Free Energies in the Condensed Phase

To introduce the effects of a condensed phase environment, we consider the solvation process for a single molecule as a thermodynamic cycle composed of 

The type of construction given in Fig. 10a, which utilizes the Hess s law of constant heat summation, can serve as a means of quantitatively analyzing the thermodynamics of solvation. Further, this view of the solvation process provides a method for considering different standard states. For nonionic species a commonly used standard state is infinite dilution. Although activities become infinite for ions in this limit, it is still a useful reference state because the analytic Debye-Hiickel limiting law is valid in this regime.168 

This implies that when the pure liquid solvent is chosen as the reference state, only terms involving canonical averages over the potential energy of interaction between the solute and the solvent (plus the changes in the internal free energies discussed in the previous section) contribute to the free energy of solvation at infinite dilution. At finite concentration, the solute-solute interaction terms have to be considered as well. 

The full calculation of the ligand-protein interaction free energy in solution, as diagrammed in Fig. 10b, corresponds to the determination of the free energy of solvation of the three separate species and the evaluation of the appropriate difference i.e., 

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Equation 5.19 relates the molecular energy states of the primed and unprimed isotopomers in condensed and vapor phase to VPIE. The correction terms account for the difference between the Gibbs and Helmholtz free energies of the condensed phase, and vapor nonideality. The comparison is between separated isotopomers at a common temperature, each existing at a different equilibrium volume, V or V, and at a different pressure, P or P, although AV = (V — V) and AP = (P — P) are small. Still, because condensed phase Q s are functions of volume, Q = Q(T,V,N), rigorous analysis requires knowledge of the volume dependence of the partition function, and thus MVIE, since the comparisons are made at V and V. That point is developed later. 

Much of chemistry occurs in the condensed phase solution phase ET reactions have been a major focus for theory and experiment for the last 50 years. Experiments, and quantitative theories, have probed how reaction-free energy, solvent polarity, donor-acceptor distance, bridging stmctures, solvent relaxation, and vibronic coupling influence ET kinetics. Important connections have also been drawn between optical charge transfer transitions and thennal ET. 

Bash, P.A., Field, M.J.,Karplus, M. Free energy perturbation method for chemical reactions in the condensed phase A dynamical approach baaed on a combined quantum and molecular dynamics potential. J. Am. Chem. Soc. 109 (1987) 8092-8094. 

In this section we will briefly review some of the main different approaches that have been used up to now in order to evaluate the potential energy of each configuration in a Monte Carlo run. As we have already stated, the force fields that describe intra- and intermolecular interactions are at the heart of such statistical calculations because the free energy differences that we want to evaluate are directly dependent on the changes of those interactions. In fact, the important advances of the last ten years in the power of computer techniques for chemical reactions in the condensed phase, that we have mentioned in the Introduction, have been due, to a great extent, to the continual evolution in force fields, with added complexity and improved performance. 

An external magnehc held does not create vortex lines because currents induced by gradients of the phase of kA are absent [see Eq. (19)]. To determine the response of the CEL condensate to rotahon we shall consider its free energy in the frame of reference, rotahng with a constant angular velocity lJ. This energy is dehned as follows... 

Equation A1.3 shows that isotope effects calculated from standard state free energy differences, and this includes theoretical calculations of isotope effects from the partition functions, are not directly proportional to the measured (or predicted) isotope effects on the logarithm of the isotopic pressure ratios. Rather they must be corrected by the isotopic ratio of activity coefficients. At elevated pressures the correction term can be significant, and in the critical region it may even predominate. Similar considerations apply in the condensed phase except the fugacity ratios which define Kf are replaced by activity ratios, a = Y X and a = y C , for the mole fraction or molar concentration scales respectively. In either case corrections for nonideality, II (Yi)Vi, arising from isotope effects on the activity coefficients can be considerable. Further details are found in standard thermodynamic texts and in Chapter 5. 

We define the cohesive energy Ecoh (Johansson, Skriver ) as the difference between the energy of an assembly of free atoms in their ground state (see Table 1 of Chap. A) and the energy of the same assembly in the condensed phase (the solid at 0 °K), (this definition yields a positive number for Ecoii). It coincides with the enthalpy of sublimation AHj (see Chap. A) (which is usually extrapolated at room temperature). 

A large number of studies have been devoted to measuring the ionization potential in the liquid and the solid phases (see Refs. 179-181, 189, 190). Some of these results are presented in Table VII, from which one can see that for most substances V0 is negative, and so the ionization potential in the condensed phase is smaller than the photoemission threshold Eph. However, for some substances (for instance, for n-pentane, /i-decane, and neon), VQ is positive, meaning that in this case it is more advantageous, from the energy point of view, for an electron to make a transition into vacuum than to remain in a quasi-free state. 

Note that this equation relates the free energy change in the condensed phase to die fugacities of the vapour phase. For substances which have rather low vapour pressures,... 

The other approach to calibration, by comparison with experimental data, also presents difficulties. Here, the problem is in ensuring that the calculations relate correctly to the experiment, which, in this field, most frequently refers to finite-temperature free energy differences in the condensed phase. As well as obtaining accurate electronic energy differences, therefore, zero-point energy (zpe), thermal or enthalpic (H), entropic (S), and medium ef-... 

Bash, P.A., Field M.J. and Karplus M., Free Energy Perturbation Method for Chemical Reactions in the Condensed Phase A Dynamical Approach Based on a Combined Quantum and Molecular Mechanics Potential. J. Am. Chem.Soc. (1987) 109 8092-8094. 

As has been discussed above, molecular clusters produced in a supersonic expansion are preferred model systems to study solvation-mediated photoreactions from a molecular point of view. Under such conditions, intramolecular electron transfer reactions in D-A molecules, traditionally observed in solutions, are amenable to a detailed spectroscopic study. One should note, however, the difference between the possible energy dissipation processes in jet-cooled clusters and in solution. Since molecular clusters are produced in the gas phase under collision-free conditions, they are free of perturbations from many-body interactions or macro-molecular structures inherent for molecules in the condensed phase. In addition, they are frozen out in their minimum energy conformations which may differ from those relevant at room temperature. Another important aspect of the condensed phase is its role as a heat bath. Thus, excess energy in a molecule may be dissipated to the bulk on a picosecond time-scale. On the other hand, in a cluster excess energy may only be dissipated to a restricted number of oscillators and the cluster may fragment by losing solvent molecules. 

Figure 7. A schematic representation of the solvent free energy functions involved in charge transfer reactions in the condensed phase. Indicated are the reaction free energy and the reorganization free energy defined in equations (22) and (27). The activation free energy and the photo absorption energy hco discussed below are also indicated.

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