Carbon dioxide in an ionic liquid: Structural and rotational dynamics · 2016-03-23 · Ionic liquids (ILs), which have widely tunable structural motifs and intermolecular interactions

THE JOURNAL OF CHEMICAL PHYSICS 144, 104506 (2016)

Carbon dioxide in an ionic liquid: Structural and rotational dynamicsChiara H. Giammanco,a) Patrick L. Kramer,a) Steven A. Yamada, Jun Nishida, Amr Tamimi,and Michael D. Fayerb)

Department of Chemistry, Stanford University, Stanford, California 94305, USA

(Received 16 January 2016; accepted 16 February 2016; published online 11 March 2016)

Ionic liquids (ILs), which have widely tunable structural motifs and intermolecular interactionswith solutes, have been proposed as possible carbon capture media. To inform the choice of anoptimal ionic liquid system, it can be useful to understand the details of dynamics and interactionson fundamental time scales (femtoseconds to picoseconds) of dissolved gases, particularly carbondioxide (CO2), within the complex solvation structures present in these uniquely organized materials.The rotational and local structural fluctuation dynamics of CO2 in the room temperature ionic liquid1-ethyl-3-methylimidazolium bis(trifluoromethylsulfonyl)imide (EmimNTf2) were investigated byusing ultrafast infrared spectroscopy to interrogate the CO2 asymmetric stretch. Polarization-selectivepump probe measurements yielded the orientational correlation function of the CO2 vibrationaltransition dipole. It was found that reorientation of the carbon dioxide occurs on 3 time scales:0.91 ± 0.03, 8.3 ± 0.1, 54 ± 1 ps. The initial two are attributed to restricted wobbling motionsoriginating from a gating of CO2 motions by the IL cations and anions. The final (slowest) decaycorresponds to complete orientational randomization. Two-dimensional infrared vibrational echo(2D IR) spectroscopy provided information on structural rearrangements, which cause spectraldiffusion, through the time dependence of the 2D line shape. Analysis of the time-dependent 2DIR spectra yields the frequency-frequency correlation function (FFCF). Polarization-selective 2D IRexperiments conducted on the CO2 asymmetric stretch in the parallel- and perpendicular-pumpedgeometries yield significantly different FFCFs due to a phenomenon known as reorientation-inducedspectral diffusion (RISD), revealing strong vector interactions with the liquid structures that evolveslowly on the (independently measured) rotation time scales. To separate the RISD contribution tothe FFCF from the structural spectral diffusion contribution, the previously developed first orderStark effect RISD model is reformulated to describe the second order (quadratic) Stark effect—thefirst order Stark effect vanishes because CO2 does not have a permanent dipole moment. Throughthis analysis, we characterize the structural fluctuations of CO2 in the ionic liquid solvation envi-ronment, which separate into magnitude-only and combined magnitude and directional correlationsof the liquid’s time dependent electric field. This new methodology will enable highly incisivecomparisons between CO2 dynamics in a variety of ionic liquid systems. C 2016 AIP PublishingLLC. [http://dx.doi.org/10.1063/1.4943390]

I. INTRODUCTION

Levels of carbon dioxide in the atmosphere are quicklyrising and interest is growing in technologies for mitigation,either through a reduction in production or through capture.1

Forays into removing CO2 directly from the atmosphere withcurrent technology have been disappointing and fall woefullyshort of sufficient capacity. Thus, the solution that remains isto capture the gas before it enters the atmosphere.2 There havebeen many systems proposed for the capture of carbon dioxide,including various solvents, amines, polymers, metal organicframeworks, and ionic liquids (ILs).3,4 Current DOE costand efficiency targets for the removal of the carbon dioxideproduced from power plant emissions5 cannot be achievedwith the leading amine-based capture technology.6 The focusthen must shift to more efficient removal and the developmentof materials and conditions to optimize the process.

a)C. H. Giammanco and P. L. Kramer contributed equally to this work.b)Author to whom correspondence should be addressed. Electronic mail:

[email protected]

For post-combustion capture of CO2 to be efficient, thecapturing medium must be reused many times. If CO2 isbound irreversibly, an impossibly large amount of materialwould be required to capture the 2.2 × 109 metric tons ofcarbon dioxide produced in 2013 in the US alone.7 Thus,the capture material must be able to take out carbon dioxidefrom the waste gas stream selectively and in high yield,release the gas later for collection, and then be recycled.Typically, a pressure or temperature swing is employed torelease the absorbed gas. Ideally, it would take as little energyas possible to recover the CO2 and the capture mediumwould continue working and not degrade or evaporate overmany cycles. Many molecular solvents employed to datecan remove carbon dioxide effectively, but they require highpressures inaccessible in post-combustion waste gas streamsand can suffer from evaporative losses.

Room temperature ionic liquids (RTILs) are salts thatremain molten at or below room temperature and offermany advantages for carbon capture. As they consist ofdiscrete cations and anions, RTILs often have incredibly

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104506-2 Giammanco et al. J. Chem. Phys. 144, 104506 (2016)

low vapor pressures and so do not evaporate readily.Their chemical properties are also highly tunable, withconstituent ions that are readily functionalized, and hencecan be made task-specific.8–11 Even non-optimized, atatmospheric pressures, several ionic liquids, particularly thosebased on the imidazolium cation, show high solubility andselectivity for CO2.4,12–14 Some forays have been madetowards functionalizing ILs for carbon capture through theaddition of amine groups that can chemically bond thecarbon dioxide.15,16 However, the thermodynamic propertiesof these systems have not been thoroughly explored oroptimized.17,18 To search for an optimal solvent, it is importantto understand what is contributing to the carbon dioxidesolvation structures and dynamics so that those propertiescan be tuned and exploited. 1-ethyl-3-methylimidazoliumbis(trifluoromethylsulfonyl)imide (EmimNTf2, see Figure 1)is a particularly well studied aprotic ionic liquid, which hasa high CO2 solubility,12,19 low viscosity, and good thermaland electrochemical stability.20,21 Thus, it is a good first testcase for exploring the dynamics of carbon dioxide in an ionicliquid system.

Previous studies have investigated the amount of carbondioxide various materials can hold as well as the rate of CO2translational diffusion through these materials.13,22–30 Oneother study has looked at the dynamics of the CO2 in ionicliquids using two-dimensional infrared (2D IR) measurementsof CO2’s spectral diffusion to infer structural dynamics.31 Thispaper builds upon the previous study by examining the spectraldiffusion dynamics in more detail, and additionally, exploresCO2 rotational dynamics for the first time.

Extensions of the 2D IR experiments to multiplepolarization configurations reveal that carbon dioxide in ionicliquids undergoes reorientation-induced spectral diffusion(RISD), a phenomenon that has recently been encountered forsmall molecule probes in different ionic liquid systems.32,33

FIG. 1. Structure of the ionic liquid used: 1-ethyl-3-methylimidazoliumbis(trifluoromethylsulfonyl)imide. Abbreviated to EmimNTf2 throughout thepaper.

RISD is the result of a vibrational probe experiencingstrongly directional interactions and having fast rotationaldegrees of freedom, while a component of the surroundingstructure evolves much more slowly. Reorientation of theprobe samples this directional interaction more rapidly thanthe overall structural evolution of the medium does. The2D IR measurement of the frequency-frequency correlationfunction (FFCF), which quantifies spectral diffusion ratesand fluctuation amplitudes, is directly impacted by RISD.Different polarization configurations of the excitation pulsesand emitted signal result in distinct FFCF decays because eachmeasurement configuration samples a differently weightedensemble of rotating molecules. Analysis of the total spectraldiffusion must take this RISD contribution into account toobtain the structural spectral diffusion (SSD). Measurementand a complete analysis of the reorientation dynamics aretherefore necessary for properly assigning the true structuralevolution time scales and fluctuation amplitudes present in theFFCF decays.

The organization of the paper is as follows. Theexperimental procedures for sample preparation and anoverview of the linear and nonlinear time-resolved infraredspectroscopic measurements are given in Section II. Section IIIcontains the major results of this investigation. The linearabsorption spectrum of CO2 in the IL is presented anddiscussed in Section III A. The dynamics underlyingthis IR absorption will be extracted via the time-resolvedIR techniques. Carbon dioxide’s reorientation dynamics,obtained from the polarization-selective pump-probe (PP)anisotropy measurements, are analyzed in Section III B.The non-exponential decays of the orientational correlationfunction are assigned to a single population of solvatedCO2 molecules that undergo several periods of restrictedangular diffusion (wobbling-in-a-cone)32,34–39 prior to afreely diffusive randomization of orientational coordinates.Polarization-selective 2D IR spectral analysis, which yieldsthe spectral diffusion dynamics of CO2 in the ionic liquid, ispresented in Section III C. Frequency evolution dynamics ona range of time scales, from sub-picosecond to hundreds ofpicoseconds, are evident. The FFCFs from 2D IR spectrain the parallel and perpendicular polarization geometriesdisplay clearly distinct spectral diffusion dynamics, a clearsign of RISD. For more straightforward comparison to othertechniques, such as molecular dynamics (MD) simulations, weconstruct the isotropically averaged two-dimensional spectraand compute the FFCF, which still contains contributionsfrom RISD. In Section IV, RISD theory is briefly reviewedwith the Stark effect model for vibrational frequency couplingto an internal electric field. While the Stark model waspreviously developed to first order in the electric field,this first-order term vanishes for the highly symmetric CO2molecule, which does not have a permanent dipole moment.Thus, the equations for polarized FFCF observables arederived with the second order (quadratic) Stark effect. Armednow with an appropriate model for the SSD and RISD, theexperimental FFCF results are decomposed in Section V toreveal their structural contributions, with the orientationalcontribution fixed by the measured pump-probe anisotropydata. Concluding remarks, with an eye towards applying


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these theoretical developments and results to further exploreCO2 dynamics in various ionic liquid (or other complexmaterial) environments, are given in Section VI. Details ofthe calculations for RISD theory with the second order Starkeffect model, including the particular assumptions under whichthe results in Section IV are applicable, are provided in theAppendix.

II. EXPERIMENTAL PROCEDURES

A. Sample preparation

The RTIL 1-ethyl-3-methylimidazolium bis(trifluoromethylsulfonyl)imide (EmimNTf2) was purchasedfrom IoLiTec Ionic Liquids Technologies Inc., dried for aweek under vacuum and light heating to about 60 C, andstored in a nitrogen glove box. The water content was below10 ppm, as determined by coulometric Karl Fischer titration.13CO2 (99% purity) was purchased from Icon Isotopes andused without further purification. The isotopically labeledcarbon dioxide was gently bubbled through the IL in a vialwith care taken to exclude atmospheric moisture. The ionicliquid with dissolved CO2 gas was sandwiched between 3 mmCaF2 windows, separated by a Teflon spacer. The samplecells were assembled in a dry atmosphere to prevent wateruptake by the ionic liquid. Linear infrared spectra were takenwith a Thermo Scientific Nicolet 6700 FT-IR spectrometer.Background spectra of the ionic liquid were taken underidentical conditions and subtracted from the carbon dioxidein ionic liquid spectra to give the contribution from the13CO2 alone. The carbon dioxide concentration was keptsufficiently low that vibrational excitation transfer betweenadjacent molecules was statistically unlikely to occur. Usingthe extinction coefficient of carbon dioxide in water40 andaluminosilicate glasses,41 the concentration was estimatedto be around 2500 ion pairs per carbon dioxide from thebackground-subtracted FTIR spectrum (Section III A).

B. Ultrafast infrared techniques

Two different experimental 2D IR setups, for whichthe layout and procedures have been described in detailpreviously,42,43 were used to record polarization-selective2D IR spectra and IR pump-probe measurements. Bothapparatuses shared the same general layout. Briefly, aTi:Sapphire oscillator/regenerative amplifier system, operatedat 1 kHz, pumped an optical parametric amplifier (OPA) basedon a BBO crystal. The output of the OPA was difference-frequency mixed in a AgGaS2 crystal to create mid-IR pulses.The mid-IR beams propagated in an enclosure that was purgedwith air scrubbed of water and carbon dioxide to minimizeabsorption of the IR. Some carbon dioxide remained despitethe purge, but the isotopic labeling shifted the probe absorptionaway from this wavelength so that the data were not distortedby atmospheric 12CO2 absorption.

The IR light was split into two beams, a stronger pump anda weaker probe, for both the pump-probe and vibrational echomeasurements. To collect pump-probe data, the pump beampolarization was rotated to 45 relative to the horizontally

polarized probe using a half wave plate and polarizer andwas chopped at half the laser repetition rate. The pump andprobe beams are focused and crossed in an approximately100 µm diameter spot at the sample. The relative pump-probe delay was set precisely using a mechanical delaystage. After traversing the sample, the probe polarizationwas resolved either parallel or perpendicular to that of thepump beam by a computer controlled rotatable polarizer. Theprobe polarization was projected back to horizontal followingthe measurement before detection. For general vibrationalecho experiments, three IR pulses of the desired polarizationsimpinge on the sample and stimulate the emission of thevibrational echo in the phase-matched direction and withpolarization(s) dictated by those of the excitation pulses. Toobtain phase information, the echo was heterodyned with afourth pulse, the local oscillator (LO). The signal (probe orecho with LO) was dispersed by a spectrograph with a 300line/mm grating and detected on a 32 pixels, liquid nitrogencooled, mercury-cadmium-telluride array.

The two setups used differed in the bandwidth supported(and thus minimum pulse length) and the pixel width ofthe detecting array. The first system42 was employed for thepump-probe measurements. It afforded greater time resolutionas it had shorter pulses (∼70 fs). This system was modifiedwith a diverging lens (f = −7.5 cm) after the spectrograph(located slightly before the exit focal plane) to further spreadout the dispersed wavelengths spatially. This setup enabled thearray detector, with its relatively wide pixels (0.5 mm width),to better sample the resolution supplied by the spectrograph.Spectral data were retrieved every 0.6 cm−1.

The second system43 was employed in the acquisition ofthe vibrational echo signal and was configured in a pump-probe geometry with a Germanium acousto-optic mid-infraredpulse shaper. Three incident pulses, the first two of whichare generated collinearly by the pulse shaper, stimulatedthe emission of the vibrational echo in the phase-matcheddirection, which is collinear with the third (probe) pulse.The probe serves as the LO for optical heterodyne detectionof the echo signal. Though this system did not support asgreat an IR bandwidth as the first (and thus did not haveas short time resolution), there was significantly more powerat the vibrational probe absorption wavelengths and it wasoutfitted with an array detector with narrower pixels (0.1 mm).The narrow pixels enabled sampling the spatially dispersedspectral data with greater resolution. A width equivalent toseveral pixels separates the thin detector elements. To collectthe frequencies lost between pixels, the monochromator centerwas shifted by half the pixel-to-pixel frequency spacing. Thetwo sets of interferogram data at 32 detection frequencieseach were interlaced to yield one complete data set at 64frequencies. The pulse shaper enables phase cycling andcollection of echo interferograms in the partially rotatingframe for scatter removal and rapid data acquisition. The pulseshaper additionally allows 2D IR data to be worked up withoutthe need for phase corrections in post-processing44,45 becausethe echo signal is self-heterodyned by the collinear pulse 3,which serves as the LO. This method results in no uncertaintyin the relative timing of pulses that define coherence periodsduring which there is rapid phase evolution.43,46,47


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III. RESULTS AND DISCUSSION

A. Linear IR spectrum

Carbon dioxide has four vibrational modes: a symmetricstretch, an asymmetric stretch, and a doubly degeneratebend. In this study, the asymmetric stretch is monitored.The peak of unlabeled carbon dioxide (12CO2) shifts from2350 cm−1 in the gas phase to 2342 cm−1 when solvatedby the RTIL EmimNTf2. The vibrational probe used was13C-labeled CO2 to avoid the interference of atmosphericcarbon dioxide (predominantly 12CO2) with the signal. Theisotopically labeled 13CO2 absorbs at 2277 cm−1 in the IL (seeFigure 2). Though the unlabeled carbon dioxide is a goodprobe in itself,31,48 we have chosen to use the isotopicallylabeled species to limit the effects of atmospheric carbondioxide interference with the measurement and distortion ofthe results.

Carbon dioxide dissolves quite readily in EmimNTf2.12,19

Imidazolium-based ionic liquids typically can solubilize largequantities of the gas, comparable to or better than othercommon cations in terms of mole fraction dissolved.4,13,25,49

However, the anion is more important for the solubility.19,25

Experiments and molecular dynamics simulations show agreater affinity for and a stronger organization of the CO2molecules around the anion.19 The identity of anion, and toa lesser extent the cation, can cause the center of the linearIR absorption spectrum of CO2 to shift and broaden. Thepartial charges on the isolated carbon dioxide molecule arerather large, with the carbon having +0.70 of a charge and−0.35 for each oxygen,50 and so it associates quite readilywith the polar groups of the ionic liquid. There is verylittle volume expansion when the gas is added to the ionic

FIG. 2. Background-subtracted and normalized FTIR spectrum of 13CO2 inthe ionic liquid EmimNTf2. The major shoulder on the red side is attributedto a hot band: absorption of the asymmetric stretch when the bend is alreadythermally excited. It is displaced to the red of the main asymmetric stretchabsorption by the combination band shift. The smaller shoulder to the redof the hot band is due to a low concentration of 18OCO, which accompaniesthe isotopically enriched 13CO2. Inset shows the spectrum of EmimNTf2 withdissolved carbon dioxide prior to subtraction of the neat IL spectrum. Thelocation of the vibrational probe absorption is indicated on top of the ILbackground.

liquid.51 The anion radial distribution function changes littleupon the addition of CO2

19 suggesting that the ionic liquidstructure remains relatively unperturbed by the addition ofsmall amounts of carbon dioxide, in contrast to more typicalsolvents. To a first approximation, carbon dioxide occupiesthe spaces in the RTIL that would otherwise be vacant.51

The absorption band (Figure 2) is narrow, with a 5 cm−1

full width at half-max (FWHM). This is somewhat narrowerthan the 7 cm−1 FWHM absorption spectrum of CO2 inwater,48 but peak narrowing in an ionic liquid versus othersolvents is not unusual.36,52,53 The narrow band suggestseither that the vibrational frequency does not change greatlyas the carbon dioxide experiences different environments inthe ionic liquid or that the gas molecules are found in alimited range of configurations. To the red (low frequency)side of the line there is a noticeable shoulder, both withlabeled and unlabeled CO2. It has been shown31,54 that thisshoulder is a hot band, the result of the thermally populatedbend coupling to the asymmetric stretch; the absorption showsup red-shifted by the combination band shift. While the hotband does not contribute directly to the observed signalbecause of its redshift and low amplitude, its coupling to thefundamental transition at 2277 cm−1 does affect the analysisof isotropic population decay data. This topic was addressedin a previous investigation.54 The observation of structuraldynamics through spectral diffusion within the fundamentalband and anisotropic reorientation dynamics is unaffected bythe small isotropic population effects.36 A very small peakfurther to the red is attributed to molecules that contain an18O instead of two 16O. This is not unreasonable as the 13CO2gas is slightly enriched in 18O, according to the manufacturerspecifications. This peak is shifted from the main peak andhas a very low amplitude. Therefore, it does not interfere withthe experiments.

B. Reorientation dynamics

Polarization-selective pump-probe experiments measureboth the population relaxation (vibrational lifetime) and theanisotropy (reorientation dynamics). The polarization of thepump pulse is set at 45 relative to the horizontal probe,and the polarization of the probe signal parallel (S∥(t)) andperpendicular (S⊥(t)) to the pump is then resolved. Thesesignals can be expressed in terms of population relaxation,P(t), and the second Legendre polynomial orientationalcorrelation function, C2(t),

S∥(t) = P(t) [1 + 0.8C2(t)] , (1)S⊥(t) = P(t) [1 − 0.4C2(t)] . (2)

The population relaxation is given by,

P(t) = (S∥(t) + 2S⊥(t))/3. (3)

At early pump-probe delay times, a non-resonant signal,which tracks the pulse durations of the pump and probeinterferes with the signal from the resonant vibrations. Thus,only the data after a delay time of 300 fs is used. The 13CO2 wasfound to have a vibrational lifetime of 64 ± 2 ps, sufficientlylong that good signal to noise data can be collected over the


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longest time scales of interest for rotation and IL structuraldynamics. There is also population transfer from the shoulderhot band to the main peak that occurs with the bend relaxationtime. A full account of the population dynamics and the bendrelaxation has been presented,54 and so will not be reproducedhere. The internal vibrational excitation dynamics presentin the isotropic population decay do not interfere with theorientational relaxation measurement.36

The orientational relaxation of the molecules canbe extracted from the anisotropy data. The pump-probeorientational anisotropy is equal to the transition dipoleorientational correlation function, that is, the second Legendrepolynomial correlation function, C2(t) scaled by 0.4. It has theform,

r(t) = S∥(t) − S⊥(t)S∥(t) + 2S⊥(t) = 0.4C2(t). (4)

For perfect orientational correlation at time zero, the initialvalue is r(0) = 0.4. As the probe rotates, the orientationalcorrelation function decays and the anisotropy goes to zerofor complete orientational randomization. Fits of the curves donot achieve 0.4 (perfect correlation) when extrapolated backto time zero because of fast inertial motion, which occurson ultrafast time scales within the pulse width.55 However,by analyzing the initial deviation from 0.4, the extent of thismotion can be quantified.32,36,37,55

The anisotropy decays of 13CO2 in EmimNTf2, atdetection wavelengths spanning the entire vibrational band(see Figure 2), are presented in Figure 3. After the initialinertial drop, the curves decay in a non-exponential mannerto nearly zero by 120 ps. The anisotropy data are fit well bytriexponential decays. Because there is only one population

FIG. 3. Pump-probe orientational anisotropy plotted as color-coded solidlines for detection frequencies (numerical values in cm−1) across the band.Triexponential decay fits to the curves are shown in red. The fits, whose timeconstants were shared among all frequencies but amplitudes freely varied, arein excellent agreement with the data across the entire band. The variation inthe measured anisotropy with wavelength is attributed to different extents ofwobbling motions (i.e., varying cone angles) across the band. The inset showsthe early time data (to 5 ps) more clearly. Note that the center of the band isthe highest-valued curve and the curves descend farther from the center onboth sides.

of carbon dioxide molecules as was determined from thelinear spectra (Figure 2) and the lifetime measurementsdescribed previously, i.e., the observed band is not madeup of separate but overlapping absorption bands that resultfrom distinct independent subensembles of molecules,54 theanisotropy decays are interpreted as the time scales for twoperiods of restricted angular diffusion (wobbling-in-a-cone)and a complete rotational randomization process.34–37 Thewavelengths across the band are fit simultaneously, allowingthe amplitudes of the contributing exponentials to vary whilesharing the time constants. The fits, shown as the red lines inFigure 3, are in excellent agreement with the data. Allowingthe long time constant to vary did not improve the quality ofthe fit and made the wavelengths with lower signal (at thefar edges of the band) more susceptible to noise. Since theanharmonicity is so large (23 cm−1) compared to the peakwidth, the 1-2 transition does not interfere with measurementson low frequency side of the 0-1 band. The decay timeconstants t1, t2, and t3, which are determined self-consistentlyacross the wavelength spread, are given in Table I.

Using the method described previously,36,38 the coneangles are recovered from the amplitudes of the exponentialterms in the anisotropy decay, and the decay time constantsare associated with the correlation times of the restricteddiffusive reorientation processes. The correlation time ofthe first cone is given simply by τc1 = t1, that of thesecond cone is τc2 = (1/t2 − 1/t3)−1, and the correlation timefor the final complete orientational randomization (by freediffusion) is τm = t3. The inertial cone is an ultrafast motionwhose complete angular range is fully sampled within thepulse duration; all we can reliably conclude about its timedependence is that it must occur with a time constant muchless than about 200 fs.55

Each successive wobbling motion, enabled by a relaxationof constraints following structural relaxation, adds to the totalangular range sampled by the probes.36 The three cone angles,θin, θc1, and θc2, corresponding to the inertial ballistic motion,and first, and second diffusive wobbling periods, are displayedin Figure 4. The inertial cone half-angle is approximately20 ± 2. After the first wobbling time, the total angularrange (resulting from both the inertial and first wobblingcone38) averaged across the band is 38 ± 2. The total coneresults from the combination of the inertial, first wobbling,and second wobbling cones and is 50 ± 2 averaged overall wavelengths. These progressively larger cones, trackingthe cumulative increase in (restricted) orientational samplingby the CO2 molecule’s transition dipole, can be pulled apartinto the angular contributions from each process, as if it was

TABLE I. Time constants from multi-exponential fits and wobbling-in-a-cone analysis of the orientational relaxation of small vibrational probes inEmimNTf2.

Vibrational probe t1=τc1 (ps) t2 (ps) τc2 (ps) t3=τm (ps)

13CO2 0.91 ± 0.03 7.2 ± 0.1 8.3 ± 0.1 54 ± 1HODa . . . 4.0 ± 0.5 4.7 ± 0.7 25.1 ± 0.7Methanol-d4

a 0.8 ± 0.1 5.6 ± 0.7 6.4 ± 0.9 42 ± 1Ethanol-d6

a 0.8 ± 0.1 9.2 ± 0.9 10 ± 1 88 ± 10

aReproduced from Kramer et al.36


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FIG. 4. Cone half angles, from the wobbling-in-a-cone analysis of the ori-entational anisotropy decays of the 0-1 band, plotted versus frequency. Theinertial cone (black squares) and the second wobbling cone (green upwardtriangles) show little variation with wavelength. The first wobbling cone (redcircles) is very strongly correlated with wavelength, becoming larger almostsymmetrically with detuning from the center frequency (2277 cm−1).

unaffected by the others.38 These individual contributions canbe compared to each other in amplitude as well at differentwavelengths.36

As can be seen in Figure 4, only the first wobbling coneangle (red points) is significantly wavelength dependent. Thiswavelength dependence suggests that the specific environmentaround the CO2 molecule, which determines the vibrationalfrequency, restricts the fastest observable motions in aparticular manner. The other cones have fairly consistentsize across the band and thus are not dependent on thespecific environment of the carbon dioxide molecule withinexperimental uncertainty. These then reflect the time scales ofthe ionic liquid components, which gate the excursions of theCO2 molecule.

The time scales of reorientation can be compared to thefindings for water, methanol, and ethanol in EmimNTf2.36

These probes form hydrogen bonds with the IL anions,52,53

whereas carbon dioxide does not, and so the anglesof movement are correspondingly modified. However, thedynamics of reorientation occur on roughly the same timescales. This further supports the proposed mechanism thatsmall molecule probe reorientation is being gated by the ionicliquid motions. Given that the molecular sizes and shapes aresimilar, the identity of the probe does not have a very majoreffect on the general time scales of reorientation.

The above results can also be compared to therotational dynamics extracted from an MD simulation byHazelbaker et al.30 For the lowest CO2 concentrationin that study (which corresponds to a mole ratioof 25:1 IL ion pairs to carbon dioxide molecules),in the similar ionic liquid 1-butyl-3-methylimidazoliumbis(trifluoromethylsulfonyl)imide (BmimNTf2) at roomtemperature, the authors calculated an integrated 62 ps rotationtime constant (for C2(t)). This concentration of the carbondioxide is approaching the dilute solution limit. Thus, the MD

results can be compared to the experimentally determinedreorientation dynamics in this study, even though the latter arefrom solutions that are almost 100 times more dilute. Althoughit is for a different ionic liquid and at a higher dissolved gasconcentration, the rotational correlation time constant fromthe MD simulation and the experimental longest time constantof 54 ± 1 ps measured here are remarkably close.

The data can also be compared to predictions of rotationaldiffusion in the hydrodynamic limit, using a modified Debye-Stokes-Einstein (DSE) equation. The longest reorientationtime scale, τm, is equal to 1/6 Dr (with Dr the diffusionconstant for this final rotational process). If the diffusivemotion is well described by hydrodynamics, we can relate thetime scale to properties of the system by,56

τm =f Veffη

kBT, (5)

where T is temperature, η is viscosity, kB Boltzmann’sconstant, and Veff is the effective molecular volume. Theshape factor is f = 1 for stick boundary conditions and f (ρ)for slip conditions, where ρ is the axial ratio of the molecularellipsoid. The ρ dependence of f with the slip boundarycondition has been calculated by Youngren et al. and Huet al.57,58 We estimate the dimensions of the carbon dioxidemolecule using the known bond length, 116 pm,59 and the vander Waals radii of carbon and oxygen. The effective volumeis then Veff = 22.4 Å3, ρ = 0.58, and f = f (ρ) = 0.215 for aprolate spheroid. Using our measured value η = 35.8 cp forthe viscosity at 298 K, τstick

m = 210 ps and τslipm = 45 ps. The

stick boundary condition was originally derived for systemswhere a layer of much smaller solvent molecules adhereto the surface of a larger solute. As the solute rotates, the“stuck” layer of solvent is carried with it, and interactionswith the surrounding solvent are responsible for friction.While the physical system studied here is not consistentwith the stick model (the solute is smaller than the solvent),small solutes that have strong, specific, interactions withthe solvent sometimes yield diffusion rates consistent withstick boundary conditions. Slip boundary conditions describesituations where the only friction preventing the solute fromrotating freely is overlap of the swept volume of the solutewith regions occupied by the solvent. Because the carbondioxide is a small molecule located among the large ions thatcomprise the ionic liquid, slip boundary conditions shouldapply. Note that the experimentally measured 54 ± 1 ps isrelatively close to the slip value of 45 ps.

If the molecular motions are truly hydrodynamic, thefriction term governing both rotational and translationaldiffusion is the same. The rotational diffusion constant can becompared to the translational diffusion constant to determineif the system displays Stokes-Einstein (SE) behavior, i.e.,is actually described well by hydrodynamics. The Stokes-Einstein-Sutherland equation gives the translational diffusionconstant as,

DT =kBT6πηr

,

where r = 3/(4π) × V 1/3eff is the effective radius, and the other

values are the same as in Eq. (5). Substituting in the numbers


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from above gives DT = 3.5 × 10−11 m2/s. At room temperatureand a pressure of 1 atm, the diffusion constant of carbondioxide through EmimNTf2 is 5 × 10−10 m2/s, as measured bythin film pressure drop23 and gravimetric29 techniques. Thisis an order of magnitude faster than predicted. However,the diffusion of neutral solutes through ionic liquids isnot necessarily hydrodynamic.60 For small neutral solutes,according to Araque et al.,61 the diffusion is altered by thelocal friction (which is not described well by the bulk ILviscosity). In this case, the SE equation is clearly violated.Thus, while the longtime scale rotational behavior canbe approximated by hydrodynamics (i.e., the DSE equationwith slip boundary conditions) rather well, the translationaldiffusion cannot.

C. Spectral diffusion

To determine the spectral diffusion dynamics of carbondioxide in the ionic liquid, 2D IR spectroscopy experimentswere performed.44,62–64 In this third-order nonlinear timeresolved technique, three excitation pulses impinge on thesample and following the third pulse give rise to the vibrationalecho signal, which is an IR pulse emitted in the phase-matcheddirection. A fourth pulse, serving as a LO, is overlappedspatially and temporally with the signal pulse for opticalheterodyne detection, which provides the requisite phaseresolution and amplifies the small signal. The time delaybetween pulses 1 and 2 is the coherence time, τ, and thetime between pulses 2 and 3 is the population time (waitingtime), Tw. Data are collected by scanning τ, while fixing Tw ateach desired value of the waiting time, causing the vibrationalecho phase to evolve in time (τ) relative to the LO pulse. Theresulting temporal interferogram is frequency resolved anddetected on the 32-element array, providing the vertical axis,ωm, of the 2D IR spectra. At each ωm, the interferogram isnumerically Fourier transformed to give the horizontal axis,ωτ, of the 2D IR spectra.

The first and second pulses label the initial probefrequencies and create a population state. During the waitingtime Tw, the structure of the liquid evolves, which causes thevibrational frequencies to shift as the particular interactions ofeach molecule contributing to the inhomogeneous absorptionline evolve. Note that any change in the vibrational frequencyof the probe (spectral diffusion) will contribute regardlessof whether it is from the structural evolution of the matrixsurrounding the probe or from reorientation of the probe ina quasi-static environment. The Tw period is ended by thearrival of the third pulse, which also stimulates the emissionof the vibrational echo pulse. The echo contains informationon the final frequencies of the vibrational oscillators.

The two dimensional spectrum then can be understoodby reading off which frequencies were initially absorbedfrom the ωτ axis and which frequencies were emitted onthe ωm axis. At times before spectral diffusion is complete,the 2D band shows up as elongated along the diagonal—corresponding to a vibrational oscillator only being ableto sample frequencies close to its initial frequency, butnot the entire inhomogeneously broadened absorption line.When correlation between the starting frequencies and ending

frequencies has been lost, spectral diffusion is complete,and the peak appears symmetrically rounded. The limit ofcomplete spectral diffusion is only experimentally accessibleif the vibrational lifetime is sufficiently long compared tothe slowest spectral diffusion time scale. Thus, the structuraldynamics of the liquid can be extracted from the change inshape of the 2D IR spectra as a function of Tw.

The amplitudes and time scales of spectral diffusion arequantified by the FFCF. The FFCF is the joint probabilitythat a vibrational oscillator with an initial frequency will havethat same frequency at a later time, averaged over all of theinitial frequencies in the inhomogeneous spectral distribution.The Center Line Slope (CLS) method63,64 is used to extractthe FFCF from the Tw-dependent shape of the 2D IR spectra.The CLS has been previously shown to be mathematicallyequivalent to the normalized Tw-dependent portion of theFFCF.63,64 The complete FFCF is modeled with the form,

C(t) = ⟨δω(t)δω(0)⟩ = δ(t)T2+

i

∆2i exp(−t/τi), (6)

where ∆i and τi are the frequency fluctuation amplitudeand time constant, respectively, of the ith component. Acomponent of the FFCF is motionally narrowed and is a sourceof homogeneous broadening in the absorption line shape if∆iτi < 1. In this case, it is not possible to determine ∆i andτi separately. The homogeneous contribution to the FFCF isgiven by the first term on the right hand side of Equation(6). The homogeneous portion of the absorption spectrum hasa pure dephasing linewidth given by Γ∗ = ∆2τ/π = 1/(πT∗2 ),where T∗2 is the pure dephasing time. The total homogeneousdephasing time that is measured, T2, also depends on theorientational relaxation and vibrational population relaxationand is given by,

1T2=

1T∗2+

12T1+

13Tor

, (7)

where T1 and Tor are the vibrational lifetime and orientationalrelaxation time. The homogeneous contribution to the lineshape is a Lorentzian with FWHM Γ = 1/(πT2). The ∆i inEq. (6) are standard deviations of the individual Gaussiancontributions to the inhomogeneous linewidth.

If the probe only experienced structural fluctuations ofthe surrounding fluid without undergoing independent motion(reorientation), the structural fluctuations would completelydetermine the FFCF. However, recent polarization-selective2D IR experiments have shown (using an ionic liquid as thesolvent) that the probe itself can rotate on a time scalefast enough that the surrounding fluid acts as a quasi-static interaction potential, such that the rotation of theprobe contributes significantly to changes in its frequency,independent of fluid structural fluctuations.32,33 This RISDcontribution, when it occurs as it does here (see below)needs to be accounted for when analyzing the total FFCF toextract the rates and amplitudes of spectral diffusion processescaused by liquid structure rearrangements. The presence ofRISD causes the measured spectral diffusion to proceed fasterthan the contribution from the structural spectral diffusionalone. If all of the spectral diffusion already occurs faster than


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the rotation times including orientational wobbling, no RISDeffect will be observed.

To determine if RISD is occuring, two polarizationconfiguration vibrational echo experiments were conducted.⟨X X X X⟩ and ⟨X XYY ⟩ polarization configuration 2D IRspectra were taken, where an X indicates a beam polarizedhorizontally and Y is the orthogonal vertical polarization.⟨X XYY ⟩ means the first two pulses are vertically polarizedand the third pulse and the detected echo polarization havehorizontal polarization. When all frequency fluctuations arecaused by liquid structure evolution, the ⟨X X X X⟩ 2D spectraare sufficient to quantify the spectral diffusion; in this case,the time dependence of the ⟨X XYY ⟩ spectra will be identicalto that of the ⟨X X X X⟩ spectra. This has been shown to be thecase in bulk water, for example, from polarization-selective2D IR experiments on the O–D stretch of dilute HOD inH2O.33 However, once rotation becomes significantly fasterthan some structural evolution time scales, the experimentally

observed spectral diffusion is no longer the same for thedifferent polarizations.

Significant RISD contributions have been observedwith small molecule probes in ionic liquids, in particular,EmimNTf2.65 Therefore, measurements were taken to seeif RISD occurred in the carbon dioxide–IL system. 2Dinfrared spectra at several waiting times in both the parallelpumped, ⟨X X X X⟩, and perpendicular pumped, ⟨X XYY ⟩,configurations are displayed in Figure 5. As Tw is increased,there is a clear change in both sets of 2D band shapesfrom generally correlated along the diagonal (dashed lines inFigure 5) to progressively more rounded as spectral diffusionoccurs over longer time intervals.

The Tw-dependent progress of spectral diffusion for bothpolarizations is determined quantitatively using the CLSmethod.63,64 The center line slope curve is the normalizedFFCF; the difference from 1 at Tw = 0 is determined by thehomogeneous contribution to the absorption spectrum and can

FIG. 5. 2D IR spectra of 13CO2 inEmimNTf2 for the ⟨XXXX⟩ (left col-umn) and ⟨XXYY ⟩ (right column)polarization configurations at differentwaiting times. Note that as the wait-ing time increases the spectra becomemore rounded and less elongated alongthe diagonal (black dashed line) due tospectral diffusion.


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FIG. 6. Spectral diffusion of carbon dioxide’s asymmetric stretch inEmimNTf2, as quantified by the Center Line Slope (CLS) of 2D IR spectrataken in the ⟨XXXX⟩ (black dots) and ⟨XXYY ⟩ (red dots) polarizationconfigurations. Inset shows the CLS of the isotropic 2D spectra constructedas ⟨XXXX⟩+2⟨XXYY ⟩ from the correctly normalized components (seeSection III C). Solid lines are independent biexponential fits to the three decaycurves (results in Table II).

be used to determine the homogeneous linewidth. The resultsof the CLS analysis are shown in Figure 6 (points). Loss offrequency correlation is apparent on several time scales inthe normalized FFCF, with complete randomization occurringwithin 200 ps. Furthermore, it is clear that the ⟨X X X X⟩(black dots) and ⟨X XYY ⟩ (red dots) CLS data sets are notthe same; the ⟨X XYY ⟩ CLS data decay more rapidly with Tw

than the ⟨X X X X⟩ data. This is precisely the trend anticipatedfor systems whose spectral diffusion has rotation-inducedcomponents.32,33

While having the ⟨X X X X⟩ and ⟨X XYY ⟩ polarizationconfiguration data will be critical to the following analysis, itis also useful to consider the isotropic 2D spectral data. Theisotropically averaged spectra and corresponding FFCF areuseful for comparison to MD simulations. When no laser fieldpolarization factors are included in extracting observablesfrom a simulation, the simulation results will still include theRISD, but the values are isotropically averaged. These datacould be acquired experimentally as a separate experimentusing magic angle polarization but it can also be constructedfrom the ⟨X X X X⟩ and ⟨X XYY ⟩ data, in analogy with pump-probe S∥ and S⊥ as given in Eq. (3). To add the spectratogether, they must have the correct relative amplitudes. Sincethe echo experiments were taken sequentially, under differentlaser conditions and with different polarizers, there is noguarantee that they have the same overall normalization andcan be simply added together. To obtain the correct relativeamplitudes, the projection of the full 2D IR spectrum onto theωm axis was compared to the polarization pump-probe data,⟨X X X X⟩ with S∥ and ⟨X XYY ⟩ with S⊥. The projection and PPdata must be the same, up to an overall normalization factor,from the projection-slice theorem.45 Comparing the amplitudeof the projection to the pump-probe data at multiple waitingtimes (typically earlier Tw spectra with good signal to noise)yielded a best-fit scaling constant, which was then used to

scale the entire 2D IR spectra set. Following the correct scalingof the amplitudes, the 2D spectra were linearly combined as⟨X X X X⟩ + 2⟨X XYY ⟩, to produce the isotropic spectrum fromwhich the isotropic CLS curve was obtained. The isotropiccurve is shown in the inset of Figure 6.

FFCF analysis was performed on the fits to the CLSof all 3 configurations and is summarized in Table II. AllCLS curves fit to biexponentials. The fits are the solidcurves through the data points in Figure 6. Due to thedifferent polarization weightings of the RISD contribution,the amplitudes and time constants generated from these fitsare not the same. This demonstrates how important it is toidentify systems that display RISD and account for it. If onlythe ⟨X X X X⟩ configuration data had been collected (as is oftendone when the spectral diffusion dynamics are investigated),the time scale of the fast process would be reported astwice as slow as in the isotropic case. If these numbers areused to validate a simulation, this discrepancy could lead toerroneous conclusions. In addition, it is possible to separatethe polarization dependent FFCFs shown in Figure 6 intocontributions from structural spectral diffusion and rotationinduced spectral diffusion, permitting the time dependence ofthe structural evolution of the system to be explicated. This isdone below.

By separating the RISD contribution to the FFCF fromthe structural spectral diffusion, more can be learned aboutthis system. A previously developed theory for calculatingthe polarization-selective FFCF observables, making use of astraightforward model,32,33 suggests that the data could be fitto separate the contributions of structural spectral diffusionfrom the reorientation of the probe in these decays. However,applying this theory to the spectral diffusion of carbon dioxiderequires care. The theory,33 which has been successful in aprevious analysis of polarization-dependent spectral diffusiondata,32 assumes that the polarization-dependence of the time-dependent 2D line shapes emerges from a polarization-dependent FFCF entering the normal nonlinear responsefunction calculations.66 This variable correlation functionis a polarization-weighted FFCF in which the ensembleaverage includes weighting by the vibrational probe transitiondipole’s overlap with the four electric field polarization vectorsinvolved in the excitation of the molecule and emission of thesignal field.33 The previously developed model assumed thatthe vibrational transition in question coupled to a quasi-staticelectric field, induced by the surrounding solvent, via the

TABLE II. Complete FFCF parameters that include the combined effects ofstructural spectral diffusion and reorientation induced spectral diffusion.

Polarization Γ (cm−1)a ∆1 (cm−1) τ1 (ps) ∆2 (cm−1) τ2 (ps)

⟨XXXX⟩ 3.4 ± 0.1 0.94 ± 0.03 13.0 ± 0.7 1.11 ± 0.03 58 ± 2⟨XXYY ⟩ 3.6 ± 0.1 0.98 ± 0.03 6.2 ± 0.7 1.04 ± 0.03 44 ± 3Isotropicb 3.5 ± 0.1 0.89 ± 0.03 6.6 ± 0.7 1.13 ± 0.03 45 ± 2

aΓ= 1/(πT2) is the FWHM of the Lorentzian homogeneous linewidth, with T2 thetotal dephasing time. The frequency fluctuation amplitudes ∆1 and ∆2 are standarddeviations of the Gaussian inhomogeneous line shapes, which are convolved for thetotal inhomogeneous contribution ∆2

total=∆21+∆

22. All three configurations give a total

inhomogeneous linewidth of ∼5.6 cm−1 (given as FWHM= 2.355×∆total).bCalculated by constructing the isotropic data from the correctly normalized parallel andperpendicular 2D IR spectra and then obtaining the CLS.


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first order Stark effect.32,33 For molecules with a permanentdipole moment, the use of the first order Stark mechanismhas worked well in reproducing 2D IR data in a varietyof simulations.67–74 A frequency shift due to the first orderStark effect requires the molecule to have a permanent dipole.Because carbon dioxide is a linear molecule with a reflectionplane perpendicular to the axis, no such dipole is present.To quantitatively decompose the observed polarization-dependent CLS decays into their structural and rotationalcontributions, a new model for evaluation of the polarization-weighted FFCF observables was developed based on thesecond order (quadratic) Stark effect.

IV. RISD THEORY AND FFCF DECOMPOSITIONFOR THE SECOND ORDER STARK EFFECT

A molecule in a condensed phase system can experiencehighly directional interactions with its surroundings, whichare a major factor in determining the instantaneous vibrationalfrequency (frequency within the inhomogeneous absorptionspectrum) of an infrared absorption mode of interest. Spectraldiffusion occurs at least in part due to the time-evolutionof these directional interactions with the surroundings.Fluctuations of the liquid structure will modify the directionalcoupling with the vibrational transition; these dynamicschange the frequency and are referred to as SSD. TheSSD fluctuation amplitudes and time scales give valuableinformation about the structural changes of the liquid or othermaterial in which the vibrational probe is located.

The probe molecule’s reorientation will modify thedirectional couplings, and hence vibrational frequencies(RISD), without the necessity of structural changes in theenvironment (assuming the environment is not considerablymodified by the molecular rotation itself). The RISD processis illustrated conceptually in Figure 7 for the CO2 molecule.As the molecule reorients, its frequency changes, but itscontribution to the 2D IR vibrational echo signal decreases

FIG. 7. Illustration of the RISD process for a linear, inversion-symmetricmolecule, carbon dioxide (CO2). The molecule experiences a relativelyslowly evolving electric field that, through the second order Stark effect, isin part responsible for inhomogeneous broadening. Rotation of the moleculechanges both the orientation of the transition dipole moment (parallel tothe molecular axis) and the difference polarizability tensor relative to theelectric field created by the environment, altering its vibrational frequency.The contribution of the frequency change to ⟨XXXX⟩ is reduced while thecontribution to ⟨XXYY ⟩ is increased.

in the ⟨X X X X⟩ case and increases for ⟨X XYY ⟩. Our modelfor strongly directional interactions that can lead to RISD,and hence differences between 2D IR observables in differentpolarization configurations, is the Stark effect due to the netelectric field induced at the position of the molecule by itssurroundings.32,33,67

The shift of a particular molecule’s vibrational transitionfrequency ω due to the Stark effect, expanded up to secondorder in the field, is,

∆ωE = ω − ω0 = −∆µ · E −12

E · ∆α · E, (8)

where ω0 is the natural oscillator frequency in the absenceof electric fields, E is the net electric field vector at theposition of the molecule, ∆µ is the dipole moment differencebetween the first excited state (n = 1) and ground state(n = 0), and ∆α is the difference in polarizability tensorbetween the 1 and 0 states.75–77 When vibrational excitationresults in a non-zero change in dipole moment, the ∆µterm (first order Stark effect) is typically dominant and theweaker ∆α term (second order Stark effect) can often beignored.76 Previous theoretical treatments of RISD in 2D IRspectroscopy, showing excellent agreement with experimentalobservations of hydrogen-bonded hydroxyls in an ionic liquid,have focused on the first order Stark effect’s contribution toinhomogeneous broadening.32,33 However, a molecule withinversion symmetry, e.g., carbon dioxide, cannot have apermanent dipole moment and its vibrational modes willnot have difference dipoles, ∆µ. The second order Starkeffect is a conceptually straightforward model, which canlead to directional couplings of these molecules’ vibrationalfrequencies to their environments. In particular, second orderStark shifts are naturally invariant to 180molecular rotations,a requisite feature of any appropriate model of the frequencyfluctuations for molecules (CO2) that have such symmetry.The polarization-weighted frequency-frequency correlationfunctions (PW-FFCFs)33 for the second order Stark model,which describe the polarization-selective time-dependent 2DIR band shapes as discussed in Section III C, are derived inthe Appendix. The results are briefly discussed here.

Unlike first order Stark effect coupling, the secondorder model permits the separation of the scalar magnitudeand vector direction of the field fluctuations. Rotation ofa molecule will only affect the vector fluctuations, sincethe magnitude fluctuations are the same regardless of theprobe molecule’s orientation. For a particular polarizationconfiguration, denoted p, the PW-FFCF within the secondorder Stark model is given in general by,

Cp(t) = ⟨δω(t)δω(0)⟩p = Fs(t) + Fv(t)R(2)p (t), (9)

where Fs(t) is the correlation function of the scalar-coupledfrequency fluctuations, Fv(t) is the correlation function ofthe vector frequency fluctuations, and R(2)

p (t) is an RISDcorrelation function whose form is given below in Eqs. (12).The scalar-coupled correlation function is scaled by severalconstants that depend on the coupling and has the form,

Fs(t) = λ22

(1 + 2ax

3

)2 (E2(t)E2(0) − ⟨E2⟩2

). (10)


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The on-axis and off-axis components of the differencepolarizability (see Eq. (A2) and discussion in the Appendix),∆αz and ∆αx = ∆αzax, respectively, determine the couplingstrength of the electric field to the vibrational frequency. Notethat, for the CO2 molecule, ∆αx is expected to be smallerin magnitude than ∆αz, with either the same or oppositesign. The second order Stark tuning rate has been defined asλ2 = −∆αz/2. Only the magnitude, E(t), of the fluctuatingsample electric field vector E(t) appears in Eq. (10); fieldorientational dynamics and molecular probe rotation do notaffect the scalar part of the PW-FFCF. The scalar FFCFtherefore reports on a subset of purely structural dynamics inthe material.

The correlation function of vector frequency fluctuations,appearing as the second term in (9): Fv(t)R(2)

p (t), has beenfactored into the product of a structural correlation functionFv(t) and an RISD correlation function R(2)

p (t), similarly toprevious results for PW-FFCF’s with a first order Stark effectmodel (that contain only a vector term).32 The vector-coupledSSD correlation function is also scaled by the couplingconstants and has the form,

Fv(t) =4λ2

2(1 − ax)29

×E2(t)E2(0)D2

m0(αt, βt, γt)D2∗m0(α0, β0, γ0) , (11)

where (αt, βt, γt) is the orientation of the field vector at timet and (α0, β0, γ0) is its orientation at time zero. The Dl

mn areWigner rotation matrices, which are proportional to sphericalharmonics when one index is zero as in Eq. (11).78 The anglebrackets ⟨· · · ⟩ denote an isotropic ensemble average in Eqs.(10) and (11); these correlation functions are independentof the laser field polarizations because the molecular probeorientation does not appear. Because any axis system canbe chosen to perform the isotropic average, the value of m(which can be 2, 1, 0, −1, or −2) does not affect the resultin (11). From the scaling constant in Eq. (11), we see that ifax = 1 (equal on- and off-axis difference polarizabilities), thevector frequency fluctuations vanish and the FFCF consists ofonly the scalar term, (10). If this was the situation for CO2in EminNTf2, the polarization dependence shown in Figure 6would not occur.

The second factor in the vector FFCF term is a normalized,polarization-weighted, correlation function of the vibrationaltransition dipole orientation, as defined in general in Eq. (A19).The different weighting of frequency fluctuations felt bydifferent polarization-selected ensembles of vibrational probemolecules is entirely captured by the factor R(2)

p , with p theparticular polarization configuration of interest. Observationsof RISD through a difference in FFCF with polarization areenabled by the R(2)

p factors of the vector coupled FFCF.The RISD factors appearing in the vector part of Eq. (9)

were evaluated using the analytical methods detailed pre-viously33 for the three polarization configurations of interestin this work. We obtain the expressions,

R(2)parallel(t) =

1175

28 + 215C2(t) + 72C4(t)

1 + 4/5C2(t), (12a)

for the ⟨X X X X⟩ configuration,

R(2)perpendicular(t) =

1175

−14 + 155C2(t) − 36C4(t)

1 − 2/5C2(t), (12b)

for the ⟨X XYY ⟩ configuration, and

R(2)isotropic(t) = C2(t), (12c)

for the overall isotropic average or magic angle configuration.In Eqs. (12), C2 and C4 are the orientational correlationfunctions of order 2 and 4, respectively, of the vibrationalprobe transition dipole moment.

The RISD correlation decay factors R(2)p (t) as given

in Eqs. (12) are displayed in Figure 8 using a freediffusion model for the orientational motion. In this case,Cl(t) = exp(−l(l + 1)Dt), with D the rotational diffusionconstant. The second order orientational correlation functionC2(t) is 5/2 times the anisotropy decay, which is measured inpolarization-selective PP experiments.79 In the isotropicallyaveraged case, the normalized RISD function decays asC2(t), which is exactly the same time-dependence as theobserved reorientation in a PP measurement. The parallel andperpendicular cases, as displayed in Figure 8, decay somewhatmore slowly and more quickly than C2(t), respectively, andnotably reach offsets of +0.16 and −0.08 in the limit of infinitedelay time (in the absence of structural spectral diffusion).The frequencies of the probe molecules weighed by theinput/output polarizations will not completely randomize evenafter a long time. This offset can be qualitatively understood inthe following manner. The largest frequency jump possible isobtained by rotating a molecule 90 between time 0 and timet. Assume the molecule was initially oriented parallel to thepump polarization at time zero. After the rotation, at the latertime t, it contributes maximally to the ⟨X XYY ⟩ signal, but notat all to the ⟨X X X X⟩ signal. Hence, the parallel configurationtends towards residual correlation at long times, while theperpendicular configuration actually yields anti-correlation.

FIG. 8. The normalized, polarization-dependent parts of polarization-weighted FFCFs, in the absence of structural fluctuations. Decays are shownfor the isotropic (black, solid), parallel or ⟨XXXX⟩ (red, dash), and perpen-dicular or ⟨XXYY ⟩ (blue, dash-dot) configurations. Reorientation is assumedto be freely diffusive, with diffusion constant D. The decay in the isotropiccase is simply the orientational correlation function C2(t), which is obtaineddirectly from the pump-probe anisotropy.


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On the other hand, the isotropically averaged result C2(t),which lies between these two extreme cases, decays to zero atlong times.

As is evident from the form of Eq. (9), rotation of themolecule creates a distinction between structural fluctuationsthat only modify the square of the field magnitude, andthose that cause fluctuations of both field magnitude squaredand direction. While the presence of both a polarization-independent scalar term and polarization-dependent vectorcontribution in Eq. (9) extends the nature of the analysisof experimental 2D IR data, reorientation-induced spectraldiffusion actually grants an opportunity to learn more detailedinformation about the fluctuations of the medium when thevibrational probe has second order Stark coupling to itssurroundings. In contrast, systems with first order Stark effectcoupling contain only a polarization-dependent vector term inthe PW-FFCFs and therefore enable the determination of onlya single structural correlation function.32

To confirm the presence of vector coupling between themolecular transition and the internal electric fields, 2D IRspectra should be collected and the line shapes analyzedfor at least two different polarization configurations, such as⟨X X X X⟩ and ⟨X XYY ⟩ (as was done for 13CO2 in EmimNTf2in the present investigation). Assuming the form Eq. (9) forthe FFCF, Fs and Fv are the same for each polarizationof the observable (e.g., CLS). Measurement of the secondorder orientational correlation function C2(t) in a polarization-selective PP experiment is sufficient to completely determinethe R(2)

p (t) factor for each polarization configuration of the 2DIR data using Eqs. (12), once C4(t) is calculated from C2(t) asdiscussed below.32,33 For the isotropic case, the PP measuredC2(t) is the RISD function necessary in Eqs. (9) with (12c).

Constructing the l = 4 orientational correlation function,C4(t), from a fit to the experimentally measured l = 2 correla-tion decay requires a model. The mathematical techniquesfor relating orientational correlation functions of differentorder have been discussed in detail.32 For free diffusion(i.e., a single exponential C2(t) decay), the transformationis straightforward; all orientational correlation functionsare the single exponential forms Cl(t) = exp(−l(l + 1)Dt).Multiple exponential anisotropy decays, for a single ensembleof reorienting molecules (transition dipoles), are commonlyanalyzed within the wobbling-in-a-cone model (seeSection III B).32,34–37,39 A fit to the pump-probe anisotropydecay (or l = 2 orientational correlation function) is de-composed into the product of several wobbling correlationfunctions: single exponential decays to a plateau (squareof the generalized order parameter). These are followed bya free diffusion final reorientation factor, which is slowerthan all the wobbling components. The order parametergives the hard-cone wobbling half angle θ0. The decay timeconstant, τ(l)eff , together with the cone angle allow determinationof the rotational diffusion constant Dw for each wobblingregime.36–38 The equations relating the order parameter andwobbling time constant to the cone angle and diffusionconstant for C4(t) have been presented recently.32 Eachwobbling and free diffusion correlation function factor in C2(t)is transformed into the equivalent l = 4 correlation function,and the total C4(t) is the product of these individual factors.

V. RISD ANALYSIS FOR CO2 IN EmimNTF2

Figure 9 displays the CLS data (points, same as Fig. 6)from the ⟨X X X X⟩ and ⟨X XYY ⟩ vibrational echoes as wellas fits to these data (solid curves). The fits use our RISDmodel result, Eq. (9), with the R(2)

p (t) supplied by Eq. (12a)for ⟨X X X X⟩ or (12b) for ⟨X XYY ⟩. Time parameters andcone angles from the anisotropy curves, C2(t), (Section III B)and the expressions for C4(t) given previously32 in terms ofthe measured C2(t) were used to construct the polarization-dependent RISD factors as discussed in Section IV. Whilethe scalar and vector components of the structural FFCF aregiven in Eqs. (10) and (11), neither the initial magnitudes ofthe scalar and vector electric field correlations nor the Starkcoupling constants (λ2 and ax) are known a priori, as discussedfurther in the Appendix. The complete scaling prefactors inEqs. (10) and (11) can be useful if some of these valuesare known through independent measurements or theoreticalcalculations. Therefore, the SSD correlation functions, Fs(t)and Fν(t), are modeled as independent multiexponentials.The RISD factors are known, and it is important to notethat there are no adjustable parameters in these terms givenEqs. (12a) and (12b). The two curves shown in Figure 9are fit simultaneously. The values of the parameters in themultiexponential functions for the scalar and vector SSDcorrelation functions used to fit the (normalized) FFCF dataare freely varied. In the simultaneous fit, Fs(t) and Fν(t) arethe same for the parallel and perpendicular data sets.

Trying various multiexponential forms for the scalar andvector SSD fitting function, the best results were obtainedwith the scalar SSD, Fs, modeled with a biexponential andthe vector SSD, Fv, as a single exponential. This is theminimum number of exponentials necessary for the fit tocompletely describe the data. Adding more exponentials tothese functions does not yield additional unique time constants

FIG. 9. CLS data points for the ⟨XXXX⟩ and ⟨XXYY ⟩ 2D IR polarizationconfigurations (same as Fig. 6 data) with the simultaneous fit to both data sets(solid lines), incorporating reorientation-induced spectral diffusion (RISD)from coupling of the vibrational transition to the solvent electric field via thesecond order Stark effect. The RISD terms (see Eq. (9) with Eqs. (12a) and(12b)), have no adjustable parameters. The fits determine the time dependenceof the structural fluctuations.


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TABLE III. Amplitudes (Ai) and time constants (τi) from the fit of thepolarization-weighted normalized FFCFs with Eq. (9). The results are theeffect of structural spectral diffusion with the contributions from reorientationinduced spectral diffusion removed.

SSD Correlation function A1 τ1 (ps) A2 τ2 (ps)

Fs (scalar) 0.09 ± 0.02 20 ± 5 0.07 ± 0.02 77 ± 18Fv (vector) 0.36 ± 0.01 57 ± 5 . . . . . .

or a substantially better fit, which demonstrates that thefunctions chosen for Fs and Fv are sufficient, and adding morefree variables over parameterizes the fit. As can be seen inFigure 9, the fits to both polarization data sets are very good. Asingle set of parameters for Fs and Fv yields fits to both curves.It is important to note again that R(2)

p (t) in Equations (9) and(12) are completely determined by the separate pump-probeanisotropy measurements. There are no adjustable parametersin R(2)

p (t). The fit parameters are given in Table III.Let us now examine the time scales of the structural

spectral diffusion that are obtained from the fits that separatethe SSD from the RISD. The scalar part of the correlationfunction is composed of two exponentials with decay timeconstants of 20 ± 5 ps and 77 ± 18 ps. These processescontribute nearly equally to the total scalar amplitude. Thevector process has a time constant of 57 ± 5 ps and isresponsible for roughly two thirds of the total structuralspectral diffusion. The long time constants for the twocorrelation functions are similar, and in fact, the error barsoverlap. Therefore, it is possible that both contributions resultfrom the same IL motions. The same fluctuation can alterboth the magnitude and direction of the solvent’s internalelectric field at the CO2 position. If the times are not identical,that is not surprising. The scalar and vector SSD factors aredifferent correlation functions. Even if they report on the samedynamical process, the resulting correlation time need not beidentical.

The shorter 20 ps time constant is unique, having nocorresponding component in the vector fluctuations. Oneway this could emerge is if the frequency shifts from aclass of field magnitude fluctuations are exactly counteractedby (necessarily correlated) field direction fluctuations. Thescalar part, reporting only on magnitude fluctuations, woulddecay, but the vector part could show far less frequencydecorrelation to the point that this motion is not observableas a distinct spectral diffusion time scale. Another possibilityis that the additional time scale is caused by a processwhose effect on the vibrational frequencies is unrelated to theStark effect (or directional interactions in general). Non-vectorstructural fluctuations that induce frequency fluctuations areindistinguishable from the second order Stark contribution tothe scalar SSD correlation function (see the Appendix). Itis possible that the shorter dynamic time scale of 20 ps isreporting on isotropic dynamics, perhaps related to densityfluctuations in the liquid,80 that are not directional in theircoupling to the vibrational transition.

These SSD numbers can also be compared to the isotropicFFCF parameters discussed above and summarized in Table II.Recall that the RISD contribution was not separated from this

data; the isotropic FFCF decays on a faster time scale than thestructural correlation functions alone. The fast time constant of6.6 ps in the isotropic data is significantly faster than the actualfastest structural spectral time scale of 20 ps. This serves as afurther evidence that RISD must be accounted for to correctlydetermine the time scales of structural fluctuations. Given thatthe scalar and vector components can be separated and thetime scales of these motions determined, it will be importantto use this analysis when comparing different experimentalsystems and conditions to assess which motions are affected.The vector and scalar SSD correlation functions, Eqs. (11)and (10), are quantities that could be computed directly fromMD trajectories (up to scale factors for the Stark couplingstrengths), which may allow precise assignment of themolecular motions contributing to the various experimentallydetermined time scales and fluctuation amplitudes.

One of the assumptions that is part of the RISD analysisis that the orientational relaxation of the small probe moleculeitself does not modify its surroundings, causing structuralspectral diffusion. This is a reasonable assumption given thesize of CO2 relative to Emim+ and NTf2

−, but there is also someexperimental evidence that supports this assumption. Thewobbling-in-a-cone orientational relaxation time constants(see Table I) are much faster than the fastest componentof the structural spectral diffusion (see Table III). If theseorientational motions induced changes in the structure of thesurrounding IL, then there would be faster components in theSSD than are observed. Such fast SSD dynamics would nothave identical time constants as the wobbling time constantsas these are two distinct correlation functions, but the lack offast SSD dynamics is consistent with CO2 reorientation notinducing extensive structural changes in the surrounding IL.

Optical heterodyne detected optical Kerr effect (OHD-OKE)81 measurements were used to measure the bulkorientational relaxation time of EmimNTf2. The time constantfor complete orientational randomization of the liquid is380 ps, which is much longer than the slowest componentof the SSD, 77 ps. This comparison shows that it is notnecessary to entirely randomize the bulk liquid structureto have complete spectral diffusion, that is, to completelysample all structures that give rise to the frequencies in theinhomogeneously broadened absorption line. Thus, the ILstructures relevant in determining a particular CO2 molecule’ssolvation environment may not depend on the bulk liquid’sglobal structure.

VI. CONCLUDING REMARKS

Here, we have described the dynamics of carbon dioxidein EmimNTf2, a promising room temperature ionic liquidsolvent for carbon capture. Reorientation is found to occur on3 time scales corresponding to two different wobbling motionswith time constants of 0.91 ± 0.03 and 8.3 ± 0.1 and a longtime complete reorientation time constant of 54 ± 1 ps.

The spectral diffusion of the carbon dioxide hasbeen partitioned into its structural and rotational dynamiccontributions using RISD analysis. Since the molecule isinversion-symmetric and hence has no permanent dipole


2016 18:00:44


moment, the previously existing RISD theory (first order Starkeffect) could not be used to describe the observed behavior.In fact, attempts to fit the CO2 data with the first order Starkeffect RISD theory gave poor results. New equations weredeveloped utilizing a second order Stark effect model forthe molecular interactions with the sample electric field. Thesecond order Stark effect provides the unique opportunityto distinguish contributions to spectral diffusion from liquidstructural reorganization processes that are scalar-coupled andvector-coupled. The experiments presented here revealed thereorientation and structural fluctuations of a carbon dioxideprobe in the ionic liquid EmimNTf2 and lay out the methodsand equations for analyzing similar systems. Future work willelucidate changes at different temperatures, chain lengths, andcations and in other ionic liquid families entirely, using thesedynamics as a benchmark. The highly detailed dynamicalinformation that these analysis methods extract from CO2 inIL systems will provide very precise comparisons and maylead to structure-dynamics-function relationships useful fordeveloping applications.

ACKNOWLEDGMENTS

This work was funded by the Division of ChemicalSciences, Geosciences, and Biosciences, Office of BasicEnergy Sciences of the U.S. Department of Energy throughGrant No. DE-FG03-84ER13251 (C.H.G., P.L.K., M.D.F.,and the shorter pulse ultrafast IR spectrometer.) This materialis also based upon work supported by the Air Force Officeof Scientific Research (AFOSR) under AFOSR Grant No.FA9550-16-1-0104 (S.A.Y., J.N., A.T., M.D.F., and pulseshaping ultrafast IR spectrometer). P.L.K., J.N., and A.T.acknowledge support from Stanford Graduate Fellowships.

APPENDIX: EVALUATION OF SECOND-ORDERSTARK EFFECT FFCF EXPRESSIONS

The time-dependent frequency shift caused by the secondorder Stark effect is written as,

∆ωE(t) = −12

E(t) · ∆α(t) · E(t), (A1)

where the net electric field and difference polarizability areboth time-dependent because they are expressed in the labframe (in which the polarizations of the four laser fieldsinvolved in a 2D IR experiment are fixed). Two additionalcoordinate systems will be useful in this work, the molecularframe and the sample field frame. The difference polarizabilityis best expressed in the molecular frame (x, y, z), where it isa time-independent diagonal rank 2 tensor,

∆α =

∆αx

∆αy

∆αz

= ∆αz

ax

ay

1

≡ ∆αz∆A. (A2)

We consider the z-axis to be the molecule’s majoraxis (along the vibrational displacement coordinate) andfocus on the normalized polarizability difference tensor∆A, chosen because we anticipate ∆αz to be the majorcontributor. The remaining terms are ax = ∆αx/∆αz anday = ∆αy/∆αz.

In the lab frame, the fluctuating electric field vector isE(t) = E(t)e(t), with E the field magnitude and e the unitvector specifying its direction. Both of these will evolve ina generally correlated manner due to the molecular motionsin the medium in which the vibrational probe is embedded.We choose to begin our calculation of the observables usingthe sample field frame (XF,YF, ZF), which is defined suchthat e(t) = kF, the unit vector along the ZF axis. The fieldframe considers the time-varying electric field to have amagnitude E(t) directed along the ZF axis. Let the Eulerangles (φF(t), θF(t), χF(t)) specify the orientation of themolecular frame relative to the field frame at the time t.With Φ(φ,θ, χ) the direction cosine matrix and Φ−1(φ,θ, χ)its inverse (transpose),78 the difference polarizability in thefield frame is,

∆α(t) = ∆αz [Φ (φF(t), θF(t), χF(t))]×∆A

Φ−1 (φF(t), θF(t), χF(t)) . (A3)

The virtue of the field frame is that the (generally complicated)scalar product between the ∆α(t) tensor and the E(t) vectorsappearing in Eq. (A1) reduces to a relatively simple form withonly 3 terms,

∆ωE(t) = −12

E2(t)kF · ∆α(t) · kF

= −∆αz

2E2(t)cos2(θF(t))+axsin2(θF(t)) cos2(φF(t))

+ aysin2(θF(t))sin2(φF(t)) . (A4)

The frequency shift can be cast in its most usefulmathematical form by making use of the sphericalharmonics Ym

l, assuming the standard conventions.78 Define

ΩF = (θF, φF) as a shorthand for the first two molecular frameto field frame transformation angles, which uniquely specifythe orientation of the molecular z-axis in the sample fieldframe. It is straightforward to show that,

cos2(θF) = 1/3 + 4/3π/5Y 0

2 (ΩF),sin2(θF)cos2(φF) = 1/3 − 2/3

π/5Y 0

2 (ΩF)

+

2π/15Y 2

2 (ΩF) + Y−22 (ΩF)

and

sin2(θF)sin2(φF) = 1/3 − 2/3π/5Y 0

2 (ΩF)

−

2π/15Y 2

2 (ΩF) + Y−22 (ΩF) .

Finally, defining an effective second order Stark tuning rateλ2 = −∆αz/2, we have,


2016 18:00:44


∆ωE(t) = λ2E2(t)(

1 + ax + ay

3+

(1 −

ax + ay

2

)× 4

3

π

5Y 0

2 (ΩF(t)))+ λ2E2(t)(ax−ay)

×

2π15

Y 2

2 (ΩF(t)) + Y−22 (ΩF(t)) . (A5)

The time-dependent second order Stark shift in Eq.(A5) can evolve due to changes in the field magnitudeE and in the orientation between field and molecularframes, ΩF. Furthermore, both changes in direction of theelectric field vector e and reorientation of the molecule’sz-axis cause changes in ΩF. The latter dynamics areresponsible for reorientation-induced spectral diffusion,while the field magnitude and direction changes are thesources of structural spectral diffusion within the presentmodel.

While the treatment has been quite general so far, wecan gain some additional simplification of the Stark shiftfunctional form with the approximation, ax = ay. The focusof this work is on the vibrations of CO2, a linear molecule,for which ax = ay clearly holds in Eq. (A2). Even insomewhat more complex molecules this can be a reasonableapproximation.82 For these cases,

∆ωE(t) = λ2E2(t)(

1 + 2ax

3+ (1 − ax) 4

3

π

5Y 0

2 (ΩF(t))).

(A6)

The final term of Eq. (A5) has been eliminated and the numberof parameters reduced.

The relevant ensemble averages for the PW-FFCFs wereperformed in the lab frame. Therefore, our final expressionfor the second order Stark frequency shift was written inthis frame. We proceed in a similar manner to the previoustreatment of the first order Stark case.32,33 Suppose the Eulerangles which transform the sample field frame into the labframe at time t are (α1, β1, γ1) and that the orientation of themolecular frame’s z-axis in the lab frame at the same time isΩ1. Then, using the Wigner rotation matrices Dl

mn,78 Eq. (A6)is transformed to,

∆ωE(t) = λ2E2(t)(

1 + 2ax

3+ (1 − ax) 4

3

π

5

×m

D2m0(α1, β1, γ1)Ym

2 (Ω1)+-. (A7)

As shown in (A7), the vibrational frequency of the probefor a given electric field magnitude is uniquely determinedby (1) how the probe is oriented in the lab frame (Ω1)and (2) how the field surrounding the probe is oriented inthe lab frame (α1, β1, γ1). These two orientations determinethe relative orientation between the probe and the field asappears in the second order Stark shift (A1), which is thesole source of spectral diffusion in the present model. Both ofthese orientations need to sample all angles in the isotropic

ensemble averaging when the PW-FFCFs are calculatedbelow.

Because the expectation value of spherical harmonicsvanishes for angular momentum l > 0, the average of thesecond order Stark shift in (A7) is,

⟨∆ωE(t)⟩ = λ21 + 2ax

3⟨E2⟩. (A8)

The transition has a net frequency shift (to new center ωE0 )

because of the sample generated field in addition to time-dependent fluctuations with zero average. Hence, we canwrite the frequency fluctuation as,

δω(t) = ω(t) − ωE0 = ∆ωE(t) − ⟨∆ωE(t)⟩

= λ21 + 2ax

3E2(t) − ⟨E2⟩ + λ2E2(t) (1 − ax)

× 43

π

5

m

D2m0(α1, β1, γ1)Ym

2 (Ω1), (A9)

which satisfies the standard requirement ⟨δω(t)⟩ = 0. Notethat the frequency fluctuation contains two terms: one thatdepends only on the time-varying electric field magnitudesquared and one that additionally depends on both the fieldframe and molecular frame orientations. Both terms depend onE2(t). The first term is indistinguishable from scalar sources offrequency fluctuations, e.g., due to density fluctuations that arenot caused by the Stark effect.32,80 For the second, “vector”term, the sample generated electric field orientation entersthrough the rotation matrices D2

m0, while the molecular probeorientation affects the spherical harmonics Ym

2 . Analogousto Eq. (A9), at time t = 0, we assume the field frame hasorientation (α0, β0, γ0) and the molecular frame has orientationΩ0, relative to the lab frame.

In calculating the required frequency time correlationfunctions, we will use the notation ⟨· · · ⟩p to representa polarization-weighted ensemble average, where the p-polarization configuration weighting factors are the sameas those used for the theory of RISD with the first order Starkeffect.33 Averages that do not depend on the molecular axisorientations Ω0 or Ω1 reduce to ordinary isotropic ensembleaverages, written as ⟨· · · ⟩. Two important assumptions arenecessary for evaluation of the PW-FFCFs for second orderStark coupling. First, we assume the vibrational transitiondipole moment is parallel to the molecular z-axis, whichcontains the major contribution to the difference polarizability.For the asymmetric stretching mode of carbon dioxide thisis clearly true. Therefore, the molecular frame orientation Ω1(or Ω0) in Eq. (A9) is also the transition dipole orientation,which enters the polarization-dependent weighting factors.33

Second, the sample’s electric field magnitude and directionwill be taken to be statistically independent of the vibrationalprobe’s orientational coordinates, which should be applicablefor a small molecule reorienting in a slowly evolving electricfield with contributions from many surrounding moleculesor molecular ions.67,83 The reorientation does not necessarilyoccur by free diffusion, and we will not explicitly considerany net alignment of the vibrational probe with the field.

We calculate the polarization-weighted FFCF as follows.First, we write,


2016 18:00:44


Cp(t) = ⟨δω(t)δω(0)⟩p= λ2

2

(1 + 2ax

3

)2 (E2(t) − ⟨E2⟩)(E2(0) − ⟨E2⟩)p

+ λ22(1 − ax)2 16π

45

E2(t)E2(0)

m,n

D2m0(α1, β1, γ1)D2

n0(α0, β0, γ0)Ym2 (Ω1)Y n

2 (Ω0)p

+ 2λ22(1 + 2ax)(1 − ax)

343

π

5

E2(t) E2(0) − ⟨E2⟩

m

D2m0(α1, β1, γ1)Ym

2 (Ω1)p

= λ22

(1 + 2ax

3

)2 (E2(t)E2(0) − ⟨E2⟩2

)+ λ2

2(1 − ax)2 16π45

m,n

E2(t)E2(0)D2

m0(α1, β1, γ1)D2n0(α0, β0, γ0) Ym

2 (Ω1)Y n2 (Ω0)p

+ 2λ22(1 + 2ax)(1 − ax)

343

π

5

m

E2(t) E2(0) − ⟨E2⟩ D2

m0(α1, β1, γ1) Ym2 (Ω1)p, (A10)

where the averages of statistically independent quantitieswere factored in the final equality. The final term containsaverages including the field frame orientation, (α1, β1, γ1), andmolecular frame orientation, Ω1, at a single time point, t. Inan isotropic medium, there are no special orientations, so thisterm (i.e., the summation as a whole) vanishes. The scalarterm (first line in the final equality) is in its simplest form.The correlation function of squared electric field magnitudesdecays to ⟨E2⟩2 at long times, so the scalar term decays to

zero. To further simplify the vector term (second term in finalequality), we use the orthogonality condition,

A1A0Dlm,n(α1, β1, γ1)Dk

m′,n′(α0, β0, γ0)∝ δm,−m′δn,−n′,

(A11)with A1 and A0 scalar functions evaluated at the same (time)points as the Euler angles indexed by 1 and 0, respectively.32

With Eq. (A11) and standard relations for Wigner rotationmatrices and spherical harmonics,78 Equation (A10) becomes,

Cp(t) = λ22

(1 + 2ax

3

)2 (E2(t)E2(0) − ⟨E2⟩2

)+ λ2

2(1 − ax)2 16π45

m

E2(t)E2(0)D2

m0(α1, β1, γ1)D2m0∗(α0, β0, γ0)

Ym

2 (Ω1)Ym∗2 (Ω0)p. (A12)

As mentioned above, every axis system is equivalent in anisotropic medium. The isotropic correlation function includingthe field frame orientation in Eq. (A12) should therefore beindependent of the m-index in the rotation matrices.84 Wedefine the vector correlation function of the field squared as,

H(t) = E2(t)E2(0)D2

m0(α1, β1, γ1)D2m0∗(α0, β0, γ0)

, (A13)

and factor H(t) out of the sum in Eq. (A12) to obtain,

Cp(t) = λ22

(1 + 2ax

3

)2 (E2(t)E2(0) − ⟨E2⟩2

)+ λ2

2(1 − ax)2 H(t)16π45

m

Ym

2 (Ω1)Ym∗2 (Ω0)p.

(A14)

In H(t), the m values can be 2, 1, 0, −1, or −2. Like an atomicd orbital, for which the energy (a scalar) depends on l but not

m, isotropic quantities involving D2m0 do not depend on the

value of m.To evaluate the sum of polarization weighted correlation

functions of spherical harmonics appearing in (A14), theprobability evolution Green’s function for the orientation ofthe molecular probe is expanded as a sum over all products ofspherical harmonics,

G(Ω1, t |Ω0) =∞l=0

lm=−l

cml (t)Yml (Ω1)Ym∗

l (Ω0).

The expansion coefficients are,

cml (t) = Cl(t) = ⟨Pl(µ(t) · µ(0))⟩ , (A15)

the lth order Legendre polynomial orientational correlationfunctions of the transition dipole unit vector µ.33,79,85 With theGreen’s function and transition dipole–laser field overlap


2016 18:00:44


factors79,86 written in terms of spherical harmonics, theintegrations over initial and final orientations necessary forevaluation of the polarization-weighted average in Eq. (A14)are straightforward to perform analytically.33,78 Using thesame analytical computation methods as detailed previouslyfor the first order Stark effect PW-FFCFs,33 we calculate theaverages in Eq. (A14) for the ⟨X X X X⟩ (polarization of thefirst two pulses parallel to the third pulse and echo pulse),⟨X XYY ⟩ (polarization of the first two pulses perpendicularto the third pulse and echo pulse), and isotropic (the firsttwo pulses at the magic angle to the third pulse and theecho, isotropically averaged) polarization configurations. It isimportant to note that the isotropically averaged configurationdoes not eliminate RISD contribution from the observeddynamics. We first write,

m

Ym

2 (Ω1)Ym∗2 (Ω0)p =

54π

R(2)p (t), (A16)

where R(l)p (t) in general is the normalized, polarization-

weighted, spherical harmonic correlation function sum fororder l (discussed below). The particular results of evaluatingR(2)p (t) are,

R(2)parallel(t) =

1175

28 + 215C2(t) + 72C4(t)

1 + 4/5C2(t), (A17a)

for the ⟨X X X X⟩ configuration,

R(2)perpendicular(t) =

1175

−14 + 155C2(t) − 36C4(t)

1 − 2/5C2(t), (A17b)

for the ⟨X XYY ⟩ configuration, and

R(2)isotropic(t) = C2(t), (A17c)

for the overall isotropic average or magic angle configuration.Some comments are warranted on the general form

of RISD factors such as those in Eqs. (A17). In third-order nonlinear optical experiments (such as 2D IR), whichinvolve four field-matter interactions, C2(t) can always appearin the normalization constant for the weighting factorsin a polarization-weighted average.33 Hence, the secondorder orientational correlation function can appear in thedenominator of any RISD factor. Beyond this, consider thecorrelation functions of spherical harmonics of order l with(un-normalized) transition dipole–laser field overlap factors.It is not difficult to show, using the selection rules for integralsof spherical harmonics,78 that only orientational correlationfunctions of order l, l + 2, and |l − 2| can appear in general.

The normalized R(2)p functions above, calculated for

particular polarizations p, appeared naturally in the evaluationof Eqs. (A14) and (A16). In general, such normalized,polarization-selectively averaged RISD functions can bedefined for any order l > 0. The interpretation of theR(l)p functions is straightforward: if a system’s frequency

fluctuation is given by δω(l)(t) = ∆ × Y 0l(ΩF(t)) (angle

expressed in the sample field frame, with ∆ the characteristicfluctuation amplitude) and all changes in ΩF are due tomolecular reorientation, then it can be shown that the

PW-FFCF is,

⟨δω(l)(t)δω(l)(0)⟩p = ∆2

4πR(l)p (t), (A18)

using the orthonormality property of the spherical harmonics.We have made the general definition,

R(l)p (t) =

m

Yml(Ω1)Ym∗

l(Ω0)p

m

Yml(Ω1)2

p

, (A19)

with angles written in the lab frame, and identify Eq. (A16) asa special case of (A19). For this class of models, all spectraldiffusion is due to RISD from the orientational motion of theprobe.33 The normalized results from Eqs. (A18) and (A19)appear as multiplicative factors in more complete models ofthe FFCF that include structural spectral diffusion (e.g., timevariation of ∆).32

The normalized polarization-dependent decay functionsfor l = 2, which appeared in Eqs. (A16) and (A18), arepresented in Figure 8 for the three cases in Eqs. (A17) using apurely diffusive reorientation model and no structural spectraldiffusion (see Section IV). The R(1)

p (t) factors for the samepolarization configurations as in Eqs. (A17), using a first orderStark model (Y 0

1 potential in Eq. (A18)), have been reportedpreviously.32,33 The decay time constants for the first orderStark effect are approximately a factor of 3 times longer thanthose presented here for the Y 0

2 potential in Eqs. (A17) andFigure 8.

We return now to the full second order Stark model, withboth RISD and SSD, which resulted in Eq. (A14). Substitutionof the key intermediate result, Eq. (A16) with Eqs. (A17), givesthe complete expression for the PW-FFCF,

Cp(t) = λ22

(1 + 2ax

3

)2 (E2(t)E2(0) − ⟨E2⟩2

)+

4λ22(1 − ax)2

9H(t)R(2)

p (t), (A20)

with the H(t) vector field-squared correlation function definedin Eq. (A13). In the full correlation function, the normalizedRISD functions discussed above are modified by the H(t)factor to decay more quickly due to the fluctuations of thesample’s electric field caused by SSD. Added to this term isthe scalar correlation function of the field squared, with itslong-time offset subtracted. If the vibrational transition dipoleis fixed in space (no molecular reorientation), the R(2)

p factoris fixed at unity and the result is simply the sum of bothcorrelation functions of the electric field squared.

In the analysis of experimental spectral diffusion data,such as the decay of the CLS,63,64 the quantities λ2, ax,the initial values of the scalar and vector field squaredcorrelation functions in Eq. (A20) are generally unknown.For the restrictive assumptions of complete independenceof electric field orientation and magnitude, as well asGaussian fluctuation dynamics of the magnitudes, the initialvalues of the two structural correlation functions can bedetermined, and they can both be written in terms of the fieldmagnitude correlation function ⟨E(t)E(0)⟩ and the secondLegendre polynomial orientational correlation function ofthe field unit vector. In this case, both ax and λ2

2⟨E2⟩2, in


2016 18:00:44


principle, can be determined from the data. Additionally,spectral diffusion in systems with both first and secondorder Stark effects could be analyzed with relatively fewfree parameters, allowing determination of the relative Starkcoupling constants. However, the decoupling of a liquid orother sample’s electric field orientation and magnitude isunlikely to hold in practice.

The restrictive assumptions above that simplify thePW-FFCF do not apply for the experimental investigationpresented here. In general, we can write the polarization-weighted frequency-frequency correlation function in Eq.(A20) in the most condensed form as,

Cp(t) = ⟨δω(t)δω(0)⟩p = Fs(t) + Fv(t)R(2)p (t). (A21)

The R(2)p (t) factors are given in Equations (A17). Here,

Fs(t) and Fv(t) (Eqs. (10) and (11) in the main text) arethe correlation functions for the structural dynamics forvibrational probes that are scalar and vector coupled to themedium electric field, respectively. Sums of exponentials havebeen used extensively as models of FFCFs and are appropriatefor these structural correlation functions.

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