A class of simple expressions of increasing accuracy for the free‐energy difference between two states is derived based on numerical thermodynamic integration. The implementation of these formulas requires simulations of the initial and final (and possibly a few intermediate) states. They involve higher free‐energy derivatives at these states which are related to the moments of the probability distribution of the perturbation. Given a specified number of such derivatives, these integration formulas are optimal in the sense that they are exact to the highest possible order of free‐energy perturbation theory. The utility of this approach is illustrated for the hydration free energy of water. This problem provides a quite stringent test because the free energy is a highly nonlinear function of the charge so that even fourth order perturbation theory gives a very poor estimate of the free‐energy change. Our results should prove most useful for complex, computationally demanding problems where free‐energy differences arise primarily from changes in the electrostatic interactions (e.g., electron transfer, charging of ions, protonation of amino acids in proteins).

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