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Substituent Effects on the Vibronic Coupling for the Phenoxyl/Phenol Self-Exchange Reaction† Michelle K. Ludlow, Jonathan H. Skone, and Sharon Hammes-Schiffer* Department of Chemistry, PennsylVania State UniVersity, 104 Chemistry Building, UniVersity Park, PennsylVania 16802 ReceiVed: July 9, 2007; In Final Form: August 17, 2007
The impact of substituents on the vibronic coupling for the phenoxyl/phenol self-exchange reaction, which occurs by a proton-coupled electron transfer mechanism, is investigated. The vibronic couplings are calculated with a grid-based nonadiabatic method and a nuclear-electronic orbital nonorthogonal configuration interaction method. The quantitative agreement between these two methods for the unsubstituted phenoxyl/phenol system and the qualitative agreement in the predicted trends for the substituted phenoxyl/phenol systems provides a level of validation for both methods. Analysis of the results indicates that electron-donating groups enhance the vibronic coupling, while electron-withdrawing groups attenuate the vibronic coupling. Thus, if all other aspects of the reaction are the same, then electron-donating groups will increase the rate, while electronwithdrawing groups will decrease the rate. Correlations between the vibronic coupling and physical properties of the phenol are also analyzed. Negative Hammett constants correspond to higher vibronic couplings, while positive Hammett constants correspond to similar or slightly lower vibronic couplings relative to the unsubstituted phenoxyl/phenol system. In addition, lower bond dissociation enthalpies, ionization potentials, and redox potentials, as well as higher pKa values, tend to correspond to higher vibronic couplings relative to the unsubstituted phenoxyl/phenol system. The observed trends enable the prediction of the impact of general substituents on the vibronic coupling, and hence the rate, for the phenoxyl/phenol self-exchange reaction. The fundamental physical insights obtained from these studies are applicable to other proton-coupled electron transfer systems.
I. Introduction Proton-coupled electron transfer (PCET) reactions are essential for a wide range of chemical and biological processes, such as electrochemistry,1 photosynthesis,2-7 respiration,8,9 and enzyme reactions.10-14 A variety of theoretical methods have been developed to study PCET reactions in which an electron and a proton are transferred in a single step.15-24 The fundamental mechanism for these types of reactions involves reorganization of the solvent and protein environment, as well as the solute in some cases, to enable quantum mechanical tunneling of the hydrogen nucleus in conjunction with electron transfer. PCET reactions are often vibronically nonadiabatic with respect to the solvent and protein environment because the vibronic coupling between the reactant and product mixed electron-proton vibronic wave functions is much less than the thermal energy kBT. In this case, the rate of the reaction for each pair of vibronic states is proportional to the square of the vibronic coupling.15,16,20,21 As a result, the magnitude of the vibronic coupling significantly impacts the rates, kinetic isotope effects,andtemperaturedependencesofgeneralPCETreactions.25-27 In a previous paper, we implemented a semiclassical approach to calculate the vibronic couplings for the phenoxyl/phenol and benzyl/toluene self-exchange reactions.28 Prior density functional theory studies designated the phenoxyl/phenol reaction as PCET and the benzyl/toluene reaction as hydrogen atom transfer because the electron and proton are transferred between different †
Part of the “James T. (Casey) Hynes Festschrift”. * To whom correspondence should be addressed. E-mail: shs@ chem.psu.edu.
sets of orbitals for the former and between the same sets of orbitals for the latter.29 In our approach, all electrons and the transferring hydrogen nucleus are treated quantum mechanically, and the vibronic coupling is defined as the Hamiltonian matrix element between the reactant and product mixed electronproton vibronic wave functions. We calculated the vibronic couplings using a general semiclassical analytical expression,22 as well as analytical expressions in the limits of electronically adiabatic and nonadiabatic proton tunneling. Our calculations indicated that both of these self-exchange reactions are vibronically nonadiabatic with respect to a solvent environment at room temperature, but the proton tunneling is electronically nonadiabatic for the phenoxyl/phenol reaction and electronically adiabatic for the benzyl/toluene reaction. Thus, our analysis provided a new diagnostic for differentiating between the conventionally defined PCET and hydrogen atom transfer reactions. For both systems, the vibronic coupling decreased with the proton donor-acceptor distance, and the replacement of hydrogen with deuterium decreased the magnitude of the vibronic coupling and enhanced the decay with distance. We have also developed the nuclear-electronic orbital (NEO) method for calculating the vibronic couplings corresponding to these types of systems.30-32 In the NEO approach, the electrons and transferring proton are treated on equal footing with molecular orbital techniques. Mixed nuclear-electronic wave functions are calculated by solving a mixed nuclear-electronic time-independent Schro¨dinger equation using a variational procedure. In the two-state NEO-NOCI (nonorthogonal configuration interaction) approach,31 the vibronic coupling is
10.1021/jp0753474 CCC: $40.75 © 2008 American Chemical Society Published on Web 10/16/2007
Vibronic Coupling for Phenoxyl/Phenol Self-Exchange
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calculated in terms of the Hamiltonian matrix element between two localized nuclear-electronic wave functions obtained at the NEO-HF (Hartree-Fock) level. The advantages of the NEONOCI approach are the computational efficiency and the potential for systematic improvement by enhancing the basis sets and number of configurations. Benchmarking calculations have shown that the NEO-NOCI method provides accurate hydrogen tunneling splittings for the [He-H-He]+ model system for a range of He-He distances.31 In this paper, we apply both the semiclassical grid-based nonadiabatic method and the NEO-NOCI method to the phenoxyl/phenol self-exchange reaction and examine the impact of substituents on the vibronic coupling for this reaction. From a methodological perspective, our objective is to compare the two methods and identify the range of applicability for each method. From a chemical perspective, our objective is to understand how the vibronic coupling is influenced by the nature of the substituent (i.e., electron-donating or electron-withdrawing) and the position of the substituent (i.e., ortho, para, or meta). The analysis provides insight into the fundamental physical principles dictating the vibronic couplings for PCET reactions. The results provide experimentally testable predictions of the trends in the rates for substituted phenoxyl/phenol systems. The paper is organized as follows. Section II summarizes the two methods for calculating vibronic couplings and provides the computational details. Section III presents the Results and Discussion. The conclusions are presented in section IV.
Figure 1. Optimized geometry for the unsubstituted phenoxyl/phenol system. (a) The proton donor-acceptor O-O distance R is indicated. The pairs of substitution positions maintaining C2 symmetry are also indicated, where m, p, and o refer to meta, para, and ortho, respectively. (b) The COOC dihedral angle θ is indicated.
to the ground states, and M and Ω are the effective mass and frequency associated with the proton donor-acceptor motion. Here, the vibronic coupling is assumed to have the general form Vµν ) V(0) µν exp[-RµνδR], where δR is the deviation of the proton donor-acceptor distance from its equilibrium value. When only the nonadiabatic transition between the two ground states is included, the KIE can be approximated as27
II. Theory and Methods We have developed a theoretical formulation for general PCET reactions and derived vibronically nonadiabatic rate expressions.19-21 In this formulation, the PCET reaction occurs between two diabatic electronic states, denoted I and II, representing the localized electron transfer states. The transferring electron is localized on the donor for diabatic state I and on the acceptor for diabatic state II. The proton vibrational wave functions are calculated for each diabatic electronic state, leading to a set of reactant and product proton vibrational wave functions (II) denoted φ(I) D and φA , respectively. The vibronic coupling is defined to be the Hamiltonian matrix element between the reactant and product mixed electron-proton vibronic wave functions. The overall reaction is vibronically nonadiabatic with respect to the solvent or protein environment when this vibronic coupling is much less than kBT. In this case, the rate of reaction for each pair of vibronic states is proportional to the square of the vibronic coupling. Consequently, the vibronic couplings significantly impact the magnitudes and temperature dependences of the rates and kinetic isotope effects (KIEs). As derived previously21,27 using a series of well-defined, physically reasonable approximations, the rate of a general PCET reaction can be expressed as
k)
∑µ Pµ ∑ν
2 |V(0) µν |
exp p
[ ]x [ 2kBTR2µν
π
MΩ2
(λµν + λR)kBT
×
]
(∆G0 + λµν + ∆µν)2
exp -
4(λµν + λR)kBT
(1)
where the summations are over reactant and product vibronic states, Pµ is the Boltzmann probability for the reactant state µ, λµν is the reorganization energy, λR ) p2R2µν/2M, ∆G0 is the reaction free energy for the ground states, ∆µν is the difference between the product and reactant vibronic energy levels relative
KIE ≈
2 |V(0) H | 2 |V(0) D |
{
exp
2kBT
MΩ2
}
(R2H - R2D)
(2)
where the H and D subscripts refer to hydrogen and deuterium, respectively. These equations for the rate and KIE illustrate that the calculation of the vibronic coupling is essential for predicting the magnitudes and temperature dependences of the rates and KIEs. The objective of this paper is to examine the effects of substituents on the vibronic couplings for the phenoxyl/phenol self-exchange reaction. For simplicity, here we consider the tunneling between only the ground state reactant and product mixed electron-proton vibronic states. The reactant and product states refer to the mixed electron-proton vibronic quantum states in which the electron and proton are localized on the donor in the reactant state and on the acceptor in the product state for a fixed geometry of all other nuclei. For the vibronic coupling calculations, all nuclei are fixed at the transition state geometries, except for the transferring hydrogen, which is treated quantum mechanically. In general, the motions of the other nuclei could contribute to the vibronic coupling, but including these effects is beyond the scope of this paper. Moreover, our goal is not to provide quantitatively accurate results for these specific systems but rather to illustrate the general trends. Therefore, we utilize moderate levels of electronic structure theory that provide physically reasonable results. The quantitative accuracy of the results can be improved by using a larger basis set and including dynamical electron correlation. In our systematic study, each substituent was placed in the para, ortho, and meta positions on both rings to maintain C2 symmetry, as illustrated in Figure 1. The transition state geometries were optimized with density functional theory (DFT) using the B3LYP functional33,34 and the 6-31G* basis set35 using the Gaussian03 package.36 All transition states were confirmed to have a single imaginary frequency. For the ortho and meta
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substitutions, two different transition state geometries were optimized, and the lowest-energy transition state was used in the calculations except for o-NH2, where the transition state with slightly higher energy was used to avoid hydrogen bonding between the NH2 and the oxygen atoms. In all cases, the proton donor-acceptor O-O distance was R ) 2.40 ( 0.01 Å. We found that the three-dimensional vibronic couplings were so small at this distance that the effects of the substituents were often smaller than the numerical errors in the calculations. Therefore, to examine the trends in the vibronic couplings in a numerically meaningful way, we translated the rigid donor and acceptor molecules along the donor-acceptor axis to obtain an O-O distance of R ) 2.30 Å for each transition state structure. We confirmed that the one-dimensional vibronic couplings exhibit the same trends within the numerical accuracy at these two proton donor-acceptor O-O distances. A. Grid-Based Nonadiabatic Method for Calculating Vibronic Couplings. In a previous paper,28 we calculated the vibronic couplings for the phenoxyl/phenol reaction with the semiclassical approach developed by Georgievskii and Stuchebrukhov.22 We showed that the proton motion occurs in the electronically nonadiabatic limit, where the proton tunneling time is much less than the electronic transition time so the electronic states do not have enough time to mix completely during the proton tunneling process. In this limit, the vibronic coupling V(na) DA can be expressed as the product of the electronic (II) coupling VET and the Franck-Condon overlap 〈φ(I) D |φA 〉 of the reactant and product proton vibrational wave functions ET (I) (II) V(na) DA ) V 〈φD |φA 〉
(3)
The input quantities for this vibronic coupling expression can be calculated with electronic structure methods. In this paper, we present the vibronic couplings for the substituted phenoxyl/phenol systems calculated with the nonadiabatic expression given in eq 3. We obtained the electronically adiabatic ground- and excited-state potential energy curves along the hydrogen coordinate by calculating the state-averaged CASSCF(3,6) energy for the hydrogen positioned at discrete grid points along the axis connecting the donor and acceptor atoms. The active space was chosen to ensure that the character of the orbitals in the active space was conserved along the hydrogen coordinate and the electronic ground state was qualitatively similar to the ROHF ground state. The 6-31G basis set37 was used for all ROHF and CASSCF calculations to enable the efficient calculation of three-dimensional potential energy surfaces. We determined that the ground- and excited-state potential energy surfaces are qualitatively similar for the 6-31G and 6-31G* basis sets. These CASSCF calculations were performed with the Gaussian03 package.36 The quantities in the expression for the vibronic coupling given in eq 3 were determined from the CASSCF potential energy curves, which are depicted in Figure 2 for the unsubstituted phenoxyl/phenol system. The electronic coupling VET is half of the splitting between the two electronically adiabatic CASSCF potential energy curves at the midpoint between the donor and acceptor atoms. A valence bond model was used to fit the state-averaged CASSCF potential energy curves for the purpose of obtaining the two localized electronically diabatic potential energy curves, which are depicted in Figure 2. The details of the valence bond models are given in the Supporting Information. The one-dimensional hydrogen vibrational wave functions were calculated for the diabatic potential energy curves using the Fourier grid Hamiltonian method38,39 with 128 grid
Figure 2. (a) State-averaged CASSCF ground- and excited-state electronically adiabatic potential energy curves along the transferring hydrogen coordinate for the unsubstituted phenoxyl/phenol system. The coordinates of all nuclei except the transferring hydrogen correspond to the transition state geometry. The proton donor-acceptor distance is R ) 2.40 Å. The CASSCF results are depicted as open circles that are blue for the ground state and red for the excited state. The black dashed lines represent the diabatic potential energy curves corresponding to the two localized diabatic electron transfer states I and II. The mixing of these two diabatic states with the electronic coupling VET leads to the CASSCF ground- and excited-state electronically adiabatic curves depicted with solid colored lines following the colored open circles. The solid colored lines and the black dashed lines are nearly indistinguishable because the adiabatic and diabatic potential energy curves are virtually identical except in the transition state region. (b) Diabatic potential energy curves corresponding to the two localized diabatic electron transfer states I and II and the corresponding proton (II) vibrational wave functions φ(I) D (blue) and φA (red) for the phenoxyl/ phenol system. Since this reaction is electronically nonadiabatic, the vibronic coupling is the product of the electronic coupling VET and the overlap of the reactant and product proton vibrational wave functions (II) 〈φ(I) D |φA 〉. These figures are reproduced, with permission, from ref 28.
points spanning 2.0 Å. The Franck-Condon overlap in eq 3 is the overlap between the proton vibrational wave functions for the two diabatic potential energy curves, as depicted in Figure 2b. To study the impact of the three-dimensional character of the hydrogen vibrational wave function, we also calculated threedimensional potential energy surfaces for the hydrogen at the ROHF level for the unsubstituted phenoxyl/phenol system. We obtained the electronically diabatic three-dimensional potential energy surfaces by calculating the ROHF energy for a threedimensional grid with 32 grid points per dimension spanning half of the proton donor-acceptor axis, fitting the data points to an analytical functional form (i.e., a fourth-order polynomial), and using the analytical functional form to generate the full potential energy surface for a grid with 64 points per dimension. The three-dimensional hydrogen vibrational wave functions were calculated for the ROHF potential energy surfaces using the Fourier grid Hamiltonian method.38,39 We used the GAMESS electronic structure program40 for the three-dimensional calculations. Due to numerical difficulties and the computational
Vibronic Coupling for Phenoxyl/Phenol Self-Exchange expense associated with the three-dimensional grid-based nonadiabatic calculations, we did not perform them for the substituted phenoxyl/phenol systems. B. Nuclear-Electronic Orbital Method for Calculating Vibronic Couplings. The vibronic couplings can also be calculated with a three-dimensional treatment of the transferring hydrogen nucleus using the nuclear-electronic orbital (NEO) method.30,31 This method treats specified nuclei on the same level as the electrons and provides mixed nuclear-electronic wave functions through the solution of a mixed nuclearelectronic time-independent Schro¨dinger equation with molecular orbital techniques. Both electronic and nuclear molecular orbitals are expressed as linear combinations of Gaussian basis functions, and the variational method is used to minimize the energy with respect to all molecular orbitals, as well as the centers of the nuclear basis functions. When the NEO approach is applied to hydrogen transfer systems, the transferring hydrogen nucleus and all electrons are treated quantum mechanically. The NEO approach has been implemented in the GAMESS electronic structure program.40 The calculation of delocalized, bilobal hydrogen wave functions for hydrogen tunneling systems within the NEO framework is challenging due to the importance of electron-proton correlation. Typically, the transferring hydrogen atom for hydrogen tunneling systems is represented by two basis function centers to allow delocalization of the hydrogen vibrational wave function.32 For a symmetric system, the exact nuclear wave function will be delocalized equally over both basis function centers. The variational NEO-HF solution, however, corresponds to a nuclear wave function localized on one of the basis function centers.41 This nonphysical localization of the nuclear density at the NEO-HF level arises from the neglect of electron-proton correlation.41 Inclusion of sufficient electron-proton correlation with the NEO-full CI method enables the calculation of delocalized, symmetric nuclear wave functions, but this approach is not computationally practical for most chemical systems. We have developed the NEO-NOCI method for calculating delocalized hydrogen wave functions and the corresponding tunneling splittings.31 The NEO-NOCI method takes advantage of the localization of the variational NEO-HF wave functions. In the two-state NEO-NOCI approach, the ground- and excitedstate delocalized nuclear-electronic wave functions are expressed as linear combinations of two nonorthogonal localized nuclear-electronic wave functions obtained at the NEO-HF level. The tunneling splitting is determined by the energy difference between these two delocalized vibronic states. The hydrogen tunneling splittings calculated with the NEO-NOCI approach for the [He-H-He]+ model system with a range of fixed He-He distances are in excellent agreement with NEOfull CI and three-dimensional Fourier grid calculations.31 The NEO-NOCI method is robust and computationally efficient, and it can be applied to a wide range of chemical systems. This method can also be used to calculate vibronic couplings for PCET reactions. In the application of the NEO-NOCI method to PCET systems, only the transferring hydrogen nucleus is treated quantum mechanically on the same level as the electrons, and the total wave function is expressed as a linear combination of two localized NEO-HF wave functions. Specifically, |ΨI〉 ) ΦeI (re)ΦpI (rp) corresponds to the electron and proton localized on the donor, and |ΨII〉 ) ΦeII(re)ΦpII(rp) corresponds to the electron and proton localized on the acceptor. Here, the e and p subscripts and superscripts denote electrons and proton, respectively. The delocalized ground- and excited-state NEO-
J. Phys. Chem. B, Vol. 112, No. 2, 2008 339 NOCI wave functions are linear combinations of these localized states, and the coefficients of these wave functions are determined by solving a 2 × 2 matrix equation. For symmetric systems, the vibronic coupling is half of the difference between the ground- and first-excited vibronic state energy and can be expressed analogously to electron transfer couplings as42,43
V(NEO) ) DA
HI,II - SI,IIHI,I (1 - S2I,II)
(4)
where HI,I ) 〈ΨI|H|ΨI〉 and HII,II ) 〈ΨII|H|ΨII〉 are the energies of the localized NEO-HF solutions, HI,II ) 〈ΨI|H|ΨII〉 ) HII,I is the off-diagonal Hamiltonian matrix element, and SI,II ) 〈ΨI|ΨII〉 ) SII,I is the overlap between the localized NEO-HF solutions. This equation for the coupling can be generalized to nonsymmetric systems by replacing HI,I with (HI,I + HII,II)/2, as obtained by symmetric orthogonalization of the Hamiltonian matrix.42 In this paper, we use the NEO-NOCI method to calculate the vibronic couplings for the substituted phenoxyl/phenol systems. We used the 6-31G electronic basis set and the QZSPDN′ nuclear basis set,31 which is a quadruple-ζ, s, p, double-ζ, d nuclear basis set, for all NEO calculations. The transferring hydrogen atom was represented by two basis function centers, where each center contained both nuclear and electronic basis functions. The two centers were constrained to be equidistant from the midpoint of these symmetric systems. The basis function center separation was optimized variationally at the NEO-NOCI level for both the ground and excited states. Figure 3 depicts the localized NEO-HF states and the delocalized NEO-NOCI ground and excited vibronic states for the unsubstituted phenoxyl/phenol system. III. Results and Discussion As shown in Figure 1, the phenoxyl/phenol transition state has C2 symmetry. The proton donor-acceptor O-O distance is R ) 2.40 Å, and the COOC dihedral angle is θ ) 138° for the unsubstituted phenoxyl/phenol system. Each substituent is placed in the para, ortho, and meta positions of both rings to maintain the C2 symmetry, as depicted in Figure 1a. Table 1 provides the proton donor-acceptor O-O distance, R, and the dihedral angle, θ, for all of the optimized transition states. The complete optimized structures are provided in Supporting Information. Although R ≈ 2.40 Å for all systems, the dihedral angle varies significantly. This variation in θ is due to a combination of electronic and steric effects. As discussed above, the proton donor-acceptor distance is decreased to R ) 2.30 Å for our calculations to enhance the magnitude of the vibronic couplings and thereby enable a numerically meaningful analysis of the trends. For comparison, the grid-based nonadiabatic vibronic couplings calculated at the transition state structures with R ≈ 2.40 Å are provided in Supporting Information. This table indicates that the qualitative trends in the substituent effects on the vibronic couplings are similar for both distances. Figure 2 depicts the potential energy curves along the transferring hydrogen coordinate and the overlap between the reactant and product proton vibrational wave functions for the unsubstituted phenoxyl/phenol system.28 The CASSCF electronically adiabatic ground- and excited-state curves, as well as the electronically diabatic curves corresponding to the two electron transfer states I and II, are depicted in Figure 2a. The diabatic states correspond to fixed electronic wave functions associated with the hydrogen bonded to the donor atom (I) or to the acceptor atom (II), and the mixing of these two diabatic
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Figure 3. Mixed nuclear-electronic wave functions calculated with the NEO approach for the unsubstituted phenoxyl/phenol system. (a) The two localized NEO-HF nuclear-electronic wave functions, where the highest-energy doubly occupied electronic molecular orbital and the proton molecular orbital are depicted. (b) The delocalized NEO-NOCI ground- and first-excited vibronic state nuclear-electronic wave functions, where the highestenergy doubly occupied electronic molecular orbital and the proton molecular orbital are depicted. These figures were generated with MacMolPlot.48
TABLE 1: Proton Donor-Acceptor O-O Distance R and COOC Dihedral Angle θ for the Series of Substituted Phenoxyl/Phenol Systems substituent
R (Å)
θ
p-F m-F o-F p-CN m-CN o-CN p-NO2 m-NO2 o-NO2 p-OH m-OH o-OH p-NH2 m-NH2 o-NH2
2.40 2.40 2.40 2.39 2.40 2.40 2.39 2.40 2.40 2.39 2.40 2.40 2.40 2.41 2.41 2.40
138 138 148 48 152 149 180 156 144 154 136 135 140 130 141 129
states with the appropriate coupling VET leads to the CASSCF electronically adiabatic ground- and excited-state curves. Since this reaction is electronically nonadiabatic, the vibronic coupling is the product of the electronic coupling between the diabatic states I and II and the overlap of the reactant and product proton vibrational wave functions corresponding to these diabatic states. This overlap is depicted in Figure 2b. Figure 3 depicts the two localized NEO-HF nuclear-electronic wave functions and the delocalized NEO-NOCI nuclearelectronic wave functions for the unsubstituted phenoxyl/phenol system. For the localized NEO-HF states, the highest-energy doubly occupied electronic molecular orbital is localized mainly on the conjugated π system of the donor (I) or acceptor (II) ring, and the proton molecular orbital is localized near the donor oxygen (I) or acceptor oxygen (II). The delocalized NEO-NOCI states are mixtures of these two localized states. For both the ground- and first-excited NEO-NOCI vibronic states, the highest-energy doubly occupied electronic molecular orbital is delocalized over the conjugated π systems of both rings. The ground NEO-NOCI vibronic state corresponds to a symmetric bilobal proton molecular orbital, and the first-excited NEONOCI vibronic state corresponds to an antisymmetric bilobal
TABLE 2: Vibronic Couplings for the Unsubstituted Phenoxyl/Phenol System R (Å)
a V(na) DA (1D)
b V(na) DA (3D)
c V(NEO) DA
2.30 2.40
33.4 4.47
15.3 2.86
17.7 1.10
a One-dimensional grid-based nonadiabatic vibronic coupling in cm-1. b Three-dimensional grid-based nonadiabatic vibronic coupling in cm-1. c Three-dimensional NEO vibronic coupling in cm-1.
proton molecular orbital. The NEO-NOCI vibronic coupling is half of the splitting between the energies corresponding to the delocalized NEO-NOCI ground- and first-excited vibronic states, which is equivalent to the expression in eq 4 for this symmetric system. The calculated vibronic couplings for the unsubstituted phenoxyl/phenol system are given for R ) 2.30 and 2.40 Å in Table 2. These vibronic couplings were calculated with the onedimensional and three-dimensional grid-based nonadiabatic methods and the three-dimensional NEO-NOCI method. For all methods, the vibronic coupling is significantly smaller for the greater proton donor-acceptor distance. Moreover, the threedimensional treatment of the transferring hydrogen nucleus decreases the vibronic coupling by approximately a factor of 2. The vibronic couplings calculated with the three-dimensional grid-based nonadiabatic method and the NEO-NOCI method are in excellent agreement. The absolute differences are ∼2 cm-1, and the percentage difference is ∼10% for the shorter distance. This agreement provides a level of validation for both methods. Table 3 provides the vibronic couplings for the substituted phenoxyl/phenol systems calculated with the one-dimensional grid-based nonadiabatic method and the three-dimensional NEONOCI method. The effects of the three-dimensional versus the one-dimensional treatment of the transferring hydrogen nucleus may vary for the different substituents; therefore, a global comparison between the two methods for all systems is not warranted. For each substituent, however, the trends in the vibronic couplings for substitutions at the para, meta, and ortho positions are the same for the one-dimensional grid-based nonadiabatic method and the three-dimensional NEO-NOCI
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TABLE 3: Vibronic Couplings and Associated Quantities for the Series of Substituted Phenoxyl/Phenol Systems at R ) 2.30 Å substituent
barriera
VETb
overlapc
d V(na) DA
e V(NEO) DA
p-F m-F o-F p-CN m-CN o-CN p-NO2 m-NO2 o-NO2 p-OH m-OH o-OH p-NH2 m-NH2 o-NH2 p-NH2f m-NH2f o-NH2f
12.7 12.7 13.3 13.2 12.0 13.8 11.7 13.4 14.3 12.5 10.9 13.0 12.4 8.38 7.84 7.64 10.8 9.89 9.69
836 869 674 1094 579 790 551 675 844 476 789 771 696 702 436 539 623 390 495
4.00 × 10-2 3.85 × 10-2 3.85 × 10-2 3.23 × 10-2 5.03 × 10-2 3.33 × 10-2 5.55 × 10-2 4.90 × 10-2 3.08 × 10-2 4.88 × 10-2 5.93 × 10-2 3.83 × 10-2 4.61 × 10-2 9.74 × 10-2 1.37 × 10-1 1.34 × 10-1 4.58 × 10-2 7.37 × 10-2 7.24 × 10-2
33.4 33.5 26.0 35.3 29.1 26.3 30.6 33.1 26.0 23.2 46.8 29.5 32.1 68.3 59.9 72.0 28.5 28.8 35.9
17.7 17.6 16.1 18.7 18.4 17.0 20.4 17.6 16.4 16.6 20.3 16.0 17.3 24.3 32.3 41.0 6.25 10.3 13.7
a Barrier in kcal/mol for the CASSCF ground electronic state. Electronic coupling, which is half of the splitting in cm-1 between the ground and excited CASSCF electronic states. c Franck-Condon overlap between the reactant and product proton vibrational wave functions for the diabatic electronic states. d One-dimensional grid-based nonadiabatic vibronic coupling in cm-1. e Three-dimensional NEO vibronic coupling in cm-1. f These values were calculated for R ) 2.35 Å.
b
method, with the exception of NO2 and NH2 discussed below. Note that the variation of the vibronic coupling with ring position does not exhibit any consistent general trends. Moreover, the impact of the substituents on the magnitude of the vibronic coupling is rather modest for all substituents except for NH2, which significantly increases the vibronic coupling. The minor discrepancies between the two methods for NO2 and NH2 have been analyzed. In the case of NO2, substitutions at the meta and ortho positions lead to similar vibronic couplings, and the order is different for the two methods. This discrepancy may arise from numerical error or differences between the one-dimensional and three-dimensional treatments of the hydrogen nucleus. In the case of NH2, the order of the vibronic couplings for substitutions at the para and meta positions is interchanged. Note that the magnitude of the vibronic coupling is significantly larger and the proton transfer barriers are significantly lower for the NH2-substituted systems than those for all other systems studied. We used the formalism devised by Georgievskii and Stuchebrukhov22 to determine that these systems are still predominantly in the electronically nonadiabatic proton tunneling regime for which eq 3 is valid, suggesting that the discrepancy may be due to limitations of the NEO-NOCI method. Previously, we showed that the current implementation of the NEO-NOCI method is accurate in the relatively deep tunneling regime and becomes less accurate for systems with lower proton transfer barriers and greater vibronic couplings.31 To enable a comparison between these two methods in a more appropriate regime for the NH2-substituted systems, we calculated the vibronic couplings at R ) 2.35 Å for these systems. As shown in Table 3, the trends for the onedimensional grid-based nonadiabatic method and the threedimensional NEO-NOCI method are the same at this distance. Overall, the qualitative agreement between the grid-based nonadiabatic and NEO-NOCI methods provides a level of validation for both methods. The results indicate that a threedimensional treatment of the hydrogen nucleus is desirable for
the calculation of quantitatively accurate vibronic couplings. We emphasize that the NEO-NOCI method is significantly more computationally efficient and straightforward to implement than the three-dimensional grid-based nonadiabatic method. As mentioned above, however, the current implementation of the NEO-NOCI method has been shown to be accurate in the relatively deep tunneling regime but less accurate for lower proton transfer barriers. These difficulties arise mainly from the inadequate treatment of electron-proton correlation in the localized NEO-HF wave functions that are used as the basis states for the two-state NEO-NOCI calculations. Currently, we are extending the NEO-NOCI method to improve its accuracy and increase its range of applicability. In particular, we are developing methods that include additional electron-proton and electron-electron correlation. We analyzed the impact of substituents on the vibronic coupling in the context of the electron-withdrawing and electrondonating nature of the substituents. Figure 4a illustrates that the electron-donating substituents significantly increase the vibronic coupling, whereas the electron-withdrawing substituents slightly decrease the vibronic coupling relative to the unsubstituted phenoxyl/phenol system. Thus, if all other aspects of the reaction are the same, then electron-donating groups will tend to increase the PCET rate, while electron-withdrawing groups will tend to decrease the PCET rate. As indicated by the values given in Table 3, the electronic couplings alone do not reflect these trends. These data illustrate that the FranckCondon overlap between the reactant and product proton vibrational wave functions plays an important role in dictating these trends. For example, the NH2 substituent is the strongest electron-donating group studied, and these systems exhibit the largest vibronic couplings. All of the NH2-substituted systems have larger Franck-Condon overlaps than the other systems, but the electronic couplings span a wide range. Thus, calculation of the full vibronic coupling is necessary to predict physically meaningful trends. We also analyzed the correlations between the vibronic coupling and physical properties such as the Hammett constant, bond dissociation enthalpy (BDE), ionization potential, redox potential, and pKa. The correlations between the vibronic coupling and these physical quantities are depicted in Figure 4b-f. This figure indicates that negative Hammett constants correspond to higher vibronic couplings, while positive Hammett constants correspond to similar or slightly lower vibronic couplings relative to the unsubstituted phenoxyl/phenol system. Moreover, lower BDEs, ionization potentials, and redox potentials, as well as higher pKa values, tend to correspond to higher vibronic couplings relative to the unsubstituted phenoxyl/phenol system. The impact of electron-donating groups on the physical properties of the phenol molecule is well understood. Electrondonating groups hinder the removal of the proton from the phenol, therefore decreasing the Hammett constant and increasing the pKa. Electron-donating groups facilitate the removal of an electron from the phenol, therefore decreasing the ionization potential. Similarly, electron-donating groups hinder the reduction of the phenol, therefore decreasing the redox potential. Finally, electron-donating groups also decrease the BDE for hydrogen atom removal due to stabilization of the phenoxyl radical and destabilization of the phenol.44-47 Note that hydrogen atom removal, rather than single-proton or electron removal, is most relevant to the phenoxyl/phenol self-exchange reaction. The direct connection of these physical quantities to the vibronic coupling for the phenoxyl/phenol self-exchange reaction
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Figure 4. Vibronic couplings of substituted phenoxyl/phenol systems (a) arranged from highest to lowest in magnitude, (b) as functions of Hammett constant σm,p,49 (c) as functions of bond dissociation enthalpy (BDE),50 (d) as functions of ionization potential (IP),50 (e) as functions of redox potential (E) at pH ) 7,51 and (f) as functions of pKa.52,53 The vibronic couplings are calculated with the one-dimensional grid-based nonadiabatic method at R ) 2.30 Å. Electron-donating substituents are denoted with open diamonds, electron-withdrawing substituents are denoted with filled diamonds, and the unsubstituted phenoxyl/phenol system is denoted with a red filled square. The m-OH substituent is considered to be electron withdrawing because of its positive Hammett constant.
is not straightforward because all of these quantities are determined for the isolated phenol. The phenoxyl/phenol selfexchange reaction involves the net transfer of a hydrogen atom from the phenol to the phenoxyl within a hydrogen-bonded phenoxyl/phenol complex. A relevant phenol property is the BDE for hydrogen atom removal because both an electron and a proton are being removed from the phenol, as is the case in the self-exchange reaction. A smaller BDE is expected to be associated with a lower hydrogen transfer barrier, which would lead to a larger Franck-Condon overlap between the reactant and product proton vibrational wave functions and hence a greater vibronic coupling. Indeed, three of the substituted species with lower BDEs (p-NH2, p-OH, and m-NH2) have significantly greater Franck-Condon factors and vibronic couplings than those of the other species. In general, however, the vibronic couplings are dictated by a complex interplay among the various electronic and nuclear interactions.
IV. Conclusions In this paper, we calculated the vibronic couplings for a series of substituted phenoxyl/phenol self-exchange reactions. The vibronic couplings significantly impact the rates and kinetic isotope effects, as well as the temperature dependences, of general PCET reactions. Thus, the development and benchmarking of efficient methods for calculating these vibronic couplings is essential for the study of PCET reactions. The quantitative agreement between the three-dimensional grid-based nonadiabatic vibronic coupling and the NEO-NOCI vibronic coupling for the unsubstituted phenoxyl/phenol system provides a level of validation for both methods. The qualitative agreement in the trends for the substituted phenoxyl/phenol systems predicted by these two methods provides additional validation. We analyzed the underlying physical principles dictating the impact of substituents on the vibronic coupling for the phenoxyl/ phenol self-exchange reaction. Our analysis indicates that
Vibronic Coupling for Phenoxyl/Phenol Self-Exchange electron-donating groups enhance the vibronic coupling, and electron-withdrawing groups attenuate the vibronic coupling. Thus, if all other aspects of the reaction are the same, then electron-donating groups will increase the PCET rate, while electron-withdrawing groups will decrease the PCET rate. Our calculations illustrate that the electronic couplings alone do not reflect these trends, but rather the Franck-Condon overlap between the reactant and product proton vibrational wave functions must also be considered. Furthermore, we studied the correlations between the vibronic couplings and various physical properties of the phenol. We found that negative Hammett constants correspond to higher vibronic couplings, while positive Hammett constants correspond to similar or slightly lower vibronic couplings relative to the unsubstituted phenoxyl/phenol system. In addition, lower BDEs, ionization potentials, and redox potentials, as well as higher pKa values, tend to correspond to higher vibronic couplings relative to the unsubstituted phenoxyl/ phenol system. The trends observed in our calculations enable the prediction of the impact of general substituents on the vibronic coupling, and hence the rate, for the phenoxyl/phenol self-exchange reaction. Such predictions can be tested experimentally and may be extended to related systems. Moreover, the fundamental physical insights obtained from these studies are applicable to other PCET systems. Acknowledgment. We thank Alexander Soudackov and Mike Pak for helpful discussions regarding this work. This work was supported by NSF Grant CHE-05-01260, AFOSR Grant No. FA9550-07-1-0143, and NIH Grant GM56207. Supporting Information Available: Structures and energies for optimized geometries of substituted phenoxyl/phenol systems; details for the valence bond models used to fit the CASSCF electronically adiabatic potential energy curves along the proton coordinate; and one-dimensional grid-based nonadiabatic vibronic couplings and associated quantities for the transition state structures of the substituted phenoxyl/phenol systems. This material is available free of charge via the Internet at http://pubs.acs.org. References and Notes (1) Costentin, C.; Robert, M.; Saveant, J.-M. J. Electroanal. Chem. 2006, 588, 197. (2) Babcock, G. T.; Barry, B. A.; Debus, R. J.; Hoganson, C. W.; Atamian, M.; McIntosh, L.; Sithole, I.; Yocum, C. F. Biochemistry 1989, 28, 9557. (3) Okamura, M. Y.; Feher, G. Annu. ReV. Biochem. 1992, 61, 861. (4) Tommos, C.; Tang, X.-S.; Warncke, K.; Hoganson, C. W.; Styring, S.; McCracken, J.; Diner, B. A.; Babcock, G. T. J. Am. Chem. Soc. 1995, 117, 10325. (5) Hoganson, C. W.; Babcock, G. T. Science 1997, 277, 1953. (6) Hoganson, C. W.; Lydakis-Simantiris, N.; Tang, X.-S.; Tommos, C.; Warncke, K.; Babcock, G. T.; Diner, B. A.; McCracken, J.; Styring, S. Photosynth. Res. 1995, 47, 177. (7) Blomberg, M. R. A.; Siegbahn, P. E. M.; Styring, S.; Babcock, G. T.; Akermark, B.; Korall, P. J. Am. Chem. Soc. 1997, 119, 8285. (8) Babcock, G. T.; Wikstrom, M. Nature 1992, 356, 301. (9) Malmstrom, B. G. Acc. Chem. Res. 1993, 26, 332. (10) Siegbahn, P. E. M.; Eriksson, L.; Himo, F.; Pavlov, M. J. Phys. Chem. B 1998, 102, 10622. (11) Blow, D. M. Acc. Chem. Res. 1976, 9, 145. (12) Ramaswamy, S.; Eklund, H.; Plapp, B. V. Biochemistry 1994, 33, 5230. (13) Ren, X. L.; Tu, C. K.; Laipis, P. J.; Silverman, D. N. Biochemistry 1995, 34, 8492.
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