Umbrella sampling

Systems in which an energy barrier separates two regions of configuration space may suffer from poor sampling.

In Metropolis Monte Carlo runs, the low probability of overcoming the potential barrier can leave inaccessible configurations poorly sampled—or even entirely unsampled—by the simulation.

The standard Boltzmann weighting for Monte Carlo sampling is replaced by a potential chosen to cancel the influence of the energy barrier present.

The Markov chain generated has a distribution given by with U the potential energy, w(rN) a function chosen to promote configurations that would otherwise be inaccessible to a Boltzmann-weighted Monte Carlo run.

Series of umbrella sampling simulations can be analyzed using the weighted histogram analysis method (WHAM)[2] or its generalization.

Subtleties exist in deciding the most computationally efficient way to apply the umbrella sampling method, as described in Frenkel and Smit's book Understanding Molecular Simulation.