Effective Rotational Constants Research Articles

QUIJOTE discovery of the cation radicals HC5N+ and HC7N+

We present the discovery with the QUIJOTE line survey of the cations HC5N+ and HC7N+ in the direction of TMC-1. Seven lines with half-integer quantum numbers from J = 25/2–23/2 to 37/2–35/2 have been assigned to HC5N+ and eight lines from J = 55/2–53/2 to 71/2–69/2 to HC7N+. Both species have inverted 2Π ground electronic states with very good estimates for their B0 and ASO constants based on optical observations. The lines with the lowest J of HC5N+ exhibit multiple components due to the hyperfine structure introduced by the H and N nuclei. However, these different components collapse for the higher J. No hyperfine structure is found for any of the lines of HC7N+. The derived effective rotational and distortion constants for HC5N+ are Beff = 1336.662 ± 0.001 MHz and Deff = 27.4 ± 2.6 Hz, while for HC7N+ they are Beff = 567.85036 ± 0.00037 MHz and Deff = 4.01 ± 0.19 Hz. From the observed intensities, we derived Trot = 5.5 ± 0.5 K and N = (9.9 ± 1.0) × 1010 cm−2 for HC5N+, while we obtained Trot = 8.5 ± 0.5 K and N = (2.3 ± 0.2) × 1010 cm−2 for HC7N+. The HC5N/HC5N+, C5N/HC5N+, C5N−/HC5N+, HC7N/HC7N+, HC5N+/HC7N+, and C7N−/HC7N+ abundance ratios are 670 ± 80, 4.8 ± 0.8, 1.2 ± 0.2, 1000 ± 150, 4.2 ± 0.5, and 2.2 ± 0.2, respectively. We have run chemical modelling calculations to investigate the formation and destruction of these new cations. We find that these species are mainly formed through the reactions of H2 and the cations C5N+ and C7N+, and by the reactions of H+ with HC5N and HC7N, while they are mostly destroyed through a reaction with H2 and a dissociative recombination with electrons. Based on the underestimation of the abundances of HC5N+ and HC7N+ by the chemical model by a factor ∼20, we suggest that the rate coefficients currently assumed for the reactions of these cations with H2 could be too high by the same factor, something that will be worth investigating.

A simple model for high rotational excitations of molecules in a superfluid

Recently it became possible to study highly excited rotational states of molecules in superfluid helium through nonadiabatic alignment experiments (Cherepanov et al 2021 Phys. Rev. A 104 L061303). This calls for theoretical approaches that go beyond explaining renormalized values of molecular spectroscopic constants, which suffices when only the lowest few rotational states are involved. As the first step in this direction, here we present a basic quantum mechanical model describing highly excited rotational states of molecules in superfluid helium nanodroplets. We show that a linear molecule immersed in a superfluid can be seen as an effective symmetric top, similar to the rotational structure of radicals, such as OH or NO, but with the angular momentum of the superfluid playing the role of the electronic angular momentum in free molecules. The simple theory sheds light onto what happens when the rotational angular momentum of the molecule increases beyond the lowest excited states accessible by infrared spectroscopy. In addition, the model allows to estimate the effective rotational and centrifugal distortion constants for a broad range of species and to explain the crossover between light and heavy molecules in superfluid 4He in terms of the many-body wavefunction structure. Some of the above mentioned insights can be acquired by analyzing a simple 2 × 2 matrix.

Microscopic diffusion coefficients of dumbbell- and spherocylinder-shaped colloids and their application in simulations of crowded monolayers.

We explore the diffusion properties of colloidal particles with dumbbell and spherocylinder shapes using a hydrodynamic bead-shell approach and additional Brownian dynamics (BD) simulations. By applying the bead-shell method, we determine empirical formulas for the microscopic diffusion coefficients. A comparison of these formulas and established experimental and theoretical results shows remarkable agreement. For example, the maximum relative discrepancy found for dumbbells is less than 5%. As an application example of the empirical formulas, we perform two-dimensional (2D) BD simulations based on a single dumbbell or spherocylinder in a suspension of spheres and calculate the resulting effective long-time diffusion coefficients. The performed BD simulations can be compared to quasi-2D systems such as colloids confined at the interface of two fluids. We find that the effective diffusion coefficient of translation mostly depends on the sphere area fraction ϕ, while the effective diffusion coefficient of rotation is influenced by the aspect ratio and ϕ. Furthermore, the effective rotational diffusion constant seems to depend on the particle shape with the corresponding implementation of the interactions. In the resolution limit of our methods, the shape-dependent differences of the microscopic diffusion coefficients and the long-time diffusion constant of translation are negligible in the first approximation. The determined empirical formulas for the microscopic diffusion coefficients add to the knowledge of the diffusion of anisotropic particles, and they can be used in countless future studies.

Synchrotron-Based High Resolution Far-Infrared Spectroscopy of trans-Butadiene.

The high resolution far-infrared spectrum of trans-butadiene has been reinvestigated by Fourier-transform spectroscopy at two synchrotron radiation facilities, SOLEIL and the Canadian Light Source, at temperatures ranging from 50 to 340 K. Beyond the well-studied bands, two new fundamental bands lying below 1100 cm-1, ν10 and ν24, have been assigned using a combination of cross-correlation (ASAP software) and Loomis-Wood type (LWWa software) diagrams. While the ν24 analysis was rather straightforward, ν10 exhibits obvious signs of a strong perturbation, presumably owing to interaction with the dark ν9 + ν12 state. Effective rotational constants have been derived for both the v10 = 1 and v24 = 1 states. Since only one weak, infrared active fundamental band (ν23) of trans-butadiene remains to be observed at high resolution in the far-infrared, searches for the elusive gauche conformer can now be undertaken with considerably greater confidence in the dense ro-vibrational spectrum of the trans form.

Pure rotational spectrum of HCI

The 101–000 and 202–101 rotational transitions of a monohalomethylene, HCI, are observed in a discharge plasma of diiodomethane by Fourier-transform microwave (FTMW) and FTMW-microwave double-resonance spectroscopy. The effective rotational constant (B+C)/2, the centrifugal distortion constant ΔJ, and the quadrupole coupling constant of the iodine nucleus are determined from the observed line frequencies. The nuclear spin-rotation coupling constant Cbb is required to be included into the set of fitting parameters for a satisfactory analysis of the observed line frequencies.

Ab initio structure and vibration-rotation dynamics of the formyl and isoformyl cations, HCO+/HOC.

Accurate structure and potential energy surface of the formyl and isoformyl cation system, HCO+/HOC+, in its ground electronic state X̃ 1Σ+ have been determined from ab initio calculations using the coupled-cluster approach in conjunction with the correlation-consistent basis sets up to septuple-zeta quality. Both the isomers are confirmed to be linear at equilibrium, with the total energy minimum of HOC+ lying 14 120 cm-1 above that of HCO+ and the HCO+ → HOC+ isomerization energy barrier being 26 870 cm-1 (in the Born-Oppenheimer approximation). The equilibrium structural parameters for HCO+ are estimated to be re(HC) = 1.0919 Å and re(CO) = 1.1058 Å, whereas those for HOC+ are estimated to be re(HO) = 0.9899 Å and re(CO) = 1.1544 Å. The vibration-rotation energy levels were predicted for various isotopologues using a variational approach and compared with the experimental data. For the spectroscopically well characterized formyl cation, the observed vibration-rotation energies and the effective rotational constants are reproduced to within about 2.3 cm-1 and 1.7 MHz, respectively. The role of the core-electron correlation, higher-order valence-electron correlation, scalar relativistic, and adiabatic effects in determining the structure and vibration-rotation dynamics of both the isomers is discussed.

Isotopologue consistency of semi-empirically computed infrared line lists and further improvement for rare isotopologues: CO2 and SO2 case studies

The semi-empirical molecular rovibrational IR line lists, such as ExoMol, TheoReTs, and Ames, combine the experimental accuracy and theoretical power to reach better than 0.1 cm−1 accuracy for line positions and better than 80–90% agreement for line intensities. The quality of these existing semi-empirical IR lists allows further improvements of intensity and line positions for those unobserved minor isotopologues. This paper presents our new BTRHE (Best Theory + Reliable High-resolution Experiment) strategy implementation. For line intensity, the isotopologue consistency and the patterns of mass dependence in the Ames-296 K SO2 and CO2 IR lists are quantitatively presented along the mass-inverse coordinates. The consistency and patterns are better than those in existing experimental data. The methodology proposed here can be used to identify inconsistencies, outliers, and mistakes in intensities, and help improve Effective Dipole Model (EDM) and molecular IR databases. We call for an experimental study on the 50006 and 60007 bands of CO2 628. For line position predictions, a simple approach combining the variational IR line lists with Effective Hamiltonian (EH) model may refine the effective rotational constants A0/B0/C0 and quartic centrifugal distortion constants of minor isotopologues. The prediction accuracy may be improved by two orders of magnitude, i.e. reaching 0–5 MHz prediction accuracy in the range of J < 20–30, Ka < 10–20, and 0.01–0.02 MHz accuracy for A0/B0/C0. Several important factors have been systematically investigated and discussed, e.g. convergence, uncertainties, higher order terms, fixing EH parameters, mass coordinates, etc. A microwave (MW) line set consisting of 644,636 strong transitions (at 296K) for all 30 isotopologues and corresponding refined EH(Ames) parameters are reported in the supplementary material. This approach may be easily extended to rovibrational bands, hot bands, and other molecular systems.

Calculation of the vibration–rotational transition intensities of water molecules trapped in an argon matrix: stretching O–H vibrations spectral region

ABSTRACTThe frequencies and intensities of vibration–rotational transitions of water molecules in an argon matrix were calculated for temperatures of 6 and 30 K. The rigid asymmetric top approximation was used with available literature values of the effective rotational constants in the ground and excited vibrational states. The calculations were carried out by taking into account the existence of a non-equilibrium population distribution between the rotational levels of ortho- and para-water isomers. It was assumed that the temperature relaxation of the population of rotational levels is independent of the ortho- and para-isomers. Comparison of the results of the theoretical calculations with experimental literature data shows good agreement for the majority of the rotational structure lines for symmetric and antisymmetric stretching vibrations both in the frequency values and in the values of the relative intensities.

Strongly aligned molecules inside helium droplets in the near-adiabatic regime.

Iodine (I2) molecules embedded in He nanodroplets are aligned by a 160 ps long laser pulse. The highest degree of alignment, occurring at the peak of the pulse and quantified by ⟨cos2𝜃2D⟩, is measured as a function of the laser intensity. The results are well described by ⟨cos2𝜃2D⟩ calculated for a gas of isolated molecules each with an effective rotational constant of 0.6 times the gas-phase value and at a temperature of 0.4 K. Theoretical analysis using the angulon quasiparticle to describe rotating molecules in superfluid helium rationalizes why the alignment mechanism is similar to that of isolated molecules with an effective rotational constant. A major advantage of molecules in He droplets is that their 0.4 K temperature leads to stronger alignment than what can generally be achieved for gas phase molecules-here demonstrated by a direct comparison of the droplet results to measurements on a ∼1 K supersonic beam of isolated molecules. This point is further illustrated for a more complex system by measurements on 1,4-diiodobenzene and 1,4-dibromobenzene. For all three molecular species studied, the highest values of ⟨cos2𝜃2D⟩ achieved in He droplets exceed 0.96.

Microwave spectra of the SiH4-H2O complex: A new sort of intermolecular interaction

Microwave spectral patterns observed for the silane-water complex were found much different from those of the methane-water complex. The SiH4-H2O complex is likely to have a tightly bound structure. The effective rotational and centrifugal distortion constants: B = 3621.1193 (45) MHz and DJ = 49.84 (30) kHz led to the distance between the Si and O atoms in the complex to be 3.3 Å, much shorter than the C and O separation in the CH4-H2O of 3.7 Å, and to the silane-water stretching force constant and stretching frequency to be 2.88 N/m and 65 cm−1, respectively, which are to be compared with 1.52 N/m and 55 cm−1 of the CH4-H2O. The characteristic features of the spectra observed for the main species 28SiH4-H2O are common to those of isotopic species: 29SiH4-H2O, 30SiH4-H2O, 28SiH4-H218O, 28SiH4-D2O, 29SiH4-D2O, 30SiH4-D2O, 28SiH4-HDO, 29SiH4-HDO, 30SiH4-HDO, 28SiD4-H2O, 28SiD4-D2O, and 28SiD4-HDO. The observed spectra also indicate that the silane executes a threefold internal rotation about one of its four Si–H bonds, while the C2 symmetry axis of the water is bent away from the internal-rotation axis. An internal axis method analysis yielded an estimate of the internal-rotation potential barrier V3 to be 140 ± 50 cm−1, and those based on diagonalization of a principal axis method Hamiltonian matrix and on the extended internal axis method resulted in V3 ranging from 180 to 100 cm−1, depending on the isotopic species studied. All the measurements were done by using a pulsed nozzle Fourier transform microwave spectrometer, and the spectral assignments were made with the aid of the Stark effect, which yielded the dipole moment to be 1.730 (10) D. Transitions in higher energy states of the SiH4 internal rotation were observed, clearly resolved from the main lines, when the carrier gas was replaced from Ar to Ne.

Lamb-dip spectroscopy of the C-N stretching band of methylamine by using frequency-tunable microwave sidebands of CO2 laser lines.

Lamb-dip spectroscopy of the C−N stretching band of methylamine has been systematically extended to P-, Q-, and R-branch by using microwave sidebands of a large number of CO2 laser lines as frequency-tunable infrared sources in a sub-Doppler spectrometer. Lamb-dip signals of more than 150 spectral lines have been observed with a resolution of 0.4 MHz and their frequencies have been precisely measured with an accuracy of ±0.1 MHz. More than 30 closed combination loops have been formed, which unambiguously confirm the assignments. For over 150 vibrational excited levels in 27 substates, refined term values have been determined and expanded in J(J + 1) power-series to determine the substate origins and the effective rotational constants. For transitions with Aa torsion-inversion symmetry in torsional state υt = 0, 57 K-doublet lines displaying asymmetry splittings have been observed and the splitting constants for levels with K = 1, 2, and 3 in the excited states have been determined. Our results provide accurate experimental information for spectroscopic studies of the interesting vibrational perturbations and intermode interactions related to the C−N stretching mode, directly support astronomical surveys, and are very relevant in practice to identification and frequency determination of the CO2-laser-pumped far-infrared laser lines of methylamine.

Microwave and Quantum Chemical Study of Intramolecular Hydrogen Bonding in 2-Propynylhydrazine (HC≡CCH2NHNH2).

The microwave spectrum of 2-propynylhydrazine (HC≡CCH2NHNH2) was investigated in the 23-124 GHz spectral interval. The spectra of two conformers denoted I and II were assigned. I is the lower-energy form, and relative intensity measurements yielded an internal energy difference of 3.0(4) kJ/mol between I and II. The spectra of the ground and five vibrationally excited states were assigned for I, whereas only the spectrum of the ground vibrational state was assigned for II. Both I and II are each stabilized simultaneously by two intramolecular hydrogen bonds. The first of these hydrogen bonds is formed between the hydrogen atom of the -NH- part of the hydrazino group, and the second internal hydrogen bond is formed between one of the hydrogen atoms of the -NH2 part. The π-electrons of the triple bond is thus shared by these two hydrogen atoms. The shortest contact between a hydrogen atom of the hydrazino group and the π-electrons of the ethynyl group is found in lower-energy conformer I. The conformational properties of 2-propynylhydrazine were explored by MP2/cc-pVTZ and CCSD/cc-pVQZ calculations. The CCSD method predicts that seven rotameric forms exist for this compound. Five of these rotamers are stabilized by internal hydrogen bonding. The simultaneous sharing of the π-electrons of the triple bond by two hydrogen atoms occurs only in Conformers I and II, which are predicted to be the two forms with the lowest energies, with I 2.52 kJ/mol lower in energy than II. The effective rotational constants of the ground vibrational states of I and II were predicted by a combination of MP2 and CCSD calculations, whereas centrifugal distortion constants and vibration-rotation constants were calculated by the MP2 method. The theoretical spectroscopic constants are compared with the experimental counterparts. It is concluded that more refined calculations are necessary to obtain complete agreement.

Temperature dependence of the intensity of the vibration-rotational absorption band ν2 of H2O trapped in an argon matrix

Using two sets of effective rotational constants for the ground (000) and the excited bending (010) vibrational states the calculation of frequencies and intensities of vibration-rotational transitions for J″=0−2; and J′=0−3; was carried out in frame of the model of a rigid asymmetric top for temperatures from 0 to 40K. The calculation of the intensities of vibration-rotational absorption bands of H2O in an Ar matrix was carried out both for thermodynamic equilibrium and for the case of non-equilibrium population of para- and ortho-states. For the analysis of possible interaction of vibration-rotational and translational motions of a water molecule in an Ar matrix by 3D Schrödinger equation solving using discrete variable representation (DVR) method, calculations of translational frequencies of Н2О in a cage formed after one argon atom deleting were carried out. The results of theoretical calculations were compared to experimental data taken from literature.

Superfluidity of helium-4 around a Mg11 cluster

We present results for 4He solvation of a Mg11 cluster obtained with path integral Monte Carlo simulations. Using a pair-wise additive model for the helium-cluster interaction, we calculate energies, 4He densities and the superfluid response of 4He to a rotation of the Mg11 cluster which has symmetric top symmetry. We find a significant superfluid fraction in the first 4He solvation layer around both axes of the cluster well before 4He covers the whole cluster. Based on the superfluid fraction, we estimate that Mg11 in a large 4He nanodroplet would have an effective rotational constant between 60% and 70% of the gas phase rotational constant for rotation about the axis with larger moment of inertia. The dependence of the effective rotational constants on 4He cluster size N is similar to that of previously studied molecules with much larger gas phase rotational constants, with an initial decrease and a turn-around (at N = 13) when part of the 4He density becomes superfluid.

The formic acid-nitric acid complex: microwave spectrum, structure, and proton transfer.

Rotational spectra are reported for seven isotopologues of the complex HCOOH-HNO3 in a supersonic jet. The system is planar and bound by a pair of hydrogen bonds, much like the more widely studied carboxylic acid dimers. Double proton exchange interconverts the system between a pair of equivalent structures, as revealed by a splitting of the a-type spectrum that disappears when one of the hydrogen bonding protons is replaced by deuterium. The observation of relative intensities that are consistent with nuclear spin statistics in a symmetric and antisymmetric pair of tunneling states provides additional evidence for such a motion. The observed splittings in the pure rotational spectrum are 1-2 orders of magnitude smaller than those recently reported in the pure rotational spectra of several related carboxylic acid dimers. This is a curious difference, although we note that because the observed spectra do not cross the tunneling doublet, the splittings are a measure of the difference in effective rotational constants for the two states, not the tunneling frequency itself. The observed rotational constants have been used to determine an accurate vibrationally averaged structure for the complex. The two hydrogen bond lengths, 1.686(17) Å and 1.813(10) Å for the hydrogen bonds involving the HNO3 and HCOOH protons, respectively, differ by 0.127(27) Å. Likewise, the associated oxygen-oxygen distances determined for the parent species, 2.631 and 2.794 Å, differ by 0.163 Å. These results suggest that the double proton transfer is necessarily accompanied by substantial motion of the heavy atom frame, and thus this system, in principle, provides an excellent prototype for multidimensional tunneling processes. Ab initio calculations of the binding energy and the barrier height are presented. Excellent agreement between the calculated equilibrium structure and the experimental, vibrationally averaged structure suggests that the vibrational wave function is not highly delocalized in the region between the equivalent potential wells. (14)N nuclear quadrupole hyperfine structure is interpreted in terms of the degree to which the HNO3 releases its proton in either of the equivalent potential energy minima.

A benchmark study of molecular structure by experimental and theoretical methods: Equilibrium structure of thymine from microwave rotational constants and coupled-cluster computations

Accurate equilibrium, re, structures of thymine have been determined using two different, and to some extent complementary techniques. The composite ab initio Born–Oppenheimer, re(best ab initio), structural parameters are obtained from the all-electron CCSD(T) and MP2 geometry optimizations using Gaussian basis sets up to quadruple-zeta quality. The semi-experimental mixed estimation method, where internal coordinates are fitted concurrently to equilibrium rotational constants and geometry parameters obtained from a high level of electronic structure theory. The equilibrium rotational constants are derived from experimental effective ground-state rotational constants and rovibrational corrections based on a quantum-chemical cubic force field. Equilibrium molecular structures accurate to 0.002Å and 0.2° have been determined. This work is one of a few accurate equilibrium structure determinations for large molecules. The poor behavior of Kraitchman’s equations is discussed.

Accurate Determination of the Deformation of the Benzene Ring upon Substitution: Equilibrium Structures of Benzonitrile and Phenylacetylene

Accurate equilibrium, re, structures of the monosubstituted benzene molecules benzonitrile, C6H5CN, and phenylacetylene, C6H5CCH, have been determined using two different, to some extent complementary techniques. The semiexperimental, r(e)(SE), structural parameters are the result of a least-squares fit to equilibrium rotational constants derived from experimental effective ground-state rotational constants and rovibrational corrections based principally on an ab initio cubic force field. The composite ab initio Born-Oppenheimer, r(e)(BO), structural parameters are obtained from frozen-core and all-electron MP2 and the CCSD(T) geometry optimizations using Gaussian basis sets up to quintuple-zeta quality. The DFT(B3LYP) method, with two different Gaussian basis sets, 6-31G* and 6-311+G(3df,2pd), was used to calculate the cubic force field employed during the r(e)(SE) structure determination. With the 6-31G* basis set, the error of the rovibrational correction is to a large extent random, whereas with the 6-311+G(3df,2pd) basis set it is mainly systematic. As shown here, systematic errors do not have a significant effect on the accuracy of the derived structure; the quality of the structural fit, however, is sensitive to the true accuracy of the ground-state rotational constants. An even more important general conclusion of this study is that the addition of extra rotational constants from multisubstituted species does not seem to improve the accuracy of the r(e)(SE) structures, quite in contrast to the highly desirable availability of data corresponding to all singly substituted species.

Rotational spectroscopic study of hydrogen cyanide embedded in small 4He clusters

High resolution microwave spectra of the a-type, J = 1-0, transitions of He(N = 1-6)-H(12)C(14)N, He(N = 1-6)-H(13)C(14)N, He(N = 1-6)-H(12)C(15)N, He(N = 1-7)-D(12)C(14)N, and He(N = 1-6)-D(13)C(14)N clusters produced in a supersonic jet expansion were measured and analyzed. The resulting effective rotational constants, B(eff), initially decrease with the number of the attached helium atoms before reaching a minimum at N = 3 helium atoms for all isotopologues. The subsequent increase in B(eff) for N ≥ 4 is indicative of the onset of microscopic superfluidity. Comparison of our experimental B(eff) constants with those from quantum Monte Carlo simulations [A. A. Mikosz, J. A. Ramilowski, and D. Farrelly, J. Chem. Phys. 125, 014312 (2006)] reveals a nearly congruent trend in B(eff) for N up to 6. Analysis of the hyperfine structure of the (14)N containing isotopologues yielded a gradual incremental increase in the magnitude of χ(aa) and <P(2)(cos θ)> for N = 1-6, which suggests the internal rotation of the HCN molecule is becoming increasingly hindered.

Observation of the linear C2H2–N2 van der Waals complex in the 2CH range using CW-CRDS

CW-cavity ring down spectroscopy was used to record the spectrum of 12C 2H 2-N 2 in a free jet expansion around the 2CH acetylene band. Twenty lines were assigned thanks to combination differences, which were combined with literature values leading to the retrieval of ground state rotational constants consistent for a broader set of J″. Effective rotational constants, the band origin and the estimation of the lifetime of the perturbed upper state are determined. © 2012 Elsevier B.V. All rights reserved.

Conformational and vibrational studies of isomeric hydrogen cyanide tetramers by quantum chemical methods

The results of structural studies and detailed harmonic and anharmonic vibrational analysis on two hydrogen cyanide (HCN) tetramers diaminomaleonitrile (DAMN) and diaminofumaronitrile (DAFN), which are important molecules for understanding the chemistry of interstellar space and nitrile rich environments, are being reported on the basis of density functional theory using second-order perturbation theory. Both the molecules are found to have C1 symmetry. While all the heavy atoms of DAMN lie in the same plane (maximum deviation 6°), the two nitrogen atoms in DAFN are out of plane by about 15°. The two amino groups are tetrahedral and do not have significant bond angle anisotropy. Detailed conformational studies are reported on the two molecules and their possible rotational isomers are identified. Complete vibrational analysis based on harmonic and anharmonic frequencies, intensity of infrared and activity of Raman bands and potential energy distribution over the internal coordinates has been provided for the two molecules. Affect of hydrogen bonding on molecular geometry and frequencies of the NH stretch modes has been studied by calculations on the dimers of the two molecules. A close agreement has been observed between the experimental and calculated frequencies. Vibrational–rotational constants such as rotational constants in the ground vibrational state (A0, B0, C0) and the effective rotational constants (Ae, Be and Ce), including terms due to quartic centrifugal distortion constants, rotation–vibration coupling constants, Wilson and Nielsen's centrifugal distortion constants have been calculated using B3LYP and B97-1 functionals and 6-31G**, 6-311+G** and TZVP basis sets.