Free energy perturbation error estimation

Check. Use the Crooks relation (5.35) to check whether the forward and backward work distributions are consistent. Check for consistency of free energies obtained from different estimators. If the amount of dissipated work is large, caution may be necessary. If cumulant expressions are used, the work distributions should be nearly Gaussian, and the variances of the forward and backward perturbations should be of comparable size [as required by (5.35) for Gaussian work distributions]. Systematic errors from biased estimators should be taken into consideration. Statistical errors can be estimated, for instance, by performing a block analysis. 

What level of inaccuracy can be expected for a simulation with a certain sample size N1 This question can be transformed to another one what is the effective limit-perturbation Xf or xg in the inaccuracy model [(6.22) or (6.23)] To assess the error in a free energy calculation using the model, one may histogram / and g using the perturbations collected in the simulations, and plot x in the tail of the distribution. However, if Xf is taken too small the accuracy is overestimated, and the assessed reliability of the free energy is therefore not ideal. In the following, we discuss the most-likely analysis, which provides a more systematic way to estimate the accuracy of free energy calculations. 

This equation is exact, but since the thermal average has to be estimated by a finite sample, statistical errors will grow along the magnitude of the perturbation AU. As shown by Lu et al. perturbation schemes with better performance can be formulated [159,160], The final simulation and calculation of properties are done in the free-energy optimal geometry thus obtained. 

The energies and properties of a confined atom smoothly approach the unconfined atom values as the box radius R becomes very large, so it is of interest to know the behaviour of these small departures from the free atom values for large R. In practice, it is desirable to have reliable analytical estimates of the departures, rather than calculate them directly as the difference of two, relatively large, quantities, a process which becomes increasingly prone to cancellation, and other, errors as R increases. The problem here is to treat a (small) perturbation of the boundary conditions a more traditional approach, in which the perturbation would be in the potential, cannot be used because, although the corrections to the properties are small, the perturbation itself is infinite for r > R. 

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