Using the method of weighted residuals to compute potentials of mean force

We propose a general framework for approximating the potential of mean force (PMF) along a reaction coordinate in conformational space. This framework, based on the method of weighted residuals, can be viewed as a generalization of thermodynamic integration and direct histogram methods. Using weighted residuals allows for higher-order approximations to the PMF in the form of a global spectral method or a finite element method. In addition, the higher degree of continuity provided by spectral and higher-order elements makes weighted residual methods an attractive choice for use in tandem with biasing force methods. As an analysis tool, the weighted residuals framework provides a context for direct comparison of thermodynamic integration and histogram based methods. For validation of the new method, numerical experiments are performed on two systems: a simple double-well and alanine dipeptide in vacuum. Comparisons between the new weighted residual methods, thermodynamic integration, and WHAM are performed. When configuration space is perfectly sampled the high-order weighted residual methods are found to exhibit exponential convergence. For more realistic sampling, the weighted residual methods performed comparably to the other two. However, results suggest that spectral type methods are more robust with respect to parameter choices describing the solution space.