Heterodyned 3D IR spectroscopy

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Chemical Physics 341 (2007) 95–105

Heterodyned 3D IR spectroscopy

Feng Ding, Martin T. Zanni *

Department of Chemistry, University of Wisconsin-Madison, Madison, WI 53711, United States

Received 5 December 2006; accepted 7 June 2007Available online 15 June 2007

Abstract

Heterodyned 3D IR spectra are reported for three model compounds: W(CO)6, azide in an ionic glass, and an iridium metal dicar-bonyl. The 3D spectra are generated using a fifth-order pulse sequence whose signal is Fourier transformed along three separate coher-ence evolution times. Because the pulse sequence rephases static vibrational frequencies into a photon echo, the 3D IR spectra exhibitspherical features for homogeneously broadened vibrational modes and elongated cigar-like features for inhomogeneously broadenedsystems. The 3D spectra can be visualized as a series of planes appearing at the overtone and combination band frequencies. Each planecontains a 2D correlation map. In a typical 2D IR spectrum, both diagonal peaks and cross peaks appear in the same correlation map,but in 3D IR spectroscopy the diagonal and cross peaks are separated into different 2D planes, greatly improving the spectral resolution.Simulations of the 3D IR spectra are also presented and future improvements to the technique are discussed. This work demonstratesthat increased control over vibrational coherences enhances the resolution and information content of multidimensional infraredspectroscopies.� 2007 Elsevier B.V. All rights reserved.

Keywords: 2D IR; 3D IR; Ultrafast; Infrared spectroscopy

1. Introduction

Infrared spectroscopy is an old and powerful technique.For decades it has been used to identify molecular com-pounds, study peptide and protein structures, recognizeDNA secondary and tertiary conformations, and monitorkinetic processes such as chemical reactions [1,2]. The use-fulness of infrared spectroscopy stems from its sensitivityto chemical functional groups, environments, and anhar-monic couplings that are responsible for the chemicallycharacteristic infrared band frequencies and linewidths.In more recent times, infrared spectroscopy has begun toevolve beyond simple linear absorption [3–16]. Using fem-tosecond phase-locked pulses and heterodyned signals aswas done in the optical regime [17,18], new non-linear tech-niques are providing additional and more quantitativeinformation than can be obtained by linear spectroscopy.

0301-0104/$ - see front matter � 2007 Elsevier B.V. All rights reserved.

doi:10.1016/j.chemphys.2007.06.010

* Corresponding author.E-mail address: [email protected] (M.T. Zanni).

For example, 2D IR spectra can be generated from a pulsesequence of three femtosecond pulses (Fig. 1a). Typicalspectra exhibit cross peaks with splittings that provideoff-diagonal anharmonicities and intensities that give theangles between two coupled transition dipoles [5,6]. Theseare insightful quantities because their values are closelyrelated to molecular structure [19–21]. Furthermore, theanti-diagonal widths of the 2D IR lineshapes can be usedto measure the time-evolution of the vibrational eigenstatefrequencies, thereby uncovering the environmental dynam-ics that create the infrared linewidths [22–25]. Besides pro-viding this additional information, 2D IR spectroscopyalso has enhanced spectral resolution. Linear infrared spec-tra are often congested by overlapping resonances. Byspreading a linear infrared spectrum into a second dimen-sion, 2D IR spectra improve the resolution and thus arebetter suited for identifying compounds and monitoringstructures.

With the obvious improvements of 2D IR over linearFTIR, extending infrared spectroscopy into a third dimen-sion should have additional benefits. In fact, there have

mailto:[email protected]

Fig. 1. (a) Third and (b) fifth-order pulse sequences used to collect 2D IR and 3D IR spectra, respectively. kn are the wavevectors and the relative time-delays are tn. Coherence evolution times are represented by oscillatory shapes whose relative frequencies illustrate the difference between 1Q and 2Qcoherences. Population or ‘‘waiting times’’ are shown as exponential decays. The emitted field (solid) is time-resolved with the local oscillator (LO) pulse.The seven Feynman diagrams that describe the fifth-order responses for a (c) single oscillator and (d) two-oscillator systems, respectively.

96 F. Ding, M.T. Zanni / Chemical Physics 341 (2007) 95–105

been several theoretical articles proposing various 3D IRtechniques [25–27]. 3D IR spectroscopies will retain theadvantages inherent to 2D IR spectroscopy, but can alsobe used to probe non-Gaussian frequency fluctuations[25,27], measure higher lying vibrational states [11,28], bet-ter characterize the anharmonic curvatures of the potentialsurface [29], correlate multiple eigenstates, and furtherimprove the spectral resolution. In principle, any plot ofthree variables is 3D, but only a specific type of 3D spec-troscopy can accomplish all of the above. For example, a3D plot can be made from a series of third-order 2D IRspectra collected as a function of the ‘‘waiting’’ time t2

(Fig. 1a), which can either evolve as a coherence or a pop-ulation depending on the phase matching conditions andthe time-ordering of the pulses [30–33]. This type of 3Dplot will not probe higher lying vibrational states nornon-Gaussian frequency fluctuations because there are stillonly three interactions between the laser pulses and thesample. In the applications described here, we define 3DIR spectroscopy as the result of a 3D Fourier transformof a signal arising from at least a fifth-order resonant inter-action between the infrared electric fields and the sample.In the case where each pulse interacts with the sample once,this requires that at least five pulses generate the 3D IR sig-nal, such as the pulse sequence shown in Fig. 1b. With five-pulse interactions, higher vibrational states can be accessedthan with three-pulse interactions and there exists up totwo waiting times (t2 and t5). 3D IR spectra need not be

collected solely in the time-domain, but can be mea-sured using frequency scanned pulses like was done withcoherent anti-stokes Raman spectroscopy or in the firstimplementation of 2D IR spectroscopy that used hole-burning [3,34].

In this paper, we report heterodyned 3D IR spectra ofthree model systems: W(CO)6 in hexane, azide in an ionicglass, and the iridium dicarbonyl: dicarbonylacetylaceto-nato iridium(I) in a mixture of hexane and chloroform thatwe refer to as Ir(CO)2. These three systems were chosen asthe first systems to study using 3D IR spectroscopy becausetheir carbonyl vibrational eigenstates, vibrational dynamicsand couplings are well-understood, and thus the 3D IRspectra can be simulated from known parameters.W(CO)6 has three degenerate carbonyl anti-symmetricstretches and homogeneous vibrational dynamics in hexane[35,36]. In contrast, the anti-symmetric stretch of azide isextremely inhomogeneously broadened because of stronginteractions with the ions in the glasseous solvent [11,28].The differences in vibrational dynamics lead to very differ-ent 3D shapes for azide and W(CO)6. Ir(CO)2 is homoge-neously broadened like W(CO)6, but has two IR-activecarbonyl stretches that are coupled [29]. The coupling givesrise to cross peaks in a typical 2D IR spectrum. In the 3DIR spectrum reported here, the cross peaks appear in differ-ent planes from the diagonal peaks, demonstrating theenhanced spectral resolution of 3D versus 2D IRspectroscopy.

F. Ding, M.T. Zanni / Chemical Physics 341 (2007) 95–105 97

Heterodyned 3D IR spectroscopy requires pulse trainswith highly accurate phase control. Phase drift during col-lection of 2D IR spectra causes aberrations which distortspeak shapes and degrades the quality of the spectra. 2D IRspectrometers have been stabilized both passively andactively [37–40]. Passive stabilization includes using diffrac-tive optics and active stabilization has been accomplishedwith HeNe tracer beams and piezo electric mirrors to mea-sure and compensate for pathlength drift. However, bothof these methods only phase stabilize the relative phaseof pairs of pulses. The phase requirement for the 3D IRspectra reported here are more stringent because the com-bined phase of all the pulses needs to be stabilized and thespectra take longer to collect. To meet this requirement, wefollow a procedure that we recently published to periodi-cally measure a reference phase during the experiment thatis used to correct the time-domain data [41]. The result ishigh-quality 3D IR spectra.

The spectra presented here are the simplest examples of3D IR spectroscopy. 3D IR spectra contain both real andimaginary components. We report absolute value spectraas well as real spectra. The real spectra have better resolu-tion than the absolute value spectra and are more informa-tive because they also contain phase information about thepeaks, but like 2D IR spectra, require that the phase be cal-ibrated. Calibration of 2D IR spectra is not trivial [42] and3D IR spectra no less so. Future work will address methodsfor phasing 3D IR spectra. In this paper, we phase thespectra by comparison to simulations. It should also bepossible to generate absorptive spectra by summing multi-ple 3D IR spectra that are properly phased like is done in2D IR spectroscopy [43]. But even with these improve-ments yet to be made, the enhanced resolution of 3D IRover 2D IR spectroscopy is apparent.

2. Materials and methods

Femtosecond mid-IR pulses are generated by a home-built OPA that is pumped by a commercial Ti: Sapphirelaser system which generates <100 fs pulses at a repetitionrate of 1 kHz [11]. The infrared pulses are then split intofour beams, three of which serve as excitation pulses(�500 nJ each, 150 cm�1 bandwidth) and the fourth asthe local oscillator (<5 nJ). The pulses are tuned to the fun-damental frequency of the sample. The three excitationbeams intersect the sample in an equilateral triangle config-uration with wavevectors k1, k2, and k3 that arrive at thesample in that order. The time-dependent electric field ismeasured in the ks = �2k1 + k2 + 2k3 direction using bal-anced heterodyne detection with a fourth local oscillatorpulse called kLO. The first and third pulses both interactwith the sample twice, creating a fifth-order pulse sequence,analogous to using a pump-probe or two-pulse photonecho geometry to measure a third-order signal. The timedelays between the five pulses are labeled t1–t5 witht1 = t4 = 0 (Fig. 1b). To collect 3D IR spectra, the t2, t3

and t5 time-delays are incremented and 3D Fourier trans-

forms are performed. The delay times are controlled usingpairs of anti-reflection coated ZnSe wedges cut at 5 ± 0.2�[41]. A pair is inserted into each of the k1, k3 and kLO beampaths.

The 3D IR spectra were measured for t5 = 0–3950 fs in89 fs steps, t2 = 0–2039 fs in 38 fs steps, and t3 = 0–3318 fs in 79 fs steps for W(CO)6 in hexane. For azide inglass, the step sizes were t5 = 0–3822 fs in 89 fs steps,t2 = 0–1784 fs in 38 fs steps, and t3 = 0–3620 fs in 76 fssteps. For Ir(CO)2, t5 = 0–3822 fs in 89 fs steps, t2 = 0–1784 fs in 38 fs steps, and t3 = 0–3620 fs in 77 fs. At eacht3 time, unwanted phase drift was monitored by movingt5 and t3 to zero and rescanning t2. Each 1D scan in the3D data set was scaled and phase adjusted using the t2 ref-erence scan following our previously published methods[41]. At these step sizes, each 3D IR spectrum consistedof about 100,000 data points and took approximately 6 hto collect. Window functions that were used previously infifth-order 2D IR spectra[29] were applied to the 3D IRspectra of Ir(CO)2 for reasons that are discussed below.The resolution of spectra depends on the scan time for eachaxis. With the scan times used here (2039–3950 fs), the fre-quency resolution falls between 2.6 and 1.4 cm�1.

The 3D IR spectra are taken for three different samples:W(CO)6 in hexane, azide in glass and Ir(CO)2 in hexaneand chloroform (1:4). The azide, W(CO)6 and Ir(CO)2 werepurchased from Aldrich. W(CO)6 in hexane and Ir(CO)2 inhexane and chloroform (1:4) are held between two CaF2

plates with a spacing of 56 lm at a concentration corre-sponding to an optical density of 0.3 and 0.4, respectively.The azide in glass sample is made of �5 mM azide solvatedin the ionic glass 3KNO3:2Ca(NO3)2 (Tg = 60 �C), which isheld between two CaF2 plates separated by �50 lm for anOD of 0.3. No signal is observed in the fifth-order phasematching direction for cells containing only the solvent.

3. Experimental 3D IR spectra

In this section, we report the 3D IR spectra of W(CO)6,azide in an ionic glass and Ir(CO)2. The spectra have beencollected using a two-quantum (2Q) photon echo pulsesequence that we have used previously to generate fifth-order 2D IR spectra [11,28,29]. The pulse sequence isshown schematically in Fig. 1b and described in detail inSection 8 below. The oscillatory and exponential decay fea-tures in Fig. 1b represent coherence and population evolu-tion times. In this sequence, the first two pulse interactionscreate coherences between the ground vibrational state(t = 0) and the overtone and combination (t = 2) bands.The third pulse converts this 2Q coherence into a 1Q coher-ence by converting either the ground state or the overtone/combination band to a fundamental, thus roughly halvingthe frequency of the oscillations. The fourth pulse stops thecoherence by creating a population followed by the finalpulse that creates a second 1Q coherence with oppositephase, causing the photon echo. Thus, a Fourier transformalong the coherence evolution times t2, t3 and t5 produces a

Table 1Peak labels and positions of W(CO)6 and azide

Labels W(CO)6 Azide FeynmanpathsPeak positions

of (±3.0 cm�1)(presented as(x2,x3,x5))

Peak positionsof (±3.0 cm�1)(presented as(x2,x3,x5))

A (3952.0,1983.0,1983.0) (4168.0,2097.0,2097.0) R1, R2

B (3952.0,1983.0,1969.0) (4168.0,2097.0,2071.0) R4

C (3952.0,1969.0,1983.0) (4168.0,2071.0,2097.0) R3

D (3952.0,1969.0,1969.0) (4168.0,2071.0,2071.0) R5, R6

E (3952.0,1969.0,1954.0) (4168.0,2071.0,2045.0) R7

Listed in the fourth column are the Feynman Paths that contribute to thepeaks, and these Feyman Paths are presented in Fig. 1c.


3D IR spectrum that correlates the overtone and combina-tion bands along one axis to the fundamentals along theother two. In the figures that follow, the 3D IR spectraare plotted with volumes that represent a surface at a givensignal intensity reported in the figure captions. 2D cutsthrough the 3D IR spectra are also shown, which have20 contour lines from minimum to maximum unless other-wise specified.

4. W(CO)6 in hexane

Shown in Fig. 2a is the absolute value 3D IR spectrumof W(CO)6 in hexane. It consists of five spheres that lie in aplane at x2 = 3952 cm�1, which is the first overtone fre-quency of W(CO)6. The surfaces of these spheres are plot-ted at intensities equal to 30% of the largest intensity in thespectrum. Thus, the spheres have different sizes dependingon the chosen value of the plotted intensity. Three of thespheres are negative (A, C and E) and the other two arepositive (B and D). The phases are not apparent in theabsolute value spectrum, but do appear in the real part(not shown). Shown in Fig. 2b is a planar cut throughthe five spheres. From this figure it is clear that the spheresappear along two lines with frequencies x3 = 1983 cm�1

and 1969 cm�1, which are the fundamental and the funda-mental shifted by the overtone anharmonicity of W(CO)6.Along x5 the peaks are separated by the first and secondanharmonicities with peak A appearing at the fundamentalfrequency along both the x3 and x5 axes. Thus, this pulse

Fig. 2. (a) Experimental fifth-order 3D IR spectrum of W(CO)6 in hexane. (balong x3 axis. (d) Simulated fifth-order 3D IR spectrum of W(CO)6 in hexane. Tof 30% of the maximum intensity. All spectra are absolute value plots.

sequence generates a 3D IR spectrum that correlates thefrequencies of the overtone eigenstates along the x2 axisto the fundamentals along the x3 and x5 axes. This datawas reported previously as a demonstration of our passivephase correction procedure [41]. Listed in Table 1 are thefrequencies for each peak, and a diagram of the eigenstatesare shown in Fig. 5a.

5. Azide in an ionic glass

Fig. 3a and b show the absolute value and real 3D IRspectra for the antisymmetric stretch of azide in an ionicglass. Similar to W(CO)6, five volumes appear in the spec-tra at the fundamental and overtone frequencies. However,the volumes are now elongated parallel to the cubic

) 2D IR plane at 3952 cm�1 along x2-axis. (c) 2D IR plane at 1969 cm�1

he experimental and simulated spectra are drawn with an isosurface value

Fig. 3. Experimental (a) absolute value and (b) real fifth-order 3D IR spectrum of azide in glass. Simulated (c) absolute value and (d) real fifth-order 3DIR spectrum of azide in glass. The absolute value spectra are drawn with an isosurface value of 40% of the maximum intensity. Blue volumes representpositive intensities while red is for negative. In the real spectra, the positive volumes use an isosurface value of 54%, and the negative volumes are at 35%.(For interpretation of the references in colour in this figure legend, the reader is referred to the web version of this article.)


‘‘diagonal’’, which we define as 2x2 = x3 = x5. Since the3D IR spectra reported here are collected with a rephasingpulse sequence, a photon echo is generated. Thus, the spec-tra are line narrowed, and the elongation is a consequenceof the fact that the system is inhomogeneously broadened.The fifth volume does not appear in these plots because it ismuch weaker than the others, but it is observed at lowerisosurface values (not shown). Listed in Table 1 are theobserved frequencies for each peak, and a diagram of theeigenstates are shown in Fig. 5a.

6. Ir(CO)2 in chloroform and hexane (1:4)

Fig. 4a shows the 3D IR spectrum of the metal dicar-bonyl Ir(CO)2. Ir(CO)2 has two IR active carbonyl modes,the symmetric and asymmetric stretches. The 3D pulsesequence accesses vibrational states up to the second over-tone and combination bands, which we refer to as the 3Qstates (See Fig. 5b for an energy level diagram), but thespectra can be mostly understood by considering just thefundamentals, the first overtone and combination bands.The symmetrical and anti-symmetrical stretch modes eachproduces five peaks along the cubic diagonal that resemblethe peaks of W(CO)6. We refer to these as diagonal peaks.Peaks off of the diagonal also appear in the spectrum,which are the cross peaks. In the 2D version of this fifth-order spectrum, all of the cross peaks lie in the same plane[29]. In the 3D version, the cross peaks are separated into

three planes at x2 = 3987.5, 4049.0 and 4137.5 cm�1 thatcorrespond to the first overtone and combination bands.

Shown in Fig. 4b are 2D cuts through the 3D IR spectraat these three frequencies. In the top cut atx2 = 4137.5 cm�1, the five diagonal peaks (A–E) appearalong the diagonal near the fundamental symmetrical fre-quency x = 2074.0 cm�1. In addition, cross peaks appearin the upper left corner of the 2D cut (G and F), one ofwhich correlates the asymmetric and symmetric stretches(F). In the middle cut at x2 = 4049 cm�1, only cross peaksappear in the data; these peaks include both asymmetricand symmetric stretch transition dipoles in their pathways(I-Q). Finally, the bottom cut at x2 = 3987.5 cm�1 isshown. This cut is similar to the top cut, but the crosspeaks appear in the opposite quadrant. Table 2 lists the fre-quencies for each observed peak.

7. Theory and simulations

The theoretical formalism necessary to simulate the 3DIR spectra has been presented in our earlier papers[11,28,29]. Our earlier papers focused on fifth-order 2DIR spectra, so we rewrite the formalism here in the contextof 3D IR spectroscopy. Except for W(CO)6, the eigenstateenergies and vibrational dynamics that dictate the peakpositions and their shapes have been measured previouslyand are used in the 3D IR simulations presented here[11,28,29]. We divide the Section 8 into two subsections

Fig. 4. (a) Experimental fifth-order 3D-IR spectrum of Ir(CO)2. The spectrum is drawn with an isosurface value of 5% of the maximum intensity. (b) 2Dcuts through the 3D IR spectra at (i) 3987.5 (ii) 4049.0 and (iii) 4137.5 cm�1 along the x2 axis. The cut at 4049.0 cm�1 is drawn with 10 contour lines fromminimum to maximum and the intensity is multiplied by 5. (c) The same three 2D cuts presented in one plane. (d) Simulations of the 2D cuts presented in(b).


representing the two classes of molecules studied here: asingle oscillator system and a coupled oscillator system.(W(CO)6 is strictly not a single oscillator system. However,the solvent/solute interactions are weak enough that itsthree asymmetric stretches are degenerate and indistin-guishable for the purposes of this paper.)

7.1. Single oscillator system: W(CO)6 and azide

The time-dependent polarization P(t) created by thelaser pulses incident on the sample can be expressed as aperturbative expansion with respect to the electric field:

P totðtÞ ¼ P ð1ÞðtÞ þ P ð3ÞðtÞ þ P ð5ÞðtÞ þ � � � / iRð1ÞðtÞEðtÞþ i3Rð3ÞðtÞE3ðtÞ þ i5Rð5ÞðtÞE5ðtÞ þ � � � ð1Þ

where Rn(t) is the time-dependent nth-order response of thesystem to the laser field En(t). t is an absolute time variable.In pulsed infrared spectroscopy, En(t) consists of a series ofinfrared pulses with wavevectors kn and separated by rela-tive time-delays tn, e.g. E5(t1, t2, t3, t4, t5). Most 2D IR spec-troscopies are collected using the third-order responseR(3)(t), but the 3D IR spectra must be generated from atleast a fifth-order response R(5)(t) according to the defini-tion in the Introduction. There are many fifth-order re-

sponses that can be generally classified into a series ofpulse sequences, each one of which generates a unique typeof 3D IR spectrum. Time delays, phase matching, and/orphase cycling [18,27,44,45] can be used to uniquely measurethe desired 3D IR spectrum.

The 3D IR spectra reported here are generated from afifth-order 2Q rephasing pulse sequence that is the sum ofseven individual responses from the perturbative expansionof the time-dependent density matrix Rð5Þðt1; t2; t3; t4; t5Þ ¼P

nRð5Þn ðt1; t2; t3; t4; t5Þ. The seven responses can be repre-sented using Feynman paths shown in Fig. 1c. The mea-sured signal is generated by heterodyning the emittedelectric field, which is proportional to Sð5Þðt5; t1; t2;t3; t4Þ / i6

PnRð5Þn ðt1; t2; t3; t4; t5Þ and it is given by

Sð5Þðt5; t1; t2; t3; t4Þ¼�1

7eix10t1 eið2x10�Dð1ÞÞt2 ½2F ð5Þð01j02j01j00j10Þ

þF ð5Þð01j02j12j11j10Þe�iDð1Þt3

þF ð5Þð01j02j01j11j21ÞeiDð1Þt5

þ2F ð5Þð01j02j12j22j21Þe�iDð1Þt3 eiDð1Þt5

þF ð5Þð01j02j12j22j32Þe�iDð1Þt3 eiðDð1ÞþDð2ÞÞt5 �� eix10t3 e�ix10t5 ð2Þ

Fig. 5. Energy level diagrams of (a) a single oscillator system: W(CO)6 or azide in glass, and (b) a coupled oscillator system: Ir(CO)2. The notation used torefer to the eigenstate modes are given on the left and the experimentally determined eigenstate energies are given on the right.

Table 2Peak labels and positions of Ir(CO)2

Labels Positions of the peaks (±3.0 cm�1) (presented as (x2, x3, x5)) Feynman paths

A (3987.5, 2000.0, 2000.0), (4137.5, 2074.0, 2074.0) R1, R2

B (3987.5, 2000.0, 1987.5), (4137.5, 2074.0, 2063.5) R4

C (3987.5, 1987.5, 2000.0), (4137.5, 2063.5, 2074.0) R3

D (3987.5, 1987.5, 1987.5), (4137.5, 2063.5, 2063.5) R5, R6

E (3987.5,1987.5, 1975.0), (4137.5,2063.5, 2046.0) R7

F (3987.5, 2000.0, 2074.0), (4137.5,2074.0, 2000.0) R1, R2

G (3987.5, 2000.0, 2049.0), (4137.5,2074.0, 1975.0) R4

H (3987.5, 1975.0, 2049.0) R5, R6

I (4049.0, 2000.0, 2000.0), (4049.0, 2074.0, 2074.0) R1, R2

J (4049.0, 2000.0, 1987.5), (4049.0, 2074.0, 2063.5) R4

K (4049.0, 2049.0, 2049.0) R5, R6

L (4049.0, 2000.0, 2074.0), (4049.0, 2074.0, 2000.0) R1, R2

M (4049.0, 2000.0, 2049.0), (4049.0, 2074.0, 1975.0) R4

N (4049.0, 1975.0, 2074.0), (4049.0, 2049.0, 2000.0) R3

O (4049.0, 1975.0, 2063.5), (4049.0, 2049.0, 1987.5) R5, R6

P (4049.0, 1975.0, 2049.0) R5, R6

Q (4049.0, 2049.0, 1953.0) R7

Listed in the third column are the Feynman Paths that contribute to the peaks, and these Feynman Paths are presented in Fig. 1d.


when all of the laser pulses have identical polarizations.F(5)(ij|kl|mn|op|qr) describes the dynamical response of the

system and contains the transition dipole amplitude factors(Fig. 1c). The coefficient 1/7 arises from the orientational

Table 3Dynamical parameters used in the simulations

Homogeneous time (fs) Inhomogeneous width (cm�1)

W(CO)6 6500 ± 500 1.0 ± 0.5Azide 3900 ± 500 15.0 ± 2.0Ir(CO)2 2200 ± 500 1.0 ± 0.5


response, which is assumed to be separable from the vibra-tional degrees of freedom. S(5)(t5; t1, t2, t3, t4) oscillates withthe three frequencies x10, x21 = x10 � D(1), and x32 =x10 � D(1) � D(2), where x10 is the fundamental frequency,D(1) is the anharmonic shift between t = 0 � 1 andt = 1 � 2 transitions, and D(2) is the second anharmonicshift defined as the difference in frequency between thet = 1 � 2 and t = 2 � 3 transitions. Fig. 5a is the energylevel diagram of a single oscillator system. For a Morseoscillator, D(1) = D(2). The signal is collected in the time-do-main as a function of t2, t3 and t5, and the 3D IR spectrumgenerated from a 3D Fourier transform.

The 3D lineshapes of the peaks in the 3D IR spectra aredetermined by the terms F(5)(ij|kl|mn|op|qr) that account forfrequency fluctuations and population relaxation. The fre-quency fluctuations are generally described with a fre-quency–frequency correlation function, but if thedynamics lie in the Bloch limit, the correlation functioncan be approximated by a homogeneous lifetime and aninhomogeneous distribution [46]. Azide in glass is well-described by Bloch dynamics [11,28] as is W(CO)6 in hex-ane which has a nearly ideal Lorentzian lineshape. As aresult, we write the fifth-order dynamical response as

F ð5Þð01j02j01j00j10Þ ¼ F ð5Þð01j02j01j11j10Þ¼ F ð5Þð01j02j12j11j10Þ

¼ 2jl10j6 exp � 1

T 2

ðt1 þ 4t2 þ t3 þ t5Þ�

� 1

2r2

10ðt1 þ 2t2 þ t3 � t5Þ2�

ð3Þ

F ð5Þð01j02j01j11j21Þ ¼ F ð5Þð01j02j12j11j21Þ¼ F ð5Þð01j02j12j22j21Þ¼ �2F ð5Þð01j02j01j00j10Þ ð4Þ

F ð5Þð01j02j12j22j32Þ ¼ 3F ð5Þð01j02j01j00j10Þ ð5Þwhere T2 is the homogeneous dephasing time which in-cludes population relaxation, and r10 is the width of theGaussian inhomogeneous distributions. Their relativeintensities are set by the product of the transition dipoleslmn between the vibrational states m and n in each stepof the Feynman path. Eqs. (2)–(5) uses the harmonicapproximation for l21 ¼

ffiffiffi2p

l10 and l32 ¼ffiffiffi3p

l10, whichpredicts that the relative intensities of the peaks are�1:2:�1 for the x10, x21, and x32 peaks respectively.

The simulated 3D IR spectra of W(CO)6 is shown inFig. 2d. Each of the five spheres can be attributed to oneor two of the seven Feynman paths from Fig. 1c, and theassignments are made in Table 1. All seven paths oscillatewith the overtone frequency (x02 = 3952 cm�1) during t2,which is why all five spheres lie in the same plane. Threeof the Feynman diagrams oscillate with x01 and four withx21 during t3, which is why the five peaks lie along tworows. The lowest frequency peak E corresponds to a x23

transition. According to Eq. (2), the peaks should have rel-ative intensities of �2:2:�1:4:�3 for peaks A:B:C:D:E, but

the experimental intensities are instead �2.8:1.9:�1:2.7:�1.After taking into account the laser pulse bandwidth, wefind that these intensities can be simulated with roughlyequal transition dipole strengths between the first threevibrational levels (e.g. l21 = 1.1 l10 = l32). In principle,electrical anharmonicity could cause large deviations fromthe scaling laws predicted by the harmonic approximationfor transition dipoles used in Eq. (2), but it seems morelikely that the vibrational transitions are partially saturatedsince the transition dipole of W(CO)6 is extremely large(�1 D) [47,48]. Experiments at reduced pulse intensitiescould test this hypothesis, although we do not investigatethis effect further. We also note that when this pulsesequence was used previously to generate a fifth-order sig-nal and study possible cascading effects [11,28,29], the t3

time-delay was not incremented. As a result, the 3D IRspectrum was collapsed into 2D and not all five peakscould be separately observed. Thus, the 3D IR experimentresolves more of the Feynman pathways.

The simulated 3D IR spectrum of azide in the ionic glassis shown in Fig. 3c and d. Like W(CO)6, five volumesappear, although they are stretched along the cube diago-nal. The elongated volumes reflect the fact that azide inglass is extremely inhomogeneously broadened. Previouswork found that strong solute/solvent interactions betweenazide and the ions in the glass caused substantial amountsof linewidth broadening (17 cm�1) with little spectral diffu-sion because of the essentially static glasseous environment[11,28]. In contrast, the homogeneous linewidth is only1.6 cm�1. Since this 3D IR spectrum was generated froma photon echo pulse sequence, the inhomogeneous broad-ening is removed along the x3 and x5 anti-diagonals to givethe homogeneous linewidth. A 3D IR spectrum collectedwith a non-rephasing pulse sequence would be spherical.Other rephasing pulse sequences would narrow the vol-umes along the other directions and proper summationsof 3D spectra would give absorptive peaks [25].

The simulated spectra agree with the experimental datavery well. The simulations use parameters within the errorbars of our previous publications [11,28,29] and are listedin Table 3. Our previous work on azide in this glass sug-gested that D(1) 5 D(2) and that the D(2) anharmonicitywas partially correlated to the frequency fluctuations ofx01. Unfortunately, the frequency resolution of these datais insufficient to investigate these observations further.

7.2. Two oscillator system: Ir(CO)2

The 3D IR spectra of multiple oscillator systems can besimulated from a generalized treatment of the one oscilla-


tor system [29]. Shown in Fig. 1d are seven Feynman dia-grams with indices i, j, k, and l representing the 1Q, 2Qand 3Q manifolds of the vibrational eigenstates (seeFig. 5b). In a normal mode basis consisting of a symmetricand asymmetric stretch (s and a), the 1Q eigenstates areindexed by i and j which have fundamental frequenciesEa and Es. The 2Q states are indexed by k and includethe first overtone and combination band states. Thesestates have energies Ea+a = 2Ea � Da, Es = 2Es � Ds andEa+s = Ea + Es � Das, respectively. In a similar manner,the 3Q states are labeled by l, and include the second over-tone states (Ea+a+a = 3Ea � 2Da � D2a; Es+s+s = 3Es �2Ds � D2s) and the second combination bands (Ea+a+s =2Ea + Es � D2as; Es+s+a = Ea + 2Es � D2sa). In this nota-tion, Da (Ds) and Das are the two-quantum diagonal andoff-diagonal anharmonicities, respectively. The three-quan-tum off-diagonal anharmonicities are D2as and D2sa.D2a(D2s) is the anharmonicity of the second overtone band.Fig. 5b is the energy level diagram of a coupled oscillatorsystem. When i = j = 1, k = 2, and l = 3, these Feynmandiagrams reduce to those in Fig. 1c for a single oscillatorsystem.

To calculate the 3D IR spectrum, the same procedure asfor the one-oscillator system is followed, except that theindices are iterated through the larger manifolds of states,resulting in a larger number of Feynman Pathways. Thepeaks are assigned Feynman pathways in Table 2. The nineeigenstate energies and the dynamical parameters weobtained by iteratively fitting the simulated spectrum tothe experiment fall within the error bars of our previousstudies on Ir(CO)2. Three slices along x2 at 3990 cm�1,4049 cm�1 and 4137 cm�1 of the simulated spectra andthe experimental spectra are shown in Fig. 4c. The simu-lated peaks agree with the experiment to within 3 cm�1.

8. Discussion

The three model compounds reported here provide thebasis to interpret 3D IR spectra. The pulse sequences cho-sen for this study separate the peaks into planes that lie atthe overtone and combination band frequencies. Eachplane contains a 2D IR spectrum that is like a traditional2D correlation plot between the fundamental frequencies.However, in a traditional 2D IR spectrum the cross peaksare often obscured by the much more intense diagonalpeaks because both cross peaks and diagonal peaks lie inthe same plane. In this 3D IR pulse sequence, the diagonalpeaks are now separated into different planes than the crosspeaks, thereby diminishing the diagonal/cross peaks over-lap and greatly improving the spectral resolution. Newcross peaks also appear in the overtone planes that arenot observed in traditional 2D IR spectra. These arisebecause the fifth-order pulse sequence interacts with thesample five times rather than 3, thus producing additionalpeaks. Systems with three coupled oscillators will exhibitcross peaks generated from cubic anharmonicities thatare not directly measurable in 2D IR spectra.

We also note that care needs to be taken when analyzingthe 3D spectrum in terms of 2D planes. When the planesare spaced closer than the peak widths, volumes can leakacross adjacent planes, which can confuse the analysis. Inthe 3D IR spectra of Ir(CO)2 reported here, windowfunctions are used to minimize the leakage caused by longtails of the 3D peaks. The best resolution would comefrom absorptive 3D IR spectra that would have the mostcompact 3D peaks and the least amount of leakage. Elon-gated spectra caused by inhomogeneous dynamics suchas for azide in glass (Fig. 3) will always span a range offrequencies, in which case a full 3D visualization isimperative.

In addition to the distribution of peaks, the 3D shapes ofthe peaks contain information on the vibrational dynamicsof the system. The pulse sequence used here starts in a 2Qcoherence that is then rephased in two steps to create a pho-ton echo. Time-domain examples of this 2Q echo have beenreported previously [11,28]. As a result of the echo nature ofthis pulse sequence, homogeneous dynamics like those inW(CO)6 appear as spheres while systems with inhomoge-neous broadening are line-narrowed to produce elongatedshapes that look like cigars. The spectra reported here, how-ever, do not fully utilize the abilities of 3D IR spectroscopyin this regard. It is now well-known that 2D IR spectra canbe used to measure the two-point frequency correlationfunction that describes the time-evolution of the evolvingfrequency distribution [22,23,46]. The frequency correlationfunction is measured by monitoring the peak shapes as afunction of the t2 waiting time (see Fig. 1a) [49]. Theextracted frequency correlation function is accurate whenthe vibrational dynamics have a Gaussian distribution offrequencies, but does not work for non-Gaussian lineshapessuch as asymmetric lineshapes [46]. Asymmetric lineshapescan be caused by non-Condon effects, two or more overlap-ping bands, or a continuous distribution of states. In thesecases, a single two-point frequency correlation functionis not adequate. The spectra can still be simulatedfrom first-principles ab initio/molecular dynamics calcula-tions, but the frequency fluctuations themselves cannot beaccurately extracted directly from the data. However, itshould be possible to extract three-point frequency correla-tion functions directly from experimental 3D IR spectracollected as a function of two waiting times [25,27]. Thefifth-order pulse sequence used here cannot be used in thisregard because it only has a single waiting time (t4, seeFig. 1b), but other 3D pulse sequences do have this ability.Implementing such pulse sequences will require an experi-mental setup with five individual IR pulses so that all 5time-delays can be incremented. Furthermore, the amountof time required to collect a 3D IR spectrum will need tobe greatly decreased because many 3D spectra will needto be measured. This improvement in speed might be gainedby using an array detector to collect the spectra, using com-puter generated pulse sequences [50–52], up-converting theIR echo to a visible one [53], or perhaps recording accor-dion type spectra [54].

Fig. 6. Simulated fifth-order 3D-IR spectrum of Ir(CO)2 for the polarization conditions (a) h�45�, 45�,0�, 0�, 0�, 90�i and (b)h17.9�,11.1�, 94.1�,�48.2�, 61�, 0�i, respectively. The spectra are drawn with isosurface values of 30% and 75% of the maximum intensity, respectively.


Another future improvement will be to use IR pulsesequences with controllable polarizations. The orienta-tional response for fifth-order pulse sequences has beenreported [29,55]. Polarization control over all five pulseswill enable much more sophisticated control over peakintensities than is available in 2D IR spectroscopy. In 2DIR spectroscopy, polarizations can be used to measurethe angles between coupled oscillators as well as suppressthe diagonal peaks to decrease spectral congestion[29,56]. Control over spectral properties is even moreadvanced in 3D IR spectroscopy. For example, a simulated3D IR spectrum of Ir(CO)2 is shown in Fig. 6a generatedfrom a polarization condition h�45�, 45�, 0�, 0�, 0�, 90�i,where the angles correspond to the five pulses and the localoscillator, respectively. This polarization conditionremoves all polarizations of the peaks in the overtoneplanes, leaving just the plane of cross peaks (compare toFig. 4a). Another example is shown in Fig. 6b for thepolarization condition h17.9�, 11.1�, 94.1�,�48.2�, 61�, 0�i.This condition suppresses all of the peaks from the 3DIR spectrum except for H and I. Peak H appears at the fun-damental frequencies of the two coupled oscillators and isthus particularly useful for correlating coupled vibrationalmodes. At lower isosurface values, additional weak peaksoccur from pathways that contain forbidden transitions.Polarization control in 3D IR spectroscopy is more power-ful than in 2D, because the Feynman pathways (Fig. 1d)sample a larger number of transition dipoles and thus havea higher fidelity with respect to polarization.

9. Conclusions

In conclusion, 3D IR spectroscopy is demonstratedusing three model compounds that illustrate the typicalcharacteristics expected for 3D IR spectra. These charac-teristics include homogeneous dynamics, inhomogeneousbroadening, and coupled transition dipoles. While moreadvanced pulse sequences are possible that will generateabsorptive features, measure non-Gaussian vibrational

dynamics, and enable sophisticated polarization control,it is already clear that the additional spectral resolutionof 3D over 2D IR spectroscopy will be useful in resolvingcongested infrared spectra. Extending infrared spectros-copy from 1D into 2D and now to a third dimension fol-lows the historical progression of multidimensional NMRspectroscopy. While there are many differences betweenspin dynamic and vibrational dynamics, this work showsthat manipulation of vibrational coherences leads toenhanced control over infrared spectral content [57] inanalogy to the improvements that spin control enhancesNMR spectroscopy. It is unlikely that infrared spectros-copy will ever reach the absolute structural resolution ofNMR, but multidimensional IR spectroscopies can be usedto follow structural changes over much faster timescalesand now with higher spectral resolution than before.

Acknowledgements

This research is supported by the National ScienceFoundation (Grant No. 144-NL24), the Beckman Re-search Institute and the Packard Foundation.

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Heterodyned 3D IR spectroscopy

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