R proton localizations inside the reactant and product electronic states bring about their tiny overlap and correspondingly small vibronic coupling. In other words, V12Sp 1a,2b can even be drastically smaller sized than the thermal energy due to the compact value of Sp . That is, PCET happens in the vibronically 1a,2b nonadiabatic regime. For quite massive reaction free energies between 1a and 1b and among 2a and 2b, PESs I and II cut down to parabolas that describe EPT or HAT. Conversely, ifEarly applications and extensions of Marcus theory to proton and atom transfer reactions had been described by Sutin230,231 and by Marcus.232 In actual fact, Marcus’ analysis starts using the ET cross-relation inside the weak-coupling limit. Take into consideration the self-exchange weak-overlap ET reactionsOx n + Red n Red n + Ox n (n = 1, two)(6.1)plus the cross-reactionOx1 + Red 2 Red1 + Ox(6.2)By assuming the following 1350653-20-1 custom synthesis approximation for the reorganization energy related using the cross-reaction:232,= 12 = 21 = 11 + 22(six.3)the rate constants knn (n = 1, 2) and k12 for eqs 6.1 and six.two are associated byk12 (k11k 22K12f12 )1/(6.4)whereK12 = k12 k(6.five)could be the equilibrium continuous for the ET cross-reaction. f12 is often close to unity,7 andln f12 = – (GR 2 (ln K12)2 = 2kBT four ln(k11k 22/n 2)(six.six)Equations 6.4 and six.six hold for el 1 and neglect the work terms linked with forming the precursor and successor complexes, except that their difference is usually integrated in thedx.doi.org/10.1021/cr4006654 | Chem. Rev. 2014, 114, 3381-Chemical Testimonials reaction cost-free energy GR(see ref 7). Indeed, the price constants from the bimolecular reactions in solution must be inserted into eq 6.4 for comparison to experiments. These rate constants include the operate terms wr and wp that happen to be needed to bring the reactants and items for the mean charge donor- acceptor distance inside the activated complex. Within this perspective, the ET cross-reaction rate is7,122,233-G k12 = elZ exp – kBTReview(6.7)In eq 6.7, the power ratio in the exponent is written on a per molecule basis, Z is really a bimolecular collision frequency, along with the activation free of charge power isG = w r + GR 1 +(six.8a)withGR = G+ w p – w r(six.8b)In eq six.8b, Gis the “standard” free energy of reaction7,122,236 for separated reactants and items.237,238 Assuming that Z is approximated effectively by the geometric imply from the homologous frequencies for the self-exchange reactions, Z11 and Z22, and that eq six.3 holds, eqs six.4 and 6.six are replaced by239 (see also the Supporting Information)k11k 22K12f 1/2 12 k12 el W12 (el)11(el)(six.9a)withwr + wp – w – w 11 22 W12 = exp – kBT(6.9b)andln f12 = – (G+ w p – w r)2 2kBT=1 4 ln k11k 22 + Z2(el)11(el)(ln K+wr – wp kBT)applicable to “strong-overlap” reactions like atom and proton transfers and to strong-overlap ET.232 Extending the Marcus theory for ET in between weakly interacting redox partners to proton and atom transfer reactions232 requires reconsideration of your assumptions leading for the totally free power factor in the Marcus rate expression. Mainly Coulombic perform is performed to bring reactants together,233 however the major contribution towards the totally free power barrier for ET is supplied by readjusting bond lengths and angles in the reactants (i.e., inner-sphere contributions) and by the reorientation of solvent molecules (outer-sphere contribution). The vibrational motion on the ligands along with the dielectric relaxation in the solvent polarization via many degrees of freedom, inside the linear response regime, result in the parabolic PFESs.