Nonparametric High Resolution Coherent 3D Spectroscopy as a

Oct 18, 2018 - A new method for generating High Resolution Coherent 3D (HRC3D) ... 2D slices in 3D space are combined to make a 3D rotational pattern...
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Nonparametric High-Resolution Coherent 3D Spectroscopy as a Simple and Rapid Method for Obtaining Excited-State Rotational Constants Thresa A. Wells, Victoria J. Barber, Muhire H. Kwizera, Patience Mukashyaka, and Peter C. Chen* Chemistry Department, Spelman College, Atlanta, Georgia 30314, United States

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S Supporting Information *

ABSTRACT: A new method for generating high-resolution coherent 3D (HRC3D) spectra has been developed that is based on the nonparametric fourwave mixing process MENS (multiply enhanced nonparametric spectroscopy). The resulting spectra have rotational patterns that are different from those produced previously using the parametric four-wave mixing process CARS. A change in the rotational pattern facilitates a new approach to scanning where orthogonal 2D slices in 3D space are combined to make a 3D rotational pattern. This 3D rotational pattern may then be used to calculate rotational constants for levels in the excited electronic state and upper regions of the ground electronic state. Unlike previous forms of HRC3D spectroscopy, this new approach provides a stand-alone rapid and simple tool for the rotational analysis of electronic spectra without the need for obtaining peak positions or molecular constants from other (1D or 2D) forms of spectroscopy.

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INTRODUCTION Coherent multidimensional spectroscopy is a vibrant and growing field that has attracted much attention because of the technique’s ability to overcome long-standing limitations of conventional 1D spectroscopy12.3 The vast majority of applications involve recording 2D spectra, while exploratory work in the third dimension is ongoing. 3D techniques have the potential for being even more powerful than 2D techniques, but experimental challenges include the time required for acquiring additional dimensions of data and maintaining phase stability for time-domain approaches.4 High-resolution coherent 3D (HRC3D) spectroscopy is a technique that is useful for overcoming the severe spectral congestion that is pervasive in rotationally resolved electronic spectra. Like 3D NMR, HRC3D spectroscopy uses expansion into the third dimension in order to improve spectral resolution over its 2D counterpart. High-resolution coherent 2D (HRC2D) spectroscopy provides advantages over conventional high-resolution 1D spectroscopy because it automatically sorts peaks in 2D space by quantum number, selection rule, and species.3,5 The expansion from 1D to 2D also increases the distance between peaks and therefore reduces congestion. However, large and complex molecular systems often yield 2D spectra that are congested by thousands of peaks; therefore, further expansion into the third dimension is needed to overcome persistent congestion problems. Published in 2013, the first paper on HRC3D spectroscopy describes the ability to further reduce spectral congestion by 2 orders of magnitude. The method also provides species selectivity; specific planes within 3D space can contain peaks that are exclusively from one species in a mixture of isotopologues.6 © XXXX American Chemical Society

In the past, all HRC2D and HRC3D spectra have been recorded in the frequency domain using a CARS-like four-wave mixing (FWM) process (Figure 1A). However, other FWM

Figure 1. FWM diagrams: the first two (A,B) describe parametric processes that cannot be distinguished using 2D spectroscopy but can be distinguished using 3D spectroscopy. The last three diagrams (C− E) describe three different temporal pathways for the nonparametric FWM process MENS.

processes used for multiresonant nonlinear spectroscopy are available for generating coherent 2D and 3D spectra. In fact, the proposed FWM process (see Figure 1B) was originally misidentified in earlier papers on HRC2D spectroscopy7−10 because different FWM processes can produce similar patterns in 2D spectroscopy. This mistake was discovered after the Received: September 4, 2018 Revised: October 15, 2018 Published: October 18, 2018 A

DOI: 10.1021/acs.jpca.8b08640 J. Phys. Chem. A XXXX, XXX, XXX−XXX

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Figure 2. Predicted rotational patterns in the 2D slices for CARS (top) and MENS (bottom), assuming that B is similar in value for all four levels and that the FWM processes are triply resonant.

development of HRC3D spectroscopy,3,11 which can better distinguish between FWM processes that produce similar rotational and vibrational patterns in 2D spectra. Despite its improvement in resolution, selectivity, and process identification, HRC3D spectroscopy can be challenging to implement for several reasons. First, generating a 3D spectrum requires significantly more data and more acquisition time compared to 2D spectra. Using multichannel or multiplex detection, data acquisition may be carried out using a step− scan−scan approach, where two frequencies are scanned and a third frequency is stepped. Each stepped value produces a 2D slice in 3D space, and these 2D slices may be stacked to create a complete 3D plot. However, if the recording time for each 2D slice is tens of minutes, then the time required to collect hundreds or thousands of these slices can be prohibitive. Second, frequency domain coherent 3D spectroscopy offers

many choices regarding the possible experimental parameters. Collecting HRC3D spectra requires identifying the best FWM process to use, the types of lasers to use (e.g., narrow-band, broad-band, tunable), which lasers to scan and which to step, and the optimum multidimensional peak patterns for obtaining the desired information. The large number of choices can appear to be daunting. Third, previous versions of HRC3D based upon the parametric CARS process were designed as a way to further decongest HRC2D spectra by expanding the spectral information to a third dimension. Therefore, analysis of the peaks relied on recording both 2D and 3D spectra, as well as using molecular constants from the literature. The technique would be much more useful if spectral analysis could be carried out without relying on additional information from 1D and 2D spectroscopies. B

DOI: 10.1021/acs.jpca.8b08640 J. Phys. Chem. A XXXX, XXX, XXX−XXX

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The Journal of Physical Chemistry A The purpose of this work is to demonstrate the first use of a nonparametric FWM process for generating HRC3D spectra, to describe how MENS HRC3D spectra are different from those produced using CARS and to explore how this new approach can overcome some of the challenges that exist with CARS-based HRC3D spectroscopy. The results show a specific way to run a MENS-based HRC3D experiment that quickly and easily provides the molecule’s rotational constants for all three upper levels as well as the rotational quantum number for the ground level. The time and amount of data required are approximately 2−4 orders of magnitude less than those of conventional ways of generating 3D spectra, and analysis of the peaks may be carried out without relying on information from 1D and 2D spectroscopies.

where Γaji = Γ*ji − Γ*ja − Γ*ai − Γaa, and Γ*ji is the pure dephasing rate. The first term in the bracket for the MENS eq 4 is independent of dephasing while the last two terms are dephasing-dependent. The last two terms are responsible for the additional dephasing-induced peaks that have been observed in 1D coherent nonparametric spectra under highpressure conditions.12 In the absence of such large dephasing conditions, these two dephasing-dependent terms may be neglected and the signal is determined by the first term in the bracket. The CARS and MENS processes could be used to probe similar levels (b, c, and d) if the frequencies ω1 and ω2 in CARS were used for ω2 and ω3 in MENS and the remaining input frequencies were adjusted so that level d was the same for both processes. However, ω1 in MENS is anti-Stokes relative to all other beams, while the MENS ω4 frequency is Stokes of ω1. By contrast, the ω4 frequency for CARS is anti-Stokes to all other beams. A Stokes output signal is much more susceptible to interference from Raman and fluorescence than an antiStokes output signal. Through careful choice of ω3, however, ω4 can be produced in a region where fluorescence and Raman emission can be avoided (e.g., the near-infrared region). Another difference is that CARS and MENS produce different kinds of rotational patterns in the 2D slices produced during the step−scan−scan process. In order to facilitate multichannel detection, one of the three input fields must be broad-band. If the detected wavelength (ω4) is assigned to the x-axis, the remaining two input fields can be either scanned (yaxis) or stepped (z-axis). The result is six different types of scans for each FWM process in HRC3DS. Figure 2 shows the resulting patterns predicted for these six different kinds of scans, and the patterns for CARS and MENS clearly differ. These simulations are based upon the assumption that the rotational selection rule is ΔJ= ±1 for each photon, that the rotational constants are approximately the same for all levels (i.e., B′ ≈ B″), and that the FWM process is triply resonant. Resonant peaks are produced when the degree of resonance equals or exceeds the dimensionality; therefore, in 3D space, true peaks are only produced when all levels in the FWM diagram are real (i.e., no virtual levels) and the process is triply resonant. The triply resonant requirement may be satisfied if levels a and b are in the ground electronic state while levels c and d are in an excited electronic state; doing so causes each photon to connect levels from different electronic states, thereby bypassing the Δν = ±1 harmonic oscillator selection rule. The simulations also show that for each scan the rotational pattern depends on the value of the stepped laser. In some cases, the scan can produce two different kinds of patterns, one when the stepped laser is resonant with a P-type transition and the other when the stepped laser is resonant with an R-type transition. Both cases create a three-peak pattern with the shape of a triangle. In all other cases, the stepped laser can be resonant with four different possibilities, with the notation m, n, p, or r, as defined in the figure legend. In this case, the resulting rotational patterns consist of only one or two peaks. Figure 2 shows that the MENS process has four scan types that yield three-peak patterns, while the CARS process only has two. The three peak (triangular) patterns are more useful than the one- or two-peak patterns for several reasons. First, three-peak patterns are more unique: the three points can form different kinds of triangles (acute, obtuse, right) with many

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THEORY FWM processes may be divided into two categories: parametric (where the starting and ending levels are identical) and nonparametric (where the final level is different from the starting level). Nonparametric processes can provide some unique capabilities, such as line-narrowing in inhomogeneously broadened systems12 and sensitivity to collisional and pressureinduced dephasing in gaseous and condensed phase systems.13,14 Nonparametric techniques are also more complicated to model because they involve three separate time-ordered pathways. Figure 1C−E shows an example of the three corresponding FWM diagrams used to describe three separate time-ordered pathways for the multiply enhanced nonparametric spectroscopy (MENS) process. MENS was originally developed by Wright and co-workers15−17 for linenarrowing experiments and to avoid fluorescence interference.18 Other nonparametric FWM processes have also been used to produce coherent 2D spectra in condensed phase samples19−23 and demonstrate line-narrowing.24 The equation describing the parametric CARS process (shown in Figure 1A) can be written as25 μμμ μ χ (3) = ∑ ac cb bd da ΔcaΔbaΔda (1) where, for the ji transition, μji is the transition dipole moment, Δji is the resonance denominator (Δji = ωji − ωlasers - iΓji), ωji is the difference in frequency between levels j and i, ωlasers is the combination of laser frequencies required for achieving resonance, Γji is the line width, and the summation is over all contributing states. For the nonparametric process MENS (shown in Figure 1C−E), contributions from the three temporal pathways are added to give ÄÅ É μab μac μcb μdb μab μac μcb μdb ÑÑÑÑ ÅÅÅ μab μac μcb μdb (3) ÑÑ χ = ∑ ÅÅÅ + + ÅÅÇ ΔcaΔcbΔdb ΔcaΔdaΔdb ÑÑÑÖ −Δ*baΔcbΔdb (2)

The line width depends on both the population relaxation and pure dephasing Γji =

Γjj + Γii 2

+ Γ*ji

(3)

After expanding and making certain substitutions, the MENS equation can be rewritten in the following form Ä ÑÉ μ μ μ μ ÅÅÅ 1 i Γ adb iΓ adc ÑÑÑ ÑÑ χ (3) = ∑ ab ac cb db ÅÅÅÅ + − ΔdbΔdc ÑÑÑÖ Δ*caΔda ÅÅÇ Δ*ba Δ*baΔdb (4) C

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because it requires much less data and time than the otherwise time-consuming approach of taking hundreds or thousands of parallel slices and stacking them to make a complete 3D plot. Another advantage of the 3D rotational pattern is that it does not suffer from the problem of mistakenly identifying fake peaks that are due to taking a 2D slice through a doubly resonant resonance line. The maximum number of resonances that can be achieved using a FWM process is three. In such a case, all levels (a, b, c, and d) are real. For a doubly resonant FWM process, one of these levels (either b, c, or d) is virtual. If a level is virtual, the corresponding resonance denominator will be a large real number, thereby reducing the size of resulting signal, removing selectivity, and affecting the structure of the resonance feature. For HRC3DS, a fully resonant FWM process can produce individual peaks in 3D space, but a doubly resonant FWM process will produce resonant lines that propagate through 3D space. This situation is analogous to that in HRC2DS, where a triply or doubly resonant process can produce peaks but a singly resonant process produces resonant lines that can propagate through 2D space. In order to produce true peaks, the number of resonances must equal or exceed the number of dimensions in a multidimensional technique. If the doubly resonant lines intersect the plane of the 2D slice, then the points of intersection may contain features that appear to be peaks in the 2D slice. Figure 4 shows some patterns containing these fake peaks for the MENS scans involved in producing the 3D rotational pattern shown in Figure 3, if the FWM process is doubly resonant. Many of these patterns are resonance lines that will not be mistaken for 3D peaks. For the 1,4 scan with ω2 stepped, however, fake peaks occur when level c is virtual. The doubly resonant lines responsible for these fake peaks are perpendicular to the 2D slice. Therefore, these peaks are easily identified as being fake because their position does not change when ω2 is changed. The fake peaks for the 2,4 scan with ω1 stepped and level d as virtual are more difficult to discern because the doubly resonant lines are not perpendicular to the 2D slice. Therefore, the coordinates for these peaks change when ω1 is changed. Unlike true triply resonant peaks, however, their position changes will be continuous rather than discrete.

different kinds of orientations. By contrast, one-peak patterns are not unique at all, and two-peak patterns can only yield three possible variations: vertical, horizontal, or diagonal line orientation. Second, triangles produced by three-peak patterns have measurable heights and widths that can have simple algebraic relationships with the molecules’ rotational constants and rotational quantum numbers. Furthermore, two of the triangle-producing MENS scans involve the same axes in 3D space (ω1, ω2, and ω4): the 1,4 scan with ω2 stepped and the 2,4 scan with ω1 stepped. Experimentally, the advantage of using these two scans is that one of these two types of scans can be immediately followed by the other. During data analysis, the resulting 2D slices will be orthogonal to each other and can therefore be combined on the same 3D plot. The result is a three-dimensional rotational pattern that consists of six peaks after combining three orthogonal planes, as shown in Figure 3. For example, in the

Figure 3. Rotational pattern in 3D space that consists of six peaks. It includes two right triangles (blue/yellow and red/green) and two triangles that are acute or obtuse (blue/red and yellow/green).

laboratory, if one carries out a 1,4 scan and identifies a threepeak pattern (e.g., the red and blue peaks in Figure 3), one can set ω1 to be resonant with each of these peaks and then scan ω2 to create additional orthogonal 2,4 spectra. If one sets ω1 to a value that is resonant with the two red peaks, then a 2,4 scan will produce a slice that contains one additional related peak (marked green in Figure 3). If one then carries out a second 2,4 scan with ω1 set to the ω1 value that is resonant with the blue peak, then the resulting 2,4 slice will contain two additional related peaks (marked yellow in Figure 3). Using this approach, only one 1,4 scan and two 2,4 scans are needed to create the 3D rotational pattern shown in Figure 3. The ω1 values for both 2,4 scans are determined by the ω1 value for the red and blue peaks in the 1,4 spectrum. Alternatively, a 3D rotational pattern could be recorded by taking an initial 2,4 scan and using that to set the ω2 values for two 1,4 scans. After these three slices are recorded, analysis of the peak positions can be used to determine the rotational constants for the three upper levels (b, c, and d) as well as the rotational quantum number for level a. Four or more independent measurements can lead to the four independent equations needed to determine Bb, Bc, Bd and J″, and these measurements involve the heights and widths of the triangles that comprise the rotational pattern in Figure 3. An example is provided in the Results and Discussion section. This approach is attractive

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EXPERIMENTAL METHODS Frequency domain spectra were recorded using injection seeded Nd:YAG-pumped q-switched nanosecond lasers with a repetition rate of 10 Hz and homodyne multichannel detection. The experimental setup was similar to that described previously8 with a few minor changes. The instrumentation included a Coherent ScanmatePro dye laser with rhodamine 590 dye (range = 557−585 nm, line width of 0.15 cm−1) pumped by the second harmonic from a Spectra-Physics Lab 170 Nd:YAG laser. The second harmonic beam from a Spectra-Physics GCR-230 Nd:YAG laser was also used to pump a Spectra-Physics Cobra dye laser that contained a mixture of rhodamine B and sulforhodamine 640 dye for a spectra range of 590−610 nm (line width of 0.06 cm−1). It was also used to pump a home-built broad-band degenerate OPO26 with its BBO crystals tilted to generate light from 1030 to 1100 nm. The beams were combined in time and space, using collinear phase matching, and focused using a 0.5 m FL lens into an evacuated glass sample cell that contained iodine vapor. Iodine was selected as the sample molecule because of the availability of high-quality molecular constants in the literature. D

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Figure 4. Predicted rotational patterns in the 2D slices for the triply and doubly resonant MENS scans needed to create the 3D rotational pattern in Figure 3. These patterns are for the case where all B’s have a similar value.

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RESULTS AND DISCUSSION Spectra were recorded that illustrate the ability of 3D spectroscopy to overcome congestion compared to 2D spectroscopy. Seven spectra with different stepped laser wavelengths were acquired with dimensions of 6 nm × 16 nm, and none of them contained more than eight triangles (24 peaks). This peak density is at least two of orders of magnitude lower than that normally encountered in HRC2D spectra. Figure 5 shows the three spectra that were extracted from larger plots so that they would correspond to the three 2D slices shown in Figure 3. Figure 5a shows the first spectrum in this series and was recorded by setting ω2 to 597.3769 nm and scanning ω1 from 562.432 to 561.433 nm. This spectrum shows six peaks that form two acute triangles. These two triangles involve different sets of rotational and vibrational quantum numbers and are unrelated; they both just happen to share a similar resonance wavelength (597.3769 nm) for the stepped field. The lower unmarked triangle corresponds to an R-type triangle and has the shape predicted in Figures 2 and 4 for a MENS 1,4 triply resonant scan where ω2 is stepped and ω3 is broad. The other triangle (with points connected by lines and circles around each point) is a P-type triangle that looks slightly different from the shapes predicted in Figures 2 and 4. This triangle is acute rather than obtuse because the rotational constant for the excited B state of iodine is so much smaller than that for the ground X state that the relative order of the m- and r-type resonances are switched. Therefore, for I2, the observed order along the x-axis (ω4) is nprm (from low to high) instead of npmr, and the P-type triangle is acute. The y-axis position of the three circled peaks in Figure 5a are 561.902 and 561.682 nm. These two values were used as the two stepped ω1 values for Figure 5b,c. After setting ω1 to one of those two values, ω2 was scanned from 597.882 to 596.888 nm. The circled peaks in Figure 5a also appear (from a

The molecule’s relatively small rotational constants also make it useful for testing the ability of the technique to overcome severe congestion in polyatomic molecules. The resonances being used are from the molecule’s well-known B3Πu+ → X1Σg+ transition involving a selection rule of ΔJ = ±1. Four optics (CVI Y1 mirror, Thorlabs 950 SWP filter, Semrock 577 notch filter, and a CVI 850 LWP filter) were used to remove the pump beams, and the resulting FWM signal was focused into a 1.25 m monochromator (SPEX 1250m). The monochromator was equipped with a 1200 g/mm grating and a CCD with 2048 columns of pixels that were 13 μm wide. In order to acquire step−scan−scan spectra, one of the dye laser’s wavelengths was scanned over a distance of 1 nm at a rate of 0.001 nm per second, and the intensity of the generated ω4 light at each wavelength was acquired and recorded using the CCD at a rate of 1 spectrum per second. The monochromator was centered at approximately 933 nm. The total amount of time required to collect each spectrum was less than 20 min. Additional slices could then be recorded after changing the other dye laser’s wavelength and running another scan. A sixth-order polynomial fitting function was used to improve the wavelength calibration of the CCD. This fitting function was checked against seven lines from a neon lamp, and the resulting uncertainty was approximately 0.007 nm, or 0.08 cm−1. The accuracy of the ScanmatePro dye laser was checked with a second 1.25 m monochromator equipped with a 2400 g/mm grating and a CCD pixel size of 13 μm. The accuracy of the Cobra dye laser was determined using a Bristol 821 wavemeter with a specified accuracy of within ±0.02 cm−1. The positions of the peaks were determined using a modified version of nmrDraw with a multidimensional peak picker that provided the coordinates for the tallest peaks in the spectra. E

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may be the case for systems where severe spectral congestion in 1D or 2D spectroscopy has prevented analysis and/or when new molecular species are believed to be detected. The first step for this method was to use the coordinates for the six points to determine the height and width of the three triangles marked in Figure 5a−c. These widths and heights are equal to the following for a molecule that behaves like a rigid rotor Bb ″(4J ″ − 2) = the width of the triangle in Figure 5a Bd ′(4J ″ + 2) = the height of the triangle in Figure 5a

Bb ″(4J ″ − 2) = the width of the triangle in Figure 5b Bc ′(4J ″ + 2) = the height of the triangle in Figure 5b Bb ″(4J ″ + 6) = the width of the triangle in Figure 5c Bc ′(4J ″ + 2) = the height of the triangle in Figure 5c

The above list provides four independent equations because the first and third equations are similar and the fourth and sixth are similar. The equations containing a Bb″ were simultaneously solved to yield a value of J″ = 58 and Bb″ = 0.03873 cm−1. Substitution of J″ into the remaining equations yielded Bc′ = 0.03244 cm−1 and Bd′ = 0.02993 cm−1. The accuracy of these values is limited because this simple method does not take into consideration higher-order terms such as centrifugal distortion. For a well-characterized molecule like iodine, equations with a large set of Dunham coefficients may be available.27,28 A comparison between the experimentally measured peak positions and the predicted positions calculated by using a full set of Dunham coefficients is given in Table 1. The experimental and simulated values agree with each other. The identified rotational and vibrational quantum numbers for the six points are consistent with each other. The discrepancies between simulated and experimental values are small and insignificant when taking into consideration the line widths of the lasers and the accuracy of the instrumentation used to calibrate them. Higher-order molecular constants that incorporate correction terms like centrifugal distortion and rotation−vibration interaction could be determined by acquiring and then analyzing additional adjacent triangles associated with neighboring values of the ground-state rotational constant J″. Such triangles could be acquired by recording additional scans with different stepped laser frequencies.

Figure 5. (a−c) Three HCR3D scans (from top to bottom): (a) 1,4 scan with ω2 set to 597.3769 nm, (b) 2,4 scan with ω1 set to 561.902 nm, and (c) 2,4 scan with ω1 set to 561.682 nm.

different perspective in 3D space) in Figure 5b,c, where they are circled using the same colors (red and blue) shown in Figures 3 and 5a. The coordinates for the top of each peak were used to obtain approximate values for the ground state rotational quantum number J″ and the three rotational constants Bb″, Bc′, and Bd′. The purpose of this exercise was to develop a method for quickly determining such constants for future applications where such information is not known about the molecule. Such

Table 1. Comparison between the Experimentally Determined Coordinates of the Six Points Shown in Figure 3 and the Calculated Values from Simulations Based upon Constants in the Literature (refs 27 and 28) frequency

type

experimental (cm−1)

calculated (cm−1)

exp. − calc. (cm−1)

J″

J′

ν″

ν′

ω1 ω1 ω2 ω2 ω4 ω4 ω4 ω4

P R P R P P R R

17791.80 17798.81 16735.29 16742.89 10716.37 10718.66 10725.59 10727.56

17791.65 17798.63 16735.33 16742.95 10716.45 10718.62 10725.60 10727.52

0.15 0.18 −0.04 −0.06 −0.08 0.04 −0.01 0.04

70 70 70 70 72 70 70 68

69 71 69 71 71 69 71 69

1 1 1 1 39 39 39 39

22 22 11 11 22 22 22 22

F

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CONCLUSIONS High-resolution coherent multidimensional spectroscopy is an effective tool for overcoming severe spectral congestion that is often observed in rotationally resolved electronic spectra. Expansion to the second dimension generates patterns that can organize and sort peaks into easily identifiable patterns. However, congestion often remains a problem in highresolution coherent 2D spectra. HRC3D spectroscopy based upon CARS was developed in order to solve this problem. In order to analyze the resulting data, however, the original approach required information from conventional 1D spectra and coherent 2D spectra. Furthermore, acquiring a large set of step−scan−scan spectra and then stacking them to create a 3D spectrum requires considerable data and acquisition time. MENS is an attractive alternative FWM process to CARS for creating HRC3D spectra because it can be used as a quick, simple, single way to obtain the experimental data needed to calculate rotational constants and quantum numbers. CARS and MENS produce different kinds of rotational patterns that depend on a number of experimental parameters, including the degree of resonance and which lasers are scanned, stepped or broad-band. For a simple diatomic molecule with a selection rule of ΔJ = ±1, the rotational patterns produced by MENS include twice as many three-peak rotational patterns as CARS, and two types of scans involve the exact same three frequency axes. These two scan types can therefore be used to rapidly and easily create 2D slices that can then be plotted orthogonally in 3D space. The resulting 3D rotational pattern contains six peaks and can be produced with just three scans. The coordinates of these six peaks can quickly and easily be used to calculate the rotational constants for all three upper levels in the FWM diagram and the ground-state rotational quantum number for a rigid rotor molecule. By comparison, the more conventional step−scan−scan approach for generating three 2D slices could require hundreds or thousands of scans in order to generate enough slices to create a 3D spectrum with enough information to perform such calculations. The roughly 2−4 orders of magnitude reduction in required data and acquisition time makes this new approach much more attractive for future applications such as the study of mixtures where unknown or previously unidentified molecules may be present. If needed, additional scans could also be recorded to acquire additional adjacent triangles that could then be used for calculating higher-order correction terms needed to account for effects such as centrifugal distortion and rotation−vibration interaction. Prior work suggests that the method described here should also work for polyatomic molecules. Polyatomic molecules have additional rotational quantum numbers, rotational constants, and vibrational modes. Unlike diatomic molecules, their spectra may be perturbed by conical intersections. HRC3DS spectra recorded using a parametric process (CARS) show that the rotational patterns produced by the polyatomic molecule NO2 are the same kind of triangles produced by the diatomic molecule Br2.29 The severe spectral congestion found in the 1D and HRC2D spectra of NO2 is absent in the molecule’s HRC3DS spectra, even though the molecule’s density of states is unusually high due to a series of low-lying conical intersections. Furthermore, the selectivity made possible by setting the stepped laser wavelength to match a specific resonance and the requirement that the FWM process be triply resonant limit the observed peaks to those

involving specific rotational quantum numbers and specific pairs of rovibrational levels. For example, if the stepped laser is sufficiently narrow to select a single resonance in a prolate symmetric rotor that has a selection rule of ΔJ = ±1 and ΔK = 0, the resulting rotational pattern should consist of a limited number of triangles. Additional scans with additional fixed laser wavelengths may be needed in order to probe different vibrational modes, K values, and rotational constants (e.g., A, B, and C for an asymmetric rotor). If the selection rules involve a greater number of resonances (e.g., ΔJ = 0, ±1) the observed rotational pattern may include more than three peaks. In a paper6 on HRC3DS, the number of peaks in the rotational pattern was predicted to increase from 3 to 6−7 under such conditions. Even though the resulting patterns may be more complicated, the general principle of collecting spectra and then switching the scan type to record orthogonal slices that involve the same rotational quantum numbers should still provide an effective way to reduce the number of spectra and acquisition time needed to calculate molecular constants. The exact patterns and equations for these more complicated cases will need to be developed on a case-by-case basis, but the total number of different possible rotational patterns that could be observed and used for this new method should be limited.

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ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jpca.8b08640. Additional independent equations for calculating higherorder Dunham coefficients from adjacent triangles in the 3D rotational pattern (PDF)

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Phone: 404-270-5742. ORCID

Peter C. Chen: 0000-0001-5481-2635 Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This work was supported by National Science Foundation Grant CHE-1608010. The authors wish to acknowledge Nihal Jemal for her assistance in the development of the theory.

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REFERENCES

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DOI: 10.1021/acs.jpca.8b08640 J. Phys. Chem. A XXXX, XXX, XXX−XXX