Large Centrifugal Distortion Research Articles

Solution of the Rovibrational Schrödinger Equation of a Molecule Using the Volterra Integral Equation

By using the Rayleigh-Schrödinger perturbation theory the rovibrational wave function is expanded in terms of the series of functions ϕ0,ϕ1,ϕ2,…ϕn, where ϕ0 is the pure vibrational wave function and ϕι are the rotational harmonics. By replacing the Schrödinger differential equation by the Volterra integral equation the two canonical functions α0 and β0 are well defined for a given potential function. These functions allow the determination of (i) the values of the functions ϕι at any points; (ii) the eigenvalues of the eigenvalue equations of the functions ϕ0,ϕ1,ϕ2,…ϕn which are, respectively, the vibrational energy Ev, the rotational constant Bv, and the large order centrifugal distortion constants Dv,Hv,Lv….. Based on these canonical functions and in the Born-Oppenheimer approximation these constants can be obtained with accurate estimates for the low and high excited electronic state and for any values of the vibrational and rotational quantum numbers v and J even near dissociation. As application, the calculations have been done for the potential energy curves: Morse, Lenard Jones, Reidberg-Klein-Rees (RKR), ab initio, Simon-Parr-Finlin, Kratzer, and Dunhum with a variable step for the empirical potentials. A program is available for these calculations free of charge with the corresponding author.

Strong-field impulsive alignment in the presence of high temperatures and large centrifugal distortion

We achieve impulsive alignment in a $400\phantom{\rule{0.3em}{0ex}}\mathrm{K}$ sample of diatomic iodine. This alignment results in an ensemble of coherent rotational wave packets with revival structures at 220 and $440\phantom{\rule{0.3em}{0ex}}\mathrm{ps}$. The revivals contain many more oscillations than are typically seen in rotational wave packets. The high-temperature sample contains a very large distribution in $J$, with strong centrifugal distortions. We find that the high initial temperature, rather than the strength of the laser impulse, gives rise to the centrifugally distorted revival oscillations. We present polarization measurements of this rotational wave packet together with semiclassical simulations showing the importance of the thermal distribution of population in both rotational and vibrational states. Our results show that excitation of coherent rotations is possible even in a system where the thermal energy is large compared to the energy exchanged with the laser field.

High resolution spectroscopy of H2 12C16O in the 1.9 to 2.56 µm spectral range

Infrared spectra of H2CO covering the 1.9 µm–2.5 µm spectral domain have been recorded at high resolution (0.005 cm−1) using Fourier transform spectroscopy leading to the observation and analysis of the ν1 + ν6, ν2 + ν4 + ν6, 2ν3 + ν6, ν3 + ν5, ν1 + ν2, ν2 + ν5, 2ν2 + ν6 and 3ν2 bands. The line frequencies were calculated using effective (empirical) Hamiltonian models, which account for the main Coriolis and vibrational interactions. Using an interactive scheme it was possible to least-squares fit the observed energy levels to within a few thousandths of a cm−1. The observed–calculated differences do not match the spectral precision (∼0.0008 cm−1), but, given the congestion in the spectrum resulting from the density of the vibrational states as well as the large centrifugal distortion and Coriolis and anharmonic coupling effects, we believe that a reasonable agreement was obtained. From the fittings the following band centers were derived where the expanded uncertainties are one standard deviation (i.e. k = 1). Finally a number of line intensities were measured having uncertainties of about 25%, which permits the determination of the main terms appearing in the expansion of the transition moments. A comprehensive list of line frequencies and intensities has been generated.

Rotational spectrum and structure of Si3.

The rotational spectrum of a pure silicon cluster, the Si3 trimer, has been observed for the first time. From the rotational constants of the normal and the 29Si and 30Si isotopic species, a precise geometrical structure has been derived: the trimer is an isosceles triangle with a bond to the apex Si of length 2.177(1) A and an apex angle of 78.10(3) degrees. The substantial inertial defect and fairly large centrifugal distortion suggest that the molecule possesses a shallow bending potential. Si3 is a good candidate for astronomical detection because radio lines of comparably massive silicon molecules (e.g., SiC2, SiC4, and SiS) are readily observed in at least one astronomical source. The rotational spectra of Si6, Si9, and even larger polar silicon clusters may be detectable with the present technique, as well as similar germanium clusters.

Rotational spectrum and theoretical structure of the carbene HC4N

Following a high-level coupled cluster calculation, the rotational spectrum of the bent HC4N singlet carbene was detected in a supersonic molecular beam by Fourier transform microwave spectroscopy. The three rotational constants, the leading centrifugal distortion constants, and two nitrogen hyperfine coupling constants were determined to high accuracy. The rotational constants agree with those calculated ab initio to better than 0.5%. Like the isoelectronic C5H2 carbene of similar structure, HC4N was found to have fairly large centrifugal distortion and a large inertial defect. The calculated dipole moment of HC4N is 2.95 D.

FOURIER-TRANSFORM ABSORPTION SPECTRUM OF THE H 217O MOLECULE IN THE 9711–11 335 cm -1 SPECTRAL REGION: THE FIRST DECADE OF RESONATING STATES

Fourier transform spectra of 18O-enriched, 17O-enriched, and natural water vapor recorded between 9600 and 11 400 cm -1 have been analyzed to assign the H 2 17O spectral lines. More than 1000 transitions were finally assigned to the H 2 17O isotopic species leading to 420 precise experimental energy levels of the (0 0 3), (2 0 1), (1 2 1), (1 0 2), (3 0 0), (2 2 0) vibrational states. Rotational, centrifugal distortion, and resonance coupling parameters have been derived from the fit of the experimental energy levels to an effective Hamiltonian based on Padé–Borel approximants well suited to describe the large centrifugal distortion in H 2O. The resulting rms deviation is 0.013 cm -1 with 97 varied parameters.

Interaction of benzene and halogens in the gas-phase: rotational spectrum of C6H6···ClF

Ground-state rotational spectra of three isotopomers, C6H6···35ClF, C6H6···37ClF and C6D6···35ClF, of a complex formed by benzene with chlorine monofluoride have been observed by pulsed-nozzle, Fourier-transform microwave spectroscopy. Chemical reaction between the two components was precluded by use of a fast-mixing nozzle. A vibrational satellite associated with a low lying state (v=1) was also observed for the C6H6···35ClF and C6D6···35ClF isotopomers. The spectra were established to be of the symmetric-top type, but with large centrifugal distortion effects associated both with the frequencies of the unperturbed line centres and with the Cl nuclear quadrupole hyperfine structure. In the ground state, these effects were reflected in a large centrifugal distortion constant DJK, in higher order distortion constants of significant magnitude, and in a strong dependence of the Cl nuclear quadrupole coupling constant on the quantum number K. In the v=1 state, the constant DJK and the centrifugal distortion terms χK and χD that affect the nuclear quadrupole hyperfine structure were a similar magnitude to those of the v=0 state, but of opposite sign. The nuclear quadrupole coupling constants, χR, appropriate to the J=0, K=0 states, and the changes in the rotational constants B0 with isotopic substitution, were interpreted in terms of a model for the complex in which, at equilibrium, the ClF axis is inclined at an angle ϑ≈14° to the C6 axis of benzene, with the two axes intersecting at the ClF centre of mass. The Cl atom lies closer to the benzene ring than does the F atom. The extrapolation of the ClF internuclear axis intersects the plane of the benzene ring at a point (*) that lies at a distance of ca. 0.24 Ainside the ring from the centre of a C–C bond. The conformation with ϑ=0° (ClF axis and benzene C6 axis coincident) corresponds to a potential energy maximum, concentric with which is an approximately circular valley corresponding to ϑ≈14°. This type of potential energy surface is consistent with the symmetric-top nature of the observed spectra, with the relatively small energy separation of the v=0 and v=1 states, and with large, but oppositely signed, centrifugal distortion effects in the two states originating in a Coriolis interaction between them. The distance r(*···Cl)=3.313 Awas also determined.

Microwave spectrum of silylidene H2CSi

Rotational spectral lines of the H2CSi molecule are observed in the millimeter-wave and submillimeter-wave regions. The molecule is produced in a glow-discharge plasma of a gaseous mixture of SiH4 and CO. Rotational constants and centrifugal distortion constants have precisely been determined from observed frequencies of 32 a-type transitions with J=5–4 to J=12–11 and Ka=0 to Ka=4. From the observed inertial defect, the vibrational frequency of the CH2 rocking mode is estimated to be as low as 331 cm−1, which is consistent with the large quartic centrifugal distortion constants, DJK and d2. Higher-order centrifugal distortion constants up to the octic terms are necessary to obtain a good fit between the observed and calculated frequencies within experimental uncertainties. The low vibrational frequency and the necessity of the higher-order centrifugal distortion terms indicate a floppy nature of the CH2 rocking mode. A preliminary radioastronomical search for H2CSi in space has been carried out toward a few astronomical sources without success.

Rotational spectra and structures of the C6H6–HCN dimer and Ar3–HCN tetramer

A comparative study has been made of the rotational properties of C6H6–HCN and Ar3–HCN, observed with the Balle/Flygare pulsed beam, Fourier transform microwave spectrometer. C6H6–HCN is found to be a prolate symmetric top and Ar3–HCN an oblate one, both with the H in the middle. The rotational constants B0, DJ, and DJK of the parent species are 1219.9108(4) MHz, 1.12(3) kHz, and 18.32(8) kHz for C6H6–HCN, and 886.4878(1) MHz, 10.374(2) kHz, and 173.16(1) kHz for Ar3–HCN. Rotational constants are reported for the isotopic species C6H6–H13CN, -HC15N, and 13CC5H6–HC15N, and for Ar3–HC15N and -DCN. Analysis of the 14N hyperfine interaction χ finds its projection on the figure axis to be −4.223(4) MHz in C6H6–HCN and −1.143(2) in Ar3–HCN. They correspond to average projection angles θ between the HCN and figure axes of 15.2° and 45.3°, respectively. A pseudodiatomic analysis of the rotational constants gives the c.m. to c.m. distance to be 3.96 Å in C6H6–HCN and 3.47 Å in Ar3–HCN. While the rotational properties of C6H6–HCN are ‘‘normal,’’ those of Ar3–HCN display a long list of ‘‘abnormalities.’’ They include a J-dependent χ(14N) similar to that of Ar–HCN; a very large projection angle θ; large centrifugal distortion including higher-order terms in HJ and HJK; splitting of the K=3 transitions into J-dependent doublets; and the ready observation of an excited vibrational state. These behavioral differences are related qualitatively to the interaction surfaces for the two clusters, calculated with the molecular mechanics for clusters (MMC) model, and discussed. The potential minimum for C6H6–HCN is smooth, circular, steep except for a flat bottom, and deep (1762 cm−1). That for Ar3–HCN is tricuspid, with large gullies, and shallow (507 cm−1). In addition to the dispersion forces, the dominant interaction forming C6H6–HCN is between the benzene quadrupole moment and the HCN dipole moment, a strong 4-2 potential. That in Ar3–HCN is polarization of the spherical Ar by the HCN dipole and quadrupole moments, a weak 0–2,4 potential.

Microwave spectrum and rotational isomerism of gaseous nitrosoethane, CH3CH2NO

The microwave spectrum of gaseous nitrosoethane has been measured in the 7–40 GHz region. The monomeric vapour is shown to exist as two rotational isomers, one the cis form with a planar heavy-atom structure, the other a staggered gauche conformer. The microwave spectrum of gauche-nitrosoethane shows large centrifugal distortion shifts and additional splitting arising from tunnelling through the trans barrier. Relative intensity measurements show the cis to be more stable than the gauche form by 2.1 ± 0.4 kJ mol–1. The dipole moments were determined through the Stark effect: µa= 2.316 (2), µb= 0.623 (4) and µt= 2.398 (2) D (1 D ≈ 3.33564 × 10–30 C m) for the 15N cis, and µa= 2.288 (4), µb= 0.814 (5), µc= 0.460 (9) and µt= 2.471 (4) D for the 15N gauche form. Nitrogen-14 quadrupole coupling constants have been determined to be χaa=–2.525 (56) and (χbb–χcc)=–7.311 (97) MHz for the cis, and (χbb–χcc)= 3.518 (40) MHz for the gauche conformer.Molecular structures have been calculated for both conformers on the basis of the oxygen-18, nitrogen-15 and main species data. The ∠ CCN angle opens up by 8° on going from gauche- to cis-nitrosoethane. The barrier hindering internal rotation of the methyl group is the same for both conformers within experimental error, V3= 10.9 ± 0.3 kJ mol–1 for the cis form and 10.9 ± 0.3 kJ mol–1 for the gauche conformer.

Model calculations on the ground vibrational state of Ar–HCN

The Ar–HCN complex exhibits unusually wide amplitude bending along with large isotope and centrifugal distortion effects. A model in which Ar–HCN tunnels between linear and T-shaped configurations is able to quantitatively account for most of these experimental observations. A parametrized model potential is used and solutions are obtained variationally using a form for the wave function which is arbitrary in the bending angle but Gaussian in the stretching coordinate. The accuracy of the variational solution is checked against results from a two dimensional numerical relaxation procedure. In the final potential, the center of mass separation is about 4.62 Å in the linear, global minimum and 0.86 Å less at the T-shaped configuration. The potential is such that, in the ground vibrational state, the T-shaped region is not classically allowed but the wave function penetrates significantly into this region. This tunneling is the origin of the large isotope effects. The combination of the tunneling with contraction in bond length between the linear and T-shaped structures leads to the large centrifugal effects.

J dependence of χa(14N) for the Ar–HCN dimer: Coupling of stretching and bending in the potential function

The previously reported rotational spectrum of Ar–HCN [J. Chem. Phys. 81, 4922 (1984)] has shown the weakly bound dimer to be highly nonrigid. Superficially linear, the dimer has several anomalies, including large centrifugal distortion and an unexpectedly large bending amplitude of the HCN. We here describe high-resolution rotational spectra which identify another anomaly. The 14N hyperfine interaction constant of the dimer increases linearly with J(J+1) for Ar–HCN, 36Ar–HCN, and Ar–DCN, indicating a decrease in the average HCN bending amplitude (θ). For Ar–HCN this is from 30.97° for J=0 to 30.17° for J=5. At the same time, there is an increase in the average Ar to HCN c.m. separation R from 4.3433 to 4.3496 A˚. The cause of this behavior and of the other anomalies is found to be the shape of the potential function as calculated with a largely classical electrical model employing low-order moments and multipole polarizabilities. The calculated potential surface exhibits strong coupling between radial and angular motions, with smaller angles favored at larger R. There is an axial hump in the potential curve generated by rotating the HCN through the linear structure, and the height of the hump is sensitive to R. In contrast, a hump is not found in the equilibrium region of the potential surface for Ar–HF. The difference between the two surfaces arises from the combination of their similar electrical interactions with different hard-wall effects. Basically, the differences in the shapes of HF (nearly spherical) and HCN (cigar shaped) give interaction surfaces that cause the properties of Ar–HCN to contrast with those of Ar–HF. An important part of the constrast is the fact that the hydrogen halides are unusual in being more spherical than most other highly polar molecules.

Direct IR laser absorption spectroscopy of jet-cooled CO2HF complexes: Analysis of the ν1 HF stretch and a surprisingly low frequency ν6 intermolecular CO2 bend

High sensitivity, tunable laser direct absorption methods are exploited to obtain high resolution IR spectra (Δν≲0.001 cm−1) of weakly bound CO2HF complexes in a pulsed supersonic slit jet expansion. Transitions from the ground vibrational state corresponding to a single quantum excitation of the ν1 HF stretch are observed and analyzed with a semirigid linear molecule Hamiltonian. The observed increase in both B (+1.75%) and D (+55%) upon ν1 excitation is inconsistent with the commonly used diatomic approximation, and is not possible to rationalize for a nearly linear upper state geometry with small amplitude zero point motion of the intermolecular CO2 bend coordinate. We consider an alternative centrifugal straightening mechanism which predicts large centrifugal distortion effects due to end over end rotation of a complex with a nonlinear vibrationally averaged geometry in a weak bending potential. In support of this interpretation, hot band spectra are observed arising from bend excited complexes significantly populated in the 16 K expansion; intensity based estimates of the internal excitation indicate a surprisingly low bend frequency of 10±5 cm−1. A preliminary analysis of the spectra as l doubling in a Π←Π vibrational hot band for a linear equilibrium geometry is presented. An alternative interpretation of the spectrum as asymmetry doubling of a K=1←1 rotational hot band for a bent geometry is also considered. This latter interpretation is more consistent with the data and predicts a CO2 bend angle in the complex for K=1 between 25° and 30°. Linewidths for the upper vibrational states in CO2HF exhibit homogeneous broadening 3–4 times in excess of the apparatus resolution. Voigt analysis of the absorption line shapes indicates linewidths (FWHM) of 136±16 MHz and independent of J state; this corresponds to a relaxation lifetime of 1.1±0.2 ns for HF in the complex.

The molecular beam spectrum and the structure of the hydrogen fluoride dimer

The molecular beam spectrum of the hydrogen fluoride dimer has been investigated in detail. Microwave and radiofrequency transitions of (HF)2, HFDF, and (DF)2 are reported for both the Ka =0 and Ka =1 rotational levels. The results provide information on the structure of the dimers and on the nature of the intermolecular potential energy surface. The average F–F distance is shown to be approximately 2.78 Å but with a probably significantly shorter equilibrium distance. The nonhydrogen bonded hydrogen atom is bent 63°±6° from the F–F axis. In addition, analysis of the perpendicular electric dipole moment indicates a nonlinear hydrogen bond. The large centrifugal distortion effects and the unusual quantum mechanical tunneling provide a crude model for the potential surface associated with the hydrogen bond; the effective bond stretching force constant is (1.36±0.03)×104 dyn cm−1.

Microwave and radiofrequency Stark spectrum of ArHCN: A highly nonrigid molecule

The rotational spectrum of the weakly bound complex ArHCN has been observed using molecular beam electric resonance spectroscopy. The spectrum is superficially characteristic of that of a linear molecule with both unusually large centrifugal distortion (requiring a J6 dependent distortion term to fit the data) and an unexpectedly large bending amplitude. The spectroscopic constants are The centrifugal distortion constant DJ is remarkably large and abnormally sensitive to isotopic substitution. Using the usual model, the stretching and bending force constants obtained from these data are an order of magnitude smaller than those similarly computed for the hydrogen halide complexes of argon. The calculated stretching and bending frequencies are 10 cm−1, predicting that excited vibrational levels should be populated in the beam. Three transitions have been observed which appear to correspond to an excited vibrational level of ArDCN, but poor signal-to-noise has prohibited their unambiguous assignment. While seemingly, the data support a hydrogen bonded structure for this complex, they require an anomalously short bond length of 2.7 A, despite the apparent weakness of the bond. The vibrationally averaged molecular geometry and force constants are seen to be unexpectedly sensitive to isotopic substitution. Our conclusion is that the observed behavior of this system is best interpreted in terms of strong coupling between the radial and angular degrees of freedom associated with the weak bond. The amplitude of oscillation in the two coordinates is large, making the ‘‘structure’’ of this complex nebulous.

8 3Kr nuclear quadrupole coupling, microwave spectrum, and structure of KrHCN

Rotational spectra have been observed for ten isotopic species of KrHCN using the pulsed Fabry–Perot Fourier transform technique. The 83Kr nuclear quadrupole coupling constant has been measured in 83KrHC 14N, 83KrHC 15N, and 83KrDC 14N. Values of the rotational constants B̄0, centrifugal distortion constants DJ and H, 14N and 83Kr nuclear quadrupole coupling constants are These constants are consistent with a linear or near-linear configuration, with a Kr–HCN center-of-mass separation of 4.54 Å, and the Kr atom located 27° off the HCN figure axis on average, where the vertex of this angle is placed on the HCN center-of-mass. KrHCN exhibits unusually large centrifugal distortion in comparison to previously studied Kr–hydrogen halide systems. We find that the measured 83Kr nuclear quadrupole coupling constants in 83KrHC 14N and 83KrDC 14N are consistent with a long range polarization model previously used to explain values of χKr in KrH(D)Cl and KrH(D)F.

Ground state spectroscopic constants of isothiocyanic acid, HNCS, from its microwave and millimeter wave spectra combined with far infrared data

Abstract The microwave and millimeter wave spectra of isothiocyanic acid, HNCS, in the ground vibrational state have been investigated in the frequency region 8–300 GHz. The a-type R-branch transitions have been assigned up to J = 25 and Ka = 4, and the a-type qQ1 branch transitions up to J = 45. No b-type transitions could be identified in the frequency region covered. The far infrared data reported by Krakow, Lord, and Neely [J. Mol. Spectrosc., 27, 148 (1968)] were combined with our millimeter wave data in order to determine reliable spectroscopic constants. The rotational Hamiltonian, Watson's formalism with S reduction, has been extended empirically to higher order to facilitate the fitting of the large centrifugal distortion effects. The obtained constants are: A = 1357.3 GHz; B = 5883.4627 MHz; C = 5845.6113 MHz; D J = 1.19393 kHz; D JK = −1025.37 kHz; D K = 51.57 GHz; d 1 = −13.781 Hz; d 2 = −4.59 Hz. The 14N quadrupole coupling constant has also been determined: χaa = 1.114 MHz.

Microwave spectrum, conformation, barrier to internal rotation, centrifugal distortion, and dipole moment of methylpropargyl ether

The microwave spectrum of methylpropargyl ether, CH 3OCH 2CCH, has been investigated in the 11.9–26.5 GHz region. Only the gauche rotamer with a dihedral angle of 68° ± 2° from the syn position was assigned. Other forms are not present in concentrations exceeding 10 % of the total. The barrier to internal rotation of the methyl group was determined to be 2512 ± 75 cal mol −1. The dipole moment components are μ a = 0.290 ± 0.003 D, μ b = 0.505 ± 0.012 D, and μ c = 1.016 ± 0.003 D. The total dipole moment is 1.171 ± 0.013 D. Extensive centrifugal distortion analyses have been carried out for the ground as well as for two vibrationally excited states. For the ground state, transitions up to J = 77 were assigned and a large centrifugal distortion exceeding 9 GHz enabled the determination of accurate quartic and significant sextic distortion coefficients.

The Microwave and Millimetre Wave Spectrum, Molecular Constants, Dipole Moment, and Structure of Isocyanic Acid, HNCO

The microwave and millimetre wave spectra of six isotopic species of isocyanic acid, HNCO, have been measured. The molecule is a very slightly asymmetric prolate rotor. Besides the a type R branches, previously reported for two species, we have assigned for the first time a type Q branches and also some intense b type P and R branch transitions. Accurate values of the rotational constants have been determined. The rotational constant A0 is particularly large, with the result that the molecule undergoes very large centrifugal distortion, and terms up to the 12th power in the angular momentum are required to fit the spectrum. 14N nuclear quadrupole hyperfine splitting has been measured and accurate values for both χaa and (χbb − χcc) have been obtained, the latter for the first time. From the Stark effect of the b type transitions we have found the b component of the dipole moment to be 1.35 ± 0.1 D, an order of magnitude larger than previously supposed, and have used it to reinterpret earlier data indicating a variation of the a component with Ka. An improved molecular structure has been obtained. Astrophysical consequences of the large b component of the dipole moment are discussed. A table of transitions of potential astrophysical interest is presented.

Rotational analysis of the 5933 Å band of NO 2

By exciting NO 2 with a narrow band (0.035 cm −1 FWHM) tunable dye laser in the region of 5935 Å, an extensive analysis of the K = 0, 1, and 2 subbands of a vibronic transition has been carried out. The analysis of this transition proves that the upper state has the electronic symmetry 2 B 2. The origin of the analyzed band is placed at 16 849.48 cm −1 and the following molecular constants provide a best fit to the measured line positions: A v = 7.85 cm −1, B v = 0.451 cm −1, C v = 0.394 cm −1, and ϵ bb + ϵ cc = −0.0877 cm −1. The band exhibits a number of obvious perturbations with a marked intensity falloff for increasing N and K values. Large centrifugal distortion constants are required, apparently caused by N-dependent perturbations. A large inertial defect, Δ = 3.26 amu A ̊ 2 , is also found. By attributing the inertial defect to the vibrational dependence of A v , we estimate the geometry for the NO 2 2 B 2 excited state extrapolated to the vibrational origin to be r 0 = 1.31 A ̊ and θ 0 = 111°.