Thermodynamic perturbation

Thermodynamic perturbation is a technique to evaluate free energy differences between two systems with Hamiltonians based on the expres- 

This free energy difference can be expressed as an ensemble average using 

The ensembles needed to apply the perturbation technique may be obtained from molecular dynamics or Monte Carlo simulations. Meaningful results from perturbation calculations are obtained when the probability function Tt(p, q ) is accurately sampled in the regions of phase space where A3 f(p, q ) is nonnegligible. In practical terms, this means that those regions of phase space should be adequately sampled in the reference system. 

Interchanging the labels leads to the mirror expression for the free energy difference given by Eq. [17], illustrating how the same free energy difference may be obtained from an ensemble that was generated for system 1  

This equation is equivalent to Eq. [17]. In practice, both equations can be used to check the accuracy of the calculation. The difference in results obtained using both equations can serve as an indication of the adequacy of sampling of the ensembles. When significantly different results are obtained, one or both of the ensembles did not sample phase space sufficiently to be representative for the reference and perturbed Hamiltonians. 

The exponential in this equation involves the difference of two energies, rather than an energy itself, and as long as this is sufficiently small compared with k T, a typical simulation run is able to provide a good estimate of the difference in Helmholtz energy of A and B using eq. (11.28). 

If the energy difference is larger we can introduce a number of intermediate states between A and B by using a coupling parameter X (0 X 1) such that 

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Larsen B, Rasaiah J C and Stell G 1977 Thermodynamic perturbation theory for multipolar and ionic fluids Mol. Phys. 33 987... 

Wertheim M S 1987 Thermodynamic perturbation theory of polymerization J Chem. Phys. 87 7323... 

Truncating this series after the first derivative and integrating provides the basis for the hermodynamic integration approach. Moreover, if the Taylor series expansion is continued intil it converges then Equation (11.45) is equivalent to the thermodynamic perturbation brmula, so providing a link between the two approaches. In practice, it is always necessary... 

III- slow growth expression can be derived from the thermodynamic perturbation expres sion (Equation (11.7)) if it is written as a Taylor series ... 

Free energy calculations rely on the following thermodynamic perturbation theory [6-8]. Consider a system A described by the energy function = 17 + T. 17 = 17 (r ) is the potential energy, which depends on the coordinates = (Fi, r, , r ), and T is the kinetic energy, which (in a Cartesian coordinate system) depends on the velocities v. For concreteness, the system could be made up of a biomolecule in solution. We limit ourselves (mostly) to a classical mechanical description for simplicity and reasons of space. In the canonical thermodynamic ensemble (constant N, volume V, temperature T), the classical partition function Z is proportional to the configurational integral Q, which in a Cartesian coordinate system is... 

To obtain thermodynamic perturbation or integration formulas for changing q, one must go back and forth between expressions of the configuration integral in Cartesian coordinates and in suitably chosen generalized coordinates [51]. This introduces Jacobian factors... 

Finally, the associative term is computed by using generalizing thermodynamic perturbation theory. One then obtains [38]... 

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. 

It is now well-established that for atomic fluids, far from the critical point, the atomic organisation is dictated by the repulsive forces while the longer range attractive forces serve to maintain the high density [34]. The investigation of systems of hard spheres can therefore be used as simple models for atomic systems they also serve as a basis for a thermodynamic perturbation analysis to introduce the attractive forces in a van der Waals-like approach [35]. In consequence it is to be expected that the anisotropic repulsive forces would be responsible for the structure of liquid crystal phases and numerous simulation studies of hard objects have been undertaken to explore this possibility [36]. 

Tobias, D. J. Brooks III, C. L., Calculation of free energy surfaces using the methods of thermodynamic perturbation theory, Chem. Phys. Lett. 1987,142, 472-476... 

Thermodynamic perturbation theory represents a powerful tool for evaluating free energy differences in complex molecular assemblies. Like any method, however, FEP has limitations of its own, and particular care should be taken not only when carrying out this type of statistical simulations, but also when interpreting their results. We summarize in a number of guidelines the important concepts and features of FEP calculations developed in this chapter ... 

This takes the conventional form of standard thermodynamic perturbation theory, but with the decisive feature that interactions with only one molecule need be manipulated. Here (.., )r indicates averaging for the case that the solution contains a distinguished molecule which interacts with the rest of the system on the basis of the function AUa, i.e., the subscript r identifies an average for the reference system. Notice that a normalization factor for the intramolecular distribution cancels between the numerator and denominator of (9.22). 

Free energy calculations rely on a well-known thermodynamic perturbation theory [6, 21, 22], which is recalled in Chap. 2. We consider a molecular system, described by the potential energy function U(rN), which depends on the coordinates of the N atoms rN = (n, r2,..., r/v). The system could be a biomolecule in solution, for example. We limit ourselves to a classical mechanical description, for simplicity. Practical calculations always consider differences between two or more similar systems, such as a protein complexed with two different ligands. Therefore, we consider a change in the system, such that the potential energy function becomes ... 

Applications of Thermodynamic Perturbation Formulas Ligand Binding... 

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