The Potential Distribution Theorem

The PDT expression for the chemical potential of a classical molecular species is 

The chemical potential pa on the left is the full chemical potential including ideal and excess parts. In this chapter we will scale the chemical potentials by (3 and often refer to this unitless quantity as the chemical potential. 3/ia yields the absolute activity. The first term on the right is the ideal-gas chemical potential, where pa is the number density, Aa is the de Broglie wavelength, and q 1 is the internal (neglecting translations) partition function for a single molecule without interactions with any other molecules. 

At equilibrium, the chemical potential for a given molecular species is constant throughout the system. The two terms on the right-hand side of (11.4) can vary in space, however, so as to add up to a constant. In an inhomogeneous system, the number density and excess chemical potential adjust so as to yield the same constant chemical potential. Due to the local nature of the excess chemical potential, it is reasonable to define an excess chemical potential at a single point in space and/or for a single molecular conformation [29]. That excess chemical potential then determines 

A useful view of averages can be expressed via the PDT. The configurational average of the quantity F is 

The normalization integrals for the averages in the numerator and denominator cancel each other, leaving the traditional expression for the thermal average of F with the distinguished molecule present in the solution. This expression for the average will prove helpful several times below. The PDT is discussed extensively in Chap. 9, and in [29], 

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We present a molecular theory of hydration that now makes possible a unification of these diverse views of the role of water in protein stabilization. The central element in our development is the potential distribution theorem. We discuss both its physical basis and statistical thermodynamic framework with applications to protein solution thermodynamics and protein folding in mind. To this end, we also derive an extension of the potential distribution theorem, the quasi-chemical theory, and propose its implementation to the hydration of folded and unfolded proteins. Our perspective and current optimism are justified by the understanding we have gained from successful applications of the potential distribution theorem to the hydration of simple solutes. A few examples are given to illustrate this point. 

Equations (2) and (3) relate intermolecular interactions to measurable solution thermodynamic properties. Several features of these two relations are worth noting. The first is the test-particle method, an implementation of the potential distribution theorem now widely used in molecular simulations (Frenkel and Smit, 1996). In the test-particle method, the excess chemical potential of a solute is evaluated by generating an ensemble of microscopic configurations for the solvent molecules alone. The solute is then superposed onto each configuration and the solute-solvent interaction potential energy calculated to give the probability distribution, Po(AU/kT), illustrated in Figure 3. The excess... 

A virtue of the potential distribution theorem approach is that it enables precise assessment of the differing consequences of intermolecular interactions of differing types. Here we use that feature to inquire into the role of electrostatic interactions in biomolecular hydration. 

With applications to protein solution thermodynamics in mind, we now present an alternative derivation of the potential distribution theorem. Consider a macroscopic solution consisting of the solute of interest and the solvent. We describe a macroscopic subsystem of this solution based on the grand canonical ensemble of statistical thermodynamics, accordingly specified by a temperature, a volume, and chemical potentials for all solution species including the solute of interest, which is identified with a subscript index 1. The average number of solute molecules in this subsystem is... 

We noted above that the potential distribution theorem benefits from the fact that only local information on solute—solvent interactions is needed for AU in Eq. (5). Here we develop that idea into the quasichemical description for local populations involved in Eq. (25) (Pratt and LaViolette, 1998 Pratt and Rempe, 1999 Hummer etal., 2000 Pratt et al., 2001). 

The theories of hydration we have developed herein are built upon the potential distribution theorem viewed as a local partition function. We also show how the quasi-chemical approximations can be used to evaluate this local partition function. Our approach suggests that effective descriptions of hydration are derived by defining a proximal... 

Nearly 10 years after Zwanzig published his perturbation method, Benjamin Widom [6] formulated the potential distribution theorem (PDF). He further suggested an elegant application of PDF to estimate the excess chemical potential -i.e., the chemical potential of a system in excess of that of an ideal, noninteracting system at the same density - on the basis of the random insertion of a test particle. In essence, the particle insertion method proposed by Widom may be viewed as a special case of the perturbative theory, in which the addition of a single particle is handled as a one-step perturbation of the liquid. 

Computing thermodynamic properties is the most important validation of simulations of solutions and biophysical materials. The potential distribution theorem (PDT) presents a partition function to be evaluated for the excess chemical potential of a molecular component which is part of a general thermodynamic system. The excess chemical potential of a component a is that part of the chemical potential of Gibbs which would vanish if the intermolecular interactions were to vanish. Therefore, it is just the part of that chemical potential that is interesting for consideration of a complex solution from a molecular basis. Since the excess chemical potential is measurable, it also serves the purpose of validating molecular simulations. 

In this chapter, we discuss, exemplify, and thus support the assertions that the potential distribution theorem provides ... 

Potential distribution methods are conventionally called test particle methods. Because the assertions above outline a general and basic position for the potential distribution theorem, it is appropriate that the discussion below states the potential... 

Background Notation and Discussion of the Potential Distribution Theorem... 

Here we establish notation that is integral to this topic in the course of discussion of basic features of the potential distribution theorem (PDT). 

The potential distribution theorem has been around for a long time [13-17], but not as long as the edifice of Gibbsian statistical mechanics where traditional partition functions were first encountered. We refer to other sources [10] for detailed derivations of this PDT, suitably general for the present purposes. 

Our point of view is that the evaluation of the partition function (9.5) can be done by using any available tool, specifically including computer simulation. If that computer simulation evaluated the mechanical pressure, or if it simulated a system under conditions of specified pressure, then /C,x would have been determined at a known value of p. With temperature, composition, and volume also known, (9.2) and (9.1) permit the construction of the full thermodynamic potential. This establishes our first assertion that the potential distribution theorem provides a basis for the general theory of solutions. 

Beck, T. L. Paulaitis, M. E. Pratt, L. R., The Potential Distribution Theorem and Models of Molecular Solutions, Cambridge University Press Cambridge, 2006... 

Most free energy and phase-equilibrium calculations by simulation up to the late 1980s were performed with the Widom test particle method [7]. The method is still appealing in its simplicity and generality - for example, it can be applied directly to MD calculations without disturbing the time evolution of a system. The potential distribution theorem on which the test particle method is based as well as its applications are discussed in Chap. 9. 

Here, we report some basic results that are necessary for further developments in this presentation. The merging process of a test particle is based on the concept of cavity function (first adopted to interpret the pair correlation function of a hard-sphere system [75]), and on the potential distribution theorem (PDT) used to determine the excess chemical potential of uniform and nonuniform fluids [73, 74]. The obtaining of the PDT is done with the test-particle method for nonuniform systems assuming that the presence of a test particle is equivalent to placing the fluid in an external field [36]. 

This relation is the potential distribution theorem [73, 74], which gives a physical interpretation of the cavity function in terms of the chemical potential, and the excess interaction generated by the test particle, Y j>2 u(rv)> yia the ensemble average of its Boltzmann factor. In numerical simulation, the use of such a test-particle insertion method is of prime importance in calculating the cavity function at small distances and particularly at zero separation. Note that if the particle labeled 1 approaches the particle labeled 2, a dumbbell particle [41] is created with a bond length L = r2 n corresponding to a dimer at infinite... 

The initial introduction of conditional probabilities is typically associated with the description of independent events, P A, B) = P A)P B) when A and B are independent. Our description of the potential distribution theorem will hinge on consideration of independent systems first, a specific distinguished molecule of the type of interest and, second, the solution of interest. We will use the notation ((... ))o to indicate the evaluation of a mean, average, or expectation of ... for this case of these two independent systems. The doubling of the brackets is a reminder that two systems are considered, and the subscript zero is a reminder that these two systems are independent. Then a simple example of a conditional expectation can be given that uses the notation explained above and in Fig. 1.8 ... 

Since the density p appears in a dimensionless combination here, the concentration dependence of the chemical potential comes with a choice of concentration units. The first term on the right side of Eq. (3.1) expresses the colligative property of dilute solutions that the thermodynamic activity of the solute, is proportional to its concentration, p. The excess chemical potential accounts for intermolecular interactions between the solution molecules, and is given by the potential distribution theorem (Widom, 1963 1982) ... 

In the context of van der Waals theory, a and b are positive parameters characterizing, respectively, the magnitude of the attractive and repulsive (excluded volume) intermolecular interactions. Use this partition function to derive an expression for the excess chemical potential of a distinguished molecule (the solute) in its pure fluid. Note that specific terms in this expression can be related to contributions from either the attractive or excluded-volume interactions. Use the Tpp data given in Table 3.3 for liquid n-heptane along its saturation curve to evaluate the influence of these separate contributions on test-particle insertions of a single n-heptane molecule in liquid n-heptane as a function of density. In light of your results, comment on the statement made in the discussion above that the use of the potential distribution theorem to evaluate pff depends on primarily local interactions between the solute and the solvent. 

Note how averages associated with the system including the solute of interest look from the perspective of the potential distribution theorem. The averaging ((... ))o does not involve the solute-solution interactions, of course. Those are averages... 

If molecular densities were determined on the basis of Eq. (3.38), atomic densities might be evaluated by contraction of those results. Equation (3.38) provides a derivation of the previously mentioned conditional density of Eq. (3.4). This point hints at a physical issue that we note. As we have emphasized, the potential distribution theorem doesn t require simplified models of the potential energy surface. A model that implies chemical formation of molecular structures can be a satisfactory description of such molecular systems. Then, an atomic formula such as Eq. (3.35) is fundamentally satisfactory. On the other hand, if it is clear that atoms combine to form molecules, then a molecular description with Eq. (3.38) may be more convenient. These issues will be relevant again in the discussion of quasi-chemical theories in Chapter 7 of this book. This issue comes up in just the same way in the next section. 

The joint probability density for the positions of two specific atoms can be evaluated using the potential distribution theorem formula for averages, Eq. (3.24), p. 42 ... 

Classical statistical mechanical theory is, for the most part, adequate for the solutions treated in this book (Benmore et al, 2001 Tomberli et al, 2001), as has been discussed more specifically elsewhere (Feynman and Hibbs, 1965). It is important to distinguish that issue of statistical mechanical theory from the theory, computation, and modeling involved in the interaction potential energy U N). The potential distribution theorem doesn t require specifically simplified forms for /(2V) on grounds of statistical mechanical principal simplifications can make calculations more practical, of course, but those are issues to be addressed for specific cases. 

The point to be underscored is that many available approximate solutions for p (TV, TV ) of Eq. (3.65), e.g. (Gomez and Pratt, 1998), with the Boltzmann-Gibbs treatment of heavy-particle exchange, can be applied to evaluation of the potential distribution theorem. The most important physical requirement is that such a model gracefully adapt to the classical limit, because that is the most important physical limit for molecular solutions. 

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