Electron-Transfer Dynamics in a Zn-Porphyrin-Quinone Cyclophane

Article pubs.acs.org/JPCB

Cite This: J. Phys. Chem. B 2018, 122, 3854−3863

Electron-Transfer Dynamics in a Zn-Porphyrin-Quinone Cyclophane: Effects of Solvent, Vibrational Relaxations, and Conical Intersections William W. Parson*,† †

Department of Biochemistry, University of Washington, Seattle, Washington 98195, United States S Supporting Information *

ABSTRACT: Rate constants for photochemical charge separation and recombination in a zinc-porphyrin-benzoquinone cyclophane are calculated by an approach that was developed recently to include effects of vibrational dephasing and relaxation and to reduce the dependence on freely adjustable parameters. The theory is extended to treat the rate of vibrational relaxation individually for each vibrational sublevel of the initial charge-transfer product. Quantum-mechanical/ molecular-mechanical simulations of the reactions in iso-octane, toluene, dichloromethane, and acetonitrile suggest that charge separation occurs at conical intersections in the two more polar solvents, but at avoided crossings in the nonpolar solvents. In agreement with experimental measurements, however, the calculated rate constants for charge separation are similar in polar and nonpolar solvents. Charge recombination to the ground state is found to have electronic coupling factors smaller than that of charge separation and to be affected more strongly by interactions with the solvent.

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INTRODUCTION Complexes of quinones with porphyrins provide instructive models for photosynthetic charge separation.1−19 In an important series of studies, Staab, Michel-Beyerle, and coworkers9,10,12,13,16 examined the dynamics of light-driven electron transfer in Zn-porphyrin-benzoquinone cyclophanes with various substituents on the porphyrin and quinone rings (Figure 1). The charge-separation kinetics occurred on a subpicosecond time scale, and were surprisingly insensitive to the polarity and the dielectric relaxation time of the solvent. Changes in the substituents on the porphyrin or quinone ring also had relatively little effect on the rate. Charge recombina-

tion reactions in zinc-free analogs occurred on the time scale of 10−10 s in nonpolar solvents, and somewhat more rapidly in dichloromethane and acetonitrile.13,15 Michel-Beyerle, Staab, and their colleagues9,13,16 rationalized these observations with the aid of a widely used expression for nonadiabatic electrontransfer coupled to a single, representative vibrational mode:20,21 ∞

ket =

[−(ΔG + λs + nhν)2 /4λskBT ]

(1)

Here, ket is the electron-transfer rate constant, V is the electronic-interaction matrix element that couples the diabatic initial and final states, λs is the solvent reorganization energy, ν and λv are the frequency and reorganization energy of the vibrational mode, and ΔG is the standard free energy difference between the initial and final states. The factors following 2πV2/ ℏ on the right-hand side of eq 1 represent the Franck− Condon-weighted density of vibronic states in which the reactant and product are degenerate. Michel-Beyerle et al.9,12,13,16 obtained the free energy change ΔG by measuring delayed fluorescence from the zinc-free cyclophanes and correcting for effects of the zinc on the oxidation potential of the porphyrin; they then optimized the parameters V, ν, λv, and λs to fit their data. Although eq 1 was able to reproduce the measured rate constants for both charge separation and charge recombination,

Figure 1. Model of 5,15-[3,6-dimethyl-p-benzoquinone-1,4-diylbis(4,1-butanediyl-2,1-benzeno)]-2,3,7,8,12,13,17,18-octamethylporphyrin zinc (PQC).12 The figure was made with VMD23 using the structure at the end of a 5 ns trajectory in the ground state (see the Methods Section). The zinc atom is represented in gray and other atoms in CPK colors; hydrogens are omitted for clarity. © 2018 American Chemical Society

(λ /hν)n 2πV 2 (4πλskBT )−1/2 e−λ v / hν ∑ v exp ℏ n! n=0

Received: January 30, 2018 Revised: March 9, 2018 Published: March 20, 2018 3854

DOI: 10.1021/acs.jpcb.8b01072 J. Phys. Chem. B 2018, 122, 3854−3863

Article

The Journal of Physical Chemistry B

Figure 1, with the aid of an expression that was developed recently to incorporate a dependence on the rates of vibrational relaxations and dephasing:22,30

this required using values for some of the adjustable parameters that might not be physically realistic. For example, acetonitrile was assigned a solvent reorganization energy smaller than that of hexane. The greater dipole moment of acetonitrile would be expected to lead to a correspondingly larger reorganization energy. In addition, eq 1 assumes that ket increases quadratically with |V|. This relationship will hold only as long as |V|2 is small relative to k23ℏ2/2T2, where T2 is the effective time constant for decay of coherence between the reactant and the initial product state, and k23 is the effective rate constant for relaxations that transfer energy to vibrational modes that are not coupled to electron transfer.22 The values of |V| that were used (e.g., 133 cm−1) might exceed this limit.22 Equation 1 also neglects the possible dependence of |V| on the reaction coordinate, and Häberle et al.16 assumed further that |V| was the same for charge separation and recombination and for the presence and absence of zinc. Worth and Cederbaum24 and Dreuw et al.25 suggested that photochemical charge separation in Zn-porphyrin-quinone cyclophanes occurs at a conical intersection between the lowest excited singlet π−π* state and the charge-transfer (CT) state. They proposed that this intersection lies along a vibrational mode that modulates the distance between the porphyrin ring and the quinone. When applied to a computer model lacking the chains that attach the quinone to the porphyrin in PQC, a combination of CASSCF and TDDFT quantum calculations found an intersection of the diabatic π−π* and CT states at a point where the Zn was about 2.5 Å from the nearest O of the quinone.25 This is in good accord with the crystal structure12 of the tetramethyl-tetraethylporphyrin analog of PQC, which has a Zn−O distance of 2.532 Å. Calculations with continuum solvent models indicated that interactions with a polar solvent would stabilize the CT state strongly relative to the π−π* state. However, the authors did not present rate constants for electron transfer. Borrelli, Domcke, and co-workers26,27 have described quantum calculations on a Mg-porphyrin-benzoquinone complex in which the quinone and porphyrin rings lie in perpendicular planes with one of the quinone’s oxygens about 4.5 Å from the porphyrin. This model also lacked the chains that constrain cyclophane structures. The total intramolecular vibrational reorganization energy (λv) for electron transfer to the quinone in this complex was calculated to be 4328 cm−1, with vibrational modes of the quinone making the dominant contribution (3557 cm−1). Borrelli and Domcke26 found that, if ΔG was set equal to −λv in a vibronic model with five harmonic modes, electron transfer could occur on a subpicosecond time scale by electronic coupling (i.e., possibly at an avoided crossing rather than a conical intersection). They noted that energy would have to move to other vibrational modes in order to prevent recurrences of the starting state, but did not address how rapidly this process would occur. Effects of the solvent also were not considered. In related work, Borrelli and Peluso28,29 calculated the spectrum of Franck−Condon-weighted density of states for electron transfer from bacteriopheophytin to ubiquinone in photosynthetic reaction centers of Rhodobacter sphaeroides. Although this system differs from the Mg-porphyrin-benzoquinone complex in a number of ways, λv was found to be very similar (∼4300 cm−1), with vibrational modes of the quinone again making the largest contributions. The present study re-examines the dynamics of electron transfer in the Zn-porphyrin-benzoquinone cyclophane of

⟨kcalc⟩1 =

∑ i,j

k 23 |Vi , j(0)|2 Fi , j |Vi , j(0)|2 Fi , j + k 23ℏ2 /2T2

Pi , j(0) (2)

In this expression, ⟨kcalc⟩1 denotes the electron-transfer rate constant averaged over configurations of the reactant state (| 1⟩), here the lowest π−π* excited singlet state of PQC); | Vi,j(0)| is the root-mean-square value of the electronic interaction matrix element that couples |1⟩ to the initial CT product state (|2⟩), averaged over configurations in which the diabatic energy gap between vibrational sublevel i of |1⟩ and sublevel j of |2⟩ (xi,j) goes to zero; Fi,j is the Franck−Condon factor for transitions between i and j, weighted with a Boltzmann factor reflecting the relative population of sublevel i; Pi,j(0) is the probability of finding the system in a configuration with an energy gap xi,j = 0; and T2 and k23 are, as above, the effective time constant for decay of coherence between |1⟩ and |2⟩ and the mean rate constant for vibrational relaxations from |2⟩ to a more stable product state (|3⟩). The same expression can be used to treat charge recombination from |3⟩ to the ground state (|0⟩). In addition to handling the approach to a limiting rate when | V| becomes large, this treatment is largely free of adjustable parameters. In favorable situations, values for all the terms that are needed can be obtained from quantum calculations and molecular-dynamics simulations, with some shifting of the calculated energies to match experimentally measured values if necessary. The treatment has worked well for reactions of free radicals in organic solvents,30 and in a simpler form that neglected excited vibrational states, for reactions of tryptophan residues and other indole derivatives in peptides and water.22 However, the dependence of the relaxation rate constant k23 on the composition of the product’s vibrational sublevel j has not been considered in detail. PQC presents a more challenging test of the theory because its large electronic coupling factor makes the calculated rate constant more sensitive to the treatment of vibrational relaxations and dephasing. PQC also raises the question of whether eq 2 applies to transitions at conical intersections as well as at avoided crossings.

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METHODS Molecular models of the Zn-octamethylporphyrin-dimethylbenzoquinone cyclophane shown in Figure 1 (PQC) were embedded in a bath of iso-octanol, toluene, dichloromethane, or acetonitrile, which was constrained to a sphere with a radius of 20 Å. Following equilibration of the solvent by 520 ps of molecular dynamics (MD) at 293 K, trajectories were continued in either the ground electronic state (|0⟩), the lowest π−π state (|1⟩), or the lowest CT state (|3⟩) in 1 fs steps using the quantum-mechanics/molecular-mechanics (QM/ MM) programs ENZYQ and INDIP.31 These programs combine a semi-empirical (restricted Hartree−Fock) quantum mechanical approach based on QCFF/PI32,33 with MD routines based on ENZYMIX.34 See the Supporting Information for details on the porphyrin bond-stretching and bending parameters and the sigma charges of PQC and the solvents (Tables S1 and S2 and Figure S1). Wave functions, energies, and charges of the 44 π-atoms of PQC were evaluated quantum mechanically on each step of the trajectory. The π−π* and CT excited states considered contributions from nine singly excited 3855


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porphyrin bond lengths and angles in the MD force field had little effect on the calculated energies.

electronic configurations. Structures were saved for analysis every 100 fs in trajectories lasting 5 ns, or for higher resolution of autocorrelation functions, every 1 fs in trajectories lasting 200 ps. INDIP optimized induced dipoles31,35−37 separately for the reactant and product states, but these made only negligible contributions to the calculated energy gaps. Electronic coupling factors were calculated as described.31,38 Dipole moments were calculated using the uncorrected quantum charges and were averaged over 1000 configurations during QM/MM trajectories in toluene; they do not include induced dipoles. Franck− Condon factors were calculated by Manneback’s39 recursion formulas.

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RESULTS AND DISCUSSION Probability Distributions, Free Energies, and Reorganization Energies. Figure 2 shows calculated probability

Figure 3. Probability of finding various values of the 0−0 energy gap for charge recombination from the lowest CT state to the ground state during trajectories of PQC in iso-octane (A), toluene (B), dichloromethane (C), or acetonitrile (D) and in either the CT state (3, cyan curves) or the ground state (0, black). The bin width 2ε is 200 cm−1.

Figure 4A shows the calculated solvent reorganization energies for charge separation, λs = (⟨ΔE00⟩1 − ⟨ΔE00⟩3)/2, where ⟨ΔE00⟩1 and ⟨ΔE00⟩3 are the mean values of ΔE00 during trajectories propagated in the π−π* and CT states, respectively.40 The reorganization energies for charge recombination, calculated similarly as (⟨ΔE00⟩3 − ⟨ΔE00⟩0)/2 from trajectories in the CT (|3⟩) and ground states (|0⟩), also are shown, and are very similar to those for charge separation. As one would expect, the polar solvents, dichloromethane and acetonitrile, have considerably larger reorganization energies than those of toluene and iso-octane. Figure 4B shows the standard free energy changes calculated as ΔG°1,3 = (⟨ΔE00⟩1 + ⟨ΔE00⟩3)/2 for charge separation and ΔG°3,0 = (⟨ΔE00⟩3 + ⟨ΔE00⟩0)/2 for recombination.40 The figure also includes experimental values for charge separation reported by Heitele et al.12,13,16 The calculated values of ΔG°1,3 for the nonpolar solvents are close to zero, again in good accord with Dreuw et al.’s25 calculations. The experimental values are more negative by about 5000 cm−1 in these solvents and by about 2000 cm−1 in acetonitrile and dichloromethane. Differences between the calculated and experimental values are not unexpected, because the corrections that enter into the latter might not capture all of the interactions involving the Zn, and because measurements of delayed fluorescence could probe the CT state at a different time relative to the relaxations that follow charge separation. Is is clear that the present calculations also do not necessarily give very accurate energies because they underestimate the energy of the porphyrin’s lowest π−π* state. PQC has a weak “Q” absorption band near 576 nm and a fluorescence peak near 636 nm,12 which together put the 0−0 excitation energy for the first excited state at approximately 16 500 cm−1 above the ground state. The ENZYQ calculations underestimated this energy by about 4000 cm−1. Nguyen et al.41 found that TDHF-INDO/S calculations underestimated

Figure 2. Probability of finding various values of the 0−0 energy gap from the lowest π−π* state to the lowest CT state during QM/MM trajectories of PQC in iso-octane (A), toluene (B), dichloromethane (C), or acetonitrile (D). The trajectories were propagated for 5 ns in either the π−π* state (1, red curves) or the CT state (3, cyan), and structures were saved every 100 fs for analysis. The bin width 2ε for the distributions is 200 cm−1. See Figure S2 for plots with higher resolution. The sharp dips in some of the curves at ΔE00 ≈ 0 are discussed below.

distributions (P(ΔE00)) of the 0−0 energy gap (ΔE00) for electron transfer from the lowest excited π−π* state of the porphyrin to the quinone in PQC. The gap between the reactant and product states depends on the solvent and on whether the QM/MM trajectory is run in the ground, π−π*, or charge-transfer (CT) state. The energy gaps in the nonpolar solvents, toluene and iso-octane, cluster around values close to zero, in good agreement with the calculations by Dreuw et al. for PQC in vacuo.25 The energies are more negative in trajectories run in the CT state in dichloromethane or acetonitrile. Because the same force field was used for all of the trajectories, the differences probably reflect changing interactions with the solvent rather than intramolecular vibrational reorganizations. Similar probability distributions for charge recombination from the CT state to the ground state are shown in Figure 3. Reasonable modifications of the 3856


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(see below), the potential surfaces in polar solvents evidently include conical intersections. Although these intersections could lie along intramolecular vibrational modes that change the Zn−O distance as Dreuw et al.25 suggests, the additional coordinates that make conical intersections possible in this system appear to depend on interactions with the solvent. Warshel and Chu42 and Burghardt and Hynes43 have emphasized the importance of the solvent in creating conical intersections for photoisomerization of protonated retinyl Schiff bases. Electronic Interaction Matrix Elements. Figure 5 shows the calculated electronic coupling factors (|V|) for charge

Figure 4. Calculated solvent reorganization energies (λs, A) and standard free energy changes (ΔG°, B) for charge separation (purple squares and red circles, π → CT) and recombination (cyan triangles and blue diamonds, CT → ground) in (left to right) iso-octane, toluene, dichloromethane, and acetonitrile, plotted as functions of the solvent’s dipole moment. Trajectories were run with two values of the force constant for stretching the Zn−O bond: 100 kcal mol−1 Å−2 (red circles and cyan triangles) and 400 kcal mol−1 Å−2 (purple squares and blue diamonds); the results are essentially the same. The black circles are experimental values of ΔG° for charge separation.12,13,16 Figure 5. RMS electronic interaction matrix elements for charge separation (|1⟩ → |2⟩, red curves) and recombination to the ground state (|3⟩ → |0⟩, cyan) calculated during 5 ns trajectories of PQC in iso-octane (A), toluene (B), dichloromethane (C), and acetonitrile (D). Matrix elements for charge separation were evaluated in PQC’s lowest π−π* state (|1⟩) and considered configuration interactions of the quinone’s LUMO with nine molecular orbitals of the porphyrin (HOMO−3 through LUMO+3 before excitation). Those for charge recombination were evaluated in the lowest CT state (|3⟩) and considered only interactions of the HOMO of the quinone anion with the LUMO of the porphyrin cation. The bin width 2ε is 200 cm−1. The dashed black curves are least-squares polynomial fits that neglect points where the small probability of finding the system makes the calculations of |V| increasingly noisy.

the Q-band excitation energy for Zn porphyrin in methanol by a comparable amount (3500 cm−1), while TDDFT calculations overestimated the energy. Dreuw et al. obtained more accurate excitation energies with DFT/BLYP and SVWN methods.25 Although the absolute energies calculated here for the π−π* state thus are too low, and the same is likely to be true for the CT state, previous work encourages the supposition that the calculated differences between the π−π* and CT energies, and the effects of solvents on these differences, probably are more reliable.22,30,31 Because ΔE00 is evaluated after diagonalization of the Fock matrix, it is not the same as the 0−0 energy gap between the diabatic states that figure in eq 2 (x0 ≡ x0,0). However, ΔE00 and x0 should converge when x0 is large relative to the interaction matrix element |V|. The overall shapes of the probability distributions in Figures 2 and 3 can be described well by Gaussian functions except in the regions around ΔE00 = 0, where P(ΔE00) in some cases dips sharply below the values expected for a Gaussian. These deviations can be seen more clearly by narrowing the bin width and expanding the energy scale of the plots as illustrated in Figure S2. Since they are significant only where ΔE00 crosses zero, they can be ascribed reasonably to avoided crossings between the potential surfaces of the adiabatic reactant and product states rather than to local minima. They are much more prominent in the nonpolar solvents, iso-octane and toluene, than in dichloromethane or acetonitrile (Figure S2). Because P(ΔE00) remains well above zero as ΔE00 crosses zero in the polar solvents, while the electronic coupling between the diabatic states remains nonzero

separation and recombination as functions of ΔE00. The coupling factors for charge separation are considerably larger than those for recombination, and they depend strongly on the energy gap. The dashed lines in Figure 5 are polynomial fits, which are meaningful only where the probability of finding the system is substantial (cf. Figures 2 and 3). Qualitatively similar dependences of |V| on ΔE00 were seen in previous studies of charge separation in aqueous peptides,22,31 but not for chargeshift reactions of some organic free radicals.30 Vibronic Coupling and Vibrational Relaxations. Figure 6 displays the autocorrelation function of the time-dependent ΔE00 during a ground-state trajectory of PQC in iso-octane. Similar plots from trajectories in the other solvents are shown in Figure S3. The initial decay component with a time constant of about 6 fs (Figure 6A, insert) can be ascribed to the 3857


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Figure 7. Fourier transforms of the energy-gap autocorrelation functions from trajectories of PQC in the ground state (A) and the lowest CT state (B), both in iso-octane. The autocorrelation function had a total of 8192 time points, giving the Fourier transform a spectral resolution of 2 cm−1. The amplitudes are normalized relative to the strongest peak.

Figure 6. Autocorrelation functions of the time-dependent ΔE00 during a trajectory of PQC in the ground state in iso-octane displayed on several different time scales. The autocorrelation function is normalized to 1.0 at zero time. The solid curve in the inset in panel A is a least-squares fit of the first 21 data points to the function a1exp(−k1t) + a2sin(ν1t + ϕ), which returned the values a1 = 0.778 ± 0.024, k1 = 160 ± 19 ps−1, a2 = 0.296 ± 0.030, ν1 = 246 ± 17 ps−1, and ϕ = 2.25 ± 0.20 ps. Structures were saved for analysis at 1 fs intervals in a trajectory lasting 200 ps.

⎡⎛

kafd(xo) =

∑ ⎢⎢⎜⎜∑ x′

⎣⎝

i ,j

⎤ ⎞ ⎟G(x′ − x )⎥ o ⎥ ⎟ |V (xo)| Fi , j(xo) + k 23ℏ /2T2 ⎠ ⎦ k 23 |V (xo)|2 Fi , j(xo) 2

2

(3)

Here Fi,j(x0) is the Franck−Condon factor for a pair of states (vibrational sublevels i of |1⟩ and j of |2⟩) that are quasidegenerate when the 0−0 energy gap between diabatic states | 1⟩ and |2⟩ is in the small interval between x0 − ε and x0 + ε, weighted as in eq 2 to reflect the relative population of sublevel i; G(x′ − xo) is a lineshape function; and V(x0), k23, and T2 are as defined above. The electron-transfer rate constant then becomes

dephasing of vibrations that modulate the gap at different frequencies.22,44 These modulations are damped with a dominant time constant of about 4.6 ps (Figure 6B), which probably reflects relaxation from the initial CT state (|2⟩) to a more stable product (|3⟩) as energy diffuses to vibrational modes that are not coupled to the reaction.22,44 Additional slower decay components are seen in some of the autocorrelation functions in the two polar solvents (Figure S3). These slower components are not averaged reliably in the relatively short (200 ps) trajectories used to obtain the autocorrelation functions, but probably have little bearing on the initial charge-separation dynamics. Autocorrelation functions of ΔE00 from trajectories propagated in the CT state (not shown) were very similar to those for the ground state. Because the oscillatory components of the autocorrelation functions are similar in all of the solvents (Figure S3), they evidently stem mainly from intramolecular vibrational modes of PQC. In addition to numerous minor peaks with energies below 1000 cm−1, Fourier transforms of the autocorrelation functions have prominent peaks in the regions of 500, 1000, 1300, 1600, 1800, and 3300 cm−1 (Figure 7). Comparable features are seen in the spectra of calculated intramolecular vibronic coupling factors for the reduction of benzoquinones.26,29 Electron Transfer in a Multimode Harmonic Model. Equation 2 can be put in a more practical form by defining a weighted Franck−Condon-density function of the diabatic 0−0 energy gap, kafd(x0):

⟨kcalc⟩1 =

∑ kafd(xo)P(xo) xo

(4)

in which P(x0) is the probability of finding the diabatic 0−0 energy gap in the interval x0 ± ε during a QM/MM trajectory in |1⟩. The interaction matrix element |Vi,j(0)| of eq 2 is written as a function of x0 in eq 3 to recognize the dependences on ΔE00 illustrated in Figure 5. This is a significant approximation, since a given value of x0 represents many vibrational states of both the solvent and the electron carriers. In particular, it assumes that the electronic coupling of the zero-point sublevels of |1⟩ and |2⟩ can be used for all sublevels i and j. The approximation, nevertheless, seems preferable to using a constant value of |V| at all energies. The choices of ε and the lineshape function G are arbitrary but relatively unimportant, as long as ε is sufficiently small and G(x′ − xo) is sufficiently narrow relative to the distribution function P(x0). In principle, the actual lineshape function also depends on i and j. Evaluation of kafd is simplified considerably if the manifold of vibronic states that are coupled to electron transfer is represented by a model with a set of harmonic vibrational modes whose energies and vibronic coupling factors correspond to peaks in the Fourier transform of the autocorrelation 3858


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The Journal of Physical Chemistry B function of the energy gap.30,44 Table 1 presents the energies (hνi) and vibronic coupling factors (si) for such a model with

T2 was taken to be 6.25 fs (1/k1 in Figure 6A) in all cases. For Figure 8A, the relaxation parameter k23 was fixed at 0.44 ps−1 for all of the vibrational levels of the product. This is twice the rate constant for the damping of the oscillations in Figure 6B (2/4.6 ps), and is an estimate of the mean rate constant for equilibration of the n = 0 (zero-point) and n = 1 levels of a vibrational mode, which occurs approximately twice as rapidly as the overall damping.44 In Figure 8B, k23 was evaluated individually for each of the product’s initial vibrational states as described below. For curves 1, 2, and 4 of Figure 8A,B, |V| was fixed at 250, 500, and 750 cm−1, respectively. For curve 3, |V| was treated as the function of ΔE00 shown with a dashed line in Figure 5C for charge separation in dichloromethane. Increasing |V| raises kafd(x0) and shifts the peak to more negative values of the energy gap. A Gaussian with σ = 200 cm−1 was used as the lineshape function G. To evaluate k23(j) for individual vibrational states (Figure 8B), the sum of the rate constants for transitions of state j to and from all other vibrational states of |2⟩ was written as44,45

Table 1. Eight-Mode Harmonic Vibrational Model mode (i)

hνi (cm−1)

si

1 2 3 4 5 6 7 8

100 200 500 1000 1300 1600 1800 3300

0.30 0.30 0.70 0.60 0.20 1.20 0.40 0.10

eight vibrational modes. The coupling factors are scaled so that Σihvisi = λv, where λv is the calculated vibrational reorganization energy in the Mg-porphyrin-benzoquinone and bacteriopheophytin-ubiquinone systems analyzed by Borrelli et al.26,27 (4300 cm−1). Figure 8 shows kafd(x0) for the 8-mode model with several different treatments of |V| and k23. The dephasing time constant

8

k 23(j) = k 23(0) ∑

⎧ ⎪

∑ ⎨[δn ,n + 1nmj + δn ,n − 1nmk]

k≠j m=1

⎪

⎩

j m

[exp(εkj /kBT ) + 1]−1

k m

j m

k m

⎫

⎪ k μ⎪

∏ δn ,n ⎬ j μ

μ≠m

⎭

(5)

In this expression, k23(0) is the mean rate constant for equilibration of the n = 0 and n = 1 levels of a vibrational mode, taken again to be 0.44 ps−1 for all of the modes; njm is the initial excitation level of mode m in vibrational state j of |2⟩ (with j ranging from 0 to 14 in the present implementation); εkj is the difference between the total energies of the vibrational states j and k; and delta-function δs,t is 1 if s = t and zero otherwise. Equation 5 allows transitions between vibrational states only if the excitation level of a single mode changes by ±1, while the levels of all of the other modes remain the same. It also makes the simplifying assumption that vibrational transitions are linearly coupled to fluctuating fields with a broad power spectrum, rather than being dominated by vibrational resonances at discrete frequencies, and it does not accord special relaxation mechanisms to conical intersections. The mean k23(j) for states with a given total vibrational energy (⟨k23(Evib)⟩) increases with the energy as shown in Figure S4, partly because the factors njm and nkm tend to grow, and partly because the thermal factor [exp(εkj /kBT) + 1]−1 favors relaxations to lower levels with dissipation of heat to the surroundings. Ivanov and co-workers46−49 have used a

Figure 8. Function kafd(x0) for the 8-mode model (Table 1) calculated with relaxation rate constant k23 either fixed at 0.44 ps−1 for all vibrational levels of the initial product (A) or evaluated individually for each vibrational state as described by eq 5 (B). The dephasing time constant T2 was 6.25 fs, and the electronic coupling matrix element |V| was either 250 (red curve 1), 500 (cyan curve 2), 750 cm−1 (dark blue curve 4), or represented by the function shown with the dashed curve in Figure 5C for charge separation in dichloromethane (magenta curve 3). The calculations included the 1.24 × 106 vibrational sublevels of |2⟩ with energies up to 15 100 cm−1 above the zero-point level and a Boltzmann distribution of sublevels of |1⟩ with vibrational energies up to 1000 cm−1 at 293 K.

Table 2. Calculateda and Observedb Rate Constants for Charge Separation solvent c

iso-octane toluene dichloromethane acetonitrile

kcalc(0)/s−1 0.82 0.31 0.98 0.89

× × × ×

1012 1012 1012 1012

kcalc(1)/s−1 2.03 0.84 1.91 1.63

× × × ×

kcalc(2)/s−1

1012 1012 1012 1012

3.34 1.47 2.37 2.61

× × × ×

1012 1012 1012 1012

kcalc(3)/s−1 4.46 2.30 4.19 3.69

× × × ×

1012 1012 1012 1012

kobs/s−1 2.21 2.02 2.06 2.55

× × × ×

1012 1012 1012 1012

Four calculated values of the rate constant are given for each solvent: for kcalc(0), P(x0) was represented by a Gaussian fit to the ΔE00 distribution shown with red symbols (curve 1) in the panel for that solvent in Figure 2; for kcalc(1), kcalc(2), and kcalc(3), the calculated 0−0 energy gaps were shifted by −1000, −2000, and −3000 cm−1, respectively. The electronic coupling factor |V(x0)| was represented by the polynomial function shown with a dashed curve through the red symbols (|1⟩ → |2⟩) in the panel for that solvent in Figure 5 and fixed at its final value on either side of the indicated range. The values of T2 and k23 given in the legend to Figure 8B were used for all four solvents. bThe observed rate constant (kobs) was taken to be a1/τ1 + a2/τ2, where τ1, τ2, a1, and a2 are the time constants and fractional amplitudes reported16 for its two exponential components. c kobs in this row was measured in n-hexane.16 a

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The Journal of Physical Chemistry B Table 3. Calculateda Rate Constants for Charge Recombination solvent

kcalc(0)/s−1

kcalc(1)/s−1

kcalc(2)/s−1

kcalc(3)/s−1

iso-octane toluene dichloromethane acetonitrile

× × × ×

× × × ×

× × × ×

× × × ×

2.23 2.66 1.72 1.30

10

10 1010 1011 1011

1.33 2.02 1.68 1.44

10

10 1010 1011 1011

8.39 1.52 1.56 1.41

9

10 1010 1011 1011

5.48 1.09 1.39 1.24

9

10 1010 1011 1011

kcalc(4)/s−1 3.42 7.42 1.20 1.00

× × × ×

109 109 1011 1011

Five calculated values of the rate constant are given for each solvent: kcalc(0) was obtained by using a Gaussian function fit to the calculated ΔE00 distribution shown with cyan symbols (curve 3) in the panel for that solvent in Figure 3; for kcalc(1), kcalc(2), kcalc(3), and kcalc(4), the calculated energy gaps were shifted by −1000, −2000, −3000, and −4000 cm−1, respectively. The electronic coupling factor |V(x0)| was represented by the polynomial function shown with a dashed curve through the cyan symbols (|3⟩ → |0⟩) in the panel for that solvent in Figure 5 and fixed at its final value on either side of the indicated range. See Table 2 and Figure 8 for other details. a

reactions depend strongly on the solvent. The probability distributions in Figure 2, or the free-energy functionals derived from them [Δg(x0) = −kBTln(P(x0))], can be compared with the corresponding functions for the charge-shift reactions (see Figures 2−4 and S2−S5 of ref 30). In both cases, polar solvents stabilize the product of the reaction during QM/MM trajectories propagated in the final state (|3⟩). Trajectories run in the reactant state, however, give different results. Whereas polar solvents shift P(ΔE00) for |1⟩ in PQC only slightly (Figure 2), the probability functions for |1⟩ in the charge-shift reactions move to considerably higher energies.30 The difference can be attributed to the fact that the π−π* state in PQC is uncharged and has a dipole moment of only about 0.6 D, and so it interacts only weakly with the solvent, while the reactants in the charge-shift reactions are anionic. One consequence of the different nature of the reactant states is that polar solvents make ΔG°1,3 more negative for charge separation in PQC (Figure 4B), but have little effect on ΔG°1,3 in the charge-shift reactions.30 Equation 1 and the classical Marcus equation55 could have difficulty dealing with solvent effects in some systems because, apart from problems associated with |V| and the need to adjust multiple parameters phenomenologically, they focus on the overall free energy change and reorganization energy rather than on P(x0) for the reactant state. Although calculated values of the free energy changes and reorganization energies are presented above for purposes of discussion, eqs 2−5 do not use any of these quantities. Charge-Recombination Rate Constants. Rate constants for charge recombination were calculated in the same manner as for charge separation by using Gaussian fits to the probability distributions for the CT state in Figure 3 with the corresponding coupling factors from Figure 5. The calculated rate constants (Table 3) are smaller than those for charge separation by factors of 10 to 100. Unfortunately, there are no experimental values for comparison here, because these have been reported only for zinc-free analogs of PQC. As noted above, errors in the calculated energy of the CT state relative to the ground state probably are larger than errors in the energy difference between the π−π* and CT states. There is, however, a qualitative difference between the calculated rate constants for charge separation and recombination. While making the energy gap more negative increases the calculated rate constant for charge separation in all four solvents (Table 2), it decreases the rate constant for recombination in nonpolar solvents (Table 3). Charge recombination in nonpolar solvents thus can be assigned mainly to the left-hand side of the kafd(x0) plots in Figure 8. This limb of the kafd plots corresponds to the “inverted” region of the Marcus curve,20,21,55 where decreasing of the Franck−Condon factors slows the reaction even though

treatment of vibrational relaxations that is similar to eq 5 but does not include the thermal factor, which is needed for microscopic reversibility. The leveling off of ⟨k23(Evib)⟩ at high energies seen in Figure S4 probably reflects the limited number of vibrational modes in the model and the truncation of njm and nkm at 14. However, the reliability of eq 5 becomes increasingly dubious at high energies, where anharmonicity would cause mixing of the individual normal modes. Inspection of Figure 8A,B shows that evaluating k 23 individually for each vibrational state increases kafd(x0) at moderately negative values of x0, depending on the magnitude of |V|. The effect can be substantial if |V| is large, but diminishes at both positive and strongly negative values of x0, where the thermally weighted Franck−Condon factors become small. Charge-Separation Rate Constants. Table 2 presents rate constants for charge separation calculated by eqs 3−5 for the 8-mode harmonic model along with the observed rate constants for comparison. The polynomial functions shown in Figure 5 are used for the electronic coupling factors, P(x0) is represented by Gaussian functions fit to the probabilitydistribution functions (P(ΔE00)) for the π−π* state in Figure 2, and k23 is evaluated individually for each vibrational state. To examine the effect of uncertainties in the calculated energies, the 0−0 energies in the P(x0) and |V(x0)| functions were shifted by various amounts ranging from 0 to −3000 cm−1. The rate constants obtained using the calculated energies agree with the measured values to within about a factor of 3, and shifts of the energies by only about 2000 cm−1 suffice to make the two identical. The overall agreement seems remarkably good (and perhaps partly fortuitous), considering the approximations that remain in the theory. The agreement between the calculated and measured rate constants for charge separation holds to about the same accuracy for the reactions in dichloromethane and acetonitrile as for those in iso-octane and toluene. This is noteworthy because Figures 2 and S2 suggest that the reactions occur at conical intersections in the two polar solvents but at avoided crossings in the nonpolar solvents. Since the measured chargeseparation rates are similar in polar and nonpolar solvents, the results call into question the view that conical intersections provide especially efficient funnels for photochemical electron transfer.24,50−52 Although it would be inappropriate to draw broad conclusions from the small set of reactions considered here, eqs 3−5 appear to handle conical intersections as well as avoided crossings. It is instructive to compare the solvent dependence of photochemical charge separation in PQC with that of groundstate, charge-shift reactions in which an electron moves from one organic group to another on a hydrocarbon scaffold.30,53,54 Unlike charge separation in PQC, the rates of the charge-shift 3860


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and conformational dynamics of a system with minimal recourse to adjustable parameters that, though conceptually useful, may not have much quantitative significance. The theory now has been extended to include a dependence of the relaxation rate constant k23 on the initial vibrational state of the CT product. The results obtained with PQC provide insights into the different effects of solvents on charge-separation, charge-recombination, and charge-shift reactions, and they show that eqs 2−5 can be used for systems with strong electronic coupling, as well as for systems with weaker coupling, and for reactions that occur at conical intersections, as well as at avoided crossings. The same expressions should apply to reactions that depend on conformational changes in the reactant or slow motions of the solvent, although longer QM/MM trajectories probably would be needed to capture P(x0) reliably for such systems.

electron transfer becomes more favorable thermodynamically. Numerous workers have noted that charge recombination to the ground state typically is considerably slower than photochemical charge separation, and that this could reflect the unfavorable Franck−Condon factors associated with strongly negative energy gaps. In PQC, however, the lower recombination rate appears to stem at least partly from smaller values of |V| (Figure 5), rather than exclusively from less favorable Franck−Condon factors. While increasing the polarity of the solvent has little effect on the calculated rate constant for charge separation (Table 2), it increases that for the recombination to the ground state (Table 3). This is qualitatively in accord with experimental observations on zinc-free analogs of PQC13 and rigid porphyrin-quinone dyads, both with and without zinc.14 Conversely, shifting the energy gap has less effect on the calculated rate constant for charge recombination. These differences are explained by the fact that, unlike the relatively nonpolar π−π* state, the CT reactant state for the recombination has a dipole moment of about 3.0 D. Polar solvents tend to stabilize the CT state, making ΔE00 less negative (Figure 3). Sufficiently strong interactions with the solvent can move charge recombination to the region of the peak or the right-hand limb of the kafd(x0) plot (Figure 8), which corresponds to the “normal” region of the Marcus curve.20,21,55 A low rate of charge recombination then must be attributed primarily to smaller values of |V| (Figure 5). Additional Comments and Conclusions. The weighted Franck−Condon factors Fi,j(x0) in eq 3 were calculated on the assumption that |1⟩ had reached a Boltzmann distribution of vibrational states at the temperature of the QM/MM simulation. This assumption is potentially problematic because vibrational equilibration in porphyrins typically requires about 10 ps,56−58 which is considerably longer than the time scale of electron transfer. Because the probability functions P(x0) shown in Figures 2 and S2 represent averages over ensembles that have been in the π−π* state for times up to 5 ns, they also might not give proper weighting to excited vibrational states that are populated transiently by the excitation flash. However, departures from the vibrational equilibrium probably are not a major concern for the reactions considered here, because the 575 nm excitation flashes that were used experimentally to promote PQC to the π−π* state left the excited molecule with very little excess vibrational energy.16 In situations when vibrational disequilibrium needs to be treated, one can average probability functions from multiple short trajectories, starting with projections to the π−π* state from random configurations of the ground state.31,59 Equation 3 also could be averaged over a time-dependent distribution of the vibrational states in a density-matrix model of the system.44,60 From the probability functions shown in Figures 2 and S2, one would expect charge separation in PQC to have an activation energy on the order of 100 cm−1 (0.012 eV). However, the rate was measured only at 293 K,16 and the simulations described here were all performed for this temperature. The probability function P(x0), T2, k23, the lineshape function G(x′ − xo), and the weighting factor for the relative population of vibrational sublevel i could have a variety of dependences on the temperature. The use of eqs 2−5 for treating the temperature dependence of electron-transfer rates therefore remains to be explored. In conclusion, the approach used here appears to offer a powerful way to relate electron-transfer rates to the structure

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ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jpcb.8b01072. Atomic sigma charges for PQC; numbering scheme for PQC, and angles and bond lengths in the zinc-porphyrin force field; sigma charges of solvent atoms; highresolution plots of 0−0 energy probability distributions; autocorrelation functions of energy gaps in four solvents; and mean relaxation rate constant for individual vibrational states as a function of the vibrational energy (PDF)

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]; Phone: (206) 523-0142. ORCID

William W. Parson: 0000-0003-1805-8000 Notes

The author declares no competing financial interest.

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ACKNOWLEDGMENTS I thank Bill Hazelton, Maria-Elisabeth Michel-Beyerle and Arieh Warshel for helpful comments on the manuscript. This work was not supported by external funding.

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ABBREVIATIONS CASSCF, complete active space self-consistent field; CT, charge-transfer; DFT/BLYP, density functional theory using the Becke−Lee−Yang−Parr functional; HOMO, highest occupied molecular orbital; LUMO, lowest unoccupied molecular orbital; PQC, 5,15-[3,6-dimethyl-p-benzoquinone1,4-diylbis(4,1-butanediyl-2,1-benzeno)]-2,3,7,8,12,13,17,18-octamethylporphyrin zinc; QM/MM, quantum mechanics/ molecular mechanics; RMS, root-mean-square; TDDFT, timedependent density-functional theory; TDHF-INDO/S, timedependent Hartree−Fock theory with intermediate neglect of differential overlap; QCFF/PI, quantum consistent force field for π electrons; SVWN, density functional theory using the Slater−Vosko−Wilk−Nusair functional

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REFERENCES

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Electron-Transfer Dynamics in a Zn-Porphyrin-Quinone Cyclophane

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