Electronic resonance with anticorrelated pigmentvibrations drives photosynthetic energy transferoutside the adiabatic frameworkVivek Tiwari, William K. Peters, and David M. Jonas1

Department of Chemistry and Biochemistry, University of Colorado, Boulder, CO 80309-0215

Edited* by James T. Hynes, University of Colorado, Boulder, CO, and approved November 28, 2012 (received for review June 29, 2012)

The delocalized, anticorrelated component of pigment vibrationscan drive nonadiabatic electronic energy transfer in photosyntheticlight-harvesting antennas. In femtosecond experiments, this energytransfermechanism leads to excitationof delocalized, anticorrelatedvibrational wavepackets on the ground electronic state that exhibitnot only 2D spectroscopic signatures attributed to electronic coher-ence and oscillatory quantumenergy transport but also a cross-peakasymmetry not previously explained by theory. A number of anten-nas have electronic energy gaps matching a pigment vibrationalfrequency with a small vibrational coordinate change on electronicexcitation. Such photosynthetic energy transfer steps resemble mo-lecular internal conversion through a nested intermolecular funnel.

2D spectroscopy | exciton | Förster resonance energy transfer

Photosynthesis, which powers life on our planet, is initiatedwhensunlight is captured by antenna proteins containing light-ab-

sorbing pigments. The antenna protein positions the pigments tocouple them and alters their electronic energies to direct electronicenergy transfer to a reaction center, which stores this energychemically. Photosynthetic energy transfer is remarkably fast andefficient, often with quantum yields equal to one within experi-mental error (1, 2). Even with atomic resolution structures ofseveral antennas and advances in quantum chemistry that providethe electronic structure and couplings between pigments (3), fullunderstanding of the energy transfer mechanism and design prin-ciples has remained elusive.Femtosecond 2D electronic spectroscopy (4) has revealed not

only energy transfer pathways (2, 5) but also oscillations indicativeof quantummechanical coherence (2, 6) (with a single systemhavingtwo or more distinct properties at the same time, like Schrödinger’scat) that can persist throughout the energy transfer process (2Dsignatures of photosynthetic energy transfer). This coherence spansmore than one pigment in the antenna (6, 7). Curiously, it has manysignatures in commonwith coherence between electronic states, butthe longest lifetimes at cryogenic temperatures are typical of co-herence between vibrational states, inspiring studies to elucidate itsorigin and role in photosynthetic light harvesting (8–12).Following Förster (13), electronic energy transfer is convention-

ally considered in an adiabatic framework [the Born–Oppenheimerapproximation (14)], where the motions of fast electrons are sepa-rable from slow vibrations. In this framework, each electronic statehas a potential energy surface on which molecules vibrate, andnonadiabatic changes in electronic state occur only when themolecules vibrate to a place on the potential surface where theadiabatic approximation breaks down.Within the adiabatic frame-work, an energy transfer step can be either adiabatic (Förster’sstrong coupling) or nonadiabatic (Förster’s very weak coupling).Although Förster recognized that the adiabatic framework wouldbe appropriate for electronic energy transfer in the very weak andstrong coupling limits but not necessarily in between the limits (13),he also argued that the very weak and strong coupling limits over-lapped for systems with continuous spectra (such as antennas).Conical funnels (15), where electronic potential energy surfa-

ces approach so closely that the adiabatic approximation breaksdown, play an important role in photochemistry (15, 16) and of-ten funnel molecules to lower-energy electronic states (internal

conversion). A conical funnel may be either a conical intersectionbetween adiabatic potential surfaces or a near miss, but it mustallow a change in electronic state before vibrational equilibration(15). Recent experiments on molecules (17) have found thatfemtosecond pulse-driven passage through a conical funnel gen-erates vibrational wavepackets on the ground electronic state thatexhibit signatures of coherence between the excited electronicstates connected by the funnel. In these experiments, quantumvibrational wavepacket width promotes nonadiabatic motionthrough the funnel, and signatures of electronic and vibrationalcoherence on the excited state can be rapidly suppressed by weaknonadiabatic coupling (17). This observation naturally raises thequestion of whether similar nonadiabatic dynamics drive photo-synthetic energy transfer.In antennas, several reported 2D signatures of photosyn-

thetic energy transfer match calculations for coherence be-tween excited electronic states [electronic coherence (EC)]and oscillating electronic-state populations [quantum trans-port (QT)] (10, 18). Antenna 2D spectra have overlappingexcitonic resonances with oscillatory phase-twisted 2D peak-shapes extending beyond resonance; methods for locatingpeaks and determining phase relationships are contested (19,20). Some 2D signatures are incompatible with the usual adi-abatic [Franck–Condon (FC)] (14) excitation of vibrationalwavepackets. Although QT was proposed to explain oscillatorydiagonal peaks (18), others argued that picosecond beat decayindicates a vibrational origin with amplitudes enhanced byvibrational–electronic coupling (21). Here, we show that oneshould expect small-amplitude vibrational wavepackets inwhich two pigments vibrate out of phase, that they give rise tononadiabatic vibrational electronic mixing between the excitedstates, and that this mixing leads to delocalized anticorrelatedvibrational wavepackets on the ground electronic state thatexhibit the reported signatures. The anticorrelated vibrationalwavepackets also generate signatures that have not been ex-plained by previous models incorporating EC, QT, or vibrations.The model reproduces these signatures when the frequency ofa FC active vibration is nearly resonant with the donor–acceptorelectronic energy gap; this vibrational-electronic resonance drivesenergy transfer through nonadiabatic dynamics entirely outsidethe adiabatic framework.

ModelThe dimer model used for this study starts with two pigments(A and B), each with an identical intramolecular harmonic

Author contributions: V.T., W.K.P., and D.M.J. designed research; V.T. and W.K.P. per-formed research; V.T., W.K.P., and D.M.J. analyzed data; and V.T., W.K.P., and D.M.J.wrote the paper.

The authors declare no conflict of interest.

*This Direct Submission article had a prearranged editor.

Freely available online through the PNAS open access option.

See Commentary on page 1148.1To whom correspondence should be addressed. E-mail: David.Jonas@Colorado.EDU.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1211157110/-/DCSupplemental.

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vibration. Using energy in frequency units, the Hamiltonian forpigment A is

HA =12ωA

�q2A + p2A

�IA +

�EA −ωAdAqA

���A��A��; [1]

where IA is the identity operator, qA and pA are the dimensionlessposition and momentum operators of the vibration with frequencyωA, jAi is the excited electronic state with vertical excitation energyEA, and dA is the FC displacement of the vibration (14). The Ham-iltonian for pigment B has the same form but with a vertical exci-tation energy difference, Δ = (EB −EA). If either pigment is excited,the two pigments interact through a Coulombic coupling (3), whichat long range can be calculated by the transition dipole approxima-tion; therefore, Hdimer = HA + HB + JðjAihBj+ jBihAjÞ, where thecoupling J is assumed to be independent of the intramolecularvibrational coordinates. The two pigment system has a single elec-tronic ground state (no pigments excited), two singly excited elec-tronic states (one for each pigment), and a doubly excited electronicstate (both pigments excited). The electronic states of coupled pig-ment systems are referred to as (Frenkel) excitons. The singly ex-cited (single-exciton) electronic states are delocalized to someextent by the coupling, and the energy of the doubly excited statediffers from EB + EA by the biexciton binding energy.The above Hamiltonian is standard in energy transfer (2). An

exact nonadiabatic treatment of the resulting states, their con-sequences in the 2D spectrum (including signals from the groundelectronic state), and role in photosynthetic energy transfer arepresented here. For a nonadiabatic treatment, it is useful to projectthe localized intramolecular vibrational coordinates into correlatedand anticorrelated vibrational coordinates, q+ = ðqA + qBÞ=21=2 andq− = ðqA − qBÞ=21=2, that are delocalized over both pigments. För-ster (13) recognized that the delocalized, anticorrelated vibration,because it tunes the electronic energy gap between pigments, isthe relevant vibrational coordinate for driving energy transfer atall coupling strengths, regardless of the extent of electronic de-localization. In the language of conical intersections (16), q− is thetuning coordinate g. In a delocalized, correlated intramolecularvibration, every bond length and angle vibrates with exactly thesame phase on both molecules; in the delocalized, anticorrelatedintramolecular vibration, every bond in one molecule contractswhile it expands in the other (and every bond angle bends in theopposite sense). Proteins can alter pigment vibrations (22); in thiscase, correlated and anticorrelated vibrations need not be normalmodes or vibrate with equal magnitude on two coupled pigments.This change of vibrational coordinates does not assume thatvibrations are delocalized but reveals that only the projection ofa vibration onto the delocalized, anticorrelated coordinate drivesand can be driven by nonadiabatic mixing. The nonadiabaticinteractions are thusmore clearly seen by writing the single-excitonHamiltonian as H1 = Hcorr + Hint, with

Hcorr =�1=2

�h�EA +EB

�+ω

�q2+ + p2+

�−

ffiffiffi2

pωdq+

iI1 [2]

and

H int =�1=2

�ω�q2− + p2−

�I1

+

2666664

−Δ2−ωdq−ffiffiffi

2p

�jAihAj JjAihBj

JjBihAj+Δ2+ωdq−ffiffiffi

2p

�jBihBj

3777775: [3]

H1 uses a basis where the vibrational coordinate is delocalizedand the electronic states are localized. Because Hcorr dependsonly on q+ and Hint depends only on q−, the adiabatic correlated

dynamics are exactly separable from the nonadiabatic anticorre-lated dynamics. Thus, a correlated vibration, although it disruptsthe quantum-phase relationship between the ground electronicstate and both single-exciton states, does not alter the potentialenergy difference between single-exciton states, does not disrupttheir quantum-phase relationship, and plays no role in nonadia-batic dynamics. [That is why correlated vibrations oscillate co-herently on the excited state throughout the energy transferprocess (23, 24).] In contrast, an anticorrelated vibration affectsthe excited states of pigments A and B oppositely in H int, tunestheir potential energy difference, disrupts their quantum-phaserelationship, and drives nonadiabatic dynamics. Both coordinatesaffect the absorption lineshape and 2D peakshapes. Because theanticorrelated vibration is delocalized over both coupled pig-ments, it cannot drive or be driven by nonadiabatic dynamics fora single-pigment system, explaining the results in refs. 6 and 7. Incontrast to the adiabatic approximation, which first solves for elec-tronic states of the dimer at fixed vibrational coordinates, the non-adiabatic treatment simultaneously solves for mixed vibrational–electronic states (SI Text).Model parameters are very roughly based on one pair of

excitons in the Fenna–Matthews–Olson (FMO) complex fromgreen sulfur bacteria (1). Its pigment, bacteriochlorophyll a (BChla) (22, 25), has FC active vibrations with frequency ω≈200 cm−1

and a stabilization energy of ð1=2Þωd2 ≈ 5 cm−1. In the dimermodel, the transition dipoles of the two pigments are perpen-dicular. The coupling is J = 66 cm−1 [typical in FMO (2)]. Thedifference in electronic energy between the two pigments, Δ, hasa static distribution with average hΔi = 150 cm−1 (also typical) andvariance σ2Δ. hΔi and J are chosen so that the average energygap between excitons (hΔEX i= h2½ðΔ=2Þ2+ J2�1=2i) matches thevibrational energy hΔEX i≈ω. Based on the excitonic peaks inFMO absorption (26), such a resonance occurs between the firstexciton and an exciton ∼200 cm−1 higher; additional resonancesseem likely (Table S1). The exciton splitting inhomogeneity,σΔEX = 26 cm−1, is slightly smaller than the value determined at19 K from the anisotropy for a pair of excitons separated by∼150 cm−1 in FMO (27). Because correlated vibrations do notaffect the nonadiabatic dynamics, the damping of correlatedvibrations can be incorporated through nonlinear responsefunctions (17). A critically damped Brownian oscillator witha frequency of 70 cm−1 and a stabilization energy of 30 cm−1

represents the correlated component of the low-energy pho-non sideband (this stabilization energy reproduces the Stokes’shift at 80 K) (2).Fig. 1 shows that the vibrational and electronic states of Hint in

the adiabatic approximation gradually change electronic char-acter with the anticorrelated vibrational coordinate. The deriv-atives of these adiabatic wavefunctions lead to nonadiabaticcouplings that must be included to calculate the exact states (16,17). The exact nonadiabatic states (which are both vibrationallyand electronically mixed) exhibit a rapid variation in their elec-tronic character over the entire range of coordinates, althoughthe adiabatic energy levels are reasonably accurate. This break-down of the adiabatic framework affects both the 2D spectra andthe photosynthetic energy transfer process.With a constant Coulombic coupling between pigments, the

adiabatic potentials for excitons in Fig. 1 do not intersect, andthere is no conical intersection (pigments must translate or rotaterelative to each other by more than the protein allows to send thecoupling to zero). However, the nonadiabatic changes in electroniccharacter with anticorrelated vibrational coordinate indicatea funnel. Because Eq. 2 indicates that the potentials are identicalin correlated vibrational coordinates, extending the curves in Fig.1 into 2D surfaces reveals a nonconical kind of funnel, in whichadiabatic potential surfaces arising from two different moleculesare nested inside each other in all intramolecular coordinates;the adiabatic framework breaks down everywhere inside thisnested intermolecular funnel. Nesting requires that the vi-brational displacement be small compared with the zero-point

1204 | www.pnas.org/cgi/doi/10.1073/pnas.1211157110 Tiwari et al.

amplitude, whereas nonadiabatic coupling requires vibrationaldisplacement or vibrational-dependent coupling.2D spectra are generated by crossing three noncollinear fem-

tosecond pulses (wave vectors ka, kb, and kc) inside the sample,measuring the four-wavemixingfield radiatedwithwavevector ks=kc+ kb− ka as a function of τ (the delay between pulses a and b) andt (the time after pulse c) at a fixed T (the interval between thesecond and third pulses), and Fourier-transforming with respect toτ and t. For scans over all τ, the real part of the 2D spectrum(absorptive 2D spectrum) reveals net changes in sample absorptionas a function of both the excitation (ωτ) and detection (ωt) fre-quencies. Some of the coherence signatures in 2D spectroscopydepend on whether τ is positive (pulse a is before pulse b forrephasing 2D spectra) or negative (pulse b is before pulse a fornonrephasing 2D spectra) and the physically meaningful relativesigns of the frequencies. Fig. 2 shows an all-parallel pulse polari-zation 2D spectrum for the dimermodel at T= 0 (it is simpler thanthe 2D spectrum of FMO, which involves 7 single-exciton statesand 21 biexciton states). Oscillations in the marked 2D peaks asa function of T indicate quantum coherence. 2D spectra generallycontain contributions from reduced ground-state absorption, ex-cited-state stimulated emission, and excited-state absorption (total2D spectra).In a 2D experiment, the first pulse pair can create vibrational

coherence on the ground state (by stimulated Raman scattering)along with both vibrational and electronic coherence on the excitedstate (4). An adiabatic picture predicts that vibrational coherencewill only cause weak modulations of the 2D spectra because ofsmall FC vibrational displacements in antenna pigments. Fur-thermore, the adiabatic approximation predicts that vibrationalcoherence will have the same polarization signatures as electronic-state populations. The nonadiabatic wavefunctions illustrated inFig. 1 generate dramatically different behavior. Stimulated Ramanexcitation of ground-state vibrational wavepackets can involve setsof transition dipoles identical to those sets involved in the

excitation of EC. This electronically enhanced contribution to 2Dspectra can be appreciated from Fig. 3, which uses wave-mixingdiagrams (SI Text) to show the states and fields for four terms in thenonlinear optical response from the nonadiabatically coupledstates of Hint. The first (leftmost) frequency is ωτ (excitation), andthe last (rightmost) frequency is ωt (detection). Pathways corre-sponding to the diagonal peaks (DPs; DP1 and DP2) and thecross-peaks (CPs; CP12 and CP21) are arranged accordingto their position in the 2D spectrum shown in Fig. 2. Forthe pathway contributing to CP12, extensive mixing betweenv- = 0 on B and v− = 1 on A (illustrated in Fig. 1) along theanticorrelated coordinate allows vibrational coherence on theground state to be excited exclusively through strong electronictransitions with Δv− = 0 basis state character; the frequencies(arrow lengths) and transition dipole directions (orange vs. bluearrows using the color scheme in Fig. 1) are all the same as thosefrequencies and directions for excited-state EC (Fig. S1). Theother paths are similar but not fully electronically enhanced ateach step (black arrows); for example, in DP1 and CP21, in-teraction with pulse b depends on nonresonant mixing of jv− = 0ion A and its vibrational overlap with jv− = 1i on G. As a result,ground-state vibrational wavepackets closely mimic signatures ofexcited-state electronic wavepackets. In particular, the transitiondipole sequences in Fig. 3 show that nonadiabatic excitation ofanticorrelated ground-state wavepackets will generate oscillationsthat survive the polarized pulse sequence used on LightHarvestingComplex II (LHCII) from green plants (28).

ResultsThree predicted signatures of EC between excited states in 2Dspectra have been reported for antennas. Signature 1, diagonalpeak amplitude beating, was first reported in the absorptive 2Dspectra of FMO (29); Fig. 4 shows that signature 1 also arisesfrom ground-state vibrations for both diagonal peaks in the2D spectra. Signature 2, a negative cross-peak CP12 beating fre-quency (for positive ωt) in the rephasing 2D spectrum, has been

−2 −1 0 1 2−200

020

040

0

q− == ((qA −− qB)) 2

Ene

rgy

((cm

−−1))

−− ||A ⟩⟩||v == 0⟩⟩

−− ||B ⟩⟩||v == 0⟩⟩

−− ||A ⟩⟩||v == 1⟩⟩

−− ||B ⟩⟩||v == 1⟩⟩

−− ||A ⟩⟩||v == 2⟩⟩

||A ⟩⟩||B ⟩⟩

((||A ⟩⟩ ++ ||B ⟩⟩)) 2

((||A ⟩⟩ −− ||B ⟩⟩)) 2

Fig. 1. Isolated pigment (dashed potential curves and levels), coupled adi-abatic (solid potential curves and levels), and exact nonadiabatic (levels notattached to any curve) excited-state vibrational–electronic levels for Hint ofthe dimer model with ΔEX =ω. v- is the quantum number of the anti-correlated vibration. Color shows the electronic-state character, which isa linear combination of the two pigment states, at each coordinate (SI Text).The electronic character does not change for isolated pigments and variesslowly for adiabatic states. Rapid changes in electronic character for thenonadiabatic levels over the entire coordinate range indicate a completebreakdown of the adiabatic framework.

11.0 11.2 11.4 11.6 11.8 12.0ωt /2πc (1000 cm-1)

11.0

11.2

11.4

11.6

11.8

12.0

-ωτ /

2πc

(100

0 cm

-1)

T=0

o

o x

x

-1.0

-0.5

0.0

0.5

1.0

Fig. 2. Real part of the ground electronic-state contribution to therephasing 2D electronic spectrum for the dimer model with (EA + EB)/2 =11,574 cm−1 at a temperature of 80 K. Here, the vertical axis is the excitationfrequency −ωτ, and the horizontal axis is the detection frequency ωt. Theamplitude of the 2D spectrum for each frequency pair is indicated by colorand contours at the 0%, 2%, 4%, 6%, 8%, and 10–90% levels. Positive andnegative contours are solid and dashed, respectively. The waiting time is T =0 fs. The 2D spectra are dominated by four resolved peaks, which oscillate inamplitude and shape with T. Diagonal peak maxima are marked by ○ (lowerleft, DP1; upper right, DP2); cross-peak maxima are marked by x (upper left,CP12; lower right, CP21).

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reported for PC645 (the phycocyanin PC645 antenna from themarine cryptophyte Chroomonas CCMP270) (30); Fig. 3 showsthat signature 2 is also reproduced for nonadiabatic excitation ofground-state vibrations (SI Text and Fig. S1). Signature 3, cross-peak amplitude modulations that occur only in rephasing and notin nonrephasing 2D spectra, can arise from EC (30) without QT.Ref. 30 reports CP12 beats with a signal to noise ratio of 2.4:1 inrephasing 2D spectra but reports no CP12 beats (signal to noiseratio = 1.2:1) in the nonrephasing 2D spectra of PC645. Fig. 4,Upper vs. Lower compares the beating amplitudes in the rephas-ing and nonrephasing 2D spectra—CP12 beats are approximatelyfive times weaker in the nonrephasing 2D spectrum, which isalso consistent with experiment. For nonadiabatic excitation ofground-state vibrations, CP12 beats should be detectable withhigher signal to noise nonrephasing 2D spectra. The ground-statevibrational origin for these three signatures naturally explainstheir robust behavior at physiological temperature, which wasrecently reported for FMO (31) and PC645 (30).A ∼180° phase difference between opposite cross-peak beats

has been reported in rephasing 2D spectra of PE545 (the phy-coerythrin PE545 antenna from the marine cryptophyte Rhodo-monas CS24) and attributed to EC (32). Contradicting thisinterpretation, for isolated peaks probed on center, Butkus et al.(20) show that opposite 2D cross-peaks oscillate in phase forboth EC and adiabatic excitation of vibrations. A 160° phasedifference arises from nonadiabatic excitation of anticorrelatedground-state vibrations (Fig. 4, Upper).

The calculations predict another significant experimental dif-ference. According to models of EC (2) and QT (10), oppositecross-peaks (related by reflection across the 2D diagonal, such asCP12 and CP21) are expected to oscillate with equal amplitudes ina rephasing 2D spectrum (figure 5, top in ref. 10). In contrast, Fig.4, Upper shows that anticorrelated ground-state vibrations createmuch stronger oscillations on CP12 (with excitation frequencyjωτj> detection frequency jωtj) than CP21. Diagrams (Fig. 3) showthat CP12 involves resonant vibrational–electronic coupling be-tween excited states at each step, whereas the weaker peak doesnot. Only beats for CP12, the cross-peak that the nonadiabaticanticorrelated vibration model predicts to be stronger, have beenreported for FMO (18, 19) and PC645 (30). Figure 3b in ref. 32shows both cross-peaks for PE545—they have asymmetric beatamplitudes with the stronger beats as calculated here.A fourth 2D signature, oscillations of diagonal peaks with phase

∼90° behind CP12 oscillations in the rephasing 2D spectrum (10,18), has been interpreted as evidence of QT, because EC alonecannot explain oscillations of the diagonal peaks in a rephasing 2Dspectrum. It is not clear what determines the experimental-phaserelationship in QT models (ref. 18 discusses a different phase re-lationship than shown in figure 5 in ref. 10). Fig. 4, Upper showsa 120° phase relationship betweenCP12 andDP2 near vibrational–electronic resonance. Ground-state vibrational wavepacket oscil-lations do not require a close resonance between intramolecularvibrational and electronic energy difference frequencies, but theyare enhanced within a broad maximum (∼130-cm−1 width) aroundresonance (Fig. S2). However, away from resonance, where non-adiabatic vibrational–electronic mixing is reduced and energytransfer between the pigments is presumably less efficient, thecalculations show a 180° phase relationship between CP12 and thediagonal beats (Fig. S2). Therefore, a 90° phase relationship would

Fig. 3. Wave-mixing pathways (SI Text) for the oscillatory ground-state 2Dsignal (D3 in ref. 4) showing resonant enhancement by nonadiabatic cou-pling of vibrational and electronic levels. For each diagram, the vertical axisis energy and time runs from left to right (neither drawn to scale). Thepathways are arranged to correspond with peaks in the rephasing 2Dspectrum (Fig. 2). Delocalized, anticorrelated vibrational levels on eachelectronic state are indicated by solid lines for v- = 0, dashed lines for v- = 1,and dotted lines for v- = 2; their orange and blue colors indicate localizedelectronic basis states on pigments A and B, respectively. As in Fig. 1, res-onant pairs of levels couple to form the nonadiabatic states (the first pair isroughly ½jAijv− = 1i± jBijv− = 0i�=21=2). The orange (blue) vertical lines in thefigure represent field–matter interactions using the A (B) electronic char-acter of a mixed level, with no change in v-, yielding a vibrational overlapintegral approaching one. Thus, CP12 is fully electronically enhanced atevery step, with all frequencies and transition dipole directions matchingthose frequencies and directions for purely electronic coherence (Fig. S1).Vertical lines in black represent weaker field–matter interactions—theseinteractions have small vibrational overlap or lack vibrational–electronicresonance. As a result, oscillations of DP1, DP2, and CP21 are not fullyelectronically enhanced.

Fig. 4. Absolute amplitudes and relative-phase relationships between thediagonal (DP1 and DP2) and cross-peaks (CP12 and CP21) as a function ofwaiting time T in the rephasing (Upper) and nonrephasing (Lower) groundelectronic-state contributions to the 2D spectra for the dimer model. Thetransients are offset (additive constants in boxes) to show phase relation-ships but not multiplicatively scaled. The peaks oscillate at a vibrationalfrequency of 200 cm−1, which is in resonance with the excitonic energy gap.In the rephasing 2D spectra, the cross-peak beats are 160° out of phase witheach other, CP12 beats are 120° ahead of the diagonal peaks, and CP12beats are ∼14 times stronger than CP21.

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suggest that electronic energy transfer involves nonadiabatic vi-brational–excitonic resonance in FMO.

DiscussionAlthough peak overlap in the experimental 2D spectra of an-tennas can distort the amplitude and phase relationships foundhere for resolved peaks, the above observations suggest that ef-ficient energy transfer proceeds through nonadiabatic interactionbetween two excitons that is resonantly enhanced by an FC activevibration of the monomeric pigment. The delocalized, anti-correlated vibration of the pigments may be the key component ofthe bath that exchanges energy with the electronic system in QTmodels. When the resonant mechanism is operative, the averageenergy gap between excitons should match a pigment vibrationalfrequency in the fluorescence or resonance Raman spectrum.Seven of nine off-diagonal beating frequencies reported for FMOmatch FC active vibrational frequencies of BChl a (Table S1). Thetwo frequencies that do not match have only been reported inref. 19. Furthermore, the vibrational frequencies that match off-diagonal beat frequencies in FMO are remarkably stable withrespect to structural and isotopic perturbations (Table S1). Turneret al. (30) remark that pigment vibrational frequencies are close tofrequencies in rephasing 2D spectra of PC645.The calculations above all show the contribution to the 2D

spectrum from ground-state vibrations. The ground-state signal isdriven only by initial dynamics on the excited states while stillcoherent with the ground state, and therefore, it can be accuratelycalculated without including slower relaxation between excitedstates. Decay of coherence between the ground and excited statesis included through damped correlated vibrations and gives rise tothe antidiagonal width of the diagonal peaks (for the real ab-sorptive 2D spectrum of the dimer model, this ∼80-cm−1 width atwaiting time T = 0 seems roughly comparable with the width forthe resolved lowest exciton peak in the 2D spectrum of FMO)(29). EC plays no role in the timescales over which oscillations onthe ground electronic state persist; they decay through vibrationaldephasing on the ground electronic state.Given that the key vibrational–electronic resonance between

two excitons is established by the electronic absorption spectrumof FMO (26) and the fluorescence spectrum of BChl a (25), theabove signatures of delocalized anticorrelated vibrations shouldbe present in the 2D spectra of FMO (18, 31). They also seem toarise in antennas from cryptophyte marine algae (30, 32), whichharvest different wavelengths. The breakdown of the adiabaticframework drastically alters the modeling and interpretation ofantenna 2D spectra. Fig. 5 shows coherence signatures from thetotal 2D spectrum, which includes excited-state contributions (butwithout any relaxation of population or coherence between ex-cited states). The experimentally reported phase relationshipsand asymmetrical cross-peak oscillation signatures in Fig. 4 areobscured by excited-state coherences. DP2 has a dominant os-cillation with ∼1-ps period arising from the splitting between itsnearly degenerate nonadiabatic levels (second and third levels inFig. 1). DP1 has its deepest oscillations at the excitonic splittingΔex in the nonrephasing 2D spectrum; these oscillations arisefrom EC and decay within ∼300 fs in Fig. 5. Comparison with Fig.4 shows that amplitudes for ground- and excited-state beats at thevibrational–excitonic frequency are comparable after ∼300 fs.Broadly, three types of coherence at a common frequency

Δex ∼ ω arise with resonant nonadiabatic coupling, and there-fore, there are roughly three coherence decay timescales. First,2D signatures of excited-state coherence at Δex (for resonance,Δex ∼ ω) disappear on a timescale dictated by anticorrelated in-homogeneity and coupling; these signatures are gone by ∼300 fsfor the dimer model (σΔEX = 26 cm−1) and disappear by ∼200 fsfor σΔEX = 34 cm−1 [the anisotropy beat decay for this excitonsplitting inhomogeneity is ∼180 fs in FMO (27)], although themodel does not include any coherence decay for an individualdimer. Second, relaxation of excited-state coherence at the vibra-tional frequency ω (for resonance, ω ∼ Δex) may have aspects ofboth vibrational and electronic coherence decay (perhaps this

relaxation generates EC signatures identical to those signaturesalready present on the ground state—if not, this coherenceprobably decays before ground-state signatures are seen). Third,coherent vibrations on the ground state (and possibly correlatedvibrations on the excited state) at ω should decay with typical vi-brational timescales (approximately picoseconds). As a result,ground-state vibrational coherence is likely to survive longest,with frequencies that are nonadiabatically enhanced by vibra-tional–excitonic resonance generating the most persistent ob-served signatures. For antennas, the timescale on which theexcited-state coherence signatures in Fig. 5 decay and give way tothe ground-state coherence signatures in Fig. 4 is not yet clear,but it could be as short as ∼200 fs for FMO at 80 K.Signatures found here may not be unique to nonadiabatic vi-

brational–excitonic resonance; for example, asymmetries acrossthe 2D diagonal can occur for FC excitation of vibrations onelectronic excitation (33). Models for antennas are needed toquantitatively test how much of the beating can be accounted forby resonant electronic enhancement of ground-state vibrations.Polarization sequences (28) may provide additional signatures(17). By analogy to the two-color pump-probe experiments thatVos et al. (23) used to distinguish between ground- and excited-state vibrations, two-color 2D spectra with pulses a/b of one colorand pulse c of another color could separately probe for ground-and excited-state beating.For energy transfer outside the adiabatic framework, the pro-

tein has at least four ways to control energy transfer: first, it cancontrol the electronic coupling; second, it can control the exci-tonic energy gap; third, it can control the vibrational displacement(22); and fourth, it might control how the coupled vibration dis-sipates energy. Comparing two related phycocyanins, Womickand Moran (34) have calculated that a vibrational–excitonic res-onance in one speeds up energy transfer. Resonance with an FCactive vibration (or group of vibrations) may provide a way for theprotein to select the energy acceptor for each donor. The firstthree control mechanisms dictate the strongest nonadiabaticinteractions and establish a nested intermolecular funnel; the

Fig. 5. Absolute amplitudes and relative-phase relationships between thediagonal and cross-peaks as a function of waiting time T in the totalrephasing (Upper) and nonrephasing (Lower) 2D spectra for the dimermodel, including ground-state bleaching, excited-state emission, and ex-cited-state absorption for zero biexciton binding energy. Vertical scales canbe compared directly with Fig. 4. As long as the irregular excited-stateoscillations persist, the phase relationships in Fig. 4 will be obscured, andCP21 beats will have roughly the same amplitude as CP12 beats.

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fourth mechanism may dictate how population relaxes down tothe lowest, electronically decoupled v- = 0 level, which completesthe electronic energy transfer process (as long as Zωvib

~> kBT)(35). Vibrational relaxation on the ground electronic state of theisolated pigments is relevant to this completion; in a fully local-ized basis, energy transfer from the v = 0 level of excited pigmentB can leave pigment A electronically excited with v = 0 andpigment B in its ground electronic state with v = 1 (a possibilityexcluded by approximations in refs. 21, 34, and 35).

ConclusionsReported signatures of photosynthetic energy transfer in the 2Delectronic spectra of antennas (Fig. S3 and Table S2) point tononadiabatic vibrational–electronic mixing resonantly enhancingthe amplitude of delocalized, anticorrelated vibrational wave-packet motion on the ground electronic state, which is likely tooutlive electronic and vibrational–electronic coherence. Further-more, the present mechanism predicts an asymmetry betweenopposite cross-peak oscillations (found for an antenna fromcryptophyte marine algae, PE545) but not explained by EC or QT.The mechanism also predicts a reduced amplitude cross-peakbeating in nonrephasing 2D spectra that is consistent with exper-iment but has not yet been reported. Although quantitativemodelswith nonadiabatic vibrational–electronic resonance, electronicdecoherence, and relaxation under physiological conditions areneeded to develop a deeper picture of and perspective on thisenergy transfer mechanism, it is remarkable how closely beat fre-quencies in the 2D spectra of the FMO complex match the fre-quencies of FC active skeletal vibrations in its BChl a pigment.

Thus, although additional studies are needed, it seems to us likelythat resonant nonadiabatic coupling plays a role in photosyntheticenergy transfer and that vibrational–electronic resonances innested intermolecular funnels are an important design principle.

MethodsNonadiabatic dynamics are fully incorporated using quantum states of thedimer Hamiltonian in sum-over-states formulas for the nonlinear opticalresponse (4). 2D spectra also reflect interactions with the bath, which shouldbe decomposed into correlated and anticorrelated parts. To approximatethe anticorrelated component of both the phonon sideband and static in-homogeneities in the protein, the nonadiabatic problem is solved formembers of an ensemble with a static Gaussian distribution of electronic siteenergy differences (σΔ = 34 cm−1), and the nonlinear responses are added.The temperature was fixed at 80 K to match experiments on FMO. The sum-over-states response was multiplied by the correlated Brownian oscillatorresponse, and 2D Fourier transform (FT) spectra were calculated (SI Text)using a 3D FT algorithm. Calculation of a waiting time series of 2D spectratook ∼3 h on two hex-core 2.8-GHz Intel Westmere processors.

ACKNOWLEDGMENTS. We thank Greg Scholes and Greg Engel for usefuldiscussion and sharing unpublished data. We thank Donatas Zigmantas andDavid Bocian for helpful discussions of their work. We thank David Yarkonyfor helpful comments on a draft manuscript. This material is based on worksupported by National Science Foundation Grant CHE-1112365. This workused the Janus supercomputer, which is supported by National ScienceFoundation Grant CNS-0821794 and the University of Colorado at Boulder.The Janus supercomputer is a joint effort of the University of ColoradoBoulder, the University of Colorado Denver, and the National Center forAtmospheric Research.

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