Complex hyperfine-fine structure overlapping in the microwave spectrum of 3,4-lutidine†

Eléonore Antonelli ^a and Ha Vinh Lam Nguyen *^ab
aUniv Paris Est Creteil and Université Paris Cité, CNRS, LISA, F-94010 Créteil, France. E-mail: lam.nguyen@lisa.ipsl.fr
^bInstitut Universitaire de France (IUF), F-75231 Paris, France

First published on 2nd July 2025

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1. Introduction

Molecular jet Fourier transform microwave (FTMW) spectroscopy is a powerful tool for investigating gas-phase molecular properties with high precision. It enables the measurement of rotational transitions, providing insights into molecular structure,^1–9 internal dynamics,^10–13 chiral discrimination^14–16 and noncovalent interactions.¹⁷ A key area of research in FTMW spectroscopy involves quantum tunneling effects, which arise from hindered internal motions such as methyl internal rotation,^18–21 proton inversion^22–26 and ring puckering.^27,28 When molecular configurations can interconvert through such motions, quantum tunneling manifests in the spectrum as line splittings.²⁹ These splittings are highly sensitive to the underlying potential energy surface as well as the steric and electrostatic forces that shape it. The exceptional resolution of FTMW spectroscopy allows for the precise determination of these splittings, offering valuable information about the molecular structure and internal dynamics.³⁰

In recent years, the study of coupled large amplitude motions (LAM) involving two inequivalent methyl internal rotors has gained attention in microwave spectroscopy.³¹ These quantum tunneling effects cause each rotational transition to split into a quintet consisting of the (σ₁σ₂) = (00), (01), (10), (11), and (12) torsional components.³² The splitting depends i.a. on the barrier hindering the methyl torsion which typically lies between 0 cm⁻¹ in the case of a free rotator and ~1200 cm⁻¹ in the case of a rigid-rotor approximation. In general, the higher the barrier, the smaller the splitting. For molecules such as o-xylene (molecule (1) in Fig. 1), 3,4-dimethylfluorobenzene (2), 3,4-dimethylanisole (3) and 3,4-dimethylbenzaldehyde (4), the barriers to internal rotation of the adjacent methyl groups are in the intermediate range due to steric hindrance, i.e. 430.00(37) cm⁻¹ and 467.90(17) cm⁻¹ for the m- and p-methyl groups in syn-3,4-dimethylanisole (3a), respectively, 499.64(26) cm⁻¹ and 533.54(22) cm⁻¹ for anti-3,4-dimethylanisole (3b),³² 456.20(13) cm⁻¹ and 489.78(15) cm⁻¹ for 3,4-dimethylfluorobenzene (2),³³ 508(1) cm⁻¹ and 551(9) cm⁻¹ for syn-3,4-dimethylbenzaldehyde (4a), 454(2) cm⁻¹ and 481(5) cm⁻¹ for anti-3,4-dimethylbenzaldehyde (4b)³⁴ and 521(18) cm⁻¹ for xylene (1).³⁵ Although the resulting torsional splittings are small, in order of less than 1 MHz, they are resolvable thanks to the high experimental accuracy of molecular jet FTMW instruments. If a nitrogen atom is present in addition, the nuclear spin I = 1 of the ¹⁴N nucleus introduces in addition to the fine torsional splittings hyperfine splittings²⁹ in the same order of magnitude, further increasing the spectral complexity. We are interested in investigating the microwave spectrum of 3,4-lutidine (34LUT) to explore this complexity which challenges the resolution of the molecular jet FTMW technique and the spectral assignment. The experimental work was supplemented by quantum chemical calculations.

2. Quantum chemical calculations

2.1. Geometry optimisations

Geometry optimisations were performed using the Gaussian16 program³⁶ at three levels of theory. Two second-order Møller–Plesset perturbation theory (MP2)³⁷ calculations were conducted with Pople's basis sets 6-311++G(d,p) and 6-31G(d,p).³⁸ Additionally, density functional theory (DFT) calculations were performed using the B3LYP^39,40 functional with Grimme's dispersion correction⁴¹ and Becke–Johnson damping,⁴² also in combination with the 6-311++G(d,p) basis set. These levels of theory are chosen because they have been shown to provide reliable theoretical rotational constants useful as spectral assignment guidance for many aromatic rings.^43–47 The predicted equilibrium rotational constants B_e and dipole moment components are presented in Table 1. The vibrational ground state rotational constants B₀ and quartic centrifugal distortion constants were obtained from anharmonic frequency calculations and are also given.

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Fig. 2 illustrates the geometry of the 34LUT optimised at the B3LYP-D3(BJ)/6-311++G(d,p) level of theory. The nuclear coordinates in the inertial principal axis system are provided in Table S1 of the ESI.† To assess whether other levels of theory are suitable for predicting the rotational constants of 34LUT and similar molecules, additional geometry optimisations were performed for benchmarking purposes. The methods tested include MP2, as well as several DFT methods such as B3LYP including Grimme's dispersion correction, Becke–Johnson damping and CAM (Coulomb Attenuation Method) functional,⁴⁸ Truhlar's M06-2X,⁴⁹ Minnesota MN15,⁵⁰ Head-Gordon's ωB97X-D,⁵¹ and PBE0.⁵² The benchmarking results are summarised in Table S2 of the ESI.†

2.2. Methyl internal rotations

The barriers to the internal rotation of the two methyl groups were calculated at the B3LYP-D3(BJ)/6-311++G(d,p), MP2/6-311++G(d,p) and MP2/6-31G(d,p) levels of theory by varying the dihedral angles α₁ = ∠(C₂, C₃, C₇, H₁₀) and α₂ = ∠(C₅, C₄, C₈, H₁₄) in 10° steps while optimising all other molecular parameters. The predicted V₃ potential values for the methyl group in the meta position are 470.2 cm⁻¹, 461.0 cm⁻¹ and 470.5 cm⁻¹, respectively, and those for the para position are 382.1 cm⁻¹, 373.8 cm⁻¹ and 378.4 cm⁻¹. These values indicate intermediate barriers.

The coupling between the two inequivalent methyl torsions was investigated through a two-dimensional potential energy surface (2D-PES) as a function of the dihedral angles α₁ and α₂. These angles were varied in a 10° grid while all other molecular parameters were optimised at the B3LYP-D3(BJ)/6-311++G(d,p) level of theory. The potential energy points were parameterised using the following 2D Fourier expansion:

	(1)

Using the Fourier coefficients, the 2D-PES was drawn as a colour contour plot visualised in Fig. 3. As indicated by the values of the Fourier coefficients in eqn (1), the coupling term cos(3α₁)cos(3α₂) between the two methyl torsions contributes less than 4% of the V₃ potential terms, suggesting a relatively weak interaction between the two methyl rotors. The 2D-PES obtained at the MP2/6-31G(d,p) and MP2/6-311++G(d,p) levels are similar and are illustrated in Fig. S1 of the ESI.†

2.3. ¹⁴N nuclear quadrupole coupling constants (NQCCs)

The NQCCs of the ¹⁴N nucleus in 34LUT were determined using Bailey's approach⁵³ which has been widely applied for predicting ¹⁴N NQCCs in various molecules.^54–58 The geometry optimised at the MP2/6-311++G(d,p) level was used as the base for electric field gradient (EFG) calculations, and a calibration factor of −4.599 MHz a.u.⁻¹, as recommended for π-conjugated systems, was subsequently applied.⁵⁹ The resulting NQCCs are χ_aa = −3.315 MHz, χ_bb = 0.101 MHz, χ_cc = 3.214 MHz and χ_ab = 2.710 MHz. Due to symmetry, the NQCCs χ_ac and χ_bc are both zero. While similar values can be obtained directly from quantum chemical calculations, the resulting NQCCs vary substantially depending on the level of theory employed (see for example Table 1). In contrast, Bailey's method offers greater robustness by calibrating against a broad set of experimental data. This makes it less sensitive to the computational method and more reliable for providing values that consistently match experimental observations. For this reason, it is particularly valuable in studies like the present one on 34LUT for resolving a complex hyperfine structure.

3. Microwave spectroscopy

3.1. Measurements

A commercial sample of 34LUT was obtained from TCI Europe, Zwijndrecht, Belgium, and used as received. A survey scan of 34LUT was recorded between 9.8 and 12.0 GHz using a pulsed molecular jet cavity-based FTMW spectrometer called the “big cavity”.⁶⁰ The scan was acquired as a series of overlapping spectra, each consisting of 50 co-added free induction decays (FIDs), with a step size of 250 kHz. Fig. 4 presents a segment of this scan. For sample introduction, a few drops of 34LUT were applied to a piece of pipe cleaner and inserted into a steel tube near the nozzle. Helium, maintained at a constant pressure of 2 bar, was used as the carrier gas to create the supersonic expansion. The mixture of 34LUT and helium was expanded into the cavity hold at an approximate pressure of 10⁻⁷ mbar.

After the spectrum has been initially assigned, high resolution measurements for lines observed not only in the survey scan but also outside the original scan range, covering frequencies between 2 and 20 GHz, were performed using the tone-excitation mode of the newly developed Passage And Resonance In Synergy (PARIS) spectrometer.⁶¹ This mode of the PARIS spectrometer operates with a Fabry–Pérot resonator in a Coaxially Oriented Beam-Resonator Arrangement (COBRA) configuration, revealing Doppler doublets for each transition. In this setup, the sample was placed in a dedicated holder positioned within the gas line, allowing helium at a constant pressure of 1 bar to flow over it and facilitate vaporization. The instrumental resolution was 2 kHz. However, due to line broadening from unresolved splittings caused by the combined effects of methyl internal rotation and ¹⁴N nuclear quadrupole coupling, the measurement accuracy was estimated to be slightly larger, around 4 kHz. This corresponds to about 1/10 of the average full width at half maximum (FWHM). A representative spectrum of the 4₀₄ ← 3₀₃ transition is displayed in Fig. 5.

For some measurements near the spectrometer frequency range limits, i.e. below 4 GHz and close to 20 GHz, helium was replaced with neon to optimise performance, because neon offers better cooling efficiency in supersonic expansions, which was particularly helpful for observing the lowest-J transitions, e.g., 1₀₁ ← 0₀₀. Additionally, measurements at the frequency range limits of our spectrometer are often less stable and exhibit reduced line intensity. In these conditions, neon provided narrower linewidths and reduced line overlap compared to helium.

3.2. Spectral assignments and fits

To begin the spectral assignment, we used the survey spectrum recorded between 9.8 and 12.0 GHz. As a first step, the effects of methyl internal rotation and ¹⁴N nuclear quadrupole coupling were neglected, and the molecule was treated as a rigid rotor. Quantum chemical calculations, with the results in Table 1, indicated that a-type transitions should be the most intense, while b-type transitions were weaker, and c-type transitions should be absent. Using the rotational constants obtained at the MP2/6-311++G(d,p) level, a rigid-rotor spectrum was predicted with the XIAM program⁶² and compared to the experimental survey spectrum. The rigid-rotor assignment was straightforward.

We then considered the effects of the two methyl internal rotations and ¹⁴N nuclear quadrupole coupling by extending the rigid-rotor fit. Initial values for the V₃ potentials and the angles between the methyl rotor axes and the principal a-axis ∠(i,a) were taken from B3LYP-D3(BJ) calculations, while the NQCCs were those calculated using Bailey's method. The predicted splittings showed that the (00), (01), (10), (11) and (12) torsional components overlapped significantly with the ¹⁴N quadrupole hyperfine structure. This overlap was particularly evident in the three strongest hyperfine components of each rotational transition. However, two weaker hyperfine components, located at either side of the central strong hyperfine triplet (see Fig. 4 and 5), were still observable for many transitions. These weaker components played a crucial role in identifying the fine splitting pattern associated with the methyl internal rotations. Once this fine structure was assigned, it was systematically applied to the overlapping strong hyperfine triplet. Ultimately, almost all hyperfine components and methyl torsional splittings could be successfully assigned. In total, 680 transitions were identified and included in a XIAM fit, yielding a standard deviation of 5.5 kHz. The final fit was achieved using only the three rotational constants, three quartic centrifugal distortion constants, the two V₃ terms, the two angles ∠(i,a) and the NQCCs χ_aa and χ₋ = χ_bb − χ_cc without the need for additional coupling or higher-order terms in the Hamiltonian. To minimise the correlation with V₃, the internal rotational constant F₀ was fixed at 160 GHz, a typical value for methyl groups.⁶³ Though the standard deviation is still slightly higher than the measurement accuracy estimated at 4 kHz, we did not attempt to reduce this further, since the splittings between the (11) and (12) torsional components could not be resolved for all observed transitions.

The obtained molecular parameters are summarised in Table 2, and the complete list of all fitted frequencies along with their residuals is given in Table S3 of the ESI.†

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4. Results and discussion

Using the XIAM program, a total of 680 transitions were assigned and fitted with a root-mean-square deviation of 5.5 kHz. Though the standard deviation of the fit is still slightly higher than the measurement accuracy of 4 kHz, due to the overlapping fine-hyperfine structures, we consider that the chosen Hamiltonian model implemented in XIAM, which includes only the essential parameters, i.e. the rotational constants, centrifugal distortion terms, torsional barriers, angles between the internal rotor axes and the a-principal axis and NQCCs, is sufficient to accurately describe the spectral features of 34LUT.

A benchmark of the rotational constants across various tested levels of theory, presented in Table S2 of the ESI,† shows that both the ab initio MP2 and DFT methods yield equilibrium rotational constants in good agreement with the experimental ground state values, with all deviations below 2%. The three levels of theory selected in Section 2.1, B3LYP-D3(BJ)/6-311++G(d,p), MP2/6-311++G(d,p) and MP2/6-31G(d,p), each show deviations below 0.2%. Among them, MP2/6-31G(d,p) performs particularly well, consistent with prior benchmarks on several aromatic ring containing molecules.^64–67 Therefore, we continue to recommend this level for calculating rotational constants as guidance for spectral assignment in microwave studies of lutidine derivatives. The basis sets including diffuse function (+ or ++) perform comparably to those without. The 6-31G(d,p) basis set also performs excellently when combined with the B3LYP-D3 or B3LYP-D3(BJ) methods. Additionally, the PBE0-D3 method with the 6-311G(3df,3pd) basis set, with or without diffuse functions, is also a suitable choice.

Although the experimental rotational constants correspond to the vibrational ground state and should, in principle, be compared with zero-point vibrationally corrected (anharmonic) constants, they often agree better with the equilibrium constants calculated from simple geometry optimisations. This apparent paradox is a well-known phenomenon, arising from a fortunate compensation of errors between neglected vibrational effects and other approximations in the calculations. Obtaining accurate anharmonic corrections requires sufficiently high-level methods that are computationally demanding and not always necessary when the primary goal is to obtain reliable predictions for guiding spectral assignments. In this context, using equilibrium rotational constants from moderate levels of theory remains an effective approach.

The barriers to methyl internal rotation in 34LUT are 510 cm⁻¹ for the methyl group in the meta position and 426 cm⁻¹ for the one in the para position. These values fall within the intermediate range of 400–600 cm⁻¹, similar to those observed in related molecules such as o-xylene (1),³⁵ 3,4-dimethylfluorobenzene (2),³³syn- and anti-3,4-dimethylanisole (3),³² as well as syn- and anti-3,4-dimethylbenzaldehyde (4)³⁴ (see Fig. 1). In o-xylene, both methyl groups are equivalent and have the same barrier of 521 cm⁻¹. For other molecules, the methyl group in the meta position has a lower barrier than the one in the para position. The opposite trend is observed in 34LUT.

We tried to explain this with a combination of electrostatic and steric effects. For 3,4-dimethylfluorobenzene and 3,4-dimethylanisole, the fluorine and methoxy substituents are highly electronegative and located outside the benzene ring. Their ortho/para-directing effects increase electron density at these positions, leading to higher barriers for the para methyl group. In 3,4-dimethylbenzaldehyde, the oxygen atom in the aldehyde substituent is positioned away from the methyl groups, minimizing electrostatic effects. In this case, steric interactions dominate, with the meta methyl group experiencing a higher barrier due to its proximity to the aldehyde group, a trend that is even more pronounced in the syn conformer. For 34LUT, the nitrogen atom exhibits a meta-directing effect, increasing electron density at the meta position (see the charge distribution in Fig. 1),⁶⁸ and thereby raising the barrier for internal rotation. Because nitrogen is part of the ring and not an external substituent, electrostatic effects are stronger than in 3,4-dimethylfluorobenzene and 3,4-dimethylanisole. This explains why the difference in methyl rotation barriers is more pronounced in 34LUT than in these other systems.

34LUT is a planar molecule with its symmetry plane perpendicular to the c-axis. The experimental value of χ_cc of 3.3480 MHz can be directly compared to those of related pyridine derivatives, i.e., the picolines.^69–71 To understand the effect of two methyl groups on the electric field at the nitrogen atom, 34LUT is compared with 2-picoline, 3-picoline and 4-picoline, whose χ_cc values are 3.22 MHz, 3.50 MHz and 3.78 MHz, respectively, as shown in Fig. 6. Though the χ_cc values are relatively close, the position of the methyl group has a noticeable effect: the values increase progressively from 2-picoline to 3-picoline and 4-picoline, and the value of 34LUT falls between those of 2-picoline and 3-picoline. Since the components of the NQCC tensor are directly proportional to those of the EFG tensor, this trend reflects an increasing electron density around the nitrogen nucleus along the series from 2-picoline to 34LUT, to 3-picoline and then to 4-picoline. To support this interpretation, we performed natural bond orbital (NBO) calculations and extracted the Mulliken charges on the nitrogen atom. The results show a consistent trend: the nitrogen carries charges of −0.066, −0.107 and −0.150 for 2-picoline, 3-picoline and 4-picoline, respectively, in line with the increasing χ_cc values. The presence of two methyl groups at the meta and para positions in 34LUT obviously reduces the electron density at the nitrogen compared to a single methyl group substitution in either the meta or para-position. While some slight deviation is observed in the trend with the Mulliken charge of −0.117, this level of variation is acceptable given the known limitations of charge partitioning calculations. We therefore rely primarily on the experimental χ_cc values, which provide a more reliable and direct measure of the EFG at the nitrogen nucleus.

Conclusion

The rotational spectrum of 34LUT has been successfully assigned and analysed. The presence of two inequivalent methyl groups combined with the ¹⁴N nuclear quadrupole coupling leads to complex spectral splitting patterns, which were fully resolved and interpreted. Among the benchmarked computational levels, we recommend MP2/6-31G(d,p) for calculating rotational constants as guidance for spectral assignment in microwave studies of lutidine derivatives. The observed inversion in barrier heights, where the meta methyl group has a higher barrier than the para methyl group, contrasts with previous studies on similar molecules and is attributed to the meta-directing effect of the nitrogen atom within the aromatic ring. This study highlights the importance of both steric and electrostatic contributions in determining the internal rotation dynamics of substituted aromatic molecules.

Author contributions

The manuscript was written with the contributions of all authors. All authors approved the final version of the manuscript.

Conflicts of interest

There are no conflicts of interest to declare.

Data availability

The data supporting this article have been included as part of the ESI.†

Acknowledgements

This work was supported by the European Union (ERC, 101040480-LACRIDO). Views and opinions expressed are however those of the authors only and do not necessarily reflect those of the European Union. Neither the European Union nor the granting authority can be held responsible for them. E. A. thanks the Université Paris-Est Créteil for a demi-bourse de thèse.

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Footnote

† Electronic supplementary information (ESI) available: Nuclear Cartesian coordinates, basis set variation, Fourier coefficients of the potential curves and the 2D-PES, potential energy surfaces depending on the dihedral angles α1 and α2 calculated at the MP2/6-311++G(d,p) and MP2/6-31G(d,p) levels of theory and frequency list. See DOI: https://doi.org/10.1039/d5cp01704a

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