Determining Free-Energy Differences Through Variationally Derived Intermediates

[Image: see text] Free-energy calculations based on atomistic Hamiltonians and sampling are key to a first-principles understanding of biomolecular processes, material properties, and macromolecular chemistry. Here, we generalize the free-energy perturbation method and derive nonlinear Hamiltonian transformation sequences yielding free-energy estimates with minimal mean squared error with respect to the exact values. Our variational approach applies to finite sampling and holds for any finite number of intermediate states. We show that our sequences are also optimal for the Bennett acceptance ratio (BAR) method, thereby generalizing BAR to small sampling sizes and non-Gaussian error distributions.