Atic PT and, all round, vibronidx.doi.org/10.1021/cr4006654 | Chem. Rev. 2014, 114, 3381-Chemical Evaluations cally nonadiabatic electron-proton transfer. This can be since the nonadiabatic regime of ET implies (a) absence of correlation, in eq five.41, in between the vibrational functions n that belong to diverse electronic states sufficiently far from the intersections among electron-proton PESs and (b) little transition probabilities close to these intersections which might be determined by the smaller values on the vibronic couplings. This means that the motion along the solvent coordinate isn’t limited for the ground-state vibronic adiabatic surface of Figure 23b. Even though eq five.40 allows a single to speak of (electronically) nonadiabatic ET, the combined impact of Vnk and Sp around the couplings of eq 5.41 nk doesn’t let a single to define a “nonadiabatic” or “vibrationally nonadiabatic” PT. This really is in contrast together with the case of pure PT in between localized proton vibrational states along the Q coordinate. Hence, a single can only speak of vibronically nonadiabatic EPT: this really is proper when electronically nonadiabatic PT takes location,182 because the nonadiabaticity on the electronic dynamics coupled with PT implies the presence with the electronic coupling Vnk within the transition matrix element. five.three.2. Investigating Coupled Electronic-Nuclear Dynamics and Deviations in the Adiabatic Approximation in PCET Systems by way of a Simple Model. Adiabatic electron-proton PESs are also shown in Figure 23b. To construct mixed electron/proton vibrational adiabatic states, we reconsider the form of eq 5.30 (or eq five.32) and its option in terms of adiabatic electronic states and the Heptadecanoic acid Autophagy corresponding vibrational functions. The off-diagonal electronic- nuclear interaction terms of eq 5.44 are removed in eq five.45 by averaging more than a single electronic adiabatic state. However, these terms couple various adiabatic states. In truth, the scalar multiplication of eq 5.44 on the left by a distinct electronic adiabatic state, ad, shows that the conditionad [-2d(x) + G (x)] (x) = 0 x(5.47)will have to be satisfied for any and so that the BO adiabatic states are eigenfunctions of the full Hamiltonian and are hence options of eq five.44. Indeed, eq five.47 is normally not happy exactly even for two-state 2883-98-9 site models. This is observed by using the equations inside the inset of Figure 24 with all the strictly electronic diabatic states 1 and 2. Within this very simple one-dimensional model, eqs 5.18 and five.31 cause the nuclear kinetic nonadiabatic coupling termsd(x) = – V12 2 d 2 = x two – x1 d12 x 2 – x1 12 two (x) + 4V12(five.48)(5.43)andad G (x)Equation 5.43 would be the Schrodinger equation for the (reactive) electron at fixed nuclear coordinates within the BO scheme. As a result, ad is the electronic component of a BO solution wave function that approximates an eigenfunction in the total Hamiltonian at x values for which the BO adiabatic approximation is valid. In actual fact, these adiabatic states give V = E, but correspond to (approximate) diagonalization of (eq 5.1) only for compact nonadiabatic the full Hamiltonian kinetic coupling terms. We now (i) analyze and quantify, for the very simple model in Figure 24, functions of the nonadiabatic coupling in between electronic states induced by the nuclear motion which can be crucial for understanding PCET (for that reason, the nonadiabatic coupling terms neglected inside the BO approximation will probably be evaluated in the analysis) and (ii) show how mixed electron-proton states of interest in coupled ET- PT reactions are derived in the.