Convergence of free energy

Bitetti-Putzer, R. Yang, W. Karplus, M., Generalized ensembles serve to improve the convergence of free energy simulations, Chem. Phys. Lett. 2003, 377, 633-641... 

Direct Calculations of Free Energy Converge Very Siowiy... 

As we have already pointed out, the theoretical basis of free energy calculations were laid a long time ago [1,4,5], but, quite understandably, had to wait for sufficient computational capabilities to be applied to molecular systems of interest to the chemist, the physicist, and the biologist. In the meantime, these calculations were the domain of analytical theories. The most useful in practice were perturbation theories of dense liquids. In the Barker-Henderson theory [13], the reference state was chosen to be a hard-sphere fluid. The subsequent Weeks-Chandler-Andersen theory [14] differed from the Barker-Henderson approach by dividing the intermolecular potential such that its unperturbed and perturbed parts were associated with repulsive and attractive forces, respectively. This division yields slower variation of the perturbation term with intermolecular separation and, consequently, faster convergence of the perturbation series than the division employed by Barker and Henderson. 

To illustrate how stratification works in the context of free energy calculations, let us consider the transformation of state 0 into state 1 described by the parameter A. We further assume that these two states are separated by a high-energy barrier that corresponds to a value of A between Ao and Ai. Transitions between 0 and 1 are then rare and the free energy estimated from unstratified computer simulations would converge very slowly to its limiting value, irrespective of the initial conditions. If, however, the full range of A is partitioned into a number of smaller intervals, and... 

It has been mentioned that perhaps the greatest limitation to the precision of free energy calculations to date has been the often-inadequate sampling of a representative set of configurations of the system. Increases in computer power of course increase the radius of convergence of such calculations. Such increases come not only from the Moore s Law improvements in hardware, but also from algorithmic... 

Convergence properties of free energy calculations Alpha-cyclodextrin complexes as a case study. J. Am. Chem. Soc. 116 6293 (1994). 

Figure 2. Energies (upper panel) and temperatures (lower panel) of the 30 replica modified parallel tempering simulation of the trp-cage protein reported in the text. The dotted line in the upper panel corresponds to the estimate of the global optimum of the free energy (obtained independently). The lower panel demonstrates a rapid equilibration of the temperatures during the simulation. The upper panel demonstrates the convergence of the energy and the rapid exchange of information between the different replicas as discussed in the text.

PROBLEM L1.6 How does convergence of the sum under the influence of only the retardation function R (l ) create the appearance of the 1/Z3 variation of free energy ... 

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