Vibrational Relaxations and Dephasing in Electron-Transfer Reactions

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Vibrational Relaxations and Dephasing in Electron-Transfer Reactions William W. Parson J. Phys. Chem. B, Just Accepted Manuscript • DOI: 10.1021/acs.jpcb.6b08803 • Publication Date (Web): 18 Oct 2016 Downloaded from http://pubs.acs.org on October 19, 2016

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The Journal of Physical Chemistry

Vibrational Relaxations and Dephasing in Electron-Transfer Reactions

William W. Parson*†

†

Department of Biochemistry, University of Washington, Seattle, WA 98195

email: [email protected] phone: (206) 523-0142

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ABSTRACT: The rates of nonadiabatic electron-transfer reactions depend on four main factors: the probability of finding the system in a conformation in which the reactant and product states have the same energy, the electronic coupling that drives oscillations between the two diabatic states, the dephasing that damps these oscillations, and the vibrational or electronic relaxations that trap the product state by transferring energy to the surroundings. This paper develops a simple expression that combines these factors in a relatively realistic manner. Values for all the parameters in the expression can be obtained from microscopic quantum-mechanical/molecular-mechanical simulations. The theory is tested by calculations of the rates of electron transfer from excited indole rings to a variety of acceptors in peptides and indole-acrylamide compounds. For the systems that are studied, the theory gives considerably better agreement with experiment than expressions that do not consider the rates of vibrational relaxations and dephasing.

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INTRODUCTION Although dynamical effects of solvent motions on electron-transfer reactions have been widely discussed, an efficient but realistic treatment of these effects has remained elusive. This paper presents a simple expression that deals with vibrational dephasing and relaxations for nonadiabatic reactions, and that can be implemented by quantum-mechanical/molecularmechanical (QM/MM) simulations. The theory is tested by calculating rate constants for electron transfer from excited indole rings to a variety of acceptors in peptides and acrylamide compounds and comparing the results with experimental data from the literature.

RESULTS AND DISCUSSION 1. Theory. Let ρ be the reduced density matrix for an ensemble of systems in which an initial vibronic state, |1〉, evolves into a charge-transfer (CT) state, |2〉, which then relaxes to a stable product, |3〉. The stochastic Liouville equation1-4 gives the time dependence of ρ as ⁄ = i⁄ℏ , + ,

(1)

where V is the Hamiltonian of the diabatic basis states, and the relaxation matrix R describes stochastic interactions with the surroundings. We assume that interconversion of |1〉 and |2〉 is driven only by the off-diagonal elements of V (V12 and V21), with the contributions of R to this step being negligible. If we neglect the relaxation to |3〉 for the moment, expanding Eq. (1) for the two-state system of |1〉 and |2〉 gives ⁄ = i⁄ℏ −

(2)

for the population of |2〉 (ρ22). For coherence element ρ12, we obtain ⁄ = i⁄ℏ − + − − / ,

(3)

where T2 is the time constant for the decay of phase coherence between |1〉 and |2〉. T2 here is given by 1/T2 = -R12 = 1/T2* + 1/2T1, in which T2* and T1 are time constants for, respectively, 3 ACS Paragon Plus Environment


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pure dephasing and longitudinal relaxations, the latter including both the stochastic interconversions of |1〉 and |2〉 that we have assumed to be negligible and the conversion to |3〉 that remains to be considered. Although more comprehensive treatments replace the relaxation matrix element R12 by a time-dependent function with Gaussian and imaginary components,4-10 the simple factor -1/T2 suffices for many purposes and will be used here. The quantity − in Eq. (3) is the difference between the total energies of |2〉 and |1〉, including the internal energies of the electron carriers, interactions of the electron carriers with the solvent, and interactions of induced dipoles within the solvent. Call this energy gap x. If |V12| > |V12|, which we can view as a requirement for nonadiabticity. In addition, Eqs. (9) and (10) assume that the system explores the relevant conformational space rapidly relative to the electron-transfer reaction. Some light-driven 6 ACS Paragon Plus Environment

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reactions, such as the initial charge-separation steps in photosynthetic reaction centers, probably occur before vibrational equilibration is complete, and thus may be too rapid for this assumption to hold.12, 44 Reactions that require the reactants to diffuse together or to undergo rare conformational fluctuations can have multiphasic kinetics with slow phases that Eqs. (9) and (10) also would not capture.45-47 Callis et al.48 have discussed how a distribution of relaxation times could affect the kinetics of tryptophan photooxidation in proteins.

2. Implemenatation. Microscopic QM/MM simulations of a system in state |1〉 can provide values for all the parameters in Eq. (10). Supplementary Figures S2-S4 show representative plots of the time-dependent energy gaps in simulations of electron transfer from excited indole rings to various acceptors in peptides and acrylamide compounds. I simulated eighty-six such reactions, propagating each of the QM/MM trajectories for 3 ns in 1-fs steps at 280 K with the programs ENZYQ and INDIP36, 49 and recording x(t) and |V(t)| at intervals of 100 fs.50 The calculations included induced electric dipoles optimized separately for the reactant and product state of each reaction.36 To find P(0), the values of x from each trajectory were sorted into bins with a width 2ε = 200 cm-1, which is the full width at halfmaximum of the Lorentzian distribution of energies associated with a relaxation time T2 = 26.5 fs. The energy gaps for some of the reactions did not provide P(0) directly because x(t) remained above zero throughout the trajectory. In these cases, P(0) was estimated from plots of the free energy functional15, 16, 18, 19, 23, 51 g(x) = -kBT ln[P(x)/Pm].

(11)

Here Pm is the fraction of the time that the gap is within ±ε of its most probable value (i.e., the probability that the system is at the free-energy minimum for |1〉), P(x) is the fraction of the time the gap is within ±ε of value x, kB is the Boltzmann constant and T is the temperature. Fitting g(x) to a polynomial function of x and taking the zero-order term gives the activation free energy (∆G‡), and P(0) then is P(0) = Pm exp(-∆G‡/kBT).

(12)

|V(0)| was obtained similarly by fitting plots of |Vrms(x)|. 7 ACS Paragon Plus Environment


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Representative plots of g(x) and |V(x)| are shown in Supplementary Figures S5 and S6. While some of the free-energy functionals have canonical parabolic shapes, others have irregular shapes with multiple local minima. Table S1 gives the calculated values of ∆G‡, P(0) and |V(0)| for all the reactions. The vibrational relaxation rate constant k23 can be obtained from the autocorrelation function of x (C(t)). Since C(t) depends on both pure dephasing and vibrational relaxations, I fit the section between 0.1 and 50 ps to a double-exponential function plus a constant, and took the slower exponential component to represent vibrational relaxations. Its time constant (1/k23) was typically in the range of 5 to 10 ps (Table S2). Dephasing requires recording the energy gap at shorter time intervals. For this, I ran separate 30-ps trajectories with x recorded at intervals of 1 fs. I fit the first 200 fs of C(t) from these simulations to two exponentials plus a constant, and took the time constant of the faster exponential component to represent T2. It typically was on the order of 10 fs (Table S2). Figure S7 shows representative plots of the two regions of C(t). Identification of the sub-picosecond component of C(t) with dephasing and the slower exponential component with k23 rests on studies of a density-matrix model of a system that included three electronic states, five harmonic vibrational modes, and a thermal bath.12 C(t) for that model had two main exponential components with time constants comparable to those seen in the present work. Increasing the rate constant for thermal equilibration of the two lowest levels of a vibrational mode (a free parameter in the model) sped up the slower component with little effect on the faster one, indicating that the slower component reflects vibrational thermalization while the faster one probably stems mainly from pure dephasing of fields oscillating at different frequencies. In recent studies using ring-polymer molecular dynamics, the probability of recrossing of electron-transfer trajectories at the transition point was found to decay with a sub-picosecond time constant comparable to the fast component seen here.52 MD simulations of water have resolved two components with time constants of about 10 fs and 1 ps,42 the faster of which seems likely to be mainly pure dephasing. The constant terms in the double-exponential fits used in the present work represent further, slower relaxations that probably have little bearing on the electron-transfer dynamics. However, C(t) also can include Gaussian and sinusoidal components that the present analysis neglects and that could distort the results. In addition, contributions from vibrational 8 ACS Paragon Plus Environment

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relaxation could contaminate the exponential component ascribed to dephasing, and vice versa. In many cases, uncertainties in k23 probably are the main difficulty in implementing Eq. (10). Column 4 in Table S2 gives the ratio of h/T2 to |V(0)|, which the steady-state treatment offered above assumes to be >> 1. The ratio is greater than 10 for all the reactions considered here, and greater than 20 for all but a few. The steady-state approximation thus appears to be acceptable for these reactions. Column 5 gives 2|V(0)|2T2/ℏ2, the rate constant for return from |2〉 to |1〉 at points where x = 0 (Eq. (9)). This is greater than k23 for most of the reactions, indicating that a return to |1〉 competes effectively with relaxation to |3〉. The last two columns of Table S2 give the rate constant 〈k13〉1 calculated by Eq. (10) for each of the reactions, along with the summed value of 〈k13〉1 for each system. Figure 2 compares the final sums with experimental values derived from measured fluorescence yields or lifetimes.36, 53 Most of the calculated rate constants are of the proper order of magnitude. Although Eq. (10) neglects transitions to excited vibrational levels of |2〉, the agreement with experiment seems remarkably good, considering that the treatment has no adjustable parameters other than those embedded in the QM/MM protocol.

3. Comparison with the Marcus Equation. For comparison with Eq. (10), Fig. 3 shows the rate constants calculated by the Marcus equation13, 24, 54 '3# =

1

42 //

ℏ 5'6

2

1 78 2

− Δ: ‡⁄'6 ,

(13)

in which λ is the reorganization energy of the reaction. Here |V| usually is assumed to be independent of x. If the free energies of the reactant and product states are parabolic functions of x with the same curvature, the activation free energy in Eq. (13) can be obtained from the relationship

∆G‡ = (∆Go + λ)2/4λ,

(14)

where ∆Go again is the standard free energy change.24 Because many of the reactions considered in the present work had non-parabolic free-energy functionals, and because |V| 9 ACS Paragon Plus Environment


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sometimes varied significantly with x, the calculations shown in Fig. 3 used the values of ∆G‡ and |V(0)| from Table S1. A parabolic approximation was, however, used for the reorganization energy:

λ ≈ (λ1 + λ3)/2,

(15a)

5< = 〈〉* − 〈〉 '= ⁄2> were obtained from the QM/MM trajectories described above. For 〈x〉3 and >* , I recorded x(t) at 100-fs intervals in a separate 3-ns trajectory for the product state of each reaction. Values of 〈x〉1, 〈x〉3, λ1, and λ3 for all the reactions are collected in Table S3. The Marcus equation overestimates the rates for all the reactions considered here (Fig. 3). Although errors in λ could contribute to the discrepancies, Eq. (13) has a relatively weak dependence on this parameter. Overestimates of |V| also are unlikely to account for the disagreement, because ab initio calculations48 with three different basis sets give values for |V| that are, if anything, somewhat larger than those obtained by the semiempirical procedure in ENZYQ. The problem, more likely, lies in neglecting the limits imposed by vibrational relaxation when |V| is large. As might be expected in view of the non-parabolic free-energy curves, rate constants calculated with Eq. (13) but with ∆G‡ obtained by Eq. (14) were widely scattered and were not well correlated with the experimental values (not shown). Finding >* and 〈x〉3 from a trajectory in the product state requires deciding how much of the trajectory to include in the analysis. At short times after the MM force field is changed and the initial structural accommodations are underway, the energy gap and variance are not yet representative of |3〉. Long times, on the other hand, can allow additional relaxations that contribute little to the competition between k21 and k23 (see Supplementary Fig. S3 for examples). The values of λ3 and 〈x〉i in Table S3 were averaged over a 2.95-ns period beginning 50 ps after projection from state 1. Reducing the time allowed for slow relaxations would decrease λ, resulting in even larger overestimates of the electron-transfer rates in Fig. 3. Note that errors in λ3 and 〈x〉3 do not affect Eq. (10), where all the parameters are obtained from a trajectory in |1〉. Neither ∆Go nor λ is needed there, and there are no assumptions about the shapes of the free-energy curves. 10 ACS Paragon Plus Environment

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4. Transitions to Excited Vibrational Levels. For most of the reactions considered here, the mean energy gap 〈x〉1 is greater than zero (Table S3). The exceptions that fall in the “Marcus inverted region”24, 54, 56 involve electron transfer from tryptophan to protonated histidine residues in two β-hairpin peptides (HP-8H+ and HP-10H+ in Table S3) and transfer to an amide group in a TrpCage peptide (TC-K8A). Equation (10) would be expected to underestimate the rate of electron transfer if x(t) frequently goes below zero, because it neglects transitions to excited vibrational sublevels of the product. To include transitions to other vibrational sublevels in an approximate manner, one could replace P(0) by the probability of finding x in a larger region that extends below zero. A function that fell off quickly at positive values of x and more gradually at negative values was used for this purpose in previous studies.36, 49 Because the breadth and shape of this function could not be extracted uniquely from QM/MM simulations, they had to be adjusted phenomenologically. More rigorously, one can expand Eq. (10) to

〈'* 〉 = ∑D,'

2 '23 @A0B@ CD,' 2

@A0B@

2

CD,' + '23 ℏ 022

.D,' 0 ,

(16)

where Fj,k is the Franck-Condon factor for transitions between vibrational sublevel j of |1〉 and sublevel k of |2〉, and Pj,k(0) is the mean probability that j and k are degenerate to within ±ε. In principle, the energies and Franck-Condon factors for all the vibrational modes that are coupled to an electron-transfer reaction can be obtained from a Fourier transform of the autocorrelation function C(t).12, 17, 51, 55, 57 However, the Fourier transform usually contains such a dense forest of modes that dissecting the contributions to x(t) from individual modes requires replacing the actual system by a model with a small set of effective modes. Note also that a sum of Boltzmann-weighted Franck-Condon factors cannot be factored out of Eq. (16), as often is done in treatments that do not consider the dependence of the rate on k23 and T2.13, 15, 58-65

Including transitions to excited vibrational levels therefore remains a challenge.

Although contributions from intramolecular vibrational modes sometimes are treated separately in order to include excited vibrational levels in the Marcus equation, comparisons of the calculated rate constants with experiment47, 66-68 have required adjusting one or more 11 ACS Paragon Plus Environment


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free parameters and usually have neglected transitions to higher electronic states. In the two tryptophan-histidinium peptides studied here, where 〈x〉1 for forming the lowest electronic CT state was negative, the next higher CT state had a more favorable P(0) and was found to be a major product (Tables S1 – S3). Formation of higher electronic states has been discussed previously for other systems,69, 70 and when it occurs, relaxations to lower electronic states could augment the vibrational relaxations described by k23. Rapid transfer of an electron to a secondary acceptor seems likely to play such a role in photosynthetic reaction centers.71-73

5. Quenching of Tryptophan Fluorescence. Because the test systems considered here involve photoexcitation of tryptophan and related molecules, the calculations bring out several points concerning these particular systems. Reference to Table S2 shows that electron transfer to the amide group on the carboxyl side of a tryptophan residue was always faster than transfer to the amide on the amino side. As Callis and coworkers have discussed,48, 64 this preference often results more from a favorable P(0) than from strong electronic coupling. In the TrpCage peptides, however, |V(0)| also was much higher for transfer to the amide on the carboxyl side (Table S1). In some of the peptides, electron transfer to another amide group also had a favorable P(0) and was found to be at least as important as transfer to either of the tryptophan’s amides (see, for example, amide a9 in hairpin 8N). Transfer to protonated histidines and some other side chains also was rapid in the hairpin peptides, in agreement with previous work.36, 65, 74, 75 Protonated histidine was less effective in the villin headpiece, where transfer to the carboxyl-side amide had a smaller k23 but more favorable values of both P(0) and |V(0)| (Tables S1 and S2). These conclusions are difficult to test, because the measured fluorescence yields or lifetimes provide experimental values only for the sum of the rate constants for transfer to all the available acceptors. The two non-peptide systems that were studied (acrylyl-TA and acrylyl-HTPI) are covalently-linked adducts of indoles with acrylamide. Eftink et al.,53 who first described these compounds, surmised reasonably that their exceptionally fast electron transfer resulted from strong through-bond electronic coupling. The present work points instead to a combination of a favorable P(0) and high k23. The calculated electronic coupling factors for acrylyl-TA and acrylyl-HTPI actually are relatively small (Table S1), probably because the accepting molecular orbitals are spread over atoms that are farther from the indole ring. As noted above, 12 ACS Paragon Plus Environment

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however, the semiempirical procedure used here tends to give smaller values for |V| than ab initio methods,48 and might underestimate the coupling in some cases.

ASSOCIATED CONTENT Supporting Information The supporting information is available free of charge via the Internet at http://pubs.acs.org. Discussion of the relationship of k13 to the time required for |3〉 to attain half its final population (Fig. S1); representative plots of x(t) (Figs. S2-S4), g(x) (Fig. S5), |V(x)| (Fig. S6), and autocorrelation functions (Fig. S7); tables of calculated quantities for all the reactions (Tables S1-S3).

AUTHOR INFORMATION Corresponding Author *Email: [email protected]. Phone: 206-523-0142. Notes The author declares no competing financial interest. ACKNOWLEDGEMENTS I am indebted to Patrik Callis for many fruitful discussions, exchanges of computational results, incisive criticisms of earlier versions of the theory, and ab initio calculations of the structures of Ac-HTPI and Ac-TA in their CT states. These structures were used to define force fields for the simulations that provided >* and 〈x〉3 for the acrylamide compounds. I also thank Arieh Warshel, William Hazelton, Oleg Prezhdo and Ross McKenzie for helpful suggestions. The work presented here was not supported by external funding.



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FIGURE LEGENDS Figure 1. Dependence of 〈k13〉1/P(0) on |V(0)|, as calculated by Eq. (10) with k23 = 0.20 ps-1 and 1/T2 = 50 ps-1 (purple), 100 ps-1 (cyan), 200 ps-1 (green) and 500 ps-1 (red). Figure 2. Correlation of the sum of the electron-transfer rate constants calculated by Eq. (10) for each peptide or acrylamide adduct ( ∑ 〈k13〉1 from Table S2) with the observed rate constant (kobs) obtained from the measured fluorescence lifetime for indole-acrylamide adducts,53 or from the fluorescence yield (Φf) for hairpin and TrpCage peptides36 and the villin headpiece.76 Fluorescence yields were converted to rate constants by the relationship Φf = kr/(kr + knr + kobs), where kr, kobs and knr are, respectively, the rate constants for radiation (kr = 4x107 s-1), all the available electron-transfer reactions, and other nonradiative processes (knr = 8x107 s-1). The values for kr and knr are taken from Yu et al.77 and are assumed to be the same for all the systems. The experimental uncertainties in kobs are on the order of ±10%. Note that the ordinate and abscissa scales are logarithmic. The dashed diagonal line represents perfect agreement. The smallest calculated rate constant, which has the greatest disagreement with experiment, is for the β-hairpin peptide with tyrosine at position 8.

Figure 3. Correlation of the observed electron-transfer rate constant with the rate constant calculated by the Marcus equation. ∆G‡ and |V(0)| are from Table S1, and the reorganization energies from Table S3. The calculated value is the sum of kET (Eq. (13)) for all the reactions of each peptide or acrylamide adduct; kobs is the same as in Fig. 2. Note that the logarithmic ordinate and abscissa scales are compressed relative to those in Fig. 2. The dashed diagonal line represents perfect agreement.


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(30) (31) (32)

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Vibrational Relaxations and Dephasing in Electron-Transfer Reactions

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