Average molecular partition function

When treating polyatomics it is convenient to define an average molecular partition function, In = (lnQ)/N, for an assembly of N molecules. In the dilute vapor (ideal gas) this introduces no difficulty. There is no intermolecular interaction and In = (In Q)/N = ln(q) exactly (q is the microcanonical partition function). In the condensed phase, however, the Q s are no longer strictly factorable. Be that as it may, continuing, and assuming In = (In Q)/N, we are led to an approximate result which is superficially the same as Equation 5.10,... 

In Eq. 8, the denominator, named z, stands for the molecular partition function. It can be rigorously calculated for simple molecules using experimental spectroscopic data and is related to the average number of states that are accessible to a molecule at a given temperature. 

In order to calculate the partition function and molecular averages, a 4x4 statistical weight matrix for the fth residue is formulated to correlate the states of residues i-1, i, and / + 1 of the polymer chain ... 

The conformational properties of an uncharged molecular chain are well described by a (discrete) semiflexible chain model [33]. The chain is comprised of mass points, each one may represent several monomers, at positions r, (z =0,..., N). The (average) length of a bond is l. The partition function of such a chain is given by... 

Recent years have seen the extensive application of computer simulation techniques to the study of condensed phases of matter. The two techniques of major importance are the Monte Carlo method and the method of molecular dynamics. Monte Carlo methods are ways of evaluating the partition function of a many-particle system through sampling the multidimensional integral that defines it, and can be used only for the study of equilibrium quantities such as thermodynamic properties and average local structure. Molecular dynamics methods solve Newton s classical equations of motion for a system of particles placed in a box with periodic boundary conditions, and can be used to study both equilibrium and nonequilibrium properties such as time correlation functions. 

Statistical mechanics forms the foundation of the methodological developments of the free energy difference techniques, providing the link between macroscopic, measurable quantities of chemical systems, and the detailed, microscopic description of the molecular system. The thermodynamic quantities of interest are expressed in terms of ensemble averages, phase space probabilities or partition functions, all of which eventually are determined by the system Hamiltonian. The main difficulty in practical calculations does not lie in... 

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