Monte Carlo approach to the evaluation of the configura - tional free energy of chains

The application of a Monte Carlo approach for computing the configurational free energy of chains is described. Whereas the effectiveness of the previous Monte Carlo methods is limited to the systems in which the energies of the stares (configurations) are not far from zero, the present approach can yield accurate estimates of the free energy of the systems with large configurational energy. The method utilises the relationship: where AGM is the free energy difference between a pair of systems A and E and (M(UB-UA))A is the canonical ensemble average of the Metropolis function over A, UA and U, being the energies of a stare (configuration) corresponding to A and E respectively. (M(UB-UA))A is estimated from the configuration samples generated by a method described previously. Calculations of the free energy as a function of bond flexibility are performed for a chain of twenty bonds confined to a tetrahedral lattice. During the past two decades or so Monte Carlo methods have shown remarkable success in the computation of the ensemble means relating to such properties as configurational energy, configurational heat capacity and pressure (Wood 1968, La1 1971). However, since a Monte Carlo sample contains only a small fraction of the total configurations assumed by a system, the applicability of the method in the evaluation of the configurational partition function (Q) has been severely limited. Recently Valleau and co-workers and Bennett carried out studies on models for monatomic fluids with the main objective of enhacing the scope of the Monte Carlo approach in the evaluation of the partition function (e.g. Torrie and Valleau 1974, 1977, Bennett 1976). Their investigations indicate that it is feasible to estimate the ratio of the partition functions of a pair of systems; the computation is greatly facilitated if there exists sufficient overlap in the canonical ensembles corresponding to the two systems. The viability of a Monte Carlo scheme for calculating the partition function ratio depends essentially on the existence of a relationship between such a ratio and the ensemble means that are amenable to accurate estimation. Following Bennett consider a pair of systems, A and B, with configurational energies UA and U,, respectively, corresponding to a point q in the configuration phase space. Now let there be a function f (x) such that f (x) / f (-x) = e-'.