A Geometric Approach to Constrained Molecular Dynamics and Free Energy

Abstract

We consider a molecule constrained to a hypersurface S in the conguration space R^m. In order to derive an expression for the mean force acting along the constrained coordinate we decompose the molecular vector field, and single out the direction of the respective coordinate utilising the structure of affine connections. By these means we reconsider the well-known results derived by Sprik et al. [1] and Darve et al. [2]; we gain concise geometrical insight into the different contributions to the force in terms of molecular potential, mean curvature, and the connection 1-form of the normal bundle over the submanifold S. Our approach gives rise to a Hybrid Monte-Carlo based algorithm that can be used to compute the averaged force acting on selected coordinates in the context of thermodynamic free energy statistics.