Thermodynamic integration , free-energy

T. P. Straatsma and J. A. McCammon, Ghem. Phys. Lett., 167, 252 (1990). Free Energy Thermodynamic Integrations in Molecular Dynamics Simulations Using a Non-iterative Method to Include Electronic Polarization. 

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. 

Next, an exploration of thermodynamics and equilibrium, based on a conceptual understanding of entropy and Gihbs free energy. This integrated presentation lays a common foundation for these concepts and provides a basis for understanding the origin and form of the equilibrium constant and the behavior of equilibrium systems. 

The most accurate route to the thermodynamic properties from the SSOZ equation seems to be the energy equation." The integral from which the internal energy is obtained (see Eq. (2.3.1)) seems to be relatively insensitive to errors in the predicted site-site correlation functions. It might on this basis be reasonably assumed that calculations of the Helmholtz free energy via integration of the Gibbs Helmholtz equation... 

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]. 

Straatsma, T.P, Berendsen, H.J.C. Free energy of ionic hydration Analysis of a thermodynamic integration technique to evaluate free energy differences by molecular dynamics simulations. J. Chem. Phys. 89 (1988) 5876-5886. 

Fig. 6. Free energies of hydration calculated, for a series of polar and non-polar solute molecules by extrapolating using (3) from a 1.6 ns trajectory of a softcore cavity in water plotted against values obtained using Thermodynamic Integration. The solid line indicates an ideal one-to-one correspondence. The broken line is a line of best fit through the calculated points.

The overall free energy can be partitioned into individual contributions if the thermo-lynamic integration method is used [Boresch et al. 1994 Boresch and Karplus 1995]. The itarting point is the thermodynamic integration formula for the free energy ... 

Appendix 11.1 Calculating Free Energy Differences Using Thermodynamic Integration... 

Mitchell M J and J A McCammon 1991. Free Energy Difference Calculations by Thermodynamic Integration Difficulties in Obtaining a Precise Value. Journal of Computational Chemistry 12 271-275. 

The electrostatic free energy contribution in Eq. (14) may be expressed as a thennody-namic integration corresponding to a reversible process between two states of the system no solute-solvent electrostatic interactions (X = 0) and full electrostatic solute-solvent interactions (X = 1). The electrostatic free energy has a particularly simple form if the thermodynamic parameter X corresponds to a scaling of the solute charges, i.e., (X,... 

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... 

The free energy is the most important equilibrium thennodynamic function, but other quantities such as the enthalpy and entropy are also of great interest. Thermodynamic integration and permrbation fonnulas can be derived for them as well. For example, the derivative of the entropy can be written [24]... 

The free energy difference between two stable conformations can be obtained by a thermodynamic integration approach [38,39]. Let q and represent the centers of the two corresponding energy wells. The free energy derivative is seen to be... 

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