Activation free energy, classical

Figure A3.8.3 Quantum activation free energy curves calculated for the model A-H-A proton transfer reaction described 45. The frill line is for the classical limit of the proton transfer solute in isolation, while the other curves are for different fully quantized cases. The rigid curves were calculated by keeping the A-A distance fixed. An important feature here is the direct effect of the solvent activation process on both the solvated rigid and flexible solute curves. Another feature is the effect of a fluctuating A-A distance which both lowers the activation free energy and reduces the influence of the solvent. The latter feature enliances the rate by a factor of 20 over the rigid case.

The expression in Eq. (10) for the exponent in Eq. (9) is quite similar to that for the activation free energy in electron transfer reactions derived by Marcus using the methods of nonequilibrium classical thermodynamics8 ... 

Smaller values of the activation free energy due to (i) the distortion of the shape of the free energy surfaces and (ii) the increase of the resonance splitting, AJF, of the potential free energies for the classical subsystem due to the increased overlapping of the wave functions of the quantum particles. 

The height of the potential barrier separating the initial and final states of the nuclear subsystem decreases and, hence, the Franck-Condon factor increases (Fig. 6). In the classical limit, this results in a decrease of the activation free energy. 

Equation 5.7 Classical Marcus relationship for the activation free energy of electron transfer... 

Nucleation Rate. The value of AGmax is often considered to be an activation free energy for nucleus formation. Moreover, it is generally assumed that there is also a certain barrier to incorporation of a molecule into an embryo, owing to diffusional resistance this is usually expressed in a molar activation free energy for transport AG. Classical nucleation theory then stipulates that the nucleation rate (number of nuclei formed per unit time per unit volume) will be given by... 

Figure 18. PI-QTST activation free-energy curves as a function of the proton asymmetric stretch coordinate for a A-H-A proton transfer model (see Ref. 77). The solid line depicts the classical free-energy curve for the solute in isolation with a rigid A-A distanee, while the dotted line is the quantum free energy for the rigid, isolated solute with a fully quantized proton. The long-dashed line is the quantum free-energy curve for the isolated solute in which the A-A distance is allowed to fluctuate. The dot-dashed and short-dashed lines depict the quantum free-energy curves for the rigid and flexible solutes, in the polar solvent.

Figure 20. Activation free energy along the asymmetric FT coordinate for the aqueous-phase HjO -HjO dimer at 300K. The result for classical hydrogen nuclei is shown by the solid curve, while the quantum centroid free energy is given by the dashed curve.

Figure 21. Adibatic solvent activation free-energy curves for the Fe /Fe electron transfer reaction with a platinum electrode at 300 K calculated obtained using the model of Ref. 50, path-integral quantum transition-state theory, and umbrella sampling. The solid line depicts the quantum adiabatic free-energy curve, while the dashed line depicts the curve in the classical limit. In both cases, the left-hand well corresponds to the Fe stable state, while the right-hand well is the Fe " stable state.

The semi-classical temperature dependence of electron transfer rate constants has been inferred from Equations (l)-(3). However, it is frequently observed that these equations are more successful at describing the overall behavior of the activation free energy, AG a than that of the component activation enthalpy, and entropy, A5 da, terms. Newton has suggested... 

---