Microwave spectroscopy of ternary and quaternary van der Waals clusters

Regularities and Anomalies in Neon Matrix Shifts of Hydrogen-Bonded O-H Stretching Fundamentals

O-H bond stretching vibrations in hydrogen-bonded complexes embedded into cryogenic neon matrices are subtly downshifted from cold gas phase reference wavenumbers. To the extent that this shift is systematic, it enables neon matrices as more universally applicable spectroscopic benchmark environments for quantum chemical predictions. Outliers are indicative of either an assignment problem in one of the two cryogenic experiments or they reveal interesting dynamics or structural effects on the complexes as a function of the environment. We compile 6 literature-known pairs of experimental data in jet and neon matrix expansions and realize a 6-fold expansion of that number through targeted matrix isolation and/or slit jet expansion spectroscopy presented in this work. In many cases, the neon matrix shift is less than the uncertainty of the currently best-performing blind quantum chemical predictions for the gas phase, but in specific cases, it may exceed the currently achievable theoretical accuracy. Some evidence for a positive correlation of the matrix shift with the hydrogen bond shift is found, similar to observations for helium nanodroplets. Outliers in particular for water acting as a donor are discussed, and in a few cases they call for a future reinvestigation. Substantial improvement in the correlation of the matrix shift with the hydrogen bond shift is achieved for ketone monohydrates by removing a vibrational resonance. New insights into nitrile hydration isomerism are obtained, and the linear OH stretching spectrum of the jet-cooled ammonia-water complex is presented for the first time. Vibrational spectroscopy in weakly perturbing solid rare gas quantum matrices for the benchmarking of gas phase theory and future explicit theoretical treatments of the quantum matrix environment to better understand the outliers are both encouraged.

Rotational Spectroscopy of the Acetone-Water Complex: Large Amplitude Motions

Acetone (CH3COCH3), the simplest ketone, has recently attracted considerable attention for its important role in atmospheric chemistry and in the formation of ices in extraterrestrial sources that contain complex organic molecules. In this study, we employed a combination of experimental rotational spectroscopy and quantum chemistry calculations to investigate the structure and dynamics of the acetone-water complex. Our aim was to understand how non-covalent interactions with water affect the methyl internal rotation dynamics of acetone, and how water-centered large amplitude motions alter the observed physical properties compared to those predicted at the equilibrium position. Detailed rotation-tunneling analyses of acetone-H2O and -D2O reveal that the interactions with water disrupt the equivalence of the two methyl rotors, resulting in a noticeably lower methyl rotor barrier for the top with the close-by water compared to that of free acetone. The barrier for the methyl group further from water is also lower, although to a lesser degree. To gain further insights, extensive theoretical modelling was conducted, focusing on the associated large amplitude motions. Furthermore, quantum theory of atoms in molecules and non-covalent interactions analyses were utilized to visualize the underlying causes of the observed trends.

Low-Temperature Gas-Phase Kinetics of Ethanol-Methanol Heterodimer Formation

The structures of gas-phase noncovalently bound clusters have long been studied in supersonic expansions. This method of study, while providing a wealth of information about the nature of noncovalent bonds, precludes observation of the formation of the cluster, as the clusters form just after the orifice of the pulsed valve. Here, we directly observe formation of ethanol-methanol dimers via microwave spectroscopy in a controlled cryogenic environment. Time profiles of the concentration of reagents in the cell yielded gas-phase reaction rate constants of k_Me-g = (2.8 ± 1.4) × 10^-13 cm³ molecule^-1 s^-1 and k_Me-t = (1.6 ± 0.8) × 10^-13 cm³ molecule^-1 s^-1 for the pseudo-second-order ethanol-methanol dimerization reaction at 8 K. The relaxation cross section between the gauche and trans conformers of ethanol was also measured using the same technique. In addition, thermodynamic relaxation between conformers of ethanol over time allowed for selection of conformer stoichiometry in the ethanol-methanol dimerization reaction, but no change in the ratio of dimer conformers was observed with changing ethanol monomer stoichiometry.

Hydrogen bonding networks and cooperativity effects in the aqueous solvation of trimethylene oxide and sulfide rings by microwave spectroscopy and computational chemistry

The intermolecular interactions responsible for the microsolvation of the highly flexible trimethylene oxide (TMO) and trimethylene sulfide (TMS) rings with one and two water (w) molecules were investigated using rotational spectroscopy (8-22 GHz) and quantum chemical calculations. The observed patterns of transitions are consistent with the most stable geometries of the TMO-w, TMO-(w)₂, and TMS-w complexes at the B2PLYP-D3(BJ)/aug-cc-pVTZ level and were confirmed using spectra of the ¹⁸O isotopologue. Due to its effectively planar backbone, TMO offers one unique binding site for solvation, while water can bind to the puckered TMS ring in either an axial or equatorial site of the heteroatom. In all clusters, the first water molecule binds in the σ_v symmetry plane of the ring monomer and serves as a hydrogen bond donor to the heteroatom. The second water molecule is predicted to form a cooperative hydrogen bonding network between the three moieties. Secondary C-H⋯O interactions are a key stabilizing influence in trimers and also drive the preferred binding site in the TMS clusters with the axial binding site preferred in TMS-w and the equatorial form calculated to be more stable in the dihydrate. Using an energy partition scheme from the symmetry-adapted perturbation theory for the O, S, and Se containing mono- and dihydrates, the intermolecular interactions are revealed to be mainly electrostatic, but the dispersive character of the contacts is enhanced with the increasing size of the ring's heteroatom due to the key role of longer-range secondary interactions.

Rotational Spectrum and Molecular Structures of the Binary Aggregates of 1,1,1,3,3,3-Hexafluoro-2-propanol with Ne and Ar

The structures and binding topologies of two binary van der Waals complexes 1,1,1,3,3,3-hexafluoro-2-propanol (HFIP)···Ne and ···Ar were investigated. The rotational spectra of these two complexes including several isotopic species containing ²⁰Ne, ²²Ne, ⁴⁰Ar, ¹³C, and hydroxyl D were measured using a chirped pulse Fourier transform microwave spectrometer and a cavity-based Fourier transform microwave spectrometer. While HFIP was shown to exist in both the gauche and trans configurations based on previous reports, the rare gas atom is predicted to attach to HFIP in several different binding topologies, leading to a total of nine possible structural isomers for each complex. Only one isomer was detected for each species, and it corresponds to the most stable one predicted, based on the comparison of the experimental rotational constants and electric dipole moment components with the theoretical predictions and on the isotopic data. We applied quantum theory of atoms in molecules (QTAIM) and electrostatic potential calculations to examine the different rare gas binding sites and to explore the nature of the interactions in these two complexes and several previously reported alcohol···Ar complexes. The effects of fluorination are also discussed by comparison with the binary complexes of isopropanol···Ne and ···Ar.

Many-Body Dispersion

A broad range of approaches to many-body dispersion are discussed, including empirical approaches with multiple fitted parameters, augmented density functional-based approaches, symmetry adapted perturbation theory, and a supermolecule approach based on coupled cluster theory. Differing definitions of "body" are considered, specifically atom-based vs molecule-based approaches.

Structural and dynamical features of the 2,2,2-trifluoroethanol⋯ammonia complex

The preferred conformation of trifluoroethanol⋯ammonia is established experimentally and the effects of large vibrational motions on its properties evaluated.

Rotational spectra of van der Waals complexes: pyrrole-Ne and pyrrole-Ne2

The van der Waals 1 : 1 and 1 : 2 adducts between the aromatic molecule pyrrole (Pyr) and the rare gas atom neon (Ne) have been investigated using a combination of chirped pulse Fourier transform microwave spectroscopy and quantum-chemical calculations. Rotational spectra of two and three isotopologues of Pyr-Ne and Pyr-Ne2, respectively, arising from the combinations of the 20Ne and 22Ne isotopes, were identified and a partial rs structure determined. Unusual spectral intensities have been observed with a significant enrichment of heavier isotopic species in the jet molecular expansion. The observed rotational constants of Pyr-Ne are consistent with a nearly symmetric prolate top with the Ne atom located above the plane of pyrrole. The trimer presents C2v symmetry with the Ne atoms located one on each side of the ring plane. The experimental 14N nuclear quadrupole coupling constants have been determined for all the isotopologues of Pyr-Ne and Pyr-Ne2 complexes. Similar values to those of isolated pyrrole have been found, which suggests that the electrical gradient field of pyrrole does not change much upon complexation. The observed spectroscopic parameters have been compared with those of other aromatic-rare gas complexes.

Rotational spectrum and structure of 2-chlorothiophene and its complex with argon

2-chlorothiophene and its van der Waals complex with argon were studied by supersonic-jet Fourier transform microwave spectroscopy. Rotational measurements of parent and the mono-substituted ³⁷Cl, ³⁴S and ¹³C isotopologues of the monomer allowed its precisely structural determination. Rotational spectra of ³⁵Cl and ³⁷Cl isotopologues were observed and assigned for the van der Waals complex of 2-chlorothiophene with Ar. Ab initio calculations carried out at the MP2/6-311++G(d,p) level of theory complement the experimental studies. Spectroscopic and theoretical results support a conformation in which the Ar atom locates above the plane of aromatic ring and toward the substituted carbon atom. The distance between the centers of mass of 2-chlorothiophene and argon is 3.589 Å. The Non-Covalent Interaction analysis and Symmetry-Adapted Perturbation theory were performed to reveal the nature of the non-covalent interaction within the complex.

The rich conformational landscape of perillyl alcohol revealed by broadband rotational spectroscopy and theoretical modelling

We explore the conformational landscape of perillyl alcohol in order to properly account for the sources of the conformers observed.

A microwave spectroscopic and ab initio study of keto–enol tautomerism and isomerism in the cyclohexanone–water complex

Experimental structure and keto–enol conversion barrier of cyclohexanone–water from microwave spectroscopy and ab initio calculations.

Hydration of the simplest α-keto acid: a rotational spectroscopic and ab initio study of the pyruvic acid–water complex

Non-covalent interactions analysis of hydrogen bonding in the pyruvic acid water complex.

Microwave spectroscopic detection of flame-sampled combustion intermediates

Microwave spectroscopy was used to detect and identify combustion intermediates after sampling out of laboratory-scale model flames.

Tunnelling and barrier-less motions in the 2-fluoroethanol–water complex: a rotational spectroscopic and ab initio study

Rotational spectrum of 2-fluoroethanol–water reveals interesting water and methyl internal rotation tunneling and barrier-less motions in the hydrogen-bonded complex.

Rotational spectroscopy of the methyl glycidate–water complex: conformation and water and methyl rotor tunnelling motions

Rotational transitions of methyl glycidate–water exhibit relatively large water tunnelling splittings, a surprise considering that water is quite tightly bound.

Contrasting Effects of Water on the Barriers to Decarboxylation of Two Oxalic Acid Monohydrates: A Combined Rotational Spectroscopic and Ab Initio Study

Using rotational spectroscopy, we have observed two isomers of the monohydrate of oxalic acid, the most abundant dicarboxylic acid in the atmosphere. In the lowest-energy isomer, water hydrogen-bonds to both carboxylic acid groups, and the barrier to decarboxylation decreases. In the second isomer, water bonds to only one carboxylic acid group, and the barrier increases. Though the lower barrier in the former is not unequivocal evidence that water acts as a photocatalyst, the higher barrier in the latter indicates that water acts as an inhibitor in this topology. Oxalic acid is unique among dicarboxylic acids: for the higher homologues calculated, the inhibiting topology of the monohydrate is lowest in energy and most abundant under atmospheric conditions. Consequently, oxalic acid is the only dicarboxylic acid for which single-water catalysis of overtone-induced decarboxylation in the atmosphere is plausible.

Rotational spectroscopy of the atmospheric photo-oxidation product o-toluic acid and its monohydrate

o-Toluic acid, a photo-oxidation product in the atmosphere, and its monohydrate were characterized in the gas phase by pure rotational spectroscopy. High-resolution spectra were measured in the range of 5-14 Hz using a cavity-based molecular beam Fourier-transform microwave spectrometer. Possible conformers were identified computationally, at the MP2/6-311++G(2df,2pd) level of theory. For both species, one conformer was identified experimentally, and no methyl internal rotation splittings were observed, indicative of relatively high barriers to rotation. In the monomer, rocking of the carboxylic acid group is a large amplitude motion, characterized by a symmetrical double-well potential. This and other low-lying out-of-plane vibrations contribute to a significant (methyl top-corrected) inertial defect (-1.09 amu Å(2)). In the monohydrate, wagging of the free hydrogen atom of water is a second large amplitude motion, so the average structure is planar. As a result, no c-type transitions were observed. Water tunneling splittings were not observed, because the water rotation coordinate is characterized by an asymmetrical double-well potential. Since the minima are not degenerate, tunneling is precluded. Furthermore, a concerted tunneling path involving simultaneous rotation of the water moiety and rocking of the carboxylic acid group is precluded, because the hilltop along this coordinate is a virtual, rather than a real, saddle-point. Inter- and intramolecular non-covalent bonding is discussed in terms of the quantum theory of atoms in molecules. The percentage of o-toluic acid hydrated in the atmosphere is estimated to be about 0.1% using statistical thermodynamics.

Rotational spectroscopy of methyl benzoylformate and methyl mandelate: structure and internal dynamics of a model reactant and product of enantioselective reduction

Pure rotational spectra of a prototypical prochiral ester, methyl benzoylformate (MBF), and the product of its enantioselective reduction, (R)-(-)-methyl mandelate (MM), were measured in the range of 5-16 GHz, using a cavity-based molecular beam Fourier-transform microwave spectrometer. Potential conformers were located using density functional theory calculations, and one conformer of each species was identified experimentally. The minimum energy conformer of MBF, in which the ester group is in a Z orientation, was observed for the first time. Based on an atoms-in-molecules analysis, MBF contains a weak CH···O=C hydrogen bond between the carbonyl oxygen atom of the ester group and the nearest hydrogen atom of the aromatic ring. In the minimum energy conformer of MM, the ester group is oriented to accommodate a hydrogen bond between the hydrogen atom of the hydroxyl group and the carbonyl oxygen atom (OH···O=C), rather than the sp(3) oxygen atom (OH···O-C). For both species, splittings of the rotational transitions were observed, which are attributed to methyl internal rotation, and the orientations and barrier heights of the methyl tops were determined precisely. The barrier heights for MBF and MM are 4.60(2) and 4.54(3) kJ mol(-1), respectively, which are consistent with values predicted by high-level wavefunction-based calculations. On the basis of an atoms-in-molecules analysis, we propose that destabilization of the sp(3) oxygen atom of the ester group most directly dictates the barrier height.

The benzoic acid-water complex: a potential atmospheric nucleation precursor studied using microwave spectroscopy and ab initio calculations

The pure rotational, high-resolution spectrum of the benzoic acid-water complex was measured in the range of 4-14 GHz, using a cavity-based molecular beam Fourier-transform microwave spectrometer. In all, 40 a-type transitions and 2 b-type transitions were measured for benzoic acid-water, and 12 a-type transitions were measured for benzoic acid-D2O. The equilibrium geometry of benzoic acid-water was determined with ab initio calculations, at the B3LYP, M06-2X, and MP2 levels of theory, with the 6-311++G(2df,2pd) basis set. The experimental rotational spectrum is most consistent with the B3LYP-predicted geometry. Narrow splittings were observed in the b-type transitions, and possible tunnelling motions were investigated using the B3LYP/6-311++G(d,p) level of theory. Rotation of the water moiety about the lone electron pair hydrogen-bonded to benzoic acid, across a barrier of 7.0 kJ mol(-1), is the most likely cause for the splitting. Wagging of the unbound hydrogen atom of water is barrier-less, and this large amplitude motion results in the absence of c-type transitions. The interaction and spectroscopic dissociation energies calculated using B3LYP and MP2 are in good agreement, but those calculated using M06-2X indicate excess stabilization, possibly due to dispersive interactions being over-estimated. The equilibrium constant of hydration was calculated by statistical thermodynamics, using ab initio results and the experimental rotational constants. This allowed us to estimate the changes in percentage of hydrated benzoic acid with variations in the altitude, region, and season. Using monitoring data from Calgary, Alberta, and the MP2-predicted dissociation energy, a yearly average of 1% of benzoic acid is expected to be present in the form of benzoic acid-water. However, this percentage depends sensitively on the dissociation energy. For example, when using the M06-2X-predicted dissociation energy, we find it increases to 18%.

ROVIBRATIONAL BOUND STATES OF THE Ar2Ne COMPLEX

We report exact quantum dynamics calculations of the eigenstate energy levels for the bound rovibrational states of the Ar2Ne complex, across the range of J values for which such states are observed (J = 0–35). All calculations have been carried out using the ScalIT suite of parallel codes. These codes employ a combination of highly efficient methods, including phase-space optimized discrete variable representation, optimal separable basis, and preconditioned inexact spectral transform (PIST) methods, together with an effective massive parallelization scheme. The Ar2Ne energy levels were computed using a pair-wise Aziz potential plus a three-body correction, in Jacobi co-ordinates. Effective potentials for the radial co-ordinates are constructed, which reveal important physical insight into the two distinct dissociation pathways, Ar2Ne → NeAr + Ar and Ar2Ne → Ar2 + Ne . A calculation of the bound vibrational (J = 0) levels, computed using the Tang–Toennies potential, is also performed for comparison with results from the previous literature.

Finite-temperature quantum simulations of mixed rare gas clusters

Finite-temperature quantum Monte Carlo simulations are presented for mixed neon/argon rare gas clusters containing up to n=10 atoms. For the smallest clusters (n=3) comparison with rigorous bound state calculations and experiments shows that the present approach is accurate to within fractions of wavenumbers for energies and to within a few percent or better for rotational constants. For larger cluster sizes, for which no rigorous quantum calculations are available, comparison with experiment becomes even more favorable. In all simulations accurate pair potentials for the rare gas-rare gas interactions are employed and comparison with high-level electronic structure calculations suggest that many-body interactions play a minor role. For the largest clusters investigated (Ne(4)Ar(6)) gradual melting of the neon phase is observed while the argon-phase remains structurally intact.

The structure of the NO(X (2)Pi)-N(2) complex: A joint experimental-theoretical study

We report the first measurement of the spectrum of the NO-N(2) complex in the region of the first vibrational NO overtone transition. The origin band of the complex is blueshifted by 0.30 cm(-1) from the corresponding NO monomer frequency. The observed spectrum consists of three bands assigned to the origin band, the excitation of one quantum of z-axis rotation and one associated hot band. The spacing of the bands and the rotational structure indicate a T-shaped vibrationally averaged structure with the NO molecule forming the top of the T. These findings are confirmed by high level ab initio calculations of the potential energy surfaces in planar symmetry. The deepest minimum is found for a T-shaped geometry on the A(")-surface. As a result the sum potential also has the global minimum for this structure. The different potential surfaces show several additional local minima at slightly higher energies indicating that the complex most likely will perform large amplitude motion even in its ground vibrational state. Nevertheless, as suggested by the measured spectra, the complex must, on average, spend a substantial amount of time near the T-shaped configuration.

Microwave spectrum of [1,1]-pyridine-Ne2

We investigated the rotational spectra of six isotopologues of pyridine--Ne(2), formed by combinations of two isotopes of the nitrogen atom ((14)N and (15)N) in pyridine with two isotopes of the rare gas atoms ((20)Ne and (22)Ne), by using pulsed jet Fourier transform microwave spectroscopy. We detected the C(2v) symmetry conformer, denoted as [1,1], where the Ne atoms are located one on each side of the ring plane. The [2,0] species, with the two Ne atoms on the same side of the ring, was not observed. Two structural parameters, R and theta, that localize the rare gas atoms with respect to pyridine have been determined. The (14)N nuclear quadrupole coupling constants have been obtained for the isotopologues containing this nucleus.

A single molecule as a dielectric medium

For three molecules with weak or negligible charge overlap, we prove that the three-body interaction energy obtained from quantum perturbation theory (to leading order) fits a dielectric model with a nonlocal electronic screening function. The electronic charge cloud of each molecule acts as a dielectric medium for the interaction of the remaining two with the nonlocal dielectric function epsilon(r,r') obtained by O. S. Jenkins and K. L. C. Hunt [J. Chem. Phys. 119, 8250 (2003)], by considering the charge redistribution induced in a single molecule by an external perturbation. The dielectric function depends parametrically on the coordinates of the nuclei, within the Born-Oppenheimer approximation. We also prove that the force on each nucleus in molecule A depends on intramolecular dielectric screening within A. The potential from the charge distribution of B, screened by C acting as a dielectric medium, is further screened linearly within A; and similarly, with the roles of B and C reversed. In addition, the potential due to the unperturbed charge distribution of B and the potential due to the unperturbed charge distribution of C, acting simultaneously, are screened nonlinearly within A. The results show that nonlocal dielectric theory holds on the molecular level, provided that the overlap of the electronic charge distributions is weak.

Rotational spectrum of the mixed van der Waals triad pyridine-Ar-Ne

We report the Fourier transform microwave spectra of four isotopologues of the hetero triad pyridine-Ar-Ne, formed by the combination of two isotopes of the nitrogen atom (14N and 15N) in pyridine with two isotopes of the rare gas (RG) atoms (20Ne and 22Ne), by using pulsed jet spectroscopy. We detected the conformer denoted [1,1], with the Ne and Ar atoms located one on each side of the ring plane. The [2,0] species, with the two RG atoms on the same side of the ring, was not observed. Ab initio MP2/6-311++G** calculations suggest the rotational spectrum of this species to be complicated by the presence of several almost equivalent minima, separated by very low barriers. Four structural van der Waals parameters, R(Ar), R(Ne), theta(Ar) and theta(Ne), which localize the RG atoms with respect to pyridine, are determined. The 14N nuclear quadrupole coupling constants are obtained for the isotopologues containing this nucleus.

Infrared spectra and intensities of the H2O and N2 complexes in the range of the nu1- and nu3-bands of water

The IR spectra of complexes of water with nitrogen molecules in the range of the symmetric (nu(1)) and antisymmetric (nu(3)) bands of H(2)O have been studied in helium droplets. The infrared intensities of the nu(3) and nu(1) modes of H(2)O were found to be larger by factors of 1.3 and 2, respectively, in the N(2)-H(2)O complexes. These factors are smaller than those obtained in recent theoretical calculations. The conformation of the N(2)-H(2)O complex was estimated. Spectra and IR intensities of the (N(2))(2)-H(2)O and N(2)-(H(2)O)(2) complexes were also obtained and their structures are discussed.

High-resolution microwave spectrum of the weakly bound helium-pyridine complex

High-resolution rotational spectra of the helium-pyridine dimer were obtained using a pulsed molecular beam Fourier transform microwave spectrometer. Thirty-nine R-branch (14)N nuclear quadrupole hyperfine components of a- and c-type dipole transitions were observed and assigned. The following spectroscopic parameters were obtained: rotational constants A=3875.2093(48) MHz, B=3753.2514(45) MHz, and C=2978.4366(81) MHz; quartic centrifugal distortion constants D(J)=0.124 08(55) MHz, D(JK)=0.1200(43) MHz, D(K)=-0.2451(25) MHz, d(1)=0.004 27(27) MHz, and d(2)=0.000 16(10) MHz; sextic centrifugal distortion constants H(J)=0.003 053(35) MHz, H(JK)=-0.006 598(47) MHz, and H(K)=0.004 11(59) MHz; (14)N nuclear quadrupole coupling constants chi(aa)((14)N)=-4.7886(76) MHz, chi(bb)((14)N)=1.4471(76) MHz, and chi(cc)((14)N)=3.3415(43) MHz. Our analyses of the rotational and (14)N quadrupole coupling constants show that the He atom binds perpendicularly to the aromatic plane of C(5)H(5)N with a displacement angle of approximately 7.0 degrees away from the c axis of the pyridine monomer, toward the nitrogen atom. Results from an ab initio structure optimization on the second order Moller-Plesset level are consistent with this geometry and gave an equilibrium well depth of 86.7 cm(-1).

Calculation of argon trimer rovibrational spectrum

Rovibrational spectra of Ar3 are computed for total angular momenta up to J=6 using row-orthonormal hyperspherical coordinates and an expansion of the wave function on hyperspherical harmonics. The sensitivity of the spectra to the two-body potential and to the three-body corrections is analyzed. First, the best available semiempirical pair potential (HFDID1) is compared with our recent ab initio two-body potential. The ab initio vibrational energies are typically 1-2 cm-1 higher than the semiempirical ones, which is related to the slightly larger dissociation energy of the semiempirical potential. Then, the Axilrod-Teller asymptotic expansion of the three-body correction is compared with our newly developed ab initio three-body potential. The difference is found smaller than 0.3 cm-1. In addition, we define approximate quantum numbers to describe the vibration and rotation of the system. The vibration is represented by a hyper-radial mode and a two-degree-of-freedom hyperangular mode, including a vibrational angular momentum defined in an Eckart frame. The rotation is described by the total angular momentum quantum number, its projection on the axis perpendicular to the molecular plane, and a hyperangular internal momentum quantum number, related to the vibrational angular momentum by a transformation between Eckart and principal-axes-of-inertia frames. These quantum numbers provide a qualitative understanding of the spectra and, in particular, of the impact of the nuclear permutational symmetry of the system (bosonic with zero nuclear spin). Rotational constants are extracted from the spectra and are shown to be accurate only for the ground hyperangular mode.