Coupled Large Amplitude Motions: A Case Study of the

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Coupled Large Amplitude Motions: A Case Study of the Dimethylbenzaldehyde Isomers M. Tudorie,†,⊗ I. Kleiner,*,† M. Jahn,‡ J.-U. Grabow,‡ M. Goubet,§ and O. Pirali∥,⊥ †

Laboratoire Interuniversitaire des Systèmes Atmosphériques (LISA), UMR 7583 CNRS/IPSL, Universités Paris-Est et Paris Diderot, 61 avenue du Général de Gaulle, 94010 Créteil Cedex, France ‡ Institut für Physikalische Chemie und Elektrochemie, Lehrgebiet A, Gottfried-Wilhelm-Leibniz-Universität, Callinstraße 3A, D-30167 Hannover, Germany § Laboratoire de Physique des Lasers, Atomes et Molécules, UMR 8523 CNRS, Université Lille 1, Bâtiment P5, F-59655 Villeneuve d’Ascq Cedex, France ∥ AILES beamline, Synchrotron SOLEIL, L’Orme des Merisiers Saint-Aubin, 91192 Gif-sur-Yvette, France ⊥ Institut des Sciences Moléculaires d’Orsay, UMR 8214 CNRS, Université Paris-Sud, Bât. 210, 91405 Orsay cedex, France S Supporting Information *

ABSTRACT: The microwave spectra of the 3,4- (syn and anti), 2,5- (syn), and 3,5-dimethylbenzaldehyde (DMBA) molecules have been recorded for the first time in the 2− 26.5 GHz frequency range, using the high resolution COBRAFTMW spectrometer in Hannover. The experimental assignments and fits are supplemented by ab initio quantum chemical calculations of the conformational energy landscape and dipole moment components. The analysis of the spectra of the four observed isomers, including spectroscopic constants and large amplitude motion parameters, are presented in this paper. The DMBA isomers belong to a series of similar molecules obtained formally by adding one or more methyl group(s) at the aromatic ring. These molecules serve as prototype systems for the development of the theoretical model of asymmetric top molecules having Cs symmetry while containing in addition two nonequivalent methyl tops (C3v), exhibiting different barrier heights and coupling terms. Thus, the DMBA isomers represent good species for testing the recently written two-top internal rotors BELGI program.

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INTRODUCTION Internal rotors play an important role in astrophysical and biomimetic studies, but their physical and chemical fundamental understanding is still a challenge despite several decades of work. As part of a planned series of studies on internal rotation of one or more methyl internal rotors, so far a number of papers have been published on molecules containing one methyl internal rotor using our BELGI-Cs or BELGI-C1 code,1,2 available at the PROSPE Web site.3 Concerning molecules containing two nonequivalent internal methyl tops, our method was only applied to two systems: the methyl acetate molecule (CH3COOCH3)4 and the methyl propionate molecule (CH3CH2COOCH3).5 In addition, the paper on which we based our theoretical framework dealt with another two-top molecule, the N-methylacetamide molecule (CH3NHCOCH3) studied by Ohashi et al.6 The BELGI-Cs2Tops program has been designed for fitting high resolution torsion-rotation spectra of molecules with (i) two nonequivalent methyl rotors, (ii) two different 3-fold barriers, and (iii) a plane of symmetry at equilibrium. In methyl acetate a rather low barrier to internal rotation of 101.740(30) cm−1 was found for the acetate methyl group © XXXX American Chemical Society

whereas the methoxy methyl group showed a considerably higher barrier of 422.148(55) cm−1.4 In methyl propionate, there are also two methyl groups which give rise to spectral splittings due to internal rotation, but the barriers hindering the two internal rotations are very different: for the CH3CH2COmethyl group the barrier is very high (820.46(99) cm−1) whereas for the -OCH3 methyl group, an intermediate barrier of the order of 429.324(23) cm−1 was determined.5 Some years ago, we were interested in the rotational spectrum of paratolualdehyde (CH3−C6H4−CHO)7 (pTA) which was measured using the pulsed-molecular-beam Fouriertransform microwave spectrometer in Hannover and the millimeter wave free-jet absorption spectrometer in Bologna. For pTA the barrier height is very low (about 28 cm−1), and the nearly free internal rotation of the methyl group splits each of the rotational transitions into two components of A and E symmetry. For this molecule, we were able to fit within Special Issue: Terry A. Miller Festschrift Received: July 31, 2013 Revised: September 13, 2013

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dx.doi.org/10.1021/jp407603y | J. Phys. Chem. A XXXX, XXX, XXX−XXX

The Journal of Physical Chemistry A

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anti-3,4-DMBA, show different sets of barrier heights. By changing the position of the two methyl groups on the benzene ring, and numbering the atoms with the aldehyde group on carbon atom “1”, we obtain the following cases: (i) when the two methyl groups are on the carbon atoms in positions 3 and 5, the two internal rotors are almost freely rotating and their respective torsional barriers are both rather low [53.0068 (62) cm−1 and 25.2504 (41) cm−1]; (ii) when one methyl group is attached to the carbon atom in position 2, adjacent to the aldehyde group and the other methyl group is in position 5, one internal rotor’s barrier is low [5.35 (30) cm−1 ] whereas the other barrier is rather high [565.66 (15.97) cm−1]; and (iii) when the two methyl groups are located on neighboring carbon atoms in positions 3 and 4, both barriers are high [508.12 (1.08) and 550.74 (8.82) cm−1 for the syn- and 454.06 (1.34) and 480.64 (4.35) cm−1 for the anti- isomer]. These isomers of DMBA, which have not previously been investigated at high resolution in the microwave (MW) or in the far-infared (far-IR) spectral ranges, represent an excellent test of our program. In the present study, we combined different experimental high resolution techniques (MW and farIR spectroscopy) with theoretical methods (effective Hamiltonians and quantum chemical calculations). Supersonic Jet Fourier Transform Microwave (COBRA-FTMW) spectroscopy9−11 used in our work is an excellent technique to study precisely the structure and dynamics of molecules in the gas phase. This technique allows us to observe the low-lying energy conformers and rotational levels by cooling down the molecular species, as well as to measure the line frequencies with an extremely high accuracy of a few (or less) kHz. The combination of FTMW spectroscopy with ab initio calculations has now become almost a standard in studying large organic or sizable molecules.12−14 Ab initio methods provide first estimations of the rotational constants, the molecular structure, and the barrier heights, which are prerequisites for a preliminary analysis of the spectrum for such large molecules. The use of an accurate Hamiltonian model to fit within experimental accuracy the high resolution experimental data usually permits in turn to calibrate and validate the ab initio calculations. We also recorded here far-IR measurements (50−650 cm−1) at the synchrotron radiation facility SOLEIL. It permitted us to measure the spectrum of the low-frequency vibrational modes of the DMBA isomers. The bright far-IR synchrotron continuum combined with a high resolution Fourier transform infrared (FTIR) spectrometer and a long absorption path cell permitted the observation of very weak bands whose positions represented an additional validation of our ab initio calculations. The rest of the paper is organized as follows. In the Experimental Section, we present the experimental details for the COBRA-FTMW and far-IR setups. In the Theoretical Method Section, we first discuss our quantum chemistry methods and then we briefly present our spectroscopic model and the BELGI-Cs-2Tops code. The results section is divided into (i) the ab initio results for the four DMBA isomers, (ii) the far-IR and the COBRA-FTMW spectroscopic analysis, and finally (iii) the results of the global fits of each of the four observed isomers using the BELGI-Cs-2Tops code. In the Discussion section, we compare the results of high resolution spectroscopy and ab initio calculations. The last section is devoted to our conclusions.

experimental accuracy the rotational transitions in the ground state (m = 0) and first excited state (m = 1). The principal motivation for the work on pTA is related to the question of how, and how much, information can be transmitted across single bond systems versus how and how much it can be transmitted across conjugated bond systems. pTA, as drawn in Figure 1, consists of a benzene ring with a methyl group

Figure 1. Configuration of paratolualdehyde; the a-b plane is the plane of the page.

attached at one side and an aldehyde group attached at the opposite side. If the aldehyde group is ignored, the local environment of the methyl group would lead to a 6-fold barrier to internal rotation, similar to that found in toluene.8 But, if as expected from the delocalization of the oxygen π-electron into the aromatic ring, the aldehyde group can transmit information concerning its syn-anti orientation through the benzene ring, then the methyl group would have a 3-fold barrier to internal rotation. The study performed on pTA shows it is indeed the case. Compared to pTA, the dimethylbenzaldehyde molecules presently studied contain an extra methyl group attached to the ring (see Figure 2). The main questions addressed here are how

Figure 2. Calculated structures of the Dimethylbenzaldehyde isomers: (a) syn-3,4-DMBA, (b) anti-3,4-DMBA, (c) syn-2,5-DMBA, and (d) 3,5-DMBA.

large are the internal rotation barriers, and how strongly the two methyl groups are coupled to each other. High resolution spectroscopy can contribute to the answer of these questions, as the Hamiltonian interaction terms (kinetic and potential terms) can be directly fitted. The four isomers presently studied, referred to in the following as the 3,5-DMBA, the syn-2,5-DMBA and the syn- and B



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EXPERIMENTAL SECTION COBRA-FTMW Experiment. All the measurements in the microwave range were done using the Hannover FTMW spectrometer. This supersonic jet Balle-Flygare-type Fourier transform microwave spectrometer (2−26 GHz) achieves an exceptionally narrow line width due to its COBRA implementation. The setup has been described extensively,9−11 therefore, only brief experimental details are given here. The liquid sample was inserted inside of a reservoir located at the nozzle exit and mildly heated below 323 K. Neon or Argon at 150−200 kPa was used as the carrier gas, creating a supersonic jet that was expanded along the axis of the evacuated Fabry− Perot resonators. Typically, microwave pulses of 1 mW at 1 μs length were used to polarize the sample for optimized emission signals. The resulting free-induction decay (FID) was digitized in the time domain and Fourier transformed to the frequency domain, where the molecular signal is represented as its amplitude spectrum. The resonance frequencies appear as a Doppler doublet because of the coaxial orientation of the jet and the propagation direction of the electromagnetic field. An example of different transitions for each isomer, 3,5-, syn-2,5-, syn-3,4, and anti-3,4-DMBA, are given in Figure 3. Far-Infrared Setup at the Synchrotron SOLEIL. The FTIR spectra of the DMBA isomers were recorded using a Bruker IFS-125 HR Fourier transform spectrometer. The far-IR

continuum source was the synchrotron radiation extracted by the AILES beamline at the synchrotron SOLEIL.15 The experimental setup was already described elsewhere.16 Briefly, samples at their equilibrium vapor pressures at room temperature (7, 10, and 9 Pa of 3,4-DMBA, 2,5-DMBA, and 3,5-DMBA, respectively) were injected in the White-type absorption cell equipped with 50 μm thick polypropylene windows. The absorption path length was set to 150 m. A 4.2 K liquid helium cooled Si-bolometer detector and a 6 μm thick mylar beamsplitter were used. The spectra are Fourier transforms of 220 and 350 coadded interferograms recorded at a resolution of 0.5 and 0.001 cm−1, respectively.

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THEORETICAL METHODS Computational Details. All the calculations were carried out using the Gaussian 09 software package.17 Combinations of the MP2 method or the B3LYP functional with the correlationconsistent polarized triple valence basis set (cc-pVTZ) of Dunning18 or the 6-311++G(3df,2p) Gaussian basis set19 were chosen because they are well-known to offer the best compromise between calculation time and accuracy of the results for medium size molecules. The geometries were fully optimized using the tight convergence option. The QST2 procedure as implemented in Gaussian 09 was used to find the transition states (TS) between the stable conformers. The zero point energy (ZPE) corrections were estimated from harmonic frequency calculations at the same level of theory as for the geometry optimization. The negative frequency occurring when calculating the harmonic frequencies on a TS geometry (which actually confirms a correct optimization at a saddle point of the potential energy surface) has been taken into account in the ZPE corrections. Finally, the 1D cuts of the potential energy surface (PES) were made by rotating the dihedral angle corresponding to the internal rotations of the methyl groups, by steps of 3°, all other internal coordinates being reoptimized at each step.

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SPECTROSCOPIC MODEL AND CODE The theoretical model and the BELGI-Cs-2Tops program have been described in great detail elsewhere.4,6 We emphasize here only a few points. Just as the one-top program (BELGI-Cs), we take advantage of solving the torsion-rotation Hamiltonian in two steps,20 where the first step deals with the torsion-Krotation part of the problem and the second step deals with all the rest. However, in the BELGI-Cs-2Tops program, we differ from ref 20 by using a modified or “quasi” principal-axismethod (PAM)6 instead of the rho-axis-method (RAM),1,21 which requires placing the z axis nearly parallel to the top axis. The PAM (or “quasi-PAM”) was chosen under the two assumptions that (i) most molecules with two nonequivalent CH3 tops will be large enough so that the ρ value for each top will be relatively small and (ii) the PAM is well suited to problems with small ρ values.21 Those two assumptions turned out to be true here, since for all the isomers of the DMBA molecules, the ρ values are around 0.01. The BELGI-Cs-2Tops program used here is most closely related to that applied only once in the literature for a treatment of the microwave spectrum of N-methylacetamide,6 but to speed up the process, we have implemented in our code a two step diagonalization procedure22 similar to that used in the BELGI-one top code.1,2 We also use a truncation in the

Figure 3. Examples of the COBRA-FTMW spectra obtained at Hannover for three isomers of DMBA. The molecular transitions are labeled using the usual J′Ka′Kc′ ← J″Ka″Kc″ of the asymmetric rotor notation and also using the symmetry species for two-top internal rotors, in the G18 permutation-inversion group.6 Because of the large spectral range needed to visualize the internal rotation splittings, the 3,5-DMBA spectrum is shown as a low resolution high speed scan, whereas for syn-3,4-DMBA and syn-2,5-DMBA, the spectra are shown at high resolution. 3,5-DMBA (upper left box) illustrates the case with two low barriers and syn-3,4-DMBA (bottow left box) the case with two high barriers. For syn-2,5-DMBA we have two categories of splittings, one category belongs to the high barrier case (the A-E2 splitting in the right lower panel), whereas the other category belongs to the low barrier case (the E1 splitting in the right upper panel). Transitions belonging to the E3 and E4 symmetry species are also shown. C



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number of basis functions retained after the first diagonalization step. The basis set for our first (torsional) diagonalization step consists of products of exponentials of the form (2π)−1 exp[(3k1+σ1)iα1] exp[(3k2+σ2)iα2] where the integers |k1| and |k2| can be both chosen as long as they are less than a basis set cutoff parameter ktronc, which is set to 10 in our calculation. The Hamiltonian is diagonalized as separate blocks, each characterized by one of the five (σ1, σ2) pairs (0, 0), (1, 0), (0, 1), (+1, +1), and (+1, −1), corresponding to the symmetry species A, E1+, E2+, E4+, and E3+, respectively (see Table 3 of ref 4.). Only the lowest order pure torsional operators of the Hamiltonian are considered in this first step (see eq 3 of ref 4.): Htor =

∑

Table 1. Calculated Equilibrium Rotational Constants, Projections of the Permanent Electric Dipole Moment on the Principal Inertial Axes, and Relative Energies of the 3,4DMBA Conformers

2

[FP i αi + (1/2)V3i(1 − cos 3αi)] + F12pα1 pα 2

i = 1,2

+ V12s sin 3α1 sin 3α2 + V12c(1 − cos 3α1) (1 − cos 3α2)

(1)

Following this first diagonalization step, we normally keep the lowest 42 = (2 × ktronc + 1) × 2 torsional energy levels and wave functions for use in the second step. In this second step, we diagonalize the rest of the Hamiltonian: Hrot + Htor‐rot

(2)

method

B3LYP/gaussa

B3LYP/ccpVTZ

MP2/ccpVTZ

Ae/MHz Be/MHz Ce/MHz μa/D μb/D relative energiesb/ cm−1 TS C3c TS C4c TS syn-anti

2700.411 909.895 686.299 3.89 1.48

syn-3,4-DMBA 2698.809 910.695 686.651 3.65 1.62

2706.057 911.168 687.395 3.57 0.56

442 (490) 425 (455) 3225 (3134)

Ae/MHz Be/MHz Ce/MHz μa/D μb/D relative energiesb/cm−1 TS C3c TS C4c

2947.627 860.170 671.337 4.41 0.96 45 (68) 422 (435) 382 (392)

445 (491) 426 (453) 3352 (3243) anti-3,4-DMBA 2948.310 860.427 671.529 4.22 0.93 58 (68) 427 (438) 386 (394)

460 (519) 447 (506) 3031 (2940) 2963.258 860.137 672.132 4.51 1.81 52 (44) 440 (499) 413 (471)

a gauss = the 6-311++G(3df,2p) basis set. bValues in parentheses are ZPE corrected. cThe numbering of the atoms is consistent with Figures 2a and 2b.

We can construct these Hamiltonian terms by taking symmetry-allowed Hermitian products of a rotational factor (chosen from operators of the form JxpJyqJzr, where p, q, and r are integer exponents) and a torsional factor for each top i = 1,2 (constructed in turn from products of operators of the form Pαi p , cos 3qαi, sin 3rαi, where p, q, and r again represent integers).

the syn and anti orientations corresponds to the aldehyde group perpendicular to the ring plane; the two methyl groups are 3fold tops with a succession of equilibrium (EqS) and TS every 60°. Interestingly, the barrier heights are slightly different if the methyl group is attached to the third carbon atom (denoted C3) or to the fourth carbon atom (denoted C4) and both barriers are also different if the aldehyde group is in a syn or anti orientation. Although these differences are close to the expected calculation error, they might be explained in terms of information transfers across the aromatic ring, because of the π-electron delocalization, so that the asymmetry of the aldehyde group has an influence on the torsional motion of the methyl groups. In other words, there might be an indirect coupling between the functional groups that goes beyond a simple repulsive interaction. 2,5-DMBA Isomer. As for 3,4-DMBA, 2,5-DMBA (shown in Figure 2c) presents two stable conformations and a high TS with syn, anti and perpendicular orientations of the aldehyde group (the energy of the TS relative to the syn conformer is 2593 cm−1 without and 2435 cm−1 with the ZPE correction, respectively). However, because of repulsive forces with the hydrogen atoms of the adjacent methyl group (denoted C2, see Figure 2c), the anti conformer has a high relative energy with respect to the syn so that it is not expected to be observed in our spectra (the energy of the anti relative to the syn is 354 cm−1 without and 344 cm−1 with the ZPE correction, respectively). For the same reason, the C2 methyl group behaves as a standard 3-fold top with a relatively high barrier (599 cm−1 without and 594 cm−1 with the ZPE correction, respectively), a hydrogen atom in the ring plane and pointing away from the aldehyde group being the EqS. Concerning the methyl group attached to the fifth carbon atom (denoted C5), the absence of nearby functional groups makes it free from

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AB INITIO CALCULATIONS 3,4-DMBA Isomer. 3,4-DMBA (see Figure 2) was chosen as the first calculated isomer because some experimental data which may be used as reference is available in the literature.23 The expected steric repulsion between the two methyl groups, attached to two successive carbon atoms of the benzene ring, should result in a relatively rigid structure, easier to estimate from ab initio calculations. Indeed, all geometry optimizations led to two stable conformations with a Cs point group and a syn or anti orientation of the aldehyde group (see Figures 2a and 2b, respectively). For both conformations, the methyl groups have one hydrogen atom in the symmetry plane (i.e., the benzene ring) in opposite directions. Relative energies, equilibrium rotational constants, and projections of the permanent electric dipole moment on the principal inertial axes are gathered in Table 1. Very slight variations of these results with the methods are observed. However, accounting for large amplitude motions in semirigid molecules usually needs sophisticated post-treatments of high level calculations (see for example the studies of Senent et al.24). Therefore, the combination of the highest level MP2/cc-pVTZ, having a reasonable computational time in the present case, has been chosen for all subsequent calculations. In addition, the very good agreement between experimental and calculated frequencies in the far-IR range (see Supporting Information, Table S1) gives good confidence in the quality of the results. Explorations of the PES have shown that the three groups attached to the benzene ring behave as standard internal rotors with relatively high barriers: the TS (Transition State) between D



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the same side of the benzene ring) remains at exactly 120°. These nonequivalent barriers arise from the fact that the TS corresponding to a rotation of the C5 group passing by an eclipsed position with the C2 group (denoted C5e, see Figure 4b) is at a higher energy than the TS passing by a staggered position (denoted C5s). To estimate the couplings between the two internal rotors, a 2D PES along the C2 and C5 group torsions should ideally be computed, but the calculation time at this level would be unreasonably long (a 1D curve over 120° with 3° steps takes about 2 weeks). The B3LYP/cc-pVTZ combination, which gave results for the 3,4-DMBA isomer very similar to those from the MP2/cc-pVTZ, appeared at first as a good candidate to lower the computational cost. As can be seen in Figure 4a, however, this method completely failed in locating the EqS by calculating the energy minima at the C5s TS positions and producing the energy curve of a 3-fold top. This failure was further confirmed by the determination of a negative harmonic frequency at its calculated energy minimum, one characteristic of a TS structure. It emphasizes the fact that much care should be taken when using DFT methods to investigate floppy large amplitude motions. Thus, the MP2 level being mandatory, exploration of the coupled PES has been limited to a second scan along the C5 group torsional angle as before, but with the C2 group fixed at its TS position instead of its EqS position. Comparison of the two curves (see Figure 4b) shows a shift of the EqS positions, a higher C5e TS barrier, and a lower C5s TS barrier when the C2 group is at its TS position. Although the slight shift of a few degrees in torsional angle is most probably within the calculation error, the barrier height differences are large enough (about 50%) to be taken into account. Just as for 3,4-DMBA, these energy differences might be considered as evidence of an energy transfer between the two functional groups across the benzene ring. 3,5-DMBA Isomer. The more symmetrical arrangement of the functional groups around the benzene ring (see Figure 2d) leads to equivalent syn and anti conformers. Equally, rotations of the methyl groups of 60° result in mirror images of the original conformations. Thus, it may be considered that only one stable conformer exists for the 3,5-DMBA isomer (see Figure 2d). As for the C5 group of 2,5-DMBA, both methyl groups (denoted C3 and C5) are free from direct interactions because of the absence of adjacent functional groups, so that the four EqS correspond to a hydrogen atom of each methyl group perpendicular to the ring plane. The quasi-equivalence of the four EqS is confirmed by the calculations, giving energy differences of only about 1 cm−1 when the hydrogen atoms of the two methyl groups are on the same or opposite side of the ring plane, which is within the expected calculation error. Here again, the internal rotors are 6-fold tops with low asymmetric barrier heights, the EqS appearing roughly every 60° and the TS with eclipsed positions (denoted C3e and C5e) at higher energies than the TS with staggered positions (denoted C3s and C5s) (see Figure 5). Here again, the different behaviors of the two methyl torsions in addition to the different barrier heights in the staggered and eclipsed positions in both cases tend to indicate the possibility of energy transfer between the three functional groups attached to the ring.

direct interactions. As a consequence, the EqS corresponds to a hydrogen atom perpendicular to the ring plane (see Figure 2c), that is similar to toluene.8,25 Exploration of the PES along the C5 methyl group torsion has led to several interesting results. Calculations were performed from 0° to 120° and then translated by 120° and 240° to display a complete rotation (0° to 360°) in Figure 4.

Figure 4. Calculated potential energy curve (step of 3°) along the dihedral angle of the C5 methyl group (internal rotation coordinate) of the syn-2,5-DMBA isomer. (a) Comparison between calculations at the MP2 (white circles) and B3LYP (black triangles) levels. (b) Curves at the MP2/cc-pVTZ level along the same coordinate but with the other methyl group (C2) at equilibrium (white circles) and at the transition state (black squares) position. Pictures represent the positions of C5 (rotating anticlockwise) relative to C2 at its equilibrium position, the bold gray line corresponding to the benzene ring.

The C5 top actually rotates in a 6-fold well, with six equivalent minima and two sets of three equivalent barriers. The EqS appearing approximately every 60° (see white circles curves in Figure 4), with one hydrogen alternately above and below the benzene plane, are either the same or mirror images of each other. The presence of nonequivalent barriers causes the separation between adjacent minima to alternate between a larger separation of about 70° (through high barriers) and a smaller separation of about 50° (through low barriers), while the separation between next-nearest exact minima (hydrogen at

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SPECTRAL ANALYSIS Far-IR Spectra. Because of the congestion of the spectra of such medium size molecules, the rotational structure of the farIR bands could not be resolved even at the maximum resolution of the spectrometer. Only jet-cooled conditions,

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Figure 5. Calculated potential energy curve (step of 3°) for the 3,5DMBA isomer along the dihedral angle of (a) the C3 methyl group and (b) the C5 methyl group (internal rotation coordinates), the other methyl group being at its equilibrium position in both cases. Figure 6. Far-infrared spectra recorded at 0.5 cm−1 resolution of (a) 3,4-DMBA, (b) 2,5-DMBA, and (c) 3,5-DMBA samples at equilibrium vapor pressures at room temperature.

reducing the number of lines and their Doppler broadening, might make possible the observation of a rotationally resolved spectrum, but in the present case the band intensities are probably too weak for the present experimental capabilities (as an example, the detection limit of the Jet-AILES apparatus26 in the far-IR range is estimated to about 20 km·mol−1 for the integrated intensities as provided by Gaussian). Therefore, the spectra recorded at 0.5 cm−1 resolution offered the best compromise between accuracy of the band center position and the signal-to-noise ratio (S/N). The spectra, displayed in Figure 6, show several assignable bands which are clearly different from one isomer to the other. Indeed, the far-IR spectral range is a fingerprint region, since the low frequency vibrational modes are very particular to a given molecule. The observed and calculated band centers for the four isomers with their attributions based on the ab initio calculations are given in Supporting Information, Table S1. It is noteworthy that for such medium size molecules with very low frequency modes, the maximum of the observed broad bands might be shifted because of the presence of overlapping hot bands (see for example recent far-IR studies of naphthalene27,28), so the accuracy of the band centers is estimated to be several wavenumbers.

COBRA-FTMW Spectra. A total of 211, 410, 145, and 270 measurements for 3,5-, syn-2,5 and syn- and anti-3,4-DMBA respectively were included in separate fits, with an estimated measurement uncertainty of 0.5 kHz. The coverage of the data sets and the quality of the fits are presented in Table 2. Examples of different transitions for each isomer of DMBA measured at Hannover are given in Figure 3. It is quite clear from the figure how the different isomers show various internal rotation patterns: the 3,5-DMBA isomer (two low barriers) have 5 well resolved components for the 808-707 transitions belonging to the A transitions (A1 and A2 are overlapped at the scale of the figure), the E1 transition, which is split from the A transition by about 25 GHz because of the higher internal rotation barrier of about 53 cm−1 and the E2 transition, which is split from the A transition by about 50 GHz because of the lower barrier of about 25 cm−1. The E3 and E4 transitions, whose splittings occur because of the coupling between the two tops, are separated by almost 200 GHz. For other isomers the separation between the various components are totally different. The first isomers we analyzed were syn- and antiF



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prevents us from observing any transitions that are not within the ground torsional state of the molecule, that is, there is no torsional excitation in either of the two methyl tops. For this reason, as for our study of methyl acetate,4 a number of important rotation-torsion interaction terms could not be unambiguously determined (see below). The maximum J value treated for the 3,5-DMBA isomer was 11, which corresponds to an excitation energy in this degree of freedom of (1/2)(B + C) J(J + 1) ≈ 3.9 cm−1. The maximum value of Ka = 5 for the Atype lines and Ka = 3 for the E-type lines was determined by the available assigned transitions. The symmetry species coverage is rather evenly distributed for 3,5-DMBA. It can be seen from Table 2 that there are roughly equal numbers of MW transitions for each of the five symmetry species in the permutation-inversion group G18, that is, A (A1 and A2 species can be grouped together), E1, E2, E3, and E4. For fitting the data of the 3,5-DMBA conformer using our BELGI-Cs-2Tops we applied the same strategy as in the case of methyl acetate. As mentioned above, this isomer had the most difficult spectrum to assign because of the large splittings due to the two low internal rotation barriers. The usual molecular rotational constants A, B, C and the associated five centrifugal distortion constants were first fitted. The “internal rotation constants” F1 (for top 1) and F2 (for top 2) were fixed to values calculated from our ab initio equilibrium geometry, and the corresponding 3-fold potential constants V31 (for top 1) and V32 (for top 2) were fitted. The internal rotation constant F12, (multiplying the top-top kinetic energy interaction operator p1p2 ≡ pα1pα2 in eq 1)4,6 was fixed to its calculated value from the ab initio geometry while the two top-top interaction constants in the potential energy (V12s and V12c) were fitted. The Supporting Information, Table S2 gives the 21 molecular constants varied in our final fit of the 211 lines, as well as the three fixed parameters (F1, F2, F12). To compare the parameters fitted by the BELGi-Cs-2Tops code with the ab initio results, we convert the fitting constants of Supporting Information, Table S2 into constants in the principal axis system (PAM) as described in ref 6. They are presented and compared to the ab initio results in Table 3. It can be seen that the agreement is very satisfactory for this isomer. Global Fit of the syn-2,5-DMBA Isomer. The strategy for fitting the data set is the same as for 3,5-DMBA. The global standard deviation is again very close to the experimental accuracy for the 410 lines included in the fit (see Table 2) with the maximum J and Ka values treated equal to 14 and 5, respectively. In the Supporting Information, Table S3 we present the parameters used by BELGI-Cs-2Tops. We were able to determine 18 parameters, among which four are connected to the low barrier (V31 =5.35 (30) cm−1 and its J(J + 1), K2, (Jx2 − Jy2) dependences, that is, V31J, V31K, V31BC, respectively) and one parameter is connected to the structural parameters of top 1 (r1J). Like for 3,5-DMBA, we had to fix the F1 and F2 internal rotation constants for each top as well as the kinetic interaction term F12. Since the barrier height for top 2 is much higher (565.66 (15.97) cm−1), the internal rotation splittings are very small and it was therefore not possible to determine all the torsional parameters for that top with high accuracy. As a consequence, the q2 and r2 parameters (which multiply the operators Jz p2 and Jx p2 relative to the top 2, respectively) are less well determined. Finally, only one interaction term in the potential (V12c) could be determined; V12s was not determined and its value was fixed to zero.

Table 2. Overview of the FTMW Data Set Coverage and Fit Qualities Γa

#b

rmsc

A E1 E2 E3 E4 total

67 35 40 38 31 211

1.9 2.8 3.8 3.6 5.2 3.4

A E1 E2 E3 E4 total

94 80 87 73 76 410

A E1 E2 E3 E4 total

36 27 28 27 27 145

A E1 E2 E3 E4 total

56 53 54 53 54 270

Jmaxd

Kmaxd

11 11 11 11 11

5 3 3 3 3

3,5-DMBA

2,5-DMBA 2.8 14 4.4 14 3.3 14 3.6 14 3.9 14 3.7 syn-3,4-DMBA 1.5 15 1.2 12 1.8 12 2.5 12 1.6 12 1.8 anti-3,4-DMBA 1.3 15 1.7 15 1.3 15 1.8 15 2.0 15 1.7

5 5 5 5 5

3 3 3 3 3

3 3 3 3 3

a

Symmetry species of the lines in each row. bNumber of lines in each data group. cRoot-mean-square deviation of each data group in kHz. d Largest J and Ka values in each data group

3,4-DMBA because their internal rotation splittings are very small as both have high barriers hindering the internal rotation of the two methyl groups, above 400 or even 500 cm−1, and therefore their patterns are easily recognizable (see Figure 3, bottom left). Then we moved on to the analysis of syn-2,5DMBA which has one low and one high barrier as an intermediate case. Finally we carried out the much harder analysis of 3,5-DMBA, which has two low barriers hindering the internal rotation of the methyl groups and presents the largest internal rotation splittings. However, the results for 3,5-DMBA give the most determinable parameters, so we first describe the global fits on this isomer. In the following sections, we discuss the different global fits we performed on the DMBA isomers, using the BELGI-Cs2Tops code described in the theoretical section. For each case, we used as initial guesses the values of the rotational constants (A, B, C) and torsional parameters (F1, F2, F12, V31, V32, q1, q2, r1, and r2, where F1, F2, F12, V31, and V32 are defined in eq 1, the q1 and q2 the parameters multiply the Jz p1 and Jz p2 operators, r1 and r2 multiply the Jx p1 and the Jx p2 operators, respectively)4 all derived from the structures obtained by ab initio calculations. Global Fit of the 3,5-DMBA Isomer. An overview of the data set obtained for 3,5-DMBA is given in the top part of Table 2. The rotational quantum number and symmetry species coverage of the transitions in the fit correspond to the instrument coverage. However, the use of jet-cooled conditions G



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Table 3. Two Sets of Molecular Parameters for 3,5-DMBA in the PAM System, One Derived6 from the Least-Squares Fitted Parameters in the Supporting Information, Table S2, the Other from the Ab Initio Structure parameter

BELGI-Cs-2Tops

ab initioa

parameter

unit

BELGI-Cs-2Tops

ab initioa

APAM BPAM CPAM

MHz MHz MHz

1751.470(12) 1095.5972(30) 679.654595(90)

1759.903 1098.115 681.904

APAM BPAM CPAM

MHz MHz MHz

2499.09 (3.12) 989.83 (16) 714.6513 (11)

2510.233 993.053 717.810

f1 = F1 Iα1 V31 V61e

cm−1 amu Å2 cm−1

5.4b 3.1402c 53.0068 (62) 0.fixed

5.4 3.0983 39d 3d

f1 = F1 Iα1 V31 V61e


5.33b 3.1828c 5.35 (0.30) 0fixed

5.33 3.126 18d 3d

f2 = F2 Iα2 V32 V62e


5.5b 3.1101c 25.2504 (41) 0.fixed

5.5 3.071 19d 10d

f2 = F2 Iα2 V32


5.47b 1.3790c 565.66 (15.97)

5.47 3.195 594d

f12= F12 V12s V12c

cm−1 cm−1 cm−1

−0.15f 0. 7.69 (1.12)

−0.15 −5 10

ρ1a ρ1b ρ1 ρ2a ρ2b ρ2

unitless

0.014210 (23)g −0.0026803 (49)g 0.014461(24)g −0.0059 (11)g 0.00136 (24)g 0.0060 (11)g

0.014045 −0.002618 0.014287 - 0.014134 0.002855 0.014420

θ ∠(i1,a) ∠(i1,b) ∠(i1,c)

rad degrees

0.00484(23)h 25.465i 64.534j 90.00

0.0001650 25.248 64.775 90

30.235i 59.765j 90.00

27.050 62.951 90

−1

f12 = F12 V12s V12c

cm cm−1 cm−1

−0.03 0.727 (18) −8.8990 (35)

−0.03


unitless

0.00265682(15)g −0.0066015(13)g 0.00976611(15)g 0.00779774(85)g 0.00465464(51)g 0.00908132(99)g

0.00270 −0.00652 0.00706 0.00777 0.00459 0.00903

rad degrees

−0.004746(10) 75.8694i 14.1305j 90.0

θ ∠(i1,a) ∠(i1,b) ∠(i1,c) ∠(i2,a) ∠(i2,b) ∠(i2,c) a

unit

Table 4. Two Sets of Molecular Parameters for syn-2,5DMBA in the PAM System, One Derived6 from the LeastSquares Fitted Parameters in the Supporting Information, Table S3, the Other from the Ab Initio Structure

f

43.6594i 46.3406j 90.0

h

−0.004689 75.512 14.511 90.0

∠(i2,a) ∠(i2,b) ∠(i2,c)

43.453 46.552 90.0

a

Calculated equilibrium parameters at the MP2/cc-pVTZ level. Parameter held fixed in the fit, see the Supporting Information, Table S3. cThe internal rotation inertia moments Iα1 and Iα2 are derived from Iα1 = 505379.076/F01, where F01 = AB/[ρ1a2B2 + ρ1b2A2]1/2 and similar equations for Iα2. dEnergies are ZPE corrected. e The potential barrier is V(α) = V31 (1 − cos3α1) + V61 (1 − cos6α1). f See eq 10, eq 13, and eq 14 of ref 6. gValues derived from eq 11 and eq 12 of ref 6. hAngle about the y-axis, which relates the a, b, c PAM axes to the x, y, z axes. iAngle between the methyl top axis and the PAM a-axis. jAngle between the methyl top axis and the PAM b-axis: ∠(i,b) = 90° − ∠(i,a). b

b

Calculated equilibrium parameters at the MP2/cc-pVTZ level. See eq 10, eq 13, and eq 14 of ref 6. Parameters kept fixed in the fit (see the Supporting Information, Table S2). cThe internal rotation inertia moments Iα1 and Iα2 are derived from Iα1 = 505379.076/F01, where F01 = AB/[ρ1a2B2 + ρ1b2A2]1/2 and similar equations for Iα2. dEnergies are ZPE corrected. eThe potential barrier is V(α) = V31 (1 − cos3α1) +V61 (1 − cos6α1). fSee eq 10, eq 13, and eq 14 of ref 6. gValues derived from eq 11 and eq 12 of ref 6. hAngle about the y-axis, which relates the a, b, c PAM axes to the x, y, z axes. iAngle between the methyl top axis and the PAM a-axis. jAngle between the methyl top axis and the PAM b-axis: ∠(i,b) = 90° − ∠(i,a).

As already mentioned above, we note that for this isomer the ab initio potential calculation at the MP2/cc-pVTZ level requires in its fit not only V31 but also a V61 term (second term in the potential Fourier expansion) as illustrated in Figure 4. This term is estimated to be 3 cm−1. However, our attempt to fix V61 to this value in the BELGI-Cs-2Tops code did not result in any improvements (or significant changes in the V31 term), so we fixed V61 to zero in our fit. Concerning the interaction terms V12s and V12c, theoretical values can be estimated from the variations of the barriers to internal rotation of the C5 group when the C2 group is at its EqS or its TS (see Figure 4b). As for methyl acetate,29 the difference may be approximated for both terms by V(TS,TS) − (V(EqS,TS) + V(TS,EqS)), where the first and second

In Table 4, we compare the values obtained by BELGI-Cs2Tops for the syn-2,5-DMBA isomer with those obtained by ab initio calculations. The values of the structural angles (∠(i,a), ∠(i,b), and ∠(i,c)) and the values of the ρ components for top 1 (derived from the experimental splittings corresponding to the low barrier) are in very good agreement with the ab initio calculations. Not surprisingly, the corresponding parameters for top 2 (derived from the experimental splittings corresponding to the high barrier and obtained from the q2 and r2 parameters using eq 11 and eq 12 of ref 6) are not as well determined. They differ from the ab initio values by about 11% for the angles of the methyl group. H



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Table 6. Two Sets of Molecular Parameters for anti-3,4DMBA in the PAM System, One Derived6 from the LeastSquares Fitted Parameters in the Supporting Information, Table S5, the Other from the Ab Initio Structure

variables of the potential V(1st,2nd) correspond to the state of the C2 and C5 group, respectively. These values (see Table 4) combined with the experimental ones provide reasonable qualitative estimations of V12s and V12c (see discussion section hereafter). Global Fit of the syn- and anti-3,4-DMBA Conformers. The Supporting Information, Tables S4 and S5 gives the 13 and 14 molecular constants varied in our final fits of 145 and 270 lines of syn- and anti-3,4-DMBA, respectively, to reach a standard deviation of 1.7 kHz. Tables 5 and 6 present comparisons with the ab initio results for the structure and barrier heights of the two conformers. It can be seen that when both barriers are high and nearly equal to each other, structural parameters, such as the direction of both methyl groups in the

parameter

Table 5. Two Sets of Molecular Parameters for syn-3,4DMBA in the PAM System, One Derived6 from the LeastSquares Fitted Parameters in the Supporting Information, Table S4, the Other from the Ab Initio Structure parameter

unit

BELGI-Cs-2Tops

ab initioa

APAM BPAM CPAM

MHz MHz MHz

2686.08 (5.54) 908.66 (68) 684.9108(4)

2706.057 911.168 687.395

f1 = F1 Iα1 V31


5.49b 3.156c 508.12(1.08)

5.49 3.124 506d

f2 = F2 Iα2 V32


5.44b 3.781c 550.74(8.82)

5.44 3.126 519d

f12= F12 V12s V12c


−0.095e 0. 0.

−0.095

unit

BELGI-Cs-2Tops

ab initioa

APAM BPAM CPAM

MHz MHz MHz

2934.98 (3.08) 858.32 (15) 669.79937(15)

2963.258 860.137 672.132

f1 = F1 Iα1 V31


5.46b 2.980c 454.06 (1.34)

5.46 3.124 471d

f2 = F2 Iα2 V32


5.49b 2.784c 480.64 (4.35)

5.49 3.124 499d

f12 = F12 V12s V12c


−0.12e 0. 0.

−0.12


unitless

0.01624 (27)f −0.001752 (44)f 0.01633(27)f −0.01054 (52)f −0.003585 (89)f 0.01113(52)f

0.017550 −0.001525 0.01762 −0.0131 −0.0037 0.01365

θ ∠(i1,a) ∠(i1,b) ∠(i1,c)

rad degrees

−0.00069 (18)g 20.25h 69.75i 90.00

−0.00167 16.64 73.35 90.0

49.30h 40.70i 90.00

44.20 45.79 90.0

∠(i2,a) ∠(i2,b) ∠(i2,c) a

Calculated equilibrium parameters at the MP2/cc-pVTZ level. Parameter held fixed in the fit, see Supporting Information, Table S5. cThe internal rotation inertia moments Iα1 and Iα2 are derived from Iα1 = 505379.076/F01, where F01 = AB/[ρ1a2B2 + ρ1b2A2]1/2 and similar equations for Iα2. dEnergies are ZPE corrected. eSee eq 10, eq 13, and eq 14 of ref 6. fValues derived from eq 11 and eq 12 of ref 6. gAngle about the y-axis, which relates the a, b, c PAM axes to the x, y, z axes. h Angle between the methyl top axis and the PAM a-axis. iAngle between the methyl top axis and the PAM b-axis: ∠(i,b) = 90° − ∠(i,a). b


unitless

0.01666 (20) 0.00065 (20)f 0.01788(27)f 0.0111 (11)f 0.00566 (26)f 0.0125(10)f

0.01671 0.00024 0.01672 0.00883 0.00479 0.01004

θ ∠(i1,a) ∠(i1,b) ∠(i1,c)

rad

−0.00649 (69)g 6.6032h 83.3968i 90.00

2.471 87.532 90.0

33.6723h 56.3276i 90.00

31.8340 58.1700 90.0

∠(i2,a) ∠(i2,b) ∠(i2,c)

f

molecule and the components of the ρ vector, are not well determined and differ from the ab initio calculations. Supporting Information, Tables S6−S9 give the observed values minus the calculated values for each isomer and each symmetry species. The deviations of each symmetry group (a few kHz) are rather close to the experimental accuracy which is estimated to be better than 1 kHz, even though some of the E species show slightly worse observed minus calculated values than the A species.

a

Calculated equilibrium parameters at the MP2/cc-pVTZ level. Parameter held fixed in the fit, see the Supporting Information, Table S4. cThe internal rotation inertia moments Iα1 and Iα2 are derived from Iα1 = 505379.076/F01, where F01 = AB/[ρ1a2B2 + ρ1b2A2]1/2 and similar equations for Iα2. dEnergies are ZPE corrected e See eq 10, eq 13, and eq 14 of ref 6. fValues derived from eq 11 and eq 12 of ref 6, fixed in the fit. gAngle about the y-axis, which relates the a, b, c PAM axes to the x, y, z axes. hAngle between the methyl top axis and the PAM a-axis. iAngle between the methyl top axis and the PAM b-axis: ∠(I,b) = 90° − ∠(I,a). b

■

DISCUSSION The present analysis using the BELGI-Cs-2Tops code for the four observed DMBA isomers is quite instructive. The high resolution data can serve as a guideline for validating the quantum chemical calculations, as well as for validating the BELGI-Cs-2Tops method. When going from 3,5-DMBA (two I



Article

low internal rotation barriers below 50 cm−1) to syn-2,5-DMBA (one low barrier around 5 cm−1 and one high barrier above 565 cm−1) and to syn- and anti-3,4-DMBA (two high barriers around 450−550 cm−1), different cases occur. The rotational constants derived from experimental results agree with the ab initio calculations by better than 0.5% for the two first, and by better than 1% for the two others. This is a very satisfactory agreement, given the facts that (i) ab initio values are calculated at equilibrium whereas the values derived from experiments are determined in the ground vibrational state and (ii) the structure of the entire molecule is slightly modified during the motions of the internal rotors, which is a dynamical effect not taken into account by the calculations. The internal rotation parameters (potential barrier heights V31 and V32 of the tops, angles between the symmetry axes of the two tops and the principal axes, and moment of inertia) are also in good agreement with the ab initio values. It is noteworthy that the accuracy of the values derived from the experiments depends on the magnitude of the internal rotation splittings: the smaller the splittings are (i.e., the higher the potential barriers are), the less determinable are the internal rotation parameters, and the most correlation effects occur. These effects presumably arise because the smaller splittings can be determined to fewer significant digits than the larger splittings, and they therefore contain less information. This can be seen clearly in the case of the two 3,4DMBA conformers, where the angles between the methyl top axis and the PAM a and b axes for both tops (∠(i1,a) and ∠(i1,b), ∠(i2,a) and ∠(i2,b)) agree only within a few degrees with the ab initio values, as well as in the case of the angles ∠(i2,a) and ∠(i2,b) for the highest barrier methyl top group of syn-2,5-DMBA. In addition, the value of the ρ2 parameter for syn-2,5-DMBA corresponding to the highest barrier top (calculated using eq 11 of ref 6 from the fitted values of q1, q2, r1, and r2, but also from the fixed values of F1, F2, F12, see Table 4) differs from its ab initio value by more than a factor of 2. The value of the fitted parameter q2 (0.067 (12) cm−1, see Supporting Information, Table S3) is also quite different from q1 (−0.15263 (18) cm−1) for this isomer. This is the consequence of an important correlation between the internal rotation parameter q2 and V32. Fixing the q2 value to +0.15 cm−1 (which corresponds to its ab initio calculated value) leads to a fit of similar quality, but the value of V32 goes up to 649 cm−1, which is too high to be considered as a reasonable value. Concerning the interaction terms for syn-2,5-DMBA, the value obtained from the difference between the heights of the highest barrier (TS C5e) is in better agreement with the V12c term deduced from the fits (10 vs 7.69 cm−1, respectively, see Table 4) than the value obtained from the lowest barrier heights’ difference (TS C5s, −5 cm−1). Although there is no clear experimental validation of the present method, it appears to be an interesting approach to estimate ab initio values of the V12c and V12s interaction terms that should be investigated more deeply in a more favorable case than with the DMBA isomers. Decreasing this correlation between the internal rotation parameters (and also the correlation between the F1, F2 and V31, V32 parameters) when the torsional barriers are high would require measurements within or between different excited torsional states. The search specifically for vt = 1 rotational transitions for molecules of the size studied at temperatures around 50−80 K (like for paratoluadehyde7) would fit perfectly our requirement and would avoid congestion from other vibrations at low frequencies. Alternatively, the present study emphasizes the fact that spectral analysis and ab initio

calculations are complementary techniques. Indeed, when the limits of the former are reached (transition splittings lower than the experimental resolution because of high barriers), more reliable properties are obtained from the latter. Contrarily, when the values are of the order of the expected calculations errors, more reliable properties are obtained from the fits of widely split spectral transitions. For cases at the limit of one or the other technique’s capabilities, combining both types of results delivers at least upper and lower limits into which the “real” values most probably fit.

■

CONCLUSION In this work, we have studied the microwave spectra (2−26.5 GHz) of four dimentylbenzaldehyde (DMBA) molecules using the high resolution COBRA-FTMW spectrometer in Hannover, with the support of ab initio calculations. The 3,5-DMBA isomer presents two internal methyl groups hindered by low internal rotation barriers of 53.0068 (62) and 25.2504 (41) cm−1. Two interaction parameters describing the coupling of the two tops were determined, the term V12c = −8.8990 (35) cm−1, multiplying the potential term (1 − cos α1)(1 − cos α2) which corresponds to almost a third of the lowest barrier height and the term V12s = 0.727 (18) cm−1 which multiplies the (sin 3α1)(sin 3α2) term. The syn-2,5-DMBA contains one methyl internal rotor with a low barrier of V31 = 5.35(30) cm−1 and one rotor with a high barrier of V32 = 565.66 (15.97) cm−1. Only the coupling term V12C = 7.69 (1.12) cm−1 could be determined, and it has a similar magnitude as for 3,5-DMBA, but with opposite sign. The syn- and anti-3,4-DMBA conformers contain two high barriers ranging from about 450 cm−1 to 550 cm−1. For syn-2,5- and 3,4-DMBA, the small internal rotation splittings resulting from the high barriers lead to less well determined internal rotation angles and barrier heights and to some correlation between those parameters. Ab initio calculations provided in a first step a reliable initial set of parameters, which is a very helpful starting point for the spectral analysis. All the spectroscopic data were fitted within almost experimental accuracy, validating the BELGI-Cs-2Tops program recently written to deal with the rotational spectra of two nonequivalent internal rotors. In turn, the sets of parameters derived from the high resolution data served as a guideline for validating the quantum chemical calculations, performed with different methods and basis sets. Once confidence in these calculations was established, fixing some parameters to their theoretical values during the modeling process permitted in some cases to break correlations, improving the accuracy of the results. At worst, when the limits of the spectral and the ab initio approaches are reached in the same molecule, then the combination of both techniques provides at least domains into which the less determined values most probably fit.

■

ASSOCIATED CONTENT

S Supporting Information *

Table S1: Calculated harmonic frequencies (MP2/cc-pVTZ) and observed frequencies (Synchrotron SOLEIL) of the low frequency modes (

Coupled Large Amplitude Motions: A Case Study of the

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