Instantaneous free energy

The first step of the derivation involves the BO approximation separating the characteristic timescales of the electronic and nuclear motions in the system. In this step, the instantaneous free energy depending on the system nuclear coordinates q is defined by... 

The instantaneous free energy Pei(q) is the equilibrium free energy, implying equilibrium populations of the electronic states in the system. It is not suitable for describing nonequilibrium processes with nonequilibrium populations of the ground and excited states of the donor-acceptor complex. 

To derive the instantaneous free energies E, one needs an explicit model for a dipolar polarizable solute in a dipolar polarizable solvent. This need is addressed by the Drude model for induced solute and solvent dipole... 

Nonconserved Fields and the Allen-Cahn Equation. Nonconserved order parameters (sueh as the state of order itself as introduced in eqn (12.22)) can have complex spatial distributions that evolve in time. The governing equation in this case is provided by the Allen-Cahn equation. To obtain this equation using the ideas introduced above, we hark back to the L2 norm. It is assumed that the instantaneous free energy of the system can be written down as a functional of the order parameter and its gradients. For example, in the context of the Allen-Cahn equation, the relevant free energy functional is... 

In the deformed state, the variables in the Hamiltonian change from ( R , r ) to ( R , Ar ). However, the distribution p( r ) of finding the topology r depends solely on how the material is made instantaneously at thermal equilibrium (i.e., at constant temperature T, pressure p, etc.) i.e., p( r ) does not depend on the external deformation tensor A. Then, the final answer for the free energy of the deformed network is... 

Note that AA — xg > 0 in (6.36) is the average dissipated work of the NEW calculation in the complementary g) direction. Equation (6.36) shows that, given the same number of switching paths (/V), a simulation with a slower switch leads to a smaller variance in the free energy, since the value of A A — xg becomes smaller. Also note that AA — xg has its maximum value in the FEP case (instantaneous switch), and is zero when the switching path is reversible. 

We now analyze the chemical species prevalent in water at these extreme conditions by defining instantaneous species based on the O-H bond distance. If that distance is less than a cut-off value rc, we count the atom pair as being bonded. Determining all bonds in the system gives the chemical species at each point in time. Species with lifetimes less than an O-H bond vibrational period (10 fs) are transient and do not represent bound molecules. The optimal cut-off rc between bonded and nonbonded species is given by the location of the maximum in the free energy surface.83... 

RELATIVE SUPERSATURATION. The free energy that drives nucleation and crystal growth is directly related to a substance s relative supersaturation (RS), and the latter is a unitless parameter equal to [X]instantaneous/... 

Arnett and coworkers later examined the reaction of lithium pinacolone enoiate with substituted benzaldehydes in THE at 25 °C. The determination of the heat of reaction indicated that the Hammett p value for the process is 331. Although the aldol reaction was instantaneous in THF at 25 °C, the reaction with o- or p-methylbenzaldehyde could be followed using a rapid injection NMR method in methylcyclohexane solvent at —80 °C. Application of Eberson s criterion based on the Marcus equation, which relates the free energy of ET determined electrochemically and the free energy of activation determined by kinetics, revealed that the barriers for the ET mechanism should be unacceptably high. They concluded that the reaction proceeds via the polar mechanism . Consistent with the polar mechanism, cyclizable probe experiments were negative . The mechanistic discrepancy between the reactions of benzaldehyde and benzophenone was later solved by carbon kinetic isotope effect study vide infraf. ... 

Figure 1. Nonequilibrium free energy as a function of instantaneous solvent polarization for the ground electronic state S0 and the excited state Si of an ideal probe. In this example the equilibrium solvent polarization in S, (X q) is larger than in S0 (Ar q) because the dipole moment is larger in Si than S0. Reprinted from Ref. 31 with permission, from J. Chem. Phys. 88, 2372 (1988). Copyright 1988, American Physical Society.

---