In terms of scale, we believe it is useful to delineatetransfer in light harvesting complexes from excitation migration and trapping in the photosynthetic unit. We suggest that the designation “excitationtransport” is preferably reserved for the excitation migration throughout the photosynthetic unit, while the rapidequilibration and flow in an isolated complex istransfer. Calculations at present have mainly focused onin isolated light harvesting complexes because there exists the opportunity to examine deeplymechanisms. The consequences of more sophisticatedand the deviations they will predict from random walk models on longer length scales, for example, for photosynthetic units (>100 chromophores) are unknown. As we discussed previously,the average effect is likely difficult to discern, but mechanisms at play in tails of the distributions of trapping times will be interesting. We can summarize this viewpoint by stating that an outstanding challenge is to connect details ofat the isolated light harvesting protein level to the collective function of many proteins.

It is generally thought thatmoves more or less by a random walk, historically based on early models forhoppingand Pearlstein's hypothesis in the 1960s that coherent transfer should not dramatically change the limits of excitation diffusion length.However, such conclusions are subject to revision, especially as we develop this field of research in terms ofand experimental probes of the mechanisms ofRecently, research has suggested that a balance between(quantum effects) and decoherence (loss of quantum effects) is optimal for excitation transport.In that case, the random walk model must be modifiedand more sophisticatedofare required.Likewise any quantummade must be appropriate for open systems; delocalizationsuch as the Inverse Participation Ratio are not appropriate when decoherence is taken into account.

The size of the photosynthetic unit can change depending on whether or not growth is light limited. The essential principle is that thecenters should be photo-excited at a rate balanced with the rate of linear electron flow from HO to NADP. The right balance inhibits backand thus losses, downstream from the photo-initiation processes. A large photosynthetic unit helps maintain the rate of photosynthetic production under low light conditions because it provides thecenter with an increased absorption cross section. Light can be limiting, for example in microbial mats,turbid water,competition in aquatic communities,or at great depths in the sea.A further interesting complexity is that not only the size, but also the organization of light harvesting complexes withcenters has been documented to change depending on light conditions.

The photosynthetic unit typically comprises about 200–400 light harvesting chromophores associated with eachcenter, depending on organism and growth conditions.The upper limit to the size of a photosynthetic unit is dictated by interplay of two primary factors: the rate of excitation transfer and thelifetime.The minimum size of a photosynthetic unit is controlled by a delicate balance ofabsorption, conversion, and dissipation of excess excitations. The physical size and layout of a supercomplex of the photosynthetic unit in the vicinity of photosystem II in higher plants is shown in Fig.. Experiments have estimated there are about 200 chlorophylls percenter under normal growth conditions.In purple bacteria, a combination of site-energy tuning of chromophores by the protein and excitonic effects creates a naturalgradient directing excitations towards thecenter (specifically to the special pair). Imaging of photosynthetic membranes of purple bacteria by atomic force microscopy reveals the arrangement of light harvesting complexes (LH2) andcenters clearly,while the ratio of LH2 tocenters can also beIt is found that 2–7 LH2 complexes are associated with eachcenter, meaning there are ∼100–200 bacteriochlorophyll molecules percenter. Despite this low ratio of antenna molecules tocenters, Voshave shown that excitation migrates over about 1000 bacteriochlorophylls—in other words, about 25 photosynthetic units are connected.

The layout of light-absorbing molecules in the photosynthetic unit, Fig., may appear like sprawling labyrinths of chromophores with little order. Nevertheless, short-lived electronic excitations (lifetimes of 1–5 ns) traverse these molecularly vast landscapes with high efficiency to sensitizecenters and initiate photosynthetictransduction. The typical photosynthetic unit is an excellent example of finely tuned adaption to limiting conditions. Chromophores are organized so there is optimal harvesting of light across a relatively broad spectral range. Optimal in this case means within possible constraints of resources available—light, nutrients, etc.

Theseenable us to detect how entanglement among multiplediminishes during dynamics evolution, for example, four-site entanglement decays quickly, followed by three-site entanglement, and so on.We can thereby map out in considerable detail how excitation is delocalized in a multichromophore aggregate.

We have recently demonstrated that it is also possible tomultipartite entanglement in mixed states analytically, once more by exploiting the relationship between purity, entanglement, and the statistical moments.that are a function of the statistical momentscan be applied to the mixed state in question, as well as to a reference state σ with the same level of purity. In order tothe-partite entanglement in a mixed state one can use the equationwhere τis a function of the statistical moments up to orderas described in Ref.. The challenge is in deriving the reference state σ, which should have the maximum level of (–1)-partite entanglement for that level of purity. Thisreturns values of statistical correlation that are not possible for states with lower orders of entanglement at that level of entropy. Effectively, one can remove the noise from theirin order to process the signal. Much like the bipartiteknowledge of population statistics and purity within a system is all that is needed to fullythe delocalization length, at any level of decoherence.

Though suchprovide excellent information on thewithin these systems, they only do so using two-body correlations. Recently,of many-body entanglement for pure states, using higher order statistical moments than the IPR, were applied to model light harvesting complexes.Adapting theseto mixed states, which is significantly harder, finally delivers the true “delocalization length” in the systems: a precise number of sites that share quantum correlations during every step ofdespite decoherence. A computationally intense numerical method derived by Leviwould provide such a

This result could prove essential in the experimentalofin light harvest complexes; by gathering time dependent population statistics and employing a judicious assessment of entropy within a system, it could be possible to gain a strong understanding of the time dependentwithin a system, without having to do full state tomography. For example, one can fullythe purity of a state with a number ofthat scale linearly with system size.

In parallel to research on delocalization in LH2, the field of quantum computation and quantum information was being established, and with it a whole new field dedicated toentanglement in mixed states. The emergence of this field has been a great asset, and the suitability of theseto light harvesting complexes has recently been demonstrated.of bipartite entanglement such as the tangle,the negativity,and the relative entropyhave all been employed to chart the evolution ofin light harvesting systems.Interestingly, thesethe IPR, the tangle, and the relative entropy, can be related using the purityFor example, the total two-body entanglement, or bipartite entanglement,(ρ), in a system with one excitation can be written as the difference between the purity and the IPR,whereis the statistical moment of order two of the system populations, which is exactly equivalent to the IPR.

Therefore the IPR, as strictly a measure of state populations, is unsuitable in its application to mixed states. As such, it is unable to reveal the evolution of coherence within these light harvesting complexes when we are concerned with including the effects of homogeneous line broadening.

At the time the Inverse Participation Ratio (IPR), defined by Thoulesswas frequently employed as aof delocalization.Thisalso known as a second order statistical moment, looks at the variance of probabilities within aψ =+ …delocalized oversites:The IPR can only be used, however, in calculations based onFor example, whenare calculated as a function of static disorder by Monte Carlo averaging.The crucial distinction here is that eachin the ensemble is calculated and the IPR is calculated for eachand statistically averaged. That is different than first averaging thethemselves, by constructing a density matrix, then calculating the delocalization length. Consider an experiment thatthe evolution of aafter many iterations, a clear picture of the population dynamics is formed; however, the coherences between these populations get washed out due to the phase differences in averaging the slightly different states. Representing these populations and coherences in a density matrix ρ, one can see how much the populations differ from their coherences using acalled the purity. The purity is defined as the trace of the square of a density matrix,A pure state has a purity value of 1 and represents a fully coherent state that can also be written as aA mixed state represents a statistical mixture of different coherent states and will have a smaller value as low as 1/, in a system of size, if the state is fully mixed. In information terms, a decrease in purity leads to an increase in entropy.

The 2007 report by Fleming and co-workers ignited research about quantumin light harvesting complexes.The foundation for this new experimental insight was a body of literature examiningdelocalization in photosynthesis. A European Science Foundation workshop (organized by Leonas Valkunas and Rienk van Grondelle) held in Birstonas, Lithuania in 1996 highlighted the excitement about coherent delocalization ofThe book of abstracts notes that “Major questions concern…the time over which the excitation must be considered as coherent….” The topic of discussion was largely Cogdell and co-workers’ reportof an atomic resolution crystal structure for the peripheral light-harvesting complex LH2 from the purple anoxygenic bacterium.)strain 10050. Papers associated with this workshop highlight studies of delocalization in LH2 and theofdelocalization. They are collected in volume 101, issue 37 (1997) of the

From a physical point of view, reasons behind changes in delocalization includelocalization—the interplay of the Stokes shift in the site basis anddelocalization with a lesser nuclear reorganization—or an explicit dependence ofresonance on various nuclear coordinates (). It is clear that experiments that can follow or characterize delocalization, with femtosecond resolution, after photoexcitation will give insights into some interesting photophysics. Perhaps such experiments can even reveal howwavepackets evolve in time, although achieving state or process tomography for disordered condensed phase systems will be challenging.Another approach is to devise andmetrics ofdelocalization.

delocalization, if strong, can be indicated by spectral features that are perturbed, shifted, or split.Scaling of transition strength or nonlinear susceptibility is useful when comparativecan be made.Similarly, changes in the radiative rate (“superradiance”) are quite sensitive to delocalization.Coherent superpositions of states that contribute oscillating cross-peaks to two-dimensional electronic(2DES) data have recently been proposed to reveal delocalization and coherences—and particularly how they dephase.Another probe ofdelocalization can beabsorption in pump-probe and 2DES experiments.Two-exciton states, lying at approximately twice thenecessarily accompanystates. Whilestates are defined in the basis of permutations of excitation among the interacting molecules, two-exciton states are defined as permutations of two excitations.absorption reveals theto two-exciton electronic transition(s).

At the foundation of the most interestingeffects in light harvesting systems are molecularMolecularstates (which is what we mean by “exciton” in this context) are electronicwhere theis shared by two or more molecules. The electron density remains localized on the molecules individually, but the transition density (see Ref.) for theis a well-defined superposition of molecule-localized transition densities. In the kinds ofthat has interested researchers recently, this delocalization evolves in time. From a definition point of view, that means we cannot find a basis rotation among localized and delocalized representations that consistently diagonalizes the density matrix as the dynamics evolve after photo-excitation. Thus we find off-diagonal density matrix elements that signal quantum mechanical

C. Intramolecular vibrations, delocalization, and coherence

spectroscopy. High frequency modes (compared to kT/h) are clearly evident as vibronic progressions in absorption and emission spectra. Torsional modes that have a frequency change from ground to excited state can broaden absorption spectra substantially. 110 122, 054501 (2005). 110. G. Heimel, M. Daghofer, J. Gierschner, E. List, A. Grimsdale, K. Müllen, D. Beljonne, J. L. Brédas, and E. Zojer, J. Chem. Phys., 054501 (2005). https://doi.org/10.1063/1.1839574 theory for electronic energy transfer employs a separation between electronic and nuclear factors and thereby naturally includes vibronic details of molecular spectroscopy in the Förster spectral overlap. 26 54, 57– 87 (2003). 26. G. D. Scholes, Annu. Rev. Phys. Chem., 57–(2003). https://doi.org/10.1146/annurev.physchem.54.011002.103746 energy conservation and therefore decides the rate of energy transfer. 111 111. I. B. Berlman, Energy Transfer Parameters of Aromatic Compounds ( Academic Press , New York , 1973). energy transfer (exciton relaxation) in the intermediate to strong electronic coupling regimes, especially for interpretation of coherent oscillations observed in two-dimensional electronic spectroscopy, see Refs. 76 11, 20130901 (2014). 76. F. Fassioli, R. Dinshaw, P. C. Arpin, and G. D. Scholes, J. R. Soc., Interface, 20130901 (2014). https://doi.org/10.1098/rsif.2013.0901 112–116 9, 113– 118 (2013). 112. A. W. Chin, J. Prior, R. Rosenbach, F. Caycedo-Soler, S. F. Huelga, and M. B. Plenio, Nat. Phys., 113–(2013). https://doi.org/10.1038/nphys2515 3, 2029 (2013). 113. A. Chenu, N. Christensson, H. F. Kauffmann, and T. Mancal, Sci. Rep., 2029 (2013). https://doi.org/10.1038/srep02029 587, 93– 98 (2013). 114. V. Butkus, D. Zigmantas, D. Abramavicius, and L. Valkunas, Chem. Phys. Lett., 93–(2013). https://doi.org/10.1016/j.cplett.2013.09.043 The nature of coherences in the B820 bacteriochlorophyll dimer revealed by two-dimensional electronic spectroscopy ,” Phys. Chem. Chem. Phys. (in press). 115. M. Ferretti, V. Novoderezhkin, E. Romero, R. Augulis, A. Pandit, D. Zigmatas, and R. van Grondelle, “,” Phys. Chem. Chem. Phys. (in press). https://doi.org/10.1039/c3cp54634a Intramolecular radiationless transitions dominate exciton relaxation dynamics ,” Chem. Phys. Lett. (in press). 116. C. Jumper, J. Anna, A. Stradomska, J. Schins, M. Myahkostupov, V. Prusakova, D. Oblinsky, F. N. Castellano, J. Knoester, and G. D. Scholes, “,” Chem. Phys. Lett. (in press). https://doi.org/10.1016/j.cplett.2014.03.007 117 5, 3012 (2014). 117. E. J. O’Reilly and A. Olaya-Castro, Nat. Commun., 3012 (2014). https://doi.org/10.1038/ncomms4012 Intramolecular vibrations play a marked role in molecularHigh frequency modes (compared to) are clearly evident as vibronic progressions in absorption and emission spectra. Torsional modes that have a frequency change from ground tocan broaden absorption spectra substantially.Försterfor electronicemploys a separation between electronic and nuclear factors and thereby naturally includes vibronic details of molecularin the Förster spectral overlap.That is of profound quantitative importance because it is often vibronic overlap of the red tail of the donor fluorescence spectrum with the vibronic progression of the acceptor absorption spectrum that enablesconservation and therefore decides the rate ofRecent work has highlighted the importance of understanding the role of intramolecular vibrations inrelaxation) in the intermediate to strong electronic coupling regimes, especially for interpretation of coherent oscillations observed in two-dimensional electronicsee Refs.and, and references cited therein. In a striking recent report, O’Reilly and Olaya-Castro show that it is the intramolecular nuclear coherences in combination with electronic coherences that ensure the coherences are definitively quantum in nature.

A common theoretical framework for going beyond Förster theory is to diagonalize the electronic Hamiltonian, then add coupling to low frequency bath nuclear degrees of freedom perturbatively (e.g., Redfield theory) or nonperturbatively. Despite the recent leaps in theoretical models for energy transfer, it is not obvious that roles of high frequency intramolecular vibrations—that are so important in the Förster spectral overlap—are appropriately captured in present theories for intermediate or strong coupling. We would like to stimulate discussion about the limitations of the electronic exciton basis, or the polaron basis, versus a vibronic basis.

118–120 135, 154311 (2011). 118. C. G. Heid, P. Ottiger, R. Leist, and S. Leutwyler, J. Chem. Phys., 154311 (2011). https://doi.org/10.1063/1.3652759 136, 174308 (2012). 119. P. Ottiger, S. Leutwyler, and H. Köppel, J. Chem. Phys., 174308 (2012). https://doi.org/10.1063/1.4705119 131, 204308 (2009). 120. P. Ottiger, S. Leutwyler, and H. Köppel, J. Chem. Phys., 204308 (2009). https://doi.org/10.1063/1.3266937 measured high resolution spectra, free from an “environment” that would cause line broadening and contribute to exciton relaxation. Therefore these experiments reveal, in essence, a zeroth-order description of the vibronic exciton levels that may be a good basis for theories of exciton transfer in the condensed phase. The main finding of these studies is that it is essential to include vibrations in the basic model because the electronic transition strength is distributed over the entire vibronic manifold, leading to a substantial reduction in exciton interactions. The electronic coupling is adjusted by Franck-Condon factors, so that exciton splittings—even of the origin bands—are significantly smaller than predicted from electronic structure calculations. Leutwyler and co-workers call this striking reduction of electronic coupling as vibronic quenching. The vibronic ladder of states measured for a molecular aggregate is thus very different from that calculated under the crude assumption that nuclei are frozen. Leutwyler and co-workers have studied model molecular dimers in the gas phase.Theyhigh resolution spectra, free from an “environment” that would cause line broadening and contribute torelaxation. Therefore these experiments reveal, in essence, a zeroth-order description of the vibroniclevels that may be a good basis foroftransfer in the condensed phase. The main finding of these studies is that it is essential to include vibrations in the basic model because the electronic transition strength is distributed over the entire vibronic manifold, leading to a substantial reduction ininteractions. The electronic coupling is adjusted by Franck-Condon factors, so thatsplittings—even of the origin bands—are significantly smaller than predicted from electronic structure calculations. Leutwyler and co-workers call this striking reduction of electronic coupling as. The vibronic ladder of statesfor a molecular aggregate is thus very different from that calculated under the crude assumption that nuclei are frozen.

exciton delocalization with respect to intramolecular geometry distortions, Fig. 2 theories for energy transfer beyond the Förster limit is difficult. Owing to the opposite way, the potentials are displaced in the example shown in Fig. 2 exciton is delocalized. At larger displacements in either direction along the x-axis, the diabatic curves have an energy gap much greater than the electronic coupling, so the lowest energy excited state is localized on just one molecule. The breakdown in the Born-Oppenheimer approximation produces an interesting coordinate-dependence todelocalization with respect to intramolecular geometry distortions, Fig.. This effect nicely shows why including intramolecular modes inforbeyond the Förster limit is difficult. Owing to the opposite way, the potentials are displaced in the example shown in Fig., typical of an antisymmetric normal mode of a molecular dimer, degeneracy, and near degeneracy of the diabatic potentials occurs only near the ground state equilibrium geometry, where the diabatic curves cross. At these coordinates, therefore, theis delocalized. At larger displacements in either direction along the-axis, the diabatic curves have angap much greater than the electronic coupling, so the lowestis localized on just one molecule.

exciton delocalization is both the challenge for theory—it directly reflects the breakdown of the Born-Oppenheimer approximation—and a fascinating target for experiments to probe. Vertical transitions probed by linear spectroscopy are useful starting points, particularly when vibronic transitions are well resolved. On the other hand, it would be very revealing if delocalization as a function of the nuclear coordinate axis could be traced out. A possible approach is to use nuclear wavepackets generated by femtosecond laser pulses 121 14, 368– 375 (1981). 121. E. Heller, Acc. Chem. Res., 368–(1981). https://doi.org/10.1021/ar00072a002 spectroscopic probe. The coordinate dependence of thedelocalization is both the challenge for theory—it directly reflects the breakdown of the Born-Oppenheimer approximation—and a fascinating target for experiments to probe. Vertical transitions probed by linearare useful starting points, particularly when vibronic transitions are well resolved. On the other hand, it would be very revealing if delocalization as a function of the nuclear coordinate axis could be traced out. A possible approach is to use nuclear wavepackets generated by femtosecond laser pulsesas a localizedprobe.

excited state absorption (ESA) indicates exciton delocalization, by detecting the exciton to two-exciton transitions, it could be used as an interesting probe for coherence in 2DES. Figure 3 Chroomonas sp. CCMP270). Partly connected with the excitonic dimer located in the center of the PC645 complex, the 2D spectra show an off-diagonal cross-peak that oscillates remarkably as a function of waiting time. 75 463, 644– 648 (2010). 75. E. Collini, C. Y. Wong, K. E. Wilk, P. M. G. Curmi, P. Brumer, and G. D. Scholes, Nature (London), 644–(2010). https://doi.org/10.1038/nature08811 77,78 14, 4857– 4874 (2012). 77. D. B. Turner, R. Dinshaw, K. K. Lee, M. S. Belsley, K. E. Wilk, P. M. G. Curmi, and G. D. Scholes, Phys. Chem. Chem. Phys., 4857–(2012). https://doi.org/10.1039/c2cp23670b 2, 1904– 1911 (2011). 78. D. B. Turner, K. E. Wilk, P. M. G. Curmi, and G. D. Scholes, J. Phys. Chem. Lett., 1904–(2011). https://doi.org/10.1021/jz200811p Considering thatabsorption (ESA) indicatesdelocalization, by detecting theto two-exciton transitions, it could be used as an interesting probe forin 2DES. Figureshows a selected 2DES spectrum recorded for PC645 (sp. CCMP270). Partly connected with the excitonic dimer located in the center of the PC645 complex, the 2D spectra show an off-diagonal cross-peak that oscillates remarkably as a function of waiting time.This oscillation cannot be explained solely by vibrational coherences.Note also that the cross-peak coincides with an ESA, although we cannot distinguish whether the oscillations are present only in the ESA and/or the underlying bleach.

wavefunction varies strongly. Specifically, the exciton delocalization depends on this nuclear coordinate like the example shown in Fig. 2 3(b) energy of the vibration and the electronic coupling between the molecules is smaller than the energy difference between the absorption bands. It is interesting to hypothesize a scenario where the ESA oscillates—essentially blinks on and off—as a nuclear wavepacket travels to and fro along a nuclear coordinate for which the electronicvaries strongly. Specifically, thedelocalization depends on this nuclear coordinate like the example shown in Fig.. To illustrate, consider the coordinate of an intramolecular vibrational mode sketched in Fig.. This particular example has some specific features: two interacting molecules have electronic absorption bands offset by approximately theof the vibration and the electronic coupling between the molecules is smaller than thedifference between the absorption bands.

We imagine that if the short laser pulse generates a wavepacket starting at a coordinate in the region indicated as “localized,” the probe will not detect ESA because the electronic coupling is small compared to the zeroth-order energy difference between electronic transitions on the two molecules at this geometry. Now imagine the wavepacket evolving along the coordinate axis until it reaches the region indicated as “delocalized.” At this point, the zeroth-order electronic transitions of the two molecules are almost degenerate (because the electronic energy gap is matched by the vibrational energy) and exciton states will prevail. The probe will now detect ESA as an exciton to two-exciton transition. The oscillating nuclear wavepacket will produce a modulation in the probe because the ESA is modulated by exciton delocalization that depends on nuclear coordinate. A modulation of the transition moments would also be evident and detected in anisotropy measurements, which might be a more sensitive probe. If such a scenario can be identified it will help to map out the intramolecular coordinate dependence of exciton delocalization. Perhaps this is what we have detected for PC645?

122 118, 1296– 1308 (2014). 122. S. D. McClure, D. B. Turner, P. C. Arpin, T. Mirkovic, and G. D. Scholes, J. Phys. Chem. B, 1296–(2014). https://doi.org/10.1021/jp411924c spectroscopy to study a photosynthetic cryptophyte antenna complex, PC577 isolated from Hemiselmis pacifica (CCMP 706), Fig. 4 excited electronic state. The formalism developed in the past for coherent superpositions of vibrational modes (wave packets) was useful for interpreting how the amplitude and phase of the oscillations vary with probe wavelength (λ probe ) in the transient absorption data. 123 96, 6147– 6158 (1992). 123. W. T. Pollard, S. L. Dexheimer, Q. Wang, L. A. Peteanu, C. V. Shank, and R. A. Mathies, J. Phys. Chem., 6147–(1992). https://doi.org/10.1021/j100194a013 Quantitative approaches are needed to disentangle the effects of many vibrational degrees of freedom. In recent work,we used broad-band femtosecond pump-probeto study a photosynthetic cryptophyte antenna complex, PC577 isolated from(CCMP 706), Fig.. This light harvesting complex has a peculiar open structure, so unlike PC645 it does not incorporate a strong excitonic dimer. Analysis of vibrational wave-packet dynamics showed the oscillations are contributed by superpositions of levels in theThe formalism developed in the past for coherent superpositions of vibrational modes (wave packets) was useful for interpreting how the amplitude and phase of the oscillations vary with probe wavelength (λ) in the transient absorption data.Direct Franck-Condon excitation of the ground-state equilibrium population with coherent, broadband light generates a superposition of multiple, closely spaced vibrational levels along different vibrational coordinates of the molecule. This superposition of vibrational levels is non-stationary under the Hamiltonian of the system and launches a wave packet that explores the multi-dimensional phase space of the excited vibrational coordinates.

energy spacing between the ground-state and excited-state surfaces, the position of the wave packet as a function of pump-probe time delay is evident in the transient absorption spectra, 124 75, 3410– 3413 (1995). 124. C. J. Bardeen, Q. Wang, and C. V. Shank, Phys. Rev. Lett., 3410–(1995). https://doi.org/10.1103/PhysRevLett.75.3410 probe . A key feature is that the phase of the oscillations undergoes a π radians shift at a probe wavelength (λ probe ) corresponding to the global minimum of the potential-energy surface. 125 363, 320– 325 (1993). 125. M. H. Vos, F. Rappaport, J.-C. Lambry, J. Breton, and J.-L. Martin, Nature (London), 320–(1993). https://doi.org/10.1038/363320a0 4(b) probe = 640 nm, which corresponds to the maximum of the fluorescence spectrum. That shows that the wavepacket was produced on the excited electronic state. The reason for the node is that the wavepacket passes through this point twice per period of oscillation rather than just once like at the other values of λ probe . The wave packet oscillates about the minimum of the multi-dimensional potential-energy surface and, based on thespacing between the ground-state andsurfaces, the position of the wave packet as a function of pump-probe time delay is evident in the transient absorption spectra,resolved by λ. A key feature is that the phase of the oscillations undergoes a π radians shift at a probe wavelength (λ) corresponding to the global minimum of the potential-energy surface.For example, in Fig., the oscillations clearly decrease in amplitude and abruptly change phase around λ= 640 nm, which corresponds to the maximum of the fluorescence spectrum. That shows that the wavepacket was produced on theThe reason for the node is that the wavepacket passes through this point twice per period of oscillation rather than just once like at the other values of λ