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

explicitly, we replace it by a continuous electric field that represents a statistical average over all solvent degrees of freedom at thermal equilibrium. This field is usually called the ‘reaction field’ in the regions of space occupied by the solute, since it derives from reaction of the solvent to the presence of the solute. The electric field at a given point in space is the gradient of the electrostatic potential φ at that point, and the work required to create the charge distribution may be determined from the interaction of the solute charge density ρ with the electrostatic potential according to

The charge density ρ of the solute may be expressed either as some continuous function of r or as discrete point charges, depending on the theoretical model used to represent the solute. The polarization energy, GP, discussed above, is simply the difference in the work of charging the system in the gas phase and in solution. Thus, in order to compute the polarization free energy, all that is needed is the electrostatic potential in solution and in the gas phase (the latter may be regarded as a dielectric medium characterized by a dielectric constant of 1).

11.2.1The Poisson Equation

At the heart of all continuum solvent models is a reliance on the Poisson equation of classical electrostatics to express the electrostatic potential as a function of the charge density and the dielectric constant. The Poisson equation, valid for situations where a surrounding dielectric medium responds in a linear fashion to the embedding of charge, is written

where ε is the dielectric constant of the medium. Insofar as continuum solvation involves representing the solute explicitly and the solvent implicitly, the charge distribution of the solute is thought of as being inside a cavity that displaces an otherwise homogeneous dielectric medium. Thus, there are really two regions, one inside and one outside the cavity, in which case the Poisson equation is properly written as

The Poisson equation is valid under conditions of zero ionic strength. If dissolved, mobile electrolytes are present in the solvent, the Poisson–Boltzmann (PB) equation applies instead

where q is the magnitude of the charge of the electrolyte ions, λ is a simple switching function which is zero in regions inaccessible to the electrolyte and one otherwise, and κ2

where I is the ionic strength of the electrolyte solution. The inverse of κ is also called the Debye length.

Thus, in order to determine the electrostatic potential in solvents containing either nonelectrolytes or electrolytes, we need only solve Eqs. (11.5) or (11.6), respectively, using the known charge density of the solute and some cavity defining how the dielectric constant varies about the solute. As differential equations go, Eq. (11.5) is straightforward, but Eq. (11.6) is fairly unpleasant. As a result, it is often simplified at low ionic strength by using a truncated power expansion for the hyperbolic sine, giving the so-called linearized PB equation

Note that it is fairly common in the literature for continuum solvation calculations to be reported as having been carried out using Poisson–Boltzmann electrostatics even when no electrolyte concentration is being considered, i.e., the Poisson equation is considered a special case of the PB equation and not named separately.

For certain ideal cavity shapes, the relevant PB equations have particularly simple analytic solutions. While such ideal cavities are not typically to be expected for arbitrary solute molecules, consideration of some examples is instructive in illustrating how more sophisticated modeling may be undertaken by generalization therefrom.

11.2.1.1 Ideal cavities

Consider a conducting sphere bearing charge q, which may be taken as an approximation to a monatomic ion. The charge on such an object spreads out uniformly on the surface, and the charge density at any point on the surface may thus be expressed as

q

ρ (s) = (11.9) 4π a2

where s is a surface point and a is the radius of the sphere. So, in order to evaluate Eq. (11.3), we will need to integrate only on the surface of the sphere (since the charge density is zero everywhere else). To determine the electrostatic potential at the surface we must approach from the outside (the dielectric constant of a conductor is infinite and the electrostatic potential everywhere inside is zero, so there is a formal discontinuity in the potential at the surface). From the outside, the electrostatic potential is well known to be equivalent to that for a point charge q at the origin, giving the central field result

where ε is the exterior dielectric constant. Taking r on the surface of the sphere, i.e., |r| = a, Eq. (11.3) becomes

As the square of the charge, the dielectric constant, and the ionic radius must all be positive, work must be expended to charge the sphere, but the work is less for higher exterior dielectric constants, as expected. Recalling that the polarization energy is the difference in the required work in the gas phase and solution, we may write

which is the so-called Born equation for the polarization energy of a monatomic ion in atomic units.

If instead of carrying a charge, our sphere appears to be characterized by a perfectly dipolar distribution having dipole moment µ, an analogous analysis provides

which is the so-called Kirkwood–Onsager equation in atomic units.

An important difference between the Born and Kirkwood–Onsager formulae is that the former depends on the charge, which is a property of the system restricted to integral values, while the latter depends on the dipole moment, which can potentially vary in different environments. In the context of quantum mechanical calculations, let us define the Kirk- wood–Onsager polarization energy operator by invoking µ as the dipole moment operator in Eq. (11.13). In that case, the Schrodinger¨ equation in solution becomes

+

where H is the usual gas-phase Hamiltonian. Written in this fashion, the components of the second term on the l.h.s. that precede the final dipole moment operator may be regarded as the reaction field.

Equation (11.14) is an example of a non-linear Schrodinger¨ equation. It can be solved in the usual HF fashion by construction of a Slater determinant formed from MOs ψ that are optimized using a modified Fock operator according to

+

´

where Fi is the usual gas-phase Fock operator for MO i (Angyan´ 1992). A critical feature of Eq. (11.15) is that it involves an additional level of iteration compared to the standard HF approach. Not only must the final wave function render the density matrix and Fock

operator stationary, but it must also lead to a stationary dipole moment. Solution of the HF equations (or the equivalent Kohn–Sham DFT equations) in such a fashion, where accounting for solvation leads to a non-linear Schrodinger¨ equation, is referred to as a self-consistent reaction field (SCRF) calculation.

Inspection of Eq. (11.14) should make it clear that the manner in which the dipoledependence enters into the equations will lead to an increase in dipole moments in increasingly polar solvents. As noted in Section 11.1, the increase in the dipole moment in such an SCRF formalism provides an energy lowering that is counterbalanced by an increase in the energy computed from the ‘usual’ Hamiltonian H (the first operator on the l.h.s.) so that a stationary solution is reached when additional distortion costs associated with H exactly balance additional energy lowering associated with further increasing the dipole moment.

In describing the results from SCRF calculations, it is useful to keep careful track of the various components of the energy. The electrostatic component of the solvation free energy is the difference between the energy in the gas phase and the energy in solution. This may be written

where the difference between the first and third expectation values on the r.h.s. in the first line of Eq. 11.16 defines the distortion energy EEN, which must be positive since (gas) minimizes H . The ‘EN’ subscript on this term emphasizes it is associated with the electronic and nuclear components of the total energy; in the absence of any geometry reoptimization, the N subscript is superfluous. As written, Eq. (11.16) mixes potential and free energies, but we will ignore this issue for now.

The Kirkwood–Onsager equations can be generalized to include multipole moments higher than the dipole, leading to the expression

where each component m of every molecular multipole M of order l interacts with the reaction field, which is itself expressed as a multipole expansion equal and opposite to the molecular multipoles, through the reaction field factors f that carry the dependence on dielectric constant and cavity radius. In principle, the multipole expansion may be carried out to infinite order, but in practice, some judicious choice of l is made in Eq. (11.17) to keep things tractable. A fairly typical choice is l = 6 (note that l = 0 and l = 1 define the Born and Born–Kirkwood–Onsager (BKO) approaches, respectively).

The simplicity of the BKO approach to computing polarization free energies led to its widespread use for the qualitative analysis of solvation effects on various properties for many years (including in the absence of any explicit theoretical calculations). For quantitative purposes, however, it suffers from a number of undesirable features. One such feature is the slow nature of the convergence of Eq. (11.17) with respect to l. Table 11.2 lists GEP

Table 11.2 GEP values (kcal mol−1) for trans 1,2-dichloroethane as a function of the truncation point in the multipole moment expansiona

a From Christiansen and Mikkelsen 1999

values for trans 1,2-dichloroethane computed at the MCSCF level for various choices of l. Note that, since the trans conformation has no dipole moment by symmetry, a simple BKO calculation must predict a polarization free energy of zero, which represents a very large error. Including the quadrupole moment captures 50 percent of the total, but further convergence initially proceeds slowly (note that there is no requirement for convergence to proceed in a monotonic fashion), and it is not until l = 8 that the result is converged to within about 5 percent. Since 1,2-dichloroethane overall has a rather simple electronic distribution, it is disturbing to consider how much larger l may have to be to accurately treat more complex molecules.

Worse still, however, is that even well-converged values are unlikely to be meaningful in the absence of the solutes in question being well described as spherical. When they are not, and very few molecules are, the value that should be chosen for the radius a is highly ambiguous. Since the dipole term has an inverse cubic dependence on this parameter, small variations can have large effects on solvation free energies, and the literature is replete with examples where obviously non-physical values have been chosen, rendering interpretation of subsequent results highly suspect.

This situation can be somewhat ameliorated by choosing a regular ellipsoid instead of a sphere for the solute cavity. In that case, Eq. (11.17) can still be solved in a simple fashion, with the reaction field factors depending on the ellipsoidal semiaxes (Rinaldi, Rivail, and Rguini 1992). However, while this is clearly an improvement on a spherical cavity, the small number of solutes that may be well described as ellipsoidal does not make this a particularly satisfactory solution.

So, while derivations of SCRF theory using ideal cavities are very useful for conceptual purposes, they are insufficiently accurate for all but the most crude analyses. Modern applications of continuum models almost invariably use arbitrary cavity shapes, typically constructed from overlapping atomic spheres, and we turn to examples of these models next.

11.2.1.2 Arbitrary cavities

The concept of molecular shape with which most chemists are comfortable is almost certainly that represented by space-filling models constructed from the overlap of atomic spheres

having appropriate van der Waals radii. For such arbitrary, lumpy cavities, analytic solutions of the PB equation are no longer possible, and the reaction field must be determined numerically. The approach taken by most classical PB software – classical implying the charge distribution is not allowed to change – is formally to

1.Divide space according to a three-dimensional grid.

2.Define the molecular cavity and assign gridpoints the appropriate dielectric constant – in classical calculations, the interior is often assigned a dielectric constant between two and four to mimic solute polarizability.

3.‘Discretize’ the solute charge distribution onto interior grid points using some algorithm – e.g., divide every atomic partial charge equally over the nearest grid point and its 14 nearest neighbors.

4.Determine the electrostatic potential at each grid point by numerical solution of the PB equation; this process is typically iterative.

5.Once the potential is available, evaluate Eq. (11.3) as a pointwise sum over points carrying non-zero charge.

There are many technical challenges associated with this process that are worth keeping in mind. Like any numerical method, it is most successful when the density of discrete points is very high. However, as we are working with three-dimensional space, an order of magnitude decrease in the spacing between adjacent points increases the total number of points by three orders of magnitude, making the solution of the PB equation much more computationally demanding. So-called ‘focusing’ methods have been developed to try to move from coarse grids to finer grids in an efficient manner. With most grid densities in everyday use, the density remains sufficiently coarse that different orientations of the solute in space can give rise to non-trivially different values for GP. Reported values are sometimes averaged over several random orientations.

A related issue is that the potential can be very sensitive to grid points very near the cavity surface, where the dielectric constant is changing instantaneously. By construction, the cavity is actually defined only to within the grid-point spacing. The van der Waals radii defining the cavity surface that determines whether a given gridpoint is inside or outside the solute may either be chosen arbitrarily or optimized for a particular computational model (see, for example, Banavali and Roux 2002).

The primary area where classical PB equations find application is to biomolecules, whose size for the most part precludes application of quantum chemical methods. The dynamics of such macromolecules in solution is often of particular interest, and considerable work has gone into including PB solvation effects in the dynamics equations (see, for instance, Lu and Luo 2003). Typically, force-field atomic partial charges are used for the primary solute charge distribution.

It is noteworthy that with biomolecules it is often the electrostatic potential itself that is of primary interest, not its use to compute solvation free energies. Since the PB potential is presumably a more accurate picture of the potential in solution than one that would be

derived from a vacuum calculation, as described in Chapter 9, the method is often employed for this purpose. When potentials are visualized on the molecular van der Waals surface, many enzymes, for instance, show large regions of uniformly positive or negative potential, suggesting preferred binding sites for ligands having opposite charge, or channels for directing in substrates having opposite charge, etc.

Rather than solving the PB equation on a three-dimensional grid, the differential equation can be recast into a boundary element problem by representing the potential using a charge density spread over the molecular surface (see, for instance, Zauhar and Varnek, 1996). To make the calculation more convenient, the surface is usually tesselated into spherical triangles, and the charge density on each element is collapsed into a point charge in the center of the triangle. The charge –potential integral of Eq. (11.3) is thus replaced by a sum over charge –charge interactions. As a solution of what amounts to a surface integral instead of a volume integral, this procedure is somewhat less sensitive to numerical noise, but still requires some care to ensure sufficiently small surface tesserae are employed. A problem of some concern can arise when the centroids of spherical triangles associated with two different atoms are very near one another in space. In that case, the short-range charge – charge interaction can be so large as to introduce significant instabilities. As a result, some procedures delete regions of the surface near where atomic spheres overlap and accept a reduced accuracy in being able to represent the potential as a consequence.

Coming back to quantum mechanical continuum models, in the most general sense we now seek to solve the non-linear Schrodinger¨ equation

where V is a general reaction field inside the cavity that depends upon . As shown for the special case of the Kirkwood–Onsager model above, when is expressed as a Slater determinant, the orbitals minimizing Eq. (11.18) can be determined from

where F is the Fock operator. Entirely analogous expressions exist for DFT.

In formalism, this is really no different than the classical situation just described, except that the electronic-charge distribution is continuous, as opposed to discretized, and the non-linear character of the equations introduces an iterative component to the SCRF procedure that goes hand in hand with permitting relaxation of the charge distribution. That being the case, the methods used to represent the reaction field are essentially the same as those used in the classical situation. For example, SCRF schemes solving for the reaction field on a three-dimensional grid have been described by both Chen et al. (1994) and Tannor et al. (1994).

Perhaps the most widely used scheme for SCRF implementations of the Poisson equation is the surface area boundary element approach. This was first formalized by Miertus, Scrocco, and Tomasi (1981), and these authors referred to their construction as the polarized continuum model (PCM). While that name continues to find ample use in the literature, MST (the initials

of the authors’ last names) finds roughly equal usage, and some authors use PCM to refer generically to any continuum SCRF scheme.

A number of variations on the PCM formalism have appeared since its first publication. Some are purely technical in nature, designed to improve the computational performance of the method, e.g., an integral equation formalism for solving the relevant SCRF equations which facilitates computation of gradients and molecular response properties (IEF-PCM; Cossi et al. 2002), an extension to permit application to infinite periodic systems in one and two dimensions (Cossi 2004), and an extension to liquid/liquid and liquid/vapor interfaces (Frediani et al. 2004). Others reflect differences in how the molecular cavity is defined. For the most part, Tomasi and co-workers have maintained a strategy where the cavity is constructed from overlapping atomic spheres having radii 20% larger than their tabulated van der Waals radii, with a special distinction being made between ‘polar’ and ‘non-polar’ hydrogen atoms. As an alternative, Foresman et al. (1996) suggested defining the cavity as that region of space surrounded by an arbitrary isodensity surface, i.e., a surface characterized by a constant value of the electron density. That surface can either be located from the gas-phase density, and held fixed (IPCM) or determined self-consistently, adding yet another iterative level to the SCRF process (SCIPCM). Part of the motivation for these latter two modifications was to decrease the number of cavity parameters from one per atom to one total. However, insofar as the modeling of a molecular solvent by a continuum is by nature a fictional construct, it is not obvious that such a decrease in parameters can be regarded as a virtue. A further discussion of cavity definitions is deferred to Section 11.4.1, and it suffices to note here that the IPCM and SCIPCM methods tend to be considerably less stable in implementation than the original PCM process, and can be subject to erratic behavior in charged systems, so their use cannot be recommended (Cossi et al. 1996).

A third possibility that has received extensive study in the SCRF regime is one that has seen less use at the classical level, at least within the context of general cavities, and that is representation of the reaction field by a multipole expansion. Rinaldi and Rivail (1973) presented this methodology in what is arguably the first paper to have clearly defined the SCRF procedure. While the original work focused on ideal cavities, this group later extended the method to cavities of arbitrary shape. In formalism, Eq. (11.17) is used for any choice of cavity shape, but the reaction field factors f must be evaluated numerically when the cavity is not a sphere or ellipsoid (Dillet et al. 1993). Analytic derivatives for this approach have been derived and implemented (Rinaldi et al. 2004).

Most of the models described above have also been implemented at correlated levels of theory, including perturbation theory, CI, and coupled-cluster theory (of course, the DFT SCRF process is correlated by construction of the functional). Unsurprisingly, if a molecule is subject to large correlation effects, so too is the electrostatic component of its solvation free energy.

Note that, insofar as all of the above models simply represent alternative mathematical approaches to solving the Poisson equation, in the limit of converging them with respect to grid density, tesserae density, multipole expansion, etc., they should all give identical