Temperature Dependence of the Rate of Intramolecular Electron

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Temperature Dependence of the Rate of Intramolecular Electron Transfer William W. Parson J. Phys. Chem. B, Just Accepted Manuscript • DOI: 10.1021/acs.jpcb.8b06497 • Publication Date (Web): 04 Sep 2018 Downloaded from http://pubs.acs.org on September 5, 2018

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The Journal of Physical Chemistry

Temperature Dependence of the Rate of Intramolecular Electron Transfer

William W. Parson*†

†

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

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

ACS Paragon Plus Environment

The Journal of Physical Chemistry 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

ABSTRACT: Quantum mechanical/molecular mechanical simulations are used to explore the temperature dependence of intramolecular electron-transfer rates in systems that represent both the “normal” and “inverted” regions of the Marcus curve. The treatment uses an approach that includes effects of vibrational relaxations and dephasing and is largely free of adjustable parameters. Effects of temperature on the distribution of the energy gap between the reactant and product (P(xo)), the electronic-interaction matrix element, and the rates of dephasing and vibrational relaxations are considered. The simulations reproduce the measured rate constant and temperature dependence well for photochemical charge separation in a porphyrin-benzoquinone cyclophane and for a ground-state charge-shift reaction in a biphenylyl-androstane-naphthylyl radical. They overestimate the rate of the charge-shift reaction in a biphenylyl-androstanebenzoquinone adduct, but are in accord with the observation that this reaction is almost independent of temperature. Arrhenius plots of rate constants calculated with various P(xo) distributions show that the apparent activation enthalpy depends on whether or not P(xo) shifts with temperature.

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INTRODUCTION In contrast to the rates of endergonic reactions, which typically increase with temperature in accord with the Arrhenius equation, the rates of exergonic intramolecular electron-transfer reactions sometimes remain nearly constant or even decrease. Explanations of this behavior often focus on the magnitude of the reorganization energy (l, the change in free energy associated with relaxation of the products from the initial to the final nuclear configuration), relative to the change in standard free energy (DG). Intramolecular electron-transfer reactions that occur in the “inverted region” of the Marcus curve,1-5 where DG + l < 0, tend to be insensitive to temperature. This observation is explained by the notion that, whereas the reactants in endergonic reactions require thermal excitation to access the lowest vibrational level of the products, the reactants in an exergonic reaction can have productive nuclear overlaps with excited vibrational states of the products.6-11 The importance of vibrational overlaps can be formulated quantitatively by eq 1, in which electron transfer is coupled to a harmonic intramolecular vibrational mode with frequency n and reorganization energy lv:6-10 𝑘"# =

%&' ( ℏ

4𝜋𝜆- 𝑘. 𝑇

01/% 04v /67 𝑒

4v /67 8 D 𝑒𝑥𝑝 9EF 9!

− ∆𝐺 + 𝜆- + 𝑛ℎ𝜈 % /4𝜆- 𝑘. 𝑇 .

(1)

Here ket is the electron-transfer rate constant, V is the electronic-interaction matrix element that mixes the diabatic initial and final states, and ls is a portion of the reorganization energy that is treated classically and usually is attributed mainly to the solvent. The term 𝑒 04v/67

4v /67 8 9!

is the

Franck-Condon factor for transitions from the reactant’s lowest vibrational level to vibrational level n of the product. According to eq 1, the activation free energy for forming the product in vibrational level n is (DG + ls + nhn)2/4ls, which goes to zero for n = -(DG + ls)/hn. Since n cannot be negative, this matching occurs only if DG + ls ≤ 0. Changes of V, DG or ls with temperature can give nonlinear Arrhenius plots and reactions that speed up at low temperatures.12-14 This treatment has been used to rationalize the temperature dependence of many intramolecular electron-transfer reactions. In pioneering experiments, Miller, Closs and colleagues attached a biphenylyl group and a benzoquinonyl, naphthylyl or other electron-

3



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accepting group to a rigid hydrocarbon (androstane) spacer, converted one of the groups to a free radical by pulse radiolysis, and measured the rate of electron transfer between the two groups.1519

Transfer from a biphenylyl to a naphthylyl group, for which DG + ls was judged to be

positive, sped up with increasing temperature, while transfer to a benzoquinonyl group, with DG + ls negative, was almost independent of temperature.17-18 In other important experiments, Michel-Beyerle, Schaab and colleagues studied photochemical charge separation and recombination in porphyrin-quinone cyclophanes.20-23 The rate of electron transfer from the excited porphyrin to a benzoquinonyl group changed only slightly with temperature. Waskasi et al.13 described a fullerene-porphyrin dyad in which the charge-recombination rate rose about 3fold with decreasing temperature down to 150 K before dropping off at lower temperatures. Reactions that speed up at low temperatures are well known in photosynthetic reaction centers, with electron transfer from bacteriopheophytin to ubiquinone and the back reaction from ubiquinone to the special pair of bacteriochlorophylls in bacterial reaction centers being examples.24-26 In spite of its conceptual importance, eq 1 has several limitations. It considers only a single vibrational mode, and it becomes unreliable if |V| is large enough so that ket is limited by the rate of vibrational relaxations that take the initial product state out of resonance with the reactant.27 In addition, the flexibility afforded by its five adjustable parameters (DG, n, lv, ls and V) could allow the experimental data for some systems to be fit with values that have little physical significance. The danger of overfitting the data grows if the reactant and product are assumed to have non-parabolic potential surfaces, or if DG, V or ls is given a dependence on temperature. For a more general treatment that considers vibrational relaxations and dephasing, and that is less dependent on adjustable parameters, the rate constant can be written27-29 𝑘"# ≈

OP 𝑘HIJ (𝑥L )

𝑃 𝑥L ,

(2)

with 𝑘HIJ (𝑥o ) =

O[

𝑖,𝑗

2 𝑘23 (𝑗) 𝑉(Oo ) 𝐹𝑖,𝑗 (Oo ) 2

𝑉(Oo) 𝐹𝑖,𝑗 (Oo) + 𝑘23 (𝑗)ℏ2 2𝑇2

𝐺 𝑥′ − 𝑥o .

(3)

In eq 2, P(xo) is the probability of finding the diabatic zero-zero energy gap (DE00) between the reactant (|1ñ) and the initial product (|2ñ) within a small span of energies (± e) around the value xo. In eq 3, Fi,j(xo) is the Franck-Condon factor for a pair of states (vibrational sublevels i of the

4


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reactant and j of the product) that are quasi-degenerate when DE00 = xo, weighted with a Boltzmann factor for the relative population of sublevel i at a specified temperature; G(x¢-xo) is a lineshape function for DE00; T2 is the effective time constant for decay of coherence between |1ñ and |2ñ; k23(j) is the rate constant for relaxations that convert |2ñ from vibrational level j to a more stable vibronic state (|3ñ) by dissipating energy to vibrational modes that are not coupled to electron transfer; and V(xo) is the RMS value of the electronic matrix element, with the recognition that this could vary with xo. In favorable situations, all the quantities needed to apply eqs 2 and 3 to a particular system can be obtained from quantum calculations and quantum mechanical/molecular mechanical (QM/MM) simulations, although approximations in the quantum treatment can necessitate some shifting of the calculated energies. This approach has been used to calculate rate constants for electron transfer in a variety of systems at ambient temperatures.27-29 It appears to be suitable for reactions with |V| ranging from below 10 cm-1 to more than 700 cm-1, and for at least some reactions that take place at conical intersections as well as those that occur at avoided crossings. It is not immediately clear whether eqs 2 and 3 can reproduce the temperature dependences of reactions in both the inverted and “normal” regions of the Marcus curve, since these expressions differ from eq 1 in numerous ways. The present study examines this question. Three representative systems are considered: transfer of an electron from the biphenylyl radical to the benzoquinonyl group in a biphenylyl-androstane-benzoquinonyl adduct (BIP-BQO) in methyltetrahydrofuran falls in the inverted region,15, 19 while transfer to the naphthylyl group in a biphenylyl-androstane-naphthylyl adduct (BIP-NAP) occurs in the normal region;15, 19 photochemical charge separation in the porphyrin-quinone cyclophane 5,15-[p-benzoquinone1,4-diylbis(4,1-butanediyl-2,1-benzeno)]-10,20-diphenylporphyrin (HPQ) in dichloromethane occurs close to the peak of the Marcus curve, where DG + l = 0.21 Structures of the three molecules are shown in Supporting Information Figure S1. The treatment used here assumes that the reactant establishes a Boltzmann distribution of vibrational energies rapidly on the time scale of electron transfer. Photochemical reactions that precede vibrational thermalization require a different approach such as a density-matrix treatment if the initial system is significantly out of equilibrium.30-32 Although photochemical charge separation in HPQ is rapid enough to challenge this assumption, the excitation pulses that

5



were used experimentally for this system provided little excess vibrational energy,21 and the P(xo) distribution functions in the ground and excited states differ only slightly (see below). METHODS Quantum mechanical/molecular mechanical simulations and calculations of Franck-Condon factors and electronic-interaction matrix elements were performed as described,27-29, 33 using the restricted Hartree-Fock (RHF) programs ENZYQ and INDIP for photochemical charge separation in HPQ, and their unrestricted Hartree-Fock (UHF) versions (ENZYQ-uhf and INDIP-uhf) for the reactions of the BIP-BQO and BIP-NAP radicals. Each system included an all-atom bath of iso-octane, dichloromethane or 2-methyltetrahydrofuran composed of the number of solvent molecules required to fill a sphere with a radius of 23.0 Å to the proper density at atmospheric pressure and 296 K. The volume of the sphere was adjusted as necessary to give the measured densities34-36 of the solvents at other temperatures. The atomic charges, polarizabilities, dipole moments and MM parameters for the solvents were the same as used previously.28-29, 33 The atomic charges of 2-methyltetrahydrofuran were from Tan et al.37 Fields from the induced dipoles of solvent atoms were evaluated at each step, and interactions with these fields contributed to the energies of the subsystem that was treated quantum mechanically. The classical energies of changing the induced dipoles also were included in DE00. Forces from interactions with induced dipoles were represented implicitly in the permanent charges of the solvent atoms, and indirectly through the changing charge-charge interactions of quantum atoms with the solvent, but were not evaluated explicitly.33 The extent to which this simplification affects calculated values of ket remains to be explored. The solvated system was equilibrated by a 10 ps QM/MM trajectory at 293 or 296 K, followed by 0.5 ns at 340 K. Trajectories lasting 4 ns then were run in both the reactant and product electronic states at decreasing temperatures from 340 to 180 K. Trajectories of 2 ns also were run for HPQ in the ground state at some temperatures. Each of the trajectories below 340 K was preceded by an equilibration period of 10 ps starting with the coordinates recorded at 0.5 ns at the previous temperature. Structures were saved every 100 fs for analysis in most of the simulations, and for higher time resolution, at 1 fs intervals in simulations of HPQ in iso-octane. The iso-octane series consisted of five trajectories of 200 ps at each temperature, starting from coordinates recorded after 1.0, 1.5, 2.0, 2.5 and 3.0 ns in the longer trajectories. As explained

6


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previously,28 the UHF programs used for BIP-BQO and BIP-NAP required separate trajectories with an unpaired electron on the electron donor or acceptor. F Relaxation rate constants 𝑘%\ and 1/T2 were obtained from autocorrelation functions of the

time-dependent energy gap essentially as described,27-29 with the following systemization of the F procedure for extracting 𝑘%\ . Since the autocorrelation function A(t) oscillates between positive

and negative values as it decays to noise and small oscillations at long times (Figure S2), |A(t)| first was transformed to the monotonically decreasing function of time 𝐼 𝑡- = 𝐼_ − #b F

𝐴 𝑡 𝑑𝑡, where Im is the value of the integral

#c F

𝐴 𝑡 𝑑𝑡 at the end of the autocorrelation

function (tm ≈ 33 ps). The last 2/3 of I(ts) (ts ~11 to 33 ps) was fit to a straight line, I(ts) = a + bts, and the difference, I(ts) - a - bts, then was fit to an exponential, Dexp(-krts) plus a constant, for ts F from 1 fs to tm. 𝑘%\ was taken to be 2kr.

RESULTS AND DISCUSSION Probability Distributions, Free Energy Changes and Reorganization Energies. Figure 1 shows the distributions of the calculated 0-0 energy gaps for charge-separation in HPQ and the charge-shift reactions of BIP-NAP and BIP-BQO at various temperatures between 180 and 340 K. The panels in the upper row (A, C and E) represent trajectories on the potential surface of the reactant electronic state (|1ñ), and those in the lower row (B, D and F) represent trajectories on the potential of the product (|3ñ). To match conditions that were used experimentally,17-18, 21 the simulations of HPQ (A and B) were run in dichloromethane, and those of BIP-NAP (C and D) and BIP-BQO (E and F) in 2-methyltetrahydrofuran. Whereas the UHF calculations28 for BIPNAP and BIP-BQO (C-F) give the energy differences (xo) between the diabatic reactant and initial product states (|1ñ and |2ñ) directly, the RHF calculations for HPQ (A and B) provide adiabatic energies with an avoided crossing between the reactant and product surfaces. (Note the sharp spikes near DE00 = 0 in Figure 1A.) Previous calculations29 revealed similar avoided crossings for charge separation in a zinc-porphyrin-benzoquinone cyclophane in iso-octane or toluene but not in the more polar solvents dicloromethane and methyltetrahydrofuran, where the reactions of the zinc porphyrin appear to occur at conical intersections. The diabatic energies that enter into eqs 2 and 3 can be obtained by fitting the main peak of the adiabatic energy distribution to a Gaussian.29 In addition to this peak, the distributions of energies from

7



trajectories of HPQ in the product state have a minor peak that becomes increasingly prominent at low temperatures. Lowering the temperature sharpens the P(xo) distributions and shifts the distributions in the product state to more negative energies (Figures 1B, D and F). The P(xo) distributions from trajectories of BIP-BQO or BIP-NAP in the reactant state all shift to higher energies (Figure 1C and E), but those for HPQ in the reactant (p-p*) state hardly move (Figure 1A). This is in accord with the fact that HPQ has no net charge and only a small dipole moment in the reactant state but acquires a substantial dipole after charge separation, whereas the BIP-NAP and BIP-BQO radicals have a charge of -1 that moves from the biphenylyl group to the acceptor. The negative entropy change that opposes ordering of the solvent decreases in importance at low temperatures, allowing polar solvents to rearrange in ways that stabilize the charge-transfer product in HPQ, and stabilize either the reactant or the product in the charge-shift reactions depending on the location of the charge. HPQ

BIP-NAP

BIP-BQO

Relative Probability

1.0

A

0.8

C

E

0.6 0.4 0.2 0

-10

-5

0

5

0

5

10

15

-15

-10

-5

0

1.0

Relative Probability

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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B

0.8

D

F

0.6 0.4 0.2 0

-20

-15

-10

-5 -1

∆E00 / 1000 cm

-20

-15

-10

-5 -1

∆E00 / 1000 cm

-35

-30

-25

-20 -1

∆E00 / 1000 cm

Figure 1. Probability of finding DE00 within ±100 cm-1 of the value given on the abscissa, during QM/MM trajectories of (A) HPQ in the lowest p-p* excited state, (B) HPQ in the lowest 8


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charge-transfer state, (C) BIP-NAP with an unpaired electron on BIP, (D) BIP-NAP with an unpaired electron on NAP, (E) BIP-BQO with an unpaired electron on BIP, and (F) BIP-BQO with an unpaired electron on BQO. In A and B, DE00 is the adiabatic energy difference including configuration interactions after diagonalizing the Fock Hamiltonian; in C – F it is the difference between the diabatic energies of the highest-occupied molecular orbitals when the unpaired electron is on one or the other carrier. The energies for BIP-NAP were shifted by 2,200 to 2,300 cm-1 to align them with the free energies measured experimentally by Liang et al.;17 no shifting was used for BIP-BQO or HPQ. Temperatures from 180 to 340 K are indicated by colors from dark violet to red. See the text for other details. -----------Figure 2 offers a more quantitative account of how the peak positions (µ) and standard deviations (s) of the energy distributions change with temperature. Panels A, B and C are for charge separation in HPQ; D, E and F, for the charge-shift reaction in BIP-NAP; and G, H and I, for the charge-shift reaction in BIP-BQO. Panels C, F and I give the free energy changes (DG) and solvent reorganization energies (ls) extracted from the energy distributions, along with the sums of DG + ls. DG was calculated as (µr + µp)/2 and ls as |µr - µp|/2, where µr and µp are the means of Gaussian fits to the probability functions in the reactant and product states, respectively.38 The latter expression assumes that the reactant and product Gaussians have the same variances, which is only approximately the case; as shown in Figure 2B, E, H, the standard deviations of the Gaussians for the product states (sp) are somewhat smaller than those for the reactants (sr). Lowering the temperature increases |µr - µp| and thus increases ls for all three reactions. Stabilization of the product at low temperatures makes DG more negative for charge separation in HPQ by decreasing µp, but has little effect on DG for either of the charge-shift reactions, in which µp and µr change in opposite directions. The very weak temperature dependence of DG for BIP-BQO in methyltetrahydrofuran agrees well with experimental measurements by Liang et al.17 The approximately linear decreases in ls with temperature in all three systems are comparable to the effects observed by Vath et al.39-40 in measurements of charge-transfer absorption and emission bands, and are similar to the effect obtained with a dielectric continuum model of the solvent for BIP-BQO.17 Matyushov’s SolvMol model predicts larger, nonlinear

9



changes in ls.14, 41 As indicated above, DG + ls is positive for BIP-NAP, negative for BIP-NAP and slightly below zero for HPQ; it decreases with temperature in both BIP-BQO and BIP-NAP, but is essentially constant in HPQ. Note, however, that eqs 2 and 3 make no direct use of either

ls or DG. Panels A and B of Figure 2 also include the peak positions and widths of P(xo) from trajectories of HPQ in the ground state. These distributions peak at slightly higher energies and are slightly broader than the distributions in the excited state. The rearrangement of charge that accompanies excitation of HPQ to the p-p* state, though relatively minor, thus evidently favors the electron-transfer reaction that follows. HPQ

BIP-NAP

BIP-BQO

0

0

1

-1

3

3

2

1

1

H

1 0 -1

C

0

λs, ∆G+λs / eV

λs, ∆G+λs / eV

0

1 0 -1

F

∆G,

λs, ∆G+λs / eV

2

E

0

200

G -30

B

-2

-20

-10

-1

-1

2

σ / 1000 cm

σ / 1000 cm

-1

-15 3

D

σ / 1000 cm

A

0

-10

-2 250

300

Temperature / K

350

1 0 -1

I

∆G,

-10

µ / 1000 cm

-1

-5

µ / 1000 cm

µ / 1000 cm

-1

10

∆G,

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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-2 200

250

300

350

Temperature / K

200

250

300

350

Temperature / K

Figure 2. A, mean (µ) of Gaussian fit to the main peak of the probability function of DE00 for charge separation in HPQ, calculated from trajectories in the ground state (yellow triangles), p-p* state (red circles) or product CT state (cyan squares). B, standard deviations (s) of the

10


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Gaussians with corresponding symbols in A. C, standard free energy change (DG, blue squares), solvent reorganization energy (ls, blue circles) and DG+ls (green empty circles) calculated from

µ and s for the reactant and product trajectories in A. D, same as A for charge transfer in BIPNAP with the unpaired electron on either BIP (red circles) or NAP (cyan squares). E, same as B for charge transfer in BIP-NAP. F, same as C for charge transfer in BIP-NAP. G, same as D for charge transfer in BIP-BQO with the unpaired electron on BIP or BQO. H, same as E for BIPBQO. I, same as F for BIP-BQO. See Figure 1 and the text for details. Note that the vertical scales differ in A, D and G. -----------Along with dynamic fluctuations, the variances of the P(xo) distributions (sr2 and sp2) include static inhomogeneity reflecting barriers to transitions between different configurations of the solvent. In particular, a trajectory of finite length in the reactant state is unlikely to visit all the configurations that become important in the product state, and vice versa. One consequence of this limited sampling of configurational space is that the free energy functionals defined as gi(xo) = -kBTln[Pi(xo)], with Pi(xo) being the distribution of energy gaps during a trajectory in state i, do not conform to the idealized relationship gp(xo) = gr(xo) + xo.38 In addition, sr2 and sp2 do not necessarily go to zero with kBT, and the solvent reorganization energies calculated as ls = |µr - µp|/2 are smaller than those given by the expressions ls,r = sr2/2kBT and ls,p = sp2/2kBT, which assume that gp(xo) = gr(xo) + xo. Ghorai and Matyushov have reported on similar discrepancies between ls, ls,r and ls,p in simulations of charge-transfer reactions in polar solvents, and have related them to the reorganization entropy.42-43 Free energy functionals that do satisfy the relationship gp(xo) = gr(xo) + xo can be obtained by free-energy-perturbation calculations, and in some cases simply by combining probability distributions from trajectories in the reactant and product states.38 However, these treatments probably would not be appropriate for the problem at hand because slow relaxations to configurations that are populated significantly only after the reaction have little or no bearing on the rate constant for the forward direction. To examine whether moderate increases in the lengths of the trajectories would modify the calculated P(xo) distributions substantially, the trajectories of HPQ at 180 K were continued for an additional 4 ns; µr became more negative by 1.4%, but µp, sr and sp all changed by less than 1%.

11



Dephasing and Vibrational Relaxations. In addition to P(xo), T2, k23 and |V| in eq 3 all could depend on temperature. The Boltzmann weighting of the Franck-Condon factors of course changes with temperature, but the Franck-Condon factors themselves are less likely to change, and changes in the lineshape function G(x¢-xo) have little influence on ket as long as the linewidth remains narrower than the P(xo) distribution. Effects of temperature on T2 and k23 were examined for the charge-separation reaction of HPQ in iso-octane, where the main components of dephasing and vibrational relaxations are well resolved in autocorrelation functions of the energy gap (Figure S2). A Fourier transform of the autocorrelation function (Figure S3) provides most of the information needed to construct a model with a reduced set of harmonic modes, which F allows the vibrational relaxation rate constant k23(j) to be expressed as a function of j and 𝑘%\ , F is approximately the rate constant for equilibration of the two lowest levels of a mode.29 𝑘%\

twice the overall rate constant for decay of oscillations in the autocorrelation function.30 Such vibronic models have been described previously for BIP-BQO and BIP-NAP,28 and Table S1 presents one for HPQ. The vibrational reorganization energy of 4300 cm-1 in the HPQ model comes from quantum calculations by Borrelli et al. on other porphyrin-benzoquinone complexes.44-46 Figure S4 shows the function kafd(xo) calculated for this model at 300 K. F for HPQ in As shown in Figure 3, raising the temperature from 180 to 340 K increases 𝑘%\

iso-octane by about 25% (0.76±0.14´109 s-1K-1), and reduces the effective rate constant for dephasing (1/T2) by a similar percentage (–0.26±0.07´1012 s-1K-1). In themselves, these changes F and T2 would cause ket for HPQ to increase slightly with temperature. in 𝑘%\

12


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Page 13 of 29

0.8

0.4

1

F 𝑘%\ / ps

-

0.6

0.2

A

0

-1

250

1/T2 / ps

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


200 150 100 50 0

B 200

250 300 Temperature / K

350

F Figure 3. A, rate constant (𝑘%\ ) for equilibration of the two lowest levels of a vibrational mode,

calculated as twice the rate constant (kr) for decay of the main oscillations in the autocorrelation function of the energy gap for charge separation in HPQ in iso-octane. B, rate constant (1/T2) for decay of coherence between the reactant and initial product states for the same reaction, calculated by fitting the first 32 fs of the autocorrelation function to a sum of exponential and sinusoidal functions of time plus a constant (see the insert in Figure S2). Circles and error bars represent the mean and standard deviation of the mean of results from five 200 ps trajectories of HPQ in the lowest p-p* state. The black lines are least-squares fits to straight lines. See the text for details. ---------F The temperature dependences of 𝑘%\ and T2 were not examined closely for the charge-

separation reaction of HPQ in dichloromethane or other polar solvents, where additional, slower decay components that are unlikely to affect the electron-transfer rate complicate the

13



Page 14 of 29

autocorrelation functions. In previous studies of a related cyclophane containing a zinc porphyrin, the main components of the decay on both picosecond and sub-picosecond time scales were similar in iso-octane, toluene, dichloromethane and acetonitrile, suggesting that they depend more on intramolecular vibrations than on interactions with the solvent.29 F Moderate changes in 𝑘%\ or 1/T2 would have little effect on the BIP-NAP and BIP-BQO

reactions, where 𝑉

%

F % 0, but is almost independent of µ when µ < 0. The behavior at positive values of µ is qualitatively in accord with the semiclassical Marcus equation,1-5 in which µ = DG + l and (neglecting the weak dependence of the pre-exponential factor on T-1/2) the activation free energy DG‡ increases quadratically with |µ| on both sides of zero. But µ is not necessarily identical to DG + l. In HPQ, for example, DG +

ls » -1600 cm-1 (-0.20 eV, Figure 1C) and adding 4300 cm-1 for lv gives DG + ls + lv » +2700 cm-1, in comparison with µ » -1600 cm-1 (Figure 1A). Recent work has shown that focusing on the P(xo) distribution for the reactant state instead of on DG + l also helps to clarify effects of solvents on ket.28-29 Figure 7 illustrates the additional point that, although DH‡ becomes small when µ = 0, it does not necessarily go to zero. For HPQ, the DH‡ that remains here reflects effects of temperature on k23(0), T2, and to a lesser extent, |V| (Figures 3 and 4).

-1

6

4

‡

∆H / kcal mol

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

2

0

-8

-4

0

4

8

-1

µ / 1000 cm

Figure 7. Apparent activation enthalpy (DH‡, filled circles) for charge separation in HPQ calculated with P(xo) distribution for the reactant state shifted to put the mean of the distribution 20


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(µ) at the energy given on the abscissa. The values plotted are the slopes of lines fitted to the points in Figure S6. The solid curve is a least-squares fit to a 3rd-order polynomial for µ ≥ 0: DH‡ = a + bµ + cµ2 with a = 0.5617 ± 0.0178 kcal mol-1, b = (-1.86 ± 0.083)´10-5 kcal mol-1 cm, c = (5.56 ± 0.08)´10-5 kcal mol-1 cm2. The dashed curve is a similar fit for µ ≤ 0: a = 0.5116 ± 0.0156 kcal mol-1, b = (2.14 ± 0.73)´10-5 kcal mol-1 cm, c = (3.25 ± 0.70)´10-6 kcal mol-1 cm2. Note that µ for HPQ is independent of temperature (Figures 1A, 2A); Figure 8 illustrates the behavior of DH‡ when µ changes with temperature. -------HPQ exemplifies a system in which the mean of the distribution of P(xo) for the reactant is independent of temperature (Figures 1A, 2A). BIP-BQO and BIP-NAP introduce the complication that their P(xo) distributions shift to more negative energies with temperature (Figures 1C, 1E, 2E, 2G). If µ is below the peak of kafd(xo), as it is for BIP-BQO, the shift to more negative energies reduces the overlap with kafd(xo). Other things being equal, ket therefore falls as the temperature rises, giving the negative DH‡ illustrated by the calculated rate constants in Figure 6B. But if µ ³ 0 as in BIP-NAP, the shift in the P(xo) distribution increases the overlap with kafd(xo), raising ket and contributing to the positive DH‡ seen in Figure 6A. Figure S7 shows representative Arrhenius plots of ket calculated with the means of the P(xo) distributions for the reactant state of BIP-BQO moved to various energies, and Figure 8 presents apparent activation enthalpies derived from such plots. The transition from negative to positive DH‡ occurs at µ » -1000 cm-1. However, the exact point at which this change occurs will depend on the system. As noted above, a relaxation that decreases |V| at low temperatures would act in opposition to the shift of the P(xo) distribution and move the transition point to a more negative energy. If the P(xo) distribution shifted in the positive direction with temperature, as it might in some systems, distributions with µ £ 0 could have a positive DH‡.

21



-1

8

‡

∆H / kcal mol

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

4

0

-4 -10

-5

0

5

10

-1

µ / 1000 cm

Figure 8. Apparent activation enthalpy (DH‡) for the charge-shift reaction of BIP-BQO calculated with the means (µ) of the P(xo) distributions for the reactant state shifted to various energies. DH‡ was obtained from Arrhenius plots like those shown in Figure S7. The empty circles represent the average value of µ for the nine temperatures in each Arrhenius plot (180 to 340 K), and the horizontal error bars span the nine values. To allow for curvature of the Arrhenius plots, DH‡ was taken to be proportional to the coefficient b returned by fitting log10(ket/s-1) to a 3rd-order polynomial, a + bT-1 + cT-2. -------Because the MM programs treat forces classically, the P(xo) distributions presented here cannot be extended confidently to lower temperatures where zero-point energies of bond stretching and bending become significant relative to kBT. The fully quantum-mechanical approach described by Borrelli and Peluso26, 48 would be more appropriate for calculations of ket at temperatures below 100 K. Their approach probably could be modified along the lines of eq (3) to handle reactions with large |V| and augmented by QM-MM simulations to obtain frequencies and coupling factors for vibrational modes of the solvent.

22


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ASSOCIATED CONTENT Supporting Information Structures of HPQ, BIP-NAP and BIP-BQO (Figure S1); autocorrelation function of the energy gap for charge separation in HPQ in iso-octane (Figure S2); Fourier transform of the autocorrelation function (Figure S3); energies and vibronic coupling factors in an eight-mode harmonic model for HPQ (Table S1); kafd(xo) for this model (Figure S4); dependence of |V| on the energy gap for charge separation in HPQ (Figure S5); and Arrhenius plots of ket calculated with P(xo) distributions centered at various energies (Figures S6 and S7). AUTHOR INFORMATION Corresponding Author *Email: [email protected]. Phone: 206-523-0142. Notes The author declares no competing financial interest. ACKNOWLEDGEMENTS I thank Bill Hazelton, Arieh Warshel and the reviewers for helpful comments and suggestions. This work was not supported by external funding.

23



References 1.

Marcus, R. A. On the Theory of Oxidation-Reduction Reactions Involving Electron

Transfer I. J. Chem. Phys. 1956, 24, 966-978. 2.


Transfer. II. Applications to Data on the Rates of Isotopic Exchange Reaction. J. Chem. Phys. 1957, 26, 867-871. 3.

Marcus, R. A. Theory of Oxidation-Reduction Reactions Involving Electron Transfer.

Part 4. A Statistical-Mechanical Basis for Treating Contributions from the Solvent, Ligands and Inert Salt. Disc. Farad. Soc. 1960, 29, 21-31. 4.


Transfer. V. Comparison and Properties of Electrochemical and Chemical Rate Constants. J. Phys. Chem. 1963, 67, 853-857, 2889. 5.

Marcus, R. A. Chemical and Electrochemical Electron-Transfer Theory. Ann. Rev. Phys.

Chem. 1964, 15, 155-196. 6.

Efrima, S.; Bixon, M. On the Role of Vibrational Excitation in Electron Transfer

Reactions with Large Negative Free Energies. Chem. Phys. Lett. 1974, 25, 34-37. 7.

Ulstrup, J.; Jortner, J. The Effect of Intramolecular Quantum Modes on Free Energy

Relationships for Electron Transfer Reactions. J. Chem. Phys. 1975, 63, 4358-4368. 8.

Jortner, J. Temperature Dependent Activation Energy for Electron Transfer between

Biological Molecules. J. Chem. Phys. 1976, 64, 4660-4667. 9.

Bixon, M.; Jortner, J. Non-Arrhenius Temperature Dependence of Electron-Transfer

Rates. J. Phys. Chem. 1991, 95, 1941-1944. 10.

Marcus, R. A. Nonadiabatic Processes Involving Quantum-Like and Classical-Like

Coordinates with Applications to Nonadiabatic Electron Transfers. J. Chem. Phys. 1984, 81, 4494-4500. 11.

Yudanov, V. V.; Mikhailova, V. A.; Ivanov, A. I. Reorganization of Intramolecular High

Frequency Vibrational Modes and Dynamic Solvent Effect in Electron Transfer Reactions. J. Phys. Chem. A 2012, 116, 4010-4019. 12.

Davis, W. B.; Ratner, M. A.; Wasielewski, M. R. Conformational Gating of Long

Distance Electron Transfer through Wire-Like Bridges in Donor-Bridge-Acceptor Molecules. J. Am. Chem. Soc. 2001, 123, 7877-7886.

24


Page 24 of 29

Page 25 of 29 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


13.

Waskasi, M. M.; Kodis, G.; Moore, A. L.; Moore, T. A.; Gust, D.; Matyushov, D. V.

Marcus Bell-Shaped Electron Transfer Kinetics Observed in an Arrhenius Plot. J. Am. Chem. Soc. 2016, 138, 9251-9257. 14.

Waskasi, M. M.; Newton, M. D.; Matyushov, D. V. Impact of Temperature and Non-

Gaussian Statistics on Electron Transfer in Donor-Bridge-Acceptor Molecules. J. Phys. Chem. B 2017, 121, 2665-2676. 15.

Miller, J. R.; Calcaterra, L. T.; Closs, G. L. Intramolecular Long-Distance Electron

Transfer in Radical Anions. The Effects of Free Energy and Solvent on the Reaction Rates. J. Am. Chem. Soc. 1984, 106, 3047-3049. 16.

Closs, G. L.; Miller, J. R. Intramolecular Long-Distance Electron Transfer in Organic

Molecules. Science 1988, 240, 440-447. 17.

Liang, N.; Miller, J. R.; Closs, G. L. Correlating Temperature Dependence to Free-

Energy Dependence of Intramolecular Long-Range Electron Transfers. J. Am. Chem. Soc. 1989, 111, 8740-8741. 18.

Liang, N.; Miller, J. R.; Closs, G. L. Temperature-Independent Long-Range Electron-

Transfer Reactions in the Marcus Inverted Region. J. Am. Chem. Soc. 1990, 112, 5353-5354. 19.

Closs, G. L.; Calcaterra, L. T.; Green, N. J.; Penfield, K. W.; Miller, J. R. Distance,

Stereoelectronic Effects, and the Marcus Inverted Region in Intramolecular Electron Transfer in Organic Radical Anions. J. Phys. Chem. 1986, 90, 3673-3683. 20.

Heitele, H.; Pöllinger, K.; Kremer, K.; Michel-Beyerle, M. E.; Futscher, M.; Voit, G.;

Weiser, J.; Staab, H. A. Electron Transfer in Porphyrin Quinone Cyclophanes. Chem. Phys. Lett. 1992, 188, 270-278. 21.

Heitele, H.; Pöllinger, F.; Häberle, T.; Michel-Beyerle, M. E.; Staab, H. A. Energy Gap

and Temperature Dependence of Photoinduced Electron Transfer in Porphyrin-Quinone Cyclophanes. J. Phys. Chem. 1994, 98, 7402-7410. 22.

Pöllinger, F.; Musewald, C.; Heitele, H.; Michel-Beyerle, M. E.; Staab, H. A.

Photoinduced Electron Transfer in Cyclophane-Bridged Porphyrin-Quinone Molecules. A Subpicosecond Transient Absorption Study. Ber. Bunsen-Gesellsch. Phys. Chem. 1996, 100, 2076-2080. 23.

Häberle, T.; Hirsch, J.; Pöllinger, F.; Heitele, H.; Michel-Beyerle, M. E.; Anders, C.;

Döhling, A.; Krieger, C.; Rückemann, A.; Staab, H. A. Ultrafast Charge Separation and Driving

25



Force Dependence in Cyclophane-Bridged Zn-Porphyrin-Quinone Molecules. J. Phys. Chem. 1996, 100, 18269-18274. 24.

Parson, W. W. Flash-Induced Absorbance Changes in Rhodospirillum Rubrum

Chromatophores. Biochim. Biophys. Acta 1967, 131, 154-172. 25.

Kirmaier, C.; Holten, D.; Parson, W. W. Temperature and Detection-Wavelength

Dependence of the Picosecond Electron Transfer Kinetics Measured in Rhodopseudomonas Sphaeroides Reaction Centers: Resolution of New Spectral and Kinetic Components in the Primary Charge Separation Process. Biochim. Biophys. Acta 1985, 810, 33-48. 26.

Borrelli, R.; Peluso, A. The Temperature Dependence of Radiationless Transition Rates

from Ab Initio Computations. Phys. Chem. Chem. Phys. 2011, 13, 4420-4426. 27.

Parson, W. W. Vibrational Relaxations and Dephasing in Electron-Transfer Reactions. J.

Phys. Chem. B 2016, 120, 11412-11418. 28.

Parson, W. W. Effects of Free Energy and Solvent on Rates of Intramolecular Electron

Transfer in Organic Radical Anions. J. Phys. Chem. A 2017, 121, 7297-7306. 29.

Parson, W. W. Electron-Transfer Dynamics in a Zn-Porphyrin-Quinone Cyclophane:

Effects of Solvent, Vibrational Relaxations, and Conical Intersections. J. Phys. Chem. B 2018, 122, 3854-3863. 30.

Parson, W. W.; Warshel, A. A Density-Matrix Model of Photosynthetic Electron Transfer

with Microscopically Estimated Vibrational Relaxation Times. Chem. Phys. 2004, 296, 201-206. 31.

Parson, W. W.; Warshel, A. Dependence of Photosynthetic Electron-Transfer Kinetics on

Temperature and Energy in a Density-Matrix Model. J. Phys. Chem. B 2004, 108, 10474-10483. 32.

Mangaud, E.; de la Lande, A.; Meier, C.; Desouter-Lecomte, M. Electron Transfer within

a Reaction Path Model Calibrated by Constrained DFT Calculations: Application to MixedValence Organic Compounds. Phys. Chem. Chem. Phys. 2015, 17, 30889-30903. 33.

McMillan, A. W.; Kier, B. L.; Shu, I.; Byrne, A.; Andersen, N. H.; Parson, W.

Fluorescence of Tryptophan in Designed Hairpin and Trp-Cage Miniproteins: Measurements of Fluorescence Yields and Calculations by Quantum Mechanical Molecular Dynamics Simulations. J. Phys. Chem. B 2013, 117, 1790-1809. 34.

Pádua, A. A.; Fareleira, J. M. N. A.; Calado, J. C. G. Density and Viscosity

Measurements of 2,2,4-Trimethylpentane (Isooctane) from 198 K to 348 K and up to 100 Mpa. J. Chem. Eng. Data 1996, 41, 1488-1494.

26


Page 26 of 29

Page 27 of 29 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


35.

De Lorenzi, L.; Fermeglia, M.; Torriano, G. Density and Viscosity of 1-Methoxy-2-

Propanol, 2-Methyltetrahydrofuran, a,a,a-Trifluoroethane, and their Binary Mixtures with 1,1,1-Trichloroethane at Different Temperatures. J. Chem. Eng. Data 1996, 41, 1121-1125. 36.

Chorazewski, M.; Troncoso, J.; Jacquemin, J. Thermodynamic Properties of

Dichloromethane, Bromochloromethane, and Dibromomethane under Elevated Pressure: Experimental Results and Saft-Vr Mie Predictions. Ind. Eng. Chem. Res. 2014, 54, 720-730. 37.

Tan, R.-R.; Shen, X.; Hu, L.; Zhang, F.-S. Liquid-to-Glass Transition of Tetrahydrofuran

and 2-Methyltetrahydrofuran. Chin. Phys. B 2012, 21, 086402/1-086402/10. 38.

Warshel, A.; Parson, W. W. Dynamics of Biochemical and Biophysical Reactions:

Insight from Computer Simulations. Q. Rev. Biophys. 2001, 34, 563-679. 39.

Vath, P.; Zimmt, M. B.; Matyushov, D. V.; Voth, G. A. A Failure of Continuum Theory:

Temperature Dependence of the Solvent Reorganization Energy of Electron Transfer in Highly Polar Solvents. J. Phys. Chem. B 1999, 103, 9130-9140. 40.

Vath, P.; Zimmt, M. B. A Spectroscopic Study of Solvent Reorganization Energy:

Dependence on Temperature, Charge Transfer Distance, and the Type of Solute-Solvent Interactions. J. Phys. Chem. A 2000, 104, 2626-2633. 41.

Matyushov, D. V. Solvent Reorganization Energy of Electron Transfer in Polar Solvents.

J. Chem. Phys. 2004, 120, 7532-7556. 42.

Ghorai, P. K.; Matyushov, D. V. Solvent Reorganization Entropy of Electron Transfer in

Polar Solvents. J. Phys. Chem. A 2006, 110, 8857-8863. 43.

Matyushov, D. V. Energetics of Electron-Transfer Reactions in Soft Condensed Media.

Acc. Chem. Res. 2007, 40, 294-301. 44.

Borrelli, R.; DiDonato, M.; Peluso, A. Role of Intramolecular Vibrations in Long-Range

Electron Transfer between Pheophytin and Ubiquinone in Bacterial Photosynthetic Reaction Centers. Biophys. J. 2005, 89, 830-841. 45.

Borrelli, R.; Domcke, W. First-Principles Study of Photoinduced Electron-Transfer

Dynamics in a Mg-Porphyrin-Quinone Complex. Chem. Phys. Lett. 2010, 498, 230-234. 46.

Borrelli, R.; Thoss, M.; Wang, H.; Domcke, W. Quantum Dynamics of Electron-Transfer

Reactions: Photoinduced Intermolecular Electron Transfer in a Porphyrin-Quinone Complex. Mol. Phys. 2012, 110, 751-763.

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47.

Chaudhuri, S.; Hedström, S.; Méndez-Hernández, D. D.; Hendrickson, H. P.; Jung, K. A.;

Ho, J.; Batista, V. S. Electron Transfer Assisted by Vibronic Coupling from Multiple Modes. J. Chem. Theory Comput. 2017, 13, 6000-6009. 48.

Borrelli, R.; Peluso, A. Elementary Electron Transfer Reactions: From Basic Concepts to

Recent Computational Advances. WIREs. Comput. Mol. Sci. 2013, 3, 542-559.

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‡

∆H / kcal mol

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4

0

-4 -10

-5

0

5 -1

〈∆E00〉 / 1000 cm

TOC Graphic

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10

Temperature Dependence of the Rate of Intramolecular Electron

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