Iently smaller Vkn, a single 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 energy range. Equation 5.63 gives a straightforward, constant conversion between the diabatic and adiabatic pictures of ET within the nonadiabatic limit, exactly where the tiny electronic couplings between the diabatic electronic states trigger decoupling with the different states in the proton-solvent subsystem in eq 5.40 and of the Q mode in eq five.41a. Having said that, though compact Vkn values represent a sufficient situation for vibronically nonadiabatic behavior (i.e., in the end, VknSp kBT), the smaller overlap in between reactant and kn item proton vibrational wave functions is typically the reason for this behavior inside the time 19542-67-7 custom synthesis evolution of eq five.41.215 In truth, the p distance dependence from the vibronic couplings VknSkn is p 197,225 determined by the overlaps Skn. Detailed discussion of analytical and computational approaches to get mixed electron/proton vibrational adiabatic states is discovered in the literature.214,226,227 Right here we note that the dimensional reduction in the R,Q for the Q conformational space in going from eq 5.40 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 procedure treats R and Q on an equal footing as much as the remedy of eq five.59 (for example, e.g., in eq five.61). Then, eq 5.62 arises from averaging eq five.59 more than the proton quantum state (i.e., all round, over the electron-proton state for which eq 5.40 expresses the price of population transform), to ensure that only the solvent degree of freedom remains described in terms of a probability density. Nevertheless, although this averaging will not mean application in the double-adiabatic approximation in the common context of eqs five.40 and five.41, it leads to exactly the same resultwhere the separation of the R and Q variables is allowed by the harmonic and Condon approximations (see, e.g., section 9 and ref 180), as in eqs five.59-5.62. Inside the normal adiabatic approximation, the effective potential En(R,Q) in eq 5.40 or Ead(R,Q) + Gad (R,Q) in eq 5.59 offers the productive prospective 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 delivers a link in between the behavior from the system about the diabatic crossing of Figure 23b plus the overlap with the localized reactant and item proton vibrational states, since the latter is determined by the dominant range of distances among the proton donor and acceptor allowed by the efficient Tebufenozide Cancer possible in Figure 23a (let us note that Figure 23a is actually 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 nevertheless it also presents some essential differences that merit additional discussion. In the diabatic representation or the diabatic approximation of eq five.63, the electron-proton terms in Figure 23b cross at Q = Qt, where the prospective energy for the motion on the solvent is E p(Qt) and the localization from the reactive subsystem in the kth n or nth possible well of Figure 23a corresponds towards the exact same power. The truth is, the possible power of each nicely is offered by the typical electronic power Ej(R,Qt) = j(R,Qt)|V(R ,Qt,q) + T q| j(R,Qt) (j = k, n), and also the proton vibrational energies in both wells are p|Ej(R,Qt)|p + Tp = E p(Qt). j j j j In reference.