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

a Values in parentheses are for vertical process.

b Values differ from those originally reported by Patterson, Cramer, and Truhlar (2001) by 0.08 V per electron consumed. This difference reflects a more accurate measurement of the absolute potential of the normal hydrogen electrode as 4.36 V instead of 4.44 V since the time of that publication. See Lewis et al. (2004) and Section 11.4.1.

each level being supplemented by aqueous solvation energies computed from the SM5.42R/BPW91/DZVP//BPW91/aug-cc-pVDZ level of theory when appropriate. The reduction potentials in solution are in units of volts relative to the standard hydrogen electrode, and the authors provide a detailed appendix showing how to convert between the various standard states and conventions typically adopted in theoretical and experimental work. They note that the CCSD(T) level, combined with the continuum solvent model when needed, is a better choice than the DFT method; the mean unsigned error in predicted reduction potentials for the CCSD(T) model is 0.09 V while for the DFT model it is 0.24 V. As the DFT level does somewhat better for the gas-phase free energies of reaction than the CCSD(T) level, it appears that there is some modest cancellation of errors in the solvation free energies that improves the performance of the CCSD(T) model.

Having identified the optimal level of theory, the authors apply it to various structures, primarily stationary points on the gas-phase PES, to characterize the energetics associated with various postulated mechanistic pathways (Figure 11.13). They identify the first mechanistic step as electron transfer followed by barrierless chloride ion elimination to generate the pentachloroethyl radical (C2Cl5• ). They discount the proposed heterolysis of this radical prior to a second electron transfer on the basis of the higher energy of the products C2Cl4+• and Cl−. Instead, they find that following a second electron transfer, there is again a barrierless elimination of a second chloride anion. However, this elimination is possible either from the same carbon as the first, in which case chloro(trichloromethyl)carbene is generated, or from the other carbon, in which case perchloroethylene is generated. While the former is much less exergonic than the latter, the barrierless nature of both reactions suggests that partitioning will be controlled by complex dynamic factors.

The importance of the former reaction is that it suggests a mechanism for the creation of Cl3CCO2H as a product. Based on analysis of the computed activation free energy for the rearrangement of chloro(trichloromethyl)carbene to perchloroethylene, the authors suggest that oxygen atom transfer to the carbene from some environmental source can

Figure 11.13 Relative aqueous free energies (eV) for various species on different reductive dechlorination paths of hexachloroethane as computed by Patterson and co-workers The relative energies are properly balanced, although for simplicity spectator species are not shown in every case. What is involved in achieving this balance for energy? What about free energy?

be kinetically competitive. Such a transfer would generate the acyl chloride equivalent of Cl3CCO2H, which would hydrolyze in short order to the carboxylic acid.

This paper provides an example of how accurate continuum models can open the door to the modeling of condensed-phase processes where solvation free energies have a very large influence on reaction energetics. It additionally offers a case study of how to first choose a model on the basis of experimental/theoretical comparisons over a relevant data set, and then apply that model with a greater expectation for its utility. The generality of this approach to other (equilibrium) electrochemical reactions seems promising.

Bibliography and Suggested Additional Reading

Bashford, D. and Case, D. A. 2000. ‘Generalized Born Models of Macromolecular Solvation Effects’

Annu. Rev. Phys. Chem., 51, 129.

Cramer, C. J. and Truhlar, D. G. 1999. ‘Implicit Solvation Models: Equilibria, Structure, Spectra, and Dynamics’, Chem. Rev., 99, 2160.

Cramer, C. J. and Truhlar, D. G., Eds., 1994. Structure and Reactivity in Aqueous Solution, ACS Symposium Series 568, American Chemical Society: Washington, DC.

Li, J. Cramer, and Truhlar, D. G. 1999. ‘Application of a Universal Solvation Model to Nucleic Acid Bases. Comparison of Semiempirical Molecular Orbital Theory, Ab Initio, Hartree – Fock Theory, and Density Functional Theory’, Biophys. Chem., 78, 147.

Llano, J. and Eriksson, L. A. 2002. ‘First Principles Electrochemistry: Electrons and Protons Reacting as Independent Ions’, J. Chem. Phys., 117, 10193.

Orozco, M. and Luque, F. J. 2000. ‘Theoretical Methods for the Description of the Solvent Effect on Biomolecular Systems’, Chem. Rev. 100, 4187.

Reichardt, C. 1990. Solvents and Solvent Effects in Organic Chemistry , VCH: New York.

Roux, B. and Simonson, T., Eds. 1999. Biophys. Chem. 78(1/2), [special issue devoted to implicit solvent models].

Schutz, C. N. and Warshel, A. 2001. ‘What Are the Dielectric “Constants” of Proteins and How to Validate Electrostatic Models?’, Proteins, 44, 400.

Tapia, O. and Bertran,´ J. 1996. Solvent Effects on Chemical Reactivity , Kluwer: Dordrecht.

Winget, P., Cramer, C. J., and Truhlar, D. G. 2004. “Computation of Equilibrium Oxidation and Reduction Potentials for Reversible and Dissociative Electron-Transfer Reactions in Solution”, Theor. Chem. Acc., 112, 217.

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12

Explicit Models for Condensed Phases

12.1 Motivation

At the heart of chemistry are atoms and molecules – they are the basis set in which chemical events are expressed. While the continuum models described in Chapter 11 can be very efficient and powerful in situations where the molecular nature of a surrounding condensed phase is superfluous to the question at hand, they are unsuitable when knowledge of the explicit behavior of the surroundings is deemed to be as important as its effect on some system embedded therein.

As noted in Chapter 11, the explicit representation of a condensed phase leads to a system characterized by an enormous number of degrees of freedom. This system thus has associated with it a phase space of high dimensionality, and typically one in which there are large volumes within a few kBT of one another in energy. Properties of such a system must be determined as statistical averages over phase space, as already discussed in some detail in Chapter 3. However, Chapter 3 was concerned primarily with observables other than thermodynamic properties, e.g., radial distribution functions, electrical moments, or vibrational frequencies. Here, the initial focus will be on carrying out simulations of condensed-phase systems specifically to extract thermodynamic information, including the free energy of solvation, the importance of which has already been amply discussed in Section 11.1.

12.2 Computing Free-energy Differences

As noted previously in Chapters 3 and 10, statistical thermodynamics relates all thermodynamic observables to the partition function Q. For ease of reference, the definition of Q and the equations defining various thermodynamic variables as a function of Q, some of which have appeared previously, are as follows

Q = e−E(q,p)/ kBT dqdp (12.1)

Essentials of Computational Chemistry, 2nd Edition Christopher J. Cramer

2004 John Wiley & Sons, Ltd ISBNs: 0-470-09181-9 (cased); 0-470-09182-7 (pbk)

where U is the internal energy, P is the pressure, H is the enthalpy, A is the Helmholtz free energy, S is the entropy, and G is the Gibbs free energy (in subsequent discussion, the Gibbs free energy will be implied by the words ‘free energy’ unless the Helmholtz free energy is explicitly specified). Note that we have adopted the classical expression for Q by formulating it as a phase space integral over all spatial (q) and momentum (p) coordinates. This assumes that the energy levels, computed as the sum of kinetic and potential energy terms by the Hamiltonian H (not to be confused with the enthalpy), are sufficiently closely spaced that we may convert the sum-over-states formulation of Q (see Eq. (10.2)) into an integral.

12.2.1Raw Differences

In chemistry, one is typically interested not in absolute values of thermodynamic functions but in their changes over the course of a chemical process. Consider, for instance, if one were to be interested in the difference in U for the proton shift reaction HCN → HNC in aqueous solution. Because U is a state function, the precise path over which the reaction occurs is not important – we need only evaluate U at the two endpoints to determine the difference. If we make use of the relationship

where Eq. (3.6) for the probability P of being at a particular point in phase space has been used. To evaluate Eq. (12.9) for both the aqueous HCN and HNC systems, we might carry

out a Monte Carlo simulation of each, and determine U as an ensemble average of E over the probabilistically correct set of snapshots generated by the MC approach, i.e.,

where B is aqueous HNC and A is aqueous HCN. To carry out the simulations in a rational way, one would want to employ the same conditions in each, e.g., number of solvent molecules, size of unit cell used within periodic boundary conditions, etc.

If one follows the procedure outlined above, the results are not very satisfying. The problem is that the total energies E are large numbers. Thus, even after sampling millions of configurations, the standard deviation in each ensemble average may still be on the order of, say, 10 kcal mol−1. Taking the error in U as the RMS of the two ensemble errors in E would then imply an error of 14 kcal mol−1. Such a large error is not very useful in most instances, which is disappointing, particularly given the large investment of computational resources required to generate the ensemble averages. Note that, with ergodic trajectories, we could have taken time averages from MD simulations instead of ensemble averages from MC simulations, but the error problem would be the same.

Let us consider instead of U the quantity A. In order to engineer a probabilistic fashion to determine A we may rewrite Eq. (12.5) as

in which case the Helmholtz free energy difference may be computed as

At first glance, the situation looks if anything worse than was true for U . Now the ensemble averages are not over the total energies (already large numbers), but over exponentials of the total energies expressed in multiples of kBT ! However, as long as the two systems A

and B do not differ from one another by enormous amounts, the ratio of the two expectation values in the last line on the r.h.s. of Eq. (12.12) (which is the inverse of the equilibrium constant for the reaction A → B) is sufficiently close to unity that errors in the individual ensemble averages on the order of one or two percent have a much smaller impact on the error of the ratio than was the case for U . Moreover, one takes the natural logarithm of the ratio to compute A, so the absolute magnitude of the error is reduced still more.

Nevertheless, converging the individual expectation values to the level of a few percent error is a painfully slow task, since the reduced probabilities of high-energy points are balanced by their exponentially larger contributions to the partition function. However, one may take advantage of the relatively small difference between systems A and B to introduce a further approximation that is extraordinarily useful.

12.2.2Free-energy Perturbation

If the ensembles in Eq. (12.2), over which the property averages for systems A and B are taken, were somehow to be the same, one would be able to take advantage of the properties of exponentials to write

where we have arbitrarily chosen to label the ensemble average as having been selected based on system A. This formulation offers some enormous advantages over Eq. (12.12). One of the most important is that all contributions to the energy from solvent– solvent interactions (which are enormously dominant over solvent– solute interactions, since there are so many more solvent molecules) cancel out in the energy difference, since the ensembles are (somehow) identical.

What is meant by an identical ensemble for two different species? It is helpful to return to our specific example of HCN and HNC. To determine the proper identical ensemble for HNC based on one chosen in the usual fashion for HCN, we first stipulate that all particles that are common to the two systems, i.e., all solvent molecules, the carbon atom, and the nitrogen atom, have identical positions and momenta when we evaluate the energy in system B as when we evaluate it in A. Then, the only contribution to the energy difference in Eq. (12.13) would be the different interactions that the hydrogen atom has with all of the other atoms, based on whether it is attached to C or N (see Figure 12.1).

So, for each snapshot of the simulation that contributes to the ensemble (by either MC or MD evaluation), we compute the energy differential for all of the atoms interacting with HB rather than HA. In Figure 12.1, the particular case of one of the hydrogen atoms on a first-shell water molecule is illustrated. As this is a non-bonded interaction in each case, the contribution from HD in a simple force field might be

Figure 12.1 HCN(H) and one water molecule from the simulation box. In free energy perturbation, the simulation snapshots would be generated using the HCN system, and the energy difference for HA being present compared to HB would then be computed over that ensemble. For example, the differential interaction of the two with HD would contribute to the full difference

where the functional forms of Eqs. (2.14) and (2.22) have been chosen to compute nonbonded interactions. This protocol defines free energy perturbation (FEP; Zwanzig 1954).

There are, however, potentially rather large problems involved in the scheme outlined thus far. In HCN, for instance, the nitrile lone pair is a fair hydrogen bond acceptor, and one may imagine that a water molecule will often be found hydrogen bonded to it, say at a distance of 2 A˚ . When a snapshot containing such a hydrogen bonded water is used to generate the HCN ensemble, HB will be materialized with a normal bond distance to the nitrogen, say 1 A,˚ and this will create an HH non-bonded interaction of only 1 A˚ . Such a geometry will be extremely high in energy, so that it should contribute in only the most paltry way to any ensemble average. However, the nature of the HCN system is such that it might be expected to occur with great regularity. The HCN ensemble will therefore be a very poor source for a HNC ensemble, and the free-energy difference computed using Eq. (12.13) will be very bad.

In order to avoid this problem, the switching between the two molecules may be broken up into smaller steps using a coupling parameter λ that may take on values from 0 to 1. We then write the energy of the system as a general function of λ

which emphasizes that the endpoints are still the physically meaningful ones, but we are willing to consider, computationally at least, chimeric systems having partial HCN and partial HNC character. The utility of such systems is that we may now generalize Eq. (12.13) to

λ=0