N the theory.179,180 The identical outcome as in eq 9.7 is recovered in the event

N the theory.179,180 The identical outcome as in eq 9.7 is recovered in the event the initial and final proton states are once more described as harmonic oscillators with all the exact same frequency and the Condon approximation is applied (see also section five.3). In the DKL treatment180 it truly is noted that the sum in eq 9.7, evaluated at the diverse values of E, has a dominant contribution which is ordinarily supplied by a worth n of n such thatApart from the 54-05-7 In Vitro dependence with the power quantities on the form of charge transfer reaction, the DKL theoretical framework may be applied to other charge-transfer reactions. To investigate this point, we take into consideration, for simplicity, the case |E| . Considering the fact that p is larger than the thermal energy kBT, the terms in eq 9.7 with n 0 are negligible in comparison with those with n 0. That is an expression of the reality that a greater activation power is necessary for the occurrence of each PT and excitation of the proton to a higher vibrational level of the accepting potential effectively. As such, eq 9.7 is usually rewritten, for a lot of applications, inside the approximate formk= VIFn ( + E + n )two p p exp( – p) exp- n! kBT 4kBT n=(9.16)where the summation was extended towards the n 0 terms in eq 9.7 (along with the sign from the summation index was changed). The electronic charge distributions corresponding to A and B will not be specified in eqs 9.4a and 9.4b, except that their distinct dependences on R are incorporated. If we assume that Adx.doi.org/10.1021/cr4006654 | Chem. Rev. 2014, 114, 3381-Chemical Testimonials and B are characterized by distinct localizations of an excess electron charge (namely, they may be the diabatic states of an ET reaction), eq 9.16 also describes concerted electron-proton transfer and, additional specifically, vibronically nonadiabatic PCET, considering that perturbation theory is employed in eq 9.3. Working with eq 9.16 to describe PCET, the reorganization power is also determined by the ET. Acesulfame supplier equation 9.16 assumes p kBT, so the proton is initially in its ground vibrational state. In our extended interpretation, eq 9.16 also accounts for the vibrational excitations that may well accompany339 an ET reaction. In the event the different dependences on R of the reactant and solution wave functions in eqs 9.4a and 9.4b are interpreted as distinctive vibrational states, but don’t correspond to PT (therefore, eq 9.1 is no longer the equation describing the reaction), the above theoretical framework is, certainly, unchanged. In this case, eq 9.16 describes ET and is identical to a well-known ET rate expression339-342 that seems as a unique case for 0 kBT/ p inside the theory of Jortner and co-workers.343 The frequencies of proton vibration within the reactant and solution states are assumed to be equal in eq 9.16, despite the fact that the remedy is often extended to the case in which such frequencies are unique. In both the PT and PCET interpretations of your above theoretical model, note that nexp(-p)/n! is the overlap p in between the initial and final proton wave functions, which are represented by two displaced harmonic oscillators, 1 inside the ground vibrational state and the other in the state with vibrational quantum number n.344 Therefore, eq 9.16 is usually recast inside the formk= 1 kBT0 |W IFn|2 exp- n=Review(X ) = clM 2(X – X )two M 2 exp – 2kBT 2kBT(9.19)(M and would be the mass and frequency on the oscillator) is obtained from the integralasq2 exp( -p2 x 2 qx) dx = exp two – 4p p(Re p2 0)(9.20)2k T two p (S0n)two = (S0pn)2 exp B 20n M(9.21)Working with this typical overlap in lieu of eq 9.18 in eq 9.17a, 1 findsk= 2k T two B 0n.