Thermodynamic integration formula

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 ... 

We assume for simplicity that the solvent is pure water, and that only the water-oxygen atoms have explicit Lennard-Jones interactions with the solute (this is typical of several common water models). We have seen that AWnp can be viewed as the free energy to change A from zero to one. Therefore, a well-known thermodynamic integration formula gives... 

The free energy difference A A = Ai — Aq can be defined (and computed for a given molecular model) through a thermodynamic integration formula ... 

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... 

The best approach to evaluation of 0 along the integration path is (no surprise) thermodynamic integration. The relevant formula is easily derived from Eq. (3.3). A one-dimensional quadrature can be constructed using the derivatives that guide the integration procedure [see Eq. (3.4)] thus... 

These workers have since gone on [73,74] to conduct phase-coexistence calculations in pores using these ideas. These recent studies have many notable features, including the use of free-energy perturbation formulas in lieu of thermodynamic integration to maintain the chemical potential equality, and the introduction of a restraining field to eliminate unilateral or bilateral phase changes. 

An alternative way to calculate the free energy difference uses thermodynamic integration. The formula for the free energy difference is derived in Appendix 11.1 and is ... 

Alternatively, one may use the methods of thermodynamic integration to compute the free energy change. This technique relies on the connection formula... 

Thermodynamic integration (TI) provides an alternative way for free energy perturbation and is based on the following formula ... 

This approach may yield a result for that is superior to the simple test particle result if the overlap in the two distribution functions is significant. Contrarily, if the distributions do not overlap significantly, it is difficult to extract a reliable result from either the test particle or overlapping distribution formulae, and an alternative approach like thermodynamic integration or the staged insertion method is favored. 

The Gaussian quadrature formula was used to perform the integration. The above procedure was repeated for each isomer in the set finally, the mean thermodynamic quantities (5)-(10) were calculated. 

There is, of course, nothing like numerical agreement and, in view of the character of the approximation formula, it was not to be expected. But if equilibria of this sort are required to be calculated from thermodynamical data, it is naturally very valuable to have a simple method of obtaining an idea, at any rate, of the sign and of the approximate magnitude of the hitherto quite unknown constant of integration I this, as the above examples show, the approximation formula does provide. For the formation of hydrobromic and hydriodic acids the conditions are quite analogous to those for the formation of hydrochloric acid of this it is very easy to convince one s self (cf. 25). 

The point which is most important in connection with the question under treatment in this book is that we are able to show that there is fresh strong evidence for the logical force of our Heat Theorem. This evidence lies in the experimental proof that there is really a general significance in the formula, deduced from a somewhat questionable mechanical model of a solid body and from molecular theory, for the calculation of constants of integration not fixed by thermodynamics. 

The formulas just developed are clearly relevant to quantum dynamics, but their relevance to the Monte Carlo computation of molecular thermodynamic properties has not yet been developed. It turns out that we can develop a theory of quantum statistical mechanics [33] that is completely analogous to the Feynman path-integral version of quantum dynamics. 

Here, coordinate z is normal to the interface, and the integration is performed over the whole perturbed region. An important point in this formula is the lower integration limit, ZG. corresponding to the choice of the phase boundary , whose attribution is not self-evident. This ambiguity of the definition (66) is removed by the thermodynamics of charged interfaces based on the Gibbs electrocapillary equation (67) ... 

---