Tutorial on Umbrella Sampling Simulation with a Combined QM/MM Potential: The Potential of Mean Force for an S_N2 Reaction in Water

We present a tutorial to carry out umbrella-sampling free-energy simulations with a combined quantum mechanical and molecular mechanical (QM/MM) potential, which may also be used in a computational or biophysical chemistry curriculum for first-year graduate and undergraduate students. In this article, we choose the Type II S_N2 Menshutkin reaction between ammonia and chloromethane to construct the potential of mean force (PMF) for the reaction in aqueous solution. In this exercise, we wish to accomplish three tasks: (1) an understanding of the concept of PMF and the umbrella-sampling free-energy simulation method, (2) the use of a combined QM/MM potential in molecular dynamics simulation of chemical reactions, and (3) an understanding of solvent effects and intermolecular interactions on chemical reactions through comparison with gas-phase results. Analysis of the simulation results allows students to appreciate the difference between the transition state along a PMF from statistical simulations versus the optimized saddle point in the gas phase, the shift of transition state with respect to solvation, and the significance of electronic polarization. Through this tutorial, students will gain the necessary skills to carry out free energy simulations of chemical reactions in solution and in enzymes both in a classroom and in the laboratory.