, environmental chemists define POC as the partition constant of a solute between water and soil, where the mass of the soil is normalized by organic carbon content (such normalization has the effect of making the partition coefficient remarkably constant over wide ranges of soil types). Using Eq. (11.30), then, one can determine a free energy of transfer into the organic carbon component of soil. An SMx model trained on a data set of a few hundred molecules in order to determine the necessary bulk ‘solvent’ properties to define a soil phase has been shown to be capable of predicting POC values to within about one log unit (Winget, Cramer, and Truhlar, 2000). Thus, continuum

while GB models are very robust for the prediction of solvation free energies, they are less successful in the generation of potentials of mean force (Rankin, Sulea, and Purisima 2003). Lack of high-quality data makes it difficult to evaluate this possibility at present, although ongoing comparisons between different theoretical models are helping to further illuminate the issue (see, for example, Jayaram, Liu, and Beveridge 1998 and Gohlke and Case 2004).

Note that one feature of Figure 11.12 is a solvent-separated minimum for the X–Y pair. Insofar as solvent-separated minima involve intervening solvent molecules that typically differ significantly in their behavior from normal bulk solvent as a consequence of being isolated between the two solutes, such situations are unlikely to be handled accurately by continuum models in general.

It is sometimes the case that the structure of the first shell (or shells) of solvent is a property of primary interest for a given modeling study. It is perhaps stating the obvious to note that in such an instance, continuum models cannot be used, since by construction they ignore the molecular nature of the solvent and assume a homogeneous surrounding medium.

Of course, if one is interested only in the free-energy well associated with full complexation, many technical aspects of the calculation are simplified. The tremendous speed of continuum solvent models has made them attractive tools in evaluating solvation effects on docking, especially insofar as they permit more extensive sampling of varying target-receptor geometries to be carried out in an efficient manner (see, for example, Gouda et al. 2003; Taylor, Jewsbury, and Essex 2003; and Zoete, Michielin, and Karplus 2003).

11.4.5Molecular Dynamics with Implicit Solvent

A large fraction of the expense of a typical MD simulation involving a solute in solution (discussed in much more detail in the next chapter) is associated with the hundreds or thousands of solvent molecules that are explicitly represented in the full simulation cell. However, when the fine details of the solvation process are not of primary interest, it can be about an order of magnitude more efficient to propagate a trajectory for the solute within the context of continuum solvation. The methodology that has been most extensively explored for this process to date has tended to involve GB solvation models developed for biomolecular force fields (although PB models have also seen substantial use). To maximize speed, Born radii are computed either from a PD algorithm or are set to constant values determined from initial PB calculations (Onufriev, Case, and Bashford 2002). For truly enormous systems, additional algorithms allowing certain portions of the solute to be held frozen while others are dynamical have been described (Banavali, Im, and Roux 2002; Guvench et al. 2002).

A particular advantage of MD with implicit solvation is that solvent friction is not an issue with respect to the solute being able to explore phase space. That is, no solvent molecules need to be pushed out of the way in order for otherwise energetically accessible largescale motions to take place. As long as the energy landscape for the solute is as accurately predicted with the continuum solvent as with an explicit solvent, this feature leads to much more rapid achievement of converged sampling (Okur et al. 2003) especially when LES is used (Cheng, Hornak, and Simmerling 2004). This behavior has been successfully exploited

for modeling pathways associated with protein folding (see, for instance, Jang, Shin, and Pak 2002 and Chowdhury et al. 2003), the timescale for which would have made simulations with explicit solvent prohibitively expensive. Of course, if one is interested in the kinetics of the process in question, then removal of the solvent friction is not helpful – the implicit solvent advantage applies only to obtaining rapid equilibrium averages.

As for the quality of the energy landscape and its effect on solute dynamics, comparisons of PCA eigenvectors from simulations using either implicit or explicit solvation have been carried out for proteins (Cornell et al. 2001), DNA (Tsui and Case 2000) and RNA (Sherer and Cramer 2002) and have generally indicated high overlap between the two models. Nevertheless, some protein folding studies have identified serious deficiencies in GB landscapes that include overestimation of salt-bridge interaction energies (Zhou 2003) and a general tendency to overstabilize nucleation (Nymeyer and Garcia 2003). One alternative to propagating a trajectory using an implicit solvent model that has also been explored has been to take a trajectory generated with inclusion of explicit solvent and then post-process it to compute individual or average solvation free energies for various snapshots, whose computation would otherwise require more sophisticated simulation protocols as described in the next chapter.

Some work has also appeared describing MD with implicit solvation for solutes described at the DFT level. Fattebert and Gygi (2002) have proposed making the external dielectric constant a function of the electron density, thereby achieving a smooth transition from solute to solvent instead of adopting a sudden change in dielectric constant at a particular cavity surface. Non-electrostatic components of the solvation free energy have not been addressed in this model.

11.4.6Equilibrium vs. Non-equilibrium Solvation

Most continuum models are properly referred to as ‘equilibrium’ solvation models. This appellation emphasizes that the design of the model is predicated on equilibrium properties of the solvent, such as the bulk dielectric constant, for instance. The amount of time required for a solvent to equilibrate to the sudden introduction of a solute (i.e., the solvent relaxation time) varies from one solvent to another, but typically is in the range of molecular vibrational and rotational timescales, which is to say on the order of picoseconds.

Processes that take place on longer timescales may thus be legitimately thought of as equilibrium processes with respect to solvation. However, the question arises of how applicable continuum models are to very fast processes. For instance, Figure 11.4 describes the relationship between gas-phase and solvated reaction coordinates for a reactive process, but the average amounts of time individual molecules spend at various positions on the reaction coordinate vary considerably. In the regions of the minima, equilibrium solvation seems assured, but transition state structures in principle live for only a single vibrational period. This suggests that the solvent may not have time to fully equilibrate to the TS structure, and a continuum model were it to be applied would overestimate the solvation free energy by assuming equilibration. In addition, considerable progress has been made in the extension of GB models to systems where an implicit membrane characterized by a dielectric constant

different from the solvent is represented in both the electrostatic and non-electrostatic terms (Spassov, Yan, and Szalma 2002).

Still faster timescales are associated with phenomena like electron transfer (i.e., redox reactions) and photon absorption/emission and possible associated electronic excitation. Since these processes occur on the timescale of electronic motion, the surrounding solvent molecules may be regarded as frozen in place during the reaction, and clearly an equilibrium view of the instantaneous solvation is incorrect.

These issues will be addressed in more detail in Chapters 14 and 15. Here, we will simply note that successful extensions of continuum models to such ultrafast processes as electron transfer and photoexcitation requires that the response properties of the solvent be explicitly separated into slow and fast parts that interact with the solute over the appropriate timescales (see, for instance, Cossi and Barone 2000). This approach can also be used for dealing with non-equilibrium effects on transition states. However, unless there is a very sudden transfer of charge that takes place over a significant distance failure to account for non-equilibrium effects rarely has much consequence in estimating a free energy of activation – errors in the gas-phase potential energy difference between minima and the TS structure are typically larger in magnitude. Thus, unmodified continuum solvation models can still be quite useful in constructing diagrams like that shown in Figure 11.4 for the purpose of describing reactivity in solution.

11.5Case Study: Aqueous Reductive Dechlorination of Hexachloroethane

Synopsis of Patterson, Cramer, and Truhlar (2001) ‘Reductive Dechlorination of Hexachloroethane in the Environment. Mechanistic Studies via Computational Electrochemistry’.

Halogenated alkanes are very useful as solvents in a variety of industrial processes (at one time they were the solvents of choice for the dry cleaning of clothes, for example). The scale of their use is such that their accidental or deliberate discharge into the environment can lead to long-term contamination problems. As is true for many environmental contaminants, the molecule originally released may not be a particular danger from an environmental perspective, but some product into which it is transformed may be considerably more cause for concern.

An example is hexachloroethane (C2Cl6). In environmental aqueous phases, it typically undergoes reductive dechlorination relatively rapidly. One product of this dechlorination, produced in small amounts, is trichloroacetic acid (Cl3CCO2H), which is a regulated carcinogen in the United States. The authors studied the mechanistic aspects of C2Cl6 reductive dechlorination with both methodological and chemical goals. The methodological question involved identifying appropriate levels of theory for modeling the relevant reactions, while the chemical questions to be addressed were associated with identifying the relevant mechanistic pathways for reduction and any possible explanation for the generation of Cl3CCO2H as a product.

Various data were available for comparison in order to identify adequate theoretical levels for application. Table 11.5 illustrates the performance of two different levels of theory, CCSD(T)/aug-cc-pVDZ//BPW91/aug-cc-pVDZ and BPW91/aug-cc-pVDZ, with