  Free energy difference calculations based on atomistic simulations generally
improve in accuracy when sampling from a sequence of intermediate equilibrium
thermodynamic states that bridge the configuration space between two states of
interest. For reasons of efficiency, usually the same samples are used to
calculate the step-wise difference of such an intermediate to both adjacent
intermediates. However, this procedure violates the assumption of uncorrelated
estimates that is necessary to derive both the optimal sequence of intermediate
states and the widely used Bennett acceptance ratio (BAR) estimator. In this
work, via a variational approach, we derive the sequence of intermediate states
and the corresponding estimator with minimal mean squared error that account
for these correlations and assess its accuracy.
