Free energy calculations sampling

The fundamental issue in implementing importance sampling in simulations is the proper choice of the biased distribution, or, equivalently, the weighting factor, q. A variety of ingenious techniques that lead to great improvement in the efficiency and accuracy of free energy calculations have been developed for this purpose. They will be mentioned frequently throughout this book. 

Verkhivker, G. Elber, R. Nowak, W., Locally enhanced sampling in free energy calculations application of mean held approximation to accurate calculation of free energy differences, J. Chem. Phys. 1992, 97, 7838-7841... 

Oberhofer, H. Dellago, C. Geissler, P. L., Biased sampling of nonequilibrium trajectories. Can fast switching simulations outperform conventional free energy calculation methods, J. Phys. Chem. B 2005,109, 6902-6915... 

The sample size in a real simulation is always finite, and usually relatively small. Thus, understanding the error behavior in the finite-size sampling region is critical for free energy calculations based on molecular simulation. Despite the importance of finite sampling bias, it has received little attention from the community of molecular simulators. Consequently, we would like to emphasize the importance of finite sampling bias (accuracy) in this chapter. 

Here we discuss a set of traditional methods for combining both the forward and reverse perturbation samples to enhance free energy calculations. 

As discussed in Sect. 6.1, the bias due to finite sampling is usually the dominant error in free energy calculations using FEP or NEW. In extreme cases, the simulation result can be precise (small variance) but inaccurate (large bias) [24, 32], In contrast to precision, assessing the systematic part (accuracy) of finite sampling error in FEP or NEW calculations is less straightforward, since these errors may be due to choices of boundary conditions or potential functions that limit the results systematically. 

Consider the forward calculation as an example. Earlier in this chapter, we concluded that the most important contribution to a forward free energy calculation comes from the low-x tail of f(x)9 and its poor sampling results in the major systematic error of the calculation. To simplify further the analysis we assume that there is a limit-perturbation x f such that x > Xf is well sampled and x < Xf is never sampled. This is illustrated in Fig. 6.5. 

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...

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. 

---