Biasing the center of charge in molecular dynamics simulations with empirical valence bond models: free energetics of an excess proton in a water droplet.

Multistate empirical valence bond (EVB) models provide an accurate description of the energetics of proton transfer and solvation in complex molecular systems and can be efficiently used in molecular dynamics computer simulations. Within such models, the location of the moving protonic charge can be specified by the so-called center of charge, defined as a weighted average over the diabatic states of the EVB model. In this paper, we use first-order perturbation theory to calculate the molecular forces that arise if a bias potential is applied to the center of charge. Such bias potentials are often necessary when molecular dynamics simulations are used to determine free energies related to proton transfer and not all relevant proton positions are sampled with sufficient frequency during the available computing time. The force expressions we derive are easy to evaluate and do not create any significant computational cost compared with unbiased EVB simulations. As an illustration of the method, we study proton transfer in a small liquid water droplet consisting of 128 water molecules plus an excess proton. Contrary to predictions of continuum electrostatics, but in agreement with previous computer simulations of similar systems, we observe that the excess proton is predominantly located at the surface of the droplet. Using the formalism developed in this paper, we calculate the reversible work required to carry the protonic charge from the droplet surface to its core, finding a value of roughly 4 k(B)T.