Ensemble formula, partitioned

Extending the concept of isomerism from molecules to ensembles of molecules (EM) leads to a mathematical model of cons itutional chemistry ). An EM is described by its list of the molecular species in the form of suitable chemical formulas. For an EM one can define two types of empirical elementary formulas, on one hand the ensemble formula, on the other hand a partitioned empirical ensemble formula which consists of the molecular formulas of the molecules in the EM. 

Since isomeric molecules may differ constitutionally and stereochemically, a partitioned ensemble formula generally corresponds to more than one EM(A). The constitutional formula of an EM(A) contains the constitutional formulas of all molecules in that EM(A). An FIEM(A), the family of isomeric ensembles of molecules of an atom set A is the collection of all... 

Fig. 3. Functions in the integrand of the partition function formula Eq. (6). The lower solid curve labeled Pq AU/kT) is the probability distribution of solute-solvent interaction energies sampled from the uncoupled ensemble of solvent configurations. The dashed curve is the product of this distribution with the exponential Boltzmann factor, e AJJ/kT r the upper solid curve. See Eqs. (5) and (6).

The JE (Eq. (40)) indicates a way to recover free energy differences by measuring the work along all possible paths that start from an equihbrium state. Its mathematical form reminds one of the partition function in the canonical ensemble used to compute free energies in statistical mechanics. The formulas for the two cases are... 

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... 

Our perspective is that the PDT should be recognized as directly analogous to the partition functions which express the Gibbsian ensemble formulation of statistical mechanics. From this perspective, the PDT is a formula for a thermodynamic potential in terms of a partition function. Merely the identification of a... 

In an NPT (constant mass, pressure and temperature) ensemble of clay-water systems the Gibbs free energy, G, is related to the partition function, Q, by the formula ... 

Statistical mechanics may be used to derive practical microscopic formulae for thermodynamic quantities. A well-known example is the virial expression for the pressure, easily derived by scaling the atomic coordinates in the canonical ensemble partition function... 

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