From Fundamental Theories to Quantum Coherences in Electron

Subscriber access provided by Kaohsiung Medical University

Perspective

From Fundamental Theories to Quantum Coherences in Electron Transfer Shahnawaz Rafiq, and Gregory D. Scholes J. Am. Chem. Soc., Just Accepted Manuscript • DOI: 10.1021/jacs.8b09059 • Publication Date (Web): 09 Nov 2018 Downloaded from http://pubs.acs.org on November 10, 2018

Just Accepted “Just Accepted” manuscripts have been peer-reviewed and accepted for publication. They are posted online prior to technical editing, formatting for publication and author proofing. The American Chemical Society provides “Just Accepted” as a service to the research community to expedite the dissemination of scientific material as soon as possible after acceptance. “Just Accepted” manuscripts appear in full in PDF format accompanied by an HTML abstract. “Just Accepted” manuscripts have been fully peer reviewed, but should not be considered the official version of record. They are citable by the Digital Object Identifier (DOI®). “Just Accepted” is an optional service offered to authors. Therefore, the “Just Accepted” Web site may not include all articles that will be published in the journal. After a manuscript is technically edited and formatted, it will be removed from the “Just Accepted” Web site and published as an ASAP article. Note that technical editing may introduce minor changes to the manuscript text and/or graphics which could affect content, and all legal disclaimers and ethical guidelines that apply to the journal pertain. ACS cannot be held responsible for errors or consequences arising from the use of information contained in these “Just Accepted” manuscripts.

is published by the American Chemical Society. 1155 Sixteenth Street N.W., Washington, DC 20036 Published by American Chemical Society. Copyright © American Chemical Society. However, no copyright claim is made to original U.S. Government works, or works produced by employees of any Commonwealth realm Crown government in the course of their duties.

Page 1 of 24 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

Journal of the American Chemical Society

From Fundamental Theories to Quantum Coherences in Electron Transfer Shahnawaz Rafiq, Gregory D. Scholes* Frick Chemistry Laboratory, Princeton University, NJ 08544 USA KEYWORDS: Electron transfer, Marcus theory, Bixon-Jortner theory, Quantum coherences, Coherent transport, High-frequency vibrations, wavepackets, Donor-acceptor complexes, Biological electron transfer, Resonant vibrations, Organic photovoltaics, Coherences in materials

ABSTRACT: Photoinduced electron transfer (ET) is a cornerstone of energy transduction from light to chemistry. The past decade has seen tremendous advances in the possible role of quantum coherent effects in the light-initiated energy and electron transfer processes in chemical, biological, and materials systems. The prevalence of such coherence effects holds a promise to increase efficiency and robustness of transport even in the face of energetic or structural disorder. A primary motive of this perspective is to work out how to think about ‘coherence’ in ET reactions. We will discuss how the interplay of basic parameters governing ET reactions like electronic coupling, interactions with the environment, and intramolecular high-frequency quantum vibrations impact coherences. This includes revisiting the insights emerged from the seminal work on the theory of ET and the time-resolved measurements on coherent dynamics to explore the role of coherences in ET reactions. We conclude by suggesting that in addition to optical spectroscopies, validating the functional role of coherences would require simultaneous mapping of correlated electron motion and atomically resolved nuclear structure. INTRODUCTION Recent work has suggested that effects collectively called coherence—long thought to be irrelevant for robust function owing to their fleeting presence, might be harnessed efficiently even when electronic coupling is not such a dominant participant.1-7 Coherence refers to phase relations among the constituents of the superposition of quantum states that carries extra correlations beyond those predicted by the classical probability theory.8-9 When such phase relationships among quantities— like quantum amplitude, are retained long enough to have a mechanistic and functional relevance to the underlying process, the dynamics are rendered coherent. In relation to practical systems, when electronic states are localized, transport of energy or charge is incoherent—it hops randomly from site to site. Similarly, stochastic incoherent transport is prevalent when the electronic states are strongly delocalized over multiple sites to form extended eigenstates—that is, population relaxes incoherently owing to minimal interactions with the surroundings. These two extremes are prevalent when the electronic coupling between the sites is either much smaller or much larger than the reorganization energy of the surroundings. In these limits, optimization of the transport becomes very challenging.10-12 In the middle of the two extremes—localization and delocalization, the interplay of energy fluctuations and transport leads to a competition between the delocalization and localization, and that is where coherences seem to matter (Figure 1). The resulting dynamics are complex and could sometimes translate to

a robust and disorder resistant transport.13-14 In this regime, due to the competition of the timescales, the system exhibits coherent dynamics at short times and at later times, incoherent transport dominates. From the coupling perspective, the three regimes are analogous to the valence electron localization or delocalization in the Creutz-Taube ions.15-16 Exploration of electron transfer (ET) systems capable of retaining phase information for a practical timescale could help to direct or control charge separation and recombination. Electron transfer is a ubiquitous process whose mechanism and dynamics are relevant to applications like solar cells, molecular electronic devices, bio-sensing techniques and various photoactivated processes.17-24 Its importance is emphasized every time an organic chemist draws a reaction mechanism. Chemistry is about coercing electrons to move. The advent of ultrafast broadband electronic spectroscopy has provided new insights into the microscopic mechanisms of such a fundamental process.22, 25-28 In this perspective, we will start with a quantum/semiclassical description of ET reactions, to lay a ground work for introducing coherences in photoinduced ET reactions. Coherences will be discussed in chemical, biological, and material systems with examples cited in each section. We will explore what is meant by coherences in ET. Specifically, we will ask how to assess adiabaticity and friction (coupling of reaction coordinate to solvent and/or nuclear modes) in experimental studies of ET reactions. The role of high-frequency vibrations in ET reactions, while not a new subject, has been highlighted by recent work using coherent spectroscopies. We will examine the mechanistic role of high-frequency vibrations using wavepacket spectroscopies. We present a perspective for new experimental strategies which could impart functional validation to the coherences beyond the scepticism emerging from pure optical spectroscopies. This is a review of early steps towards working out how to design systems that uses coherence to assist in ET and inhibit back-reactions. QUNANTUM AND SEMICLASSICAL DESCRIPTION OF ELECTRON TRANSFER The pioneering work by Marcus assumes the well-known thermally activated form of transition state theory, wherein nuclear motions of the reactant molecules and the solvent are in thermal equilibrium before the reaction.29-31 Equilibrium fluctuations of the solvent dielectric polarization enables the reactant and product energies to occasionally become quasi-degenerate, so that an ET event can occur, leaving the product in an energetically rich state. The classical Marcus theory of ET 𝑘𝐸𝑇 = ĸ𝑒𝑙 𝑛 exp(( + 𝐺)2 ⁄4𝑘𝐵 𝑇)⁡generally works well for reactions where transition probability of ET is close to unity—most

ACS Paragon Plus Environment

Journal of the American Chemical Society 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

Page 2 of 24

Figure 1. Limiting cases of Transport as a function of electronic coupling. a, Weak electronic coupling: incoherent transport from one curve to other. b, Strong electronic coupling: incoherent relaxation among highly delocalized states. c, Intermediate electronic coupling: competition between the energetic disorder and the coupling could give rise to a unique transport mechanism at short times of the dynamics.

adiabatic ET reactions; where ĸ𝑒𝑙 is the electronic transmission coefficient and 𝑛 is the frequency of passage (nuclear motion) through the transition state. However, to explain ET reactions where transition probability is much less than unity—most nonadiabatic ET reactions, a quantum mechanical description to the nuclear motion is required. The transition probability in such cases was formulated in a golden-rule description, originally by Levich and Dogonadze.32 The concept of the electron tunnelling from donor to acceptor as well as the nuclear tunnelling from reactant to product were introduced. The nuclear tunneling is required particularly in the inverted regime of ET to move from the energy minimum of the donor surface to the acceptor surface due to the avoided crossing between the two surfaces. Hopfield was the first to study how nuclear tunnelling explained low-temperature biological ET kinetics by using a semiclassical treatment.33 The golden-rule treatment was also applied by Jortner and coworkers,34-35 as well as Marcus and coworkers,36 who showed the role of high-frequency quantum vibrations of the product as energy accepting modes. The golden-rule expression for the ET rate constant is 𝑘𝐸𝑇 ⁡ =

2 ℏ

⁡𝑉 2 FCWD

1

Where V is the matrix element for the electronic coupling between the initial and final states and FCWD is the Franck-Condon weighted density-of-states. In a full quantum treatment of ET, the FCWD contains a summation over all the vibronic states of the reactant, product, and the solvent vibrational modes. Knowing that solvent stochastic fluctuations occur mainly at low frequencies and can therefore be treated classically, the quantum treatment could only be applied to the highfrequency intramolecular vibrational modes. An assumption was further made to reduce the number of high-frequency vibrations to only one averaged mode of frequency ‘’ by Jortner,37 Miller et al,38 and Brunschwig and Sutin,39 which resulted in the following semiclassical Marcus equation for a nonadiabatic limit: 0→𝑛 𝑘𝑁𝐴 =

2(𝑉 0→𝑛 )2 ℏ√4𝑠 𝑘𝐵 𝑇

exp⁡(

−(𝑠 +𝐺0→𝑛 ⁡) 4𝑠 𝑘𝐵 𝑇

2

)

2

The overall ET rate is: 0→𝑛 𝑘𝐸𝑇 ⁡ = ∑𝑛 𝑘𝑁𝐴 3 0→𝑛 𝑘𝑁𝐴 is the nonadiabatic rate constant for each vibronic channel. 𝐺0→𝑛 ⁡ = 𝐺 + 𝑛ℎ𝑣

4

(𝑉 0→𝑛 )2 ⁡ = 𝑉 2 |⟨0|𝑛⟩|2 |⟨0|𝑛⟩|2 ⁡ = (𝑆 𝑛 ⁄𝑛!)exp⁡(−𝑆)

5 6 S being the electron-vibrational coupling strength, or HuangRhys factor, is defined as a ratio of the reorganization energy and frequency of the high-frequency mode⁡(𝑣𝑖𝑏 ⁄(ℎ𝑣𝑖𝑏 )), ‘n’ is the vibrational quantum number, s and vib are the solvent and vibrational reorganization energies. In the high-temperature limit, this semiclassical Marcus equation (2 and 3) reduces to a simpler and more familiar equation: 𝑘𝐸𝑇 ⁡ = ⁡

2𝑉 2 ℏ√4𝑘𝐵 𝑇

exp⁡(

−(+𝐺)2 4𝑘𝐵 𝑇

)

7

where  = s + vib is the total reorganization energy. Equation 7 is more like a classical Marcus equation. The 1950s Marcus theory assumed a nuclear frequency factor characteristic of the potential energy surface curvature. Whereas, the later theories of solvent-limited rates assumed overdamped or diffusive motion along the reaction coordinate, leading to a different prefactors of the classical and semiclassical Marcus rate equations. The dependence of the ET rate on the energy-gap leads to three distinct regimes based on the ratio of standard free energy change between the reactant and product species and the solvent reorganization energy: normal, barrierless, and inverted regimes. 40The classical and semiclassical Marcus equations both predict a symmetric quadratic logarithmic rate constant as a function of free energy gap on either side of the barrierless regime because ET occurs only at the intersection point of the two single free energy curves. However, in the quantum description (equation 2), a more effective route is nuclear tunnelling through vibrational overlaps, which can open extra channels for ET. This condition is particularly important in the inverted region, where vibrational wavefunctions of the reactant and the product states are embedded, so Franck-Condon factors are much larger than the normal region. Siders and Marcus 36 confirmed that the inclusion of nuclear tunnelling predicts an asymmetric fall off in the logarithmic rate constant in the inverted region. The existence of the inverted region and the asymmetric shape of the 𝑙𝑛(𝑘𝐸𝑇 )⁡𝑣𝑠. ∆𝐺°curve was experimentally verified by Miller et al in intermolecular charge transfer (CT) between the biphenyl radical anion and various acceptors in a rigid lowtemperature glass as well as in biphenyl radical anion connected to various acceptors by a hydrocarbon bridge. 38, 40 Later,


2

Page 3 of 24 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


Figure 2. Semiclassical Electron Transfer Models. a, Sumi-Marcus model b, Jortner-Bixon Model c, Walker-Barbara Model. These models were built on the Fermi’s Golden rule description of ET to accommodate the role of solvation dynamics, adiabaticity, high-frequency quantum vibrations in the inverted region towards ET.

researchers in the groups of S. Farid41 and H. B. Gray42-43 verified a Marcus inverted region with a symmetric 𝑙𝑛(𝑘𝐸𝑇 )⁡𝑣𝑠. ∆𝐺°⁡curve, which was an indicative of nuclear tunneling through vibrational overlaps of low-frequency modes. More recent advances in photoinduced ET are highlighted in the work by Vauthey and coworkers.22, 44-47 The semiclassical treatment works well in the weak-coupling nonadiabatic limit, where the ET process is treated within the equilibrium transition state framework. However, in the strongcoupling limit, the ET reaction proceeds adiabatically on the lower energy surface, so that the probability of ET does not directly depend on the electronic coupling, rather on the probability of the reaction coordinate to reach the barrier crossing. Therefore, instead of equilibrium contributions, the response of the solvent polarization contributes to the reaction coordinate and ET dynamics is discussed in the framework of Kramer’s theory.48-51 Kramer’s theory treats a reaction as a potential barrier crossing by stochastic motion on the potential energy surface, where in the reaction rate is inversely proportional to the dielectric relaxation time. The reaction can thus be identified as a solvent-controlled adiabatic reaction.51 The transition from nonadiabatic ET to solvent-controlled ET, was originally studied by Zusman using stochastic motion along the reaction coordinate with transition probability proportional to V2.52 Friedman and Newton,53 Sumi and Marcus54 used the first-passage approach, while Rips and Jortner adopted the real-time path integral formalism.55 Garg, Onuchic, and Ambegaokar introduced a block-summation technique to derive an expression for the rate, which bridges the nonadiabatic and the adiabatic limits through a parameter for adiabaticity.56 𝑘𝐸𝑇 ⁡ =

𝐾𝑁𝐴 1+𝐻𝐴

⁡

8

HA is the adiabaticity parameter to entertain the transition from the nonadiabatic to the solvent-controlled adiabatic ET and implicates the delocalization of the vibronic wavefunction in the intersection region, given by: 𝐻𝐴 ⁡ = ⁡

4𝑉 2 𝑠 ℏ𝑠

9

where s is the longitudinal dielectric relaxation time in the reaction coordinate. Equation 8 represents a continuous change from the non-adiabatic limit (HA1). In the solvent-controlled adiabatic limit, the rate constant can be recast into the Kramer’s like expression for adiabatic ET rate. 1



𝑘𝐴 ⁡ = ⁡ √ 𝑠  16𝑘 𝑠

𝐵𝑇

exp⁡(

−(𝑠 +𝐺)2 4𝑠 𝑘𝐵 𝑇

)

10

The above approaches connect the nonadiabatic limit— where the probability of ET is determined by the electronic tunnelling and tunnelling fluctuations, and the adiabatic limit— where it is determined by the reaction coordinate dynamics. In other words, the two limits differ in the nature of nuclear motion through the curve crossing region, which is considered to be overdamped for nonadiabatic and ballistic or diffusive for adiabatic transfer. In the adiabatic case of strong-coupling, an electron is transferred ballistically by nuclear motion through the transition state. While, in the solvent-controlled adiabatic limit, the electronic coupling may still be weak, however, the rate is affected by frictional coupling—coupling of nuclear reaction coordinate to other nuclear coordinates and solvent relaxation coordinate. Frictional coupling ensures that the time spent in the curve crossing region is long enough that every time an electron approaches the crossing region, the donor-acceptor encounter is effective, even though electronic coupling is weak. Hence, the reaction appears adiabatic owing to the nuclear control rather than electronic tunnelling. In the solvent-controlled adiabatic limit or in the adiabatic limit of strong coupling, the nuclear motions and the solvent relaxation rates thus decide the ET probability. Therefore, many theories were put forward to address the role of vibrational motion and solvation dynamics in ET. Sumi and Marcus, in 1986, used a low-frequency (classical) vibrational degree of freedom and a classical solvent degree of freedom to address the role of vibrational motion and solvation


3


dynamics in ET (Figure 2a).54 In this two-dimensional model, the ET occurs along the classical vibrational coordinate (q), and the free energy barrier in the vibrational direction is circumvented by energy fluctuations via reorganization along the solvent coordinate (X). This model, however, predicts a vanishing ET rate constant as temperature approaches zero and also propounds that the barrier must be crossed for ET to occur, implying that there is no scope for possible quantum tunnelling through the barrier.57 Jortner and Bixon, in 1988, while retaining the classical solvent degree of freedom to account for dielectric relaxation, incorporated high-frequency quantum vibrations of the donor and acceptor centres as the intramolecular vibrational coordinate coupled to the ET process (Figure 2b).58 High-frequency vibrations were shown to influence ET dynamics in three ways: providing a multitude of parallel kinetic channels. A “polaron dressing” effect (dressing electronic coupling by the nuclear Franck-Condon factor) reduces both the non-adiabatic rate constant (1) and the adiabaticity parameter (8). Vibrational excitation reduces the effective energy gap, lowering of the activation barrier in the inverted region. The overall ET rate is a summation over all the high-frequency vibronic channels; 𝑘𝐸𝑇 ⁡ = ∑𝑛=0

0→𝑛 𝑘𝑁𝐴

11

𝑛 1+𝐻𝐴

0→𝑛 Where 𝑘𝑁𝐴 is given by equation (6) and the Franck-Condon dressed adiabaticity parameter is 𝐻𝐴𝑛 ⁡ = 𝐻𝐴 |⟨0|𝑛⟩|2 . The solvent-controlled adiabatic limit can be realized for some of the n = 0→n channels i.e. the polaron dressed adiabaticity parameter should be greater than unity for some values of n, which are denoted by {𝑛̅}. The ET rate is written as;

𝑘𝐸𝑇 ⁡ = ∑𝑛{𝑛̅}

0→𝑛 𝑘𝑁𝐴 𝑛 𝐻𝐴

+⁡ ∑𝑛≠{𝑛̅}

0→𝑛 𝑘𝑁𝐴

12

𝑛 1+𝐻𝐴

This model thus introduces nuclear tunnelling and the extent of tunnelling is decided by the vibrational mixing between the reactant and product states. Only those intramolecular modes which involve large displacements from the reactant to the product potentials need to be considered. Photoinduced ET reactions much faster than solvation dynamics have been observed experimentally.59-61 Walker et al studied ultrafast photoinduced back-electron transfer in Betaines.57 Tominaga et al found intervalence CT in mixed-valence compounds faster than the solvation dynamics.62 Kobayashi et al studied ultrafast intermolecular ET in the electron-donating solvents as fast as 100 fs though fluorescence quenching.63 Similar observations were found with Oxazines and Coumarins.59 The Sumi-Marcus model predicts a slow rate of ET reaction in these systems—all these ET reactions fall in Marcus inverted regime. The Jortner-Bixon treatment reproduces these experimental ET rates in fast relaxing solvents only, but overestimates the rate in slow-relaxing solvents, where it does not scale with solvation dynamics. Walker et al discussed how the Bixon-Jortner treatment was unable to reproduce switch from the solvation dynamical control to the vibrational control.57, 61, 64-65 Walker and Barbara, in 1992, thus came up with a hybrid model by extending the Sumi-Marcus model to incorporate the effect of high-frequency vibrations in a similar manner as that developed in the Jortner-Bixon model (Figure 2c).57 The solvent dependent rate constant was expressed as; 0→𝑛 𝑘(𝑋) ⁡ = ∑𝑛 𝑘𝑁𝐴 (𝑋) 13 0→𝑛 𝑘𝑁𝐴 (𝑋) =

42 (𝑉 0→𝑛 )2 ℎ√4𝑖 𝑘𝐵 𝑇

2

−(𝑖 +𝐺0→𝑛 (𝑋)⁡)

exp⁡(

4𝑖 𝑘𝐵 𝑇

𝐺0→𝑛 ⁡(𝑋) = 𝐺 + 𝑠 − 2𝑠 𝑋 + 𝑛ℎ𝑣

)

14 15

Page 4 of 24

The classical reorganization was split into two parts: one corresponding to the classical vibration and second to the solvent reorganization (i + s).⁡𝐺0→𝑛 ⁡(𝑋)—the effective energy gap between the ground state of reactant and the nth vibrational level of the product state, is solvent-coordinate dependent in this model. The issues of the time scale for solvent polarization that contributes to the ET reorganization energy have been much discussed by Matyushov and others. 66-69 The ET models described above established the role of vibrations in ET reactions, which can be understood more comprehensibly in the context of the proton-coupled electron transfer (PCET) reactions. In PCET reactions, while the ET can be treated as an event occurring between a localized initial and a localized final state, the proton transfer occurs between manifold of localized proton states on the initial state and a corresponding manifold of vibrational states on the final state.70 The initial state of the proton may not be thermally accessible (as the proton potential well may have a frequency of ~2000 cm-1), but with a simultaneous ET event, the solvent reorganization can provide near-degenerate multiple proton accepting levels on the final state—nested energy surfaces. For more details, Hammes-Schiffer group has developed a general theory for PCET reactions.71-77 Now that it is established that ET reactions in the intermediate or strong coupling limits depend significantly on the vibronic interactions in addition to the solvent bath polarization. Therefore, frictional coupling as explained above, plays an essential role in ET reactions. When friction is strong compared to other competing energy-scales, the reactant executes a random walk along the reaction coordinate to the barrier crossing (nonadiabatic ET).56 This is the incoherent limit. A contrary situation where there is no friction—a complete separation of the nuclear reaction coordinate from other nuclear and/or solvent degrees, should manifest as a coherent transfer (ballistic) with fastest rates of ET. However, the findings of Garg et al showed that fastest rate does not come from a frictionless limit.56 Instead, the rate increases as fiction is introduced. This is resonant with Wolynes’s work, which suggested that quantum mechanical effects, for example curve crossing, conspire with dissipation in an interesting way. He argued that friction generally increases the adiabaticity of a barrier crossing because of the increased time spent by the electron in the crossing region and multiple chances to move back and forth.78 In the regime, where friction and adiabaticity corroborate, the dynamics becomes complex and coherent interactions enabled by the reaction coordinate may matter. COHERENCE IN PHOTOINDUCED ELECTRON TRANSFER The nonadiabatic limit. In the nonadiabatic limit of Marcus’s classical theory29, the jump of electron from the reactant free energy curve to the product curve follows a Franck-Condon principle in the same way as it is obeyed in photoexcitation. In the normal regime of ET, this instantaneous transfer of the electron requires the reactant and product energies to be degenerate—driven by solvent energy fluctuations. In the inverted regime, that instantaneous transfer is achieved by invoking nuclear tunnelling to accomplish the energy matching and the phase coherence is equivalently maintained during the ET reaction. However, it is a very simplistic aspect of coherent transfer, as the coherence is maintained only for a few femtoseconds— the timescale during which the actual ET occurs. In this nonadiabatic limit, the solvent coordinate and the nuclear coordinate


4

Page 5 of 24 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


Figure 3. Wavepacket Dynamics and its Spectroscopy. a, Generation of a coherent wavepacket on electronic potentials by a broadband laser pulse. b, Motion of the wavepacket in the phase space of the electronic potential modulates the population of Cresyl Violet (in methanol) and that modulation appears as ripples in the time-domain spectroscopic signals. c, Fourier transform of the Cresyl Violet residual ripples produces frequency of the wavepacket generated along Franck-Condon active nuclear vibrations. Plots b and c are reprinted with permission from ref 103. Copyright 2016 American Chemical Society.

are completely orthogonal to each other and the reaction is promoted by the solvent dielectric relaxation. Elegant examples of such quantum coherence can be found abundantly at the molecule scale, when the relevant energy scales are very large compared to thermal energy. For example, in superexchange-mediated ET in donor-bridge-acceptor complexes, the different electronic coupling pathways across the bridge interfere quantum mechanically at the amplitude level, leading to utterly different outcome than predicted by the classical probability laws.79-86 The adiabatic limit. Quantum coherence involving electronic coupling trajectories—as described in the nonadiabatic limit— cannot be considered as a powerful tuning element, even though it is omnipresent, because it is strongly subjected to dephasing.87 One may imagine that since ET is strongly slaved to stochastic polarization fluctuations of the environment, then quantum coherences should always be subjected to strong dephasing. Some recent research suggests that quantum coherence could be facilitated by resonance with vibrational coherences—identified as vibronic coherences.6, 9, 88-91 Reimers and Hush92-93 have studied ET involving a single vibrational mode, providing insights into vibronic coupling. In the adiabatic limit, the reactant and product are coupled together to make Born-Oppenheimer adiabatic ground and excited states. Theoretical treatments by Reimers and Hush suggest that on the reactant and product sides of the reaction coordinate, the electrons are localized and vibrations are localized, but in the vicinity of the transition state the electronic wavefunction and vibrations are delocalized.94-95 Recent work has shown that ET in the inverted regime in a photosynthetic reaction center (photosystem II) is expedited when vibrational levels bridge reactant-product electronic gaps.96-99 This work highlights the importance of elucidating how molecular vibrations aid ET reactions. That issue is salient because these vibrational frequencies and the nature of those modes can be tuned by chemical structure—this giving the chemist control over this apparently powerful and exotic design ingredient.

Vibrational Coherences. The interest in the role of coherences in ET reactions was triggered by the short-pulse pumpprobe experiments of Vos and coworkers in early 1990s, in the bacterial photosynthetic reaction centres.100-102 An implicit manifestation of using short pump pulses is the generation of superposition of vibrational wavefunctions—wavepackets comprising Franck-Condon active modes (Figure 3).103 These wavepackets modulate the population of the electronic surfaces. The modulations when probed by another laser pulse, appear as oscillations on top of a smooth population decay in the timedomain signal. Oscillations are Fourier transformed to show the vibrational frequency of the Franck-Condon active modes accessed by the bandwidth of the pump pulse (Figure 3). In the photosynthetic reaction centre, the vibrational coherences along the low frequency nuclear modes surprising survived the primary ET event.100-102 This work implicated a possible functional role of vibrational coherences for ET. An outcome of that period was a better understanding of vibrational coherences in ET, succinctly pre-empted by Jortner and Bixon (discussed above).58 The Bixon-Jortner description of the role of vibrations in ET formed the basis of interpreting vibrational coherences in ET.104 The more direct questions pertaining to the role of vibrations are: are there particular vibrations that promote or attenuate the ET? Is the bath polarization coordinate orthogonal to the vibrational coordinate or is there a collective additive effect of bath polarization and intramolecular vibrations that affects the ET reaction? To answer these questions, many ET systems have been studied using ultrafast coherent pump-probe type spectroscopies in parallel with theoretical treatments to speculate the role of vibrational coherences towards the ET reactions. Coherent vibrational wavepackets generated on the reactant state are uniquely suited to report on the coupling of nuclear motions to the ET reactions, especially outside the Franck-Condon region. On one hand, vibrational coherence that is weakly coupled to the reaction coordinate does not affect the reaction and will have no bearing on the ET dynamics—hence


5


superpositions (wavepackets) involving such modes will not be affected by ET (Figure 4a). The free energy curves in such cases cross like pure diabats. On the other hand, ET aided by a nuclear mode modifies that picture. The coupled vibrational mode “nests” the free energy curves much like the way proton vibrations nest the energy curves of a proton-coupled ET reaction. Thereby, the vibration increases the number of ET channels (Figure 4b) as well as range of the activation barriers and effective reorganization energies by the permutation of offsets (Bixon-Jortner theory). This peculiar case is common to nonadiabatic and adiabatic ET reactions, though the picture gets more complicated in the adiabatic regime. The nesting of the free energy curves dephases the vibrational wavepackets during the ET reaction. To elaborate, when a superposition is generated along a coupled vibration on the reactant state, coefficients of individual vibrational levels have a set of values which define the contribution of each vibrational level to the superposition (i.e. wavepacket). Multiple crossings due to the nesting of the free energy surfaces will modify the value of the coefficients of each vibrational level as the reactant passes into the product, leading to dispersion of the wavepacket (Figure 4b). Thus, the dephasing time of the reactive mode wavepacket should correspond to the mean first passage time to the transition state, which is defined as the time needed to reach the transition state or the mixed region by a statistical ensemble of initial states. Mean first passage time is expected to be shorter than the actual ET time, as the overall ET time includes the time during which the transition state evolves into the equilibrated final state. A note on basis. We wish to point out that understanding the experimental signatures of coherence demands a general understanding of basis, because without knowing the basis of the initial state, it is challenging to identify coherence. Experiments such as two-dimensional electronic spectroscopy (2DES) detect coherence in the adiabatic basis in the Franck-Condon region. For example, in CdSe nanoplatelets, where the basis is clearly delocalized, the oscillating off-diagonal cross-peaks were assigned to an excitonic coherence (see Figure 7d-f below).87 That oscillating cross-peak indicates that the two excitonic states are coupled and share excitation. The modulation of the electronic wavepacket between the two excitonic states lives for the timescale of the optical coherence between the ground and the excitonic state. So, experiments measure coherence in the adiabatic basis, because it probes coupling between the electronic states through wavefunction delocalization. In other words, oscillations in the cross peak of 2D maps can definitively assess state coherences. However, when the dephasing is strong/rapid, the experiment will detect localized states, more like the theorist exploring a localized basis. In the case of ET reactions, the reaction coordinate follows a basis rotation, as epitomized by the method Sven Larrson used to calculate ET matrix elements.105 Calculations find that the electron is localized on the donor in the Franck-Condon region, but an external electric field can be changed, mimicking solvent reorganization, until the calculation predicts a perfectly delocalized state where the electron is shared between the donor and acceptor. Similarly, when we think about most ET reactions, an initial state is prepared in a localized reactant basis, which evolves into the transition state that is a quantum-mechanical superposition of the reactant and product states—a state best described in the delocalized basis. The transition state finally collapses into the product localized basis. In such a situation, 2DES cannot assess coherence definitively at the off-diagonal

Page 6 of 24

Figure 4. Free energy curves and intramolecular modes. The effect of coupling a vibrational mode to the ET on the wavepacket dynamics. Due to nested free energy crossings enabled by coupling a nuclear mode to the ET reaction, the wavepacket dispersion is enhanced and that translates into faster dephasing of the reactive mode wavepacket. This dephasing time could correspond to the mean first passage time to the transition state of the electron transfer reaction.

cross peaks as the coherence either exists between the electronic coupling pathways or the kinetic pathways. However, if the reactant and product states are coupled in the Franck-Condon region, then 2DES can provide definitive insights in the coherent ET dynamics. Coherences in Chemical Systems The first and foremost constraint to experimentally exploring the role of coherences in ET is to focus on systems with ET rate is faster than vibrational dephasing. Barbara and coworkers used ca. 20 fs time-resolution pump-probe spectroscopy to measure an ca. 85 fs ET time-constant corresponding to the photoinduced back-ET reaction in the mixed-valence metal dimers.106-108 Analysis of the oscillations demonstrated that the average vibrational dephasing for the observed modes is ca. 300 fs. The longer dephasing time of the vibrational coherences relative to the ET time (85 fs) indicated that coherence is maintained for some degrees of freedom during the back-ET reaction. Scherer and coworkers demonstrated that the wavepacket motion in Prussian blue intervalence CT complex originated from the ground state and the excited CT state.109 Further, it was proposed that several low-frequency vibrational modes are coupled to the CT coordinate.109 The longer lifetime of the vibrational coherences than ET could implicate that wavepacket survives the ET event and is transferred to the product surface. Wynne et al (1996) used a 40 fs, 810 nm pump pulse to create vibrational coherences in the CT state of an electron donor— acceptor pi-stacked complex between tetracyanoethylene (TCNE) and pyrene.110 Clear oscillations were observed in the ground state bleach and the stimulated emission signals during the back-ET process in 250 fs – 1.5 ps. It was found that the oscillations due to 160 cm-1 mode in the stimulated emission signal survived for the timescale of the ET and modulated the rate of the back-ET from the CT to the ground state. This vibrational wavepacket was implicated to result in coherent formation of the product state via stepwise modulations of the ground state signal.110-111 However, identifying a temporal


6

Page 7 of 24 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


modulation on the basis of visual distinction between vibrationally modulated recovery of population and sinusoidal wavepacket oscillations on the ground state is almost impossible, unless we are dealing with a diatomic molecule. For example, the familiar dissociation of Sodium Iodide studied by Zewail and coworkers.112-114 Bixon and Jortner (1997) modelled the temporal vibrational coherence effects in nonadiabatic ET dynamics in the pistacked TCNE-pyrene ET complex.104 The calculations predicted that for a nonradiative decay, the amplitude of coherences is determined by the off-diagonal matrix element—vibronic or electronic coupling between the states, and the reactant-product correlation—a function of FC vibrational overlaps between the reactant and the product states. It was proposed that in TCNE-pyrene ET complex, the most pronounced modulations originate the vibrational wavepacket motion of the initial state and not due to the vibrationally coherent product formation. The question which arises then, does a coherent excitation of a wavepacket modify the ET dynamics? A coherent excitation instantaneously prepares a non-Boltzmann distribution of an initial reactant state as superpositions of vibrational states of the Franck-Condon active nuclear modes. In the case of weak electronic coupling, due to the separation of time scales between the electronic and nuclear motion, ET occurs only after the relaxation of nuclei to the Boltzmann distribution.103 Therefore, the temporal coherences created by wavepacket motion dephase in the reactant state. On the other hand, in the intermediate regime of coupling, an ET reaction may be faster than the nuclear relaxation that dephases the wavepacket (often similar to the timescale of intramolecular vibrational relaxation), thus an ET reaction occurs from “hot” vibrational states. This will be interesting to study, but the challenge is how to analyse the effect of a non-Boltzmann distribution of initially prepared states on the ET dynamics. Mathies and coworkers studied a pi-stacked CT complex tetramethylbenzene-tetracyanoquinodimethane, using two-dimensional Raman spectroscopy and proposed that, intramolecular CT is equivalent to a conical intersection between excited and ground state.115-116 The branching space of the conical intersection is defined by the anharmonically coupled totally symmetric 323 cm-1 CCN bend—tuning mode and nontotally symmetric 1271 cm-1 C=C rocking mode—coupling mode (Figure 5a). The coherences produced along the tuning and coupling modes branch at the conical intersection and a portion of it crosses over to the ground state during the back-ET process.117-119 Betaine-30 is another example, in which photoexcitation accesses a CT state, followed by a solvent-dependent ultrafast back-ET—1.2 ps in acetonitrile and 3.1 ps in methanol. 120 The possible role of coherent vibrational motion was explored by Scholes and coworkers using a ca. 12 fs broadband pump-probe spectroscopy.121 It was found that dephasing times of certain high-frequency vibrations (1356, 1581, 1602 cm-1) correlate with the solvent-dependent ET rate, which suggests they are pertinent for the ET (Figure 5b). For example, the dephasing time constants of 1581 and 1602 cm-1 modes are 0.27 and 0.29 ps in acetonitrile, where the rate of ET is 1.2 ps and the corresponding dephasing time constants in methanol are 0.57 and 0.84 ps, where the rate of ET is 3.1 ps. Silva and coworkers reported that these high-frequency modes show frequency shifts and are anharmonically coupled to the low-frequency modes— prompting them to suggest that these modes are relevant to the

back-ET.122 Back-ET in Betaine-30 represents a radiationless crossing of the excited and the ground state potentials in the adiabatic limit.121 As per Bixon-Jortner theory, to bridge the energy gap between the two potentials, high-frequency quantum vibrations couple to the ET process. In Betaine-30, the free energy change for the product formation in acetonitrile is ca. 11,000 cm-1, so for the high frequency mode (say 1600 cm-1) to participate in the bET process, it must provide 4th or 5th vibrational quantum on the product surface (Figure 5c) to make the process barrierless (considering a total reorganization energy of ca. 3500 cm-1 for acetonitrile).121 Note that this large energy gap relative to the reorganization energy in Betaine-30 makes this a case of Marcus inverted region. A vibrational quantum (4th or 5th) that high in the potential will have a very small Franck-Condon factor associated with it. Thus, the enhancement in the rate due to the vibronic interactions is an outcome of near resonance condition between the reactant and product states but not due to the electronic coupling (ca. 2500 cm-1). The effective coupling (

From Fundamental Theories to Quantum Coherences in Electron

Recommend Documents