Rotational and Vibrational Pattern Interpretation for High-Resolution

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Rotational and Vibrational Pattern Interpretation for High-Resolution Coherent 3D Spectroscopy Benjamin R. Strangfeld,† Thresa A. Wells, and Peter C. Chen* Chemistry Department, Spelman College, Atlanta, Georgia 30314, United States S Supporting Information *

ABSTRACT: High-resolution coherent multidimensional spectroscopy provides an alternative to conventional methods for generating rotationally resolved electronic spectra of gas phase molecules. In addition to revealing information such as the relationships among peaks, it can provide clearly recognizable patterns for spectra that otherwise appear patternless due to rotational congestion. Despite this improvement, high-resolution coherent 2D spectroscopy can still exhibit congestion problems; expansion to the second dimension is often not sufficient to prevent overlapping of peaks from different patterns. A new 3D version of the technique that provides improved resolution and selectivity to help address cases with severe congestion was recently demonstrated. The experimental design and interpretation of data for the 3D technique are significantly more complicated than that for the 2D version. The purpose of this paper is to provide important information needed to plan, run, and interpret results from high-resolution coherent 3D spectroscopy experiments.

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INTRODUCTION Coherent two-dimensional1−6 and three-dimensional spectroscopy7−15 have gained much attention as analogs to multidimensional NMR schemes that reveal the relationships between peaks in optical spectroscopy. Most coherent multidimensional techniques have been developed for condensed phase systems. However, a high-resolution form16 using nanosecond lasers and frequency-domain acquisition has been created for gas phase electronic spectroscopy, in which severe rotational congestion often causes pattern obscuration. The 2D spectra produced by high-resolution coherent 2D (HRC2D) spectroscopy can display clearly recognizable patterns for systems in which other techniques yield seemingly patternless spectra. HRC2D spectra utilize 2D space to display two types of patternsa vibrational pattern and a rotational patternso that vibrational information is spatially distinct from rotational information. Within the patterns, peaks are sorted sequentially by their quantum numbers.17 The patterns for hot bands are spatially distinguishable from those for cold bands.18 Patterns from different isotopologues may also appear spatially separated.19 HRC2D is useful for recording spectra of molecules at any temperature, making it suitable for studying systems in which rotational information is needed or cryogenic preparation is not feasible. One key difference between HRC2D spectroscopy and most other forms of high-resolution spectroscopy is the fact that it relies upon resonantly enhancing a coherent optical process rather than creating an excited state population. As a result, the technique is more robust than laser-induced fluorescence, in © XXXX American Chemical Society

which intensities may be diminished by depopulating effects such as quenching and predissociation. But perhaps even more important is the fact that certain coherent processes are parametric forms of four-wave mixing (FWM). In such cases, light that is coherently generated comes from molecules in which the final level is identical to the initial level (i.e., the molecule does not experience a net gain or loss of energy from the input light fields). Unlike most other forms of conventional spectroscopy (absorption, fluorescence, etc.) the initial and final levels of a parametric FWM process will have identical vibrational and rotational quantum numbers. This fact restricts the number of levels that can participate in the creation of light and the types of peaks that subsequently appear in the coherent 2D spectrum. The resulting peaks and patterns are therefore limited in number, freer from congestion, and yet sufficiently rich in information to facilitate a detailed spectroscopic study. Despite the reduction in congestion made possible by expansion into a second dimension, all of the molecules that have been studied using HRC2D spectroscopy have yielded 2D spectra that display some degree of persistent spectral congestion. It appears that expansion into the second dimension often does not sufficiently reduce rotational congestion for gas phase electronic spectroscopy. Severe congestion can be expected to be a serious problem for systems containing mixtures, molecules at elevated temperReceived: January 21, 2014 Revised: June 1, 2014

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needed to further reduce persistent congestion problems. For example, the peaks from all isotopologues of a given species can appear in the 2D spectrum because the technique lacks the ability to provide species selectivity. Tackling this problem by increasing the dimensionality from 2D to 3D involves an increase in experimental complexity. With three orthogonal tunable axes, additional time is then needed to collect data; a single 3D spectrum consisting of 1000 × 1000 × 1000 points would require a data collection time of more than 30 years if the points are collected at a rate of one point per second. To reduce the time for data collection, our HRC2D experiments employ two broadband near-infrared beams to achieve dual-broadband multiplexing. Expansion to 3D is achieved by replacing one of these two broadband nearinfrared input fields with that from an independently tunable laser. The use of a multichannel array detector, which can be used because one of the input lasers is still broadband, significantly reduces the time required to record 3D data, but it is still considerably longer than the time required to record 2D data. Careful selection of the three input beam frequencies can reduce the number of fully resonant FWM processes and therefore help simplify analysis of the resulting HRC3D spectra. Most molecules do not have strong resonances in the nearinfrared (1000−2000 nm) region. In our experiments, only one of the input beams is narrowband, tunable, and in the UV or visible region; the other two are in the near-infrared region. This approach reduces the number of possible fully resonant FWM processes to a small number, making spectral interpretation simpler. It also ensures that the output beam can be generated in the UV or visible region, allowing use of highly efficient monochromator and CCD systems. Unlike traditional spectra that are simple xy plots (a plot of a measured quantity as a function of wavelength or frequency), high-resolution coherent multidimensional spectra display the intensity of a beam (produced by four wave mixing) as a function of two or more independent frequencies. For HRC3D spectra, the intensity of the generated beam is a function of three independent input frequencies. As a result, it is possible to use resonance patterns to distinguish among singly resonant, doubly resonant, and fully (triply) resonant FWM processes. In a conventional coherent 1D spectrum, peaks may be produced by singly resonant, doubly resonant, or triply resonant FWM processes, and determining the degree of resonance requires additional experimentation or careful line shape analysis (e.g., modeling coherence interference effects). For coherent 3D spectra, where the peaks reside in three-dimensional space (each orthogonal dimension corresponds to a unique frequency axis), the degree of resonance may be determined by the observed structure (see Table 1). Singly resonant processes should produce resonance planes that propagate through 3D space because the same resonance can be maintained by either keeping one of the four fields constant (for a resonance such as ω1 = ωlevel, where the resonance plane would be parallel to the axes for two of the fields) or by keeping the relationship between two fields constant (for a resonance such as ω1 − ω2 = ωlevel where the resonance plane would be diagonal with respect to the axes for two of the fields). Doubly resonant processes should produce resonance lines that propagate through 3D space. Only fully (triply) resonant processes can produce discrete resonance points (peaks) in the 3D space; a triply resonant process requires that all fields have the appropriate

atures, and molecules that are large or have vibronically mixed states due to conical intersections. To address severe congestion problems, a three-dimensional form of highresolution coherent multidimensional spectroscopy has recently been demonstrated.20 The purpose of high resolution coherent three-dimensional (HRC3D) spectroscopy is similar to that for 3D NMR: to use the third dimension to further reduce the peak density by several orders of magnitude. In the initial demonstration, several different patterns were observed, but the theory needed to explain them had not yet been developed. Therefore, to analyze the observed peaks in the 3D spectrum, we used literature molecular constants to create tables of transition wavelengths and then searched these tables for close matches with the experimentally observed peak positions to assign them. This approach was useful for studying bromine, but it is obviously not useful for studying molecules where the molecular constants are not known to a high degree of accuracy. Theory, combined with a systematic approach is therefore needed to plan experiments on other molecules and to interpret their spectra. Without this theory, users will unlikely be able to interpret the many different types of patterns that can be observed, identify which four-wave mixing process is responsible, and determine which species and levels are responsible for the observed peaks. For example, before we developed this theory, we were able to identify only one of several patterns observed in the HRC3D spectrum of Br2. This paper provides the background, theory, and recommended approach for planning new HRC3D experiments and analyzing the results. For HRC2D spectroscopy, the selection of laser and detection wavelengths is fairly simple. For example, to study a series of rovibronic peaks associated with transitions between the ground and an excited electronic state, one can scan the tunable laser across the red portion of a broad, congested collection of peaks while setting the monochromator to cover the blue portion. HRC3D spectroscopy is even more powerful because it can provide selectivity and further improve spectral resolution, but the experiments and analyses of results are more complicated. The experiment requires two tunable laser beams; one is scanned while the other is set (or stepped) to a resonance that can be used to achieve selectivity. The set laser can be used to achieve selectivity, but if its wavelength is not set to an appropriate resonance, the resulting 3D spectrum may be void of peaks. Furthermore, as shown later in this paper, the HRC3D spectra also have a wider variety of possible patterns, and the results can be more challenging to interpret. This paper provides solutions to these challenges and is developed for diatomic molecules; if needed, this approach and theory may be further developed for use with polyatomic molecules.

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BACKGROUND High-resolution rotationally resolved electronic spectroscopy is a powerful technique that can provide exquisitely detailed information on the structure and behavior of gas phase molecules. However, the technique is limited to very small and simple molecules; spectra produced by larger and more complicated molecules have long evaded analysis. In addition to reducing congestion by expanding into a second dimension, HRC2D can help sort peaks to facilitate their assignments. However, coherent 2D spectroscopy is not a panacea; congestion still remains a potential problem in HRC2D spectra, and the technique lacks the flexibility and selectivity B

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THEORY A common requirement in coherent spectroscopy is to identify the appropriate wave-mixing process for setting up an experiment or for interpreting experimental results. The use of three independent input beam frequencies (ω1, ω2, and ω3) to produce a FWM signal can result in up to eight possible frequency combinations (ω4 = ± ω1 ± ω2 ± ω3). For our experiments, only one of the input fields (ω1) is from a tunable visible (or UV) beam. The other two input fields are in the near-infrared region: ω2 is broadband to facilitate multiplex detection, ω3 is narrowband and tunable, and ω3 > ω2. As a result, only one frequency combination (ω4 = ω1 − ω2 + ω3) can be both phase-matched and generate light that is antiStokes of ω1, a condition that is preferred to reduce susceptibility to spectral interference from other processes, such as laser-induced fluorescence. Figure 1 shows the 10 possible FWM processes that can be used to generate coherent light at ω4 = ω1 − ω2 + ω3. Energy level diagrams 1−4 describe parametric processes (final state = initial state), diagrams 5−7 correspond to one set of nonparametric processes, and diagrams 8−10 correspond to a second set of nonparametric processes. All 10 processes have the same phase-matching geometry because they all can produce the same output frequency. However, the molecular energy levels that can provide resonance enhancements and produce peaks in the spectra are different. Therefore, the pattern of multidimensional resonance features produced by each will be unique. For a specific molecule and a given set of input wavelengths, some of the FWM processes shown in Figure 1 may be fully resonant, while others may be doubly resonant, singly resonant, or nonresonant. The FWM process responsible for generating the observed peaks must be known to design appropriate experiments and to correctly interpret results. During the development of HRC2D

Table 1. Observed Resonance Patterns for 1D, 2D, and 3D Spectroscopy with Different Degrees of Resonance Enhancementa

conventional 1D spectroscopy coherent 2D spectroscopy coherent 3D spectroscopy a

singly resonant

doubly resonant

fully resonant

peak

peak

peak

continuous line

peak

peak

continuous plane

continuous line

peak

Article

See text for explanation.

wavelengths needed to create a fully resonant four wave mixing process. Because creating and displaying the four dimensions of a 3D spectrum is a challenge, the spectra created and presented in this paper are 2D slices through the 3D spectrum, where one of the input field frequencies is held constant. If the data is displayed as a 2D slice rather than a complete 3D spectrum, additional slices may be needed to confirm the degree of resonance. For example, a doubly resonant process will produce a ridge (due to the resonance line) in a coherent 2D spectrum if the data plane contains the entire double resonance line. If the double resonance line and the plane of the 2D slice intersect, however, then the coherent 2D signal will contain a peak (at that point of intersection) rather than a line. This observation is consistent with the notion that peaks in HRC2D spectra may be caused by doubly resonant or fully resonant processes, and peaks in conventional (1D) coherent spectra may be singly resonant, doubly resonant, or triply resonant.

Figure 1. FWM diagrams that can produce an output frequency at ω4 = ω1 − ω2 + ω3. All diagrams drawn here are fully resonant (all energy levels are real). Replacement of one or two real levels by virtual levels would result in doubly resonant or singly resonant FWM processes that are weaker than fully resonant processes. Processes 1 and 4 are the most likely fully resonant candidates because molecules seldom have excited electronic levels in the near-infrared region. C

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of either ΔJ = ±1 (solid lines only) or ΔJ = 0, ±1 (solid and dotted lines). In this Figure, J represents the initial rotational quantum number, which is also the value at the end of the FWM process because the process is parametric. Because ω2 is broadband and ω4 is always spectrally dispersed and analyzed, the two choices for creating a 2D slice are to scan ω3 (while keeping ω1 fixed) or to scan ω1 (while keeping ω3 fixed). For processes 1 and 4, if ω1 is fixed, the first photon can decrease J by 1 unit (a P-type resonance), increase it by 1 unit (an R-type resonance), or leave it unchanged (Q-type resonance). Figure 3

spectroscopy, it was discovered that the FWM processes could be identified using the observed vibrational pattern.18 For example, for a typical HRC2D spectrum in which ω1 is plotted on the y-axis and ω4 is plotted on the x-axis, process 1 in Figure 1 produces a rectangular vibrational pattern (displaying both vertical and horizontal alignment), while process 3 in Figure 1 produces a parallelogram vibrational pattern with vertical and diagonal (rather than horizontal) alignment. A similar approach would be useful for HRC3D spectroscopy; if available, differences among the rotational and vibrational patterns could be used to determine which FWM process is responsible for generating the observed peaks and patterns, but first, it is important to note that most of the FWM diagrams in Figure 1 can be eliminated because they are unlikely to provide a fully resonant FWM process. Because the vast majority of molecules have excited electronic states in the UV or visible region, any resonances involving the broadband near-infrared beam (ω2) and level a will usually involve overtones or combination bands that are too weak to provide a significant resonance enhancement. Therefore, none of the nonparametric processes (5−10) will be fully resonant. Of the parametric processes, the first and fourth have the best chance of being fully resonant because it is also unlikely that the first photon (ω3) for processes 2 and 3 will be fully resonant. For process 4, each photon can cause strong interactions if the resonance occurs between a pair of levels from two different electronic states that have a reasonably large Franck−Condon factor. The first photon can match a resonance between a level in the ground electronic state and a level in an excited electronic state. The second, third, and fourth photons can each also cause resonant interactions between the ground and excited electronic states if levels a and e are in the ground electronic state and levels f and g are from an excited electronic state. For process 1, two different excited electronic states are needed to avoid the harmonic oscillator selection rule of Δv = ±1 for levels in the same electronic state. Therefore, levels f and g should involve electronic states different from level a, and level h should involve a different electronic state compared to levels f and g. The next step is to determine the rotational pattern produced by each type of FWM process. For parametric processes 1 and 4, the rotational quantum numbers can evolve according to the diagram shown in Figure 2, which is based upon a selection rule

Figure 3. Evolution diagrams of the rotational quantum numbers for process 1 (top) and the resulting rotational peak patterns (bottom). The assumptions here are that ΔJ = ±1, ω1 is fixed, and ω3 is scanned. The first evolution diagram with yellow arrows is for an R-type process, which leads to the yellow peaks; the second diagram with the red arrows is for a P-type process, which leads to the red peaks.

displays the P-type and R-type cases during the evolution of the rotational quantum number for process 1 when ω1 is fixed and ω3 is scanned for a molecule where ΔJ = ±1. For a rigid rotor in which the rotational constant B has the same value for all levels, the results show four possible values for ω3 and two possible values for ω4. Figure 4 shows comparable rotational patterns for process 4. Figure 5 shows similar information for

Figure 2. Evolution of the rotational quantum number for processes 1 and 4; J is the rotational quantum number at the initial energy level (a). For each process, the level and field (see Figure 1) are directly below the corresponding rotational quantum numbers and arrow. The solid lines are for the selection rule ΔJ = ±1, and both solid and dotted lines should be used if ΔJ = 0, ±1. D

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processes 1 and 4 and for the rotational selection rule ΔJ = 0, ±1, which leads to seven possible values for ω3 and are three possible values for ω4. These results indicate that the rotational patterns can be used to determine several important pieces of information: the FWM process, the rotational selection rule, and whether the first laser field (ω1) creates a P-type, R-type, or Q-type resonance. For example, if the resulting rotational pattern displays a series of repeating triplets and each of the three peaks in the triplet has a different y-axis (ω3) value and one of two possible x-axis values, then the FWM process is process 4 and the selection rule is ΔJ = ±1. Two of the three peaks should have the same x-axis (ω4) value; if the remaining peak has a higher frequency than the other two, then ω1 is involved in a P-type resonance, and if the remaining peak has a lower frequency than the other two, then ω1 is involved in an R-type resonance. The relative positions of the peaks along the y-axis (ω3) may be different from those shown in Figures 3−6, which are based

Figure 5. Evolution diagrams of the rotational quantum numbers for processes 1 and 4 (top), followed by the resulting rotational peak patterns for process 1 (middle) and then process 4 (bottom). The assumptions here are that ΔJ = 0, ±1; ω1 is fixed; and ω3 is scanned. Q on the y-axis of the plots indicates three different changes in the rotational quantum number: J → J, J − 1 → J − 1, and J + 1 → J + 1. Note that the distances between peaks are not drawn to scale. The red, yellow, and green peaks are the same ones shown in Figures 3 and 4; the blue peaks are new ones that occur when ΔJ = 0 is allowed.

Figure 4. Evolution diagrams of the rotational quantum numbers for process 4 (top) and the resulting rotational peak patterns (bottom). The assumptions here are that ΔJ = ±1, ω1 is fixed, and ω3 is scanned. The first diagram with the yellow arrow is for an R-type process, which leads to the yellow and green peaks; the second diagram with the red arrow is for a P-type process, which leads to the red and green peaks. (The green peaks are common to both the P-type process and the Rtype process.)

upon the assumption that the ground and excited state rotational constants are roughly equal. Furthermore, process E

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For process 4, there are also nine possible ω3 values: J + 2 → J + 1, singlet

J + 1 → J + 1, doublet J → J + 1, triplet

J + 1 → J , doublet J → J , triplet

J − 1 → J , doublet J → J − 1, triplet

J − 1 → J − 1, doublet J − 2 → J − 1, singlet Figure 6. Resonance vibrational patterns for parametric processes 1 and 4. The term “rare” indicates that triple resonances are so infrequent that a regular repeating pattern might not be produced.

Again, nine layers will be produced for each initial J value, each composed of singlets, doublets, and triplets. For process 4 with ω1 scanned and ω3 fixed, the doublets and triplets are vertically aligned. The first three ω1 resonances in the list above will create a common peak, the second three will also produce a common peak, and so will the final three. Once again, for the ΔJ = ±1 selection rule, the number of possible ω3 resonances drops to four, and the resulting patterns are all either singlets or vertically aligned doublets. Compared with an ω3 scan, ω1 scans for processes 1 and 4 result in a larger number of layers (9 layers for ΔJ = 0, ± 1 and 4 layers for ΔJ = ±1) for each initial J″ value, and they produce a wider variety of patterns for each layer. Some layers might also be more difficult to identify because they consist of singlets or doublets, which might be harder to recognize than a pattern containing three or more peaks. For example, the predicted rotational patterns for processes 1 and 4 with both types of scans and the ΔJ = ±1 selection rule are summarized in Table 2. Since repeating three-peak patterns are easier to find than

4 patterns provide an additional advantage over that of process 1: the P-, Q-, and R-type processes have some common points that may facilitate the grouping of slices that have the same initial J″ value when comparing different 2D slices of the 3D spectrum. For process 1, the P-, Q-, and R-type processes each have different ω3 (y-axis) resonances, so there are no common points other than the central Q branch. The situation is significantly different when ω3 is fixed and ω1 is scanned for processes 1 and 4. For such scans, the y-axis in the spectrum is now ω1 and the x-axis remains ω4. For process 1, use of the diagram in Figure 2 and the selection rule ΔJ = 0, ±1 leads to nine different ω3 resonance values for each initial level J value: J + 1 → J + 2, singlet

J + 1 → J + 1, doublet

Table 2. Predicted Rotational Patterns for Processes 1 and 4 and Both Types of Scans for ΔJ = ±1a

J + 1 → J , triplet J → J + 1, doublet J → J , triplet J → J − 1, doublet J − 1 → J , triplet

J − 1 → J − 1, doublet a

J − 1 → J − 2, singlet

For each of these nine values, there is only one ω1 value, but the number of ω4 values varies from 1 to 3. The result will be nine layers in the 3D plot for a given initial value of J, each with patterns of repeating singlets, doublets, or triplets. Furthermore, the peaks comprising each doublet or triplet will have identical y-axis values (i.e., they will line up horizontally). In the case of the triplets, the spacings among the three peaks are not equal. There are no peaks common to all nine layers, but the first three ω1 resonances in the list above will produce a common peak, the second three will also create a common peak, and so will the final three. In the simpler case that the selection rule is ΔJ = ±1, the number of possible ω3 resonances drops to four and the resulting patterns are all either singlets or horizontally aligned doublets.

See text.

repeating two-peak or one-peak patterns, the best general strategy for creating 2D slices appears to be keeping ω1 fixed while scanning ω3 and selecting laser wavelengths so that processes 1 and 4 are fully resonant. Furthermore, for experiments in which ω3 is fixed and ω1 is scanned, predicting an appropriate set value for ω3 can be difficult because such information is not easily obtained by looking at absorption or HRC2D spectra. For these reasons, ω3 scans are strongly recommended over ω1 scans. In addition to the rotational pattern, the vibrational pattern in the 2D slice of a HRC3D spectrum can also provide insight into the responsible FWM process. The third-order susceptibility equations for the two candidates most likely to be fully F

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Ultimately, the strongest FWM process depends upon the choice of lasers and the energy level diagrams of the molecule. For the purpose of pattern recognition, the most useful FWM process is one that produces both a rotational pattern with three peaks (as opposed to just one or two peaks) and a regular vibrational pattern (rectangular or parallelogram). The patterns that satisfy these two requirements for the choice of lasers discussed above (ω1 = ωvis, ω2 = ωbroad, and ω3 = ωNIR) are processes 1 and 4 with an ω3 scan. If this approach is taken, the vibrational and rotational patterns may then be used to determine whether process 1 or process 4 is responsible for the observed peaks. The combined rotational and vibrational patterns for the two best candidates (processes 1 and 4 with an ω3 scan) are shown in Figure 7. (The rotational patterns

resonant (processes 1 and 4) can be generated using density matrix diagrams, and the simplified expressions are χ1(3) = ξ(Δfa Δha Δga )−1 χ4(3) = ξ(Δfa Δea Δga )−1

where the ξ is the product of transition dipole moments for the four fields and the Δ’s are resonance denominators: Δea = ωea − (ω1 − ω2) − i Γea

Δfa = ωfa − ω1 − i Γfa Δga = ωga − ω4 − i Γga Δha = ω ha − (ω1 + ω3) − i Γha

where Γ is the dephasing between the two levels labeled in the subscript. Figure 6 displays the kinds of resonance lines that these resonant denominators will generate for a coherent 2D slice of the HRC3D spectrum. First consider the situation in which ω3 (narrowband NIR beam frequency) is scanned while the UV−visible input laser frequency (ω1) is fixed. If ω1 has been tuned to a resonance, then the left side of the diagram in Figure 6 is in the Δfa plane so that Δfa = −iΓfa. Both equations for processes 1 and 4 have a Δfa denominator, which results in one resonance. For process 1, two additional resonances occur at the intersection between resonance lines Δha (horizontal) and Δga (vertical). Therefore, a triple resonance situation occurs at the intersection of these two perpendicular lines within the Δfa resonance plane. In the harmonic oscillator approximation, Δha and Δga will each produce a set of somewhat regularly spaced horizontal and vertical lines (each with different vibrational quantum numbers). Therefore, the triply resonant vibrational patterns will be regularly spaced, as well, forming a rectangular grid of rows and columns. For a simple diatomic molecule, the distance between rows will be roughly equal, and the distance between columns will also be roughly equal because of the roughly equal harmonic spacing between vibrational levels. For process 4 (ω3 scanned, ω1 fixed), the resonance denominator Δea produces a diagonal resonance line in the Δfa resonance plane. Therefore, the triply resonant vibrational patterns occur at intersections between vertical double resonance lines (from Δga), diagonal double resonance lines (from Δea), and the resonance plane Δfa. The resulting parallelogram has an appearance different from that of process 1; many peaks will have the same x-axis value, but they will not have the same y-axis values (except by accidental coincidence). The resulting vibrational pattern changes if the ω1 is scanned while ω3 is fixed. If ω3 has been tuned to a resonance, then the right side of the diagram in Figure 6 (in the Δda plane, where Δda = ωda − ω3 − iΓda = −iΓda) becomes useful and processes 1 and 4 now require the overlap of three separate resonance lines rather than the overlap of two resonance lines and one resonance plane. For process 1, two resonance lines are horizontal (Δfa and Δha) and one is vertical (Δga). Therefore, triple resonances occur only when the Δfa and Δha resonance lines are on top of each other. Because this situation rarely occurs, peaks from such a process are infrequent and are irregularly spaced along the y-axis. For process 4, both Δea and Δga produce vertical resonance lines, so triple resonances from this process are also rare and irregularly spaced along the x-axis.

Figure 7. Combined rotational and vibrational patterns for processes 1 and 4 when ω3 is scanned, ω1 is fixed, and ΔJ = ±1. For the process 4, ω3 scan spectra, none of the peaks have the same y-axis values, except by coincidence.

shown in this figure are for ΔJ = ±1; a trivial substitution of the rotational patterns shown in Figure 5 can be made for cases when ΔJ = 0, ± 1). Fortunately, these two processes are easy to distinguish because of their different vibrational patterns. Process 1 produces a vibrational pattern that aligns horizontally and vertically (i.e., a rectangular grid), whereas the ω3-scanned process 4 creates a vibrational pattern made from vertical and diagonal resonance lines (i.e., a parallelogram-shaped grid). Furthermore, processes 1 and 4 yield different rotational patterns (see Figures 3 and 4). For process 4, none of the peaks have exactly the same y-axis value (except by coincidence), and two of the peaks appear in both the P-type resonance and the R-type resonance (selected by ω1). For process 1, two of the three peaks that form a “triplet” must have exactly the same yaxis value, and all of the peaks will change their position when ω1 is changed between a P-type resonance and an R-type resonance. The rotational spacing between peaks is also different for these two processes, as indicated in Figures 3 and 4. One additional benefit of HRC3D spectroscopy over HRC2D spectroscopy is that it can provide information about intermediate levels. For example, consider the intermediate level e in process 4. For an ω3 scan with a frequency equation of ω4 = ω1 − ω2 + ω3, the y-axis corresponds to ω3 and the x-axis corresponds to ω4. Rearranging the above equation yields a linear equation with a slope of 1 and a y-intercept of ω2 − ω1, which also equals ω3 − ω4. Therefore, the difference in energy between levels e and a is equal to the negative of the y-intercept of the line that runs through the vibrational patterns that align diagonally in the spectrum. This line runs through the “common points” that are G

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a dye laser. Figure 8 shows a HRC2D spectrum of Br2. The FWM process was believed to be similar to process 1 or process

colored green in Figure 4. As mentioned earlier, these points are common to both the P-type and R-type patterns for a given J″ value, whereas the other two points (red and yellow in Figure 4) appear only for the P-type or R-type triplets, respectively. The two common points are relatively easy to identify because the line forming them has a slope of m = 1, and they therefore line up with other common points that involve the same overtone or combination band (level e). A similar approach can be developed for process 1, in which the energy of level h should be equal to ω1 + ω3 for a given row. If the sample consists of a mixture, it is possible that species selectivity may be achieved by analyzing only the patterns that lie along a specific diagonal corresponding to known overtones and combination bands of the molecule of interest. This pattern-based selectivity would be in addition to that made possible by fixing the dye laser at a specific wavelength during the ω3 scan.

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EXPERIMENTAL METHODS The design of the HRC2D and HRC3D spectrometers are discussed in detail elsewhere,16,20 so only a brief description is provided here. The visible beam for ω1 was produced using a Coherent Scanmate Pro dye laser (line width of 0.15 cm−1). The broadband near-infrared beam for ω2 was produced using a broadband degenerate optical parametric oscillator (OPO) that generates light over a continuous range from 1150 to 1700 nm. The tunable near-infrared beam for ω3 was generated using the idler beam from a narrowband OPO (Spectraphysics MOPO 730, scanned from 1000 to 1100 nm, with a line width of 0.2 cm−1). The three input beams were combined using collinear phasematching, and the input beams were removed from the FWM signal using optical filters. For HRC3D spectroscopy, the three axes were the frequency of the tunable visible laser (ω1), the frequency of the narrowband tunable infrared beam (ω3), and the frequency of the detected light (ω4). For each 2D slice, however, spectra were obtained by fixing the dye laser wavelength while varying the MOPO 730 wavelength and monitoring the output with a 1.25- meter monochromator and CCD (SPEX 1250m with a 2400 g/mm grating and a CCD with 2048 columns of 13 μm wide pixels). The dependence of the signal on all three input beams was checked periodically by blocking each of the input beams and ensuring that the observed peaks disappeared. For the HRC2D spectra, the MOPO 730 signal beam was used for ω1, and the broadband OPO beams were used for ω2 and ω3. The resulting FWM signal was generated at ω4 = ω1 − ω2 + ω3 and filtered to remove the three input beams before being detected by the monochromator and CCD. The MOPO step size during the scans was 0.002 nm, and the line width was approximately 0.2 cm−1. The resulting spectra were processed using a revised version of nmrDraw that included a 2D peak picker and a Fortran program that finds all of the reproducible peaks within two sets of data run with the same parameters. To make the patterns easier to observe, only peaks above a certain intensity threshold were then plotted in the resulting spectra.

Figure 8. HRC2D spectrum of Br2. Congestion due to small molecular constants, elongated double Fortrat parabolas, and isotopologues makes it difficult to identify peaks that have the same vibrational and initial rotational quantum numbers (see text).

4 in Figure 2, except that both ω2 and ω3 were broadband. For HRC2D spectra that are relatively congestion-free, peak analysis and assignment may be carried out using recently developed pattern recognition techniques.16 These techniques involve drawing boxes with corners that lie upon peaks that have identical vibrational quantum numbers and initial rotational quantum number. Series of related boxes form double Fortrat parabolas that may resemble the shape of an “X” if the rotational constants in the upper and lower electronic states are similar. For the HRC2D spectrum shown in Figure 8, however, the level of congestion is troublesome due to several factors. First, the rotational and vibrational molecular constants are small as a result of the relatively large mass of the bromine atom. Second, the rotational constants B′ (∼0.06 cm−1) and B″ (∼0.08 cm−1) differ considerably, resulting in a rotational pattern of elongated double Fortrat parabolas. This elongation causes the parabolas associated with different vibrational quantum numbers to run into each other. Third, bromine’s 2D spectrum is further complicated because of the presence of peaks from all three naturally occurring isotopologues (79Br2, 81 Br2, and 79,81Br2). The parabolas produced by different isotopologues overlap; they are shifted in position from each other because their vibrational constants slightly differ. Finally, HRC2D spectroscopy does not provide the ability to use selectivity or other methods to address the resulting overlap and congestion problems. It also does not provide a method for verifying the accuracy of proposed assignments. Therefore, attempts to connect groups of peaks that appear to form a box can result in assignments with a high degree of uncertainty. The following example illustrates how HRC3D spectroscopy can be used to help make assignments and confirm their accuracy. Circled in Figure 8 are a few (three) HRC2D peaks with ω4 at 542.17 ±0.01 nm and ω1 at 611.82 ±0.01 nm. The two long rectangles drawn around this group of peaks show that there are several other peaks that have similar x or y resonances, indicating that they might involve the same vibrational and initial rotational quantum numbers. Without additional information, however, it would be difficult to select the four best peaks that should form a box with a high level of

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RESULTS AND DISCUSSION Bromine is a spectroscopically well-characterized molecule21−24 that serves as a useful test case. The well-known B−X transition for bromine requires light in the visible region, so achieving an electronic resonance enhancement is easily accomplished using H

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have an x-axis value of 541.90 ± 0.01 nm and a y-axis value of 611.82 ± 0.01 nm, indicating the second of the four coordinates needed for the HRC2D box. HRC3D spectroscopy can also be used to confirm the accuracy of a grouping in the HRC2D spectrum. The HRC2D spectrum indicates that several (four) choices are available for the two remaining coordinates that should form the other two corners of the box in the HRC2D spectrum. The two remaining peaks should have the same x-axes values (542.18 and 541.90 nm) but a different y-axis value. The top part of Figure 10

confidence. The top part of Figure 9 shows a HRC3D spectrum created with the dye laser (ω1) set to 611.821 (± 0.002) nm

Figure 9. Coherent 3D spectrum of bromine for which the dye laser was set to 611.821 (± 0.002) nm. For the experimental spectrum (top), differently shaped circles, boxes, or diamonds were used to indicate differently sized rotational patterns. For the simulated peaks (lower half), molecular constants for only 79,81Br2 were used, and colors were used to differentiate these three types of peaks in the simulated spectrum: the blue and red markers are for two different Rtype processes, and the green markers are for a P-type process. (See text for more details regarding the v and J values for the three different rotational patterns.)

and with a narrowband tunable near-infrared beam for ω3 that is scanned instead of the broadband near-infrared beam that was used for the HRC2D spectrum. The easily recognized patterns reveal several noteworthy features. First, the rotational patterns consist of triplets, indicating that the selection rule is ΔJ = ±1. For each of these types of triplets, none have the same y-axis values, so process 1 can be ruled out. Instead, process 4 is the likely candidate; the vibrational patterns reveal a structure that shows repeating columns, but not repeating rows, and therefore resemble parallelograms (vertical and diagonal alignment), as expected from process 4. The fact that there are at least three differently sized triplets (marked by the enclosing ovals, triangles, and rectangles in Figure 9) suggests the possibility of multiple sets of quantum numbers or isotopologues. Finally, two types of triplets (enclosed in ovals and triangles) contain a single peak on the left side, indicating that the ω1 frequency in process 4 matches a R-type resonance, whereas one type of triplet (enclosed using a diamond) appears to be a P-type resonance. The long red vertical box in Figure 9 encloses peaks that are all members of the same kind of triplet (enclosed in ovals) that also have an x-axis value (542.18 nm) similar to the circled peaks in Figure 8. As expected, the other two peaks in the triplet have matching x-axis wavelength values (541.90 nm). Consequently, this wavelength (541.90 nm) should be equal to the x-axis value for two of the four matching peaks in the HRC2D spectrum. In agreement with expectations, the HRC2D spectrum shows a small group of (three) peaks that

Figure 10. Experimental HRC3D spectrum of bromine with the dye laser set to 612.336 (±0.002) nm (top). Triangles are drawn around the observed 81Br2 patterns, rectangles are drawn around the 79Br2 patterns, and circles are drawn around the 79,81Br2 patterns. The simulated spectrum (middle) shows the peaks for the same three isotopologues, indicated by green, red, and blue markers. The bottom spectrum shows overlapping points from the top of Figure 9 (red peaks, R-type) and the top of Figure 10 (blue peaks, P-type); common peaks appear as overlapping red and blue peaks.

shows a HRC3D spectrum with the dye laser set to one of the possible y-axis values (at 612.336 nm). The fact that the common peaks at this wavelength overlap with the common peaks from when the dye laser was set to 611.821 nm confirms that these two resonance wavelengths involve the same vibrational and initial rotational quantum numbers (see the bottom spectrum in Figure 10). Figure 11 shows the final result: the box that connects the four related peaks (identical vibrational and initial rotational quantum numbers) in the HRC2D spectrum. This final step of drawing the box, along I

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method was then used to assign the quantum numbers for the triplets enclosed in ovals in Figure 10. The first step was to identify a series of triplets that lie along the same diagonal. The peaks that comprise each of these triplets should have identical values of J″ and v″ for level a as well as an identical value of v″ for level e (see Figure 1). Two of the three peaks in each triplet are “common peaks”, and a straight line with a slope of 1 can be drawn connecting all of these common peaks if both axes on the spectrum are in units of cm−1. The negative of the yintercept for the resulting diagonal line should be equal to the sum of ΔG + ΔF between levels e and a. Values of ΔG for several v″ → v″ resonances between levels a and e were then calculated. For a room temperature sample, it was reasonable to assume that several vibrational levels (v″ = 0−5 for level a) are initially populated. Two approximations were used to help identify the correct values of v″ for levels a and e. The first assumption was that ΔG > ΔF so that ΔG should be roughly comparable in size to the y-intercept. This assumption was used to calculate an approximate value of v″ for level e, which was then used to calculate an approximate value for J″ by measuring the difference in y-axis energy spacing between the two peaks in the triplet that share the same x-axis value and setting that equal to Be (4J″ − 2). (For an R-type process, the spacing should equal Be (4J″ + 6) instead). The second assumption was that the value of ΔF is not strongly dependent upon the value of J″; for Br2, ΔF changed by only 1−2 cm−1 for neighboring values of J″. The resulting values for ΔG + ΔF were then compared with the y-intercept, and the pairs of v″ values (for energy levels a and e) that fit best were identified. The values of J″ for these best fits were then refined to minimize the discrepancy. This procedure was carried out for all three isotopologues. The correct isotopologue could be easily identified because incorrect ones had simulated triplet widths (values of Bg (4J″ + 2)) that differed considerably from the observed experimental values (by several cm−1). In addition to the triplets enclosed in ovals, additional unexpected patterns (enclosed in triangles and rectangles) in Figure 10 are attributed to additional ω1 resonances due to the relatively broad line width of the dye laser (0.15 cm−1). Both the expected and unexpected patterns could be assigned. The triplets enclosed in ovals correspond to a 79,81Br2 resonance at 16326.42 cm−1, where v″ = 1, J″ = 62 and v′ = 6, J′ = 61. Eight vertical columns of triplets are due to a vibrational pattern involving v′ = 25−32 (from left to right) for level g, and four diagonal lines of triplets are observed, corresponding to v″ = 31−34 (top to bottom) for level e. The triplets enclosed in rectangles are attributed to a 79Br2 resonance at 16326.53 cm−1, where v″ = 2, J″ = 49 and v′ = 8, J′ = 48. The lower half of Figure 10 shows simulations based upon these assignments. The molecular constants used for this simulation were taken from references 22−24. Twelve columns of these triplets were simulated because of the vibrational pattern involving v′ = 29− 40, and the four diagonal lines of triplets were anticipated due to v″ = 32−35. However, only a small number (3) were actually observed, presumably because the dye laser was slightly farther off-resonance. The triplets enclosed in triangles correspond to a 81 Br2 resonance at 16326.39 cm−1, with v″ = 2, J″ = 48 and v′ = 8, J′ = 47. Twelve predicted columns were due to v′ = 29−40 and four diagonals were due to v″ = 32−35. Again, not all of these simulated triangles were observed because of being slightly off-resonance. The agreement between experimental and simulated peak positions along the x- and y-axes was typically within a few hundredths of a nanometer. The strong

Figure 11. HRC2D spectrum with a red box drawn among four peaks that were confirmed using HRC3D spectroscopy. The creation of this first box facilitates the drawing of additional boxes (similar but not identical in size), some of which are shown adjacent (with the same vibrational quantum numbers but different initial rotational quantum numbers) and farther away (different vibrational quantum numbers but same initial rotational quantum numbers as the red box and the box immediately to its left).

with comparison between resonance wavelengths in the HRC2D and HRC3D spectrum, can be used to help further specify which peaks are best when dealing with a group of several very closely spaced peaks. And the resulting box can then be used to help draw other boxes (also shown in Figure 11), thereby providing tentative assignments that may be further verified using HRC3D spectroscopy. Further analysis of the HRC3D spectrum can lead to additional information about and assignment of molecular constants, quantum numbers, competing processes, species, and intermediate levels. For example, the dimensions and shape of a triplet reveal information about the rotational constants and quantum numbers, and relatively simple relationships between the triplet dimensions and the rotational constants and quantum number can be readily exploited. The width of the triplet depends upon the two possible output (ω4) frequencies for a given initial rotational quantum number, and for process 4 (ω3 scan), these two frequencies correspond to Te′ − Te″ + G′(v′) − G″(v″) + F ′(J ″ + 1) − F ″(J ″)

and Te′ − Te″ + G′(v′) − G″(v″) + F ′(J ″ − 1) − F ″(J ″)

where J″ is the initial rotational quantum number of the molecule. If the molecule acts like a rigid rotor with a small amount of vibration−rotation interaction, then the difference between these two frequencies is [B′(J ″ + 1)(J ″ + 2) − B″(J ″)(J ″ + 1)] − [B′(J ″ − 1)(J ″) − B″(J ″)(J ″ + 1)] = B′(4J ″ + 2)

where B can be written in terms of Dunham coefficients (B = Y01 + Y11 (v + 1/2) + Y21 (v + 1/2)2 + ...). The four y-axis (ω3) resonance frequencies can be used to calculate the y-axis spacings shown in Figure 4, where the subscripts on the rotational constants indicate the corresponding level in the accompanying four-wave mixing diagram. The following J

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frequency while scanning the tunable near-infrared beam frequency produces spectra with patterns that are easier to recognize and interpret. To analyze molecules such as bromine (e.g., no excited electronic states in the near-infrared region and a selection rule of ΔJ = ±1), the FWM processes that are most likely to be successful are processes 1 and 4, which produce rotational patterns consisting of three peaks that form triplets. These two processes are easily distinguishable on the basis of their different rotational and vibrational patterns. Achieving the desired fully resonant FWM process may be easier for process 4 than process 1 because each field can create resonances between the ground electronic state and an excited electronic state; a simple system with a ground electronic state and just one bound excited electronic state would be sufficient. Unlike that for process 1, the rotational patterns for process 4 also contain common peaks that can be used to help match all spectra containing peaks with the same vibrational and initial rotational quantum numbers. When used in conjunction with HRC2D spectroscopy, HRC3D spectroscopy can help identify and confirm the four peaks in the HRC2D spectrum that have the same vibrational and initial rotational quantum numbers. The x-axis resonances for these peaks in the HRC2D spectrum should have the same value as the x-axis resonances for corresponding peaks in the HRC3D spectrum. Unlike the HRC2D spectrum, however, the HRC3D spectrum will contain a much smaller subset of the peaks as a result of the selectivity by one of the tunable lasers. For example, it is possible that all recognizable patterns will come from a single species in a mixture. The smaller subset makes it easier to recognize patterns, recognize their corresponding FWM processes, and obtain values for quantum numbers and molecular constants. HRC3D spectroscopy can also provide a means for identifying the intermediate level (h for process 1 and e for process 4) by studying the diagonals that can be drawn between triplets in HRC3D spectra. The example provided here illustrates the use of HRC3D spectra to determine quantum numbers and assign peaks when high-quality molecular constants are known. One can imagine that the reverse should also be true: determining the molecular constants should be possible if the quantum numbers are known. Obtaining a full set of hiqh-quality molecular constants will probably require the acquisition of a substantial number of 2D slices. Each slice can require an hour or more to acquire, so the time required to generate a complete 3D cube, or enough of a cube to obtain a full set of Dunham coefficients, may be considerable. Nonetheless, the additional time and effort could be justified for large numbers of molecular systems that yield spectra that are too congested for conventional and HRC2D spectroscopy.

diagonal line that runs through the spectrum was caused by unwanted stray light from the MOPO 730 signal beam during the scan. The strong vertical line was caused by unwanted stray light from nearby fluorescent lights (Hg emission at λ = 546.075 nm). For the triplets enclosed in ovals in Figure 9, the quantum numbers involved the 16340.21 cm−1 resonance between v″ = 1, J″ = 62 and v′ = 6, J′ = 63 for 79,81Br2. For this pattern, the columns correspond to v′ = 25−27 and the diagonals correspond to v″ = 31−33. Once again, additional unexpected patterns were observed as a result of the relatively wide bandwidth of the dye laser. One of these patterns (enclosed in rectangles) was another R-type process at a calculated value of ω1 = 16340.18 cm−1, corresponding to v″ = 0, J″ = 71 to v′ = 4, J′ = 72 for the same isotopologue. For this pattern, the columns correspond to v′ = 21−23 and the diagonals correspond to v″ = 30−32. An additional unexpected pattern (enclosed in diamonds) was observed and attributed to a P-type process at ω1 = 16340.36 cm−1, corresponding to v″ = 2, J″ = 43 to v′ = 8, J′ = 42, also for the same isotopologue. For this pattern, the columns correspond to v′ = 28−32, and the diagonals correspond to v″ = 32−35. All three patterns identified in Figure 9 were from the same isotopologue. For both Figures 9 and 10, the experimental results show excellent agreement with simulated results (displayed underneath) that were based on the assigned quantum numbers above. Both experimental spectra are dominated by peaks from 79,81 Br2, with initial level quantum numbers v″ = 1 and J″ = 62. Unexpected peaks from other resonances or isotopologues can be distinguished from the expected ones because the vibrational and rotational patterns are significantly different; the size of the triplets and their locations in the plot are different. Vibrational and rotational pattern recognition therefore facilitates additional selectivity by providing the ability to determine the number of simultaneously contributing ω1 resonances/ isotopologues and to classify the peaks accordingly. Pattern recognition made it possible to assign all peaks that were part of a regularly repeating pattern; peaks that were not part of a recognizable pattern remain unidentified. The use of lasers with narrower bandwidths should further improve species’ selectivity and could reduce the number of these patternless unidentified peaks.

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CONCLUSION HRC2D spectroscopy can be used to help overcome problems of congestion and pattern obscuration in high-resolution electronic spectroscopy. For many molecules, however, expansion to the second dimension may not be sufficient to overcome severe congestion. HRC3D spectroscopy may be used in conjunction with HRC2D spectroscopy to fix these severe congestion problems. In addition to using an additional dimension to increase the spacing between peaks, HRC3D spectroscopy also provides selectivity by species and quantum number(s) and the ability to confirm the accuracy of peak assignments in HRC2D spectra. These findings indicate that an effective procedure for carrying out high-resolution coherent 3D spectroscopy is to use one tunable visible/UV beam to select a resonance and two near-infrared beams, one that is scanned in frequency and the other that is broadband. This approach limits the number of possible FWM processes and also employs multichannel detection to help reduce collection times that might otherwise be prohibitively long. Fixing the tunable visible/UV beam

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ASSOCIATED CONTENT

S Supporting Information *

Included as Supporting Information are tables containing peaks assigned by vibrational and rotational quantum numbers. Both experimental and simulated peak positions are provided. This material is available free of charge via the Internet at http:// pubs.acs.org.

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AUTHOR INFORMATION

Corresponding Author

*Address: Box 307, 350 Spelman Lane, Atlanta, GA 30314. Phone: 404-270-5742. E-mail: [email protected]. K

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(19) Chen, P. C.; Gomes, M. Two-Dimensional Coherent Double Resonance Electronic Spectroscopy. J. Phys. Chem. A 2008, 112, 2999−3001. (20) Chen, P. C.; Wells, T. A.; Strangfeld, B. R. High Resolution Coherent 3D spectroscopy of Br2. J. Phys. Chem. A 2013, 117, 5981− 5986. (21) Focsa, C.; Li, H.; Bernath, P. F. Characterization of the Ground State of Br2 by Laser-Induced Fluorescence Fourier Transform Spectroscopy of the B3Π0+u−X1Σg+ System. J. Mol. Spectrosc. 2000, 200, 104−119. (22) Gerstenkorn, S.; Luc, P.; Raynal, A.; Sinzelle, J. Description of the absorption spectrum of bromine recorded by means of Fourier transform spectroscopy: the (B3Π0+u ← X1Σg+) system. J. Phys. (Paris) 1987, 48, 1685−1696. (23) Gerstenkorn, S.; Luc, P. Analysis of the long range potential of 79 Br2 in the B3Π0+u state and molecular constants of the three isotopic bromine species 79Br2, 79,81Br2, and 81Br2. J. Phys. (Paris) 1989, 50, 1417−1432. (24) Barrow, R. F.; Clark, T. C.; Coxon, J. A.; Yee, K. K. The B3Π0u+ − X1Σg+ System of Br2: Rotational Analysis, Franck−Condon Factors, and Long Range Potential in the B3Π0u+ State. J. Mol. Spectrosc. 1974, 51, 428−449.

†

(B.R.S.) Department of Chemistry and Chemical Biology, Georgia Institute of Technology, Atlanta, GA 30332. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This work was supported by National Science Foundation Grant CHE-0910232. Additional support was provided by NSF Grants EEC-0310717 and CHE- 1337522. The authors also acknowledge Jessica Robinson and Kitsi Mahasi for their assistance in generating the HRC2D spectra.

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