Potential distribution theorem derivation

As a preliminary point, we note that the decoupled averaging discussed here in classical views of the potential distribution theorem derives from the denominator of Eq. (3.17), p. 40. This is unchanged in the present quantum mechanical discussion, and thus the sampling of the separated subsystems could be highly quantum mechanical without changing those formalities. 

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. 

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... 

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... 

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. 

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. 

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. 

This will prove useful in deriving the approximations. Then we can write the quantum potential distribution theorem as... 

The strategy for our derivation will be to insert this resolution of unity, Eq. (7.13), within the averaging brackets of the potential distribution theorem, then expand and order the contributions according to the number of factors of b (j) that appear. We emphasize that physical interactions are not addressed here and that the hard-core interactions associated with discontinuity in / (j) appear for counting purposes only. 

Our discussion here explores active connections between the potential distribution theorem (PDT) and the theory of polymer solutions. In Chapter 4 we have already derived the Flory-Huggins model in broad form, and discussed its basis in a van der Waals model of solution thermodynamics. That derivation highlighted the origins of composition, temperature, and pressure effects on the Flory-Huggins interaction parameter. We recall that this theory is based upon a van der Waals treatment of solutions with the additional assumptions of zero volume of mixing and more technical approximations such as Eq. (4.45), p. 81. Considering a system of a polymer (p) of polymerization index M dissolved in a solvent (s), the Rory-Huggins model is... 

Use the potential distribution theorem and the above discussion to derive an expression for the excess chemical potential computed using the IRS idea. 

Boltzmann [3]. Boltzmann was led to thiB generalized formulation of the problem by some attempts he had undertaken (1866) 11] to derive from kinetic concepts the Camot-Clausius theorem about the limited convertibility of heat into work. In order to carry through such a derivation for an arbitrary thermal system (Boltzmann [5], (1871)) it was necessary to calculate, e.g., for a nonideal gas, bow in an infinitely slow change of the state of the system the added amount of heat is divided between the translational and internal kinetic energy and the various forms of potential energy of the gas molecule. It is just for this that the distribution law introduced above is needed. 

In the next section we shall present a simplified expansion theorem of osmotic pressure which was first obtained by McMillan and Mayer. This cluster expansion theory will be further extended in Section 3 to distribution functions, and medn results of Kirkwood and Buff will be recovered. A new and simple derivation of the cluster expansion of the pair distribution function is also given. Section 4 presents a new expression for the chemical potential of solvents in dilute solutions. Section 5 shows how the general solution theory may be applied to compact macromolecules. Finally, Section 6 deals with the second osmotic virial coefficient of flexible macromolecules and is followaJ in Sa tion 7 by concluding remarks. 

In contrast to the effective harmonic prescription for centroid-based dynamics, in CMD the force is a unique function of the system. That is, the force on a centroid trajectory at some time and position in space is not different from the force experienced by a different centroid trajectory at that same point in space but at a different time. Furthermore, the centroid trajectories are derived from the same effective potential as the one giving the exact centroid statistical distribution so that a centroid ergodic theorem will hold. The CMD approach satisfies this condition, while the analytically continued optimized LHO theory may not. Finally, CMD recovers the exact limiting expressions for globally harmonic potentials and for general classical systems. 

The Hohenberg-Kohn theorem ean be proved for an arbitrary external potential-this property of the density is ealled the v-representability. The arbitrariness mentioned above is necessary in order to define in future the functionals for more general densities (than for isolated molecules). We will need that generality when introducing the functional derivatives (p. 584) in which p(r) has to result from any external potential (or to be a v-representable density). Also, we will be interested in a non-Coulombic potential corresponding to the haniionic helium atom (cf. harmonium, p. 589) to see how exact the DFT method is. We may imagine p, which is not u-representable e.g., discontinuous (in one, two, or even in every point like the Dirichlet function). The density distributions that are not u-representable are out of our field of interest. 

To conclude, we have seen that for a given wave function and Hamiltonian, the Ehrenfest theorem can be instrumentalized to derive explicit expressions for the density and current-density distributions by rewriting it in such a way that the continuity equation results. We will rely on this option in the relativistic framework in chapters 5, 8, and 12 to define these distributions for relativistic Hamiltonian operators and various approximations of N-particle wave functions. From the derivation, it is obvious that the definition of the current density is determined by the commutator of the Hamiltonian operator with the position operator of a particle. All terms of the Hamiltonian which depend on the momentum operator of the same particle will produce contributions to the current density. In section 5.4.3 we will encounter a case in which the momentum operator is associated with an external vector potential so that an additional term will show up in the commutator. Then, the definition of the current density has to be extended and the additional term can be attributed to an (external-field) induced current density. 

The Hohenbeig and Kohn theorem (1964 Leach, 1996, pp. 528-533 Levine, 2013) states that all the properties of a molecular system in its ground state can be derived from the electron density distribution function. The total eneigy may be expressed as the sum of kinetic, potential, and exchange/correlation terms as in Equation 5.2, where p is to be understood as a function of the internal coordinates, symbolized by the vector r ... 

A fundamental equation in electrostatic theory is the Poisson equation. The Poisson equation can be derived from the Gauss theorem [245] and relates the potential to the charge density or distribution by the equation... 

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