R proton localizations in the reactant and solution electronic states lead to their compact overlap and correspondingly small vibronic coupling. In other words, V12Sp 1a,2b can even be substantially smaller than the thermal energy because of the modest value of Sp . That may be, PCET occurs inside the vibronically 1a,2b nonadiabatic regime. For quite massive reaction absolutely free energies between 1a and 1b and in between 2a and 2b, PESs I and II lower 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 truth, Marcus’ analysis begins with all the ET cross-relation inside the weak-coupling limit. Think about the self-exchange weak-overlap ET reactionsOx n + Red n Red n + Ox n (n = 1, two)(six.1)plus the cross-reactionOx1 + Red two Red1 + Ox(6.two)By assuming the following approximation for the reorganization power connected with all the cross-reaction:232,= 12 = 21 = 11 + 22(6.3)the rate constants knn (n = 1, two) and k12 for eqs 6.1 and 6.2 are Sunset Yellow FCF Formula associated byk12 (k11k 22K12f12 )1/(six.4)whereK12 = k12 k(six.5)is the equilibrium continual for the ET cross-reaction. f12 is usually close to unity,7 andln f12 = – (GR 2 (ln K12)two = 2kBT 4 ln(k11k 22/n 2)(six.six)Equations 6.four and 6.6 hold for el 1 and neglect the perform terms linked with forming the precursor and successor complexes, except that their difference might be incorporated in thedx.doi.org/10.1021/cr4006654 | Chem. Rev. 2014, 114, 3381-Chemical Evaluations reaction free energy GR(see ref 7). Certainly, the rate constants on the bimolecular reactions in remedy must be inserted into eq 6.four for comparison to experiments. These price constants include things like the operate terms wr and wp that happen to be required to bring the reactants and items for the imply charge donor- acceptor distance inside the activated complex. Within this viewpoint, the ET cross-reaction price is7,122,233-G k12 = elZ exp – kBTReview(6.7)In eq six.7, the power ratio in the exponent is written on a per molecule basis, Z is really a bimolecular collision frequency, and the activation absolutely free power isG = w r + GR 1 +(six.8a)withGR = G+ w p – w r(six.8b)In eq six.8b, Gis the “standard” free of charge power of reaction7,122,236 for separated reactants and merchandise.237,238 Assuming that Z is approximated properly by the geometric mean in the homologous frequencies for the self-exchange reactions, Z11 and Z22, and that eq six.three holds, eqs six.4 and six.six are replaced by239 (see also the Supporting Information)k11k 22K12f 1/2 12 k12 el W12 (el)11(el)(6.9a)withwr + wp – w – w 11 22 W12 = exp – kBT(six.9b)andln f12 = – (G+ w p – w r)two 2kBT=1 four ln k11k 22 + Z2(el)11(el)(ln K+wr – wp kBT)applicable to “strong-overlap” reactions for instance atom and proton transfers and to Acid-PEG2-SS-PEG2-acid References strong-overlap ET.232 Extending the Marcus theory for ET among weakly interacting redox partners to proton and atom transfer reactions232 calls for reconsideration of the assumptions major for the free of charge energy issue in the Marcus rate expression. Mostly Coulombic function is performed to bring reactants collectively,233 but the key contribution for the absolutely free power barrier for ET is supplied by readjusting bond lengths and angles inside the reactants (i.e., inner-sphere contributions) and by the reorientation of solvent molecules (outer-sphere contribution). The vibrational motion with the ligands and also the dielectric relaxation of your solvent polarization through numerous degrees of freedom, in the linear response regime, result in the parabolic PFESs.