Now contains diverse H vibrational states and their statistical weights. The above formalism, in conjunction

Now contains diverse H vibrational states and their statistical weights. The above formalism, in conjunction with eq ten.16, was demonstrated by Hammes-Schiffer and co-workers to become valid inside the much more common context of vibronically nonadiabatic EPT.337,345 In addition they addressed the computation with the PCET price parameters in this wider context, where, in contrast for the HAT reaction, the ET and PT processes frequently adhere to diverse pathways. Borgis and Hynes also created a Landau-Zener formulation for PT rate constants, ranging in the weak towards the powerful proton coupling regime and examining the case of powerful coupling in the PT solute to a polar solvent. Within the diabatic limit, by introducing the possibility that the proton is in various initial states with Boltzmann populations P, the PT rate is written as in eq 10.16. The authors present a general expression for the PT matrix element in terms of Laguerredx.doi.org/10.1021/cr4006654 | Chem. Rev. 2014, 114, 3381-Chemical Evaluations polynomials, however the identical coupling decay constant is applied for all couplings W.228 Note also that eq 10.16, with substitution of eq ten.12, or ten.14, and eq ten.15 yields eq 9.22 as a unique case.ten.four. Analytical Rate Continuous Expressions in Limiting RegimesReviewAnalytical benefits for the transition rate had been also obtained in a number of significant limiting regimes. Inside the high-temperature and/or low-frequency regime with respect for the X mode, / kBT 1, the rate is192,193,kIF =2 WIF kBT(G+ + 4k T /)two B X exp – 4kBT2 WIF kBT3 4kBT exp + + O 3kBT 2kBT (G+ + 2 k T X )two IF B exp – 4kBT2 2 2k T WIF B exp IF 2 kBT Mexpression in ref 193, exactly where the barrier prime is described as an inverted parabola). As noted by Borgis and Hynes,193,228 the non-Arrhenius dependence on the temperature, which arises from the typical squared coupling (see eq ten.15), is weak for realistic choices with the physical parameters involved inside the rate. Thus, an Arrhenius behavior from the price continuous is obtained for all sensible purposes, despite the quantum mechanical nature from the tunneling. One more substantial limiting regime is definitely the opposite of the above, i.e., the low-temperature and/or high-frequency limit defined by /kBT 1. Unique situations result from the relative values from the r and s parameters provided in eq ten.13. Two such circumstances have particular physical relevance and arise for the situations S |G and S |G . The very first condition corresponds to robust solvation by a highly polar solvent, which establishes a solvent reorganization power exceeding the difference within the absolutely free power amongst the initial and final equilibrium states of the H transfer reaction. The second a single is satisfied inside the (opposite) weak solvation regime. Inside the 1st case, eq 10.14 68181-17-9 Protocol results in the following approximate expression for the price:165,192,kIF =2 (G+ )two WIF 0 S exp – SkBT 4SkBT(10.18a)with( – X ) WIF 20 = (WIF two)t exp(ten.17)(G+ + two k T X )2 IF B exp – 4kBT(ten.18b)where(WIF 2)t = WIF two exp( -IFX )(10.18c)with = S + X + . Inside the second expression we employed X and defined within the BH model. The third expression was obtained by Hammes-Schiffer and co-workers184,197,337,345 for the sum terms in eq 10.16, beneath precisely the same circumstances of temperature and frequency, using a distinctive coupling decay continuous (and 3326-34-9 Epigenetic Reader Domain therefore a various ) for every term in the sum and expressing the vibronic coupling as well as the other physical quantities which might be involved in extra general terms suitable for.