Multistate continuum theory

Multistate Continuum Theory for Proton-Coupled Electron Transfer... 

In Figure 5a, the paraboloids correspond to electronic states aj b, a/ b, 2bj2a, and 2b/2a. This situation arises when the potential along the rp coordinate (shown in Figure 5b) has a low barrier (or no barrier) so that all vibrational states are mixtures of a and b PT states and are delocalized. Note that the minima for the Xa/Xb and Xa/Xb states are virtually identical. These examples illustrate that the multistate continuum theory provides a valuable framework for the analysis and interpretation of the wide range of behavior exhibited by PCET reactions. 

The previous work of Cukier and coworkers [7, 12] differs from the formulation described in this chapter in a number of fundamental ways. In contrast to the multistate continuum theory described in this chapter, Cukier and coworkers did not calculate mixed electronic/proton vibrational free energy surfaces as functions of two solvent coordinates. Instead, they calculated solvated proton potentials obtained by the assumption that the inertial polarization of the solvent responds instantaneously to the proton position. (This is the limit opposite to the standard adiabatic limit of the fast proton vibrational motion responding instantaneously to... 

Figure 6. Model PCET systems investigated with the multistate continuum theory.

This multistate continuum theory has been generalized for charge-transfer reactions involving Ng transferring electrons and Np transferring protons. For this general case, the solute is represented by a VB model containing VB states. The free... 

The multistate continuum theory for PCET provides a framework for the analysis of the effects of specific solute and solvent properties on the rates and mechanisms of PCET reactions. The properties of interest include the relative energies of the gas phase solute charge transfer states, the distance between the proton donor and acceptor, the distance between the electron donor and acceptor, and the solvent polarity. In Ref. [32], a comprehensive study of the effects of these physical properties on the rates, mechanisms, and kinetic isotope effects of PCET reactions is presented. Some of the predictions obtained from this study are discussed in this section. 

As shown in Fig. 16.10, the temperature dependence of the rates and KIEs predicted by the multistate continuum theory is in remarkable agreement with the experimental data [47]. The calculations indicate that the weak temperature dependence of the rates is due to the relatively small free energy barrier arising... 

The theoretical results were generated with the multistate continuum theory including the proton donor-acceptor vibrational motion. Reproduced from Ref [47]. 

Theoretical treatments of PCET reactions typically have equation (1.2) as a conceptual starting point. In Hammes-Schiffer s multistate continuum theory for PCET, the pre-exponential factor includes both electronic coupling and vibrational overlaps, and the rate is a sum over initial and final vibrational states integrated over a range of proton-donor acceptor distances. This theory has been elegantly applied to understand the intimate details of a variety of PCET reactions, but many of its parameters are essentially unattainable experimentally. 

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