The hydroxyl groups internal rotations in a methanediol molecule

a Belarusian State University, b University of Illinois, Chicago, USA *Corresponding author. E-mail address: pitsevich@bsu.by (George Pitsevich).


INTRODUCTION
A methanediol (MD) molecule is the simplest representative of diols and is used in cosmetic and food industry. It has been considered for a long time that the only equilibrium configuration [1][2][3] is realized in the MD molecule. However, Rodler [4] found out for the first time that the molecule is featured by two equilibrium configurations. The trans-configuration, in which the hydroxyl groups are oppositely directed, is much more preferable energetically than the cis-configuration, in which the O-H groups are almost parallel. A new wave of interest to the molecule was initiated recently by ascertaining the fact that MD plays an important role in the formation of atmospheric aerosols [5] as well as can be present in the interstellar space [6][7][8] and in the comets [9,10]. The search of this molecule in the cosmic space is complicated due to the absence of registered microwave and IR spectra of MD in the gas phase at present that is caused by low stability of the molecule. Indeed, the literature presents the experimental investigations of IR spectra of MD in the matrix isolation [11] as well as IR and Raman spectra in water solutions [10,[12][13][14][15][16], however the effect of the environment and possibility of formation of various associates reduces the significance of the results of these investigations for the identification of the MD molecules in the atmosphere and space. Therefore, the theoretical investigations of the spectral characteristics of this molecule are called for. MD stability in the gaseous phase was analysed for the first time within the up-to-date quantum-chemical methods in paper [7]. It was demonstrated that the molecule must be stable at temperatures below 100 K. The effect of ammonia and formic acid on MD formation process in the reaction of formaldehyde with water is investigated in papers [10] and [5], respectively. The structural and spectral characteristics of MD and its associates in the water medium are analysed theoretically in paper [17]. The effect of the solvent polarity on the ratio of MD conformers is evaluated in paper [18]. In paper [19] the barriers of internal rotation of hydroxyl groups in MD were determined for the first time, the rotational constants were calculated and the pure rotational spectrum of the molecule was simulated. The IR spectrum of two conformers of the MD molecule in the harmonic approximation and for the first time in the anharmonic one was computed and presented in paper [8]. The last two papers are of the highest interest from the standpoint of use of their results for searching MD in the atmosphere and interstellar space. However, it is obvious that the success of such search is far from being ensured without analysis of internal rotation of hydroxyl groups in the molecule. As far as we know, results of such calculations are absent in literature up to this date.
In this paper, the energies of the stationary torsional states of the MD molecule were calculated at the complete basis set (CBS) limit using the MP2/cc-pVQZ level of theory. The torsional IR spectra of MD at various temperature were computed. The values of the tunnelling frequencies for two conformers of the molecule were estimated.

CALCULATION DETAILS
The equilibrium configurations of MD are presented in Fig.1 at the global (transconformer) and local (cis-conformer) energy minima as well as in the configuration being the initial one for performing the calculation of the 2D potential energy surface (PES). The first two configurations belong to the C 2 and C S symmetry point groups. The last one belongs to the C 2V symmetry point group and represents the transition state having one imaginary frequency in the calculated IR spectrum. The hydroxyl groups rotate around the C-O bonds. Further we will be denoting the equivalent torsional coordinates as  and  . The zero values of  and  were chosen in the configuration presented on the right of Fig.1 where the hydroxyl atoms of hydrogen (Н 1 and Н 2 ) lie in the plane, which is formed by the О 5 -С 7 -О 6 atoms.   [20,21] and cc-pVQZ [22] basis set in the frame of the quantum-chemical package [23].
Configurations optimized in each node were used to calculate the kinematic coefficients using Wilson's vectors ( s -vectors) [24]. The Schrödinger equation with reduced dimensionality for the torsional vibrations of two hydroxyl groups can be written as follows [25][26][27]: The kinematic parameters can be determined from the following relations [25][26][27]:  For the MD molecule geometries that were optimized at all the grid nodes at the MP2/cc-pVQZ level of theory, potential energy values were computed at the MP2/cc-pVTZ [28] level of theory without optimization. Then energy values of every node were extrapolated over the two datasets (MP2/cc-VTZ and MP2/cc-pVQZ) to the CBS limit using known formulas [29,30].
Since the Cartesian coordinate system changed its orientation in relation to the molecule as the hydroxyl groups rotates, components of the dipole moment vector at each node of the 2D grid were recalculated for the Cartesian coordinate system related to the molecular skeleton, in which the Z axis was directed along О 5 -С 7 -О 6 angle bisector, the X axis lied in the plane of the О 5 -С 7 -О 6 atoms and the Y axis supplemented the X and Y to the right-hand triple and was, therefore, normal to the plane of three heavy atoms.
To solve the Schrödinger equation (1) the DVR method was used [31][32][33][34][35][36]. Hamiltonian matrix was calculated by the formula: Here: Squares of matrix elements of the dipole moments operator were found using [37] and the following formula: Here ,, x y z p p p are the dipole moment vector components in the molecular fixed Cartesian are the torsional wave functions in the initial (i) and final (f) states.

SYMMETRY ANALISYS
As was mention above, the MD molecule is a non-rigid object having two equilibrium configurations belonging to the C 2 and C s symmetry point groups. Both conformers can tunnel to the equivalent configurations passing through the transition state. It is natural to classify stationary torsional states using the appropriate molecular symmetry group [38][39][40]. This group includes only feasible permutation and permutation-inversion symmetry elements. One can see that these elements are E, (12)(34)(56), (34) * and (12)(56) * . These four elements form the C 2V (M) molecular symmetry group, which is isomorph to the C 2V point group symmetry [40]. It is important that all four symmetry elements are inherent to both equilibrium conformers (C 2 and C S ) of the MD molecule. In addition, one can see that MD molecule transition state has the C 2V symmetry, which is consistent with the fact that the C 2V (M) molecular symmetry group is isomorph to the C 2V point group. The irreducible representations of the C 2V (M) molecular symmetry group are shown in Table 1.
It is important to know that the symmetry elements of the 2D surfaces of the potential energy, wave functions, components of the dipole momentum and kinematic coefficients are connected with symmetry elements of the C 2V (M) group. Let's call the plane which is perpendicular to the coordinate plane ,  and intersect it along diagonal (0 0 ,0 0 -360 0 ,360 0 ) plane 1. The plane which is perpendicular to the coordinate plane ,  and intersects it along diagonal (0 0 ,360 0 -360 0 ,0 0 ) will be plane 2. The intersection of these two planes will be the C 2 symmetry axis. Then these elements will be connected with symmetry elements of the C 2V (M) group according to data from Table 2. In addition, the transformation of the torsional coordinates ,  under the action of the C 2V (M) symmetry operations includes in Table 2 too. According to the data from Table 2 one can get, for example:     3 ,, Now it is obvious that using symmetry elements of the C 2V (M) molecular symmetry group (see Table 1 [40]) and data from Gathering four characters of the irreducible representation i Г ( , ,  The two global energy minima and the two local ones corresponding to the trans-and cisconformers of MD in two structurally equivalent configurations are well visible in Fig.2. Fig.3 presents the 2D surfaces of the kinematic coefficients. It should be noted that the 2D surfaces of the kinematic coefficients F  and F  are equivalent.    Table 3 (columns 3-5). In columns 6-9 of Table 3, torsional frequencies of the transitions from the ground level are presented. We were also able to determine the torsional quantum numbers for several lowest torsional states.
The as n and s n in 10 and 11 columns of Table 3 are the quantum numbers which specify antisymmetric and symmetric torsional vibrations in trans-and cis-conformers (see the second column in Table 3). The symmetry species of the torsional wave functions are represented in the last column of Table 3. As seen from the data in Table 3 To calculate the intensities of the torsional IR spectrum, it is also necessary to consider the symmetry of the torsional wave functions with respect to the permutation of the chemically equivalent atoms in the MD molecule. Since 16 O and 12 C atoms have zero spin they can be excluded from further consideration. First of all, one has to take into account symmetry properties of the total, torsional and spin wave functions with respect to symmetry elements of the C 2V (M) molecular symmetry group which do not include the inversion operation [40,45].
These are E and (12)(34)(56) symmetry operations. Due to the last one interchanges even number of the fermion nuclei (protons) the total wave function must be symmetric with respect to these two symmetry operations and consequently has to belong to A 1 or A 2 symmetry species.
The torsional wave functions that belong to the A 1 and A 2 symmetry species are symmetric with respect to interchange two pairs of fermion nuclei while the ones that belong to the B 1 and B 2 symmetry species are antisymmetric. The reducible representation of the hydrogen's nuclear spin functions can be decomposed into irreducible representations of the C 2V (M) molecular symmetry group as follows [46]: Thereby, the torsional wave functions must be combined with the spin wave functions as it is shown in Table 4.  (12) where i g -the degeneracy of the initial torsional state value of which is consistent with the type of symmetry of the torsional wave function in accordance with the data of Tables 3 and 4, s i and s f -the initial and final state specifier value of which, as indicated in Table 4, is 1 for the А 1 и А 2 torsional states, and is -1 for В 1 и В 2 torsional states, () QT -partition function. With the use of these data, the relative intensities of the IR absorption bands conditioned by the torsional transitions in the MD molecule at different temperatures were calculated from the formula (11). Fig. 5 presents the IR spectrum of the MD molecule at the temperatures of 300 and 30 K calculated in the CBS limit. If the potential barriers are sufficiently high, it becomes possible to make the cisconformer stable at temperatures below 30 K. To estimate the height of the potential barrier in case of transition from the cis-conformer to the trans-conformer using Mathematica [37] package, the potential energy of MD was determined at every point for the values  and  specified within the range 02   with the help of interpolation. The map of gradients on the potential energy surface is presented in Fig. 6. Using Fig.6, one can find out that the minimum potential barrier, which must be overcomed by the molecule during the transition from the cis-to the trans-configuration is 327 cm -1 . However, if one takes into account zero point energy for cis-conformer (258 cm -1 ) it becomes clear that the potential barrier is only about 70 cm -1 . This fact makes the cis-conformer of the MD molecule potentially unstable. At 30 K more than 8% of the cis-conformers will have the kinetic energy equal or higher than 70 cm -1 . On the other hand, at 10 K only 0,062% of the cis-conformers will have the kinetic energy equal or higher than 70 cm -1 . Furthermore, to convert this kinetic energy to internal energy successful collision needed. However, in the low density interstellar space collisions don't happen often. Anyway, we have calculated the torsional spectrum of the cis-transformer at the temperature of 10 К which is presented in Fig.7. In this case, the 13th torsional level was considered the ground one and in the formula (12) when calculating the partition function, the summation was performed from i = 13 to i = 50 with the step of 1. While comparing the spectra in Fig.5   We also take into account that the 3rd torsional energy level is metastable since its multiplicity differs from the multiplicity of the second and ground torsional levels. This means that optical deactivation of an excited antisymmetric torsional vibration in the trans-conformer is forbidden by the selection rules inherent in the spin wave function. It is known, for example, that in the absence of external influences, water in the ortho-state can exist for a very long time [49,50,51]. With this in mind, the torsional spectrum of the trans conformer of MD was calculated for which the 3rd torsional level is ground at a temperature of 30 K. In this case, in the formula (12) when calculating the partition function, the summation was performed from i = 13 to i = 50 with the step of 1. It is presented in Fig. 8. only cis-conformer and c) both conformers, as well. Based on the torsional quantum numbers represented in Table 3