# Coupled methyl internal rotations with intermediate and low torsional barriers in 2,5-dimethylanisole investigated by microwave spectroscopy

Haoyue Sun, Lynn Ferres, Isabelle Kleiner, Ha Vinh Lam Nguyen

► To cite this version:

Haoyue Sun, Lynn Ferres, Isabelle Kleiner, Ha Vinh Lam Nguyen. Coupled methyl internal rotations with intermediate and low torsional barriers in 2,5-dimethylanisole investigated by microwave spectroscopy. The Journal of Chemical Physics, 2024, 161 (22), pp.224304. 10.1063/5.0235431. hal-04867776

HAL Id: hal-04867776

https://hal.science/hal-04867776v1

Submitted on 6 Jan 2025

HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d'enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.

<figure><img src="image_1.png" /></figure>

Distributed under a Creative Commons Attribution 4.0 International License
# Coupled methyl internal rotations with intermediate and low torsional barriers in 2,5-dimethylanisole investigated by microwave spectroscopy

Haoyue Sun,<fnref n="1" /> Lynn Ferres,<fnref n="2" /> Isabelle Kleiner,<fnref n="1" /> and Ha Vinh Lam Nguyen<fnref n="3" />,<fnref n="4" />*

We recorded and analyzed the microwave spectra of 2,5-dimethylanisole using a pulsed molecular jet Fourier transform microwave spectrometer and the newly developed Passage And Resonant-Impulse Synergy (PARIS) spectrometer across a frequency range of 2 to 26.5 GHz with support from quantum chemical calculations. Only one conformer was predicted and observed, where the methoxy group and its adjacent methyl group adopt *anti* positions. The two methyl groups, located at the *ortho* and *meta* positions of the anisole ring, exhibit internal rotation, resulting in quintet splitting of all rotational transitions. The low torsional barrier of the *m*-methyl group, amounting to 65.723611(84) cm⁻<fnref n="1" />, combined with the intermediate barrier of 451.664(19) cm⁻<fnref n="1" /> for the *o*-methyl group, presented challenges in the spectral analysis. Using the *XIAM* and *BELGI-C₃-2Tops* programs, we successfully fitted 460 torsional intrastate rotational transitions, allowing for precise determination of molecular parameters and internal rotation characteristics. The torsional barriers are compared to those in the related isomers 2,3-, 2,4-, and 3,4-dimethylanisole as well as other *o*- and *m*-substituted toluene derivatives.

## I. INTRODUCTION

Molecular structures form the basis for comprehending chemical bonds, reaction pathways, and the physical properties of substances. In recent years, both theoretical and experimental approaches to determining molecular structures have advanced remarkably. The development of numerous spectroscopic methods, including neutron scattering, nuclear magnetic resonance, microwave, infrared, and ultraviolet-visible spectroscopy, as well as X-ray crystallography and electron microscopy, has enabled the study of molecular structures with ever-increasing resolution and accuracy. Among the various techniques, microwave spectroscopy has emerged as a keystone technique for the elucidation of molecular internal dynamics beyond precisely determining molecular geometry.<fnref n="1" />,<fnref n="2" />

Determining the structures of even small molecules can be challenging when intramolecular dynamics cause energy level splittings due to tunneling effects. The most frequently observed type of these large amplitude motions

<footnote n="1">¹ Université Paris Cité and Univ Paris Est Créteil, CNRS, LISA, F-75013 Paris, France</footnote>

<footnote n="2">² Institute of Physical Chemistry, RWTH Aachen University, D-52074 Aachen, Germany</footnote>

<footnote n="3">³ Univ Paris Est Créteil and Université Paris Cité, CNRS, LISA, F-94010 Créteil, France</footnote>

<footnote n="4">⁴ Institut Universitaire de France (IUF), F-75231 Paris, France</footnote>

<footnote>* Corresponding author: lam.nguyen@lisa.ipsl.fr</footnote>
(LAMs) is internal rotation of methyl groups. Internal rotation has been extensively studied through microwave spectroscopy, addressing complex issues such as low-torsional barriers and coupled internal rotors, thereby providing a comprehensive picture of molecular behavior.³⁻⁵ While the barrier to internal rotation typically ranges from 0 cm⁻¹ (free rotator) to about 1200 cm⁻¹ (rigid-rotor approximation), there is no strict definition for a “low barrier”. One approach to categorizing barriers is as follows: > 600 cm⁻¹ as high, 200 – 600 cm⁻¹ as intermediate, < 200 cm⁻¹ as low, and < 100 cm⁻¹ as very low. When we describe barriers as “intermediate”, the methyl torsional fine splittings are moderate, and we generally expect minimal difficulty in handling methyl internal rotation effects in the spectra. “High barriers” imply that splittings are small and may not be resolvable with the resolution of the spectrometers. “Low” or “very low” barriers typically indicate larger splittings, leading us to anticipate challenges in both assigning and fitting the microwave spectra. However, these boundaries between “very low”, “low”, “intermediate”, and “high” can be fluid. For example, a barrier of 190 cm⁻¹, which falls within the “low” range, may pose no issues for spectral assignment and fitting in some molecules, while in others, it may complicate these processes.

Examples on methyl internal rotation are the investigations on *o*-, *m*-, and *p*-methylanisole,⁶⁻⁸ as well as the series of dimethylanisole where the methyl group(s) attached to the phenyl ring act(s) as internal rotor(s). In *o*-methylanisole⁶ (molecule (2) in Fig. 1) and 2,4-dimethylanisole (5),⁹ the barrier hindering the internal rotation of the methyl group at the *ortho* position of the ring is very similar in both cases (444.1 cm⁻¹ and 441.1 cm⁻¹, respectively). The value found for the methyl group at the *para* position is also very consistent in *p*-methylanisole (molecule (1) in Fig. 1, 49.6 cm⁻¹)⁸ and 2,4-dimethylanisole (5, 47.6 cm⁻¹).⁹ However, the barriers to methyl internal rotation in dimethylanisole cannot always be directly derived from those in methylanisoles. In 3,4-dimethylanisole (7),¹⁰ the methyl rotors at the *meta* and *para*-positions feature torsional barriers with values between 430 cm⁻¹ and 534 cm⁻¹, much higher than the values found for the methyl group in *m*-methylanisole (55.8 cm⁻¹ and 36.6 cm⁻¹ for the *syn* (3a) and the *anti*-conformers (3b), respectively)⁷ and in *p*-methylanisole (1, 49.6 cm⁻¹)⁸ due to steric hindrance. In 2,3-dimethylanisole (6),¹¹ the barrier of the *o*-methyl group positioned between a methoxy group and a methyl group exhibits the lowest barrier observed in the entire family (26.9 cm⁻¹), which is probably due to the electronic configuration surrounding the *o*-methyl group. To understand more about the influence of electronic effects on the methyl torsional barriers, we investigated 2,5-dimethylanisole (25DMA, molecule (4) in Fig. 1) by a combination of microwave spectroscopy and quantum chemistry, aiming at determining and comparing the barriers of the *o*- and the *m*-methyl groups with those of *o*- and *m*-methylanisole and other dimethylanisole isomers. The *m*-methyl group in 25DMA is anticipated to exhibit very low rotational
barriers, posing challenges in spectral assignment and fitting processes, while the *o*-methyl group should feature a barrier similar to that of *o*-methylanisole. The methoxy methyl group O–CH₃ has a high barrier and does not cause torsional splittings in the microwave spectra. This behavior has been consistently observed in all anisole derivatives reported in the literature, including anisole itself.¹²

<figure><img src="image_2.png" /></figure>

FIG. 1. Comparison of the barrier to methyl internal rotation in methylanisole and dimethylanisole isomers: (1) *p*-methylanisole,⁸ (2) *o*-methylanisole,⁶ (3a) *syn* *m*-methylanisole,⁷ (3b) *anti* *m*-methylanisole,⁷ (4) 2,5-dimethylanisole (this work), (5) 2,4-dimethylanisole,⁹ (6) 2,3-dimethylanisole,¹¹ (7a) *syn* 3,4-dimethylanisole,¹⁰ and (7b) *anti* 3,4-dimethylanisole.¹⁰

The methyl groups in *ortho* positions are colored blue, those in *meta* position green, and those in *para* position red. The methoxy methyl groups (black) all have a high barrier and does not cause torsional splittings in the microwave spectra.

## II. QUANTUM CHEMICAL CALCULATIONS

### A. Conformational analysis

All quantum chemical computations in this investigation were conducted with the GAUSSIAN 16¹³ package. To search for the stable conformers of 25DMA, the dihedral angle $\gamma = \angle(C_6C_1O_{10}C_{11})$ was varied in increments of 10°, corresponding to the rotation about the C₁–O₁₀ bond (see Fig. 2 for atom numbering), while all other geometry parameters were optimized at the MP2/6-311++G(d,p) level. The obtained energy data points were fitted using a Fourier expansion, with the coefficients provided in Table S-I of the Supporting Information. The resultant potential energy curve is depicted in Fig. 3, revealing a single energy minimum at $\gamma = 0°$.
<figure><img src="image_3.png" /><figcaption>FIG. 2. Optimized molecular structure of 25DMA obtained at the MP2/6-311++G(d,p) level of theory in the inertia's principal axis system.</figcaption></figure>

<figure><img src="image_4.png" /><figcaption>FIG. 3. The potential energy curve of 25DMA obtained by rotating the methoxy group about the C1–O10 bond by varying the dihedral angle $\gamma = \angle(C_6C_1O_{10}H_{11})$ in increments of 10°. The relative energies are given with respect to the lowest energy conformation with $E = -424.232876$ Hartree. Only one minimum exists at $\gamma = 0°$.</figcaption></figure>

## B. Geometry optimizations

Geometry optimizations were carried out at the B3LYP-D3BJ/6-311++G(d,p) and MP2/6-311++G(d,p) levels of theory. These two levels have been selected based on their successful application in our prior investigations of the 2,3-, 2,4-, and 3,4-dimethylanisole isomers$^{9-11}$ and other dimethyl-substituted aromatic compounds,$^{14-19}$ where they have shown reliability in determining molecular structure parameters. Table I presents the equilibrium rotational constants and predicted dipole moment components. Additionally, we performed anharmonic frequency calculations, based on the optimized structures as starting geometries, at both levels of theory to obtain ground-state rotational constants and quartic centrifugal constants. The optimized MP2/6-311++G(d,p) geometry of 25DMA is shown in Fig. 2, with atomic Cartesian coordinates available in Table S-II of the Supporting Information.
To evaluate the effectiveness of various quantum chemical levels in predicting rotational constants for assisting with microwave spectral assignments of dimethylanisole derivatives, we employed a range of quantum chemical methods and basis set combinations, applied not only for 25DMA, but also for the 2,3-, 2,4-, and 3,4-isomers. Calculations were performed using the MP2<sup>20</sup> ab initio method along with the density functionals B3LYP,<sup>21,22</sup> B3LYP-D3,<sup>23</sup> B3LYP-D3BJ,<sup>24</sup> CAM-B3LYP-D3BJ,<sup>25</sup> M06-2X,<sup>26</sup> MN15,<sup>27</sup> PBE0,<sup>28</sup> and ωB97X-D<sup>29</sup> were carried out. These methods were combined with various Pople<sup>30</sup> and Dunning<sup>31</sup> basis sets. The obtained rotational constants are compiled in Table S-III in the Supporting Information.

<table><caption>TABLE I. Equilibrium rotational constants ($A_e$, $B_e$, $C_e$), vibrational ground state rotational constants ($A_0$, $B_0$, $C_0$), centrifugal distortion constants in the symmetrically reduced Hamiltonian, and dipole moment components of 25DMA calculated at the B3LYP-D3BJ/6-311++G(d,p) and MP2/6-311++G(d,p) levels of theory.</caption><thead><tr><th>Par.</th><th>Unit</th><th>B3LYP-D3BJ</th><th>MP2</th></tr></thead><tbody><tr><td>A<sub>e</sub></td><td>MHz</td><td>1846.1</td><td>1843.9</td></tr><tr><td>B<sub>e</sub></td><td>MHz</td><td>1164.8</td><td>1163.0</td></tr><tr><td>C<sub>e</sub></td><td>MHz</td><td>723.8</td><td>722.9</td></tr><tr><td>A<sub>0</sub></td><td>MHz</td><td>1833.6</td><td>1833.6</td></tr><tr><td>B<sub>0</sub></td><td>MHz</td><td>1153.8</td><td>1151.4</td></tr><tr><td>C<sub>0</sub></td><td>MHz</td><td>717.9</td><td>717.1</td></tr><tr><td>D<sub>J</sub></td><td>kHz</td><td>0.09523</td><td>0.09444</td></tr><tr><td>D<sub>JK</sub></td><td>kHz</td><td>-0.12520</td><td>-0.12870</td></tr><tr><td>D<sub>K</sub></td><td>kHz</td><td>0.03696</td><td>0.04127</td></tr><tr><td>d<sub>J</sub></td><td>kHz</td><td>-0.03515</td><td>-0.03470</td></tr><tr><td>d<sub>K</sub></td><td>kHz</td><td>-0.01582</td><td>-0.01495</td></tr><tr><td>|μ<sub>a</sub>|</td><td>D</td><td>0.05</td><td>0.10</td></tr><tr><td>|μ<sub>b</sub>|</td><td>D</td><td>1.44</td><td>1.41</td></tr><tr><td>|μ<sub>c</sub>|</td><td>D</td><td>0.00</td><td>0.00</td></tr></tbody></table>

### C. Methyl internal rotations

To examine the LAMs in 25DMA resulting from the two inequivalent *o*- and *m*-methyl rotors, the barriers to internal rotation were calculated. We varied the dihedral angles $α_1 = ∠(C_3, C_2, C_{15}, H_{17})$ and $α_2 = ∠(C_4, C_5, C_{19}, H_{20})$ while keeping all other parameters optimized using the B3LYP-D3BJ/6-311++G(d,p) and MP2/6-311++G(d,p) levels, which were previously used for geometry optimizations. These angles correspond to rotations about the C2–C15 and C5–C19 bonds, respectively. The obtained potential energy points were parameterized using one-dimensional Fourier expansions, with the Fourier coefficients listed in Table S-IV of the Supporting Information. Utilizing these coefficients, we generated potential energy curves that depict the internal rotations of the *o*- and *m*-methyl groups, illustrated in Fig. 4 and Fig. 5, respectively. For the *o*-methyl group, we observed three-fold $V_3$ potentials with a negligible contribution from $V_6$. The parameterized $V_3$ terms were found to be 402.1 cm⁻¹
(B3LYP-D3BJ) and 443.8 cm⁻¹ (MP2). In contrast, the energy curve for the *m*-methyl group obtained at the B3LYP-D3BJ/6-311++G(d,p) level also exhibited three-fold characteristics, while the curve at the MP2/6-311++G(d,p) level displayed sixfold characteristics (see Fig. 5), with a notable *V*₆ term of 33.6 cm⁻¹ compared to the *V*₃ term of 62.4 cm⁻¹. We then repeated the calculations at the MP2/6-31G(d,p) level to verify whether the double minima observed at the MP2/6-311++G(d,p) level, but not at the B3LYP-D3BJ/6-311++G(d,p) level, are method-dependent rather than basis-set dependent. The sixfold potential is retained with a *V*₃/*V*₆ ratio of 52.8 cm⁻¹/27.0 cm⁻¹.

<figure><img src="image_5.png" /></figure>

FIG. 4. Potential energy curves for 25DMA obtained by rotating the *o*-methyl group about the C₂–C₁₅ bond, achieved by adjusting the dihedral angle α₁ = ∠(C₃, C₂, C₁₅, H₁₇) in increments of 10°. Calculations were carried out at the B3LYP-D3BJ/6-311++G(d,p) (blue curve) and MP2/6-311++G(d,p) (black curve) levels of theory. The predicted internal rotation barriers are 402.1 cm⁻¹ (B3LYP-D3BJ) and 443.8 cm⁻¹ (MP2).

<figure><img src="image_6.png" /></figure>

FIG. 5. Potential energy curves for 25DMA obtained by rotating the *m*-methyl group about the C₅–C₁₉ bond, achieved by adjusting the dihedral angle α₂ = ∠(C₄, C₅, C₁₉, H₂₀) in increments of 10°. Calculations were performed at the B3LYP-D3BJ/6-311++G(d,p) (blue curve), MP2/6-311++G(d,p) (black curve), and MP2/6-31G(d,p) (red curve) levels of theory. The *V*₃/*V*₆ ratios are 58.3 cm⁻¹/6.9 cm⁻¹, 62.4 cm⁻¹/33.6 cm⁻¹, and 52.8 cm⁻¹/27.0 cm⁻¹, respectively.
To study the interaction between the two methyl tops, we calculated two-dimensional potential energy surfaces (2D-PES) by varying the dihedral angles $\alpha_1$ and $\alpha_2$ in 10° increments, while optimizing all other geometric parameters at the B3LYP-D3BJ/6-311++G(d,p), MP2/6-311++G(d,p), and MP2/6-31G(d,p) levels of theory. The obtained energy points were parameterized using a 2D-Fourier expansion, taking into account the symmetry of the *ortho-* and *meta*-methyl tops. The corresponding coefficients can be found in Table S-V in the Supporting Information. Using these coefficients, the 2D-PES are plotted and illustrated in Fig. 6. The 2D-PES predicted at the MP2/6-31G(d,p) level of theory closely resembles that calculated at the MP2/6-311++G(d,p) level and is provided in Fig. S1 of the Supporting Information. The shapes of the minimum regions in the 2D-PES obtained from the B3LYP-D3BJ/6-311++G(d,p) level are elongated oblate instead of circular, illustrating the coupling between the two methyl internal rotors. At both MP2/6-31G(d,p) and MP2/6-311++G(d,p) levels, clear double minimum regions are observed, further demonstrating the coupling which is also reflected in the significant coupling terms $\cos(3\alpha_1)\cos(3\alpha_2)$, $\sin(3\alpha_1)\sin(3\alpha_2)$, and the $V_6$ contribution calculated for the *m*-methyl group, as summarized in Table S-V.

<figure><img src="image_7.png" /><figcaption>FIG. 6. Two-dimensional potential energy surfaces for 25DMA calculated by varying the dihedral angles $\alpha_1 = \angle(C_3C_2C_{15}H_{17})$ and $\alpha_2 = \angle(C_4C_5C_{19}H_{20})$ in 10° steps, with other geometry parameters optimized at the B3LYP-D3BJ/6-311++G(d,p) and MP2/6-311++G(d,p) levels. The numbers in the color-code ladder represent energy relative to the minimum (0%) and maximum (100%) energy points with $E_{\text{min}}$ (B3LYP-D3BJ) = -425.5597806 Hartree (0%), $E_{\text{max}}$ (B3LYP-D3BJ) = -425.5577693 Hartree (100%), $E_{\text{min}}$ (MP2) = -424.2456290 Hartree (0%), and $E_{\text{max}}$ (MP2) = -424.2432740 Hartree (100%).</figcaption></figure>
### III. MICROWAVE SPECTROSCOPY

#### A. Measurements

25DMA for synthesis was purchased from Merck, Darmstadt, Germany, and used without further purification. The liquid was applied to a pipe cleaner and inserted into a stainless steel tube upstream of the nozzle. Under a helium flow at an absolute backing pressure of 2 bar, the gaseous mixture expanded into the vacuum chamber. The spectra were recorded using a supersonic jet Fourier transform microwave spectrometer and the newly developed Passage And Resonant-Impulse Synergy (PARIS) spectrometer, covering a total frequency range from 2 to 26.5 GHz.³² A survey scan from 8.5 to 14.0 GHz was recorded, where spectral segments in steps of 0.25 MHz were overlapped. Fig. 7 displayed a portion of the recorded survey scan. All lines from the scans were re-measured at higher resolution, where they appeared as doublets due to the Doppler effect, as shown in typical high resolution spectra in Fig. 8. Due to unresolved splittings arising from the internal rotation of the methoxy methyl group, as well as proton spin-spin and spin-rotation coupling, some lines are broadened. The measurement accuracy is estimated to be 4 kHz, larger than the instrumental line-width of 2 kHz.³³

<figure><img src="image_8.png" /><figcaption>FIG. 7. A portion of the survey scan from 10250 MHz to 11050 MHz recorded by overlapping spectral segments in steps of 0.25 MHz with 50 co-added free induction decays (FIDs) for each spectral segment. The intensity is in arbitrary unit. All lines are marked with their torsional species (00), (10), (01), (11), and (12) as well as rotational quantum numbers $J'_{K_a'K_c'} \leftarrow J_{K_aK_c}$.</figcaption></figure>
<figure><img src="image_9.png" /></figure>

FIG. 8. Typical high resolution spectra of the *b*-type transition 6$_{15}$ ← 5$_{24}$ of 25DMA (also visible in the scan portion shown in Fig. 7) with the (00), (10), (01), (11), and (12) torsional components. Doppler doublets are indicated by brackets. The left hand side spectrum were recorded at 10502.50 MHz. The right hand side spectra were recorded at 10538.00 MHz (blue), 10538.75 MHz (dark red), and 10539.50 MHz (green). For each spectrum, 50 FIDs decays were co-added. The polarization frequencies are all closer to the lower frequencies of the Doppler doublets, and therefore, favoring the low frequency side.

## B. Spectral assignments and fits

Initially, the molecule was treated as a semi-rigid rotor, i.e., only the (00) species was considered. The theoretical spectrum of 25DMA was simulated using the quantum chemically predicted rotational constants and dipole moment components at the MP2/6-311++G(d,p) level of theory shown in Table I into the *XIAM* program.³⁴ Since only the dipole moment component along the *b*-axis is significantly elevated, we expected a dominant *b*-type spectrum. The rotational transitions corresponding to the *R*-branches (*J*+1)₁(*J*+1) ← *J*₀*J* and (*J*+1)₀(*J*+1) ← *J*₁*J* were assigned first, and the *K*ₐ quantum number was progressively increased. A total of 86 (00) species lines were assigned and fitted using three rotational constants and four quartic centrifugal distortion constants to a root-mean-square (rms) deviation of 3.8 kHz. No *a*- or *c*-type transitions were identified.

As a next step, we assigned the (10) symmetry species, corresponding to the *o*-methyl rotor, which has an intermediate barrier height. Consequently, the torsional splittings are relatively narrow, and both (00) and (01) species lines often appeared in the same high-resolution measurement, as shown in Fig. 8. Using the rotational constants from the rigid-rotor fit, the angle $∠(i_1,a)$ between the internal rotor axis and the principal *a*-axis obtained at the MP2/6-311++G(d,p) level, and the *V*₃ barrier of 444 cm⁻¹ obtained for the *o*-methyl group in *o*-methylanisole,⁶ we predicted the (10) torsional species frequencies. The assignment of the (10) species lines was
straightforward, leading to a *XIAM* fit that included both (00) and (10) species, still achieving an rms deviation within the measurement accuracy.

The fit was then expanded to a two-top fit to account for the torsional splittings from the *m*-methyl rotor. For this methyl rotor, larger (00)-(01) splittings are expected due to the lower torsional barrier of 50-80 cm⁻¹. The (01), (11), and (12) symmetry species were assigned together, as they are often close to each other. The assignment was based on their frequency order using *XIAM* predictions, starting with the *K*<sub>a</sub> = 0 and 1 transitions. After fitting these frequencies, additional (01), (11), and (12) lines could be identified, but the *XIAM* predictions started to fail beyond a certain number of lines. To verify the assignments, we performed separate fits for the (01), (11), and (12) species using the *SFLAMS* program,³⁵ incorporating odd power parameters in the Hamiltonian as shown in equation (1):

$$ \mathbf{H}_{\text{op}} = (q + q_J \mathbf{P}^2 + q_K \mathbf{P}_z^2) \mathbf{P}_z + (r + r_J \mathbf{P}^2) \mathbf{P}_x + \frac{1}{2} r_K \{\mathbf{P}_z^2, \mathbf{P}_x\} \quad (1) $$

With predictions from these separate fits, further lines were identified, extending both the *XIAM* and *SFLAMS* input data sets. Since *a*-type transitions are too weak to be observed, it is not possible to check the assignment using combination difference loops (Ritz cycles), and we could only verify the assignment with the *SFLAMS* fits. The molecular parameters for the five *SFLAMS* fits are provided in Table II. Finally, a global fit comprising 460 torsional lines was performed with a modified version of the *XIAM* program called *XIAM*<sub>mod</sub> that includes two more higher power parameters *D*<sub>c3K</sub> and *D*<sub>c3-</sub>,³⁵,³⁶ resulting in an rms deviation of 4.9 kHz. This deviation close to the measurement accuracy of 4 kHz is a rare observation when *XIAM* handles very low barrier internal rotation. The *BELGI-Cs-2Tops* program, employing the “quasi-principal axis method” (“quasi-PAM”), reaches the same rms deviation of 4.8 kHz. The molecular parameters in the principal axis system, determined using *XIAM* and *BELGI-Cs-2Tops*, are presented in Table III. A full list of fitted frequencies along with their residuals can be found in Table S-VI in the Supporting Information. It should be noted that only a subset of *BELGI* molecular parameters can be transformed into constants that refer to the principal axis system. The *BELGI-Cs-2Tops* parameters within the "quasi-PAM" system are provided in Table IV.
<table><caption>TABLE II. Molecular parameters of (00), (10), (01), (11), and (12) species fits of 25DMA as obtained with the program SFLAMS.</caption><thead><tr><th>Par.<sup>a</sup></th><th>Unit</th><th>(00)</th><th>(10)</th><th>(01)</th><th>(11)</th><th>(12)</th></tr></thead><tbody><tr><td>A</td><td>MHz</td><td>1855.35131(26)</td><td>1855.31724(26)</td><td>1844.24392(34)</td><td>1844.20970(34)</td><td>1844.21105(34)</td></tr><tr><td>B</td><td>MHz</td><td>1166.547898(81)</td><td>1166.541394(81)</td><td>1164.92412(11)</td><td>1164.91756(11)</td><td>1164.91804(11)</td></tr><tr><td>C</td><td>MHz</td><td>724.556223(35)</td><td>724.556210(35)</td><td>724.578756(40)</td><td>724.579016(41)</td><td>724.579101(41)</td></tr><tr><td>D<sub>J</sub></td><td>kHz</td><td>0.02711(64)</td><td>0.02641(64)</td><td>0.02939(78)</td><td>0.02980(79)</td><td>0.03010(79)</td></tr><tr><td>D<sub>K</sub></td><td>kHz</td><td>0.484(23)</td><td>0.504(23)</td><td>0.271(25)</td><td>0.233(26)</td><td>0.254(26)</td></tr><tr><td>d<sub>1</sub></td><td>kHz</td><td>-0.02387(82)</td><td>-0.02304(83)</td><td>-0.0257(10)</td><td>-0.0251(10)</td><td>-0.0258(10)</td></tr><tr><td>d<sub>2</sub></td><td>kHz</td><td>-0.00446(21)</td><td>-0.00407(21)</td><td>-0.00347(25)</td><td>-0.00331(26)</td><td>-0.00322(26)</td></tr><tr><td>q</td><td>MHz</td><td></td><td>2.0602(13)</td><td>826.18255(67)</td><td>828.12581(68)</td><td>824.10409(66)</td></tr><tr><td>q<sub>J</sub></td><td>kHz</td><td></td><td></td><td>-19.913(44)</td><td>-19.884(45)</td><td>-19.739(44)</td></tr><tr><td>q<sub>K</sub></td><td>kHz</td><td></td><td></td><td>-53.946(95)</td><td>-53.982(97)</td><td>-53.876(95)</td></tr><tr><td>r</td><td>MHz</td><td></td><td></td><td>333.00504(53)</td><td>333.81729(53)</td><td>332.18033(54)</td></tr><tr><td>r<sub>J</sub></td><td>kHz</td><td></td><td></td><td>-3.979(17)</td><td>-4.002(17)</td><td>-3.979(18)</td></tr><tr><td>r<sub>K</sub></td><td>kHz</td><td></td><td></td><td>-86.91(14)</td><td>-87.30(15)</td><td>-86.82(15)</td></tr><tr><td>N<sup>b</sup></td><td></td><td>86</td><td>86</td><td>99</td><td>95</td><td>94</td></tr><tr><td>rms<sup>c</sup></td><td>kHz</td><td>3.8</td><td>3.3</td><td>2.4</td><td>2.5</td><td>2.2</td></tr></tbody></table>

<footnote><sup>a</sup> All parameters refer to the inertia principal axis system. Watson's S reduction in I' representation was used. <sup>b</sup> Number of lines. <sup>c</sup> Root-mean-square deviation of the fit.</footnote>
<table><caption>TABLE III. Molecular parameters of 25DMA in the inertia principal axis system obtained using the XIAM<sub>mod</sub> and BELGI-C<sub>s</sub>-2Tops programs.</caption><thead><tr><th>Par.<sup>a</sup></th><th>Unit</th><th>Fit XIAM</th><th>Fit BELGI</th><th>Calc.<sup>b</sup></th></tr></thead><tbody><tr><td>A</td><td>MHz</td><td>1847.75496(30)</td><td>1843.70(17)</td><td>1843.9</td></tr><tr><td>B</td><td>MHz</td><td>1165.34766(10)</td><td>1165.623(25)</td><td>1163.0</td></tr><tr><td>C</td><td>MHz</td><td>724.552082(43)</td><td>724.76704(20)</td><td>722.9</td></tr><tr><td>D<sub>1</sub></td><td>kHz</td><td>0.02838(75)</td><td>0.0274(10)</td><td>0.09444</td></tr><tr><td>D<sub>K</sub></td><td>kHz</td><td>0.129(25)</td><td></td><td>0.04127</td></tr><tr><td>d<sub>1</sub></td><td>kHz</td><td>-0.01187(48)</td><td></td><td>-0.03470</td></tr><tr><td>d<sub>2</sub></td><td>kHz</td><td>-0.00143(12)</td><td></td><td>-0.01495</td></tr><tr><td>V<sub>3,1</sub></td><td>cm<sup>-1</sup></td><td>451.664(19)</td><td>458.72(37)</td><td>443.8</td></tr><tr><td>V<sub>3,2</sub></td><td>cm<sup>-1</sup></td><td>65.723611(84)</td><td>67.100(36)</td><td>62.4</td></tr><tr><td>F<sub>12</sub></td><td>GHz</td><td>-4.741(59)</td><td></td><td></td></tr><tr><td>ρ<sub>1</sub></td><td>unitless</td><td>0.010485674<sup>c</sup></td><td>0.010265(35)</td><td></td></tr><tr><td>ρ<sub>2</sub></td><td>unitless</td><td>0.010500760<sup>c</sup></td><td>0.01006131(59)</td><td></td></tr><tr><td>D<sub>π<sup>2</sup>J,2</sub></td><td>MHz</td><td>0.041473(33)</td><td></td><td></td></tr><tr><td>D<sub>π<sup>2</sup>K,2</sub></td><td>MHz</td><td>0.164(46)</td><td></td><td></td></tr><tr><td>D<sub>π<sup>2</sup>-,2</sub></td><td>MHz</td><td>0.282(27)</td><td></td><td></td></tr><tr><td>D<sub>c3K,2</sub></td><td>MHz</td><td>-3.12(49)</td><td></td><td></td></tr><tr><td>D<sub>c3-,2</sub></td><td>MHz</td><td>-2.72(29)</td><td></td><td></td></tr><tr><td>∠(i<sub>1</sub>,a)</td><td>°</td><td>32.6811(98)</td><td>32.64(46)</td><td>30.874</td></tr><tr><td>∠(i<sub>1</sub>,b)</td><td>°</td><td>57.3189(98)</td><td>57.56(18)</td><td>59.129</td></tr><tr><td>∠(i<sub>1</sub>,c)</td><td>°</td><td>90.0<sup>d</sup></td><td>90.0<sup>d</sup></td><td>89.634</td></tr><tr><td>∠(i<sub>2</sub>,a)</td><td>°</td><td>147.567864(40)</td><td>147.614(13)</td><td>147.362</td></tr><tr><td>∠(i<sub>2</sub>,b)</td><td>°</td><td>122.432136(43)</td><td>122.3864(85)</td><td>122.628</td></tr><tr><td>∠(i<sub>2</sub>,c)</td><td>°</td><td>90.0<sup>d</sup></td><td>90.0<sup>d</sup></td><td>89.297</td></tr><tr><td>N<sup>e</sup></td><td></td><td>460</td><td>460</td><td></td></tr><tr><td>rms<sup>f</sup></td><td>kHz</td><td>4.9</td><td>4.8</td><td></td></tr></tbody></table>

<footnote><sup>a</sup> All molecular parameters refer to the inertia principal axis system. Watson's S reduction and I' representation were used. <sup>b</sup> Calculated at the MP2/6-311++G(d,p) level of theory. The rotational constants refer to the equilibrium structure. Centrifugal distortion constants are obtained from anharmonic frequency calculations. <sup>c</sup> Derived parameters in XIAM<sub>mod</sub>. <sup>d</sup> Fixed due to symmetry. <sup>e</sup> Number of lines. <sup>f</sup> Root-mean-square deviation of the fit.</footnote>
<table><caption>TABLE IV. Molecular parameters of 25DMA referred to the quasi-PAM system obtained by the program *BELGI-Cs-2Tops*.</caption><thead><tr><th>Operator<sup>a</sup></th><th>Par.<sup>b</sup></th><th>Unit</th><th>value</th></tr></thead><tbody><tr><td><b>P</b><sub>z</sub><sup>2</sup></td><td>A</td><td>MHz</td><td>1872.63(12)</td></tr><tr><td><b>P</b><sub>x</sub><sup>2</sup></td><td>B</td><td>MHz</td><td>1170.099(18)</td></tr><tr><td><b>P</b><sub>y</sub><sup>2</sup></td><td>C</td><td>MHz</td><td>724.76704(20)</td></tr><tr><td>-<b>P</b><sup>4</sup></td><td>Δ<sub>J</sub></td><td>kHz</td><td>0.0274(10)</td></tr><tr><td>-<b>P</b><sup>2</sup><b>P</b><sub>z</sub><sup>2</sup></td><td>Δ<sub>JK</sub></td><td>kHz</td><td>-0.0440(74)</td></tr><tr><td>-<b>P</b><sub>z</sub><sup>2</sup></td><td>Δ<sub>K</sub></td><td>kHz</td><td>0.199(31)</td></tr><tr><td>-2<b>P</b><sup>2</sup>(<b>P</b><sub>x</sub><sup>2</sup> - <b>P</b><sub>y</sub><sup>2</sup>)</td><td>δ<sub>I</sub></td><td>kHz</td><td>0.00981(31)</td></tr><tr><td>-{<b>P</b><sub>z</sub><sup>2</sup>, (<b>P</b><sub>x</sub><sup>2</sup> - <b>P</b><sub>y</sub><sup>2</sup>)}</td><td>δ<sub>K</sub></td><td>kHz</td><td>0.0244(38)</td></tr><tr><td>(1/2)(1 - cos3α<sub>1</sub>)</td><td>V<sub>3,1</sub></td><td>cm<sup>-1</sup></td><td>458.72(37)</td></tr><tr><td><b>p</b><sub>1</sub><sup>2</sup></td><td>f<sub>1</sub></td><td>GHz</td><td>161.683<sup>c</sup></td></tr><tr><td><b>P</b><sub>x</sub><b>p</b><sub>1</sub></td><td>r<sub>1</sub></td><td>GHz</td><td>1.1948(46)</td></tr><tr><td><b>P</b><sub>z</sub><b>p</b><sub>1</sub></td><td>q<sub>1</sub></td><td>GHz</td><td>-3.097(13)</td></tr><tr><td>(1/2)(1 - cos3α<sub>2</sub>)</td><td>V<sub>3,2</sub></td><td>cm<sup>-1</sup></td><td>67.100(36)</td></tr><tr><td><b>p</b><sub>2</sub><sup>2</sup></td><td>f<sub>2</sub></td><td>GHz</td><td>161.683<sup>c</sup></td></tr><tr><td><b>P</b><sub>x</sub><b>p</b><sub>2</sub></td><td>r<sub>2</sub></td><td>GHz</td><td>-1.16099(21)</td></tr><tr><td><b>P</b><sub>z</sub><b>p</b><sub>2</sub></td><td>q<sub>2</sub></td><td>GHz</td><td>3.03930(17)</td></tr><tr><td><b>P</b><sub>z</sub><b>p</b><sub>2</sub><b>P</b><sub>z</sub><sup>2</sup></td><td>q<sub>2K</sub></td><td>MHz</td><td>0.01016(30)</td></tr><tr><td>(1/2)(1 - cos3α<sub>2</sub>)<b>P</b><sub>z</sub><sup>2</sup></td><td>V<sub>3,2J</sub></td><td>MHz</td><td>-0.28636(42)</td></tr><tr><td>(1/2)(1 - cos3α<sub>2</sub>)<b>P</b><sub>z</sub><sup>2</sup></td><td>V<sub>3,2K</sub></td><td>MHz</td><td>9.329(10)</td></tr><tr><td><b>P</b><sub>x</sub><b>p</b><sub>2</sub><b>P</b><sub>z</sub><sup>2</sup></td><td>r<sub>2J</sub></td><td>kHz</td><td>-0.976(58)</td></tr><tr><td>(1 - cos3α<sub>1</sub>)(1 - cos3α<sub>2</sub>)</td><td>V<sub>12C</sub></td><td>cm<sup>-1</sup></td><td>-9.62(11)</td></tr><tr><td>N<sup>d</sup></td><td></td><td></td><td>460</td></tr><tr><td>rms<sup>e</sup></td><td></td><td>kHz</td><td>4.8</td></tr></tbody></table>

## IV. RESULTS AND DISCUSSION

The microwave spectrum of 25DMA was analyzed and fitted using two programs capable of handling the internal rotation of two methyl groups, *XIAM*<sub>mod</sub> and *BELGI-Cs-2Tops* to rms deviations of 4.9 kHz and 4.8 kHz, respectively, close to the measurement accuracy of 4 kHz. This is consistent with the rms deviation of 3.8 kHz obtained for the separate fit of the (00) species, which typically only necessitate a semi-rigid rotor Hamiltonian employing quartic centrifugal distortion terms. The values for the two potential barrier heights (V<sub>3,1</sub> and V<sub>3,2</sub>) from the *XIAM*<sub>mod</sub> and the *BELGI-Cs-2Tops* fits are in reasonable agreement considering the two different Hamiltonian models. However, we note that in the *BELGI-Cs-2Tops* fit, we had to fix the value of the kinetic coupling term *F*<sub>12</sub> to zero and to float the potential interacting term *V*<sub>12C</sub> multiplying (1 - cos3α₁)(1 - cos3α₂). The fitted value of -9.62(11) cm⁻¹ is in the same order of magnitude as the value of 4.97 cm⁻¹ calculated for cos(3α₁)cos(3α₂) (see

<footnote><sup>a</sup>Operator which the parameters multiply in the program. {A, B} is the anti-commutator AB + BA. α₁ and α₂ correspond to the o- and m-methyl groups, respectively. <sup>b</sup> All parameters refer to the quasi-PAM system. <sup>c</sup> Fixed to values from the *XIAM*<sub>mod</sub> fit. <sup>d</sup> Number of lines. <sup>e</sup> Root-mean-squares deviation of the fit</footnote>
Table S-V). It is also possible to fit the $V_{cc}$ parameter multiplying $\cos(3\alpha_1)\cos(3\alpha_2)$ with $XIAM_{mod}$. In this case, we obtain a value of $-5.0902(99) \text{ cm}^{-1}$, but the parameter is strongly correlated with $V_{3,1}$ and $V_{3,2}$, making the values significantly different (446.0 cm$^{-1}$ and 56.1 cm$^{-1}$, respectively). Furthermore, $F_{12}$ cannot be fitted well and the rms deviation is around 32 kHz. If $F_{12}$ is fitted, $V_{cc}$ is very small and not determined. Therefore, we decide to present the fit without $V_{cc}$ in Table 3.

The rotational constants and internal rotation parameters obtained from the $XIAM_{mod}$ fit are considered the most physically meaningful due to the use of fewer effective parameters and the lower correlation between them. Consequently, these values are used for comparison with theoretical values. The benchmark calculations with results shown in Table S-III of the Supporting Information demonstrated that all theoretical levels used for geometry optimizations of 25DMA predicted equilibrium rotational constants in reasonable agreement with the experimental values. Specifically, the rotational constants from B3LYP-D3BJ/6-311++G(d,p) calculations, which were benchmarked for the 2,6-, 3,4-, and 2,3-isomers of dimethylfluorobenzene, performed exceptionally well for 25DMA and very well for 2,3-dimethylanisole,¹¹ 2,4-dimethylanisole,⁹ and the two conformers of 3,4-dimethylanisole.¹⁰ Consequently, we continue to recommend this level for predicting rotational constants to aid in the assignment of molecules related to dimethylanisoles.

The fitted $V_3$ potential term of the *o*-methyl rotor is close to the value of 443.8 cm⁻¹ obtained by MP2 calculations. The value of 402.1 cm⁻¹ predicted at the B3LYP-D3BJ/6-311++G(d,p) level is approximately 49.5 cm⁻¹ lower than the experimental value. However, unlike the rotational constants, calculating torsional barriers remains a challenging task. We cannot conclusively state that the MP2/6-311++G(d,p) level is more precise for calculating the methyl torsional barriers in other dimethylanisoles. For the *p*-methyl group in 25DMA, significant $V_6$ contributions are observed in both MP2 calculations, resulting in double minimum potentials, while the B3LYP-D3BJ method predicts a typical three-fold potential, as illustrated in Fig. 5. Unfortunately, the $V_6$ term cannot be experimentally determined due to the absence of excited vibrational state transitions in our supersonic jet spectrum, which prevents clarification of this discrepancy.

The barrier heights of the *o*- and *m*-methyl groups are deduced to be 451.664(19) cm⁻¹ and 65.723611(84) cm⁻¹, respectively, with $XIAM_{mod}$, showing that the torsional barriers of both methyl groups are similar to their values in the respective methylanisoles (444.1 cm⁻¹ for *o*-methylanisole (2)⁶ and 55.8 cm⁻¹ for syn *m*-methylanisole (3a),⁷ see Fig. 1 for molecule numbering). This similarity is likely due to the spatial separation between the two methyl groups on the aromatic ring.
The barrier of the *o*-methyl group is higher than that of the *m*-methyl groups, as also observed in *o*-methylanisole and *m*-methylanisole. This originates from the steric hindrance created by the adjacent methoxy group, which favors an orientation pointing away from the *o*-methyl substituent. From the values observed for *o*-methylanisole (molecule (2) in Fig. 1, 444.1 cm⁻¹), 2,4-dimethylanisole (5, 441.1 cm⁻¹),⁹ and 25DMA (4, 451.7 cm⁻¹), we conclude that the torsional barrier of the *o*-methyl group in anisole derivatives is around 444 cm⁻¹, the value observed for the parent molecule *o*-methylanisole, when no additional substituent is in close proximity. Within the intermediate barrier range, a variation of less than 10 cm⁻¹ does not cause significant differences in the torsional splittings observed in the microwave spectra. Therefore, the value of 444 cm⁻¹ is considered highly precise and more reliable than quantum chemical calculations as initial value for spectral assignment in studies of other *o*-methylanisole derivatives. The microwave spectroscopic investigation on *syn*-2,5-dimethylbenzaldehyde reveals a higher torsional barrier of 565.7 cm⁻¹ for the *o*-methyl rotor,³⁷ showing that the electronic distribution and steric influence are different in anisole and benzaldehyde. Unfortunately, microwave spectroscopic investigations on *o*-methylbenzaldehyde are not available in the literature to draw more conclusive statement.

The $V_3$ potential value of 65.723611(84) cm⁻¹ determined for the *m*-methyl group of 25DMA is in the same order of magnitude as the values of 55.7693(90) cm⁻¹ found for *syn* *m*-methylanisole (3a),⁷ 36.6342(84) cm⁻¹ for *anti*- *m*-methylanisole (3b),⁷ 47.649(30) cm⁻¹ for 2,4-dimethylanisole (5),⁹ and 49.6370(1) cm⁻¹ for *p*-methylanisole.⁸ These values fall within the range of 36 cm⁻¹ – 65 cm⁻¹. For 2,5-dimethylbenzaldehyde³⁷ and 2,5-dimethylfluorobenzene,³⁸ the *m*-methyl torsional barrier was found to be 5.35(30) cm⁻¹ and 16.386(67) cm⁻¹, respectively, which is much lower than the torsional barrier in 25DMA. Despite the same position of the methyl groups in these different molecules and the same steric free surrounding, the barrier values are significantly different from one system to another. This is probably due to the electronic environment and the electronic exchanges depending on the substituents attached to the phenyl group through the *π*-conjugated system. Further investigations are needed to yield more precise statement.

V. CONCLUSION

We investigated the microwave spectrum of 25DMA using a resonator-enhanced spectrometer and the PARIS spectrometer, spanning the frequency range of 2.0 to 26.5 GHz. Due to the internal rotation of two inequivalent methyl groups at the *ortho* and *meta* positions of the anisole ring, all rotational transitions split into quintets. Torsional transitions were analyzed and fitted using the *XIAM*<sub>mod</sub> and *BELG-C<sub>5</sub>-2Tops* programs, yielding
rms deviations of 4.9 kHz and 4.8 kHz, respectively, closely matching the measurement accuracy of 4 kHz. To confirm the assignments, the program *SFLAMS* was used for separate fits of the five torsional components. The experimental barriers to methyl internal rotation were determined as 451.664(19) cm⁻<fnref n="1" /> for the *o*-methyl group and 65.723611(84) cm⁻<fnref n="1" /> for the *m*-methyl group. A comparison of the barrier for the *o*-methyl group in 25DMA with values from *o*-methylanisole and 2,4-dimethylanisole suggests that, in anisole derivatives, the *o*-methyl barrier remains close to 444 cm⁻<fnref n="1" /> in the absence of a further nearby substituent other than the methoxy group. This study also indicates that both steric and electrostatic effects influence the internal rotation barriers, offering new insights into how structural factors shape these torsional potentials.

## SUPPLEMENTARY MATERIAL

See the supplementary materials for the nuclear coordinates of the optimized structures, basis set variation, Fourier coefficients of the potential curves and the 2D-PES, potential energy surface depending on the dihedral angles $α_1$ and $α_2$ calculated at the at the MP2/6-31G(d,p) level of theory, and frequency list.

## ACKNOWLEDGMENTS

H.S. thanks the Chinese Scholarship Councils (CSC) from the Chinese government for a PhD scholarship. This work was supported by the Agence Nationale de la RechercheANR (project ID ANR-18-CE29-0011) and 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. Part of the simulations were performed with computing resources granted by the RWTH Aachen University under the project rwth0506.

## REFERENCES

<footnote n="1">¹ W. Gordy and L. R. Cook, Microwave Molecular Spectra. In: Weissberger A, editor, Techniques of chemistry, vol. XVIII, 3rd ed. New York: John Wiley & Sons Inc. (1984).</footnote>

<footnote n="2">² J.-U. Grabow, Fourier Transform Microwave Spectroscopy Measurement and Instrumentation. In: M. Quack, F. Merkt, editors. Handbook of High-Resolution Spectroscopy, Vol. 2. Chichester, UK: Wiley (2011) Chapter 1.</footnote>

<footnote n="3">³ D. G. Lister, J. N. Macdonald, N. L Owen, Internal Rotation and Inversion: An Introduction to Large Amplitude Motions in Molecules. Academic Press: New York, NY, USA (1978).</footnote>
⁴ H. V. L. Nguyen and I. Kleiner, Phys. Sci. Rev. **7**, 679 (2020).

⁵ H. V. L. Nguyen, W. Caminati, and J.-U. Grabow, *Molecules* **27**, 3948 (2022).

⁶ L. Ferres, H. Mouhib, W. Stahl, and H. V. L. Nguyen, *ChemPhysChem* **18**, 1855–1859 (2017).

⁷ L. Ferres, W. Stahl, and H. V. L. Nguyen, *J. Chem. Phys.* **148**, 124304 (2018).

⁸ L. Ferres, W. Stahl, I. Kleiner, and H. V. L. Nguyen, *J. Mol. Spectrosc.* **343**, 44–49 (2018).

⁹ L. Ferres, W. Stahl, and H. V. L. Nguyen, *J. Chem. Phys.* **151**, 104310 (2019).

¹⁰ L. Ferres, J. Cheung, W. Stahl, and H. V. L. Nguyen, *J. Phys. Chem. A* **123**, 3497–3503 (2019).

¹¹ L. Ferres, K.-N. Truong, W. Stahl, and H. V. L. Nguyen, *ChemPhysChem* **19**, 1781–1788 (2018).

¹² B. Reinhold, I. A. Finneran, S. T. Shipman, *J. Mol. Spectrosc.* **270**, 89–97 (2011).

¹³ M. J. Frisch, G. W. Trucks, H. B. Schlegel, G. E. Scuseria, M. A. Robb, J. R. Cheeseman, G. Scalmani, V. Barone, G. A. Petersson, H. Nakatsuji, X. Li, M. Caricato, A. V. Marenich, J. Bloino, B. G. Janesko, R. Gomperts, B. Mennucci, H. P. Hratchian, J. V. Ortiz, A. F. Izmaylov, J. L. Sonnenberg, D. Williams-Young, F. Ding, F. Lipparini, F. Egidi, J. Goings, B. Peng, A. Petrone, T. Henderson, D. Ranasinghe, V. G. Zakrzewski, J. Gao, N. Rega, G. Zheng, W. Liang, M. Hada, M. Ehara, K. Toyota, R. Fukuda, J. Hasegawa, M. Ishida, T. Nakajima, Y. Honda, O. Kitao, H. Nakai, T. Vreven, K. Throssell, J. A. Montgomery, Jr., J. E. Peralta, F. Ogliaro, M. J. Bearpark, J. J. Heyd, E. N. Brothers, K. N. Kudin, V. N. Staroverov, T. A. Keith, R. Kobayashi, J. Normand, K. Raghavachari, A. P. Rendell, J. C. Burant, S. S. Iyengar, J. Tomasi, M. Cossi, J. M. Millam, M. Klene, C. Adamo, R. Cammi, J. W. Ochterski, R. L. Martin, K. Morokuma, O. Farkas, J. B. Foresman, and D. J. Fox, *Gaussian 16*, Revision B.01, Gaussian, Inc., Wallingford, CT, 2016.

¹⁴ S. Khemissi, M. Schwell, I. Kleiner, and H. V. L. Nguyen, *Phys. Chem. Chem. Phys.* **26**, 402–411 (2024).

¹⁵ S. Khemissi, V. Van, M. Schwell, I. Kleiner, and H. V. L. Nguyen, *J. Phys. Chem. A* **127**, 5779–5789 (2023).

¹⁶ S. Khemissi, A. Pérez Salvador, and H. V. L. Nguyen, *J. Phys. Chem. A* **125**, 8542–8548 (2021).

¹⁷ V. Van, T. Nguyen, W. Stahl, H. V. L. Nguyen, and I. Kleiner, *J. Mol. Struct.* **1207**, 1277–1287 (2020).

¹⁸ V. Van, J. Bruckhuisen, W. Stahl, V. Ilyushin, H. V. L. Nguyen, and I. Kleiner, *J. Mol. Spectrosc.* **343**, 121–125 (2018).

¹⁹ M. Barth, I. Kleiner, and H. V. L. Nguyen, *J. Chem. Phys.* **160**, 244303 (2024).

²⁰ C. Møller and M. S. Plesset, *Phys. Rev.* **46**, 618 (1934).

²¹ A. D. Becke, *J. Chem. Phys.* **98**, 5648–5652 (1993).

²² C. Lee, W. Yang, and R. G. Parr, *Phys. Rev. B* **37**, 785 (1988).

²³ S. Grimme, J. Antony, S. Ehrlich, and H. Krieg, *J. Chem. Phys.* **132**, 154104 (2010).

²⁴ S. Grimme, S. Ehrlich, and L. Goerigk, *J. Comput. Chem.* **32**, 1456 (2011).

²⁵ T. Yanai, D. P. Tew, and N. C. Handy, *Chem. Phys. Lett.* **393**, 51–57 (2004).

²⁶ Y. Zhao and D. G. Truhlar, *Theor. Chem. Acc.* **120**, 215 (2008).

²⁷ H. S. Yu, X. He, S. L. Li, and D. G. Truhlar, *Chem. Sci.* **7**, 5032 (2016).

²⁸ C. Adamo and V. Barone, *J. Chem. Phys.* **110**, 6158–6170 (1999).

²⁹ J.-D. Chai and M. Head-Gordon, *Phys. Chem. Chem. Phys.* **10**, 6615 (2008).

³⁰ M. J. Frisch, J. A. Pople, and J. S. Binkley, *J. Chem. Phys.* **80**, 3265 (1984).

³¹ T. H. Dunning, *J. Chem. Phys.* **90**, 1007 (1989).

³² J.-U. Grabow, W. Stahl, and H. Dreizler, *Rev. Sci. Instrum.* **67**, 4072 (1996).

³³ J.-U. Grabow and W. Stahl, *Z. Naturforsch.* **45a**, 1043–1044 (1990).
³⁴ H. Hartwig and H. Dreizler, Z. Naturforsch. **51a**, 923 (1996).

³⁵ S. Herbers, S. M. Fritz, P. Mishra, H. V. L. Nguyen, and T. S. Zwier, *J. Chem. Phys.* **152**, 074301 (2020).

³⁶ S. Herbers and H. V. L. Nguyen, *J. Mol. Spectrosc.* **370**, 111289 (2020).

³⁷ M. Tudorie, I. Kleiner, M. Jahn, J.-U. Grabow, M. Goubet, and O. Pirali, *J. Phys. Chem. A* **117**, 13636–13647 (2013).

³⁸ H. Sun, S. Khemissi, I. Kleiner, and H. V. L. Nguyen, *J. Chem. Phys.* **160**, 094302 (2024).