  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 non-linear Hamiltonian transformation sequences
for optimal sampling accuracy that differ markedly from established linear
transformations. We show that our sequences are also optimal for the Bennett
Acceptance Ratio (BAR) method, and our unifying framework generalizes BAR to
small sampling sizes and non-Gaussian error distributions. Simulations on a
Lennard-Jones gas show that an order of magnitude less sampling is required
compared to established methods.
