Successive umbrella sampling

Virnau, R Muller, M., Calculation of free energy through successive umbrella sampling, J. Chem. Phys. 2004,120, 10925-10930... 

Panel (b) illustrates the application of successive umbrella sampling [35], where each piece of the curve corresponds to several intervals used in the successive sampling. The intervals overlap just by one particle number, which is sufficient if one takes due account of detailed balance at the interval boundaries [36]. Matching the pieces at their boundaries one obtains the distribution across the entire composition range. Of course, both methods yield identical results for the final probability distribution and, provided that one starts with an appropriate modification factor, /, they require comparable amounts of CPU time. 

This formula is expected to be accurate close to qAX ). Continuing in this manner, one can obtain accurate formulas for P(q) over a broad range of q, provided the regions sampled with the successive umbrellas overlap. 

In the iterative approach using a constant bias F, the free energy barrier is reduced each successive iteration and therefore produces complete sampling of important configurations along the coupling parameter (i.e., reaction coordinates). Furthermore, more complicated umbrella potentials (e.g. see Equation 21) may also be applied with the iterative procedure. 

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