Skeletal Torsion Tunneling and Methyl Internal Rotation: The Coupled Large Amplitude Motions in Phenyl Acetate

The rotational spectrum of phenyl acetate, CH3COOC6H5, is measured using a free jet absorption millimeter-wave spectrometer in the range from 60 to 78 GHz and two pulsed jet Fourier transform microwave spectrometers covering a total frequency range from 2 to 26.5 GHz. The features of two large amplitude motions, the methyl group internal rotation and the skeletal torsion of the CH3COO group with respect to the phenyl ring C6H5 (tilted at about 70°), characterize the spectrum. The vibrational ground state is split into four widely spaced sublevels, labeled as A0, E0, A1, and E1, each of them with its set of rotational transitions and with additional interstate transitions. A global fit of the line frequencies of the four sublevels leads to the determination of 51 spectroscopic parameters, including the ΔEA0/A1 and ΔEE0/E1 vibrational splittings of ~36.4 and ~33.5 GHz, respectively. The V3 barrier to methyl internal rotation (~136 cm−1) and the skeletal torsion B2 barrier to the orthogonality of the two planes (~68 cm−1) are deduced.


Introduction
Phenyl acetate (CH 3 COOC 6 H 5 , also called phenyl ethanoate, phenol acetate, acetyloxybenzene, or acetoxybenzene) is a colorless liquid with a plastic-like odor, often used as a solvent in chemical reactions. It can be synthesized by adding acetic anhydride to phenol [1], via Baeyer-Villiger oxidation of acetophenone [2], or by decarboxylation of aspirin [3]. Currently, phenyl acetate is a drug being studied in the treatment of cancer [4]. Naturally occurring in mammals, phenyl acetate induces differentiation, growth inhibition, and apoptosis in tumor cells. Despite the great interest in medicine, knowledge of its molecular structure is very limited, though it might be helpful to develop force fields to study biological processes at a molecular scale towards understanding how phenyl acetate affects tumor cells.
Many acetates have already been studied using rotational spectroscopy, notably a series of linear aliphatic acetates starting from methyl acetate to hexyl acetate [5][6][7][8][9][10]. The methyl group in the acetyl moiety undergoes internal rotation, causing each rotational transition to split into an A-E doublet. The torsional barrier is almost invariant at approximately 100 cm −1 [7]. If a C=C double bond is attached at the α,β-position counting after the oxygen atom, the value increases to about 150 cm −1 , as in the case of vinyl acetate [11,12] and butadienyl acetate [13] where the double bond(s) are located in the OCO-plane. Surprisingly, this is not the case in isopropenyl acetate, where we observed an unusual C 1 structure. The isopropenyl group is tilted out of the OCO-plane, and a torsional barrier of 135.3498 (38) cm −1 was deduced, a value which lies between 100 and 150 cm −1 [14]. Since the molecular structure of phenyl acetate is very similar to that of isopropenyl acetate, we were interested in answering two questions: (i) Does phenyl acetate have a C s or C 1 symmetry? (ii) How high is the barrier to internal rotation of the methyl group of phenyl acetate? Though in many cases, studies on medium-sized molecules containing a phenyl ring have reported a planar heavy-atom structure as the most stable one, such as anisole [15], phenetole [16], methyl salicylate [17], or acetophenone [18], there are several molecules where at least one conformer observed in the rotational spectrum has C 1 symmetry. For example, the CH 2 OH group in benzyl alcohol [19] as well as in its derivative 3,5difluorobenzyl alcohol [20] is tilted out of the phenyl ring. In the case of 2-phenylethylamine, none of the observed four conformers possesses a structure where all heavy atoms are within the aromatic plane [21,22]. In cis-acetanilide, C 6 H 5 (NH)(CO)CH 3 , Cabezas et al. reported doublets with splittings in the order of a few tens of kHz [23]. They were interpreted to be due to the tunneling motion of the phenyl ring between two equivalent non-planar conformations, with the phenyl ring tilted out of the (NH)(CO) plane by about 40 • through a transition state where the acetamide group and the phenyl ring are perpendicular. Aviles Moreno, Petitprez, and Huet observed similar splittings in the order of about 100 kHz for E-phenylformamide, where the phenyl ring is tilted out of the (NH)(CO) plane by the same angle [24]. These splittings arise from Coriolis interactions, from which the ∆E 01 value was determined to be 3.732027 (43) GHz. In phenyl formate [25], the phenyl ring tilt angle of 72 • is remarkably larger than in cis-acetanilide and E-phenylformamide, and the ring also undergoes a tunneling large amplitude motion (LAM). The energy levels of molecules featuring such an inversion motion split into a (+) and a (−) state due to a symmetric and an anti-symmetric wave function, separated by ∆E 01 , for example in ammonia [26] (see Figure 1) [27]. In phenyl formate, the ∆E 01 value of 46.2231 (25) GHz is large, making the (+)←(−) and (−)←(+) transitions non-observable in the frequency range accessible by the spectrometer in use. However, the theoretically unsplit (±)←(±) transitions still show small splittings, referred to as v t = 0 and v t = 1, due to Coriolis interactions, which were observed and could be fitted with the spfit program [28], but many complications occurred in further assignments and fitting processes that relied on a small number of the less intense v t = 1 transitions and Coriolis splittings between the v t = 0 and 1 states.
Regarding the closely related forms of phenyl acetate and phenyl formate, we supposed a similar structure with the phenyl ring and the O-(C=O)-C plane tilted with respect to each other, enabling a skeletal tunneling motion. However, phenyl acetate features an additional methyl internal rotor, which further splits the rotational signals into an A and an E symmetry species. Thus, the A-E pairs are expected for each transition of the torsional levels, resulting in a total of four species called A0, E0 (v t = 0), and A1, E1 (v t = 1). Note that these sublevels could be labeled more properly as A + , A − , E + , and E − . The splittings are shown in the energy level scheme in Figure 1. To fit transition frequencies belonging to different phenyl tunneling states, the spfit program is well-suited. Some of its main advantages are the possibility for users to build up an individual Hamiltonian, the ability to define any desired parameters in the input, and the short calculation time. Unfortunately, spfit can only fit the A and E torsional transitions separately, and deriving the torsional barrier is a tedious task in low barrier cases [29]. The program XIAM is a counterpart, which is specialized for fitting the A-E splittings due to three-fold methyl internal rotation but is incapable of treating tunneling states arising from a two-fold potential [30]. A further separate fit was performed with the XIAM program where only the A0-E0 doublets are considered to alternatively access the methyl torsional barrier. (1) Asymmetric rigid rotor (e.g., phenetole [16]), (2) One two-fold tunneling motion (e.g., phenyl formate with the tunneling of the phenyl ring [25]), (3) One two-fold tunneling motion and a methyl internal rotation (e.g., phenyl acetate with the tunneling of the phenyl ring and the internal rotation of the acetyl methyl group, this work). Rotational a-type transitions are indicated as black arrows. Please note that, for sake of simplicity, the symmetric top J, K label is used to indicate the rotational levels, rather than the correct J, Ka, Kc one.

Quantum Chemical Calculations
Using the Gaussian 16 program package [31], the molecular structure of phenyl acetate was optimized at the MP2/6-311G++(d,p) level of theory, revealing a non-planar structure with the methyl group entgegen with respect to the phenyl ring, called conformer I, illustrated in Figure 2. A second conformer, with the methyl group zusammen with respect to the ring, was calculated much higher in energy (1347 cm −1 ) and was not considered in the present work. Similar to the case of phenyl formate, the phenyl ring is tilted out of the O-(C=O)-C plane by about 72° for conformer I. Due to the symmetry of the phenyl ring and the planarity of the acetyl moiety, four equivalent minima are possible, as shown in Figure 3, where the dihedral angle α = ∠(C13-O12-C3-C2) was varied in 10° steps, while all other geometry parameters were optimized. The obtained energy points were parametrized using a Fourier expansion, including terms with the correct symmetry of α, which are collected in Table S1 in the Supplementary Materials. The two pairs Ia/Ia * and Ib/Ib * describe two double minima in the potential energy curve, separated by a barrier of 20.65 cm −1 , while Ia and Ib, as well as Ia * and Ib * , can be converted into each other by rotating the phenyl ring by 180° with a much higher conversion barrier of 837.45 cm −1 . The geometry of the local maximum between Ia and Ia * (or Ib and Ib * ) was also optimized as a transition state, yielding a slightly higher barrier of 26.71 cm −1 . The dihedral angle β = ∠(C3-O12-C13-C14) corresponds to the orientation of the acetyl group and stays nearly invariant at 180° in the entgegen configuration, which is known to be much more stable than the zusammen one (β = 0°). The Cartesian coordinates of the Ib structure are given in Table S2 in   (1) Asymmetric rigid rotor (e.g., phenetole [16]), (2) One two-fold tunneling motion (e.g., phenyl formate with the tunneling of the phenyl ring [25]), (3) One two-fold tunneling motion and a methyl internal rotation (e.g., phenyl acetate with the tunneling of the phenyl ring and the internal rotation of the acetyl methyl group, this work). Rotational a-type transitions are indicated as black arrows. Please note that, for sake of simplicity, the symmetric top J, K label is used to indicate the rotational levels, rather than the correct J, K a , K c one.

Quantum Chemical Calculations
Using the Gaussian 16 program package [31], the molecular structure of phenyl acetate was optimized at the MP2/6-311G++(d,p) level of theory, revealing a non-planar structure with the methyl group entgegen with respect to the phenyl ring, called conformer I, illustrated in Figure 2. A second conformer, with the methyl group zusammen with respect to the ring, was calculated much higher in energy (1347 cm −1 ) and was not considered in the present work. Similar to the case of phenyl formate, the phenyl ring is tilted out of the O-(C=O)-C plane by about 72 • for conformer I. Due to the symmetry of the phenyl ring and the planarity of the acetyl moiety, four equivalent minima are possible, as shown in Figure 3, where the dihedral angle α = ∠(C 13 -O 12 -C 3 -C 2 ) was varied in 10 • steps, while all other geometry parameters were optimized. The obtained energy points were parametrized using a Fourier expansion, including terms with the correct symmetry of α, which are collected in Table S1 in the Supplementary Materials. The two pairs I a /I a * and I b /I b * describe two double minima in the potential energy curve, separated by a barrier of 20.65 cm −1 , while I a and I b , as well as I a * and I b * , can be converted into each other by rotating the phenyl ring by 180 • with a much higher conversion barrier of 837.45 cm −1 . The geometry of the local maximum between I a and I a * (or I b and I b * ) was also optimized as a transition state, yielding a slightly higher barrier of 26.71 cm −1 . The dihedral angle β = ∠(C 3 -O 12 -C 13 -C 14 ) corresponds to the orientation of the acetyl group and stays nearly invariant at 180 • in the entgegen configuration, which is known to be much more stable than the zusammen one (β = 0 • ). The Cartesian coordinates of the I b structure are given in Table S2 in  and µ c = −1.46 D. Therefore, mainly band c-type transitions are expected in the rotational spectrum, eventually accompanied by some weak a-type signals. Frequency calculations were carried out at the same level of theory, yielding an imaginary frequency describing a ring bending motion. This observation has been frequently reported in calculations using the MP2 method [32]. Repeating the geometry optimization and frequency calculations at various levels of theory shows that most levels state no imaginary frequency. calculations were carried out at the same level of theory, yielding an imaginary frequency describing a ring bending motion. This observation has been frequently reported in calculations using the MP2 method [32]. Repeating the geometry optimization and frequency calculations at various levels of theory shows that most levels state no imaginary frequency.  The dihedral angle γ = ∠(O15-C13-C14-H16) was varied in a 10° grid to calculate the torsional barrier of the methyl internal rotation. The potential curve is given in Figure 4. From the Fourier parameterization (see Table S1 in the Supplementary Materials), we determined a V3 term of 113.18 cm −1 . A very similar value of 113.40 cm −1 was deduced by optimizing the maxima as transition states [33].  calculations were carried out at the same level of theory, yielding an imaginary frequency describing a ring bending motion. This observation has been frequently reported in calculations using the MP2 method [32]. Repeating the geometry optimization and frequency calculations at various levels of theory shows that most levels state no imaginary frequency.  The dihedral angle γ = ∠(O15-C13-C14-H16) was varied in a 10° grid to calculate the torsional barrier of the methyl internal rotation. The potential curve is given in Figure 4. From the Fourier parameterization (see Table S1 in the Supplementary Materials), we determined a V3 term of 113.18 cm −1 . A very similar value of 113.40 cm −1 was deduced by optimizing the maxima as transition states [33]. The dihedral angle γ = ∠(O 15 -C 13 -C 14 -H 16 ) was varied in a 10 • grid to calculate the torsional barrier of the methyl internal rotation. The potential curve is given in Figure 4. From the Fourier parameterization (see Table S1 in the Supplementary Materials), we determined a V 3 term of 113.18 cm −1 . A very similar value of 113.40 cm −1 was deduced by optimizing the maxima as transition states [33]. To study the coupling between the two LAMs, we calculated a two-dimensional potential energy surface (2D-PES) depending on α and γ, as shown in Figure 5. The obtained energy points were also parameterized using a Fourier expansion, including terms with the correct symmetry of α and γ given in Table S3 in the Supplementary Materials. Along the γ-axis, the threefold symmetry of the methyl group can be clearly seen. For a given γ minimum, for example, γ = 0°, the four equivalent structures along the α-axis are also recognizable. The minima regions of this 2D-PES are very broad and flat. Therefore, more colors were used in the lower 50% area to enable the distinction of the four minima. At this point, it is clear that the situation observed for phenyl acetate is very similar, for the skeletal torsion, to that of phenyl formate [25]. The tunneling LAM of the skeletal motion causes low-lying excited torsional states, which are most probably observable in the rotational spectrum. To calculate these tunneling states, a two-rigid-top model was applied, as described in Ref. [25], using the following Hamiltonian: To study the coupling between the two LAMs, we calculated a two-dimensional potential energy surface (2D-PES) depending on α and γ, as shown in Figure 5. The obtained energy points were also parameterized using a Fourier expansion, including terms with the correct symmetry of α and γ given in Table S3 in the Supplementary Materials. Along the γ-axis, the threefold symmetry of the methyl group can be clearly seen. For a given γ minimum, for example, γ = 0 • , the four equivalent structures along the α-axis are also recognizable. The minima regions of this 2D-PES are very broad and flat. Therefore, more colors were used in the lower 50% area to enable the distinction of the four minima.  To study the coupling between the two LAMs, we calculated a two-dimensional potential energy surface (2D-PES) depending on α and γ, as shown in Figure 5. The obtained energy points were also parameterized using a Fourier expansion, including terms with the correct symmetry of α and γ given in Table S3 in the Supplementary Materials. Along the γ-axis, the threefold symmetry of the methyl group can be clearly seen. For a given γ minimum, for example, γ = 0°, the four equivalent structures along the α-axis are also recognizable. The minima regions of this 2D-PES are very broad and flat. Therefore, more colors were used in the lower 50% area to enable the distinction of the four minima. At this point, it is clear that the situation observed for phenyl acetate is very similar, for the skeletal torsion, to that of phenyl formate [25]. The tunneling LAM of the skeletal motion causes low-lying excited torsional states, which are most probably observable in the rotational spectrum. To calculate these tunneling states, a two-rigid-top model was applied, as described in Ref. [25], using the following Hamiltonian: At this point, it is clear that the situation observed for phenyl acetate is very similar, for the skeletal torsion, to that of phenyl formate [25]. The tunneling LAM of the skeletal motion causes low-lying excited torsional states, which are most probably observable in the rotational spectrum. To calculate these tunneling states, a two-rigid-top model was applied, as described in Ref. [25], using the following Hamiltonian: The effective torsional constant F of 0.32635 cm −1 is calculated as h/8π 2 Ic, where I = 51.654 uÅ 2 is the effective moment of inertia and c is the speed of light. I is calculated from the moments of inertia of the phenyl top I p = 121.034 uÅ 2 and of the acetyl top I ac = 90.112 uÅ 2 . Both were taken from ab initio. The Fourier expansion terms are those obtained from the parameterization of the potential energy curve given in Figure 3. Direct diagonalization of the built-up Hamiltonian matrix yields the following eigenvalues, which represent the energies of the degenerate torsional states given in parentheses: 13.801 (0), 14.823 (1), 31.931 (2), and 42.330 (3) cm −1 . The potential function in the range from 50 • to 130 • , including the torsional energy states, is illustrated in Figure 6.
The effective torsional constant F of 0.32635 cm −1 is calculated as h/8π 2 Ic, where I = 51.654 uÅ 2 is the effective moment of inertia and c is the speed of light. I is calculated from the moments of inertia of the phenyl top Ip = 121.034 uÅ 2 and of the acetyl top Iac = 90.112 uÅ 2 . Both were taken from ab initio. The Fourier expansion terms are those obtained from the parameterization of the potential energy curve given in Figure 3. Direct diagonalization of the built-up Hamiltonian matrix yields the following eigenvalues, which represent the energies of the degenerate torsional states given in parentheses: 13.801 (0), 14.823 (1), 31.931 (2), and 42.330 (3) cm −1 . The potential function in the range from 50° to 130°, including the torsional energy states, is illustrated in Figure 6. The difference between the first two levels, vt = 0 and 1, is only 1.022 cm −1 , corresponding to 30.64 GHz, which is the value of ΔE01. Calculating the Boltzmann distribution yields an N1/N0 ratio of 0.23 to 0.48 for a temperature of 1 to 2 K, i.e., the population of the first excited state vt = 1 is expected to be 23 to 48% with respect to the ground state vt = 0.

Experimental Setups
The rotational spectrum was first observed using the Bologna free jet absorption millimeter-wave spectrometer in the range from 60 to 78 GHz [34][35][36]. Initially, families of intrastate (vt = 0, 1) μc-R-type transitions such as J10,x ← (J−1)9,x and J9,y ← (J−1)8,y (with J up to 18) were measured and fitted, followed by many intrastate μa-R-type transitions and several (unexpected, because only μb is inverting) interstate μa-R-type transitions (see the explanation in a following section) [37]. The estimated uncertainty for the measurements is about 50 kHz. The sample was heated to 65-70 °C while a stream of argon with a backing pressure of Pa = 18 kPa flowed over it. The mixture was then expanded to about Pb = 0.5 Pa through a 0.3 mm diameter pinhole nozzle. The estimated rotational temperature of the molecules in the jet is 5−10 K. Later on, extensive measurements were made with two molecular jet Fourier transform microwave (MJ-FTMW) spectrometers, one in Bologna [38] (the assignment of the E0 state was made in a way similar to that described in the next section for Aachen) and one in Aachen [39]. Using the scan mode (50 co-added free induction decays per each step in a 250 kHz step width), a broadband scan was recorded from 8 to 14 GHz with the Aachen MJ-FTMW spectrometer. The substance, with a stated purity of 99%, was purchased from Sigma Aldrich, Taufkirchen, Germany, and was used without further purification. Under helium-stream, the phenyl acetate-He mixture entered the vacuum chamber under a backing pressure of 200 Pa. All signals appeared as doublets due to the Doppler effect and were re-measured with higher resolution and The difference between the first two levels, v t = 0 and 1, is only 1.022 cm −1 , corresponding to 30.64 GHz, which is the value of ∆E 01 . Calculating the Boltzmann distribution yields an N 1 /N 0 ratio of 0.23 to 0.48 for a temperature of 1 to 2 K, i.e., the population of the first excited state v t = 1 is expected to be 23 to 48% with respect to the ground state v t = 0.

Experimental Setups
The rotational spectrum was first observed using the Bologna free jet absorption millimeter-wave spectrometer in the range from 60 to 78 GHz [34][35][36]. Initially, families of intrastate (v t = 0, 1) µ c -R-type transitions such as J 10,x ← (J−1) 9,x and J 9,y ← (J−1) 8,y (with J up to 18) were measured and fitted, followed by many intrastate µ a -R-type transitions and several (unexpected, because only µ b is inverting) interstate µ a -R-type transitions (see the explanation in a following section) [37]. The estimated uncertainty for the measurements is about 50 kHz. The sample was heated to 65-70 • C while a stream of argon with a backing pressure of P a = 18 kPa flowed over it. The mixture was then expanded to about P b = 0.5 Pa through a 0.3 mm diameter pinhole nozzle. The estimated rotational temperature of the molecules in the jet is 5−10 K. Later on, extensive measurements were made with two molecular jet Fourier transform microwave (MJ-FTMW) spectrometers, one in Bologna [38] (the assignment of the E0 state was made in a way similar to that described in the next section for Aachen) and one in Aachen [39]. Using the scan mode (50 co-added free induction decays per each step in a 250 kHz step width), a broadband scan was recorded from 8 to 14 GHz with the Aachen MJ-FTMW spectrometer. The substance, with a stated purity of 99%, was purchased from Sigma Aldrich, Taufkirchen, Germany, and was used without further purification. Under helium-stream, the phenyl acetate-He mixture entered the vacuum chamber under a backing pressure of 200 Pa. All signals appeared as doublets due to the Doppler effect and were re-measured with higher resolution and under an increased number of free induction decays. The measurement accuracy of both MJ-FTMW spectrometers is about 2 kHz [40]. Eventually, line broadening occurred as a result of unresolved splittings arising from proton hyperfine structures or spin-couplings.

Assignment of the v t = 0, A Species (A0)
Using a rigid rotor model with the values for the rotational constants calculated at the MP2/6-311++G(d,p) level, the R-branch J+1 1J ← J 0J and the Q-branch J 2,J−2 ← J 1J with J ≤ 5 were assigned from the survey scan recorded with the Aachen MJ-FTMW spectrometer, independently from the assignment in the millimeter-wave range. The first fit attempts using only three rotational constants yielded a root-mean-squares (rms) deviation of 1.7 MHz. After more lines with higher J and K a from the scan were included in a fit containing 20 lines, five centrifugal distortion constants could be floated, and the rms deviation decreased to 1.24 MHz, which is still extremely large compared to the experimental accuracy of 2 kHz of the MJ-FTMW spectrometer. However, when simulating the spectrum with the fitted rotational and centrifugal distortion constants and comparing it to the measured spectrum, as shown in Figure 7, we were confident that the assignment was correct. Using predictions with the fitted parameters, we found some more lines outside the 8-14 GHz scan area and achieved a fit with 36 lines and an rms deviation of 1.3 MHz. under an increased number of free induction decays. The measurement accuracy of both MJ-FTMW spectrometers is about 2 kHz [40]. Eventually, line broadening occurred as a result of unresolved splittings arising from proton hyperfine structures or spin-couplings.

Assignment of the vt = 0, A Species (A0)
Using a rigid rotor model with the values for the rotational constants calculated at the MP2/6-311++G(d,p) level, the R-branch J+11J ← J0J and the Q-branch J2,J−2 ← J1J with J ≤ 5 were assigned from the survey scan recorded with the Aachen MJ-FTMW spectrometer, independently from the assignment in the millimeter-wave range. The first fit attempts using only three rotational constants yielded a root-mean-squares (rms) deviation of 1.7 MHz. After more lines with higher J and Ka from the scan were included in a fit containing 20 lines, five centrifugal distortion constants could be floated, and the rms deviation decreased to 1.24 MHz, which is still extremely large compared to the experimental accuracy of 2 kHz of the MJ-FTMW spectrometer. However, when simulating the spectrum with the fitted rotational and centrifugal distortion constants and comparing it to the measured spectrum, as shown in Figure 7, we were confident that the assignment was correct. Using predictions with the fitted parameters, we found some more lines outside the 8-14 GHz scan area and achieved a fit with 36 lines and an rms deviation of 1.3 MHz. This struggle in fitting a species using a rigid rotor model supplemented by centrifugal distortion corrections is extremely similar to the fitting procedure in pinacolone [41] and phenyl formate [25]. As the double minimum potential suggests a strong Coriolis coupling in phenyl acetate, a new fit was carried out using the spfit program, which can implement Coriolis cross terms. Adding E (corresponding to ΔE01), EJ (multiplying J 2 ), Fbc (multiplying JbJc + JcJb), and Fab (multiplying JaJb + JbJa) in the fit, the rms decreased to 1.7 kHz, and a total of 80 vt = 0, A species (A0) lines could be assigned in the microwave range. In addition, the assignments of the microwave lines were secured with extensive interlocking combination difference loops, as illustrated in Figure 8. The fit is shown as Fit A0 in Table S4 of   This struggle in fitting a species using a rigid rotor model supplemented by centrifugal distortion corrections is extremely similar to the fitting procedure in pinacolone [41] and phenyl formate [25]. As the double minimum potential suggests a strong Coriolis coupling in phenyl acetate, a new fit was carried out using the spfit program, which can implement Coriolis cross terms. Adding E (corresponding to ∆E 01 ), E J (multiplying J 2 ), F bc (multiplying J b J c + J c J b ), and F ab (multiplying J a J b + J b J a ) in the fit, the rms decreased to 1.7 kHz, and a total of 80 v t = 0, A species (A0) lines could be assigned in the microwave range. In addition, the assignments of the microwave lines were secured with extensive interlocking combination difference loops, as illustrated in Figure 8. The fit is shown as Fit A0 in Table S4 of the Supplementary Materials with a list of all fitted frequencies along with their residuals. (open circles) of the A0 and E0 species of phenyl acetate measured with the Aachen FTMW spectrometer. Solid lines connecting the circles indicate transitions that are checked by combination difference loops, which sum to values in good agreement with the expected measurement uncertainty of 2 kHz. These transitions were used for an initial spfit fit whose spcat predictions have guided the assignments of further lines. Note that for the E0 states, some a-type transitions are weak or fall in resonance ranges of the spectrometer and could not be measured. Therefore, some loops are not closed.

Assignment of the vt = 1, A Species (A1)
After the A0 species was assigned, we found a series of satellite lines close to the strongest A0 species lines. From our experiences with the microwave spectrum of phenyl formate [25], we suspected these transitions to be those of the vt = 1, A species (A1). In a second step, 20 A1 lines could be identified and included in a fit where the rotational and centrifugal distortion constants were fitted separately for the vt = 0 and 1 states (Fit A0/A1 MW in Table S4). The rms deviation only increased slightly to 3.6 kHz. The effective Hamiltonian: was applied where 0⟩ and 1⟩ represent the symmetric and the anti-symmetric torsional states, respectively, unifying the following operators that describe: i) the overall rotation including quartic centrifugal distortion constants: ii) the torsional splitting between the 0⟩ and the 1⟩ energy levels: iii) and the Coriolis interaction: The A0/A1 measurements in the microwave range with significantly fewer vt = 1 lines compared to the number of vt = 0 lines were then combined with those in the millimeterwave range from 60 to 78 GHz, where 40 lines were observed, almost equally distributed between the vt = 0 and 1 states. We could also assign 26 a-type interstate transitions vt = 0 ← 1 or 1 ← 0 (note that some of them are blended). The fit is shown as Fit A0/A1 MW+mmw in Table S4 of the Supplementary Materials.

Assignment of the vt = 0, E Species (E0)
Assigning the E0 state was a tedious task because the spfit/spcat program suite was not useful at the initial stage for assigning the internal rotation E species lines and the XIAM program could not deal with the tunneling motion of the phenyl ring. We first used XIAM to predict a theoretical microwave spectrum for the scan region (8-14 GHz) with that are checked by combination difference loops, which sum to values in good agreement with the expected measurement uncertainty of 2 kHz. These transitions were used for an initial spfit fit whose spcat predictions have guided the assignments of further lines. Note that for the E0 states, some a-type transitions are weak or fall in resonance ranges of the spectrometer and could not be measured. Therefore, some loops are not closed.

Assignment of the v t = 1, A Species (A1)
After the A0 species was assigned, we found a series of satellite lines close to the strongest A0 species lines. From our experiences with the microwave spectrum of phenyl formate [25], we suspected these transitions to be those of the v t = 1, A species (A1). In a second step, 20 A1 lines could be identified and included in a fit where the rotational and centrifugal distortion constants were fitted separately for the v t = 0 and 1 states (Fit A0/A1 MW in Table S4). The rms deviation only increased slightly to 3.6 kHz. The effective Hamiltonian: was applied where 0 and 1 represent the symmetric and the anti-symmetric torsional states, respectively, unifying the following operators that describe: (i) the overall rotation including quartic centrifugal distortion constants: (ii) the torsional splitting between the 0 and the 1 energy levels: (iii) and the Coriolis interaction: The A0/A1 measurements in the microwave range with significantly fewer v t = 1 lines compared to the number of v t = 0 lines were then combined with those in the millimeterwave range from 60 to 78 GHz, where 40 lines were observed, almost equally distributed between the v t = 0 and 1 states. We could also assign 26 a-type interstate transitions v t = 0 ← 1 or 1 ← 0 (note that some of them are blended). The fit is shown as Fit A0/A1 MW+mmw in Table S4 of the Supplementary Materials.

Assignment of the v t = 0, E Species (E0)
Assigning the E0 state was a tedious task because the spfit/spcat program suite was not useful at the initial stage for assigning the internal rotation E species lines and the XIAM program could not deal with the tunneling motion of the phenyl ring. We first used XIAM to predict a theoretical microwave spectrum for the scan region (8-14 GHz) with the rotational constants of Fit A0 given in Table S4 of the Supplementary Materials, the V 3 potential term, and the angles between the internal rotor axis and the principal axes of inertia obtained from calculations at the MP2/6-311++G(d,p) level. The most intense band c-type J K a K c ← J K a K c = 4 13 ← 3 03 , 5 14 ← 4 04 , 6 16 ← 5 05 , 2 20 ← 1 10 , 3 21 ← 2 11 , 2 21 ← 1 11 , 3 22 ← 2 12 lines could be identified, being of similar intensity as the corresponding A0 species. Starting from these assignments, we explicitly searched for the same combination difference loops as those found for the A0 species (see Figure 8). After a sufficient number of lines had been assigned, a separate fit of the E0 species was performed with spfit using the same parameters accounting for the ring tunneling motion (E, E J , F ab , and F bc ). In addition, two odd power order parameters, D a and D c , as well as their higher-order parameters, D aK , D cK , D cJK , and D cKK [42], are required to fit the E0 state separately, as implemented in the H op operator, as follows: We achieved an rms deviation of 10.6 kHz for 69 E0 lines. We were confident about the assignments, but due to the low number of lines, no attempts were made to further reduce this deviation by adding more parameters. The fit is shown as Fit E0 in Table S5 of the Supplementary Materials with a list of all fitted frequencies along with their residuals. To access the methyl torsional barrier, the A0 and E0 species lines were input into XIAM. Similar to the case of the A0 species, we obtained an extremely high rms deviation of 6.9 MHz. The V 3 potential was determined to be 136 cm −1 . The XIAM fit is given in Table S6 along with the observed-minus-calculated values.

Assignment of the v t = 1, E Species (E1)
The predictive power of Fit A0/A1 and Fit E0 enabled us to exclude all lines belonging to the A0, A1, and E0 species from the recorded scan. It was then possible to identify some E1 species lines corresponding to the most intense E0 lines, which are of similar intensity as those of the A1 species, as shown exemplarily for the 2 21 ← 1 11 transitions in Figure 9. Eventually, we found 17 E1 lines and fitted them together with the E0 lines using spfit. This fit is given as Fit E0/E1 in Table S5 of the Supplementary Materials. Similar to Fit A0/A1, the rotational and centrifugal distortion constants were fitted separately for the E0 and E1 species. Odd power order parameters are required, as in the case of Fit E0. They are fitted separately for the E0 and E1 species. Due to the low intensity of the E1 lines, we could not close any combination difference loops. Consequently, though we are confident about most of the E1 assignments, conclusive proof is lacking. The fit chosen for Table S5 has an rms deviation of 358.8 kHz, but we had to take out 9 E0 lines with K a = 3. Therefore, this high rms deviation may arise from the limited number of fit parameters. The frequency list is also given in Table S5. 2.2.6. Global Fitting of the A0, E0, A1, E1 Sub-States We then fit the rotational transition frequencies of all four states, A0, A1, E0, and E1, simultaneously, considering, in principle, all the interactions between the four sublevels of the vibrational ground state. A0 will interact with A1 and E0 via Coriolis and V 3 internal rotation couplings, respectively. Similarly, the other three states interact with two other states. The results of this global fit are reported in Table 1. In the fit, the value of ∆E A0/E0 was fixed at 16 GHz, corresponding to the V 3 barrier determined by XIAM. We then fit the rotational transition frequencies of all four states, A0, A1, E0, and E1, simultaneously, considering, in principle, all the interactions between the four sublevels of the vibrational ground state. A0 will interact with A1 and E0 via Coriolis and V3 internal rotation couplings, respectively. Similarly, the other three states interact with two other states. The results of this global fit are reported in Table 1. In the fit, the value of ΔEA0/E0 was fixed at 16 GHz, corresponding to the V3 barrier determined by XIAM.   a All parameters refer to the principal axis system. Watson's S reduction in I r representation was used. b Numbers of lines included in the fit. c Reduced deviation of the fit when setting the millimeter-wave and microwave measurement errors to 80 kHz and 3 kHz, respectively.
Except for D K , the values of the centrifugal distortion constants are not so homogeneous among the four states. This can be due to the fact that not all the interactions are taken into account. For example, some "local" interactions of rotational states belong to different vibrational sublevels, and their effects can be "included" in the so-called "pseudocentrifugal distortion constants", allowing accurate fittings to the experimental uncertainty of the measured frequencies. The output of the spfit program leading to the parameters of Table 1 is reported in Table S7 of the Supplementary Materials. Similar to the case of Fit E0/E1 in Table S5 of the Supplementary Materials, we could not include the 9 E0 lines with K a = 3 and also had to take out 5 E1 lines. We note that a number of v t = 0 ←→ 1 interstate µ a -R-type transitions have been used in the fit, while only µ b -type interstate transitions should be observed. These "forbidden" transitions originate from an accidental but systematic µ b -mixing of the v t = 0, J 7,x and the v t = 1, J 6,y , which have very similar energies for each J. As an example, for J = 14, the energies of the degenerate levels v t = 0, 14 7,7 , and 14 7,6 are about 303 GHz, being the same as the value of the v t = 1, 14 6,8 , and 14 6,7 degenerate levels. Their mixing will give a partial µ b -character to the reported, apparently pure µ a -type transitions. Moreover, for the E0 and E1 species, about half of the lines are intrastate µ b -transitions, while due to the kind of motion, only interstate µ b -transitions are allowed. For the E species, the K a and K c quantum numbers have no meaning for those perturbation allowed transitions, because they only indicate the order of energy.
From the value of the parameters D a or D c of the E0 state, it is possible to estimate the V 3 barrier to internal rotation of the methyl group when ac represents a molecular plane of symmetry [43][44][45]. This is not the case of phenyl acetate, but due to the low barrier to inversion at α = 90 • , the vibrational wavefunctions of v t = 0 are exactly symmetric with respect to the ac plane. In addition, when trying to fit the D b parameter, it was immediately set to zero. For this reason, we believe that the method to estimate V 3 from D a (or D c ) is applicable for phenyl acetate [29,43].
We first set a system of the following two equations: νσ λ a and (7) to calculate the first-order perturbation coefficients W (1) 0,E [46] through the relations, as follows: λ g = cos( i, g), assuming the top inertial defect I α = 3.2 uÅ 2 . Since the ac plane can be considered as a symmetry plane ( i, a) + ( i, c) = 90 • ; therefore, λ a and λ c are dependent, leading to only two unknowns. The solution of the two-equation system provides ( i, a) = 22.1 • , ( i, c) = 67.9 • , and W (1) 0,E = 0.1133. From the coefficient W (1) 0,E , the reduced barrier s estimated through Herschbach's tables is 11.5 [46]. The reduced barrier is related both to the methyl internal rotation barrier and the splitting between the A and E levels, as follows: with w v=0 1 = −0.0275 from Herschbach's tables. The achieved values are ∆ 0 = 0.5 cm −1 and thus V 3 = 143 cm −1 .

Flexible Model Calculations
The experimental value of the ∆E A0/A1 vibrational splitting can be used to determine the small barrier at α = 90 • . In order to handle a simple expression, we first defined τ = α − 90 and used the following double minimum potential: (12) where B 2 is the barrier at τ = 0 • and τ 0 is the equilibrium value of the inversion angle. Such an equation is suitable to describe our potential energy function in a small interval that includes the two minima and the "transition state". The energies and wavefunctions of vibrational states related to the skeletal torsion were calculated numerically with a suitable model [47]. We fixed the parameter τ 0 to the ab initio value, calculated B 2 , and applied Meyer's one-dimensional numerical flexible model [47]. The value ∆E A0/A1 = 36.4 GHz was reproduced when B 2 was set to 68 cm −1 . In the flexible model calculation, the τ coordinate has been considered in the ± 50 • range and solved into 41 mesh points [47]. This range is expected to reproduce the low energy part of the potential energy curve of this motion, describing the minima and the barrier at the perpendicular configuration. The value of B 2 = 68 cm −1 is considerably higher than the value obtained by the Hamiltonian in Equation (1). This discrepancy can be due to the fact that, with the numerical method, the value of the reduced constant of the motion F is calculated for every point, and it undergoes significant variations in the considered range.

Discussion
The A and E species lines belonging to the v t = 0 tunneling species of phenyl acetate were assigned. The initial assignments of both species are secured by an extensive interlocking network of combination difference loops. These A0 and E0 states could be fitted separately using the spfit program to rms deviations close to the measurement accuracy if parameters accounting for the ring tunneling motion are considered (see Tables S4 and S5 of the Supplementary Materials). This "local" approach has been proven to be very powerful as assignment guidance in many previous studies where methyl internal rotations with low torsional barriers are present [42,[48][49][50][51].
Though it is possible to derive the value of the barrier to internal rotation from the fitted odd power parameters D a and D c , the obtained barrier (V 3 = 143 cm −1 ) might not be so accurate due to several approximations [29]. The V 3 barrier was also deduced by a XIAM fit, including the A0/E0 species. Since XIAM does not take into account the interactions of A0 and E0 with A1 and E1, the standard deviation of the fit is terribly high. However, the value of V 3 = 136 cm −1 obtained from this fit is extremely close to the value of 135.3498(38) cm −1 found for isopropenyl acetate, where the isopropenyl group is also titled out of the O-(C=O)-C plane by an angle of about 70 • [14]. As summarized in a review by Nguyen and Kleiner [52], a torsional barrier of the acetyl methyl group CH 3 -COO is almost invariant at 100 cm −1 for α,β-saturated acetates [5][6][7][8][9][10]. In α,β-unsaturated acetates with the two current representatives vinyl acetate [11,12] and two isomers of butadienyl acetate [13], the conjugation over the double bond(s) increases the barrier height to 150 cm −1 . In both isopropenyl acetate and phenyl acetate, the double bond is not located in the O-(C=O)-C plane, leading to a less effective conjugation. The torsional barrier of about 136 cm −1 , between 100 cm −1 and 150 cm −1 , reflects well this observation.
The v t = 1 spectrum is significantly less intense than the v t = 0 one (see Figure 9, for example, note that the intensity is on a logarithmic scale to better visualize the weaker A1 and E1 species lines). The number of line ratio is N 1 /N 0 ≈ 1/4. The tunneling ∆E parameter could be determined and has a value of 36.4 GHz for the A0/A1 and 33.5 for the E0/E1 fit (see Table 1). Both values are close to the value of 30.6 GHz obtained from calculations using a simple two-top torsional Hamiltonian of the CH 3 -COO group and the phenyl ring. Compared to phenyl formate [25], where the ring tunneling is respected to an H-COO frame instead of the heavier CH 3 -COO frame as in the case of phenyl acetate, the Coriolis splitting ∆E 01 of 46.2231 (25) GHz of the former is significantly larger. On the other hand, the ∆E A0/A1 value of phenyl acetate is an order of magnitude larger than the value of 3.732027 (43) GHz in E-phenylformamide [24]. The smaller tilt angle of 40 • of the phenyl ring out of the (NH)(CO) plane in E-phenylformamide is mainly responsible for this Coriolis splitting being drastically lower than in phenyl formate or phenyl acetate.
The coupled LAMs observed in the microwave spectrum of phenyl acetate are very similar to those found in pinacolone [41], where in the latter molecule, a methyl internal rotation is coupled with the tunneling motion of the tert-butyl group. We show here that a model, often successful, as implemented in XIAM, fails in some specific cases. Even a very powerful and flexible program such as spfit/spcat encounters problems when tunneling motions between a double minimum potential come into play with a low-barrier methyl torsion. Kleiner's BELGI-hybrid program may be suitable to treat this problem in another global approach, taking into account the interactions of all four sub-states [53]. After the A0, A1, E0, and E1 species were assigned, all lines remaining in the broadband scan are of weak intensity and are neglected.
Supplementary Materials: The following are available online at https://www.mdpi.com/article/ 10.3390/molecules27092730/s1, Table S1: Fourier expansion of the potential energy curve of phenyl acetate given in Figures 3 and 4, Table S2: Nuclear coordinates in the principal inertial axes of conformer I b shown in Figure 2, Table S3: Fourier coefficients of the 2D-PES shown in Figure 5, Table  S4: Fitted molecular parameters of the A1 and A2 state lines using spfit and the frequency list, Table  S5: Fitted molecular parameters of the E1 and E2 state lines using spfit and the frequency list, Table  S6: Fitted molecular parameters of the A0 and E0 state lines using XIAM and the frequency list, Table  S7: Frequency list of the fit given in Table 1. Funding: This research was funded by the Agence Nationale de la Recherche ANR, grant number ANR-18-CE29-0011, by the Italian MIUR PRIN, project number 2010ERFKXL_001, and by the University of Bologna (RFO). Simulations were performed with computing resources granted by the RWTH Aachen University under the project rwth0506.