Blue Moon sampling with biased or unbiased constrained dynamics

We introduce the Blue Moon method which uses constrained molecular simulations and holonomic constraints to confine the system to the hypersurface where a prescribed reaction coordinate is constant. Blue moon sampling allows one to compute conditional expectations such as the gradient of the free energy associated with a reaction coordinate, usually referred to as the mean force. From mean force the free energy is easily obtained by thermodynamic integration. We discuss how to simplify the approach when the reaction coordinate is vectorial and there are additional molecular constraints and how to remove the bias introduced by using constrained molecular dynamics. We also discuss how to avoid the bias by substituting overdamped constrained stochastic dynamics to constrained MD. Finally we will comment on the possible combination of mean force calculations by Blue Moon & computation of minimum free energy paths.

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