Umbrella integration in two or more reaction coordinates

Umbrella integration is a method to analyze umbrella sampling simulations by calculating and integrating the mean force. Here, the method is extended to multidimensional reaction coordinates. Approximation of the probability distribution obtained from sampling by a multivariate normal distribution allows to calculate the mean force from the average and the covariance matrix of the reaction coordinate. Integration schemes of the free-energy gradient field are discussed. Integration on a real-space grid is compared to expansion of the gradient in a series of analytic functions (such as a Fourier analysis), which can be integrated, and the expansion of the gradient only at the window means in a series of analytic functions. The Fourier analysis was found particularly useful for periodic reaction coordinates, such as torsion angles. An expression is provided to calculate the Hessian of the free energy with respect to the reaction coordinates from sampling data. The utility of the method is demonstrated at the example of the free-energy surface of the alanine dipeptide in vacuum calculated with respect to the backbone torsion angles Phi and Psi. Relevance of the Jacobian term for non-Cartesian reaction coordinates is discussed.