Effect of Long-Range Corrections on Intermolecular Interactions and Vibrational Assignments of Ethylene Oxide Dimer. A Combined DFT and SQFF Study

: Theoretical structural and vibrational study for the ethylene oxide dimer have been performed by using the experimental structure determined by X-ray diffraction, the vibrational spectra and, the functional hybrids B3LYP/6-311++G** and WB97XD/6-311++G** methods. Here, the effects of dispersion on intermolecular interactions and on complete assignments of infrared and Raman spectra of dimer have been performed combining the Pulay´s scaled quantum mechanical force field (SQMFF) methodology with those two levels of calculations in order to fit the theoretical wavenumbers values to the experimental ones. Calculations including long-range corrections have revealed similar optimized energy, volume and frequencies to calculated with the B3LYP/6-31G* method, lower correlations in the geometrical parameters, higher stabilization energy, higher values in the topological parameters and higher scaled force constants than the obtained at B3LYP/6-311++G** level. Natural bond order (NBO) and atoms in molecules theory (AIM) studies with both methods reveal two types of intermolecular interactions (C-O···H and C-O···O) in the ethylene oxide dimer in accordance with the bands observed in the experimental Raman spectrum at low temperatures and with the experimental structure determined at 100 K. The nature of those interactions and their topological properties were studied by using NBO and AIM calculations. The studied properties for the ethylene oxide dimer were analyzed and compared with those obtained for the monomer. Similar assignments of the vibrational modes for dimer were obtained using the three different methods.


Introduction
Ethylene oxide, (C2H4O) is a compound broadly studied from different point of view and by using diverse spectroscopic methods (Bertie & Othen, 2011;Cant & Armstead, 1975;Bertie & Jacobs, 1978;Mills, 1965;Lowe et al., 1986;Hair, Beswick, & Janda, 1988;Winkler et al., 1999;Frimand, & Jalkanen, 2002;Bégué et al., 2007;Bernstein, & Lynch, 2009;Weiyu et al., 2009;Grabowsky et al., 2009).From a chemical point of view, the ethylene oxide monomer is a synthetic intermediate in the preparation of functional materials with desirable properties in the area of new solid polymer electrolytes which are potentially useful electrolytes to be applied in high-energy batteries (Zhang, & Wang, 2009;Polomska et al., 1999;Ali, Mohamed, & Arof, 1998;Daniel, Desbat, & Lassegues, 1988;Zhang et al., 2006;Zhang, & Wang, 2009;Oparaji, O., Zuo, & HallinanJr., 2016).The polymerization of ethylene oxide or of a mono-substituted ethylene leads to polymers in the solid state, such as the polyethylene oxide (PEO), whose physical and mechanical properties are strongly dependent of the structure.Particularly, the hydration process of a PEO tablet is of great pharmaceutical interest.The chains of a polymer in the crystalline state are aligned when the polymer is melted; the chain conformations then become disordered and, as a consequence, change the spectra.For these reasons, in many chemical industries new polymer syntheses are carefully controlled by means of infrared and Raman spectra.The infrared and Raman spectra of this epoxide in gas and liquid phases at room temperature and at low temperatures in the solid state, were published and only the assignments for the oxide as monomer were reported previously (Bertie & Othen, 2011;Cant & Armstead, 1975;Bertie & Jacobs, 1978;Mills, 1965;Lowe et al., 1986;Hair, Beswick, & Janda, 1988).The experimental electron density of ethylene oxide was derived by Grabowsky et al. (2008) from a multipole refinement highresolution X-ray diffraction data collected at 100 K and complemented by density-functional calculations at experimental and optimized geometry.On the other hand, the technique of electron momentum spectroscopy was employed to measure orbital momentum distributions for the complete valence electronic structure of ethylene oxide (Weiyu et al., 2009).From this study it is possible to derive ethylene oxide's chemically interesting molecular properties, which are subsequently compared with the results of other workers.In this context, intermolecular interactions, such as the hydrogen bonding, play an important role in the determination of the molecular structure and properties of ethylene oxide (Weiyu et al., 2009;Zhang, & Wang, 2009).At this time, the ethylene oxide as monomer was characterized both structural and vibrationally (Bertie & Othen, 2011;Cant & Armstead, 1975;Bertie & Jacobs, 1978;Mills, 1965), but the effects of dispersion on intermolecular interactions of dimer and on the vibrational assignments are not reported yet.Therefore, the aim of this paper is to report a theoretical study on ethylene oxide dimer by using experimental data existing of its structure and vibrational spectra combined with the Pulay´s scaled quantum-mechanical force field (SQMFF) methodology and the Molvib program (Pulay et al., 1983;Rauhut, & Pulay, 1995;Sundius, 2002) in order to carry out complete and reliable vibrational analyses taking into account the B3LYP/6-311++G** and WB97XD/6-311++G** methods (Becke, 1988;Lee, Yang, & Parr, 1988;Chai, & Head-Gordon, 2008).For this purpose, we optimized the dimer geometry by using both levels of calculations.Here, we present a detailed study of the vibrational spectra of the ethylene oxide dimer and propose a complete assignment of the observed bands based on a normal coordinate analysis and scaling factors together with the harmonic force fields (Pulay et al., 1983;Rauhut, & Pulay, 1995;Sundius, 2002).In addition, the spectroscopic behaviour of the ethylene oxide dimer was studied and interpreted on the basis of the molecular structure observed experimentally at 100 K (Grabowsky et al., 2009).This way, the presence of bands in the Raman spectrum in the lower wave numbers region assigned to the vibrational modes of the dimer justifies its presence at low temperature in the solid phase.In addition, the nature of the two types of intermolecular interactions found in the dimer (C-O•••H and C-O•••O) were studied by means of the natural bond orbital analysis (NBO) (Glendening et al., 1996) while their topological properties were analyzed using the atoms in molecules theory (AIM) calculations (Bader, 1990;Biegler-Konig, Schonbohm, & Bayles, 2001).Here, the theoretical molecular, vibrational and topological properties of the ethylene oxide dimer are shown and compared with those obtained experimentally for the monomer.

Computational Details
The starting point for the geometry optimization of the ethylene oxide dimer was a structure with the bonds lengths and angles taken from the experimental structure derived by Grabowsky et al. (2008) from a multipole refinement highresolution X-ray diffraction data collected at 100 K.The initial structure was modelled with the GaussView program (Nielsen, & Holder, 2008) and the calculations were made with the functional hybrids B3LYP/6-311++G** and WB97XD/6-311++G** methods (Becke, 1988;Lee, Yang, & Parr, 1988;Chai, & Head-Gordon, 2008) as implemented in the Gaussian 09 set of programs (Frisch et al., 2009).The optimized structures for the monomer (EOM) and dimer (EOD) together with the corresponding numbering of atoms for both forms and the CIF file are shown in Figure 1.
The normal internal coordinates for EOD are shown in Table S1 (Supporting Material), and they were defined as those proposed by Fogarasi et al. (1992).The harmonic force field Cartesian coordinates for the dimer that resulted from the calculations were turned into "natural" internal coordinates by the Molvib program (Sundius, 2002).The Potential Energy Distribution (PED) components larger than or equal to 10% were subsequently calculated with the Scaled Quantum Mechanics (SQM) force field.Furthermore, an NBO analysis was then performed by using the two calculation levels with the NBO 5.0 program (Glendening et al., 1996) included in the Gaussian 03 package (Chai, & Head-Gordon, 2008).The topological properties of the charge density for the dimer were computed at the same theory levels with the AIM2000 software (Bader, 1990;Biegler-Konig, Schonbohm, & Bayles, 2001).We clarify that the total energy values for EOD were not corrected for the basis set superposition error (BSSE) by the standard Boys-Bernardi counterpoise method because the energy differences with the 6-311++G** basis set are negligible, as was also observed for other dimeric species (Boys, & Bernardi, 1970;Brandán et al., 2010).

Geometry
Table 1 shows the calculated total energies, dipole moments and volume for EOD dimer by using both B3LYP/6-311++G** and WB97XD/6-311++G** methods (Becke, 1988;Lee, Yang, & Parr, 1988;Chai, & Head-Gordon, 2008) compared with the obtained for the monomer by using two methods.The dipole moment is a relevant property to understand the stability and reactivity of a compound and, therefore, a comparison of the theoretical dipole moment values for EOM and EOD are also observed in Table 1.First of all, it is observed that for both forms the highest dipole moment values and the most stable structures are obtained with the B3LYP/6-311++G** method and, the second one, the total energy for the dimer with both basis sets, is smaller than the sum of the energies of each monomer, for instance, by using the 6-31G* basis set the E value -307.5790Hartrees is more negative than 2 x (-153.7863)= -307.5726Hartrees.This possibly indicates that the presence of the dimer even in gas phase is preferable to the isolated monomer.This result is very important and it will be taken into consideration in the assignment of the vibrational spectra registered for this compound.Thus, to perform the vibrational analysis by using the most stable structure the best method is B3LYP/6-311++G**, as was also observed in other species (Scheuermann et al., 2018;Sundius, & Brandán, 2023).In addition, the B3LYP/6-31G* calculations for monomer and dimer were also presented because the WB97XD/6-311++G** level of theory for the dimer produces an energy value (-307.5730Hartrees) similar to obtained with the B3LYP/6-31G* method (-307.5790Hartrees).Moreover, the dipole moment value increases when the 6-311++G** basis set is used.When the volumes are analysed for the different methods, newly is observed that the B3LYP/6-31G* method predict practically the same V for the dimer than the WB97XD/6-311++G** level of theory.
The total energy values for the dimer uncorrected while corrected for BSSE effects using the counterpoise method with the B3LYP/6-31G* method are respectively -307.5790 and -307.5751Hartrees (Boys, & Bernardi, 1970).The BSSE value for the dimer is negligible with the larger basis set (6-311++G**), as was also observed for the dimer of 4-hydroxybenzoic acid (Brandán et al., 2020).

Geometrical parameters
Table 2 shows a comparison of the optimized geometrical parameters for EOD calculated by using the B3LYP and WB97XD methods with the corresponding experimental values which were obtained by Grabowsky et al. (2008) from a multipole refinement by X-ray diffraction data collected at 100 K and complemented by density-functional calculations at experimental and optimized geometry, as shown in Figure 1c.
A comparison of all the calculated geometrical parameters with the available experimental data in terms of root-mean-square deviation (RMSD) demonstrates a very good concordance among them.Thus, analysing the bond lengths and angles it is observed that the B3LYP method generate the lower values (0.025 Å and 0.7°) than the other one (0.096 Å and 0.8°).In the same way, the predicted dihedral angles with the WB97XD method are slightly higher than the theoretical obtained with the other method.Hence, clearly it is observed that better results are obtained with the B3LYP/6-311++G** method.Note: a This work, b Source: Grabowsky et al., 2008 Then, if the dispersion by using the WB97XD method is considered the energy value is lower and, besides, the predicted structure at this level of calculations shows lower concordances in the geometrical parameters with the corresponding experimental one than the B3LYP/6-311++G** method.In this context, the vibrational analysis and assignments of infrared and Raman spectra for the dimer must be performed with the better method, which is B3LYP/6-311++G**.

Atomic Charges, Molecular Electrostatic Potentials and Bond Orders
Atomic charges, molecular electrostatic potentials and bond orders are interesting parameters to investigate the characteristics of different bonds and those sites or regions where the reactions with different reactive take places.Thus, for the dimer the Merz-Kollman (MK), Mulliken and natural atomic population (NPA) were computed with both methods (Glendening et al., 1996;Besler, Merz, & Kollman, 1990).In Table 3 are shown these three types of charges while Figure 2 shows its different behaviours.
The detailed analysis of three charges from Fig. 2 show that with both methods the MK (blue colour) and NPA (grey colour) follow approximately a same tendency while the Mulliken charges (orange colour) calculated with the B3LYP method evidence practically the same values on the O and C atoms but with the other method they are different.Thus, with the WB97XD method the MK and Mulliken charges computed show higher differences on the O and C atoms while the three charges on H13 and H14 present the same values with this method, as observed in Table 3.On the other side, the NPA charges predict with both methods the most negative values on O atoms while the most positive on those H9, H10, H11 and H12 atoms because these atoms are involved in the formation of H bonds, as was experimentally observed (Grabowsky et al., 2008).Regarding the molecular electrostatic potentials (MEP) from Table 4 it is observed similar values for both methods although the B3LYP method shows the higher values on O atoms but lower on the C ones.However, both methods evidence the same values on the H atoms.
If now the mapped MEP surfaces are graphed for the two methods in order to see the different electrophilic and nucleophilic regions, these are those regions with blue and red colours, respectively which represent acceptors and donor's sites of H bonds.
In Figure 3 are observed the different colorations obtained for both methods.Here, on the O atoms of both graphics of Fig. 3 are observed red colours indicating the presences of lone pairs which are acceptors of H bonds while on the H7, H8, H13 and H14 atoms are observed light blue colours indicating the most labile atoms whose MEP values show less negative values with both methods (Table 4).
Then, the green colours represent inert sites where the reactions are not performed.
The analysis of the bond orders totals by atoms, expressed by Wiberg's indexes, for EOD (Table 4) shows similar values for all atoms with both methods indicating that the dispersion has not effect on the bond orders.However, the Wiberg bond index matrix in the NAO basis show eight interactions, four of them between the O1 with the H7, H9, H11, H13 atoms and other four between the O2 with the H8, H10, H12, H14 atoms.The bond orders values of those interactions are between 0.0222 and 0.0223.

NBO and AIM Studies
The stability of the EOD structures for both levels of theory were also investigated by means of NBO and AIM calculations (Glendening et al., 1996;Bader, 1990;Biegler-Konig, Schonbohm, & Bayles, 2001).The second order perturbation energies E (2) (donor  acceptor) that involve the most important delocalization are summarized in Table 5. Note: Total energy in bold letter.
For these structures, the contribution of the stabilization energies, due mainly to the ∆ETσ→ σ* charge transfer, are higher than the other ones ∆ETLP→ σ* delocalizations, being majors the values predicted for the WB97XD method.Thus, the total energy values clearly show that the dispersion has effect on stabilization energies of dimer and, that this total energy is higher for the WB97XD level of theory.
The EOD structure was also analysed by means of Bader's electron density topology analysis (Bader, 1990;Biegler-Konig, Schonbohm, & Bayles, 2001). .The localization of the Bond Critical Point (BCP) in the electron density, ρ(r) and the Laplacian values, ∇ 2 ρ(r) at these points are important for characterization of the molecular electronic structure in terms of interaction magnitude and nature.The BCP has the typical properties of the closed-shell interaction.That is, the value of ρ(r) is relatively low, the relationship |λ1|/λ3 is <1 and ∇ 2 ρ(r) is positive indicating that the interaction is dominated by the charge contraction away from the interatomic surface toward each nucleus.
The charge density and ∇ 2 ρ(r) for the calculated BCPs from the topological property analysis for the dimeric structure are shown in Table 6 while Figure 4 shows five new interactions formed.

Table 6. An Analysis of the Ring Critical Points (RCP) for Ethylene Oxide Dimer
Parameter (a.u.)    6, probably justify that this BCP can be not experimentally observed.This study evidence higher topological properties and lower distances for the dimer calculated with the WB97XD method.

Vibrational analysis
The infrared spectrum of ethylene oxide in the gas phase was taken from a previous study (Shimanouchi, 1972;NIST, n.a.), which is presented in Figure 5 compared with the corresponding predicted for both methods while the predicted Raman spectra at low temperatures between 140 and 40 cm -1 were taken from Bertie & Jacobs (1978) and they are presented in Figures 6 and 7. Here, the predicted infrared spectrum in the lower wavenumbers region (200-20 cm -1 ) can be seen in Figure 6 compared with the corresponding experimental Raman spectra between 140 and 40 cm -1 (Bertie & Jacobs, 1978).The B3LYP/6-31G* level of calculations was also included because the energy obtained at this level is approximately similar to WB97XD method.The experimental and calculated wavenumbers for the expected normal vibration modes of EOD by using the different levels of theory together with the corresponding assignments are shown in Table 7.Here, the assignments were not separate by the symmetry because these changes for some modes, as observed from the table 7.  τ, torsion; ρ, rocking; τw, twisting; a, antisymmetric; s, symmetric; s, strong; m, medium; w, weak; p,  polarized; dp, depolarized; op, out-of-phase; ip, in-phase.a This work, b From Winkler et al., 1999, c From scaled quantum mechanics force field at B3LYP/6-311++G**, d From scaled quantum mechanics force field at B3LYP/6-31G*, e From scaled quantum mechanics force field at WB97XD/6-311++G**.Sym, symmetries of vibration modes.
An agreement between theoretical and available experimental results is found, especially in the lower wavenumbers region, as seen in Figure 6.
It is important to mention that in the theoretical IR spectrum of monomer there are not observed bands in the vibrational spectra from 500 to 10 cm -1 , while in the infrared spectrum of the dimer the vibration normal modes corresponding to the dimeric species are evidenced in that region.The vibrational assignment of the experimental bands to the normal vibration modes is based on the comparison with related molecules (Bertie & Othen, 2011;Cant & Armstead, 1975;Bertie & Jacobs, 1978;Mills, 1965;Lowe et al., 1986;Hair, Beswick, & Janda, 1988;Winkler et al., 1999;Frimand, & Jalkanen, 2002;Bégué et al., 2007;Bernstein, & Lynch, 2009;Shimanouchi, 1972) and with the results of the calculations carried out in this investigation.It is important to note that some modes with Ag or Au symmetries are observed in the IR and Raman spectra probably because the involved modes are not totally antisymmetric or symmetric.The wavenumbers, infrared and Raman intensities and potential energy distribution obtained for EOD by using the B3LYP/6-31G* method is shown in Table S2.The SQM force fields for EOD can be obtained at request.The discussion of the assignments of the most important groups is presented as follows.

Assignments CH2 modes
The 3080-2800 cm -1 region is characteristic of the CH2 stretching modes (Bertie & Othen, 2011;Cant & Armstead, 1975;Bertie & Jacobs, 1978;Mills, 1965;Lowe et al., 1986;Hair, Beswick, & Janda, 1988;Winkler et al., 1999;Frimand, & Jalkanen, 2002;Bégué et al., 2007;Bernstein, & Lynch, 2009;Brandán, 2021;Iramain et al., 2022;Issaoui et al., 2017;Mortada et al., 2022), for this, the bands in both spectra between 3079 and 3044 cm -1 are assigned to the antisymmetric stretching modes while the corresponding symmetric modes bands are associated with the IR and Raman bands located between 3019 and 2913 cm -1 .Here, as predicted by the calculation, the four expected CH2 bending modes for the dimer are associated with the observed bands in both spectra between 1500 and 1451 cm -1 while the wagging modes were predicted between 1163 and 1113 cm -1 and, for this reason, they are easily assigned to the IR bands between 1168 and 1134 cm -1 .Here, only rocking and twisting modes were assigned to the IR bands respectively at 1092 and 800 cm -1 , as indicated in Table 7.

Intermonomer modes
The inter-monomers vibrational modes of the dimer usually observed in the lower wavenumbers region (Brandán et al., 2010;Mortada et al., 2022) and they are related to restricted translations or rotations of one molecule against the other (Table S2).In the 4hydroxi benzoic acid these six vibration modes are predicted by B3LYP/6-311++G** calculations between 106 and 18 cm -1 (Brandán et al., 2010).Here, the six inter-monomer modes are predicted slightly coupled and they are observed in the Raman spectrum; for this reason, they are assigned as indicated in Table 7.
The SQM B3LYP/6-311++G** calculations have no predicted the expected vibration mode at 15 cm -1 , and, for this reason, with the aid of GaussView that mode was also assigned to torsion dimer mode (Nielsen, & Holder, 2008).
Here, clearly the presence of the dimer in the solid state at 100 k justifies the observed bands in the Raman spectra at low temperatures, as reported by Bertie & Jacobs (1978).

Force field
The force constants calculated at B3LYP/6-31G*, B3LYP/6-311++G** and WB97XD/6-311++G** levels of theory EOD were estimated by using the above mentioned scaling procedure by Pulay et al. (1983) and they can be seen in Table 8.The force constants expressed in terms of simple valence internal coordinates were calculated from the corresponding scaled force field by using the Molvib program (Sundius, 2002).The calculated f(νC-H), f(νC-C), f(νC-O) and f(δ(CH2)) force constants values were compared with those obtained by Mills (1965) for the monomer and with those obtained by Bichara et al (2011) for the citric acid dimer.In general, for EOD an increase in the size of the basis set diminishes the force constants values while higher values than the other ones are predict using the WB97XD method.The different calculations procedure probably justifies the observed variations in the force constants.In this case, the values were obtained by means of the SQM procedure while Nakanaga's values were obtained by the iteration method in which the set force constants of Hirokawa were used as the starting values (Mills, 1965).Also, the lower f(νC-C) and f(νC-O) force constants values of EOD, in reference to EOM, are attributed to the different interactions that present the former, while the cyclic structure in the ethylene oxide dimer justify the variations in its force constants in relation to the force constants of the citric acid dimer.

Conclusions
In this work, theoretical molecular structures for the ethylene oxide dimer were determined by using the B3LYP and WB97XD methods and the 6-31G* and 6-311++G** basis sets.Calculations including long-range corrections by using the WB97XD/6-311++G** calculations have revealed that: (i) optimized energy, molecular volume and frequencies are similar to calculated with the B3LYP/6-31G* method, (ii) lower correlations in the geometrical parameters, (iv) higher stabilization energy, (v) higher values in the topological parameters and, (vi Such interactions are assigned to intermonomers interactions predicted at lower wavenumbers, in accordance with the bands observed in the experimental Raman spectrum at low temperatures and with the experimental structure determined at 100 K. Hence, the presence of bands associated with that stable dimeric structure was detected in the IR and Raman spectra, and complete assignments of the vibrational modes for dimer was accomplished.

Figure
Figure 1.Theoretical Structures and Numbering of Atoms for Ethylene Oxide as: a) Monomer, b) Dimer Showing the Intramolecular H-Bonds with Dashed Lines and c) Crystal Packing of Ethylene Oxide Dimer Determined by X-Ray Diffraction Source: Grabowsky et al., 2008

Figure 2 .
Figure 2. Behaviours of Atomic Charges (MK, Mulliken, NPA) for Ethylene Oxide Dimer at Different Levels of Theory

Figure 3 .
Figure 3. Calculated Electrostatic Potential Surfaces on the Molecular Surfaces of Ethylene Oxide Dimer in Gas Phase at Different Levels of Theory.Isodensity Value of 0.005

Figure 4 .
Figure 4.The Five Critical Points of the Charge Density for Ethylene Oxide Dimer Included the O---O Interaction

Figure
Figure 6.Comparison between: a) Low-Frequency Raman Spectra of Oriented Crystals of Ethylene Oxide at 22 K, Using No Analyzer and Either a Polarization Rotator in the Laser Beam (x(y, x+z)y) or b) not (x(z, x+z)y).c) Predicted Infrared Spectrum of the Ethylene Oxide Dimer at B3LYP and, d) WB97XD/6-311++G** Level of Theory in 200-20 cm -1 Region Source: Bertie & Jacobs, 1978

Figure
Figure 7.Comparison between Predicted Raman Spectra of Ethylene Oxide by Using the B3LYP/6-311++G** and WB97XD/6-311++G** Levels of Theory ) higher scaled force constants than predicted for the B3LYP/6-31G* and B3LYP/6-311++G** methods.NBO and AIM studies with both methods predict five different types of interactions in the ethylene oxide dimer, four of type C-O•••H and one C-O•••O interaction.

Table 3 . Atomic Charges # (MK, Mulliken, NPA) for Ethylene Oxide Dimer at Different Levels of Theory
Note: # Atomic units