The CH stretch overtone region (5750–6300 cm−1) of benzene and naphthalene is assigned herein using anharmonic quantum chemical computations, and the trend of how this extends to larger polycyclic aromatic hydrocarbons (PAHs) is established. The assignment of all experimental bands to specific vibrational states is performed for the first time. Resonance polyads and the inclusion of 3-quanta vibrational states are both needed to compute accurate vibrational frequencies with the proper density-of-states to match the experimental band shape. Hundreds of 3-quanta states produce the observed band structure in naphthalene, anthracene, and tetracene, and this number is expected to increase drastically for larger PAHs. The width and shape of the main peak are consistent from naphthalene to anthracene, necessitating further exploration of this trend to confirm whether it is representative of all PAHs in the CH stretch overtone region. Understanding observations of PAH sources in the 1–3 μm region from the NIRSpec instrument aboard JWST requires new computational data, and this study provides a benchmark and foundation for their computation.

Polycyclic aromatic hydrocarbons (PAHs) are found in various environments ranging from the Earth’s atmosphere to the interstellar medium, and they are known carcinogens produced in incomplete combustion processes as well as high temperature food preparation. Furthermore, their aromatic structure makes them chemically intriguing, and their stability and accessible CH stretch overtone transitions have historically been a window into studying the properties of highly vibrationally excited polyatomic molecules. PAHs have also been of significant interest to the astrochemistry community for several decades now due to their ability to readily explain astronomical IR observations, leading to their perceived ubiquity in the universe.1,2 With the launch and operation of JWST, this has only grown in interest within the past few years.

Infrared (IR) observations of many astronomical objects (e.g., protoplanetary disks, planetary nebulae, reflection nebulae, etc.) show strong bands most commonly attributed to the vibrational emission from PAHs.1 Through their various stretching and bending modes, PAHs are most notably believed to be the main carriers of the IR emission features at 3.3, 6.2, 7.7, 8.6, 11.2, 12.7, and 16.4 μm.1,3 Typically, astronomical observations of PAH IR spectra are analyzed through direct comparison with, or by fitting to, known standards. Resources such as the computed and experimental vibrational absorption spectra contained in the NASA Ames PAH IR Spectroscopic Database (PAHdb)4–7 aid in this. From these fits, the chemical and physical properties of the astronomical sources, such as the abundance of PAHs, PAH charge states, gas temperature, and radiation environment, can be deduced.8–17 These methods predominantly focus specifically on the emission at and around 3.3 μm and, more generally, between 5 and 20 μm.3,18,19 However, emission below 3.3 μm has been largely ignored due to telescope limitations and resources such as PAHdb lacking the necessary data.

The CH stretch, CH in-plane and out-of-plane bends, and CC skeletal deformation fundamentals of PAHs mainly occur in the 3–20 μm wavelength region.20,21 These vibrational modes have been the focus of the vast majority of spectroscopic studies of PAHs. Conversely, the overtones of the CH stretch fundamentals (v = 2 ← 0, v = 3 ← 0, etc.) and combination bands that include at least one quantum of CH stretch (νCH + νx) all occur in the 1–3 μm range22 and contain unique information about the astronomical PAH population. The Infrared Space Observatory (ISO)11–15,23 and AKARI24 both observed parts of the 2.5–5 μm region but lacked the spectral resolution and sensitivity to detect such weak features. With the launch of JWST, IR emission spectra have become available with unprecedented spatial and spectral resolution. Thanks to the onboard NIRSpec instrument, completely new data in the 1–3.3 μm region is now available.25–27 Because of the breakthrough jump in the quality and breadth of the NIRSpec data, an equally commensurate breakthrough in computational and experimental capabilities is required to fully utilize the value of the never-before-seen spectra.

While there is a lack of reference IR data on larger PAHs (with more than 50 carbon atoms), a small number of early studies pertaining to PAH CH overtone spectroscopy have examined small PAHs such as benzene (C6H6) and naphthalene (C10H8). In one early exploration from 1929, the experimental absorption positions of the first (v = 2 ← 0) and second (v = 3 ← 0) overtones of benzene are reported at 5950 and 8760 cm−1,28 respectively. Computational analysis of the benzene overtone series actually began in 1968,29 and a single anharmonicity constant for this molecule was reported at that time. In 1982, Reddy et al. obtained the first gas-phase IR absorption spectrum of various benzene CH stretch overtones and found the first and second CH stretch overtone band centers at 5972 and 8786 cm−1, respectively.30 A decade later, Scotoni and co-workers measured the second overtone band (v = 3 ← 0) of benzene and concluded that part of the CH oscillator strength is borrowed by states on the lower frequency side.31 

There are fewer studies on the CH stretch overtones of naphthalene. First, observation of inequivalent CH local modes via measurement of high vibrational quanta overtones of solid-state and vapor phase naphthalene was reported.32–34 Following this, Sczcepanski and Vala35–37 postulated, based on experimental absorption spectra, that PAH cations contribute significantly to astronomical IR emission based on the spectrum of the naphthalene cation. Kjaergaard et al., then, assigned seven combination bands and overtones falling between 5823 and 6133 cm−1 to the CH stretch overtones in gas-phase experiments.38,39

Now, anharmonic vibrational frequency computations are needed to fully assign the experimental spectra of benzene and naphthalene. The current state-of-the-art anharmonic methodology for the computation of IR spectra of PAHs has been validated for accuracy in the 3–20 μm region through various studies showing a mean absolute difference of between 5 and 10 cm−1 compared with experiments.21,40–52 However, these methods have yet to be used to investigate the 1–3 μm CH stretch overtone region. In this article, available experimental data are used to benchmark two current state-of-the-art second order vibrational perturbation theory (VPT2) computational methodologies as applied to the CH stretch overtone IR spectroscopy of benzene and naphthalene. In addition, detailed spectroscopic assignments beyond those previously possible are made. This paves the way for future computational investigations of PAH CH stretch overtone spectroscopy and the ability to holistically analyze and interpret JWST NIRSpec observations.27,53,54

The geometry, normal modes, and harmonic frequencies of benzene and naphthalene are determined by optimization at the B3LYP55/N07D56 level of theory using a custom integration grid consisting of 200 radial shells and 974 angular points per shell (compared to 99 radial shells and 590 angular points per shell in the default UltraFine grid)21 with Gaussian 16.57 The N07D basis set adds additional diffuse and polarization functions to the 6-31G(d) basis set, an augmentation that has been shown to increase the accuracy of anharmonic computations of large, aromatic systems such as PAHs.58 The benzene computations are performed in the D2h point group, which splits the higher symmetry D6h irreducible representation into two lower symmetry irreducible representations that cause near-degenerate normal modes. Following this, the quadratic, cubic, and quartic force constants (quartic force field; QFF) are computed via small displacements along the orthogonal normal modes. A QFF is a truncated, fourth-order Taylor series expansion of the potential surface surrounding the equilibrium geometry, following the formula:
(1)
The QFF is computed in normal mode coordinates via a linear relationship to produce a Cartesian coordinate QFF.59 In addition, semi-diagonal quartic terms are employed, as is the default in Gaussian 16, which has been shown to reduce computational cost while retaining chemical accuracy.60 

Once the QFF has been computed, two different anharmonic procedures are performed via second order vibrational perturbation theory (VPT2).61–65 The first is a straightforward 3-quantum anharmonic computation using the built-in standard VPT2 anharmonic code within Gaussian 16. The second involves the use of a locally modified version of the SPECTRO66 program to compute anharmonic frequencies of up to 2-quanta states. SPECTRO has the advantage of using resonance polyads in the anharmonic computations,67,68 which allows for the proper treatment of resonances (e.g., Fermi, Darling–Dennison) between coupled vibrational states. When two vibrational states of the same symmetry are close in frequency, they produce a near-singularity in the VPT2 treatment. These states are removed from the perturbation treatment and included in a resonance polyad matrix. The polyad matrices allow for the treatment of resonance effects while having the advantage of separating the vibrational states by symmetry, lowering the computational cost. In addition, the polyad matrices provide proper treatment of resonance chaining, a phenomenon where a vibrational state, after perturbation by a coupled state, interacts with a new vibrational state. This chaining can occur many times over, especially in spectral regions with a high density of states. Due to known trouble in accurately computing vibrational modes with frequencies below 300 cm−1, these modes are removed from the SPECTRO VPT2 treatment.

The computational stick spectrum for each method is then convolved with a Lorentzian line shape function with a full-width at half-maximum of 20 cm−1 for comparison with the experiment.

The accuracy of the 2-quanta polyad VPT2 approach using density functional theory (DFT) with the B3LYP method and N07D basis set has been well-established,41,42,44,48,50,52 and the present work will extend this to higher frequencies/shorter wavelengths. Currently, computing the anharmonic frequencies of PAHs using correlated wave function methods is inaccessible due to their high computational cost; as such, DFT methods are the most viable option. In order to further test the accuracy of this DFT-based method, Table S1 compares the harmonic frequencies of benzene computed using the DFT-based B3LYP/N07D and the wave function-based CCSD(T)-F12b/cc-pVTZ-F12 levels of theory. The B3LYP/N07D combination performs exceptionally well, with a mean absolute difference of only 5.45 cm−1, or only 0.59% compared to the CCSD(T)-F12b/cc-pVTZ-F12 level. This gives confidence in the chosen methodology, which is almost as accurate for this system as the often more accurate explicitly correlated coupled cluster method.

Figure 1 presents the room temperature, gas-phase, experimental (top panel),30 3-quanta anharmonic (middle panel), and 2-quanta anharmonic (bottom panel) IR absorption spectrum of benzene in the CH stretch first overtone (2νCH) region (5750–6300 cm−1). The experimental spectrum30 has four broad features, centered at 5820 (b), 5935 (c), 5995 (d), and 6140 cm−1 (e). The strongest feature in the experimental spectrum is centered at 5995 cm−1, whereas the most intense peak in the computational spectra is centered at 6038 and 6049 cm−1 for the 2- and 3-quanta spectra, respectively. The anharmonic computations allow spectroscopic assignment of the vibrational states that are detected by experiment. Previous work showed that the positions and intensities of the vibrational states in the 2-quanta polyad computations are expected to be more accurate than those produced by the 3-quanta standard VPT2 computations.52 This impacts the states labeled “c,” “d,” “e,” and “f,” which are strong 2-quanta combination bands. These bands are present in both computations and are the most prominent in the experiment.

FIG. 1.

Room temperature gas-phase experimental (top),30 3-quanta anharmonic (middle), and 2-quanta anharmonic (bottom) IR absorption spectrum of benzene in the CH stretch first overtone region (5750–6300 cm−1). The computational stick spectra are broadened with a Lorentzian line shape having a full-width at half-maximum of 20 cm−1. Corresponding features have been marked with matching letters.

FIG. 1.

Room temperature gas-phase experimental (top),30 3-quanta anharmonic (middle), and 2-quanta anharmonic (bottom) IR absorption spectrum of benzene in the CH stretch first overtone region (5750–6300 cm−1). The computational stick spectra are broadened with a Lorentzian line shape having a full-width at half-maximum of 20 cm−1. Corresponding features have been marked with matching letters.

Close modal

The mode labeling and corresponding frequencies of the 2-quanta polyad computations as applied to benzene are given in Table I, and those for the 3-quanta standard VPT2 results in Table II. The largest feature in the experimental spectrum is predicted to arise from the states labeled “d,” the degenerate CH stretch combination bands ν6 + ν5 and ν6 + ν4 at 6040 cm−1 in the 2-quanta calculation and 6047 cm−1 in the 3-quanta calculation; a difference of 45 and 52 cm−1, respectively. In the fundamental region (∼100–3200 cm−1), deviations on the order of 10–20 cm−1 are common.21,69 As expected, a larger difference is seen here when moving to higher-order transitions. Employing the polyad VPT2 computations, the states labeled “c” (ν5 + ν3 and ν4 + ν2) shift to a lower frequency by 31 cm−1 when compared to the standard VPT2 approach. Adding the difference of 45 cm−1 from the states “d” to the large shift of the states labeled “c” moves them to 5982 cm−1. This is ∼47 cm−1 higher in frequency than the experimental band at 5935 cm−1. Since the use of polyads in the VPT2 treatment introduces such a shift, the results are sensitive to the computational methodology, and a higher-order perturbation theory would be warranted. In addition, resonance chaining that stems from the high density of vibrational states will become even more prominent when adding the many 3-quanta states to the polyad matrix. While this could introduce additional shifts, their magnitude and direction cannot be predicted a priori.

TABLE I.

Anharmonic vibrational frequencies (in cm−1) and intensities (in km mol−1) in the 2νCH region of benzene computed via the 2-quanta polyad approach.

LabelModesAnharm. freq.Anharm. int.Expt. freq.
ν5 + ν3 6027.3 0.537 5935 
ν4 + ν2 6027.6 0.537 5935 
ν6 + ν4 6040.4 1.887 5995 
ν6 + ν5 6040.7 1.888 5995 
ν3 + ν1 6094.1 0.431  
ν2 + ν1 6094.3 0.431  
ν5 + ν2 6114.2 0.533  
ν4 + ν3 6114.3 0.533  
LabelModesAnharm. freq.Anharm. int.Expt. freq.
ν5 + ν3 6027.3 0.537 5935 
ν4 + ν2 6027.6 0.537 5935 
ν6 + ν4 6040.4 1.887 5995 
ν6 + ν5 6040.7 1.888 5995 
ν3 + ν1 6094.1 0.431  
ν2 + ν1 6094.3 0.431  
ν5 + ν2 6114.2 0.533  
ν4 + ν3 6114.3 0.533  
TABLE II.

Anharmonic vibrational frequencies (in cm−1) and intensities (in km mol−1) in the 2νCH region of benzene computed via the 3-quanta standard VPT2 approach.

ModesAnharm. freq.Anharm. int.Rel. int.
2ν11 + ν2 5748.5 0.005 0.002 
ν15 + ν7 + ν5 5804.1 0.002 0.001 
ν15 + ν8 + ν4 5804.2 0.002 0.001 
ν13 + ν7 + ν6 5807.2 0.006 0.003 
ν14 + ν8 + ν6 5807.4 0.006 0.003 
ν15 + ν7 + ν4 5809.7 0.002 0.001 
ν15 + ν8 + ν5 5809.8 0.002 0.001 
ν13 + ν8 + ν6 5810.2 0.006 0.003 
ν14 + ν7 + ν6 5810.2 0.006 0.003 
ν15 + ν7 + ν1 5814.1 0.001 0.001 
ν15 + ν8 + ν1 5814.2 0.001 0.001 
ν12 + ν9 + ν6 5833.3 0.014 0.008 
ν12 + ν10 + ν6 5833.5 0.014 0.008 
ν13 + ν7 + ν3 5840.0 0.000 0.000 
ν14 + ν8 + ν2 5840.2 0.000 0.000 
ν13 + ν8 + ν2 5840.7 0.001 0.001 
ν14 + ν7 + ν3 5840.7 0.001 0.001 
ν13 + ν7 + ν2 5843.1 0.008 0.004 
ν14 + ν8 + ν3 5843.3 0.008 0.004 
ν13 + ν8 + ν3 5848.3 0.001 0.001 
ν14 + ν7 + ν2 5848.4 0.001 0.001 
ν12 + ν9 + ν3 5868.4 0.004 0.002 
ν12 + ν10 + ν2 5868.6 0.005 0.003 
ν12 + ν9 + ν2 5869.4 0.004 0.002 
ν12 + ν10 + ν3 5869.5 0.005 0.003 
ν11 + ν9 + ν5 5873.3 0.008 0.004 
ν11 + ν10 + ν4 5873.5 0.008 0.004 
ν11 + ν9 + ν4 5874.9 0.008 0.004 
ν11 + ν10 + ν5 5875.0 0.008 0.004 
ν11 + ν9 + ν1 5884.2 0.005 0.003 
ν11 + ν10 + ν1 5884.4 0.005 0.003 
ν12 + ν7 + ν5 5964.1 0.010 0.005 
ν12 + ν8 + ν4 5964.2 0.010 0.005 
ν11 + ν7 + ν6 5968.2 0.067 0.035 
ν11 + ν8 + ν6 5968.3 0.067 0.035 
ν12 + ν7 + ν4 5969.7 0.010 0.005 
ν12 + ν8 + ν5 5969.7 0.010 0.005 
ν12 + ν7 + ν1 5973.6 0.006 0.003 
ν12 + ν8 + ν1 5973.7 0.006 0.003 
2ν9 + ν6 5983.9 0.039 0.021 
2ν10 + ν6 5984.2 0.039 0.021 
ν10 + ν9 + ν6 5985.8 0.048 0.026 
ν11 + ν7 + ν3 6000.4 0.019 0.010 
ν11 + ν8 + ν2 6000.5 0.022 0.012 
ν11 + ν7 + ν2 6005.8 0.019 0.010 
ν11 + ν8 + ν3 6005.9 0.022 0.012 
2ν9 + ν3 6018.0 0.002 0.001 
2ν10 + ν2 6018.3 0.002 0.001 
2ν9 + ν2 6019.8 0.040 0.021 
2ν10 + ν3 6020.1 0.040 0.021 
ν10 + ν9 + ν3 6020.8 0.014 0.007 
ν10 + ν9 + ν2 6020.9 0.014 0.007 
ν6 + ν5 6047.6 1.888 1.000 
ν6 + ν4 6047.6 1.888 1.000 
ν5 + ν3 6058.2 0.533 0.283 
ν4 + ν2 6058.3 0.533 0.283 
ν3 + ν1 6093.6 0.431 0.228 
ν2 + ν1 6093.7 0.431 0.228 
ν5 + ν2 6098.9 0.537 0.284 
ν4 + ν3 6099.0 0.537 0.284 
ν9 + ν7 + ν5 6126.4 0.004 0.002 
ν10 + ν8 + ν4 6126.7 0.004 0.002 
ν9 + ν8 + ν4 6127.4 0.009 0.005 
ν10 + ν7 + ν5 6127.5 0.009 0.005 
ν9 + ν8 + ν5 6131.4 0.002 0.001 
ν10 + ν7 + ν4 6131.5 0.002 0.001 
ν9 + ν7 + ν4 6133.6 0.004 0.002 
ν10 + ν8 + ν5 6133.7 0.004 0.002 
ν9 + ν8 + ν1 6136.4 0.003 0.002 
ν10 + ν7 + ν1 6136.5 0.003 0.002 
ν9 + ν7 + ν1 6137.0 0.003 0.002 
ν10 + ν8 + ν1 6137.2 0.003 0.002 
2ν7 + ν6 6236.0 0.060 0.032 
ν8 + ν7 + ν6 6236.1 0.105 0.056 
2ν8 + ν6 6236.1 0.060 0.032 
2ν7 + ν3 6268.2 0.011 0.006 
2ν8 + ν2 6268.4 0.011 0.006 
ν8 + ν7 + ν3 6273.7 0.030 0.016 
ν8 + ν7 + ν2 6273.7 0.030 0.016 
2ν7 + ν2 6279.0 0.087 0.046 
2ν8 + ν3 6279.1 0.087 0.046 
ν30 + ν6 + ν3 6464.9 0.001 0.000 
ModesAnharm. freq.Anharm. int.Rel. int.
2ν11 + ν2 5748.5 0.005 0.002 
ν15 + ν7 + ν5 5804.1 0.002 0.001 
ν15 + ν8 + ν4 5804.2 0.002 0.001 
ν13 + ν7 + ν6 5807.2 0.006 0.003 
ν14 + ν8 + ν6 5807.4 0.006 0.003 
ν15 + ν7 + ν4 5809.7 0.002 0.001 
ν15 + ν8 + ν5 5809.8 0.002 0.001 
ν13 + ν8 + ν6 5810.2 0.006 0.003 
ν14 + ν7 + ν6 5810.2 0.006 0.003 
ν15 + ν7 + ν1 5814.1 0.001 0.001 
ν15 + ν8 + ν1 5814.2 0.001 0.001 
ν12 + ν9 + ν6 5833.3 0.014 0.008 
ν12 + ν10 + ν6 5833.5 0.014 0.008 
ν13 + ν7 + ν3 5840.0 0.000 0.000 
ν14 + ν8 + ν2 5840.2 0.000 0.000 
ν13 + ν8 + ν2 5840.7 0.001 0.001 
ν14 + ν7 + ν3 5840.7 0.001 0.001 
ν13 + ν7 + ν2 5843.1 0.008 0.004 
ν14 + ν8 + ν3 5843.3 0.008 0.004 
ν13 + ν8 + ν3 5848.3 0.001 0.001 
ν14 + ν7 + ν2 5848.4 0.001 0.001 
ν12 + ν9 + ν3 5868.4 0.004 0.002 
ν12 + ν10 + ν2 5868.6 0.005 0.003 
ν12 + ν9 + ν2 5869.4 0.004 0.002 
ν12 + ν10 + ν3 5869.5 0.005 0.003 
ν11 + ν9 + ν5 5873.3 0.008 0.004 
ν11 + ν10 + ν4 5873.5 0.008 0.004 
ν11 + ν9 + ν4 5874.9 0.008 0.004 
ν11 + ν10 + ν5 5875.0 0.008 0.004 
ν11 + ν9 + ν1 5884.2 0.005 0.003 
ν11 + ν10 + ν1 5884.4 0.005 0.003 
ν12 + ν7 + ν5 5964.1 0.010 0.005 
ν12 + ν8 + ν4 5964.2 0.010 0.005 
ν11 + ν7 + ν6 5968.2 0.067 0.035 
ν11 + ν8 + ν6 5968.3 0.067 0.035 
ν12 + ν7 + ν4 5969.7 0.010 0.005 
ν12 + ν8 + ν5 5969.7 0.010 0.005 
ν12 + ν7 + ν1 5973.6 0.006 0.003 
ν12 + ν8 + ν1 5973.7 0.006 0.003 
2ν9 + ν6 5983.9 0.039 0.021 
2ν10 + ν6 5984.2 0.039 0.021 
ν10 + ν9 + ν6 5985.8 0.048 0.026 
ν11 + ν7 + ν3 6000.4 0.019 0.010 
ν11 + ν8 + ν2 6000.5 0.022 0.012 
ν11 + ν7 + ν2 6005.8 0.019 0.010 
ν11 + ν8 + ν3 6005.9 0.022 0.012 
2ν9 + ν3 6018.0 0.002 0.001 
2ν10 + ν2 6018.3 0.002 0.001 
2ν9 + ν2 6019.8 0.040 0.021 
2ν10 + ν3 6020.1 0.040 0.021 
ν10 + ν9 + ν3 6020.8 0.014 0.007 
ν10 + ν9 + ν2 6020.9 0.014 0.007 
ν6 + ν5 6047.6 1.888 1.000 
ν6 + ν4 6047.6 1.888 1.000 
ν5 + ν3 6058.2 0.533 0.283 
ν4 + ν2 6058.3 0.533 0.283 
ν3 + ν1 6093.6 0.431 0.228 
ν2 + ν1 6093.7 0.431 0.228 
ν5 + ν2 6098.9 0.537 0.284 
ν4 + ν3 6099.0 0.537 0.284 
ν9 + ν7 + ν5 6126.4 0.004 0.002 
ν10 + ν8 + ν4 6126.7 0.004 0.002 
ν9 + ν8 + ν4 6127.4 0.009 0.005 
ν10 + ν7 + ν5 6127.5 0.009 0.005 
ν9 + ν8 + ν5 6131.4 0.002 0.001 
ν10 + ν7 + ν4 6131.5 0.002 0.001 
ν9 + ν7 + ν4 6133.6 0.004 0.002 
ν10 + ν8 + ν5 6133.7 0.004 0.002 
ν9 + ν8 + ν1 6136.4 0.003 0.002 
ν10 + ν7 + ν1 6136.5 0.003 0.002 
ν9 + ν7 + ν1 6137.0 0.003 0.002 
ν10 + ν8 + ν1 6137.2 0.003 0.002 
2ν7 + ν6 6236.0 0.060 0.032 
ν8 + ν7 + ν6 6236.1 0.105 0.056 
2ν8 + ν6 6236.1 0.060 0.032 
2ν7 + ν3 6268.2 0.011 0.006 
2ν8 + ν2 6268.4 0.011 0.006 
ν8 + ν7 + ν3 6273.7 0.030 0.016 
ν8 + ν7 + ν2 6273.7 0.030 0.016 
2ν7 + ν2 6279.0 0.087 0.046 
2ν8 + ν3 6279.1 0.087 0.046 
ν30 + ν6 + ν3 6464.9 0.001 0.000 

Based on the shape of the largest band in the experimental spectrum, the “e” and “f” states are expected to contribute to the breadth of the largest feature as well as lead to the small shoulder seen on the high frequency side. In both the 2- and 3-quanta computations, these states present a distinct feature at higher frequencies. However, since both VPT2 methods already struggle to correctly predict the frequency of the “c” state within 40 cm−1, “e” and “f” are likely shifted as well.

The main intensity and structure observed in the experimental spectrum are captured well with the 2-quanta polyad approach, but to accurately describe the subtleties of the entire band, 3-quanta transitions must be considered. That is, the satellite bands “a,” “b,” “g,” and “h” all originate solely from 3-quanta states. Inspection of Table II reveals the presence of many combinations composed of three different fundamental modes, all of which include at least one CH stretching mode. Interestingly, many of the states are first overtones plus other fundamentals. Without considering the 3-quanta states, understanding the experimental spectrum would be grossly incomplete. Given the ∼40–50 cm−1 frequency difference seen here, benchmarking with other DFT methods and basis sets should be performed to identify if B3LYP is the most accurate method for the CH stretch overtone region. However, with this being the CH overtone region, the errors are expected to compound, and the ∼40–50 cm−1 frequency difference can be considered excellent performance.

Figure 2 presents the gas-phase experimental (black)30 and 3-quanta anharmonic (red) absorption spectra of benzene in the CH stretch second overtone (3νCH) region (8400–9200 cm−1), and the corresponding data are included in Table III. Considering 3-quanta transitions provides the only means to access this frequency region. The experimental spectrum presents two main peaks, one centered at 8740 and the other at 8785 cm−1. There is a small bump around 8600 cm−1. In the computed anharmonic spectrum, the strongest transitions in this region are the three-mode combination bands ν6 + ν4 + ν1 at 9034.6 cm−1 and ν6 + ν5 + ν1 at 9034.5 cm−1 that include three CH stretch fundamentals (which are expected to be able to reach 9000 cm−1). Interestingly, the vibrational states in this region contain various second overtones of the CH stretches (3νCH), as well as combination bands that include a first overtone of a CH stretch plus a fundamental of another mode (2νCH + νX). Typically, the intrinsic intensity of each overtone level, heuristically, decreases by a factor of 10.70 Second overtones are, therefore, expected to be 100 times weaker than the corresponding fundamental transition. Similarly, the first overtones are ten times weaker than the fundamentals. Their presence in the CH stretch second overtone spectrum shows just how strong these transitions are in benzene.

FIG. 2.

Room temperature gas-phase experimental (black)30 and 3-quanta anharmonic (red) IR absorption spectrum of benzene in the CH stretch second overtone region (8400–9200 cm−1). The computational stick spectrum is broadened with a Lorentzian line shape having a full-width at half-maximum of 20 cm−1.

FIG. 2.

Room temperature gas-phase experimental (black)30 and 3-quanta anharmonic (red) IR absorption spectrum of benzene in the CH stretch second overtone region (8400–9200 cm−1). The computational stick spectrum is broadened with a Lorentzian line shape having a full-width at half-maximum of 20 cm−1.

Close modal
TABLE III.

Anharmonic vibrational frequencies (in cm−1) and intensities (in km mol−1) in the 3νCH region of benzene computed via the 3-quanta standard VPT2 approach.

ModeAnharm. freq.Anharm. int.
2ν5 + ν3 9020.9 0.041 
2ν4 + ν2 9021.1 0.041 
ν6 + 2ν5 9030.7 0.005 
ν6 + ν4 9030.9 0.005 
ν6 + ν5 + ν1 9034.5 0.053 
ν6 + ν4 + ν1 9034.6 0.053 
2ν6 + ν3 9037.4 0.020 
2ν6 + ν2 9037.5 0.020 
ν6 + ν5 + ν4 9040.9 0.028 
ν5 + ν3 + ν1 9046.4 0.015 
ν4 + ν2 + ν1 9046.5 0.017 
ν6 + 2ν3 9058.8 0.004 
ν6 + ν3 + ν2 9068.9 0.021 
ν5 + ν4 + ν3 9071.7 0.016 
ν5 + ν4 + ν2 9071.8 0.016 
ν5 + ν2 + ν1 9087.1 0.015 
ν4 + ν3 + ν1 9087.1 0.017 
2ν5 + ν2 9102.3 0.005 
2ν4 + ν3 9102.3 0.005 
3ν3 9110.8 0.012 
3ν2 9110.9 0.012 
2ν3 + ν2 9130.8 0.004 
ModeAnharm. freq.Anharm. int.
2ν5 + ν3 9020.9 0.041 
2ν4 + ν2 9021.1 0.041 
ν6 + 2ν5 9030.7 0.005 
ν6 + ν4 9030.9 0.005 
ν6 + ν5 + ν1 9034.5 0.053 
ν6 + ν4 + ν1 9034.6 0.053 
2ν6 + ν3 9037.4 0.020 
2ν6 + ν2 9037.5 0.020 
ν6 + ν5 + ν4 9040.9 0.028 
ν5 + ν3 + ν1 9046.4 0.015 
ν4 + ν2 + ν1 9046.5 0.017 
ν6 + 2ν3 9058.8 0.004 
ν6 + ν3 + ν2 9068.9 0.021 
ν5 + ν4 + ν3 9071.7 0.016 
ν5 + ν4 + ν2 9071.8 0.016 
ν5 + ν2 + ν1 9087.1 0.015 
ν4 + ν3 + ν1 9087.1 0.017 
2ν5 + ν2 9102.3 0.005 
2ν4 + ν3 9102.3 0.005 
3ν3 9110.8 0.012 
3ν2 9110.9 0.012 
2ν3 + ν2 9130.8 0.004 

The overall width of the feature as observed by the experiment (FWHM ∼175 cm−1) is reproduced nicely by the anharmonic computations (FWHM ∼162 cm−1). The finer structural differences stem from the lack of access to intensity sharing via polyads in these computations, where a shift in intensities is expected to impact the relative height of the two separate peaks. In addition, resonance chaining (see Sec. II) is not captured in these computations. In particular, the group of intense states between 9025 and 9050 cm−1 will be highly influenced by resonance chaining, drastically altering the shape of the bands here. There is an average shift of about 305 cm−1 in frequency between the computations and experiments. The strongest feature in both (at 9035 cm−1 in the computational and 8785 cm−1 in the experiment spectrum) shows better agreement with a difference of 250 cm−1. The somewhat substantial difference in frequency is not surprising as it involves 3-quanta states that intrinsically lead to more error when compared to fundamental and 2-quanta overtone and combination bands.71 In addition, DFT methods are unable to treat the high symmetry group (D6h) of benzene, requiring the computations to be performed in a lower symmetry Abelian subgroup (D2h). This artificially splits otherwise degenerate features. Luckily, this is avoided for most PAHs, as they nearly all have a lower symmetry than benzene.

Figure 3 depicts the 55 °C gas-phase experimental (top panel),38 3-quanta anharmonic (middle panel), and 2-quanta anharmonic (bottom panel) IR absorption spectrum of naphthalene in the CH stretch first overtone region (5750–6300 cm−1). The anharmonic frequencies computed using the 3-quanta standard VPT2 method are, due to the large number of states, given in the supplementary material Table S2. The experimental spectrum shows a core comprised of a double peak structure. The overall shape of the band is reproduced well using both computational approaches considered in this work. However, the 3-quanta method better captures the relative intensity of the two peaks. This is because additional intense states are present in the peak labeled “d” at ∼6100 cm−1. Regarding frequency, the 2-quanta polyad frequencies have a smaller mean absolute difference of ∼ 80 and 85 cm−1, respectively, between the peaks labeled “c” and “d” when compared to the experiment.

FIG. 3.

55 °C gas-phase experimental (top),38 3-quanta anharmonic (middle), and 2-quanta anharmonic (bottom) IR absorption spectrum of naphthalene in the CH stretch first overtone region (5750–6300 cm−1). The computational stick spectra are broadened with a Lorentzian line shape having a full-width at half-maximum of 20 cm−1. Corresponding features have been marked with matching letters.

FIG. 3.

55 °C gas-phase experimental (top),38 3-quanta anharmonic (middle), and 2-quanta anharmonic (bottom) IR absorption spectrum of naphthalene in the CH stretch first overtone region (5750–6300 cm−1). The computational stick spectra are broadened with a Lorentzian line shape having a full-width at half-maximum of 20 cm−1. Corresponding features have been marked with matching letters.

Close modal

The remaining four readily identifiable features, “a,” “b,” “e,” and “f,” all stem solely from 3-quanta states. The high frequency shoulder seen in the experiment and labeled “e” originates from a group of 3-quanta states ranging from 6150 to 6175 cm−1, while the bump labeled “f” at 6135 cm−1 arises from states predicted between 6200 and 6300 cm−1. In addition, the features labeled “a” and “b,” while small, are comprised of a relatively large number of states that add up to become detectable by experiment. This, again, illustrates the great importance of considering 3-quanta states when it comes to the accurate computation of the IR absorption spectra of PAHs in the CH stretch overtone region.

Compared to benzene, the larger number of degrees of freedom in naphthalene leads to more vibrational states in the CH stretch overtone region, which creates a much broader feature. This is particularly noticeable in the composition of the main peaks. In benzene, there are four states with an intensity >0.1 km mol−1 that make up the main features (and eight total states overall between 6025 and 6125 cm−1). However, in naphthalene, this number increases to 19 states with an intensity >0.1 km mol−1 and a total of 63 states between 6025 and 6125 cm−1. Across the whole 5800–6300 cm−1 range, benzene has 80 2- and 3-quanta vibrational states, whereas naphthalene has 280. This explains both the broader feature and the smaller substructure seen in the spectrum of naphthalene when compared to benzene. The experimental spectrum of naphthalene exhibits what appears to be an underlying baseline continuum, similar to what has been observed in recent JWST PAH observations and dubbed the quasi-continuum.27 In both cases, this continuum likely arises from the large number of vibrational states present throughout the region.50 

The general shape of the CH stretch first overtone experimental absorption spectra of benzene and naphthalene is qualitatively and even semi-quantitatively reproduced by two different anharmonic quantum chemical approaches. The largest intensity modes are dominated by CH stretch overtones as well as combination bands that include CH stretches and high-intensity bending motions. Moving to larger PAHs creates more degrees of freedom and an increase in the number, density, and diversity of CH stretching modes. Overall, this will result in a broader range of CH stretch fundamental frequencies, which, in turn, will lead to a larger spread of overtone and combination bands. Anthracene and tetracene, the next in the series of linear PAHs known as the acenes and having three and four fused rings, are used as test cases. Figure 4 compares the 3-quanta anharmonic absorption spectra of benzene and naphthalene to those of anthracene and tetracene from 5800 to 6300 cm−1. The main band of anthracene is shifted to higher energy by 15 cm−1 compared to naphthalene, but the shape and width are identical. There are some minor shape differences due to the growth of satellite peaks, but the overall shape and breadth are retained. For tetracene, the main peak is shifted by 20 cm−1, but the shape exhibits both similarities and differences between naphthalene and anthracene. The double-hump structure is retained, but the increased number of CH stretches and vibrational modes in general leads to a somewhat less uniform structure with various intense peaks. The increase in the density of states with size leads to a clear shift from two distinct peaks in benzene to a more overlapped peak structure in naphthalene, an even less structured doubled peak in anthracene, and a doubled main peak pair in tetracene with a larger number of satellite peaks and higher-frequency bands beyond the main pair. Further exploration of larger PAHs with different shapes and edge structures will show if broadening will continue or if there is a maximum broadness achieved based on coincidental combination band limits. In addition, the presence of a higher density of low intensity states outside the main peak will lead to additional smaller features presenting themselves as shoulders and satellites, as shown in Fig. 4, especially for tetracene. With such large groups of coupled states, intensity redistribution will prove important, especially for the shoulder and satellite states, which will likely subsequently grow in intensity for larger PAHs. The use of resonance polyads is essential in order to properly treat this intensity sharing, which is seeing active development.

FIG. 4.

3-quanta anharmonic IR absorption spectrum of benzene (red), naphthalene (orange), anthracene (purple), and tetracene (blue) in the CH stretch first overtone region (5750–6300 cm−1).

FIG. 4.

3-quanta anharmonic IR absorption spectrum of benzene (red), naphthalene (orange), anthracene (purple), and tetracene (blue) in the CH stretch first overtone region (5750–6300 cm−1).

Close modal

The blending of features in this wavelength region may turn out to be an advantage, as studying small PAHs (with less than 30 carbons)42,72 may suffice and be representative for all PAH CH stretch overtones, an interpretation that will need to be confirmed by computing the spectra of (increasingly) larger PAHs. This potential accidental advantage would be welcomed, as the computational cost for 3-quanta computations is much greater than that for the 2-quanta case and will only be exacerbated with increasing PAH size.

See the supplementary material for the following: Harmonic vibrational frequencies for benzene computed with CCSD(T)-F12b and B3LYP, anharmonic vibrational frequencies for naphthalene using 3-quanta VPT2, and the mode numbering, symmetry, and frequencies for the fundamental vibrations of naphthalene.

V.J.E. acknowledges an appointment to the NASA Postdoctoral Program at NASA Ames Research Center, administered by the Oak Ridge Associated Universities through a contract with NASA. R.C.F. acknowledges the Mississippi Center for Supercomputing Research, supported in part by NSF Grant No. OIA-1757220. C.B. is grateful for an appointment at the NASA Ames Research Center through the San José State University Research Foundation (Grant No. 80NSSC22M0107). V.J.E., R.C.F., C.B., and L.J.A. acknowledge the support from the Internal Scientist Funding Model (ISFM) Laboratory Astrophysics Directed Work Package at NASA Ames. Computer time from the Pleiades and Aiken clusters of the NASA Advanced Supercomputer (NAS) is gratefully acknowledged.

The authors have no conflicts to disclose.

Vincent J. Esposito: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Funding acquisition (equal); Investigation (equal); Methodology (equal); Project administration (equal); Resources (equal); Software (equal); Supervision (equal); Validation (equal); Visualization (equal); Writing – original draft (equal); Writing – review & editing (equal). Ryan C. Fortenberry: Conceptualization (equal); Funding acquisition (equal); Project administration (equal); Supervision (equal); Writing – review & editing (equal). Christiaan Boersma: Conceptualization (equal); Supervision (equal); Writing – review & editing (equal). Louis J. Allamandola: Conceptualization (equal); Supervision (equal); Writing – review & editing (equal).

The data that support the findings of this study are available within the article and its supplementary material.

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