Generalized canonical partition function

As is briefly described in the Introduction, an exact equation referred to as the Ornstein-Zernike equation, which relates h(r, r ) with another correlation function called the direct correlation function c(r, r/), can be derived from the grand canonical partition function by means of the functional derivatives. Our theory to describe the molecular recognition starts from the Ornstein-Zernike equation generalized to a solution of polyatomic molecules, or the molecular Ornstein-Zernike (MOZ) equation [12],... 

Electric charge, generalized thermodynamic quantity, canonical partition function, vibrational coordinate Gas constant, resistance, radius Molar refraction... 

This principle can be put on a general footing by considering the canonical partition function. Absence of energy-entropy coupling means that this partition function is separable, i.e. it consists of the product of purely entropic and purely energetic contributions ... 

Generally, the (classical) canonical partition function Q N, V, T) can be related... 

The free energy difference methods reviewed in this chapter, unless specified otherwise, are discussed for conditions of constant volume and constant temperature (NVT). The extension to ensembles of other types is straightforward.The classical canonical partition function is determined by the classical Hamiltonian 3 6(p, q ), describing the interactions of all N particles in the system in terms of the set of generalized coordinates and conjugated momenta p. For a system with N particles at temperature T, the canonical partition function can be written as... 

Au contraire to the empirical equation of Tait for EOS predictions, theoretical models can be used but generally require an understanding of forces between the molecules. These laws, strictly speaking, need be derived from quantum mechanics. However, Lenard-Jones potential and hard-sphere law can be used. The use of statistical mechanics is an intermediate solution between quantum and continuum mechanics. A canonical partition function can be formulated as a sum of Boltzmann s distribution of energies over all possible states of the system. Necessary assumptions are made during the development of the partition function. The thermodynamic quantities can be obtained by use of differential calculus. For instance, the thermodynamic pressure can be obtained from the partition function Q as follows ... 

Irrespective of the nature of the solid, there are always species (atoms, ions and molecules) placed at the nodes of the lattice, and are animated with a motion of vibration around their equilibriiun positions. Thus, the partition function will include a contribution due to these vibrations. On the statistical level, the entities in question (atoms, ions or molecules) are considered to be localized particles and, in general, to describe the vibrations of the solid, it is sufficient to place ourselves in the context of die conventional limit case of statistics. It results from this that the contribution of the vibrations to the canonical partition function can be calculated on the basis of the atomic partition functions z by the relation ... 

We will calculate this contribution by first considering a solid as a macromolecule with TV vibrating entities, having 3AT independent vibrational degrees of freedom phonons. Two models can be used to express a contribution of these vibrations to the canonical partition function Einstein s model and Debye s, which is more general. Let us look at each of these two models in turn. 

The new internal degrees of freedom will generally tend to be vibrational degrees of freedom, and if the corresponding fundamental frequencies are Vi, V2,. .., v,. .., the contribution corresponding to the canonical partition function is ... 

We can now write formulas for the other thermodynamic functions of a general system in terms of the canonical partition function. 

The general equations for the canonical partition function and its application to thermodynamics functions are valid. From Eq. (27.4-6)... 

Consider a system of N particles with generalized coordinates and conjugate momenta p. The classical canonical partition function for the system at temperature T is given by >2... 

This equation forms the fundamental connection between thermodynamics and statistical mechanics in the canonical ensemble, from which it follows that calculating A is equivalent to estimating the value of Q. In general, evaluating Q is a very difficult undertaking. In both experiments and calculations, however, we are interested in free energy differences, AA, between two systems or states of a system, say 0 and 1, described by the partition functions Qo and (), respectively - the arguments N, V., T have been dropped to simplify the notation ... 

Because of the relationship between the different canonical ensembles the generalized partition function is best defined as the generalization of the... 

In the preceding section we have set up the canonical ensemble partition function (independent variables N, V, T). This is a necessary step whether one decides to use the canonical ensemble itself or some other ensemble such as the grand canonical ensemble (p, V, T), the constant pressure canonical ensemble (N, P, T), the generalized ensemble of Hill33 (p, P, T), or some form of constant pressure ensemble like those described by Hill34 in which either a system of the ensemble is open with respect to some but not all of the chemical components or the system is open with respect to all components but the total number of atoms is specified as constant for each system of the ensemble. We now consider briefly the selection of the most convenient formalism for the present problem. 

We shall treat more compUcated cases, such as systems with a larger number of identical or different sites, and also cases of more than one type of ligand. But the general rules of constructing the canonical PF, and hence the GPF, are the same. The partition functions, either Q or have two important properties that make the tool of statistical thermodynamics so useful. One is that, for macroscopic systems, each of the partition functions is related to a thermodynamic potential. For the particular PFs mentioned above, these are... 

It would be more circumspect to say that the partition function cannot in general be naturally or usefully expressed as a canonical average since that average inevitably falls into the category for which canonical sampling is inadequate [13]. 

Three of the eight thermodynamic potentials for a system with one species are frequently used in statistical mechanics (McQuarrie, 2000), and there are generally accepted symbols for the corresponding partition functions V[T = A = — RTlnQ, where Q is the canonical ensemble partition function ... 

We have derived a formula for the molecular partition function by considering a system containing many molecules at equilibrium with a heat bath. We can generalize our statistical mechanics by a gedanken experiment of considering a large number of identical systems, each with volume V and number of particles N at equilibrium with the heat bath at temperature T. Such a supersystem is called a canonical ensemble. Our derivation is the same the fraction of systems that are in a state with energy Et is... 

The minimization of the canonical transition state partition function as in Eq. (2.13) is generally termed canonical variational RRKM theory. This approach provides an upper bound to the more proper E/J resolved minimization, but is still commonly employed since it simplifies both the numerical evaluation and the overall physical description. It typically provides a rate coefficient that is only 10 to 20% greater than the E/J resolved result of Eq. (2.11). 

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