Testing for memory-free spectroscopic coordinatesby 3D IR exchange spectroscopyJoanna A. Borek, Fivos Perakis, and Peter Hamm1

Department of Chemistry, University of Zurich, CH-8057 Zurich, Switzerland

Edited by Charles B. Harris, University of California, Berkeley, CA, and approved June 13, 2014 (received for review April 16, 2014)

Using 3D infrared (IR) exchange spectroscopy, the ultrafast hydrogen-bond forming and breaking (i.e., complexation) kinetics of phenolto benzene in a benzene/CCl4 mixture is investigated. By introduc-ing a third time point at which the hydrogen-bonding state ofphenol is measured (in comparison with 2D IR exchange spec-troscopy), the spectroscopic method can serve as a critical testof whether the spectroscopic coordinate used to observe theexchange process is a memory-free, or Markovian, coordinate.For the system under investigation, the 3D IR results suggestthat this is not the case. This conclusion is reconfirmed by ac-companying molecular dynamics simulations, which furthermorereveal that the non-Markovian kinetics is caused by the hetero-geneous structure of the mixed solvent.

3D IR spectroscopy | solvent dynamics | ultrafast dynamics

Whenever one writes a chemical reaction equation, such asin the simplest case of an equilibrium between two states

of a molecular system

C⇌F [1]

with the corresponding Master equation for its kinetics

d½C�dt

=−kCF ½C�+ kFC½F�d½F�dt

=+ kCF ½C�− kFC½F�;[2]

one implicitly assumes Markovianity. That is, it is assumed thatthe probability to react per time unit, e.g., from C to F, scaleslinearly with concentration [C] with proportionality constant kCF,but does not depend on the history of how C has been formed.When the reaction is slow in comparison with the relaxation ofits environment, then this assumption is well justified. One canhowever think of many situations where this is not the case. Forexample, proteins are heterogeneous on very long timescales dueto their structural complexity, so enzymatic reaction catalyzed bythem might not fulfill that condition (1). Other examples arefound in glass-forming liquids close to the glass transition, whentheir relaxation covers many orders of magnitude in time (2).When the speed of a reaction approaches the ultrafast pico-

second regime, which is the topic of the present paper, then eventhe solvation time of normal liquids far away from any glasstransition may become time-limiting. For example, it has beenshown that the distance between a Na+ and a Cl− ion is not agood reaction coordinate to describe ion pair dissociation inaqueous solution (3), in the sense that the distance for which thefree energy peaks does not correspond with the transition stateof the reaction. Hidden coordinates exist, which must be solventcoordinates because there are no other degrees of freedom.Unless these solvent coordinates have not fully relaxed and losttheir memory about a reaction event, the probability for a backreaction must depend on the history. Markovianity is related toa separation of timescales between the speed of a chemical re-action of a molecular system versus that of the relaxation of its

solvent. Often, a separation of timescales goes hand-in-handwith a separation of solute and solvent, but such a system–bathapproach fails for ultrafast reactions.It is important to recognize that the question of Markovianity

is not an intrinsic property of a chemical process, but rather isrelated to its level of description and/or observability. Whenresorting to a full phase-space description of solute togetherwith the solvent as a limiting scenario, any system would beMarkovian as it follows deterministic Newtonian mechanics.However, because such a high-dimensional description is notpractical for complex solution phase systems, one typically triesto find lower-dimensional reaction coordinates to describe aprocess along which the dynamics is not necessarily Markovian.If one approaches a problem from molecular dynamics (MD)simulations, in which case full phase-space knowledge is inprinciple available in the background, one can try to design“good” reaction coordinates, where good refers to the questionof whether or not the dynamics is Markovian along these coor-dinates. This however is a notoriously difficult problem (4, 5).For protein folding, Markov state models do get close to thatpoint (6), but for ion pair dissociation discussed above (3), forexample, this goal has not been achieved because solvent coor-dinates would have to be included. In real-life spectroscopy, thesituation becomes even worse because one does not have fullphase-space knowledge and the reaction coordinate is dictatedby the spectroscopic method, with little or no freedom for theexperimenter to design a good reaction coordinate. We will callthe readout of the spectroscopy used in an experiment a “spec-troscopic coordinate.” There is no guarantee whatsoever thatsuch a spectroscopic coordinate is Markovian, and it is the topicof this paper to experimentally test whether or not it is.If a process is Markovian along a given reaction coordinate, then

the three-time-point joint probability function p(k, t2 + t1jj, t1ji, 0)to be in state k at time t2 + t1, provided the system was in state i

Significance

When the speed of a chemical reaction becomes as fast as therelaxation of its solvent, the Markovian assumption is likely tobreak down. In this case, the probability of a reaction to hap-pen at a given time point depends not only on the current stateof the molecular system itself, but also on the memory thatpersists within the solvent. The Markovian assumption is nev-ertheless essentially always made, because deviations fromMarkovianity are difficult to determine experimentally. Three-dimensional IR exchange spectroscopy provides a very sensi-tive test of Markovianity, which, in turn, opens importantinsights on how solvents influence or even control the courseof chemical reactions.

Author contributions: P.H. designed research; J.A.B., F.P., and P.H. performed research;J.A.B. and P.H. analyzed data; and J.A.B. and P.H. wrote the paper.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission.1To whom correspondence should be addressed. Email: phamm@pci.uzh.ch.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1406967111/-/DCSupplemental.

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at time 0 and in state j at time t1 (where i, j, k is either C or F),factorizes into two identical two-time-point joint probabilityfunctions p(j, t1ji, 0) (7):

pijkðt1; t2Þ≡ pðk; t2 + t1j j; t1ji; 0Þ= pðk; t2j j; 0Þ · pð j; t1ji; 0Þ: [3]

Whereas Eq. 3 is the general criterion for Markovianity, whenreducing the probability functions to functions of discrete binarycoordinates {i, j, k} = {C, F}, then Eq. 3 results in a two-statesystem Eq. 2 with exponential solutions, i.e., a Poissonian process.Two-dimensional exchange spectroscopy measures directly

the two-time-point joint probability function, and its realizationin the IR spectral range can do so on very fast, picosecond time-scales (8–20). In such an experiment, the frequency of a vibrationalmode, which can distinguish the two states C or F of a molecularsystem, is considered to be the spectroscopic coordinate. A spec-trum of the molecular system is measured twice, separated by awaiting time t, and plotted along two frequency axes ω1 and ω2(Fig. 1A). The first frequency measurement labels states C or F viaits vibrational frequency. If the waiting time t is set to zero, then allmolecules will still be in the same state C or F for the secondfrequency measurement and thus reveal the same vibrational fre-quency. Consequently, the 2D IR spectrum will contain only di-agonal peaks (Fig. 1A, Left). When increasing the waiting time t,some of the molecules will react between both measurements sothey will reveal different frequencies, giving rise to off-diagonal

peaks, or cross-peaks, in the 2D IR spectrum (Fig. 1A, Right).The time dependence of diagonal- and cross-peaks, in essence,reflects the two-time-point joint probability function p(j, tji, 0).Here, we extend exchange spectroscopy by one more fre-

quency dimension. Three-dimensional optical spectroscopy hasbeen realized for other applications both in the IR (21–26) andthe visible (27–32) (for reviews see refs. 33, 34). In 3D exchangespectroscopy, the vibrational frequency of a molecule is mea-sured three times along axes ω1, ω2, and ω3, separated by twowaiting times t1 and t2. In analogy to 2D IR spectroscopy, a 3DIR spectrum shows only the two diagonal peaks along the bodydiagonal at (ωC, ωC, ωC) and (ωF, ωF, ωF) when both waitingtimes t1 and t2 are set to zero (Fig. 1B, Top Left). When in-creasing these waiting times and exchange events takes place,cross-peaks show up. There are in total six possibilities for cross-peak in a 3D IR exchange spectrum, appearing on the off-diagonalcorners of a cube spanned by (ωC, ωF). If scanning t1 only (Fig.1B, Top Right), the (ωC, ωF, ωF) and (ωF, ωC, ωC) cross-peaksmay show up, whereas the (ωC, ωC, ωF) and (ωF, ωF, ωC) cross-peaks are revealed for t2 (Fig. 1B, Bottom Left). In addition, ifscanning both waiting times, double exchange cross-peaks (ωC,ωF, ωC) and (ωF, ωC, ωF) might become observable (Fig. 1B,Bottom Right). In full analogy to 2D exchange spectroscopy, thetime dependence of diagonal- and cross-peaks reflects the three-time-point joint probability function p(k, t2 + t1jj, t1ji, 0), which,in turn, can test Markovianity via Eq. 3. That very idea wasalready formulated for 3D-NMR exchange spectroscopy quitesome time ago (35), but, to the best of our knowledge, has not beenrealized yet for 3D NMR, and certainly not for 3D IR spectroscopy.For the purpose of this study, we chose the complexation (i.e.,

hydrogen bonding) of deuterated phenol to benzene in a ben-zene–CCl4 mixture as a model system. The spectroscopic co-ordinate to distinguish the two states of phenol, complexed (C)versus free (F), is the vibrational frequency of the OD-stretchvibration, because hydrogen bonding causes a frequency red shiftof ∼35 cm−1. Consequently, the stationary IR absorption spec-trum shows two discrete peaks centered at frequencies ωC andωF (Fig. 2), the integrated intensities of which represent theequilibrium constant. Two-dimensional IR exchange spectraof that molecular system have been presented by Fayer and co-workers (11, 12) with a hydrogen-bond dissociation time of ∼8 ps.Accompanying MD simulations (13) have suggested that the

mixed benzene–CCl4 solvent is not homogeneous and formsclusters of certain sizes that predominantly consist of eitherbenzene or CCl4. A phenol molecule thus can either be insideone of these clusters and be fully surrounded by benzene or CCl4,or at a boundary between clusters (see figure 12 of ref. 13). In thefirst case, the probability of an exchange event is expected tobe small, as the phenol molecule has to first diffuse out of thecorresponding cluster. In the second case, exchange is probablyfaster because the molecule just has to turn around. The spec-troscopic coordinate senses hydrogen bonding but cannot dis-tinguish whether the molecule has diffused into one of theseclusters or sits at a boundary. Hence, along that coordinate, theprobability of an exchange event at a given time point could beexpected to depend on the history of a particular phenol mole-cule. Two-dimensional IR exchange spectroscopy, however, isnot a sensitive probe of two-state behavior, and consequentlydeviations from it have not been discussed explicitly in refs. 11–13. In the present paper, we demonstrate that 3D IR spectros-copy is a sensitive probe of two-state dynamics, and we show thatthe hydrogen-bond exchange of phenol in benzene–CCl4 indeedis not a two-state process.

ResultsExperimental Results. Fig. 3, red contours, shows the measured3D IR spectra of phenol–benzene complexation for various t1and t2 population times. As in 2D IR exchange (11, 12), the

ωF

ω2

t1+t2t2

t1

t

ωFω1

ωC

ωC

ωF

ω2

ωFω1

ωC

ωC

FCFF

CC

CFA

B

ωC

ωF

ω1ω2

ω3

ωCωC

ωFωF

FFF

CCCωC

ωF

ω1ω2

ω3

ωCωC

ωFωF

FCC

CFF

ωC

ωF

ω1ω2

ω3

ωCωC

ωFωF

FFC CCFωC

ωF

ω1ω2

ω3

ωCωC

ωFωF

FCF

CFC

Fig. 1. Sketch of (A) diagonal- and cross-peaks in a 2D IR exchange exper-iment. At short waiting time t, only diagonal peaks (CC and FF) are present.Upon increasing t, cross-peaks CF and FC emerge. (B) Diagonal- and cross-peaks in a 3D IR exchange experiment. At short t1 and t2 population timesonly two diagonal peaks corresponding to unchanged species C and F arepresent (CCC and FFF). Upon increasing t1 and/or t2, cross-peaks emerge(CCF, FFC, FCC, CFF, CFC, and FCF). The dotted lines show the projections ofthe 3D IR peaks onto the ω12, ω13, and ω23 planes.

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spectroscopy of this molecular system is particularly simple be-cause the 1–2 and 2–3 transitions, which would complicate theinterpretation (8, 10), are shifted completely out of the spectralwindow due to the large anharmonicity of the OD vibrator. Wetherefore have to consider only the 0→1 transition in all of thefollowing discussion. The 3D data are shown only for the earliestpopulation times (t1 = 0.1, t2 = 0.1) ps, whereas for later pop-ulation times we display the projections onto the ω12, ω13, andω23 planes. In this way, we avoid hiding spectral features becauseof perspective effects and, as a side aspect, increase the signal-to-noise ratio somewhat due to further averaging along the differ-ent axes. At the same time, for the 23 = 8 peaks in a 3D IRexchange spectrum, these three projections with 3 × 22 = 12possible peaks contain the full information (Fig. 1B).

At short population times (t1 = 0.1, t2 = 0.1) ps, only two peaksare visible on the diagonal, one corresponding to free phenol(FFF) and the other to a phenol–benzene complex (CCC). Theasymmetry in peak size comes from the relatively large ratio ofextinction coefficients for the complexed and free phenol spe-cies, «C/«F = 2.3 (11). To compensate for its small extinctioncoefficient, we had chosen a relatively high concentration ofuncomplexed phenol so its peak is larger in the 1D absorptionspectrum (Fig. 2), but because the 3D IR signal size scales as «3,it ends up being smaller in Fig. 3.Upon increasing the population times t1 and t2, cross-peaks

emerge in the spectra, which appear as elongations of the di-agonal peaks because the two vibrators are rather close in fre-quencies. When increasing t1 only, as for the (3.0, 0.1)- and (6.0,0.1)-ps spectra, exchange can happen during the first and not thesecond population time. Therefore, a cross-peak FCC appears,corresponding to a free phenol that complexed during t1. Thecounterpart of that situation is the CFF cross-peak for a phenol–benzene complex that dissociated during t1. This cross-peak canbarely be seen in the (3.0, 0.1)-ps spectrum, because it is muchless intense than the FCC cross-peak due to the extinction co-efficient difference, but it becomes visible in the longer t1 = 6.0-pscase. When scanning t2 without t1 [see the (0.1, 3.0)- and (0.1,6.0)-ps spectra], exchange occurs during t2 only. This leads tocross-peaks CCF and FFC, the latter, again, visible essentiallyonly in the (0.1, 6.0)-ps spectrum.When both population times are scanned, as in the (3.05,

3.05)-ps spectrum, the cross-peaks already discussed for the (3.0,0.1)- and (0.1, 3.0)-ps spectra (CCF and FCC) now both appear.In principle, also the double exchange cross-peaks FCF and CFCshould show up in this spectrum, but they are too weak to bedetected with our current signal-to-noise ratio.

MD Results. To guide the interpretation of the experimentalresults, we performed MD simulations of the system, followingessentially the protocol of ref. 13 with adjusted concentrations

Absorbance(OD)

(cm-1)

C

F

freephenol

phenol/benzenecomplex

Absorbance(OD)

(cm-1)

C

F

freephenol

phenol/benzenecomplex

Absorbance(OD)

(cm-1)

Absorbance(OD)

(cm-1)

C

F

freephenol

phenol/benzenecomplex

Fig. 2. Absorption spectrum of deuterated phenol in a mixed benzene–CCl4solvent with two peaks: the OD-stretch vibration of a phenol-OD–benzenecomplex at 2,631 cm−1 (ωC) and that of free phenol-OD at 2,666 cm−1 (ωF).

(0.1, 6.0) ps

(0.1, 0.1) ps (0.1, 3.0) ps(3.0, 0.1) ps

2630

2670

2630

2670

2630

2670

2630

2670

2630

2670

2630

2670

ω1 (cm-1)

ω2 (cm-1)

ω3(cm-1)

ω1 (cm-1)

ω3(cm-1)

ω2 (cm-1)

(3.05, 3.05) ps

2630

2670

2630

2670

2630

2670

2630

2670

2630

2670

2630

2670

ω1 (cm-1)

ω2 (cm-1)

ω3(cm-1)

ω1 (cm-1)

ω3(cm-1)

ω2 (cm-1)

(6.0, 0.1) ps

2630

2670

2630

2670

2630

2670

2630

2670

2630

2670

2630

2670

ω1 (cm-1)

ω2 (cm-1)

ω3(cm-1)

ω1 (cm-1)

ω3(cm-1)

ω2 (cm-1)

FCC

CCF CFF

FCC

FCC

FCC

CFFFCC

FCC

CCF

CCFCCF

CCFCCF

FFCFFC

CCC

FFF

Fig. 3. 3D IR spectra of phenol–benzene complexation at various population times (t1, t2). The (0.1, 0.1)-ps spectrum is shown fully in 3D, the other spectraonly as projections onto the ω12, ω23, and ω13 planes. Experimental data are shown in red, fits to the data in black. Emerging cross-peaks are assigned.

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of the various molecular components to better match the ex-periment (Materials and Methods). Based on an order parameterthat determines whether or not the phenol molecule is hydrogenbonded at a given instant of time during the MD trajectory (SIText, section 1 and Fig. S1), we calculated the correlation function:

cðtÞ= hδHðtÞδHð0ÞihδH2i ; [4]

where H(t) is a binary function that is either 0 or 1 for free orcomplexed phenol, respectively. In the above equation, δH(t) ≡H(t) − ⟨H⟩ and ⟨H⟩ is the average of H(t). Fig. 4 shows thecorrelation function, which is surprisingly long-lived extend-ing beyond 100 ps with the present signal-to-noise ratio, anddecays in a highly nonexponential manner. That is, the blueline shows the best exponential fit ∝ exp(−t/τ), revealing a timeconstant of τ = 5 ps, and the red line a stretched exponential fit∝ expð−ðt=τÞβÞ with a stretching factor of β = 0.6 and a charac-teristic time τ = 4 ps. The latter reveals a significantly better fitbut is still not perfect, in particular for the long-time tail. In anycase, the extracted timescale is in essentially quantitative agree-ment with the experimental value (11, 12).Fig. 5A exemplifies some of the three-time-point joint prob-

ability functions pCCC, pFCC, and pCCF calculated from H(t). Asexpected, pCCC decreases as a function of t1 and t2, whereaspFCC and pCCF increase with t2 and t1, respectively. It can be seenthat pFCC and pCCF are symmetric with respect to interchangingt1 and t2, a symmetry which can be verified on general groundsfrom the time reversibility of an equilibrium ensemble. In-dependent from that, the time dependence of these three-time-point joint probability functions is rather complex and it is notobvious per se whether or not they follow two-state dynamics.

DiscussionTo test whether the experimental and MD results follow two-state dynamics, a simple test can be deduced from Eq. 3. That is,if the process was two-state (or Markovian), then one would ob-tain, e.g., for the ratio pFCC/pCCC

pFCCpCCC

≡pðC; t1 + t2jC; t1jF; 0ÞpðC; t1 + t2jC; t1jC; 0Þ

=pðC; t2jC; 0Þ · pðC; t1jF; 0ÞpðC; t2jC; 0Þ · pðC; t1jC; 0Þ=

pðC; t1jF; 0ÞpðC; t1jC; 0Þ;

[5]

which is a function of t1 only. As such, contour lines in a 2D plotof that ratio are expected to be vertical (Fig. 6, Left). In otherwords, the probability of staying in C during time t2 does notdepend on the history of the system during time t1. Likewise,the ratio pCCF/pCCC is a function of t2 only, and the contour linesare expected to be horizontal (Fig. 6, Right).

To illustrate that criterion, we first constructed a two-statemodel, which assumes that the 3D IR response can be piecedtogether from two 2D IR responses, and which includes theeffects of exchange as well as vibrational relaxation and orien-tational diffusion on the level of rate equations to describe thepeak intensities (see SI Text, sections II and III and Fig. S2 fordetails). For the parameters, we used the ones deduced from 2DIR spectroscopy (11): hydrogen-bond dissociation rate 1/kCF = 8ps, vibrational relaxation times for free and complexes phenolTF1 = 12:5  ps and 1=TC

1 = 10  ps, respectively, and orientationalrelaxation times τrF = 2:9  ps and τCr = 3:4  ps. Because we hadchosen a smaller benzene concentration to enlarge the freephenol peak, the ratio of complexed to free phenol was smaller([complex]/[free] = 0.6) than in ref. 11, and the complexationrate kFC = kCF• ([complex]/[free]) was adjusted accordingly.Fig. 5B shows three-time-point joint probability functions pCCC,

pFCC, and pCCF of the two-state model, which are equally com-plex as those from the MD simulations (Fig. 5A), but in detailquite different. Most of these differences originate from the factthat the two-state model does include in addition vibrationalrelaxation and rotational diffusion, which affect the sizes of the3D IR peaks as well. Despite the complex time dependencies ofthe three-time-point joint probability functions, the ratios pFCC/pCCC and pCCF/pCCC simplify significantly (Fig. 6C), and indeed,the contour lines are either vertical or horizontal, respectively,as expected from the Markovian criterion Eq. 5. (Very carefulanalysis in fact reveals a minor deviation from that behavior,which is not visible on the scale of Fig. 6C. This effect is relatedto orientational diffusion discussed in SI Text, section II.) Thecorresponding ratios pFCC/pCCC and pCCF/pCCC from the MDsimulation, on the other hand, deviate significantly from Eq. 5at early times (Fig. 6B). Fig. S3 shows that deviations from two-state dynamics persist also for much longer times, in agreementwith the slow tail in the correlation function (Fig. 4).With these results in mind, we return to the discussion of the

experimental results. To extract a quantity that is directly com-parable to the simulation results, we fitted the 3D IR spectrawith multivariate Gaussians for each diagonal- and cross-peak(black contour lines in Fig. 3) and thereby extracted their vol-umes (see SI Text, section IV for details). Because the variousspectra of Fig. 3 have been measured on different days, theiroverall signal intensity varies significantly and plotting the absolutevalues analogous to Fig. 5 turned out not to be meaningful.However, the ratios of Fig. 6 cancel out these day-to-day

0.1 1 10 1000

1

Cor

rela

tion

Time (ps)

Fig. 4. Correlation function as defined in Eq. 4 (black squares), togetherwith an exponential (blue line) and a stretched exponential fit (red line).

t1 (ps)

t 2(ps)

t 2(ps)

A

B

2-st

ate

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t1 (ps) t1 (ps)

pFCCpCCC pCCF

Fig. 5. Three-time-point joint probability functions pCCC, pFCC, and pCCF

deduced from (A) the MD simulation and (B) the two-state model which usesparameters deduced from 2D IR spectroscopy (11). Contour lines are in stepsof 5% of the peak signal of pCCC.

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variations, and furthermore allow one to test two-state behavior.Indeed, the contours are tilted very much like those from theMD simulation, and hence the hydrogen-bond exchange kineticsof phenol in benzene–CCl4 is concluded not to follow two-statedynamics. The essential information lies in the comparison of the(3.0, 3.0)-ps point with the (0.1, 3.0)-ps and (3.0, 0.1)-ps points,which is highlighted in Fig. 6A, Insets together with error bars. Inparticular, ratio pFCC/pCCC varies as a function of times t2beyond experimental error, contrary to what would be expectedfrom Eq. 5 if the process followed two-state dynamics. It isfurthermore reassuring that the MD simulations qualitativelyreproduce the experimental results, both in terms of the directionof the tilts as well as their size.The upward tilt of the contour lines of pCCF/pCCC (Fig. 6A,

Right) means that if the system already stayed in C for a longertime t1, the likelihood to switch during t2 is reduced. That in-terpretation is consistent with the cluster picture deduced fromMD simulations (13). That is, if a phenol molecule has alreadyproven to be complexed for a long time during t1, it likely isinside one of the benzene clusters, and thus the probability for itshydrogen bond to break is smaller. The tilt of the vertical con-tour lines of pFCC/pCCC toward the right (Fig. 6A, Left) can beexplained in a similar manner.

Concluding RemarksIn the case of two discrete states (Eq. 1), nonexponential kineticsis already a good indication of non-Markovianity, in the sensethat Eq. 2 should reveal an exponential response. The multipletimescales contained in the stretched exponential function neededto fit the two-time-point correlation function Eq. 4 (Fig. 4) re-flect the heterogeneous structure of the mixed benzene–CCl4solvent (13), with a probability for hydrogen-bond exchange thatdepends on where a phenol molecule is situated in that het-erogeneous solvent. Two-dimensional IR exchange spectros-copy, in essence, measures that two-time-point correlationfunction. However, in practice, in the presence of experimentalnoise as well as when other kinetic processes such as vibrationalrelaxation contribute, it is often very difficult to distinguish ex-ponential from nonexponential response; the data in refs. 11, 12would probably not allow one to do so. Hence the Markovianassumption is usually implicitly made, mostly because no criticaltest exists and not because it is necessarily very well justified. Inthe present paper, we demonstrate that 3D IR spectroscopyprovides a sensitive test of Markovianity by introducing a thirdtime point into the measurement. As such, the method cancompare the kinetics between two of these time points in de-pendence on the state of the system during the third time point.The analysis shows that the hydroxyl stretch frequency, which isthe spectroscopic coordinate in this case and tests whether ornot the hydroxyl group is hydrogen-bonded, is not a Markoviancoordinate.The goal in analyzing complex systems is to find observables that

make the description of a process Markovian (7). To be able to doso, one needs a test of how good a given spectroscopic coordinate is,and 3D exchange spectroscopy can achieve exactly that.

Materials and MethodsThree-Dimensional IR Setup. The 3D IR setup has been introduced previously(22, 24, 26), including an active phase stabilization based on a HeNetracer beam (36), polarization balanced detection (37), and scatteringsuppression achieved with two wobbling Brewster windows and a pho-toelastic modulator (38). The pulses were passed through individualmotorized translation stages and piezo actuators to independently con-trol the τ1, t1, τ2, t2, and τ3 delay times (where τi denotes the coherencetimes and ti the population times). The data were collected as a functionof τ1 and τ2 coherence times and frequency ω3 for a given set of pop-ulation times t1 and t2 and 2D Fourier transformed with respect to τ1 andτ2 to yield (ω1, ω2, ω3) spectra.

Due to the thick sample cell [100 μm, in comparison with 1–6 μm in other3D IR measurements (22, 24, 26)], the phasing scheme implemented pre-viously (39) could not be used. Alternatively, the phase has been de-termined such as to maximize the integral of the 3D IR spectrum. For thisparticular sample, this procedure is the most sensitive one, because onlypositive contributions are expected in the 2,600–2,700-cm−1 spectral win-dow from the 0–1 transitions, whereas transitions higher up in the vibra-tional ladder with in part opposite signs (22) are outside of that window. Abadly phased spectrum would result in the mixing in of dispersive con-tributions, i.e., it would cause negative bands to appear in the spectrum.Additionally, small phase errors induce quite significant shifts on the ω3

axis. Again, because only the 0→1 transitions contribute to the 3D IRspectrum, the peak positions can directly be compared with those in the1D absorption spectrum (Fig. 2), serving as a sensitive cross-check of thephasing procedure.

Sample Preparation. The system under investigation is phenol dissolved ina mixed benzene–CCl4 solvent. Phenol-OD was used to shift the hydroxylstretch vibration outside of the CH-stretch window. A molar ratio of 2:30:100(phenol-OD:benzene:CCl4) was used, revealing an absorbance of free andcomplexed phenol of 0.26 and 0.17 OD, respectively (Fig. 2). Phenol-OD wasobtained by deuterating phenol-OH in methanol-OD. The solution wassandwiched between two CaF2 windows with a 100-μm Teflon spacer.

MD Simulations. MD simulations were performed with essentially the samesimulation protocol as Cho and coworkers (13). That is, we used the OPLS-AA(Optimized Potentials for Liquid Simulations all atom) parametrization of

t1 (ps) t1 (ps)

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tt 2(ps)

t 2(ps)

pCCF /pCCCt 2(ps)

pFCC /pCCC

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C

A

t1 (ps)t2 (ps)

oitaroitar

t2 = 3 pst1 = 3 ps

Fig. 6. Ratios pCCF/pCCC and pFCC/pCCC for (A) experimental 3D IR data, (B) MDsimulation, and (C) two-state model. Contour lines are in steps of 0.05 and arethe same for all three plots. The dots in A (Insets) mark the positions where3D IR spectra have been taken. Ratios pFCC/pCCC and pCCF/pCCC along t2 and t1,respectively, with the corresponding other time coordinate fixed to 3.0 ps.

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CCl4 (40) and the same partial charges and Lennard-Jones parameters as ref.13 for benzene and phenol. One phenol molecule deuterated only at thehydroxyl group was dissolved in a simulation box containing 130 benzenemolecules and 498 CCl4 molecules (the benzene–CCl4 ratio was chosen smallerthan in ref. 13 to better account for the experimental conditions). MD simu-lations were performed with the Gromacs program package (41) with a time-step of 2 fs, all bonds constrained, and a time constant of 0.5 ps for thethermostat. Lennard-Jones interactions were treated with a cutoff of 1.0 nm(switching at 0.9 nm) and the long-range electrostatic forces were treated bythe particle-mesh Ewald approximation. After an initial energy minimization

and equilibration at 1 bar, production runs with a total length of 1.7 μs werecollected in the canonical (NVT) ensemble.

ACKNOWLEDGMENTS. We thank the anonymous reviewer for pointingout a critical error in the data fitting. We furthermore thank GerhardStock for insightful discussions, Minhaeng Cho and Kijeong Kwacfor making some of the force-field details of ref. 13 available to us,and Sean Garrett-Roe for help with the experimental setup in an earlystage of the project. The work has been supported by the Swiss NationalScience Foundation through the National Center of Competence andResearch (NCCR) Molecular Ultrafast Spectroscopy and Technology (MUST).

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