Spherical Top Molecules Research Articles

Jahn-Teller and L-uncoupling effects on rotational energy levels of symmetric and spherical top molecules

The effects of a small Jahn-Teller effect and of Coriolis L-uncoupling forces on the rotational energy levels of a degenerate electronic state are considered. In general the two types of mechanism make similar contributions to the energy levels, and can only be separated by the use of additional information such as the isotopic variation of the molecular parameters. The theory of the rotational energies is formulated in a way that is applicable to both symmetric and spherical top molecules. The expressions derived for symmetric tops reproduce the results of previous workers. For E or F electronic states of spherical tops, the rotational parameters are equivalent to those employed for the fundamental levels of degenerate vibrations, but they must be interpreted in terms of the form of the appropriate double or triple potential energy surface. Parameters calculated for the 3 d 2 F 2-3 s 2 A 1 transition of the ND 4 molecule from the ab initio potential surface of Havriliak and King do not agree well with the experimental values of Herzberg and Hougen.

Trends for intensities of vibrational overtones in methane

The concept of local vibrational modes is examined quantitatively for spherical-top molecules. New theoretical formulations of transition electric dipole moments are developed for diatomics and local-mode spherical tops. Transition moments inferred from high-resolution spectra of ν 3, 2ν 3 and 3ν 3 in 12CH 4 and 13CH 4 constitute the data base from which comparisons between theory and experiment, and predictions, are made.

Comparison of the symmetrized Cartesian and angular momentum basis sets to describe vibrationally excited states in spherical top molecules

Recent analyses of the 3ν3 vibrational band in two spherical tops, SF6 and SiF4, have shown the Cartesian basis to be an appropriate description for this band in the octahedral case, while the tetrahedral case is adequately represented by the angular momentum basis. Using spectroscopic constants derived from force field analyses, the F1u/F2 subblocks of the vibrational Hamiltonian matrices of the bands of four vibrational quanta and one band of five quanta 2ν23ν3 are diagonalized in each bases set for SF6, CF4, and CH4 to extend the comparison to more general vibrational levels. Both bases fail to adequately describe CF4 and CH4. The validity of the various angular momentum designations as quantum numbers is strongly dependent on the particular set of spectroscopic parameters. SF6 is poorly described by the angular momentum basis. Within a vibrational band of SF6, only a component whose vibrational motion lies along a single Cartesian axis is well described by the Cartesian basis. It is the lowest energy component, well separated in energy from the remaining components because of the single bond anharmonicity.

Depolarized light scattering studies of the collision induced polarizability anisotropy of atoms and spherical top molecules

New measurements are reported of the density dependent depolarization ratio for argon, krypton, xenon, methane and sulphur hexafluoride, and the results are analysed to provide values for the second and third depolarization virial coefficients. The relationships between the second depolarization virial coefficient, the zeroth moment of the two-body Rayleigh spectrum and the second Kerr virial coefficient are considered, and it is shown that they now provide consistent results for the collision-induced pair polarizability anisotropy. Former inconsistencies are attributed to insufficient allowance for the effects of three-body interactions. Calculations of the second and third depolarization virial coefficients based on the DID model and using the Maitland-Smith potential are in excellent agreement with the experimental results for argon, krypton and xenon.

The influence of rotation on the vibrational relaxation of the CH 4 molecule. Study of the transfers CH 4(ν 2) + Ar → CH 4(0) + Ar and CH 4(ν 4) + Ar → CH 4(0) + Ar

The influence of rotational energy transfers on the vibrational relaxation of the ν 2 and ν 4 modes of CH 4 are studied. The interaction matrix is built up in the framework of the semiclassical coupled-states approximation using the asymptotic expression of (3 j) symbols. It is assumed that the intermolecular potential is the sum of five exponential atom—atom interactions. The corresponding potential matrix is diagonalized by extending the approach of Billing to the case of spherical top molecule—atom collisions. The first-order perturbation calculation of rovibrational probabilities shows the strong influence of rotational energy transfers.

Anomalous magnetic field-induced birefringence in molecular systems with the Jahn-Teller effect

It is shown that spherical-top molecules in orbitally degenerate states possess anomalous anisotropy of electric and magnetic properties leading to temperature-dependent birefringence induced by a static magnetic field. The Cotton-Mouton constant mC is expressed in terms of anisotropic tensors of the molecular polarizability, magnetic susceptibility, magnetic moment and magneto-electric hyperpolarizability. The resulting temperature dependence of mC is different from that (quadratic in T -1) expected without the Jahn-Teller effect. This difference is defined by introducing in this paper temperature-dependent reduction factors of electronic operators, which at low temperatures coincide with the usual ones. Reduction factors are calculated for systems with G 3/2 ground electronic terms. For these cases approximate analytical expressions are obtained taking into account the vibronic coupling with e and t 2 vibrations. Numerical estimation of the expected Cotton-Mouton constant in the case of the ReF6 molecul...

Rotational excitation of CH4and H2O by slow electron impact

Cross sections for rotationally elastic and inelastic scattering of slow electrons by spherical (CH4) and asymmetric top (H2O) molecules are calculated in the adiabatic-nuclei-rotation (ANR) approximation. The elastic scattering amplitude is derived in the molecule-fixed frame and the corresponding K matrix is obtained by solving a set of coupled differential equations; in which the total interaction potential includes an ab initio molecular core potential from the ground state near Hartree-Fock wavefunctions, an exchange term in the free-electron-gas-exchange (FEGE) model plus the orthogonalisation procedure and the distortion of the target through a recently proposed nonparametric polarisation potential of Jain and Thompson (1982). Symmetry properties and the selection rules for the asymmetric rotor in general are discussed. In order to avoid convergence problems for the molecule with a permanent dipole moment, an alternative two-stage hybrid S-matrix theory of Collins and Norcross (1978) along with the closure formula of Crawford and Dalgarno (1971) is adopted to obtain converged cross sections (in which proper use of the unitarised Born approximation is made). Results are compared with recent available measurement and calculations; there is a qualitative agreement for CH4 with recent calculations performed in a different model. In the case of H2O, some difficulties are discussed in comparing directly the authors' data with recent experimental cross sections.

First quantum corrections to second virial coefficients for anisotropic interactions: Simple, corrected formulaa)

A simple formula for the first quantum correction to the second virial coefficient, valid for the interaction of any like or unlike combination of atoms, diatomics, spherical top, or symmetric top molecules is given. It is found that the commonly used formulas of Wang Chang contain an error that omits an anisotropic contribution. The sizes of the different contributions to the quantum correction are discussed. As examples, calculations for He–SF6 and He–CO2 are reported.

Pressure dependence of1H and19F chemical shifts in 1,1-difluoroethylene gas and in gaseous mixtures

We have measured the chemical shifts, δ, and ‘second virial shielding coefficient’, σ1, for gaseous 1,1-difluoroethylene and its mixture with other gases at varying pressures (∼ 10–45 atmospheres). The δ vs ρ (density) plot for 1H and 19F resonances is found to be linear within the pressure range studied. The sign of σ1 is negative. The influence of perturbers on δ and σ1 is in the following order: Xe>SF6>Kr>SiF4>CH4>CF4 (spherical top molecules) and C2H6>C2H4>HCl>CO2 (non-spherical top molecules). These trends are true for polar and non-polar solutes.

Early events and some later developments in microwave spectroscopy

The early history of microwave spectroscopy is reviewed. New directions in the field are indicated. These include: further extensions of coherent submillimeter wave spectroscopy, microwave spectroscopy of molecules in interstellar space, microwave-infrared laser double resonance, spectroscopy of ionized molecules and transient molecular radicals, studies of hydrogen-bonded molecular complexes and atom-molecule complexes, observations of “forbidden” rotational transitions in symmetric-top and spherical-top molecules, and new developments in high-temperature spectroscopy.

Calculator programs for predicting pr and qq branch separations of linear, spherical, symmetric and asymmetric top molecules

By means of calculator programs on 5 magnetic cards, the branch separations of linear, spherical, symmetric and asymmetric top molecules can be calcula

Vibrational anharmonicity in octahedralXY6molecules

Formulae for the vibrational anharmonic coefficients Xrs, Grs, Trs and Srs have been derived for the case of octahedral XY 6 spherical top molecules. The vibrational contact transformation is first presented in general form and then worked out for the specific constants of the XY 6 model. The complete set of results is given in tabular form suitable for automatic computation and the important coefficients relating to bond stretching anharmonicity (rs = 11, 12, 13, 22, 23 and 33) are also quoted as algebraic expressions. Formulae for vibration-rotation coefficients of q 2 J 2-type operators are also given as an Appendix.

Ion–polar molecule encounters

It is permissible to assume that the rate coefficient for collisions between ions and polar molecules does not depend on the moment of inertia of the latter because the rotation time is brief compared with the collision time. On taking the moment of inertia to be vanishingly small the classical collision problem can be solved exactly when the angular momentum vector is normal to the orbital plane. Use is made of the adiabatic invariance of ∮ p d q /2π in which p is an appropriate momentum and q is the conjugate coordinate. This adiabatic invariant fixes the change in the rotational energy in moving from an infinite separation to any chosen position. The average dipole orientation is thereby determined, which fixes the force acting. The potential energy function (including due allowance for the rotational energy stored) is now written down and an integral expression for the primitive rate coefficient is thence obtained. The ratio of the primitive rate coefficient to the Langevin rate coefficient depends only on the initial rotational energy and on the dimensionless parameter β = 2 αkT/D 2 , where α is the polarizability, D is the dipole moment and T is the temperature. Extensive computations have been performed. Tables are presented giving the primitive rate coefficient and also approximations to the thermally averaged rate coefficients for linear and for spherical top molecules.

Flow birefringence in gases of linear chain molecules

An alignment of the rotational angular momenta of gas molecules occurs in the presence of a gradient in the stream velocity of the gas. This can be measured as a birefringence. Measurements on gases of linear chain molecules are presented. The results confirm the general trend found earlier for diatomic and spherical top molecules, that the amount of alignment is nearly independent of the shape of the molecule.

Internal rotation and intramolecular dipolar relaxation: Effect of modulation of internuclear separation

Expressions for the spin-lattice relaxation time for a nucleus experiencing intramolecular dipolar interactions which are modulated by internal motion and by overall molecular reorientation are given for the case when the internal motion modulates the internuclear distance. Rotational diffusion models are used to describe both internal and overall rotational dynamics, and results are given for spherical and symmetric top molecules. It is shown that modulation of the internuclear distance can give rise to substantial effects on the nuclear relaxation rate in certain molecular geometries.

A semiclassical model for the vibrational excitation of spherical-top molecules in collisions with ions

A simple theory is presented to explain the previously observed single-mode vibrational excitation of the spherical-top molecules CH 4, CF 4 and SF 6 in collisions with H + and Li +. The theory is based on a three-dimensional forced-oscillator model which has been modified to take account of many independent harmonic oscillators. For small-angle collisions the linear driving forces are the dipole-, polarizability- and quadrupole-derivatives taken from IR, Raman spectroscopy and simple estimates, respectively. To explain the results at larger angles near and beyond the rainbow it has been necessary to introduce short-range repulsive forces between the ions and the outer atoms of the molecule. For small angles both the predicted first moment of the energy transfer and the time-of-flight spectra agree quantitatively with the experimental results. At large angles, for which only the first moment of the energy is available, good qualitative agreement is obtained after a slight adjustment of the potential parameters. The energy transfer as a function of time is calculated and shows a different oscillatory behavior for the proton and Li +-ion systems. Also the effect of intra-mode coupling is investigated and shown to have only a small effect on the overall energy transfer. The paper closes with a discussion of the implications of these experiments and the possible role of rotational excitation. The field strengths in these ion scattering experiments are shown to be greater than in the strongest focused Q-switched laser pulses.

Persistent Holes in the Spectra of Localized Vibrational Modes in Crystalline Solids

Persistent nonphotochemical spectral holes have been burnt at low laser intensity in the vibrational absorption line of a spherical-top molecule in alkali halide crystals. These observations complement earlier studies on electronic transitions in organic crystals and glasses and demonstrate that persistent hole burning is a general solid-state phenomenon.

Analytic estimation of almost-resonant molecular energy transfer due to multipolar potentials. II. V V relaxation of the laser levels of CF4

A quantal almost-resonant collisional energy transfer theory with a natural cutoff for the Born approximation is extended to spherical-top polyatomic molecules and used to calculate the rate from 100–400 K for the almost-resonant vibration to vibration process which depletes the upper laser level and populates the lower laser level of CF4.

Analysis of nearly degenerate bands of spherical top molecules: Matrix elements of the vibrational-rotational Hamiltonian

We formulate a general method for calculating energy eigenvalues and eigenfunctions in a simultaneous analysis of nearly degenerate vibrational-rotational bands of spherical top molecules. The basis functions are products of rigid rotor and N-dimensional harmonic oscillator eigenfunctions. General explicit expressions are derived for the matrix elements of vibrational operators in the basis of N-dimensional harmonic oscillator eigenfunctions. Using these general expressions, the matrix elements of vibrational operators in the basis of five-dimensional harmonic oscillator eigenfunctions applicable, for example, in the analysis of the nearly degenerate ν2 and ν4 fundamental vibrational-rotational bands of tetrahedral XY4 molecules like CH4, are calculated explicitly.

Anharmonic Relaxation Times of Molecular Vibrational Modes in Alkali Halide Crystals

Low-temperature infrared-saturation and hole-burning measurements have been performed on an internal vibrational mode of a spherical-top molecule embedded in a variety of alkali halide crystals. The vibrational relaxation times ${T}_{1}$ and ${T}_{2}$ satisfy the relation ${T}_{2}\ensuremath{\cong}2{T}_{1}$ but ${T}_{1}$ (\ensuremath{\leqslant} 38 nsec) is orders of magnitude smaller than expected for multiphonon decay. A study of the alkali halide dependence of these relaxation times suggests that a specific combination of internal, local, and band modes is involved in the decay process.