# Testing for memory-free spectroscopic coordinates by 3D IR exchange spectroscopy

See allHide authors and affiliations

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

## Significance

When the speed of a chemical reaction becomes as fast as the relaxation of its solvent, the Markovian assumption is likely to break down. In this case, the probability of a reaction to happen at a given time point depends not only on the current state of the molecular system itself, but also on the memory that persists within the solvent. The Markovian assumption is nevertheless essentially always made, because deviations from Markovianity are difficult to determine experimentally. Three-dimensional IR exchange spectroscopy provides a very sensitive test of Markovianity, which, in turn, opens important insights on how solvents influence or even control the course of chemical reactions.

## Abstract

Using 3D infrared (IR) exchange spectroscopy, the ultrafast hydrogen-bond forming and breaking (i.e., complexation) kinetics of phenol to benzene in a benzene/CCl_{4} mixture is investigated. By introducing a third time point at which the hydrogen-bonding state of phenol is measured (in comparison with 2D IR exchange spectroscopy), the spectroscopic method can serve as a critical test of whether the spectroscopic coordinate used to observe the exchange process is a memory-free, or Markovian, coordinate. For the system under investigation, the 3D IR results suggest that this is not the case. This conclusion is reconfirmed by accompanying molecular dynamics simulations, which furthermore reveal that the non-Markovian kinetics is caused by the heterogeneous structure of the mixed solvent.

Whenever one writes a chemical reaction equation, such as in the simplest case of an equilibrium between two states of a molecular system*C* to *F*, scales linearly with concentration [*C*] with proportionality constant *k*_{CF}, but does not depend on the history of how *C* has been formed. When the reaction is slow in comparison with the relaxation of its environment, then this assumption is well justified. One can however think of many situations where this is not the case. For example, proteins are heterogeneous on very long timescales due to their structural complexity, so enzymatic reaction catalyzed by them might not fulfill that condition (1). Other examples are found in glass-forming liquids close to the glass transition, when their relaxation covers many orders of magnitude in time (2).

When the speed of a reaction approaches the ultrafast picosecond regime, which is the topic of the present paper, then even the solvation time of normal liquids far away from any glass transition may become time-limiting. For example, it has been shown that the distance between a Na^{+} and a Cl^{−} ion is not a good reaction coordinate to describe ion pair dissociation in aqueous solution (3), in the sense that the distance for which the free energy peaks does not correspond with the transition state of the reaction. Hidden coordinates exist, which must be solvent coordinates because there are no other degrees of freedom. Unless these solvent coordinates have not fully relaxed and lost their memory about a reaction event, the probability for a back reaction must depend on the history. Markovianity is related to a separation of timescales between the speed of a chemical reaction of a molecular system versus that of the relaxation of its solvent. Often, a separation of timescales goes hand-in-hand with a separation of solute and solvent, but such a system–bath approach 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 is related to its level of description and/or observability. When resorting to a full phase-space description of solute together with the solvent as a limiting scenario, any system would be Markovian as it follows deterministic Newtonian mechanics. However, because such a high-dimensional description is not practical for complex solution phase systems, one typically tries to find lower-dimensional reaction coordinates to describe a process 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 in principle available in the background, one can try to design “good” reaction coordinates, where good refers to the question of whether or not the dynamics is Markovian along these coordinates. This however is a notoriously difficult problem (4, 5). For protein folding, Markov state models do get close to that point (6), but for ion pair dissociation discussed above (3), for example, this goal has not been achieved because solvent coordinates would have to be included. In real-life spectroscopy, the situation becomes even worse because one does not have full phase-space knowledge and the reaction coordinate is dictated by the spectroscopic method, with little or no freedom for the experimenter to design a good reaction coordinate. We will call the readout of the spectroscopy used in an experiment a “spectroscopic coordinate.” There is no guarantee whatsoever that such a spectroscopic coordinate is Markovian, and it is the topic of 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*, *t*_{2} + *t*_{1}|*j*, *t*_{1}|*i*, 0) to be in state *k* at time *t*_{2} + *t*_{1}, provided the system was in state *i* at time 0 and in state *j* at time *t*_{1} (where *i*, *j*, *k* is either *C* or *F*), factorizes into two identical two-time-point joint probability functions *p*(*j*, *t*_{1}|*i*, 0) (7):**3** is the general criterion for Markovianity, when reducing the probability functions to functions of discrete binary coordinates {*i*, *j*, *k*} = {*C*, *F*}, then Eq. **3** results in a two-state system 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 realization in the IR spectral range can do so on very fast, picosecond timescales (8⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓–20). In such an experiment, the frequency of a vibrational mode, which can distinguish the two states *C* or *F* of a molecular system, is considered to be the spectroscopic coordinate. A spectrum of the molecular system is measured twice, separated by a waiting time *t*, and plotted along two frequency axes *ω*_{1} and *ω*_{2} (Fig. 1*A*). The first frequency measurement labels states *C* or *F* via its vibrational frequency. If the waiting time *t* is set to zero, then all molecules will still be in the same state *C* or *F* for the second frequency measurement and thus reveal the same vibrational frequency. Consequently, the 2D IR spectrum will contain only diagonal peaks (Fig. 1*A*, *Left*). When increasing the waiting time *t*, some of the molecules will react between both measurements so they will reveal different frequencies, giving rise to off-diagonal peaks, or cross-peaks, in the 2D IR spectrum (Fig. 1*A*, *Right*). The time dependence of diagonal- and cross-peaks, in essence, reflects the two-time-point joint probability function *p*(*j*, *t*|*i*, 0).

Here, we extend exchange spectroscopy by one more frequency dimension. Three-dimensional optical spectroscopy has been realized for other applications both in the IR (21⇓⇓⇓⇓–26) and the visible (27⇓⇓⇓⇓–32) (for reviews see refs. 33, 34). In 3D exchange spectroscopy, the vibrational frequency of a molecule is measured three times along axes *ω*_{1}, *ω*_{2}, and *ω*_{3}, separated by two waiting times *t*_{1} and *t*_{2}. In analogy to 2D IR spectroscopy, a 3D IR spectrum shows only the two diagonal peaks along the body diagonal at (*ω*_{C}, *ω*_{C}, *ω*_{C}) and (*ω*_{F}, *ω*_{F}, *ω*_{F}) when both waiting times *t*_{1} and *t*_{2} are set to zero (Fig. 1*B*, *Top Left*). When increasing 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-diagonal corners of a cube spanned by (*ω*_{C}, *ω*_{F}). If scanning *t*_{1} only (Fig. 1*B*, *Top Right*), the (*ω*_{C}, *ω*_{F}, *ω*_{F}) and (*ω*_{F}, *ω*_{C}, *ω*_{C}) cross-peaks may show up, whereas the (*ω*_{C}, *ω*_{C}, *ω*_{F}) and (*ω*_{F}, *ω*_{F}, *ω*_{C}) cross-peaks are revealed for *B*, *Bottom Left*). In addition, if scanning both waiting times, double exchange cross-peaks (*ω*_{C}, *ω*_{F}, *ω*_{C}) and (*ω*_{F}, *ω*_{C}, *ω*_{F}) might become observable (Fig. 1*B*, *Bottom Right*). In full analogy to 2D exchange spectroscopy, the time dependence of diagonal- and cross-peaks reflects the three-time-point joint probability function *p*(*k*, *t*_{2} + *t*_{1}|*j*, *t*_{1}|*i*, 0), which, in turn, can test Markovianity via Eq. **3**. That very idea was already formulated for 3D-NMR exchange spectroscopy quite some time ago (35), but, to the best of our knowledge, has not been realized 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 benzene–CCl_{4} mixture as a model system. The spectroscopic coordinate to distinguish the two states of phenol, complexed (*C*) versus free (*F*), is the vibrational frequency of the OD-stretch vibration, because hydrogen bonding causes a frequency red shift of ∼35 cm^{−1}. Consequently, the stationary IR absorption spectrum shows two discrete peaks centered at frequencies *ω*_{C} and *ω*_{F} (Fig. 2), the integrated intensities of which represent the equilibrium constant. Two-dimensional IR exchange spectra of that molecular system have been presented by Fayer and coworkers (11, 12) with a hydrogen-bond dissociation time of ∼8 ps.

Accompanying MD simulations (13) have suggested that the mixed benzene–CCl_{4} solvent is not homogeneous and forms clusters of certain sizes that predominantly consist of either benzene or CCl_{4}. A phenol molecule thus can either be inside one of these clusters and be fully surrounded by benzene or CCl_{4}, or at a boundary between clusters (see figure 12 of ref. 13). In the first case, the probability of an exchange event is expected to be small, as the phenol molecule has to first diffuse out of the corresponding cluster. In the second case, exchange is probably faster because the molecule just has to turn around. The spectroscopic coordinate senses hydrogen bonding but cannot distinguish whether the molecule has diffused into one of these clusters or sits at a boundary. Hence, along that coordinate, the probability of an exchange event at a given time point could be expected to depend on the history of a particular phenol molecule. Two-dimensional IR exchange spectroscopy, however, is not a sensitive probe of two-state behavior, and consequently deviations from it have not been discussed explicitly in refs. 11⇓–13. In the present paper, we demonstrate that 3D IR spectroscopy is a sensitive probe of two-state dynamics, and we show that the hydrogen-bond exchange of phenol in benzene–CCl_{4} indeed is not a two-state process.

## Results

### Experimental Results.

Fig. 3, red contours, shows the measured 3D IR spectra of phenol–benzene complexation for various *t*_{1} and *t*_{2} population times. As in 2D IR exchange (11, 12), the spectroscopy of this molecular system is particularly simple because the 1–2 and 2–3 transitions, which would complicate the interpretation (8, 10), are shifted completely out of the spectral window due to the large anharmonicity of the OD vibrator. We therefore have to consider only the 0→1 transition in all of the following discussion. The 3D data are shown only for the earliest population times (*t*_{1} = 0.1, *t*_{2} = 0.1) ps, whereas for later population times we display the projections onto the *ω*_{12}, *ω*_{13}, and *ω*_{23} planes. In this way, we avoid hiding spectral features because of perspective effects and, as a side aspect, increase the signal-to-noise ratio somewhat due to further averaging along the different axes. At the same time, for the 2^{3} = 8 peaks in a 3D IR exchange spectrum, these three projections with 3 × 2^{2} = 12 possible peaks contain the full information (Fig. 1*B*).

At short population times (*t*_{1} = 0.1, *t*_{2} = 0.1) ps, only two peaks are visible on the diagonal, one corresponding to free phenol (*FFF*) and the other to a phenol–benzene complex (*CCC*). The asymmetry in peak size comes from the relatively large ratio of extinction coefficients for the complexed and free phenol species, *ε*_{C}/*ε*_{F} = 2.3 (11). To compensate for its small extinction coefficient, we had chosen a relatively high concentration of uncomplexed phenol so its peak is larger in the 1D absorption spectrum (Fig. 2), but because the 3D IR signal size scales as *ε*^{3}, it ends up being smaller in Fig. 3.

Upon increasing the population times *t*_{1} and *t*_{2}, cross-peaks emerge in the spectra, which appear as elongations of the diagonal peaks because the two vibrators are rather close in frequencies. When increasing *t*_{1} only, as for the (3.0, 0.1)- and (6.0, 0.1)-ps spectra, exchange can happen during the first and not the second population time. Therefore, a cross-peak *FCC* appears, corresponding to a free phenol that complexed during *t*_{1}. The counterpart of that situation is the *CFF* cross-peak for a phenol–benzene complex that dissociated during *t*_{1}. This cross-peak can barely be seen in the (3.0, 0.1)-ps spectrum, because it is much less intense than the *FCC* cross-peak due to the extinction coefficient difference, but it becomes visible in the longer *t*_{1} = 6.0-ps case. When scanning *t*_{2} without *t*_{1} [see the (0.1, 3.0)- and (0.1, 6.0)-ps spectra], exchange occurs during *t*_{2} only. This leads to cross-peaks *CCF* and *FFC*, the latter, again, visible essentially only 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 *CFC* should show up in this spectrum, but they are too weak to be detected with our current signal-to-noise ratio.

### MD Results.

To guide the interpretation of the experimental results, we performed MD simulations of the system, following essentially the protocol of ref. 13 with adjusted concentrations of the various molecular components to better match the experiment (*Materials and Methods*). Based on an order parameter that determines whether or not the phenol molecule is hydrogen bonded at a given instant of time during the MD trajectory (*SI Text*, section 1 and Fig. S1), we calculated the correlation function:*H*(*t*) is a binary function that is either 0 or 1 for free or complexed phenol, respectively. In the above equation, *δH*(*t*) ≡ *H*(*t*) − 〈*H*〉 and 〈*H*〉 is the average of *H*(*t*). Fig. 4 shows the correlation function, which is surprisingly long-lived extending beyond 100 ps with the present signal-to-noise ratio, and decays in a highly nonexponential manner. That is, the blue line shows the best exponential fit ∝ exp(−*t*/*τ*), revealing a time constant of *τ* = 5 ps, and the red line a stretched exponential fit *β* = 0.6 and a characteristic time *τ* = 4 ps. The latter reveals a significantly better fit but is still not perfect, in particular for the long-time tail. In any case, the extracted timescale is in essentially quantitative agreement with the experimental value (11, 12).

Fig. 5*A* exemplifies some of the three-time-point joint probability functions *p*_{CCC}, *p*_{FCC}, and *p*_{CCF} calculated from *H*(*t*). As expected, *p*_{CCC} decreases as a function of *t*_{1} and *t*_{2}, whereas *p*_{FCC} and *p*_{CCF} increase with *t*_{2} and *t*_{1}, respectively. It can be seen that *p*_{FCC} and *p*_{CCF} are symmetric with respect to interchanging *t*_{1} and *t*_{2}, a symmetry which can be verified on general grounds from the time reversibility of an equilibrium ensemble. Independent from that, the time dependence of these three-time-point joint probability functions is rather complex and it is not obvious per se whether or not they follow two-state dynamics.

## Discussion

To 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 obtain, e.g., for the ratio *p*_{FCC}/*p*_{CCC}*t*_{1} only. As such, contour lines in a 2D plot of that ratio are expected to be vertical (Fig. 6, *Left*). In other words, the probability of staying in *C* during time *t*_{2} does not depend on the history of the system during time *t*_{1}. Likewise, the ratio *p*_{CCF}/*p*_{CCC} is a function of *t*_{2} only, and the contour lines are expected to be horizontal (Fig. 6, *Right*).

To illustrate that criterion, we first constructed a two-state model, which assumes that the 3D IR response can be pieced together from two 2D IR responses, and which includes the effects of exchange as well as vibrational relaxation and orientational diffusion on the level of rate equations to describe the peak intensities (see *SI Text*, sections II and III and Fig. S2 for details). For the parameters, we used the ones deduced from 2D IR spectroscopy (11): hydrogen-bond dissociation rate 1/*k*_{CF} = 8 ps, vibrational relaxation times for free and complexes phenol *k*_{FC} = *k*_{CF}^{•} ([complex]/[free]) was adjusted accordingly.

Fig. 5*B* shows three-time-point joint probability functions *p*_{CCC}, *p*_{FCC}, and *p*_{CCF} of the two-state model, which are equally complex as those from the MD simulations (Fig. 5*A*), but in detail quite different. Most of these differences originate from the fact that the two-state model does include in addition vibrational relaxation and rotational diffusion, which affect the sizes of the 3D IR peaks as well. Despite the complex time dependencies of the three-time-point joint probability functions, the ratios *p*_{FCC}/*p*_{CCC} and *p*_{CCF}/*p*_{CCC} simplify significantly (Fig. 6*C*), and indeed, the contour lines are either vertical or horizontal, respectively, as expected from the Markovian criterion Eq. **5**. (Very careful analysis in fact reveals a minor deviation from that behavior, which is not visible on the scale of Fig. 6*C*. This effect is related to orientational diffusion discussed in *SI Text*, section II.) The corresponding ratios *p*_{FCC}/*p*_{CCC} and *p*_{CCF}/*p*_{CCC} from the MD simulation, on the other hand, deviate significantly from Eq. **5** at early times (Fig. 6*B*). Fig. S3 shows that deviations from two-state dynamics persist also for much longer times, in agreement with 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 comparable to the simulation results, we fitted the 3D IR spectra with multivariate Gaussians for each diagonal- and cross-peak (black contour lines in Fig. 3) and thereby extracted their volumes (see *SI Text*, section IV for details). Because the various spectra of Fig. 3 have been measured on different days, their overall signal intensity varies significantly and plotting the absolute values analogous to Fig. 5 turned out not to be meaningful. However, the ratios of Fig. 6 cancel out these day-to-day variations, and furthermore allow one to test two-state behavior. Indeed, the contours are tilted very much like those from the MD simulation, and hence the hydrogen-bond exchange kinetics of phenol in benzene–CCl_{4} is concluded not to follow two-state dynamics. 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. 6*A*, *Insets* together with error bars. In particular, ratio *p*_{FCC}/*p*_{CCC} varies as a function of times *t*_{2} beyond experimental error, contrary to what would be expected from Eq. **5** if the process followed two-state dynamics. It is furthermore reassuring that the MD simulations qualitatively reproduce the experimental results, both in terms of the direction of the tilts as well as their size.

The upward tilt of the contour lines of *p*_{CCF}/*p*_{CCC} (Fig. 6*A*, *Right*) means that if the system already stayed in *C* for a longer time *t*_{1}, the likelihood to switch during *t*_{2} is reduced. That interpretation is consistent with the cluster picture deduced from MD simulations (13). That is, if a phenol molecule has already proven to be complexed for a long time during *t*_{1}, it likely is inside one of the benzene clusters, and thus the probability for its hydrogen bond to break is smaller. The tilt of the vertical contour lines of *p*_{FCC}/*p*_{CCC} toward the right (Fig. 6*A*, *Left*) can be explained in a similar manner.

## Concluding Remarks

In the case of two discrete states (Eq. **1**), nonexponential kinetics is already a good indication of non-Markovianity, in the sense that Eq. **2** should reveal an exponential response. The multiple timescales contained in the stretched exponential function needed to fit the two-time-point correlation function Eq. **4** (Fig. 4) reflect the heterogeneous structure of the mixed benzene–CCl_{4} solvent (13), with a probability for hydrogen-bond exchange that depends on where a phenol molecule is situated in that heterogeneous solvent. Two-dimensional IR exchange spectroscopy, in essence, measures that two-time-point correlation function. However, in practice, in the presence of experimental noise as well as when other kinetic processes such as vibrational relaxation contribute, it is often very difficult to distinguish exponential from nonexponential response; the data in refs. 11, 12 would probably not allow one to do so. Hence the Markovian assumption is usually implicitly made, mostly because no critical test exists and not because it is necessarily very well justified. In the present paper, we demonstrate that 3D IR spectroscopy provides a sensitive test of Markovianity by introducing a third time point into the measurement. As such, the method can compare the kinetics between two of these time points in dependence on the state of the system during the third time point. The analysis shows that the hydroxyl stretch frequency, which is the spectroscopic coordinate in this case and tests whether or not the hydroxyl group is hydrogen-bonded, is not a Markovian coordinate.

The goal in analyzing complex systems is to find observables that make the description of a process Markovian (7). To be able to do so, one needs a test of how good a given spectroscopic coordinate is, and 3D exchange spectroscopy can achieve exactly that.

## Materials and Methods

### Three-Dimensional IR Setup.

The 3D IR setup has been introduced previously (22, 24, 26), including an active phase stabilization based on a HeNe tracer beam (36), polarization balanced detection (37), and scattering suppression achieved with two wobbling Brewster windows and a photoelastic modulator (38). The pulses were passed through individual motorized translation stages and piezo actuators to independently control the *τ*_{1}, *t*_{1}, *τ*_{2}, *t*_{2}, and *τ*_{3} delay times (where *τ*_{i} denotes the coherence times and *t*_{i} the population times). The data were collected as a function of *τ*_{1} and *τ*_{2} coherence times and frequency *ω*_{3} for a given set of population times *t*_{1} and *t*_{2} and 2D Fourier transformed with respect to *τ*_{1} and *τ*_{2} to yield (*ω*_{1}, *ω*_{2}, *ω*_{3}) spectra.

Due to the thick sample cell [100 μm, in comparison with 1–6 μm in other 3D IR measurements (22, 24, 26)], the phasing scheme implemented previously (39) could not be used. Alternatively, the phase has been determined such as to maximize the integral of the 3D IR spectrum. For this particular sample, this procedure is the most sensitive one, because only positive contributions are expected in the 2,600–2,700-cm^{−1} spectral window from the 0–1 transitions, whereas transitions higher up in the vibrational ladder with in part opposite signs (22) are outside of that window. A badly phased spectrum would result in the mixing in of dispersive contributions, i.e., it would cause negative bands to appear in the spectrum. Additionally, small phase errors induce quite significant shifts on the *ω*_{3} axis. Again, because only the 0→1 transitions contribute to the 3D IR spectrum, the peak positions can directly be compared with those in the 1D absorption spectrum (Fig. 2), serving as a sensitive cross-check of the phasing procedure.

### Sample Preparation.

The system under investigation is phenol dissolved in a mixed benzene–CCl_{4} solvent. Phenol-OD was used to shift the hydroxyl stretch vibration outside of the CH-stretch window. A molar ratio of 2:30:100 (phenol-OD:benzene:CCl_{4}) was used, revealing an absorbance of free and complexed phenol of 0.26 and 0.17 OD, respectively (Fig. 2). Phenol-OD was obtained by deuterating phenol-OH in methanol-OD. The solution was sandwiched between two CaF_{2} windows with a 100-μm Teflon spacer.

### MD Simulations.

MD simulations were performed with essentially the same simulation protocol as Cho and coworkers (13). That is, we used the OPLS-AA (Optimized Potentials for Liquid Simulations all atom) parametrization of CCl_{4} (40) and the same partial charges and Lennard-Jones parameters as ref. 13 for benzene and phenol. One phenol molecule deuterated only at the hydroxyl group was dissolved in a simulation box containing 130 benzene molecules and 498 CCl_{4} molecules (the benzene–CCl_{4} ratio was chosen smaller than in ref. 13 to better account for the experimental conditions). MD simulations were performed with the Gromacs program package (41) with a timestep of 2 fs, all bonds constrained, and a time constant of 0.5 ps for the thermostat. 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 by the particle-mesh Ewald approximation. After an initial energy minimization and equilibration at 1 bar, production runs with a total length of 1.7 μs were collected in the canonical (NVT) ensemble.

## Acknowledgments

We thank the anonymous reviewer for pointing out a critical error in the data fitting. We furthermore thank Gerhard Stock for insightful discussions, Minhaeng Cho and Kijeong Kwac for 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 early stage of the project. The work has been supported by the Swiss National Science Foundation through the National Center of Competence and Research (NCCR) Molecular Ultrafast Spectroscopy and Technology (MUST).

## Footnotes

- ↵
^{1}To whom correspondence should be addressed. Email: phamm{at}pci.uzh.ch.

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.

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

## References

- ↵
- ↵
- ↵
- Geissler PL,
- Dellago C,
- Chandler D

- ↵
- Best RB,
- Hummer G

- ↵
- Krivov SV,
- Karplus M

- ↵
- ↵
- Van Kampen NG

- ↵
- ↵
- ↵
- Kim YS,
- Hochstrasser RM

- ↵
- Zheng J,
- et al.

- ↵
- ↵
- ↵
- Zheng J,
- Kwak K,
- Xie J,
- Fayer MD

- ↵
- ↵
- Cahoon JF,
- Sawyer KR,
- Schlegel JP,
- Harris CB

- ↵
- ↵
- ↵
- Ji M,
- Odelius M,
- Gaffney KJ

- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- Schmidt-Rohr K,
- Spiess HW

- ↵
- ↵
- ↵
- ↵
- ↵
- ↵

## Citation Manager Formats

## Article Classifications

- Physical Sciences
- Chemistry