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. 2009 Mar 12;113(10):2117-26.
doi: 10.1021/jp809122y.

Analysis of kinetic isotope effects for proton-coupled electron transfer reactions

Sarah J Edwards  1 , Alexander V Soudackov, Sharon Hammes-Schiffer
Affiliations

Analysis of kinetic isotope effects for proton-coupled electron transfer reactions

Sarah J Edwards et al. J Phys Chem A. .

Abstract

A series of rate constant expressions for nonadiabatic proton-coupled electron transfer (PCET) reactions are analyzed and compared. The approximations underlying each expression are enumerated, and the regimes of validity for each expression are illustrated by calculations on model systems. In addition, the kinetic isotope effects (KIEs) for a series of model PCET reactions are analyzed to elucidate the fundamental physical principles dictating the magnitude of the KIE and the dependence of the KIE on the physical properties of the system, including temperature, reorganization energy, driving force, equilibrium proton donor-acceptor distance, and effective frequency of the proton donor-acceptor mode. These calculations lead to three physical insights that are directly relevant to experimental data. First, these calculations provide an explanation for a decrease in the KIE as the proton donor-acceptor distance increases, even though typically the KIE will increase with increasing equilibrium proton donor-acceptor distance if all other parameters remain fixed. Often the proton donor-acceptor frequency decreases as the proton donor-acceptor distance increases, and these two effects impact the KIE in opposite directions, so either trend could be observed. Second, these calculations provide an explanation for an increase in the KIE as the temperature increases, even though typically the KIE will decrease with increasing temperature if all other parameters remain fixed. The combination of a rigid hydrogen bond, which corresponds to a high proton donor-acceptor frequency, and low solvent polarity, which corresponds to small solvent reorganization energy, allows the KIE to either increase or decrease with temperature, depending on the other properties of the system. Third, these calculations provide insight into the dependence of the rate constant and KIE on the driving force, which has been studied experimentally for a wide range of PCET systems. The rate constant increases as the driving force becomes more negative because excited vibronic product states associated with low free energy barriers and relatively large vibronic couplings become accessible. The ln[KIE] has a maximum near zero driving force and decreases significantly as the driving force becomes more positive or negative because the contributions from excited vibronic states increase as the reaction becomes more asymmetric, and contributions from excited vibronic states decrease the KIE. These calculations and analyses lead to experimentally testable predictions of trends in the KIEs for PCET systems.

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Figures

Figure 1
Figure 1
(a) Proton potential energy curves and the associated hydrogen (solid) and deuterium (dashed) vibrational wavefunctions for the ground reactant (blue) and product (red) states. The proton potential energy curves are Morse potentials with the parameters given in the text. The proton donor-acceptor distance is 2.7 Å. (b) Square of the ratio of the hydrogen and deuterium overlaps for the ground reactant and product vibrational states 0/0 (black) and the ground reactant and first excited product states 0/1 (red) as a function of the proton donor-acceptor distance R. Note that (SH/SD)2 increases dramatically as R increases and is significantly smaller for the 0/1 pair of states.
Figure 2
Figure 2
(a) KIE as a function of proton donor-acceptor mode frequency Ω for R̅ = 2.7 Å at T = 303 K calculated with kquant (solid), khighT (dotted), and klowT (dashed) for a model system with λ = 30 kcal/mol, ΔG0 = −5 kcal/mol, and M = 100 amu. (b) KIE as a function of 1000/T for Ω = 150 cm−1 and R̅ = 2.7 Å obtained with kquant and khighT (black, lower curves) and for Ω = 850 cm−1 and R̅ = 2.6 Å obtained with kquant and klowT (red, upper curves). The line types are the same as in (a), and the two upper curves are virtually indistinguishable.
Figure 3
Figure 3
KIE as a function of frequency calculated with kquant (solid), khighT (dotted), klowT (dashed), and kUK (dot-dashed) for a model system with λ = 30 kcal/mol, ΔG0 = −5 kcal/mol, R̅ = 2.7 Å, T = 303 K, and (a) M = 100 amu and (b) M = 20 amu.
Figure 4
Figure 4
KIE calculated with kquant for a model system with R̅ = 2.7 Å, Ω = 150 cm−1, M = 100 amu, and ΔG0 = −5 kcal/mol for different values of the reorganization energy λ. In units of kcal/mol, λ = 10 (green), 20 (red), 30 (black), and 40 (blue). The four curves are virtually indistinguishable.
Figure 5
Figure 5
Free energy curves as functions of a collective solvent coordinate (center frame) with the proton potential energy curves and associated hydrogen vibrational wavefunctions on the left (reactant) and right (product) for a model system with λ = 30 kcal/mol and ΔG0 = −5 kcal/mol. The lowest three reactant vibronic states are shown in blue, and the lowest three product vibronic states are shown in red. The splittings between the free energy curves correspond to the splittings between the proton vibrational energy levels for the Morse potentials. The reorganization energy λ, driving force ΔG0, and free energy barrier ΔG00≠ for the ground reactant and product vibronic states are identified.
Figure 6
Figure 6
Free energy curves as functions of a collective solvent coordinate for hydrogen (solid) and deuterium (dashed) for a model system with λ = 30 kcal/mol and ΔG0 = 5 kcal/mol. The lowest two reactant and the lowest two product states are shown for both hydrogen and deuterium. Since the ground reactant state is chosen to have the same absolute energy for hydrogen and deuterium, the ground reactant and product states for deuterium exactly overlay those for hydrogen and therefore are not distinguishable.
Figure 7
Figure 7
KIE calculated with kquant for a model system with λ = 30 kcal/mol, ΔG0 = −5 kcal/mol, and M = 100 amu. (a) R̅ = 2.7 Å and Ω = 140 cm−1 (black), R̅ = 2.7 Å and Ω = 180 cm−1 (red), and R̅ = 2.8 Å and Ω = 140 cm−1 (blue). (b) R̅ = 2.7 Å and Ω = 150 cm−1 (black), R̅ = 2.7 Å and Ω = 300 cm−1 (red). The curves are labeled according to the R̅/Ω values in units of Å and cm−1.
Figure 8
Figure 8
Temperature dependence of the KIE calculated with kquant for a model system with λ = 3 kcal/mol, ΔG0 = −6.5 kcal/mol, R̅ = 2.7 Å, Ω = 600 cm−1, and M = 100 amu.
Figure 9
Figure 9
Driving force dependence of (a) the rate constant kHhighT, (b) the associated KIE, and (c) the contributions of pairs of reactant/product vibronic states for a model system with λ = 20 kcal/mol, R̅ = 2.7 Å, Ω = 150 cm−1, M = 100 amu, and T = 303 K. In (a) and (b), the red curve corresponds to the calculation including only the ground reactant and product vibronic states, and the black curve corresponds to the calculation that is converged with respect to excited vibronic states. In (c), the color code for the pairs of reactant/product vibronic states is as follows: 0/0 (black), 1/0 (blue), 2/0 (magenta), 0/1 (red), and 0/2 (green).
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