Thermodynamic Integration TI

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. 

Although historically less common, free energy calculations based on a different equation from classical statistical mechanics have grown in popularity in recent years. These calculations, termed Thermodynamic Integration (TI), are based on the integral... 

An alternative approach to free energy calculations is the thermodynamic integration (TI) method,18 20 which considers the ensemble average of the first derivative of the hybrid potential with respect to A at various values of A... 

Thermodynamic integration (TI) and thermodynamic perturbation (TP) have been the most widely used methodologies in the above mentioned free... 

Thermodynamic integration (TI) and free-energy perturbation (FEP) are the most widely used methods to calculate free-energy differences and are available in many commercial simulation packages. The TI method is based on the statistical connection between the Gibbs free energy, G, and the Hamiltonian, H, of the system... 

An alternative approach, called thermodynamic integration (TI) [35] can be used by introducing a continuous series of intermediates in terms of a variable k ... 

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. 

The general formalism for thermodynamic simulation methods follows from early work by Zwanzig [33], and provides a tool for the computation of thermodynamic properties, AA, AE and AS, as well as barriers for chemical processes occurring on long timescales [34]. These methods take on several guises in present implementations. The two approaches which we will describe are termed thermodynamic cycle perturbation theory (TP) [35] and thermodynamic integration (TI) [36]. Both of the methods are based on the definition of a hybrid Hamiltonian which represents some mixture of the initial state (1) and final state (2) of the system [37]. If /.represents the coordinate describing the pathway used to interconvert the two systems, then the hybrid Hamiltonian may be defined by [35, 37]. 

Calculation of the reversible work performed during the mutation A->B can be carried out using, among others. Free Energy Perturbation (FEP equation 9) and Thermodynamic Integration (TI equation 10) techniques. 

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