Iently tiny Vkn, 1 can make use of the piecewise approximation(Ek En) k ad kn (Ek En) nEp,ad(Q)(five.63)and eq five.42 is valid within each and every diabatic power variety. Equation 5.63 gives a uncomplicated, constant conversion in between the diabatic and adiabatic photographs of ET in the nonadiabatic limit, where the smaller electronic couplings among the diabatic electronic states lead to decoupling in the diverse states of the proton-solvent subsystem in eq five.40 and with the Q mode in eq five.41a. However, though tiny Vkn values represent a enough SNC80 Protocol condition for vibronically nonadiabatic behavior (i.e., ultimately, VknSp kBT), the small overlap in between reactant and kn item bis-PEG2-endo-BCN Biological Activity proton vibrational wave functions is normally the cause of this behavior in the time evolution of eq 5.41.215 In truth, the p distance dependence on the vibronic couplings VknSkn is p 197,225 determined by the overlaps Skn. Detailed discussion of analytical and computational approaches to obtain mixed electron/proton vibrational adiabatic states is found inside the literature.214,226,227 Here we note that the dimensional reduction from the R,Q to the Q conformational space in going from eq 5.40 to eq 5.41 (or from eq 5.59 to eq 5.62) doesn’t imply a double-adiabatic approximation or the choice of a reaction path inside the R, Q plane. In reality, the above process treats R and Q on an equal footing as much as the option of eq 5.59 (such as, e.g., in eq five.61). Then, eq 5.62 arises from averaging eq five.59 more than the proton quantum state (i.e., overall, more than the electron-proton state for which eq five.40 expresses the rate of population transform), so that only the solvent degree of freedom remains described in terms of a probability density. Nevertheless, though this averaging does not imply application on the double-adiabatic approximation in the general context of eqs five.40 and 5.41, it leads to the same resultwhere the separation on the R and Q variables is permitted by the harmonic and Condon approximations (see, e.g., section 9 and ref 180), as in eqs 5.59-5.62. Inside the standard adiabatic approximation, the helpful possible En(R,Q) in eq 5.40 or Ead(R,Q) + Gad (R,Q) in eq 5.59 provides the successful possible energy for the proton motion (along the R axis) at any offered solvent conformation Q, as exemplified in Figure 23a. Comparing components a and b of Figure 23 offers a link among the behavior on the system about the diabatic crossing of Figure 23b as well as the overlap on the localized reactant and product proton vibrational states, since the latter is determined by the dominant selection of distances involving the proton donor and acceptor permitted by the efficient potential in Figure 23a (let us note that Figure 23a is a profile of a PES landscape such as that in Figure 18, orthogonal to the Q axis). This comparison is related in spirit to that in Figure 19 for ET,7 but it also presents some essential variations that merit further discussion. In the diabatic representation or the diabatic approximation of eq 5.63, the electron-proton terms in Figure 23b cross at Q = Qt, where the potential energy for the motion in the solvent is E p(Qt) and the localization on the reactive subsystem within the kth n or nth potential well of Figure 23a corresponds towards the exact same energy. Actually, the possible energy of each properly is provided by the average electronic power Ej(R,Qt) = j(R,Qt)|V(R ,Qt,q) + T q| j(R,Qt) (j = k, n), plus the proton vibrational energies in both wells are p|Ej(R,Qt)|p + Tp = E p(Qt). j j j j In reference.