Optimal Intermediate Hamiltonians for Non-Equilibrium Free Energy Calculations: A Numerical Study of Markov Models

The Jarzynski relation enables the estimation of equilibrium free energy differences from non-equilibrium, finite-time switching simulations. These estimates usually converge poorly because rare trajectories dominate the exponential work average. Here, we numerically determined and explored the sequence of intermediate Hamiltonians connecting initial and final states that minimize the mean squared error (MSE) of the Jarzynski estimator and thereby enhance convergence. For discrete-time Markov models, an exact tilted-master-equation representation of the MSE in the large-sample limit, combined with automatic differentiation, enables efficient gradient-based minimization over all intermediate energies. We applied our approach to three model systems of increasing complexity: a two-state model, a double-well potential, and a shifted potential well. In all three systems, the optimal intermediate Hamiltonians jump at the initial and final times. Extensive Monte Carlo simulations show that optimal intermediates can reduce the MSE by more than an order of magnitude compared with linear and logarithmic interpolation, most strongly for large changes in the energy landscape. Remarkably, they need not dissipate less work than intermediates yielding larger errors. Our results suggest heuristics for more efficient non-equilibrium free energy calculations of realistic molecular systems: optimal intermediate Hamiltonians jump at the initial and final times; for barrier-crossing problems, the barrier should be lowered rapidly and raised again later; and minimizing dissipation does not guarantee faster convergence.