Free energy path

Finally, in Sect. 7.6, we have discussed how various free energy calculation methods can be applied to determine free energies of ensembles of pathways rather than ensembles of trajectories. In the transition path sampling framework such path free energies are related to the time correlation function from which rate constants can be extracted. Thus, free energy methods can be used to study the kinetics of rare transitions between stable states such as chemical reactions, phase transitions of condensed materials or biomolecular isomerizations. 

Figure 12. (Upper panel) Path entropy i(w) (Middle panel) path free-energy (w) = w — Ts(w), and (lower panel) Lagrange multipher X(w) equal to the inverse of the path temperature 1/7 (m ). is the most probable work value given by y(w P) = X,(rv P) = 0 or = 1 is the value of the work that has to be sampled to recover free energies from nonequilibrium work values using the JE. This is given by y(w() = l/T or d> (w() = 0 Wrev and Wdis are the reversible and average dissipated work, respectively. (From Ref. 117.)...

These equations must be solved together with the boundary conditions in Eq. (158). Note that we use the subindex (or the argument) w in all helds (k, m, y) to emphasize that there exists a solution of these fields for each value of the work w. From the entropy 5 in Eq. (159) we can evaluate the path free energy, the path temperature, and the values and introduced in Section V.A. We enumerate the different results. 

The Path Free Energy. The path free energy / = /N (Eq. (121)) is given by... 

Figure A2.5.2. Schematic representation of the behaviour of several thennodynamic fiinctions as a fiinction of temperature T at constant pressure for the one-component substance shown in figure A2.5.1. (The constant-pressure path is shown as a dotted line in figure A2.5.1.) (a) The molar Gibbs free energy Ci, (b) the molar enthalpy n, and (c) the molar heat capacity at constant pressure The fimctions shown are dimensionless...

Using the CFTI protocol, we have calculated directly both the derivative of the free energy with respect to the reaction path dA/dX and the 14 individual derivatives dA/d k, k = 1,...,14 with respect to all fixed coordinates along the path ... 

Combining Eqs. (12)-(14) yields Eq. (7) directly. Although such a free energy decomposition is path-dependent [8], it provides a useful and rigorous framework for understanding the nature of solvation and for constructing suitable approximations to the nonpolar and electrostatic free energy contributions. 

A catalyst is a substance that makes available a reaction path with a lower free energy of activation than is available in its absence. (The catalyst does not lower AG the uncatalyzed reaction path remains available.)... 

For the PPM, corresponding to the free energy of the CVM is the Path Probability Function (hereafter PPF), P t t -t- At), which is an explicit function of time and is defined as the product of three factors Pj, P2 and P3. Each factor is provided in the following in the logarithmic expression. 

In the PPF, the first factor Pi describes the statistical average of non-correlated spin fiip events over entire lattice points, and the second factor P2 is the conventional thermal activation factor. Hence, the product of P and P2 corresponds to the Boltzmann factor in the free energy and gives the probability that on<= of the paths specified by a set of path variables occurs. The third factor P3 characterizes the PPM. One may see the similarity with the configurational entropy term of the CVM (see eq.(5)), which gives the multiplicity, i.e. the number of equivalent states. In a similar sense, P can be viewed as the number of equivalent paths, i.e. the degrees of freedom of the microscopic evolution from one state to another. As was pointed out in the Introduction section, mathematical representation of P3 depends on the mechanism of elementary kinetics. It is noted that eqs.(8)-(10) are valid only for a spin kinetics. 

The path variables of the PPM corresponds to the cluster probabilities of the CVM by which the free energy is minimized to obtain the most probable state. Likewise, under a set of constraints, the PPF is maximized with respect to the path variables for each time step, which yields the optimized set of path variables. Since a set of path variables, + At), relates cluster probabilities t and at time t + At... 

---