Free Energies of Quantum Particles: The Coupled-Perturbed Quantum Umbrella Sampling Method

We introduce a new simulation method called Coupled-Perturbed Quantum Umbrella Sampling that extends the classical umbrella sampling approach to reaction coordinates involving quantum mechanical degrees of freedom. The central idea in our method is to solve coupled-perturbed equations to find the response of the quantum system's wave function along a reaction coordinate of interest. This allows for propagation of the system's dynamics under the influence of a quantum biasing umbrella potential and provides a method to rigorously undo the effects of the bias to compute equilibrium ensemble averages. In this way, one can drag electrons into regions of high free energy where they would otherwise not go, thus enabling chemistry by fiat. We demonstrate the applicability of our method for two condensed-phase systems of interest. First, we consider the interaction of a hydrated electron with an aqueous sodium cation, and we calculate a potential of mean force that shows that an e(-):Na(+) contact pair is the thermodynamically favored product starting from either a neutral sodium atom or the separate cation and electron species. Second, we present the first determination of a hydrated electron's free-energy profile relative to an air/water interface. For the particular model parameters used, we find that the hydrated electron is more thermodynamically stable in the bulk rather than at the interface. Our analysis suggests that the primary driving force keeping the electron away from the interface is the long-range electron-solvent polarization interaction rather than the short-range details of the chosen pseudopotential.