C Ib(R , Q p) c Fa(R , Q p)]VIF(12.25)exactly where Hgp is the matrix

C Ib(R , Q p) c Fa(R , Q p)]VIF(12.25)exactly where Hgp is the matrix that represents the solute gas-phase electronic Hamiltonian in the VB basis set. The second approximate expression makes use of the Condon approximation with respect for the solvent collective coordinate Qp, since it is evaluated t in the transition-state coordinate Qp. Moreover, in this expression the couplings amongst the VB diabatic states are assumed to be continuous, which amounts to a stronger application of the Condon approximation, givingPT (Hgp)Ia,Ib = (Hgp)Fa,Fb = VIF ET (Hgp)Ia,Fa = (Hgp)Ib,Fb = VIF EPT (Hgp)Ia,Fb = (Hgp)Ib,Fa = VIFIn ref 196, the electronic coupling is approximated as within the second expression of eq 12.25 plus the Condon approximation is also applied for the proton coordinate. In actual fact, the electronic coupling is computed in the value R = 0 of the proton coordinate that corresponds to maximum overlap among the reactant and product proton wave functions within the iron biimidazoline complexes studied. Hence, the vibronic coupling is written ast ET k ET p W(Q p) = VIF Ik |F VIF S(12.31)(12.26)These approximations are helpful in applications on the theory, exactly where VET is assumed to be the identical for pure ET and IF for the ET component of PCET reaction mechanisms and VEPT IF is approximated to become zero,196 since it seems as a second-order coupling within the VB theory framework of ref 437 and is hence expected to be substantially smaller sized than VET. The matrix IF corresponding towards the totally free energy in the I,F basis isH(R , Q p , Q e) = S(R , Q p , Q e)I E I(R , Q ) VIF(R , Q ) p p + V (R , Q ) E (R , Q ) F p p FI 0 0 + 0 Q e(12.27)This vibronic coupling is utilized to compute the PCET price inside the electronically nonadiabatic limit of ET. The transition price is derived by Soudackov and Hammes-Schiffer191 employing Fermi’s golden rule, with the following approximations: (i) The electron-proton absolutely free energy surfaces k(Qp,Qe) and n (Qp,Qe) I F rresponding to the initial and final ET states are elliptic paraboloids, with identical curvatures, and this holds for each pair of proton vibrational states that is involved within the reaction. (ii) V is assumed constant for each and every pair of states. These approximations had been shown to be valid for a wide selection of PCET systems,420 and in the high-temperature limit for a Debye solvent149 and within the absence of relevant Namodenoson Epigenetic Reader Domain intramolecular solute modes, they bring about the PCET rate constantkPCET =P|W|(G+ )2 exp – kBT 4kBT(12.32)exactly where P could be the Boltzmann distribution for the reactant states. In eq 12.32, the reaction totally free power isn G= F (Q p , Q e) – Ik(Q p , Q e)(Q,Qe ) p (Qp,Qe )(12.33)Beneath physically reasonable situations for the solute-solvent interactions,191,433 changes inside the no cost energy HJJ(R,Qp,Qe) (J = I or F) are roughly equivalent to changes in the possible energy along the R coordinate. The proton vibrational states that H-Arg(Pbf)-OMe Protocol correspond for the initial and final electronic states can thus be obtained by solving the one-dimensional Schrodinger equation[TR + HJJ (R , Q p , Q e)]Jk (R ; Q p , Q e) = Jk(Q p , Q e) Jk (R ; Q p , Q e)(12.28)exactly where and will be the equilibrium solvent collective coordinates for states and , respectively. The outer-sphere reorganization power linked with the transition isn n = F (Q p , Q e) – F (Q p , Q e)(12.34)The resulting electron-proton states are(q , R ; Q p , Q e) = I(q; R , Q p) Ik (R ; Q p , Q e)(12.29a)An inner-sphere contribution to the reorganization power commonly should be included.196 T.