Proton-Coupled Electron Transfer versus Hydrogen Atom Transfer...
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ARTICLE pubs.acs.org/JPCA
Proton-Coupled Electron Transfer versus Hydrogen Atom Transfer: Generation of Charge-Localized Diabatic States Andrew Sirjoosingh and Sharon Hammes-Schiffer* Department of Chemistry, 104 Chemistry Building, Pennsylvania State University, University Park, Pennsylvania 16802, United States
bS Supporting Information ABSTRACT: The distinction between proton-coupled electron transfer (PCET) and hydrogen atom transfer (HAT) mechanisms is important for the characterization of many chemical and biological processes. PCET and HAT mechanisms can be differentiated in terms of electronically nonadiabatic and adiabatic proton transfer, respectively. In this paper, quantitative diagnostics to evaluate the degree of electron-proton nonadiabaticity are presented. Moreover, the connection between the degree of electron-proton nonadiabaticity and the physical characteristics distinguishing PCET from HAT, namely, the extent of electronic charge redistribution, is clarified. In addition, a rigorous diabatization scheme for transforming the adiabatic electronic states into charge-localized diabatic states for PCET reactions is presented. These diabatic states are constructed to ensure that the first-order nonadiabatic couplings with respect to the onedimensional transferring hydrogen coordinate vanish exactly. Application of these approaches to the phenoxyl-phenol and benzyltoluene systems characterizes the former as PCET and the latter as HAT. The diabatic states generated for the phenoxyl-phenol system possess physically meaningful, localized electronic charge distributions that are relatively invariant along the hydrogen coordinate. These diabatic electronic states can be combined with the associated proton vibrational states to generate the reactant and product electron-proton vibronic states that form the basis of nonadiabatic PCET theories. Furthermore, these vibronic states and the corresponding vibronic couplings may be used to calculate rate constants and kinetic isotope effects of PCET reactions.
1. INTRODUCTION Proton-coupled electron transfer (PCET) reactions are ubiquitous throughout chemistry and biology.1-5 Concerted PCET reactions involve the transfer of an electron and a proton in a single step without a stable intermediate. The relative time scales of the electrons, transferring proton, and solvent/protein environment dictate the appropriate theoretical treatment and form of the rate constant expression. Typically, concerted PCET reactions are vibronically nonadiabatic because the quantum subsystem comprised of the electrons and transferring proton does not respond instantaneously to the solvent and protein motions. In this regime, PCET reactions are described in terms of nonadiabatic transitions between the reactant and product diabatic vibronic states, where the reactant (product) diabatic vibronic states correspond to the electron and proton localized on their donors (acceptors).4,6 The vibronic coupling is defined as the Hamiltonian matrix element between a pair of reactant and product diabatic vibronic states. In the vibronically nonadiabatic regime, the vibronic coupling is much less than the thermal energy.4,7,8 The golden rule formalism has been used to derive nonadiabatic rate constant expressions, in which each term is proportional to the square of the relevant vibronic coupling.9,10 Even within this regime, the proton transfer may be electronically adiabatic, where the electrons respond instantaneously to the proton, or electronically r 2011 American Chemical Society
nonadiabatic, where the response of the electrons is slower than the proton motion. The form of the vibronic coupling differs in these two limits.11,12 Although hydrogen atom transfer (HAT) reactions may be viewed as a subset of concerted PCET reactions, often a distinction between HAT and PCET reactions is useful for the description of chemical reactions. Traditionally, HAT reactions are defined as the simultaneous transfer of an electron and proton between the same donor and acceptor without significant molecular charge redistribution, corresponding to small solvent reorganization energies. In contrast, PCET reactions typically involve different donors and acceptors for the electron and proton and are associated with significant molecular charge redistribution, corresponding to larger solvent reorganization energies. Distinguishing between the two types of mechanisms is important because they may require different rate constant expressions. In particular, due to the negligible solvent reorganization, HAT reactions may require an explicit dynamical treatment of intramolecular solute modes rather than the Marcus theory13 description in terms of collective solvent coordinates typically applied to Received: November 24, 2010 Revised: January 12, 2011 Published: February 25, 2011 2367
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The Journal of Physical Chemistry A PCET reactions. In the literature, HAT and PCET mechanisms have been differentiated by the analysis of frontier molecular orbitals14 and topographical characteristics of the potential energy surfaces.15 We have formulated a more quantitative diagnostic for distinguishing between HAT and PCET in terms of electronically adiabatic and nonadiabatic proton transfer, respectively.12 The degree of electron-proton nonadiabaticity can be evaluated with a semiclassical formalism, where an adiabaticity parameter is defined as the ratio of the proton tunneling time and the electronic transition time.11 In the electronically adiabatic regime, the electronic transition time is much shorter than the proton tunneling time; therefore, the proton transfer occurs on the electronically adiabatic ground state surface. In the electronically nonadiabatic regime, the proton tunneling time is much shorter than the electronic transition time, and excited electronic states are involved. Our application of this semiclassical formalism to the selfexchange reactions in the phenoxyl-phenol and benzyl-toluene systems revealed that the former is electronically nonadiabatic, corresponding to PCET, while the latter is electronically adiabatic, corresponding to HAT.12 These mechanistic insights were consistent with prior analyses of the frontier molecular orbitals for these systems.14 Although the semiclassical approach is physically appealing, our previous implementation required fitting of the diabatic potential energy curves, which were represented by analytical functional forms, to electronic structure calculations of the adiabatic states. A more rigorous, well-defined method for constructing the diabatic states would be useful within this approach and, more generally, for the calculation of nonadiabatic PCET rate constants. Numerous schemes have been developed for constructing diabatic states for electron transfer reactions. Mathematically, diabatic electronic states can be defined as states for which the derivative couplings vanish at all possible nuclear configurations.16-38 Physically, the character of a diabatic state (i.e., its electronic charge density) does not change significantly as the nuclei move. For electron transfer, the two relevant diabatic states typically correspond to the electron localized on the donor and the acceptor, respectively (i.e., the reactant and product electronic states). A variety of approaches have been developed for constructing such charge-localized diabatic states for electron transfer reactions. These approaches include the minimization of derivative couplings calculated in an adiabatic basis,16,17 block diagonalization methods,18,20,24 the generalized Mulliken-Hush method23,24 and extensions with Boys localization,32,34 constrained density functional theory (DFT),30,31 and valence bond theory combined with molecular orbital or DFT methods.35,36 To our knowledge, these electronic structure approaches for constructing diabatic states have not yet been applied to PCET reactions. In this paper, we examine fundamental issues concerning the evaluation of electron-proton nonadiabaticity, the distinction between the HAT and PCET mechanisms, and the generation of charge-localized diabatic states for PCET reactions. In particular, we propose quantitative diagnostics of electron-proton nonadiabaticity that are directly accessible from standard quantum chemistry calculations. We also clarify the connection between the degree of electron-proton nonadiabaticity and the physical characteristics distinguishing HAT from PCET. Most importantly, we develop a rigorous theoretical approach to generate charge-localized reactant and product diabatic states for PCET
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reactions. In many PCET reactions, movement of the proton can induce the electron transfer reaction. Thus, we utilize the transferring hydrogen coordinate as a tool to construct chargelocalized electronic states that are diabatic with respect to this hydrogen coordinate. Specifically, we generate the charge-localized diabatic states with an adiabatic-to-diabatic transformation matrix16,17 defined to ensure that the first-order nonadiabatic coupling with respect to the transferring hydrogen coordinate vanishes. An outline of the paper is as follows. Section 2A summarizes the theoretical formalism for calculating the electron-proton vibronic states in the double adiabatic representation. Section 2B presents a theoretical approach for generating electronic states that are rigorously diabatic with respect to the onedimensional hydrogen coordinate, as well as the generation of the associated reactant and product diabatic vibronic states relevant to PCET reactions. Section 2C describes the quantitative diagnostics of electron-proton nonadiabaticity that arise from the theoretical concepts of the first two subsections. The computational methods implemented to apply these theoretical approaches to the phenoxyl-phenol and benzyl-toluene systems are discussed in section 3, and the results of these applications are presented in section 4. Specifically, section 4A describes the generation of the electronically adiabatic and diabatic potential energy curves and the calculation of the quantitative diagnostics for determining the degree of electron-proton nonadiabaticity. Section 4B presents an analysis of the adiabatic electronic states to clarify the connection between electron-proton nonadiabaticity and the physical characteristics of HAT and PCET. Section 4C presents an analysis of the diabatic electronic states for the PCET reaction to illustrate that these diabatic states exhibit physically meaningful, invariant charge localization along the transferring hydrogen coordinate. Concluding remarks are provided in section 5.
2. THEORY A. Double Adiabatic Representation. Consider a system divided into Ne electrons, Np protons, and Ns slow nuclei with coordinates re, rp, and R and masses me, mp, and {MI}, respectively, and with potential energy V(re,rp,R). The Hamiltonian for the “fast” degrees of freedom (i.e., the electron-proton subsystem) is Np X p2 2 r 0 þ He ð1Þ Hq ¼ 2mp i i0 ¼ 1
where the electronic Hamiltonian is He ¼ -
Ne X p2 2 r þ V ðre , rp , RÞ 2me i i¼1
ð2Þ
For fixed R, the eigenfunctions Φk(re,rp;R) of Hq are calculated by solving Hq Φk ðre , rp ; RÞ ¼ Ek ðRÞΦk ðre , rp ; RÞ
ð3Þ
Often, this equation is analyzed in the context of a double adiabatic basis set. For this purpose, the adiabatic electronic states for fixed (rp,R) are determined by solving He ψi ðre ; rp , RÞ ¼ εi ðrp , RÞψi ðre ; rp , RÞ 2368
ð4Þ
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The proton vibrational states for fixed R and adiabatic electronic state i are obtained by solving 0 1 Np 2 X p @r2i0 þ εi ðrp , RÞAφðiÞ μ ðrp ; RÞ 2m p 0 i ¼1 ðiÞ ¼ εðiÞ μ ðRÞφμ ðrp ; RÞ
ð5Þ
where ε(i) μ (R) is the energy of the double adiabatic electronproton vibronic state (i,μ). The double adiabatic basis functions {ζiμ(re,rp; R)} are defined as products of the adiabatic electronic and proton vibrational wave functions ζiμ ðre , rp ; RÞ ¼ ψi ðre ; rp , RÞφðiÞ μ ðrp ; RÞ
ð6Þ
The double adiabatic approximation assumes the Born-Oppenheimer separation between the slow nuclei and the fast degrees of freedom and between the electrons and the protons. In other words, the electrons and protons are assumed to respond instantaneously to motions of the slow nuclei, and the electrons are assumed to respond instantaneously to motions of the protons. In this limit, the eigenfunctions of Hq are approximated as Φk(re,rp;R) ≈ ζiμ(re,rp;R) with eigenvalues Ek(R) ≈ ε(i) μ (R). In PCET reactions, often the Born-Oppenheimer separation between electrons and quantum protons is not valid. In this case, eq 3 can be solved by expanding the eigenfunctions in the double adiabatic basis given by eq 6 X ckiμ ζiμ ðre , rp ; RÞ ð7Þ Φk ðre , rp ; RÞ ¼ i, μ Following the standard linear variational procedure, we express the Hamiltonian Hq in the double adiabatic basis and diagonalize the resulting Hamiltonian matrix to obtain the eigenfunctions and eigenvalues of eq 3. The matrix elements in this basis are given by Hiμ, jν ¼ Æζiμ jHq jζjν æep ¼ δij δμν εðiÞ μ ðRÞ -
p2 ðiÞ ðepÞ rr φðjÞ æ Æφ jd mp μ ij 3 p ν p
p2 ðiÞ ðepÞ ðjÞ Æφ jg jφν æp 2mp μ ij
ð8Þ
where
coupling terms vanish. In this case, the Born-Oppenheimer separation between the electrons and protons is valid, and the proton motion is electronically adiabatic (i.e., the electrons respond instantaneously to motions of the protons). Although the first-order nonadiabatic coupling terms can be calculated in many quantum chemistry programs, such as GAMESS,39 the second-order nonadiabatic coupling terms are more problematic because they require second derivatives.40-42 We have found that the second-order nonadiabatic couplings are often relatively large for PCET reactions and that neglecting these terms introduces significant asymmetry into the Hamiltonian matrix. Thus, we derived an alternative expression from eq 8 using the chain rule, integration by parts, and insertion of the identity operator p2 ðepÞ rr φðjÞ æ ðRÞ ½ÆφðiÞ jd Hiμ, jν ¼ δij δμν εðiÞ μ 2mp μ ij 3 p ν p ðepÞ
þ ÆφνðjÞ jdji
ðiÞ 3 rrp φμ æp þ
p2 X ðiÞ ðepÞ ðepÞ ðjÞ Æφμ jdki 3 dkj jφν æp 2mp k ð11Þ
where the summation in the last term is over all electronic states k. Assuming that the electronic basis set is complete, this expression for the matrix elements is rigorously identical to eq 8 in that no terms have been neglected, but it avoids the calculation of secondorder nonadiabatic coupling terms. Instead, this expression requires the calculation of first-order nonadiabatic coupling vectors between the states of interest, i and j, and all other electronic states, k. Typically, we can assume that for k > max{i,j}, the terms in this summation will be negligible compared to the other terms in eq 11 because the states k are much higher in energy. The main advantage of eq 11 is that it enables the calculation of vibronic state energies including both first-order and secondorder nonadiabatic couplings without the explicit calculation of the second derivative terms. A brief derivation of eq 11, as well as a comparison of implementations based on eqs 8 and 11, is provided in the Appendix. B. Diabatic Representation. As discussed in the Introduction, the theoretical description of PCET reactions is often based on the diabatic representation. In this subsection, we develop the methodology to obtain the relevant diabatic electronic states for PCET reactions. For N electronic states, we define ψB to be a column vector of the N electronic eigenfunctions {ψi(re;rp,R)} of eq 4. Then, we define an N N transformation matrix A(rp;R) such that f
ðepÞ dij ðrp ; RÞ
¼ Æψi jrrp ψj æe 8 > < Æψi jrrp He jψj æe εj - εi ¼ > : 0
ðepÞ
ð12Þ
where ξB is a column vector of functions satisfying the condition i 6¼ j i¼j
Æξi jrrp ξj æe ¼ 0
ð9Þ
is the first-order nonadiabatic coupling vector and gij ðrp ; RÞ ¼ Æψi jr2rp ψj æe
f
ξ ðre ; rp , RÞ ¼ Aðrp ; RÞψ ðre ; rp , RÞ
ð10Þ
is the second-order nonadiabatic coupling term. Here the notation Æ 3 3 3 æe and Æ 3 3 3 æp denotes integration over electronic coordinates re or proton coordinates rp, respectively. The Hamiltonian matrix expressed in the double adiabatic basis is diagonal only when the first- and second-order nonadiabatic
for all i, j
ð13Þ
Thus, the transformed electronic states {ξi(re;rp,R)} satisfy the standard definition of diabatic states with respect to the proton coordinate rp. In ref 16, Baer derived the conditions on the matrix A for the general N-state case and showed that a transformation providing diabatic states that exactly satisfy eq 13 must satisfy the curl condition when the first-order nonadiabatic coupling vector is of dimension greater than unity. For a one-dimensional proton coordinate rp, however, we can form exact diabatic electronic states using the diabatization scheme outlined by Baer.16 For simplicity, we consider the case 2369
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of N = 2 electronic states, but the extension to more electronic states is possible.29 As derived previously, the matrix A must satisfy DAt þ TAt ¼ 0 ð14Þ Drp where the matrix T has elements ðepÞ Tij ðr p ; RÞ ¼ dij ðr p ; RÞ
ð15Þ
Here, d(ep) is given by eq 9, where rrp is replaced by ∂/∂rp. ij Because eq 14 implies that A is orthogonal, we can express this matrix as 0 1 cos γ -sin γ A ð16Þ Aðrp ; RÞ ¼ @ sin γ cos γ To satisfy eq 14, γ(rp;R) must satisfy Dγ ðepÞ þ d12 ¼ 0 Drp The solution of eq 17 is of the form Z γðrp ; RÞ ¼ γðr0 ; RÞ -
ð17Þ
rp ðepÞ
d12 ðr; RÞ dr
ð18Þ
r0
where γ(r0;R) is an additive constant that must be specified at some point rp = r0. The diabatic potential energy matrix is given by W=AUA-1, where Uij = εi(rp,R)δij is the adiabatic potential energy matrix. Substituting eq 16 into this expression for W, the diabatic potential energy matrix elements are expressed as
and diagonalization of this Hamiltonian matrix provides the vibronic eigenfunctions and eigenvalues. This matrix diagonalization procedure leads to the same vibronic eigenfunctions and eigenvalues of eq 3 as those obtained using the double adiabatic basis. C. Quantitative Diagnostics of Electron-Proton Nonadiabaticity. As discussed in the Introduction, PCET reactions may exhibit electronically adiabatic or nonadiabatic proton transfer, depending on the relative time scales of the electron and proton motions. In the electronically adiabatic regime, the electrons respond instantaneously to the proton motion, and the system remains on the electronic ground state, while in the electronically nonadiabatic regime, the response of the electrons is slower than the proton motion, and excited electronic states are involved. The theoretical framework developed in the previous two subsections provides quantitative diagnostics of the electron-proton nonadiabaticity. For simplicity, we discuss the diagnostics in terms of only two electronic states (i.e., the ground and first excited adiabatic electronic states), although the extension to more electronic states is straightforward. We also consider only the two lowest-energy vibronic states and neglect mixing with higher-energy vibronic states. The extension to other pairs of vibronic states is also straightforward. A useful quantity for characterizing PCET systems is the vibronic coupling, which was defined in the Introduction to be the Hamiltonian matrix element between a pair of reactant and product diabatic electron-proton vibronic states. For a symmetric system, the vibronic coupling between the ground reactant and product diabatic vibronic states is VDA ¼
W11 ðrp , RÞ ¼ ε1 ðrp , RÞ cos2 γðrp ; RÞ þ ε2 ðrp , RÞ sin2 γðrp ; RÞ W22 ðrp , RÞ ¼ ε1 ðrp , RÞ sin2 γðrp ; RÞ þ ε2 ðrp , RÞ cos2 γðrp ; RÞ W12 ðrp , RÞ ¼ ½ε1 ðrp , RÞ - ε2 ðrp , RÞ sin ½2γðrp ; RÞ=2 ð19Þ Here, W11(rp,R) and W22(rp,R) are the diabatic electronic energies, and W12(rp,R) is the diabatic electronic coupling. Analogous to the double adiabatic vibronic basis set defined in eq 6, we define a diabatic vibronic basis set. The proton vibrational states for fixed R and diabatic electronic state i are obtained by solving ! p 2 D2 ~ ðiÞ þ Wii ðrp , RÞ j ε ðiÞ j ðiÞ μ ðrp ; RÞ ¼ ~ μ ðRÞ~ μ ðrp ; RÞ 2mp Drp2 ð20Þ where ~ε(i) μ (R) is the energy of the diabatic electron-proton vibronic state (i,μ). The diabatic vibronic basis functions {ζ~iμ(re,rp;R)} are defined as products of the diabatic electronic wave functions and associated proton vibrational wave functions ~ ζ iμ ðre , rp ; RÞ ¼ ξi ðre ; rp , RÞ~ j ðiÞ μ ðrp ; RÞ
~ iμ, jν ¼ δij δμν~ε ðiÞ j ðiÞ j ðjÞ H μ ðRÞ þ ð1 - δij ÞÆ~ μ jWij j~ ν æp
ð22Þ
ð23Þ
where the two lowest-energy “exact” vibronic state energies E1 and E2 may be calculated by solution of eq 3 in either the double adiabatic or the diabatic vibronic basis. In the double adiabatic representation, the degree of electron-proton nonadiabaticity is indicated by the magnitude of the off-diagonal terms in the Hamiltonian matrices defined equivalently by eqs 8 and 11. Another related diagnostic within the double adiabatic representation may also be useful. For electronically adiabatic proton transfer, the two lowestenergy exact vibronic states Φ1 and Φ2 possess predominantly ground electronic state character, while for electronically nonadiabatic proton transfer, these states possess more excited electronic state character. Thus, the degree of electronic nonadiabaticity for the proton transfer can be quantified by calculating the fraction of the excited electronic state character of each of these vibronic states. The contribution of the adiabatic electronic state i to vibronic state k is quantified from the coefficients in eq 7 as θi ðΦk Þ ¼
ð21Þ
Equation 3 can be solved by expanding the eigenfunctions in this diabatic vibronic basis. Expression of the Hamiltonian Hq in this basis leads to the matrix elements
E2 ðRÞ - E1 ðRÞ 2
X μ
ðckiμ Þ2
ð24Þ
An additional quantitative diagnostic of electron-proton nonadiabaticity is provided by comparison of the “full” vibronic coupling given in eq 23 with the vibronic coupling in the electronically nonadiabatic and double adiabatic limits. In the 2370
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electronically nonadiabatic limit, the vibronic coupling between the reactant and product diabatic vibronic states (1,μ) and (2,ν) is11,12 ðnaÞ
VDA ¼ Æ~ j ð1Þ j ð2Þ μ jW12 j~ ν æp
ð25Þ
This expression reduces to the familiar form of the diabatic electronic coupling multiplied by the Franck-Condon overlap between the reactant and product proton vibrational wave functions when the diabatic electronic coupling W12(rp,R) defined in eq 19 is independent of rp. In the double adiabatic limit, the vibronic coupling for a symmetric system is11,12 ðdadÞ
VDA
ð1Þ
¼
ð1Þ
ε2 ðRÞ - ε1 ðRÞ 2
ð26Þ
where ε(i) μ (R) are the energies of the double adiabatic electronproton vibronic states defined by eq 6. Thus, the double adiabatic vibronic coupling corresponds to the energy splitting between the lowest two proton vibrational states on the ground adiabatic electronic state.
3. COMPUTATIONAL METHODS We used the phenoxyl-phenol and benzyl-toluene self-exchange reactions as model systems to illustrate the theoretical concepts developed above. The transition state geometries for these systems were obtained from ref 14 using density functional theory (DFT) with the B3LYP functional43,44 and the 6-31G* basis set.45 In the present calculations, all nuclei except the transferring hydrogen atom were fixed at the transition state geometry. For each system, we calculated the two lowest-energy electronically adiabatic potential energy curves for hydrogen atom positions on a grid spanning the hydrogen donor-acceptor axis using the complete active space self-consistent-field (CASSCF) method. The CASSCF calculations were performed using the 6-31G* basis set and state-averaging over the ground and first excited electronic states with equal weighting. An active space of three electrons in six orbitals was chosen at each grid point to maintain the character of the orbitals along the hydrogen coordinate. Note that these calculations were performed at a relatively low level of theory because our objective was to utilize these systems as simple models to illustrate the qualitative features of the electronic wave functions and potential energy curves. We obtained the nonadiabatic coupling vectors directly from the CASSCF calculations. All of the CASSCF calculations were performed using the GAMESS electronic structure package.39 We calculated the proton vibrational wave functions corresponding to both the adiabatic and diabatic electronic potential energy surfaces by solving the one-dimensional Schr€odinger equations given in eqs 5 and 20, respectively. These calculations were performed with the Fourier grid Hamiltonian method46,47 using 128 grid points along the hydrogen donor-acceptor axis. For the construction of the vibronic Hamiltonians in the double adiabatic and diabatic vibronic bases given in eqs 11 and 22, respectively, 40 proton wave functions for each electronic state were included. To aid in the analysis of charge transfer properties, we calculated the dipole moments, atomic charges, and electrostatic potential maps as functions of the hydrogen coordinate for the adiabatic and diabatic electronic states. The properties of the adiabatic electronic states were calculated directly from the CASSCF wave functions with GAMESS. For the diabatic
Figure 1. Electronically adiabatic and diabatic potential energy curves as functions of the hydrogen coordinate for the (a) phenoxyl-phenol and (b) benzyl-toluene systems. The solid black curves are the ground and first excited state adiabatic energies ε1(rp,R) and ε2(rp,R), respectively, calculated with the CASSCF method. The dashed blue and red curves are the diabatic electronic energies W11(rp,R) and W22(rp,R), respectively, calculated from the expressions in eq 19 with γ(r0=0) = -π/4.
electronic states, we modified a local version of GAMESS to calculate these properties for the appropriate linear combination of configuration interaction (CI) states following the transformation given in eq 12. The atomic charges were obtained by fitting to the electrostatic potential calculated at points on the Connolly surface48 under the constraint of reproducing the total charge and dipole moment of the electronic state under consideration.49
4. RESULTS A. Electron-Proton Nonadiabaticity. Figure 1 depicts the electronically adiabatic and diabatic potential energy curves as functions of the hydrogen coordinate for the phenoxyl-phenol and benzyl-toluene systems. The solid black curves are the ground and first excited state adiabatic electronic energies ε1(rp,R) and ε2(rp,R), respectively, calculated with the CASSCF method. The blue and red dashed curves are the diabatic electronic energies W11(rp,R) and W22(rp,R), respectively, calculated from the expressions in eq 19. The electronically adiabatic curves are qualitatively similar to those presented in ref 12 but differ slightly due to use of the 6-31G* instead of the 6-31G basis set. The diabatic states in the previous work were obtained by fitting the electronically adiabatic curves with specific functional forms in an empirical valence bond potential.12 In the present work, however, the diabatic states were generated from the mathematical formulation given in section 2B, thereby avoiding the fitting procedure and the assumption of a particular functional form. In particular, the diabatic electronic wave functions were calculated with eq 12, and the corresponding diabatic electronic 2371
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Table 1. Vibronic Couplings in cm-1 for the Phenoxyl-phenol and Benzyl-toluene Systems Calculated with Various Methodsa system phenol toluene
VDA 7.3 20.9
V(dad) DA 27.6 21.0
V(na) DA 7.2 37.6
V(sc) DA 7.3 20.6
θ2(Φ1)
θ2(Φ2)
-4
2.21 10-3
-6
6.69 10-6
1.13 10
3.44 10
a
Methods: eq 23 after full basis set diagonalization to obtain VDA, eq 26 based on the double adiabatic approximation to obtain V(dad) DA , eq 25 with μ = ν = 1 in the electronically nonadiabatic limit to obtain V(na) DA , and the semiclassical formalism to obtain V(sc) DA . The last two columns provide the fractional contribution from the first excited adiabatic electronic state (i = 2) to the lowest two vibronic states (k = 1 and 2), as defined in eq 24.
Figure 3. Component of the dipole moment vector along the hydrogen donor-acceptor axis for the ground adiabatic electronic state of the phenoxyl-phenol (solid) and benzyl-toluene (dotted) systems. A positive (negative) dipole moment indicates a dipole moment vector pointing toward the acceptor (donor).
Figure 2. Component of the first-order nonadiabatic coupling vector, as defined in eq 9, between the ground and first excited adiabatic electronic states along the hydrogen donor-acceptor axis for the phenoxyl-phenol (solid) and benzyl-toluene (dotted) systems.
energies and couplings were calculated with eq 19 using the electronically adiabatic potential energy curves and the firstorder nonadiabatic coupling vectors. This procedure ensures that the diabatic states rigorously satisfy the diabaticity condition given in eq 13 along the one-dimensional hydrogen coordinate. To obtain physically meaningful diabatic states, we chose r0 = 0, corresponding to the transition state geometry, and set γ(r0) = -π/4 in eq 18. This choice ensures that the adiabatic electronic states mix maximally at the transition state geometry and that the magnitude of the diabatic electronic coupling, W12, is exactly half of the splitting between the adiabatic electronic energies at this geometry. Note that the diabatic electronic energy W11 has a minimum corresponding to the transferring hydrogen localized on the donor, while the diabatic electronic energy W22 has a minimum corresponding to the transferring hydrogen localized on the acceptor. Thus, these two diabatic states correspond to the physically meaningful reactant and product states in a PCET reaction. Table 1 presents the vibronic couplings calculated with a range of methods for the phenoxyl-phenol and benzyl-toluene systems. The full vibronic coupling, VDA, was calculated with eq 23, where the exact vibronic state energies Ek were calculated by solution of eq 3 through full basis set diagonalization of the Hamiltonian matrix. The value of VDA was found to be the same to within less than 0.1 cm-1 using either the double adiabatic vibronic basis (i. e., Hamiltonian matrix elements given by eq 11) or the diabatic vibronic basis (i.e., Hamiltonian matrix elements given by eq 22). In forming the double adiabatic vibronic Hamiltonian, we neglected the last term in eq 11 for i 6¼ j because all terms (ep) involving d(ep) k1 and dk2 vectors for k > 2 were found to be much (ep) smaller than d12 . More information about the double adiabatic
Figure 4. Partial charges determined from electrostatic potentialderived atomic charges for the ground adiabatic electronic state of the (a) phenoxyl-phenol and (b) benzyl-toluene systems. Partial charges are shown for the donor molecule (green), acceptor molecule (purple), and transferring hydrogen (gray).
vibronic Hamiltonian implementation is provided in the Appendix and Supporting Information. For comparison, we calculated the double adiabatic vibronic coupling, V(dad) DA , using eq 26 and the nonadiabatic vibronic coupling, V(na) DA , using eq 25. We also calculated the semiclassical vibronic coupling, V(sc) DA , using the methodology presented in ref 11 with parameters obtained from the exact diabatic electronic energies and couplings derived above. For both systems, the full vibronic coupling agrees well with the semiclassical vibronic coupling. For the phenoxyl-phenol system, the nonadiabatic vibronic coupling agrees well with the full vibronic coupling, and for the benzyl-toluene system, the double adiabatic vibronic coupling agrees well with the full vibronic coupling. These calculations confirm that the phenoxyl-phenol system is electronically nonadiabatic, while the benzyl-toluene system is electronically adiabatic. Thus, a comparison of the nonadiabatic and double adiabatic vibronic couplings to the full vibronic coupling 2372
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Figure 5. Electrostatic potential maps for the ground adiabatic electronic states corresponding to a density isosurface value of 0.005 for the reactant (top), transition state (middle), and product (bottom) positions of the transferring hydrogen for the (a) phenoxyl-phenol and (b) benzyl-toluene systems. Negatively and positively charged regions are indicated by red and blue coloring, respectively.
provides a diagnostic of the degree of electron-proton nonadabaticity, depending on which limiting expression agrees with the full vibronic coupling. Another quantitative measure of the degree of electronproton nonadiabaticity is the magnitude of the nonadiabatic coupling terms in eq 11. Figure 2 depicts the component of the first-order nonadiabatic coupling vector along the hydrogen donor-acceptor axis for the phenoxyl-phenol and benzyltoluene systems. The magnitude of the first-order nonadiabatic coupling is significantly greater for the phenoxyl-phenol system, supporting the identification of the phenoxyl-phenol and benzyltoluene mechanisms as electronically nonadiabatic and adiabatic, respectively. Furthermore, the degree of electron-proton nonadiabaticity can be quantified by the fraction of the excited adiabatic electronic state character of each vibronic state, as defined in eq 24. The fractional contributions of the first excited adiabatic electronic state (i = 2) to the lowest two vibronic states (k = 1 and 2) are given in Table 1. Note that the contributions of the first excited electronic state are significantly greater for the phenoxyl-phenol system than those for the benzyl-toluene system. These results are consistent with the benzyl-toluene reaction occurring on the ground adiabatic electronic state and the phenoxyl-phenol reaction involving excited electronic states. B. Distinction between PCET and HAT Mechanisms. To obtain further insight into the distinction between the PCET and HAT mechanisms, we examined the charge transfer properties for the ground adiabatic electronic states of the phenoxyl-phenol and benzyl-toluene systems. For this analysis, the donor (acceptor) is the molecule that is protonated when the hydrogen coordinate has negative (positive) values. The reactant corresponds to the protonated donor, and the product corresponds to the protonated acceptor (i.e., the two minima in Figure 1).
Figure 3 depicts the component of the dipole moment along the hydrogen donor-acceptor axis as the hydrogen is transferred. For both systems, the sign of this component of the dipole moment vector changes over the course of the reaction. The magnitude of the change in the dipole moment during the reaction, however, is significantly larger for the phenoxyl-phenol system than that for the benzyl-toluene system. This figure indicates a much greater change in the electronic charge distribution for the phenoxyl-phenol system. Figure 4 depicts the charges on the donor and acceptor molecules, as well as the transferring hydrogen, as the reaction proceeds. The charges on the donor and acceptor molecules switch signs during the reaction for the phenoxyl-phenol system but not for the benzyl-toluene system. Moreover, the changes in the charges on the donor and acceptor molecules during the reaction are much greater for the phenoxyl-phenol system than those for the benzyl-toluene system. In addition, the charge on the transferring hydrogen is more positive for the phenoxylphenol system (∼0.6 e) than that for the benzyl-toluene system (∼0.25 e). This figure illustrates that the electronic charge localizes on the donor or acceptor molecule for the phenoxylphenol system but not for the benzyl-toluene system. This figure also indicates that the transferring hydrogen has more positively charged proton character for the phenoxyl-phenol system than that for the benzyl-toluene system. This analysis is consistent with the PCET mechanism for the phenoxyl-phenol system and the HAT mechanism for the benzyl-toluene system. For the PCET mechanism, a proton (i.e., a positively charged hydrogen) transfers simultaneously with electron transfer from the donor to the acceptor molecule. For the HAT mechanism, a hydrogen atom (i.e., a nearly neutral hydrogen) transfers without significantly changing the electronic charge on the donor and acceptor molecules. 2373
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Figure 6. Component of the dipole moment vector along the hydrogen donor-acceptor axis for the ground and first excited adiabatic electronic states (solid black curves) and the diabatic electronic states ξ1 (dashed blue curve) and ξ2 (dashed red curve) for the phenoxyl-phenol system. Calculated values of the dipole moment for the diabatic electronic states around rp = 0 are omitted due to numerical noise in this region. A positive (negative) dipole moment indicates a dipole moment vector pointing toward the acceptor (donor).
Figure 7. Partial charges determined from electrostatic potentialderived atomic charges for the diabatic electronic state ξ1 of the phenoxylphenol system. Partial charges are shown for the donor molecule (green), acceptor molecule (purple), and transferring hydrogen (gray). Calculated values of the partial charges around rp = 0 are omitted due to numerical noise in this region. The corresponding plot for the diabatic electronic state ξ2 is essentially the reflection of this plot about the vertical axis corresponding to rp = 0 with the donor and acceptor partial charges interchanged (i.e., the green and purple curves interchanged).
The trends observed in Figures 3 and 4 are confirmed by the electrostatic potential maps depicted in Figure 5 for the reactant, transition state, and product positions of the transferring hydrogen. For the phenoxyl-phenol system, excess electronic charge is localized on the donor (acceptor) molecule when the proton is near the donor (acceptor). As the proton transfers from the donor to the acceptor, the electronic charge shifts from the donor to the acceptor molecule. This observation is consistent with the substantial changes in the dipole moment and the donor/ acceptor charges shown in Figures 3 and 4. In contrast, for the benzyl-toluene system, the electronic charge distribution is more evenly distributed between the donor and acceptor molecules during the entire hydrogen transfer reaction. This observation is consistent with the relatively small changes in the dipole moment and the donor/acceptor charges shown in Figures 3 and 4. These electrostatic potential maps highlight the PCET mechanism of the phenoxyl-phenol system and contrast it with the HAT mechanism of the benzyl-toluene system. The change in the electronic charge distribution during the charge transfer reaction is related to the electron-proton nonadiabaticity discussed in the previous subsection. As indicated by eq 9, the nonadiabatic coupling terms within the double adiabatic representation reflect the change in the character of the adiabatic electronic wave function along the transferring hydrogen coordinate. Thus, significant electronic charge redistribution during hydrogen transfer is associated with a large nonadiabatic coupling, which in turn is a diagnostic for substantial electronproton nonadiabaticity. Because PCET reactions are associated with greater charge redistribution than are HAT reactions, this analysis clarifies the connection between the degree of electronproton nonadiabaticity and the physical characteristics distinguishing PCET from HAT. C. Properties of Diabatic States. To illustrate the physical aspects of the diabatization scheme presented in section 2B, we examined the charge transfer properties of the diabatic electronic states and compared them to those examined in the previous subsection for the ground adiabatic electronic state. We focus on the phenoxyl-phenol system because the diabatic representation is more applicable to the PCET mechanism than to the HAT mechanism. The charge transfer properties of the diabatic electronic states for the benzyl-toluene system are provided in
Supporting Information and are qualitatively very similar to the corresponding properties of the ground adiabatic electronic state. The charge transfer properties of the diabatic electronic states are illustrated in Figures 6-8. Figure 6 depicts the component of the dipole moment along the hydrogen donor-acceptor axis for the ground and excited adiabatic electronic states (solid black curves) and the two diabatic electronic states (blue and red dashed curves). This component of the dipole moment is positive for diabatic state ξ1 and negative for diabatic state ξ2 at all positions of the transferring hydrogen (i.e., during the entire reaction). Similarly, Figure 7 illustrates that the charge on the donor molecule is negative and the charge on the acceptor molecule is positive along the entire range of hydrogen positions for diabatic state ξ1. The analogous figure for diabatic state ξ2, where the charge on the donor molecule remains positive and the charge on the acceptor molecule remains negative during the entire reaction, is provided in Supporting Information. Similar trends are exhibited in Figure 8, where the electrostatic potential maps appear virtually identical for the reactant, transition state, and product positions of the hydrogen for each diabatic state ξ1 and ξ2. These figures for the diabatic electronic states are in sharp contrast to the analogous figures for the adiabatic electronic states discussed in the previous subsection. In particular, the donor and acceptor charges depicted in Figure 4a and the electrostatic maps depicted in Figure 5a for the adiabatic electronic states show substantial changes in the electronic charge distribution during the PCET reaction. Thus, the diabatization scheme successfully transforms the adiabatic electronic states into diabatic electronic states with relatively invariant charge distributions along the hydrogen coordinate. This property of charge invariance is important when using these diabatic states in PCET theories based on the nonadiabatic golden rule formalism.4 The diabatization scheme used in these studies may be extended in several directions. For the systems studied in the present paper, the proton coordinate was defined along the axis connecting the proton donor and acceptor. In general, the onedimensional coordinate along which the nonadiabatic coupling vanishes could be chosen to be a curved path and could involve motions of other nuclei in addition to the proton, although additional complications would arise for the calculation of vibronic states from these diabatic electronic states. For example, 2374
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Figure 8. Electrostatic potential maps for the diabatic electronic states (a) ξ1 and (b) ξ2 corresponding to a density isosurface value of 0.005 for the reactant (top), transition state (middle), and product (bottom) positions of the transferring hydrogen for the phenoxyl-phenol system. Negatively and positively charged regions are indicated by red and blue coloring, respectively.
this coordinate could be chosen to be a normal-mode coordinate dominated by the proton motion. In these cases, the implementation of the diabatization scheme would require expansion of the chosen coordinate as a linear combination of Cartesian coordinates or numerical calculation of the nonadiabatic coupling along the chosen coordinate. Furthermore, for these symmetric systems, we chose r0 = 0, corresponding to the transition state geometry, and set γ(r0) = -π/4. Note that the transformed states will be rigorously diabatic (i.e., will have zero first-derivative couplings) for any choice of γ(r0), but our objective is to choose a value that provides physically meaningful charge-localized diabatic states. For asymmetric systems, we could choose r0 to be the coordinate corresponding to the maximum of d12 and set γ(r0) = -π/4 to ensure that the adiabatic states mix maximally and the diabatic states cross at the coordinate where the nonadiabatic coupling is the largest. All of these directions are interesting topics for future research.
5. CONCLUSIONS In this paper, we devised quantitative diagnostics to evaluate the degree of electron-proton nonadiabaticity in PCET systems. One diagnostic is the comparison of the vibronic coupling calculated with the full basis set diagonalization method to the vibronic couplings calculated with the double adiabatic and nonadiabatic methods. Another diagnostic is the investigation of the magnitude of the nonadiabatic coupling terms within the double adiabatic representation. A third diagnostic is the calculation of the fraction of excited electronic state character in the relevant vibronic states. Application of these diagnostics to the phenoxyl-phenol and benzyl-toluene systems confirmed that the former corresponds to electronically nonadiabatic and the latter corresponds to electronically adiabatic proton transfer.
Our calculations also clarified the connection between the degree of electron-proton nonadiabaticity and the chargetransfer characteristics distinguishing PCET from HAT. According to traditional definitions, the extent of electronic charge redistribution is significantly greater for PCET than for HAT. Thus, analysis of the dipole moment, partial atomic charges, and electrostatic potential maps for the ground state adiabatic electronic wave functions along the transferring hydrogen coordinate can be used to differentiate the PCET and HAT mechanisms. Comparison of these charge transfer properties for the phenoxyl-phenol and benzyl-toluene systems clearly designated the former as PCET and the latter as HAT. Furthermore, the significant change in charge transfer character of the ground and first excited state electronic wave functions along the transferring hydrogen coordinate is associated with a large nonadiabatic coupling between these two electronic states. In turn, this large nonadiabatic coupling is a diagnostic associated with significant electron-proton nonadiabaticity. Thus, the extent of electronic charge redistribution during the reaction is related to the degree of electron-proton nonadiabaticity, and both properties may be utilized to differentiate PCET from HAT mechanisms. Specifically, PCET and HAT mechanisms correspond to electronically nonadiabatic and adiabatic proton transfer, respectively. As a result, the nonadiabatic expression for the vibronic coupling is applicable to PCET reactions, while the double adiabatic expression for the vibronic coupling is applicable to HAT reactions. In addition, we developed a rigorous diabatization scheme for transforming the adiabatic electronic states generated from standard quantum chemistry calculations into charge-localized diabatic states for PCET reactions. These diabatic states are constructed so that the first-order nonadiabatic couplings with respect to the one-dimensional transferring hydrogen coordinate 2375
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vanish exactly. The application of this scheme to the phenoxylphenol system illustrated that the resulting diabatic states possess physically meaningful, localized electronic charge distributions that are relatively invariant along the hydrogen coordinate. These diabatic electronic states can be combined with the associated proton vibrational states to generate the reactant and product electron-proton vibronic states that form the basis of nonadiabatic PCET theories. Furthermore, these vibronic states and the corresponding vibronic couplings may be used to calculate rate constants and kinetic isotope effects of PCET reactions for comparison to experimental measurements.
’ APPENDIX: ANALYSIS OF NONADIABATIC COUPLING TERMS Equation 11 can be derived from eq 8 using the following expressions ðepÞ
ÆφðiÞ μ jdij
ðjÞ 3 rrp φν æp
ðepÞ
¼ Ædij
ðepÞ
ðepÞ
ðiÞ 3 rrp φμ æp
ðepÞ
ðiÞ 3 rrp φμ æp
ðepÞ
ðepÞ
ðiÞ ðjÞ ðjÞ jφðjÞ ν æp - Æφμ jgij jφν æp þ Æφν jdji
ðiÞ
3 rrp φμ æp ðA1Þ
P (ep) = Ærrpψi 3 rrpψjæe = k d(ep) where f(ep) ij ki 3 dkj after insertion of the identity operator. (Note that a complete electronic basis set {ψk} is assumed.) In eq A1, the first equality used the chain rule (i) for rrp(φ(j) ν φμ ), the second equality used integration by parts (neglecting the term with the proton wave function evaluated at = -d(ep) infinity) and the identity d(ep) ij ji , and the third equality used . Rearrangement of eq A1 leads to the chain rule for rrp 3 d(ep) ij ðepÞ
’ ASSOCIATED CONTENT
bS ðepÞ
ðjÞ ðiÞ ðjÞ 3 rrp ðφν φμ Þæp - Æφν jdij
ðjÞ ðjÞ ¼ -ÆφðiÞ μ jrrp 3 dij jφν æp þ Æφν jdji
¼ -ÆφðiÞ μ jfij
(Supporting Information) illustrates the asymmetry of the vibronic Hamiltonian matrix defined by eq 8 when the secondorder nonadiabatic coupling terms are neglected. This table also indicates that the magnitudes of the second-order nonadiabatic coupling terms may be comparable to the magnitudes of the firstorder nonadiabatic coupling terms in eq 8. Table S3 (Supporting (ep) terms for k > 2 are Information) illustrates that the d(ep) ki dkj much smaller than the other contributing terms in eq 11. Note that these higher-order terms can be included to achieve the desired level of convergence in the implementation of eq 11 without a substantial amount of additional effort. Overall, this example indicates that the implementation of eq 11, in which the (ep) d(ep) ki dkj terms for k > 2 are neglected, is more accurate than the implementation of eq 8, in which the second-order nonadiabatic are neglected. coupling terms g(ep) ij
ðepÞ
ðjÞ 3 rrp φν æp
ðepÞ
ðiÞ 3 rrp φμ æp
ðjÞ ðiÞ ÆφðiÞ μ jgij jφν æp ¼ -Æφμ jdij
þ ÆφðjÞ ν jdji
ðepÞ ðjÞ - ÆφðiÞ μ jfij jφν æp
Supporting Information. Adiabatic and diabatic potential energy curves shown in Figure 1 plotted with diabatic electronic coupling; analogue to Figure 7 for the second diabatic electronic state for phenoxyl-phenol system; analogues to Figures 7 and 8 depicting charge transfer properties of diabatic electronic states for benzyl-toluene system; table providing parameters obtained from the diabatic electronic states to calculate the semiclassical vibronic coupling; and tables providing a comparison of the nonadiabatic coupling terms in eqs 8 and 11 for the phenoxyl-phenol system. This material is available free of charge via the Internet at http://pubs.acs.org.
’ ACKNOWLEDGMENT We thank Alexander Soudackov, Michael Pak, and Chet Swalina for helpful discussions and advice. We gratefully acknowledge funding from NSF Grant CHE-07-49646. ’ REFERENCES
ðA2Þ
Substituting eq A2 into eq 8 leads to eq 11. A common implementation of eq 8 is to assume that the second-order nonadiabatic coupling terms (i.e., those involving g(ep) ij ) are negligible in comparison to the first-order nonadiabatic coupling terms (i.e., those involving d(ep) ij ). This approach may lead to a vibronic Hamiltonian that is not Hermitian (i.e., the vibronic Hamiltonian matrix is not symmetric) because, in (ep) (j) (j) (ep) (i) general, Æφ(i) μ |dij 3 rrpφν æp 6¼ Æφν |dji 3 rrpφμ æp. In contrast, (ep) terms are the implementation of eq 11, in which the d(ep) ki 3 dkj neglected for k > max{i,j}, still retains the symmetry of the vibronic Hamiltonian matrix. Further insight into the relative magnitudes of the terms in eqs 8 and 11P may be obtained from 2 eq A2. For example, the identity g(ep) = - k |d(ep) ii ki | leads to the (ep) (ep) 2 (ep) 2 relation |gii | g |d12 | , i = 1,2. Note that the |d12 | terms are included in the implementation of eq 11. We have analyzed the nonadiabatic coupling terms in eqs 8 and 11 for the phenoxyl-phenol system along the one-dimensional proton coordinate. The results for matrix elements pertaining to the lowest two electronic states, i,j e 2, are presented in Tables S2 and S3 of Supporting Information. Table S2
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