Free energy from molecular dynamics with multiple constraints

In molecular dynamics simulations of reacting systems, the key step to dete rmining the equilibrium constant and the reaction rate is the calculation o f the free energy as a function of the reaction coordinate. Intuitively the derivative of the free energy is equal to the average force needed to cons train the reaction coordinate to a constant value, but the metric tensor ef fect of the constraint on the sampled phase space distribution complicates this relation. The appropriately corrected expression for the potential of mean constraint force method (PMCF) for systems in which only the reaction coordinate is constrained was published recently. Here we will consider the general case of a system with multiple constraints. This situation arises when both the reaction coordinate and the 'hard' coordinates are constraine d, and also in systems with several reaction coordinates. The obvious advan tage of this method over the established thermodynamic integration and free energy perturbation methods is that it avoids the cumbersome introduction of a full set of generalized coordinates complementing the constrained coor dinates. Simulations of n-butane and n-pentane in vacuum illustrate the met hod.