Free energy perturbation and thermodynamic integration methods

The evaluation of the free energy is essential to quantitatively treat a chemical process in condensed phase. In this section, we review methods of free-energy calculation within the context of classical statistical mechanics. We start with the standard free-energy perturbation and thermodynamic integration methods. We then introduce the method of distribution functions in solution. The method of energy representation is described in its classical form in this section, and is combined with the QM/MM methodology in the next section. 

Free-Energy Perturbation and Thermodynamic Integration Methods... 

The standard and often used methods to circumvent the difficulty associated with the form of Eq. (17-30) are the free-energy perturbation and thermodynamic integration methods [13,42,43], These methods are generally applicable to free-energy... 

The primary methods for calculating free energies—free energy perturbation (FEP), thermodynamic integration (TI), and slow growth (SG)—have been extensively described in the literature. -s Since the FEP method is used in most of the calculations cited in this review, the theoretical basis of the methodology is described in this section. 

As noted above, it is very difficult to calculate entropic quantities with any reasonable accmacy within a finite simulation time. It is, however, possible to calculate differences in such quantities. Of special importance is the Gibbs free energy, as it is the natoal thermodynamical quantity under normal experimental conditions (constant temperature and pressme. Table 16.1), but we will illustrate the principle with the Helmholtz free energy instead. As indicated in eq. (16.1) the fundamental problem is the same. There are two commonly used methods for calculating differences in free energy Thermodynamic Perturbation and Thermodynamic Integration. 

This chapter reviews some theoretical aspects of the two most popular free energy difference methods, thermodynamic perturbation and thermodynamic integration, as well as assumptions and approximations made in the implementation. Advantages and disadvantages of certain implementations are discussed, and general recommendations are given for the practical application of these methods. 

This chapter has reviewed theoretical and practical aspects of thermodynamic perturbation and thermodynamic integration, two popular methods of extracting free energies from molecular simulations. These methods find broad application in molecular simulation studies of chemical and biochemical systems. The fundamental importance of free energy in physical and chemical processes will inspire further development and refinement of these techniques. With the increasing performance of new computer architectures,these free energy techniques will become even more powerful and versatile tools. 

This article describes thermodynamic perturbation and thermodynamic integration. These methods form the basis of the most popular computational techniques for free energy difference evaluation with molecular simulations. 

What has been developed within the last 20 years is the computation of thermodynamic properties including free energy and entropy [12, 13, 14]. But the ground work for free energy perturbation was done by Valleau and Torrie in 1977 [15], for particle insertion by Widom in 1963 and 1982 [16, 17] and for umbrella sampling by Torrie and Valleau in 1974 and 1977 [18, 19]. These methods were primarily developed for use with Monte Carlo simulations continuous thermodynamic integration in MD was first described in 1986 [20]. 

The next three chapters deal with the most widely used classes of methods free energy perturbation (FEP) [3], methods based on probability distributions and histograms, and thermodynamic integration (TI) [1, 2], These chapters represent a mix of traditional material that has already been well covered, as well as the description of new techniques that have been developed only recendy. The common thread followed here is that different methods share the same underlying principles. Chapter 5 is dedicated to a relatively new class of methods, based on calculating free energies from nonequilibrium dynamics. In Chap. 6, we discuss an important topic that has not received, so far, sufficient attention - the analysis of errors in free energy calculations, especially those based on perturbative and nonequilibrium approaches. 

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