Iently little 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 every diabatic energy range. Equation 5.63 provides a very simple, consistent conversion amongst the diabatic and adiabatic pictures of ET inside the nonadiabatic limit, where the little electronic couplings in between the diabatic electronic states cause decoupling on the various states with the proton-solvent subsystem in eq 5.40 and in the Q mode in eq 5.41a. Even so, although little Vkn values represent a adequate situation for vibronically nonadiabatic behavior (i.e., in the end, VknSp kBT), the tiny overlap between reactant and kn solution proton vibrational wave functions is usually the reason for this behavior in the time evolution of eq 5.41.215 In actual fact, the p distance ��-Hydroxybutyric acid Protocol dependence from 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 within the literature.214,226,227 Here we note that the dimensional reduction from the R,Q towards the Q conformational space in going from eq five.40 to eq 5.41 (or from eq five.59 to eq five.62) will not imply a double-adiabatic approximation or the selection of a reaction path in the R, Q plane. In reality, the above process treats R and Q on an equal footing as much as the solution of eq 5.59 (such as, e.g., in eq five.61). Then, eq 5.62 arises from averaging eq 5.59 more than the proton quantum state (i.e., overall, over the electron-proton state for which eq five.40 expresses the rate of population modify), to ensure that only the solvent degree of freedom remains described when it comes to a probability density. Even so, although this averaging does not mean application of your double-adiabatic approximation in the general context of eqs five.40 and five.41, it results in the identical resultwhere the separation of your R and Q variables is allowed by the harmonic and Condon approximations (see, e.g., section 9 and ref 180), as in eqs 5.59-5.62. Inside the normal adiabatic approximation, the successful prospective En(R,Q) in eq 5.40 or Ead(R,Q) + Gad (R,Q) in eq five.59 supplies the helpful possible power for the proton motion (along the R axis) at any offered solvent conformation Q, as exemplified in Figure 23a. Comparing parts a and b of Figure 23 supplies a link between the behavior on the method about the diabatic crossing of Figure 23b and also the overlap in the Cedryl acetate supplier localized reactant and solution proton vibrational states, since the latter is determined by the dominant selection of distances among the proton donor and acceptor permitted by the productive potential in Figure 23a (let us note that Figure 23a can be a profile of a PES landscape such as that in Figure 18, orthogonal for the Q axis). This comparison is related in spirit to that in Figure 19 for ET,7 but it also presents some crucial variations that merit further discussion. Inside the diabatic representation or the diabatic approximation of eq 5.63, the electron-proton terms in Figure 23b cross at Q = Qt, exactly where the possible energy for the motion of the solvent is E p(Qt) along with the localization of the reactive subsystem in the kth n or nth potential well of Figure 23a corresponds for the similar power. In actual fact, the prospective energy of each and every effectively is given by the typical 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 each wells are p|Ej(R,Qt)|p + Tp = E p(Qt). j j j j In reference.
