Convergence of single-step free energy perturbation

ABSTRACT

The convergence of free energy perturbation (Zwanzig's equation) and its non-equilibrium extension (Jarzynski's equation) is herein investigated from a practical point of view. We focus on cases where neither intermediate steps nor two-sided methods (Bennett, Crooks) can be used to compute the free energy difference of interest. Using model data sampled from several probability densities, as well as comparing results of actual free energy simulations with reference values, we find, in agreement with existing theoretical work, that systematic errors are strongly correlated with the variance in the distribution of energy differences / non-equilibrium work values. The bias metric Π introduced by Wu and Kofke (J. Chem. Phys. 121, 8742 (2004)) is found to be a useful test for the presence of bias, i.e. systematic error in the results. By contrast, use of the second-order cumulant approximation to approximate the full Zwanzig or Jarzynski equation leads to poorer results in almost all cases.

KEYWORDS:

• Free energy perturbation
• Zwanzig's equation
• Jarzynski's equation
• cumulant expansion
• QM/MM free energy simulation
• indirect cycle

Acknowledgments

We thank Fiona L. Kearns and Phillip S. Hudson for careful reading and helpful comments on the manuscript. Additionally, we would like to highlight that this material is based upon work supported by the National Science Foundation under CHE-1464946. Further, HLW would like to thank NSF for computational support via their Major Research Instrumentation Program (MRI 1531590) and via XSEDE (MCB150037).

Notes

1. While we report/utilise some results from backwards calculations, i.e., QM/MM→ MM, in this work, we stress again that this would not be feasible in most real world applications!

2. Obviously, numerical over-/underflow issues must be taken care of either by means of Equation (3(3) ) or the techniques suggested by Berg [33].

Additional information

Funding

National Science Foundation [Grant Number CHE-1464946]; Major Research Instrumentation Program [Grant Number MRI 1531590]; XSEDE [Grant Number MCB 150037].