Modeling Free Energy Errors

Perhaps the most challenging part of analyzing free energy errors in FEP or NEW calculations is the characterization of finite sampling systematic error (bias). The perturbation distributions / and g enable us to carry out the analysis of both the finite sampling systematic error (bias) and the statistical error (variance). 

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Fig. 6.5. Graphical illustration of the inaccuracy model and the relative free energy error in forward and reverse free energy calculations. A limit-perturbation Xf is adopted to (effectively) describe the sampling of the distribution the regions above x/ are assumed to be perfectly sampled while regions below it shaded area) are never sampled. We may also put a similar upper limit x f for the high-rr tail, where there is no sampling for regions above it. However, this region (in a forward calculation) makes almost zero contribution to the free energy calculation and its error. Thus for simplicity we do not apply such an upper limit here...

Tlierc are two major sources of error associated with the calculation of free energies fi computer simulations. Errors may arise from inaccuracies in the Hamiltonian, be it potential model chosen or its implementation (the treatment of long-range forces, e j lie second source of error arises from an insufficient sampling of phase space. 

The results of the simple DHH theory outlined here are shown compared with DH results and corresponding Monte Carlo results in Figs. 10-12. Clearly, the major error of the DH theory has been accounted for. The OCP model is greatly idealized but the same hole correction method can be applied to more realistic electrolyte models. In a series of articles the DHH theory has been applied to a one-component plasma composed of charged hard spheres [23], to local correlation correction of the screening of macroions by counterions [24], and to the generation of correlated free energy density functionals for electrolyte solutions [25,26]. The extensive results obtained bear out the hopeful view of the DHH approximation provided by the OCP results shown here. It is noteworthy that in... 

In this model, the finite sampling systematic error is due to the missed sampling of the important region x < Xf. The free energy estimate given by the model is... 

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. 

In practice it is helpful to know the order of magnitude of the sample size N needed to reach a reasonably accurate free energy. The inaccuracy model described above presents an effective way to relate the sample size N and the finite sampling error through perturbation distribution functions. Alternatively, one can develop a heuristic that does not involve distribution functions and is determined by exploring the common behavior of free energy calculations for different systems [25]. Although only FEP calculations are considered in this section, the analysis extends to NEW calculations. 

The second necessary ingredient in the primitive quasichemical formulation is the excess chemical potential of the metal-water clusters and of water by itself. These quantities p Wm — can typically be obtained from widely available computational packages for molecular simulation [52], In hydration problems where electrostatic interactions dominate, dielectric models of those hydration free energies are usually satisfactory. The combination /t xWm — m//, wx is typically insensitive to computational approximations because the water molecules coat the surface of the awm complex, and computational errors can compensate between the bound and free ligands. 

To summarize the attempts to refine the original LIE model, we found that an optimal equation for the binding free energy could be obtained with only one free parameter (a) and with the electrostatic coefficients (fi) derived from FEP simulations of some representative compounds in water. For the 18 compound training set that we used this model yielded a mean unsigned error of only 0.58 kcal/mol which seemed very promising. 

While the PM3-SM4 model does appear to slightly underestimate the polarity of the enol component, there is some cancellation of errors upon considering the differential transfer free energies between cyclohexane and water. As noted above, experiment indicates that the differential free energy of transfer of the dione and the enol is 3.1 kcal/mol the PM3-SM4 model predicts this value to be 2.8 kcal/mol, in excellent quantitative agreement. AM1-SM4 is less satisfactory in this regard, predicting only 1.9 kcal/mol. 

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