We recently presented a minimum variance method to calculate free energies and ensemble averages from multiple equilibrium simulations conducted at different thermodynamic states [1]. This method, based on statistical techniques for correcting biases in sampling, is useful for analyzing both simulations of phase changes and the results of single molecule experiments. This estimator gives the same results as multiple histogram methods [2,3] in the limit of vanishingly small histogram bins, but does not actually require histograms, eliminating bias due to binning. This new estimator is especially useful for highly multidimensional problems where histograms cannot be well populated. Unlike multiple histogram techniques, it also yields an expression for the statistical uncertainty of the estimates it produces.

However, this estimator (like multiple histogram techniques) requires reevaluation of the potential energy of the system at all other thermodynamic states under study. This can become a computationally unreasonable burden when there are many states, such as 3-dimensional potentials of mean force obtained through umbrella sampling or alchemical simulations with many intermediates, or in situations where the cost of obtaining the extra energy evaluations are high, such as when using polarizable force fields. This burden is especially unreasonable since many states with low overlap will contribute little to the corresponding free energy or ensemble averages. Methods such as Bennett's acceptance ratio [4,5] can be used to compute free energies between only neighboring, but this type of pairwise analysis ignores correlation between samples and is inappropriate when states overlap in phase space with many others.

To solve the problem of efficiently computing averages and free energies with large numbers of states, we derive a modification of the multistate minimum variance method that uses only states with sufficient mutual phase space overlap. For many cases of interest, this drastically decreases the number of states that must be considered with negligible loss of precision. We also present a number of test cases of this method, including free energy profiles for single molecule pulling experiments and the computation of small molecule partition coefficients.

[1] M. R. Shirts and J. D. Chodera, http://arxiv.org/abs/0801.1426 (2008)

[2] A. M. Ferrenberg and R. H. Swendsen, Phys. Rev. Lett. 63:1195-1198 (1989)

[3] S. Kumar, D. Bouzida, R. H. Swendsen, P. A. Kollman and J. M. J. Rosenberg, J. Comput. Chem., 13:1011-1021 (1992)

[4] C. H. Bennett, J. Comput. Phys. 22:245-268 (1976)

[5] M. K. Fenwick and F. A. Escobedo, J. Chem. Phys. 120:3066-3074 (2004)