Implementation of Umbrella Integration within the Framework of the Empirical Valence Bond Approach

The umbrella integration method for calculating the potential of mean force (PMF) for a chemical reaction is implemented within the empirical valence bond (EVB) framework. In this implementation, the PMF is generated along the energy gap reaction coordinate, and the biasing potential is the difference between the mapping potential, which is defined to be a linear combination of the valence bond state energies, and the EVB ground state energy. The umbrella integration method is based on the derivative of the PMF with respect to the reaction coordinate. An analytical expression for this derivative applicable to certain types of EVB potentials is presented. The advantages of the umbrella integration method are illustrated by the application of both umbrella integration and the weighted histogram analysis method to the hydride transfer reaction catalyzed by the enzyme dihydrofolate reductase. This application demonstrates that the umbrella integration method reduces the statistical errors, converges efficiently, and does not require significantly overlapping windows. A modified version of the weighted histogram analysis method that shares these advantages is also proposed and implemented.