Adiabatic free energy surfaces

Figure 12 The CT adiabatic free energy surfaces in the normal CT region. The labels and indicate two adiabatically split absorption transitions corresponding to two minima of the lower surface with the coordinates Yj and Y Ae = 0.7, AFl = 0, AEnlyJ = 0.2. The gap AEmin is the minimum splitting between the upper and lower CT surfaces (Eq. [149]).

Figure 15 Adiabatic free energy surfaces F (Y ) in the present model (solid lines, Eqs. [105] and [106]) and in the Marcus-Hush formulation (long-dashed lines, Eqs. [41] and [119]) for self-exchange CT with AF = APf = 0, X = = 1 eV, AE12 = 0.2 eV,...

Equation [139] is exact for a two-state solute, but differs from the traditionally used connection between the transition dipole and the emission intensity by the factor Vo/Vav." The commonly used combination miiVo/Vav appears as a result of neglect of the frequency dependence of the transition dipole mi2(v) entering Eq. [129]. It can be associated with the condensed-phase transition dipole in the two-state approximation." Exact solution for a two-state solute makes the transition dipole between the adiabatic free energy surfaces inversely proportional to the energy gap between them. This dependence, however, is eliminated when the emission intensity is integrated with the factor... 

Figure 5. Plot of the adiabatic free-energy surfaces against the reaction coordinate for an electron transfer reaction with AG" = 0 and //ab/- varying from 0 to 0.5.

Figure 13 The CT adiabatic free energy surfaces in the CT inverted region Ac = 0.7, AFl/ k1 = —1.0, AE12/V — 3.0. The points Y and Y+ indicate the minima of the lower and upper adiabatic surfaces, respectively. The labels bvabs/em are absorption and emission energies, and AEmjn is the minimum energy gap between the free energy surfaces (Eq. [149]).

In a series of publications, Voth and coworkers [21,24,25,26,27] explored various aspects of electron-transfer reactions. Their calculations are based on a version of the Anderson-Newns Hamiltonian presented in Sect. 1.2.4.2, which allows the direct computation of adiabatic free energy surfaces. For a Fe +/Fe " couple at a fixed distance to a Pt(lll) surface, they compared a classical and a quantized water model [24]. The quantization... 

Fig. 6 Adiabatic free energy surface for the electron transfer between two different redox couples and a metal electrode from molecular dynamic simulations [27],...

Fig. 8 Adiabatic free energy surface for simultaneous electron transfer and bond breaking from MD simulations (upper) and a linear response solvent model (lower panel) [36],...

Recently, much attention has been paid to the investigation of the role of this interaction in relation to the calculations for adiabatic reactions. For steady-state nonadiabatic reactions where the initial thermal equilibrium is not disturbed by the reaction, the coupling constants describing the interaction with the thermal bath do not enter explicitly into the expressions for the transition probabilities. The role of the thermal bath in this case is reduced to that the activation factor is determined by the free energy in the transitional configuration, and for the calculation of the transition probabilities, it is sufficient to know the free energy surfaces of the system as functions of the coordinates of the reactive modes. 

Strong interactions are observed between the reacting solute and the medium in charge transfer reactions in polar solvents in such a case, the solvent effects cannot be reduced to a simple modification of the adiabatic potential controlling the reactions, since the solvent nuclear motions may become decisive in the vicinity of the saddle point of the free energy surface (FES) controlling the reaction. Also, an explicit treatment of the medium coordinates may be required to evaluate the rate constant preexponential factor. 

Figure 23. Free energy surfaces in the adiabatic case.

Finally, on the ground-state free energy surface, the evolving adiabatic electronic wavefunction during the CCB dissociation, expressed in terms of the VB state wavefunc-tions, is... 

The mixing parameter Ae makes the CT free energy surfaces dependent on the gas-phase, adiabatic transition dipole moment. The standard extension... 

Now we can calculate the adiabatic potential free energy surfaces as a function of the solvent coordinate q and the distance to... 

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