High resolution coherent three dimensional spectroscopy of NO2 Thresa A. Wells, Angelar K. Muthike, Jessica E. Robinson, and Peter C. Chen Citation: The Journal of Chemical Physics 142, 212426 (2015); doi: 10.1063/1.4917317 View online: http://dx.doi.org/10.1063/1.4917317 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/142/21?ver=pdfcov Published by the AIP Publishing Articles you may be interested in High-resolution two-dimensional correlation spectroscopy in inhomogeneous fields: New application of intermolecular zero-quantum coherences J. Chem. Phys. 132, 134507 (2010); 10.1063/1.3374409 Analysis of polyatomic molecules using high resolution coherent two-dimensional spectroscopy: Application to nitrogen dioxide J. Chem. Phys. 129, 194301 (2008); 10.1063/1.3009265 Two-dimensional nuclear magnetic resonance correlation spectroscopy at zero field J. Chem. Phys. 111, 3077 (1999); 10.1063/1.479588 On the resonant coherent two-dimensional Raman scattering J. Chem. Phys. 109, 5327 (1998); 10.1063/1.477151 Coherent two-dimensional Raman scattering: Frequency-domain measurement of the intra- and intermolecular vibrational interactions J. Chem. Phys. 108, 1326 (1998); 10.1063/1.475505

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High resolution coherent three dimensional spectroscopy of NO2 Thresa A. Wells, Angelar K. Muthike, Jessica E. Robinson, and Peter C. Chena) Chemistry Department, Spelman College, Atlanta, Georgia 30314, USA

(Received 4 February 2015; accepted 30 March 2015; published online 20 April 2015) Expansion from coherent 2D spectroscopy to coherent 3D spectroscopy can provide significant advantages when studying molecules that have heavily perturbed energy levels. This paper illustrates such advantages by demonstrating how high resolution coherent 3D (HRC3D) spectroscopy can be used to study a portion of the visible spectrum of nitrogen dioxide. High resolution coherent 2D spectra usually contain rotational and vibrational patterns that are easy to analyze, but severe congestion and complexity preclude its effective use for many parts of the NO2 spectrum. HRC3D spectroscopy appears to be much more effective; multidimensional rotational and vibrational patterns produced by this new technique are easy to identify even in the presence of strong perturbations. A method for assigning peaks, which is based upon analyzing the resulting multidimensional patterns, has been developed. The higher level of multidimensionality is useful for reducing uncertainty in peak assignments, improving spectral resolution, providing simultaneous information on multiple levels and states, and predicting, verifying, and categorizing peaks. C 2015 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4917317]


At room temperature, gaseous polyatomic molecules yield high resolution electronic spectra that are often difficult to interpret due to the high density of peaks from large numbers of possible transitions between numerous rotational and vibrational levels in different electronic states. For very small molecules that behave like simple models, wellestablished experimental and data analysis tools are available for finding the patterns needed to assign peaks and extract information on molecular structure and behavior.1 For many molecules, however, spectral congestion and complexity can be too severe. For example, polyatomic molecules are often accompanied by severe perturbations that can cause pattern obscuration by increasing the number of peaks and affecting peak positions and relative intensities. One of the most successful techniques for dealing with spectral congestion has been laser induced fluorescence (LIF).2 By fixing a narrowband laser at a specific wavelength and spectrally analyzing the resulting luminescence, one can limit and control the number of peaks by selecting which levels will be excited and then emit light. Another widely used tool is the supersonic molecular jet, a technique that was developed in the 1970s as a way to significantly reduce congestion by cooling the molecules to cryogenic temperatures.3 As a result, the number of initially populated rotational levels and subsequent number of transitions collapse to a much smaller number, and spectral congestion is reduced to levels where peak spacings and intensities are easier to measure and be used to assign peaks. These two techniques, which are often used together, work very well for many molecular studies but provide only a partial a)Author to whom correspondence should be addressed. Electronic mail:

[email protected] 0021-9606/2015/142(21)/212426/11/$30.00

solution for others. One limitation is that most of the rotational information is lost. Another constraint is that it is not feasible to cool all samples (e.g., combustion and plasma systems). A third problem is that many phenomena (e.g., predissociation) can prevent molecules from fluorescing. A fourth concern is potential inaccuracy; assigning peaks produced by perturbed levels may require “best guesses” and involve a high degree of uncertainty. As a result, researchers are often faced with the difficult decision of whether to include questionable assignments in their published results. If available, a universal technique that could reduce spectral congestion and provide more information with a higher degree of certainty could be very useful for studying severely perturbed molecular systems that have previously resisted analysis. This paper introduces the use of coherent 3D spectroscopy as a method for studying molecules that have severely perturbed and complex spectra. The conventional NO2 spectrum has long been intensely studied by many groups and is notoriously complex.4 High Resolution Coherent 3D (HRC3D) spectroscopy is a new technique that is based on four wave mixing (FWM).5,6 For this application, each of the four photons involved in the FWM process connects an unperturbed ground state level with a perturbed excited state level. The generated FWM signal is then recorded as a multidimensional spectrum so that rotational and vibrational patterns become easy to identify. Parts of these multidimensional patterns depend upon the excited state levels, while other parts depend upon the ground electronic state levels. The high level of reproducibility and detail in the resulting patterns permits relatively easy and confirmable assignments, and the technique provides a broad range of information about both the ground and excited states. This FWM technique is also coherence-based and is therefore more universal than LIF; the technique does not rely on creating excited state populations. The technique also works well for gas phase molecules at

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any temperature and therefore can provide full rotational information for a wide range of applications and studies. BACKGROUND

Nitrogen dioxide plays a major role in the formation and destruction of ozone in the atmosphere. It absorbs light throughout the infrared, visible, and ultraviolet regions, and its unusual and infamous spectrum has long been the subject of intense study. Two atlases on the molecule have been published that show an enormous number of peaks covering an unusually broad range with few observable patterns.7,8 Over time, difficult work carried out by many groups has resulted in the identification and analysis of patterns in a small fraction of the molecule’s vast and heavily congested spectrum. The number of analyzed regions is few because established peak analysis methods make use of the intensities and spacings between observed peaks, and both of these measured quantities are perturbed in the visible region of NO2. For example, in 1975, the Field group and Zare group independently published papers9,10 on detailed room temperature LIF studies of the rovibronic patterns found at 593 nm, one of the least congested and best characterized portions of the spectrum. That same year, Levy and coworkers demonstrated the ability to dramatically reduce the number of rotational peaks using a supersonic jet and found that NO2 vibronic levels are irregularly spaced and have a density that is an order of magnitude greater than expected.3 A few years later, Zare’s group also published a detailed study of the rovibronic structure in the 612.5 nm region.11 During the 1990s, Jost, Delon, and coworkers published dozens of papers on the detailed spectroscopy of NO2 obtained from LIF and supersonic jet studies.12–15 Several of these papers include tables of ground state vibrational and excited state vibronic levels that were measured using very high resolution instrumentation (e.g., ring dye laser with a 40 MHz linewidth). The established methods used by these studies to assign peaks relied upon the intensities and spacings between observed peaks. The reason for NO2’s dense, irregular, broad, and largely patternless spectrum is a low-lying conical intersection16 between the potential energy surfaces of the ground (X 2A1) and lowest excited (A 2B2) electronic states that causes strong vibronic interactions from the near infrared (∼1000 nm) to the ultraviolet region. This mixing dramatically increases the number of observed peaks, causes the spacings between peaks to be irregular, and results in a breakdown in selection rules. The congestion is especially severe in the visible region, and the region above 17 000 cm−1 has been called the region of vibrational quantum chaos.14 In 2008, our group published a paper that introduced the use of high resolution coherent two-dimensional (HRC2D) spectroscopy for studying NO2.17 The expansion from (conventional) 1D spectroscopy to 2D spectroscopy increases the space between peaks and reduces congestion. Furthermore, cross-peaks in the off-diagonal region are produced by molecular resonances that have a specific relationship to each other (e.g., they are coupled). Several different types of instrumental techniques have been developed for generating

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coherent 2D spectra, including impulsive time domain,18 multi-resonant frequency domain,19 and multi-resonant mixed time, and frequency domain.20 HRC2D spectroscopy is a multi-resonant frequency domain form of coherent 2D spectroscopy that provides sufficient spectral resolution to resolve rotational structure for molecules in the gas phase. It makes use of a second dimension to increase the spacing between peaks by using one axis to probe one set of vibronic origins and a second orthogonal axis to probe a second set of vibronic origins. Cross-peaks arise only when double or triple resonances are achieved and when the lower level for the two vibronic resonances are identical. HRC2D spectroscopy involves overlapping three pulsed nanosecond laser (or OPO) beams to produce a FWM signal. Spectra are created by plotting the intensity of the FWM signal as a function of two of the four frequencies (e.g., the first (ω1) and the fourth (ω4)). This technique has been used to resolve peaks that would otherwise be heavily congested, sort peaks by quantum number, separate peaks from different species in a mixture of isotopologues, and record highly patterned spectra for molecules that otherwise produce seemingly patternless spectra.21–23 Peaks generated by the same vibrational and vibronic levels form a rotational pattern that resembles the shape of an “X” when, as is the case for NO2, the rotational constants of the upper and lower states are similar. Throughout the visible region, the HRC2D spectrum of NO2 is filled with X-shaped patterns that appear in repeating rows and columns, forming a vibrational pattern where the center of each X marks the approximate location of two vibronic origins, one along the x-axis and the other along the y-axis. In HRC2D spectroscopy, peaks are automatically grouped and sorted by their rotational and vibrational quantum numbers in twodimensional space. Each X is comprised of peaks in one of four branch combinations (PP, PR, RP, and RR), and the center is shifted from the vibronic origins along both axes by B′.17 Peak assignment is made convenient because of the way that the peaks appear sorted by rotational quantum number and because all peaks within the X involve the exact same upper and lower vibrational levels. Despite these successes, much of the HRC2D spectrum of NO2 is heavily congested and difficult to interpret. HRC2D spectroscopy can reduce spectral congestion and make peak assignments easy, but for NO2, the density of peaks is so high and the degree of perturbation is so severe that the resulting peak assignments are highly uncertain. One factor that contributes to the congestion is the fact that X’s with different K values overlap because they have similar x-axis and y-axis positions. NO2 has an odd number of electrons, so an additional splitting of peaks occurs due to the fine spin-rotation interaction. As a result of these two factors, the peaks from these multiple X’s overlap, raising the likelihood of misinterpretation and errors in peak assignment. In an unperturbed molecule, the congestion due to these kinds of effects can be addressed by modeling. But the perturbations for NO2 are too severe: they introduce seemingly random shifts of the position of peaks as well as extra peaks that break selection rules. The result is that there are too many peaks in the vicinity. Unfortunately, HRC2D spectroscopy and other forms of spectroscopy lack a built-in means to

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help select the correct peak and test questionable rotational peak assignments. In order to address this problem, our group has developed and explored the use of additional tools such as new multidimensional software programs designed to filter or identify candidate peaks.24 These programs can help reduce some of the congestion, but the results are usually still “best guesses” among several possible candidates and leave an unacceptably high level of uncertainty for NO2. Since existing peak assignments are then used to make new rotational assignments, inaccuracies can have a multiplicative effect if allowed to propagate in this way, resulting in a list of assignments with an unacceptable number of errors. As a result of these troubling problems and concerns that additional peak assignments might be incorrect, our group has been reluctant to publish a list of assigned peaks for NO2 since the publication of our 2008 paper. A recent advancement in the field of multidimensional spectroscopy has been the development and application of coherent three-dimensional spectroscopy.5,6,25–35 Expansion from 2D to 3D should provide the ability to reduce congestion by increasing the space between peaks. The ability of HRC3D spectroscopy to further reduce spectral congestion and provide selectivity in mixtures has recently been demonstrated with the three isotopologues of molecular bromine.5 New three dimensional pattern recognition techniques can be used to confirm the accuracy of peak assignments and to determine the FWM process, types of resonances, selection rules, quantum numbers, and molecular constants.6 Since the HRC2D instrument operates in the frequency domain, expansion to the third dimension can be made by

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simply replacing one of the broadband input fields with a narrowband tunable beam. For this 3D configuration, only one of the three input beams is in the visible or UV region; the other two are in the near infrared region. This choice minimizes the number and provides greater control over the different FWM processes that can contribute to peaks in a HRC3D spectrum. The sources for these three input beams are a tunable dye laser, a narrowband near-infrared OPO, and a broadband near-infrared OPO. The generated output beam is in the visible region at a wavelength shorter than all of the input lasers, making the technique free of interference from other light-emitting processes such as Raman or fluorescence. Figure 1 shows ten FWM diagrams that can potentially produce peaks in HRC2D and HRC3D spectra when using two near infrared beams (ω2 and ω3) and one visible beam (ω1) to create a FWM signal at ω4 = ω1 − ω2 + ω3. Other frequency combinations are possible, but they result in output wavelengths that are far away, often in regions that are not as convenient to detect. For HRC2D spectroscopy, doubly resonant and triply resonant processes can produce peaks in the 2D spectrum, but for HRC3D spectroscopy, peaks in 3D space are produced only by FWM processes that are triply resonant. In fact, for frequency-domain coherent three dimensional spectroscopy, singly resonant processes produce resonance planes, doubly resonant processes produce resonant lines, and triply resonant processes produce resonant peaks.6 Since very few molecules have their excited electronic states in the near infrared region, processes 1 and 4 are the most likely to produce the triply resonant signal needed to generate peaks in a HRC3D spectrum. Fortunately, the rotational and

FIG. 1. Wavemixing energy level diagrams for ten FWM processes that can contribute to the HRC3D signal. Levels (a)-(e) are in the ground electronic state and levels (f)-(h) are in one or more excited electronic states.

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vibrational peak patterns produced by processes 1 and 4 can be easily distinguished if the visible dye laser wavelength is held constant while the narrowband near-infrared OPO and detection wavelengths are scanned.6 Both processes 1 and 4 produce triplets in the spectrum if the molecule has a selection rule of ∆J (or ∆N) = ±1; more complicated patterns result if the selection rule is ∆J (or ∆N) = 0, ±1. The three peaks in a process 1 triplet form a right triangle; two peaks have the same x axis value and two peaks have the same y-axis value. By contrast, the triplet for process 4 does not form a right triangle; two peaks have the same x-axis value, but no peaks have the same y-axis value (unless by coincidence). Process 1 produces a vibrational pattern where multiple triangles align to form a rectangular grid comprised of rows and columns of similar triangles. In contrast, process 4 produces columns of triangles but not rows; instead the triangles are aligned along diagonal lines that have a slope of m = 1 if the x and y axes are in units of wavenumbers. Further quantitative measurement of the position of peaks that comprise the triangles can provide additional verification of the process and determination of the vibrational and rotational quantum numbers. Each photon in the process 1 and process 4 techniques connects two levels from different electronic states (in order to avoid the limiting ∆v = ±1 harmonic oscillator selection rule for levels in the same electronic state) and each plays a unique role during the formation of the HRC3D spectrum. The first photon (tunable narrowband dye laser) selects a specific resonance between a rovibrational level in the ground electronic state and a rovibronic level in an excited electronic state. The frequency of this photon determines the rotational quantum numbers K and J (or N) and the vibrational quantum numbers for both the upper and lower levels involved in the resonance. If fixed, this frequency also provides the capability for selectivity that is lacking in HRC2D spectroscopy. For process 4, the second photon comes from a near infrared broadband OPO source, and it sets up second resonances between the previously selected rovibronic level and several lower rovibrational levels in a different electronic state. If that different electronic state is the ground electronic state, then many different overtones and combination bands of the ground electronic state may be accessed by this broadband source. The use of a broadband source also facilitates multichannel detection, which is very useful in multidimensional spectroscopy for reducing the data collection time and improving the signal-to-noise ratio. And this second resonance results in a diagonal resonance line that carries information about the location of vibrations in level (e). The third photon is from a narrowband tunable near infrared beam, and it probes a third set of rovibronic levels in an excited electronic state, which may or may not be the same excited state that is involved in the first resonance. This third photon is tuned across multiple resonances during the experiment in order to create the y-axis for the multidimensional plot. The fourth and final photon is generated by the sample, and its wavelength or frequency is analyzed by a monochromator in order to create the x-axis for the plot. Detection of this fourth photon as a function of the frequency of the third photon results in the generation of peaks and patterns that appear in the HRC3D spectrum. This fourth photon connects the rovibronic

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energy levels probed by the third photon to a single final level in the ground electronic state; process 4 is a parametric process, so this final level is identical to the initial level of the first photon. Therefore, HRC3D spectroscopy is more selective than other similar incoherent spectroscopic techniques. Process 1 is slightly different from process 4: the second and third photons are switched. Since the third photon has a higher frequency than the second photon, the second resonance in process 1 involves rovibronic levels (h) that are higher than the rovibronic level (f) selected by the first photon. Therefore, the production of peaks by process 1 may require an additional excited electronic state. Both processes 1 and 4 may occur simultaneously, but one of them will likely produce larger peaks that dominate the spectrum. For example, if a second excited electronic state does not exist within sufficient proximity of the first excited electronic state, then process 1 may not be fully resonant and may therefore not generate any peaks in the 3D spectrum. In either case, the previously discussed vibrational and rotational pattern recognition techniques can be used to distinguish peaks from these two different processes. EXPERIMENTAL METHODS

The spectrometer used for these studies has been described in more detail elsewhere.5 The visible beam was produced using a Coherent Scanmate Pro dye laser (set to 612.430 nm (16 324.0 cm−1) with a linewidth of 0.17 cm−1 and a pulse energy of 1.4 mJ), and its wavelength was monitored using a Bristol 621 wavemeter. Dye laser scans with a Ne optogalvanic cell indicated that the wavemeter accuracy was to within a few picometers. The narrowband tunable infrared beam was produced using the idler beam from a Spectraphysics MOPO 730 system (tuned from 9312-9660 cm−1 with a linewidth of approximately 0.2 cm−1 and a pulse energy of 2.0 mJ). The broadband near infrared beam was produced by a degenerate OPO that emits a beam of light containing a continuous range of wavelengths from approximately 1150 to 1700 nm (approximately 5900–8700 cm−1) and a pulse energy of 3.9 mJ. All three input beams had a pulse duration of 5-10 ns. The three beams were combined in time and space using collinear phasematching and focused into a 0.5 m long glass sample cell that was evacuated and then filled with approximately 10 Torr of NO2 (purity of ≥99.5%, purchased from Aldrich and used with no additional preparation steps). Optical filters were used to remove the pump beams from the FWM beam, which was then focused onto the slits of a 1.25 m monochromator (SPEX 1250 m with a 2400 g/mm grating equipped with a CCD with 2048 columns of 13 µm wide pixels and a measured linewidth of 13 pm = 0.44 cm−1). The dye laser wavelength was fixed, the MOPO beam was scanned with a step size of 0.002 nm, and the CCD integration time was set to 10 laser shots per step. After the FWM signal was observed, each of the beams was individually blocked in order to verify that the peaks depended upon all three input beams. The dye laser was also stepped to other fixed wavelength values in order to observe the effect on the resulting spectra. The results showed dramatic changes in the number and position of peaks with small changes in the dye

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J. Chem. Phys. 142, 212426 (2015)

laser wavelength, which confirmed that they came from triply resonant processes. The resulting individual 1D spectra are stacked to create the 2D spectrum, and then multiple 2D plots may be stacked to form a 3D plot. For the purpose of convenience, the HRC3D spectra were analyzed and shown as 2D slices through the 3D plot, and a modified version of nmrDraw with a multidimensional peak picker (threshold level = 33 counts) was used to identify the locations of major peaks in the slice. The experimental HRC3D spectra in this paper are plots of points that mark the locations of these identified peaks. The linewidths of the instrumentation used for this study were not very narrow and resulted in a combined uncertainty of 0.5-0.6 cm−1 for several of the measurements. This linewidth was too broad to distinguish between rotational constants B and C for NO2. Therefore, even though NO2 is a slightly asymmetric near-prolate rotor, it is treated like a prolate symmetric rigid rotor, and the rotational terms K and Ka are used interchangeably.


Figure 2 shows a HRC3D spectrum recorded by scanning the narrowband OPO, dispersing and measuring the intensity of the generated FWM signal with the monochromator centered at 563 nm, and keeping the dye laser wavelength constant at 612.430 nm. The resulting peaks show patterns that are consistent with that predicted for triply resonant process 4. This dye laser wavelength is very close to a relatively intense vibronic origin located at 16 321.188 cm−1 (612.502 nm), measured by Jost and coworkers using laser induced fluorescence of NO2 cooled in a supersonic jet.13 Their list of vibronic origins does not include any additional vibronic origins within a range of 0.5 nm. However, earlier room temperature LIF work in this region by Zare and coworkers revealed an additional vibronic origin at 612.6 nm.11 Zare’s paper lists a total of five resonances that are within 0.1 cm−1 of the dye laser photon energy (16 324.0 cm−1). These resonances are listed below. 1. 16 323.92 cm−1, N′′Ka Kc = 21 1 to N′Ka Kc = 11 0 (P-type transition) 612.5 nm vibronic origin, 2. 16 323.98 cm−1, N′′Ka Kc = 31 3 to N′Ka Kc = 21 2 (P-type transition) 612.5 nm vibronic origin, 3. 16 323.98 cm−1, N′′Ka Kc = 141 13 to N′Ka Kc = 131 12 (P-type transition) 611.9 nm vibronic origin, 4. 16 324.02 cm−1, N′′Ka Kc = 20 2 to N′Ka Kc = 30 3 (R-type transition) 612.5 nm vibronic origin, 5. 16 323.98 cm−1, N′′Ka Kc = 22 0 to N′Ka Kc = 32 1 (R-type transition) 612.6 nm vibronic origin. The following procedure was used to analyze the HRC3D spectrum in Figure 2. 1. The first step was to identify repeating triangles. The presence of repeating triangles that consist of exactly three peaks indicates that the spectrum is dominated by resonances that follow ∆J (or ∆N) = ±1, which is consistent with that previously observed in this region.

FIG. 2. HRC3D spectrum of NO2. The peaks for the clearest triangles were used for the values in Tables II and III have been circled in yellow for R-type triangles and red for P-type triangles. Additional triangles in the spectrum were not included if it was not clear which three peaks were the best ones to select. The only P-type triangles that were selected were the ones accompanied by other similar triangles with similar x-axis values. The two vertical lines of peaks are believed to be due to one or two unidentified doubly resonant processes. Some unidentified peaks may be forbidden or due to cosmic ray events.

Each identified triangle consists of two peaks that have nearly identical x-axis values (i.e., they align vertically). The side (left vs. right) that these two vertically aligned peaks appear indicates whether the dye laser selected a P-type or R-type resonance. 2. The vibrational patterns among the triangles were then used to identify the FWM process. For process 1, the triangles should align horizontally and vertically. For process 4, the triangles should align vertically and diagonally, but not horizontally. 3. The FWM process was confirmed by examining the rotational pattern within each triangle. For each process 1 triangle, two peaks must have the same x-axis value and two peaks must have the same y-axis value, and all three peaks must form a right triangle. For process 4 triangles, two peaks should have the same x-axis value, but none of the peaks are expected to have the same y-axis value (unless by coincidence).

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Wells et al.

The spectrum in Figure 2 contains two types of triangles, both of which appear to be due to process 4. One type is slightly wider and shorter and appears to be a P-type triangle, while the other is slightly taller and narrower and appears to be an R-type triangle. In the rigid rotor approximation, the triangle widths are equal to B′(4N′′ + 2) (see Figure 3), so the difference in widths between these two types of triangles suggests that they have different starting levels in the ground state (i.e., their N′′ values for level (a) are different). Both sets of triangles are due to process 4; within each set, the triangles align vertically but not horizontally, and within each triangle, two of the peaks have the same x-axis value but none of them have the same y-axis value. This finding is noteworthy because in a previous paper,17 we had assumed that process 1 was more likely because NO2 continues to absorb even more strongly as the wavelength approaches 400 nm. These HRC3D results indicate that the primary FWM process for this range of input and output frequencies is process 4 instead. Unlike HRC2D spectroscopy where the vibrational and rotational patterns for processes 1 and 4 are identical, HRC3D spectroscopy provides the ability to distinguish between these two processes because of their different vibrational and rotational patterns. Process 4 triangles contain two diagonally aligned common peaks (shown as green peaks in Figure 3). These common peaks appear twice: once when the dye laser is fixed on a P-type transition, and once when it is fixed on the corresponding R-type transition (with the same v′′, v′, and N′′ values). These common peaks are produced when level (e) has the exact same rotational quantum number J′′ (or N′′) as that of starting level (a). Since the output frequency generated by the FWM process is ω4 = ω1 − ω2 + ω3, and ω4 is plotted on the x-axis while ω3 is plotted on the y-axis,

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a line drawn through these common peaks (see Figure 3) will have a slope of m = 1 and a y-intercept of ω2 − ω1, which is equal to ω3 − ω4. Therefore, the negative of the y-intercept is equal to the energy of the rotationless vibrations represented by level (e) in Figure 1. For this experiment, these levels correspond to high combination and overtone levels in the ground electronic state that contribute to the three required resonances for HRC3D spectroscopy. For process 4, diagonally aligned triangles may have slightly different common peak spacings; since they have different x-axis positions, the triangles involve different vibronic levels for level (g). However, triangles along the same diagonal should have identical spacings between their vertically aligned peaks (∆y = B′′(4N′′ + 6) for R-type triangles or ∆y = B′′(4N′′ − 2) for P-type triangles) because these diagonally aligned triangles involve the same pair of overtone or combination levels (represented by level (e) in Figure 1) in the ground electronic state. On the other hand, vertically aligned triangles may have slightly different values for the spacing between the two vertically aligned peaks, but the spacings between the two common peaks should be the same because vertically aligned triangles have identical excited state vibronic levels (level (g)). The remainder of the analysis procedure is based upon the process 4 diagram and is described below. Only clear and noncontroversial triangles were used for the full procedure below; triangles that had peaks that were too weak and triangles where it was not possible to determine the most appropriate peaks (points) were not included. 4. Each triangle was checked to see whether it met certain criteria: each must contain two points that have sufficiently similar x-axis coordinates and each must also consist of two

FIG. 3. Triangle rotational peak patterns for process 1 (bottom left) and process 4 (top left). A P-type triangle will include three red (and green) peaks, and an R-type triangle will consist of three yellow (and green) peaks. For the distances between peaks, N is the rotational quantum number for starting level of the FWM process at the bottom of the ground state, and the subscripts on the B’s correspond to the labeled levels in the FWM diagrams shown in Figure 1. Process 4 has green “common peaks” and diagonal lines drawn through them can be used to find related triangles (see top right). The y-intercept of these diagonal lines can be used to find the energy for level (e) vibrational origins (see text). Vertical lines can also be drawn through the peaks to find related triangles. The dotted vertical lines drawn through the centers of the triangles can be used to find the energy for level (g) vibronic origins (see text). The bottom right diagram shows possible values for N for process 4 when ∆N = ±1.

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J. Chem. Phys. 142, 212426 (2015)

diagonally aligned common peaks that are equally spaced along the x and y axes (see the top left of Figure 3). After several triangles that fit the above requirements were identified, similar triangles were found by looking directly above and below the identified ones. Vertical lines were drawn in the spectrum to help compare x-axis positions for different peaks and determine which triangles have the same vibronic origin for level (g). In order to be considered part of a column of vertically aligned triangles, all peaks in a triangle should have nearly identical x-axis coordinates as the ones already identified. Additional triangles were also found by looking diagonally from those already identified. For diagonally aligned triangles, the ∆y spacing between the two vertical points should be similar (see Figure 3). The x-axis center position of each triangle was calculated using the left and right peaks in the triangle (see the dotted vertical lines in the top right part of Figure 3). The resulting number was compared with the energy of the nearest literature vibronic origin.13 Diagonal lines were drawn through the common points of the triangles to identify which ones have the same combination or overtone vibration for level (e) (see Figure 3). For each common point, the y-intercept was calculated by subtracting the y-axis value from the x-axis value. The negatives of the averaged y-intercepts for these diagonal lines were compared with the literature vibrational origin values12 for overtones and combinations that have the appropriate symmetry (see discussion below). The distances between peaks in the clearest triangles were selected and used to determine the rotational quantum numbers and molecular constants. • The two vertically aligned points in each triangle were used to determine N′′ because their vertical spacing depends upon B′′ from level (e), which is below the region of strong perturbation. For this calculation, the value of B′′ for level e was initially assumed to be the same as that for level (a) (0.422 cm−1). Effects such as Coriolis coupling can change the value of B′′ for high overtones and combination bands, but the error introduced can be removed because the value of N′′ is subsequently rounded to the closest integer. (An alternative and more rigorous method would be to determine the value for B′′ by taking two HRC3D spectra with two different dye laser wavelengths to obtain both the P and R plots with the same starting level. One can then measure the three distances in Figure 1 that correspond to ∆y = B′′(4N′′ + 6), ∆y = B′′(4N′′ − 2), and ∆x = B′(4N′′ + 2), and solve for B′′, B′, and N′′.)

• Once N′′ is known, the horizontal and vertical spacings between the two common points in each triangle can each be used to determine B′ for level (g). All values of B′ calculated from triangles that align vertically should be identical. The value for B′′ for level (e) is also calculated, and all of these B′′ values from diagonally aligned triangles should be identical. • Identification of the rotational quantum numbers for each of the levels in the FWM process and each peak in each triangle may be obtained by using the bottom right diagram in Figure 3. For the spectrum in Figure 2, a tabulation of the determined rotational quantum numbers is included in Table I. Not all of the overtone and combination bands in the vicinity of ω1 − ω2 (level (e)) will participate in the generation of triangles; only those with an even value of v3 should be included. The other two vibrational modes of NO2 are symmetric (a1) with a selection rule between the ground and excited electronic states of ∆vi = 0, ±1, ±2, . . .. The vibrational selection rule for the v3 antisymmetric stretch (b2) is ∆v3 = 0, ±2, ±4, . . .. Therefore, if v3 = 0 for the starting level, then every level involved in FWM process 4 should have an even value for v3. The values for the vibrational origins used in this analysis came from Ref. 12. The clearest triangles were checked to determine whether their peak positions and spacings matched that expected. The peaks for these triangles are circled in Figure 2, and all of them met the following criteria. • Consistency between the vertical separation and the horizontal separation between the common points in each triangle: |∆y − ∆x| < 0.4 cm−1, • consistency between the x-axis values for the two vertical peaks within each triangle: ∆x ≤ 0.1 cm−1, • consistency in the average vertical spacing between the two vertically aligned peaks for triangles that fall along the same diagonal: ∆y ≤ 0.3 cm−1, • consistency in the x-axis coordinates for peaks that are in the same column: ∆x ≤ 0.4 cm−1, • discrepancy between K = 0 literature vibronic origins and centers of R-type triangles: ∆x ≤ 0.6 cm−1, • discrepancy between K = 0 literature vibrational origins and the y-intercepts of the R-type triangles: ∆y ≤ 0.9 cm−1, • discrepancy between the shifted (by ∆A − ∆B for K = 1) literature vibrational origins and y-intercepts of the P-type triangles: ∆y ≤ 0.9 cm−1. Triangles that met the above criteria were used to calculate the values in Tables II and III by applying step 8 of the

TABLE I. Assigned quantum numbers of the four levels in the process 4 diagram for the P-type and R-type triangles identified in Figure 2. Levels (a) and (e) are in the (unperturbed) ground electronic state, and levels (f) and (g) are perturbed due to vibronic mixing. Triangle N′′ = 2, R-type N′′ = 3, P-type

Level (a)

Level (f)

Level (e)

Level (g)

Level (a)

N′′ = 2, K′′ = 0, v′′ = 0 for all vibrational modes N′′ = 3, K′′ = 1, v′′ = 0 for all vibrational modes

N′ = 3, K′ = 0

N′′ = 2 or 4, K′′ = 0, v′′ varies (see Table III) N′′ = 1 or 3, K′′ = 1, v′′ varies (see Table III)

N′ = 1 or 3, K′ = 0

N′′ = 2, K′′ = 0, v′′ = 0 for all vibrational modes N′′ = 3, K′′ = 1, v′′ = 0 for all vibrational modes

N′ = 2, K′ = 1

N′ = 2 or 4, K′ = 1

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J. Chem. Phys. 142, 212426 (2015)

analysis procedure. For the R-type triangles, a key criterion was the ability to identify the three best points for the triangle, which often involved a check of the intensity of the peaks and comparison of the x-axis and y-axis positions and spacings with similar triangles in the same column or along the same diagonal. For the P-type triangles, an additional criterion was the existence of similar triangles along the same column (that had points with similar x-axis values). The left side of Table II lists the x-axis values for the centers of the triangles next to their closest literature vibronic origins. For the R-type triangles, the discrepancies between the center of the triangle and K = 0 literature vibronic origins are relatively small (all below 0.6 cm−1). This result suggests that for the R-type triangles, the starting level has K = 0 and v′′ = 0 for all three vibrational modes. For the P-type triangle, however, the discrepancies were about an order of magnitude larger, and several discrepancies were so similar in size to the average half spacing (∼4 cm−1) between the literature vibronic origins that they indicated a poor fit between the center of Ptype triangles and the literature K = 0 vibronic origins. The

right side of Table II shows results from using the ∆y spacing between the two vertically aligned peaks in each triangle to estimate the value for N′′. The results indicate that the two types of triangles identified in Figure 2 yielded different rotational quantum numbers: N′′ = 2 for the narrower R-type triangle and N′′ = 3 for the slightly wider P-type triangle. DISCUSSION

The quantum numbers N, v, and K for the R-type triangles are relatively easy to determine for all levels in the FWM diagram. The N′′ = 2 value for the starting level can be used with the ∆N = ±1 diagram in Figure 3 to determine the values of N for each level in the FWM diagram recorded in Table I.6 The results in Table III show how well the y-intercepts of the diagonal lines match with the literature K = 0 vibrational origins. The values for v1, v2, and v3 were taken from Ref. 12 and show that diagonals of triangles are produced only for even values of v3, which further supports the notion that v3′′ = 0 for the starting level of both the R-type and P-type triangles.

TABLE II. Results based upon the location of the peaks for clear, vertically aligned triangles that have circled peaks in Figure 2. The top part of this table is for R-type triangles and the bottom part is for P-type triangles. The vibronic origins are determined by comparison with the literature values from Ref. 13. The values for N′′ and B′ were determined using the distance between certain peaks as shown in Figure 3. N′′ was rounded to the closest integer value before calculating the experimental value for B′. Average values and standard deviations are provided when two or more triangles were vertically aligned, along with the number of corresponding triangles. For the P-type triangles, the triangle center position column provides a new list of previously undiscovered K = 1 vibronic origins. All values except for N′′ are in cm−1.

Center position of triangles in cm−1 (standard deviation, number of vertically aligned triangles)

Closest literature vibronic origin

Discrepancy (experimental-literature)

Calculated value for N′′

Average experimental value for B′ (using N′′ = 2 for the R-type and N′′ = 3 for the P-type)

Literature B′ value (from Ref. 15)

1.91 1.93 1.97 1.91 1.84 1.88 1.97 1.86 1.86 1.95 1.95 1.92 1.79 1.90 1.93

0.3761 0.4085 0.4331 0.3810 0.4187 0.4152 0.4217 0.4021 0.4355 0.3849 0.4185 0.4271 0.4114 0.3836 0.4098

0.4045 0.4084 0.4375 0.3775 0.4137 0.4362 0.4021 0.4072 0.4433 0.3882 0.4142 0.4261 0.4024 0.4103 0.3632

2.90 2.83 2.88 2.82 2.89 2.88 2.83 2.92 2.88 2.85

0.4034 0.4406 0.4498 0.3717 0.4346 0.4456 0.4109 0.3880 0.4357 0.4300

... ... ... ... ... ... ... ... ... ...

R-type triangles 17 669.06 (0.02, 2) 17 675.57 17 684.78 17 713.14 (0.014, 3) 17 722.26 17 733.92 (0.09, 4) 17 776.99 (0.04, 2) 17 787.06 (0.06, 2) 17 795.86 (0.03, 3) 17 812.68 (0.02, 3) 17 814.58 (0.01, 2) 17 833.20 (0.02, 2) 17 842.89 (0.09, 2) 17 843.70 (0.03, 2) 17 855.12

17 668.798 17 675.373 17 684.484 17 712.557 17 721.875 17 733.555 17 776.717 17 786.878 17 795.345 17 812.547 17 814.257 17 833.06 17 842.578 17 843.646 17 855.131

0.26 0.19 0.30 0.58 0.39 0.37 0.27 0.19 0.51 0.14 0.32 0.14 0.31 0.05 −0.01 P-type triangles

17 670.18 (0.03, 4) 17 699.61 (0.01, 3) 17 702.80 (0.28, 3) 17 708.96 (0.01, 2) 17 712.85 (0.01, 4) 17 723.23 (0.11, 3) 17 733.75 (0.02, 2) 17 744.22 (0.04, 2) 17 779.36 (0.04, 2) 17 808.67 (0.22, 2)

17 668.798 17 700.080 17 700.080 17 712.557 17 712.557 17 721.875 17 733.555 17 739.651 17 776.717 17 812.547

1.39 −0.47 2.81 −3.59 0.30 1.35 0.19 4.57 2.64 −3.88

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TABLE III. Results based upon the location of the peaks for clear, diagonally aligned triangles that have circled peaks in Figure 2. The top of this table is for R-type triangles, and the bottom part is for P-type triangles. The vibrational origins for level (e) were identified by comparison with the literature values from Ref. 12. For the P-type triangle, the values of v1, v2, and v3 taken from this reference were needed in order to shift these K = 0 vibrational origin values by (∆A − ∆B)K2 where K = 1, and the values of A and B were calculated using values from Ref. 14. The final column in this table indicates that the shift significantly improved the fit between observed and expected values and further supports the assignment of K = 1 for the P-type triangles. All values except for the number of triangles, and the values for v1, v2, and v3 are in cm−1. Average Y-intercept value (standard deviation, number of vertically aligned triangles)

Closest literature vibrational origin

Discrepancy (experimentliterature) Values for if K = 0 V1, V2, V3

B′′ calculated using N′′ and ∆y between the vertically aligned points in the triangle

A′′, B′′ values calculated using constants from Ref. 14

∆A − ∆B

Discrepancy (experiment-calculated) if K = 1

10.9760, 0.4325 10.5322, 0.4304 12.1954, 0.4302 9.5684, 0.4372

... ... ... ...

... ... ... ...

R-type triangles 8093.66 (0.14, 3) 8264.03 (0.25, 10) 8283.93 (0.28, 12) 8441.01 (0.19, 6)

8093.61 8264.28 8284.17 8441.44

0.05 −0.25 −0.24 −0.44

4, 4, 0 0, 7, 2 3, 6, 0 6, 1, 0

0.4071 0.4103 0.4133 0.4054 P-type triangles

8048.30 8096.22 (0.13, 4) 8120.10 (0.06) 8185.22 (0.07, 5) 8266.64 (0.13, 4) 8287.55 (0.05, 4) 8330.52 (0.18, 5)

8046.44 8093.61 8120.7 8178.27 8264.28 8284.17 8330.35

1.86 2.61 −0.60 6.95 2.36 3.38 0.17

1, 5, 2 4, 4, 0 4, 0, 2 0, 11, 0 0, 7, 2 3, 6, 0 3, 2, 2

Further support for K = 0 and v′′ = 0 is the agreement between the centers of the R-type triangles and the K = 0 literature vibronic origins. Finally, confirmation is provided by comparison to the resonances identified by Zare and coworkers; the calculated N′′ = 2 value with an R-type transition agrees well with the fourth and fifth resonances in Zare’s list. If the fifth resonance was correct, then the positions of the triangles should be shifted from the K′′ = 0 vibronic origin values due to the different (A − B)K2 values for levels (a) and (e). The absence of any discrepancy greater than 0.6 cm−1 between the center of the triangles and literature vibronic origins indicates that the correct assignment is the fourth resonance in Zare’s list (i.e., Ka = 0). For the R-type triangles, the only quantum numbers that are missing in Table I are the v′ values for levels (f) and (g); these two levels lie in the heavily perturbed region where v′ is not a good quantum number. Assigning the quantum numbers for all levels that participate in the creation of the P-type triangles requires a little more work. As was the case for the R-type triangles, the N′′ = 3 value for the starting state of the P-type triangles can be used with the ∆N = ±1 diagram in Figure 3 to determine the values for N for each level in the FWM diagram. However, K cannot be zero for the R-type triangles because odd values of N′′ should not exist for K′′ = 0 due to nuclear spin statistics (the nuclear spin of 16O is zero). The centers of the P-type N′′ = 3 triangles appear to have no apparent relationship with the literature values of K′′ = 0 vibronic origins, further suggesting that K′′ > 0 and that perturbations remove what would otherwise be a simple relationship among the K stacks. The discrepancies between the y-intercepts and the literature K = 0 vibrational origins are also about an order of magnitude greater for the P-type triangles than they were for the R-type triangles. But unlike the vibronic origins, these vibrational

0.3921 0.3967 0.4000 0.3976 0.4070 0.3992 0.4034

9.7793, 0.4320 10.9760, 0.4325 7.8601, 0.4373 15.1078, 0.4232 10.5322, 0.4304 12.1954, 0.4302 8.6969, 0.4357

1.767 2 2.963 5 −0.157 3 7.104 6 2.521 8 4.185 2 0.681 16

0.09 −0.35 −0.44 −0.16 −0.17 −0.81 −0.51

origins lie below the perturbed region and this discrepancy is significantly smaller than the typical distance between vibrational origins in this region, which is about 20-30 cm−1. Furthermore, the diagonals for the P-type triangles appeared to lie very close to those for the R-type triangles (see Figure 2). For a prolate rotor, the discrepancy between the position of level (e) for K = 0 and that for K > 0 should be equal to (∆A − ∆B)K2 where ∆A and ∆B are the differences in the constants between upper and lower levels. The changes in rotational constants A′′ and B′′ for overtones and combination bands in the ground state have been fitted using a Dunham type expansion.14 For K = 1, level (a) and level (e) should each be higher than their K = 0 counterparts by an amount equal to A − B. The calculated value of A − B for level (a) (v1 = 0, v2 = 0, v3 = 0) is 7.580 06 cm−1, and the values for level (e) depend upon the number of quanta in each of NO2’s three vibrational modes. Therefore, if K = 1, the y-intercept should be shifted by the value of ∆A − ∆B given in the seventh column of Table III. After including this shift, the y-intercepts of the P-type triangles are in reasonably good agreement with values predicted using literature vibrational origins; the results in the final column of Table III show that all discrepancies are less than 0.9 cm−1 and support the assignment of K = 1 for the P-type triangles. This result also agrees with the second resonance in Zare’s list; the other two P-type resonances in this list have the wrong N′′ values. The same approach does not work for the vibronic origins because level (g) is strongly perturbed and in the region of vibronic chaos. Attempts to shift the K = 0 vibronic origins by a constant value failed to produce a match between the centers of the P-type triangles and literature vibronic origin positions. If the K = 1 vibronic origin frequencies behave like a separate vibronic origin from the K = 0 vibronic origin and

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Wells et al.

there is no simple relationship between the two, then it makes sense to use the centers of the P-triangles to create a new list of vibronic origins for K = 1. The P-type triangle portion of the first column in Table II may therefore be regarded as a new list of experimentally observed K = 1 vibronic origin values for NO2, obtained using the center of HRC3D P-type triangles in Figure 2. Of the 25 vibronic origins found by Jost and coworkers in the range covered by our monochromator and CCD, 15 were matched to clear triangles, 1 was found but not matched due to obscuration by the vertical lines (see caption for Figure 2), and the remaining 9 were not found. Six of these last nine were the weakest vibronic origins in Jost’s list. Therefore, the sensitivity of LIF appears to be slightly better than that for HRC3D spectroscopy in this study. However, improvements in instrumentation such as using lasers with narrower linewidths and longer exposure times might yield additional triangles and provide a detection level comparable to supersonic jet LIF spectroscopy. These results show that HRC3D spectroscopy has the ability to separate features that are separated by less than the linewidth of the lasers and detection systems. According to Zare’s list of resonances, the differences between the dye laser wavelength for the P and R type triangles is only 0.04 cm−1, which is much narrower than the linewidth of the dye laser. HRC3D spectroscopy makes effective use of multidimensionality; features that have resonances that are normally too close to resolve can be distinguished by the different patterns they produce in multidimensional space.


This work describes the first application of HRC3D spectroscopy to a polyatomic molecule and demonstrates the ability of the technique to deal with strongly perturbed energy levels. In HRC3D spectroscopy, the frequency of the first photon may be stepped between scans of the third and fourth photons in order to create spectra in three dimensions. Fixing the frequency of the first photon while carrying out a scan results in a single slice that can be analyzed using a procedure similar to the one described here. Throughout that slice, the first photon is anchored on a selected resonance with specific values for K′′, K′, N′′, N′, v′′, and v′, and these values determine the location, shape, and size of the resulting patterns. If the density of peaks is high and more than one resonance is achieved, the resulting HRC3D spectrum may contain several separate patterns. For NO2 (a near prolate symmetric rotor with its strongest peaks following ∆N = ±1 and ∆K = 0 for the region being probed), each resonance produces a specific series of repeating triangles, where the width and height of the triangles depend upon N′′ and B′, and their locations along the x-axis depend upon v′′, v′, and K. More specifically, the centers of the triangles indicate the values of vibronic origins for the corresponding K value. For the spectrum shown in this paper, two different sets of triangles (each with a different size and shape) yielded two different lists of vibronic origins, one for K = 0 and one for K = 1. The list of K = 0 vibronic resonances

J. Chem. Phys. 142, 212426 (2015)

agrees with those previously published in the literature, and the list of K = 1 vibronic resonances is new. The analysis of a single HRC3D spectrum provides a considerable amount of information about both the ground and excited electronic states. This information includes the quantum numbers and molecular constants for the starting, intermediate, and highest energy levels (levels (a), (e), and (g)). Information about the fourth level (f) may be obtained from additional scans that have different fixed values for ω1. The supplemental information used to obtain these results was the ground state rotational constant B′′ and lists of vibronic and vibrational origins. Therefore, the information that can be obtained using HRC3D spectroscopy nicely complements that of existing techniques such as molecular jet LIF spectroscopy. It appears that four contributing factors make HRC3D spectroscopy well-suited for overcoming severe spectral congestion and complexity. First, as expected, the distance between peaks is increased by the expansion from 2D to 3D. But the number of peaks being analyzed is also limited by the requirement that the FWM process must be triply resonant, by the fact that the FWM process is parametric, and by the use of an analysis procedure that requires a specific peak pattern. Second, these patterns are relatively easy to identify. For an experiment involving FWM process 4, a prolate symmetric rotor, and peaks due to ∆N = ±1 and ∆K = 0, the pattern will be triangles that are relatively easy to recognize. The fact that these triangles repeat along columns and diagonals also makes them easy to find and predict. The repetitiveness helps to further reduce uncertainty when identifying triangles and their peaks. Third, the criteria for the shape and the locations of the patterns are fairly strict, making it easier to reject peaks and triangles that should not be considered. For NO2, the triangles had to satisfy a set of seven requirements listed earlier in this paper. And finally, the effect of perturbations has only a minor effect on the appearance and use of the resulting patterns. Because the patterns are multidimensional and the lasers cross back and forth between perturbed and unperturbed levels, some parts of the pattern are affected while other parts are not. Along a given vertical column of triangles in the spectrum, only one level changes (level (e)), and along a diagonal line, only level (g) changes. Therefore, as one travels along a diagonal or along a vertical line, some of the spacings between peaks will remain constant and some will change. And some of these changes will depend upon relatively stable, unperturbed, and possibly well-known ground state constants, while some will depend upon more highly perturbed and perhaps less-known excited state constants. For example, the width of the triangles depends upon the value of B′ for level (g). The width may fluctuate chaotically due to perturbations as one travels along a diagonal or from one column to another, but the width will remain constant as one travels along a vertical column. The ∆y spacing between two vertically aligned peaks in a triangle depends upon the value of B′′ for level (e). This ∆y spacing is therefore expected to remain constant as one travels along a diagonal, but its value may vary as one travels along a vertical column due to small and often more predictable changes in B′′ for different combination and overtone vibrations.

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Strong perturbations can increase the density of vibronic origins, make vibrational spacings irregular, make rotational spacings irregular, and introduce extra peaks that break selection rules. The potential effects of each of these problems on conventional spectra can preclude the analysis and assignment of peaks in other forms of spectroscopy, but the effects on HRC3D spectra appear to be relatively minor and are briefly discussed below. • Increase in vibronic origins—The spacing between the columns may become uncomfortably small for larger molecules if the density of vibronic origins increases. One possible solution might be to change the fixed dye laser wavelength so that the selected value of J′′ (or N′′) and the resulting size of the triangles is reduced. Even if the triangles run into each other, the combination of high repetitiveness and strict criteria for their peak positions should make it easier to discern individual patterns. In general, the shape of the triangle, its size, its general location, and the detailed location of the peaks within it are all useful for ensuring that only appropriate peaks are used during the analysis of the spectrum. • Irregular spacings—As described earlier, the spacings between peaks within a triangle may vary, but only limited parts of the triangles will be affected, and the patterns should still be easy to recognize. • Extra peaks—Peaks that break selection rules may appear in the HRC3D spectrum, but these peaks would be easy to neglect because the analysis procedure involves identification of triangles, not individual peaks. Once again, examination of the repetitiveness and shape of triangles, their size, general location, and the detailed location of the peaks within should make it easier to discern extra peaks from the desired triangles. These results illustrate the use of HRC3D spectroscopy as a technique that can be used to overcome problems caused by strong perturbations. Unlike other forms of spectroscopy (including coherent 2D spectroscopy) that become difficult to use under such circumstances, HRC3D spectroscopy appears to maintain its usefulness under such conditions; perturbations may slightly shift entire columns of triangles and slightly change the sizes of the triangles, but they do not affect ability to find and accurately analyze the patterns. HRC3D spectroscopy appears to be a promising new experimental tool for creating multidimensional patterns that facilitate peak assignments with a high degree of certainty, and may be useful when other techniques fail to do so.

J. Chem. Phys. 142, 212426 (2015)


The authors wish to thank Robert Field for his helpful comments on the manuscript and Kyla Ugwu for her assistance in data analysis. This material is based upon work supported by the National Science Foundation under Grant Nos. NSF CHE-0910232 and NSF CHE-1337522. 1H.

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