Vibration-rotation matrix elements for diatomic molecules; vibration-rotation interaction functions Fv′v (m) for CO

General expressions for the fourth-order vibrational matrix elements are obtained for the transitions v → v′ (v ? v′ v + 4) using a sixth-power internuclear potential and a quartic dipole moment function. The results for the dipole moment coefficients M₀ to M₄ of CO and for some transition moments R^v′_v deduced from a calculation including successively second, third and fourth order perturbation theory are compared.Using the rotational potential function expanded through the cubic term as a perturbation, we have also obtained general expressions for the vibration-rotation matrix elements. The vibration-rotation interaction functions F^v′_v (m) are calculated for the transitions v → v′ (v = 0, 10, 20 and v ? v′ ? v + 4) of CO and the coefficients C, D, E and G of the quartic polynomials fitting these functions are deduced. Taking account of uncertainties in the matrix elements R⁰₀ to R⁴₀, an error estimate for the coefficients of F²₀(m) and F³₀(m) is given.     

The effect of resonance interactions on the absorption spectra of (SF6)2 dimers in low-temperature matrices: Calculations and experiment

The IR absorption spectra of (SF₆)₂ dimers in Ar and N₂ matrices are investigated at 11 K. As a result of the resonance dipole-dipole interaction, the band of the triply degenerate vibration v ₃ is split into two components v _{X, Y} and v _Z. In comparison with the gaseous state, the splitting Δv = v _{X, Y} ? v _Z in the argon matrix decreases to 18.45 cm^?1, whereas, in the nitrogen matrix, the band v _{X, Y} is split into components v _X and v _Y, with the splitting being equal to δ ≈ 0.9 cm^?1. A model that takes into account the influence of the matrix on the spectra of dimers is developed. The model makes it possible to successively (i) calculate the resonance spectrum of an isolated dimer in terms of the model of local modes taking into account resonance interactions, (ii) determine with the help of the Monte Carlo method the structure of a matrix consisting of 512–1440 Ar (or N₂) atoms and a rigid (SF₆)₂ dimer, and (iii) take into account interactions of local dipole moments of a dimer with particles of the matrix in the approximation of dipole-induced dipole interactions. The model developed satisfactorily describes the experimental results. The calculated frequencies v _Z, v _X, and v _Y of a dimer in the matrix are shifted toward smaller frequencies as compared to the gaseous state, while the resonance splitting decreases virtually by 2 cm^?1. It is shown that, in an argon matrix with a symmetric arrangement of argon atoms nearest to a dimer, the splitting of v _{X, Y} proves to be smaller than 0.05 cm^?1. In a nitrogen matrix, this splitting increases virtually to 0.4 cm^?1.     

Study of the torsion-rotation Hamiltonian for symmetric tops using the millimeter wave spectrum of methyl silane

The torsion-rotation Hamiltonian for symmetric tops has been tested in methyl silane by combining recent anticrossing molecular beam measurements in the ground torsional state (v = 0) with pure rotational spectra taken for v as high as 4. The earlier microwave data set which consisted of J = 1 ← 0 and 2 ← 1 has been greatly extended by studying millimeter transitions for J = 4 ← 3, 5 ← 4, and 13 ← 12. An analysis of the 72 rotational frequencies for v ≤ 2 and the 15 anticrossing data for v = 0 yielded an excellent fit using 14 rotational, torsional, and distortion constants including the effective values for the A rotational constant and the barrier height V₃. No satisfactory fit could be obtained when the data set was extended to include measurements for (v = 3) or (v = 4). For each of these higher torsional levels, the difference between the observed frequencies and the predictions based on the best (v ≤ 2) constants can be expressed in terms of a shift δB_v in the B rotational constant, where δB_v is a smooth function of the torsional energy. This disagreement is of particular interest because it may result from the fact that the molecule passes from hindered to free rotation as v is increased from 2 to 4. The possibility of perturbation by a low-lying vibrational level is considered briefly. The information contained in the different types of spectra is discussed; the redundancy relations are treated and a Fourier expansion of the diagonal torsional matrix elements is introduced. For ¹²CH₃²⁹SiH₃, ¹²CH₃³⁰SiH₃, and ¹³CH₃²⁸SiH₃ pure rotational spectra for v = 0 were studied briefly in natural abundance. The results were combined with existing data for two deuterated symmetric rotors to obtain a structure based only on symmetric top rotational constants.     

High order vibrational matrix elements for diatomic molecules; dipole moment function and transition moments of CO

The fifth and sixth-order contributions to the vibrational matrix elements are obtained for the transitions v→v'(v'?v+4) using an eight power internuclear Dunham potential and a quartic power series expansion of the dipole moment function. The results for the dipole moment coefficients M₀ to M₄ of CO and the transition moments R_v^v' (with v=5, 10, 20) deduced from a calculation including succesively third, fourth, fifth and sixth-order perturbation theory are compared. The validity of these results is discussed.Using a quintic dipole moment function, general expressions for the vibrational matrix elements corresponding to the transitions v→v'(v'? v+5) are also presented and the influence of these contributions on the calculation of the dipole moment coefficients as well as the hot band transition moments of CO is shown.     

Calculation of high-order rotational centrifugal-distortion matrix elements for a Hund's case (a) basis set

The derivation of explicit expressions for the Hund's case (a) matrix elements of R^2k is discussed, where R is the mechanical rotational angular momentum operator of the molecule. A recursion relation is developed that permits matrix elements of R^2k to be expressed in terms of those of R^2(k?1), thus affording a straightforward means of calculating the case (a) matrix elements of rotational centrifugal-distortion constants D_v, H_v, L_v, M_v, etc., to an arbitrarily high order. The explicit matrix elements of L_v are listed.     

Systematics of vibrational intensities for “diatomic” molecules: Approximate formulas

Several approximations are discussed which enable one to estimate electric dipole moment transition matrix elements for diatomic molecules based on a very limited amount of experimental data. For strongly polar molecules, a simplistic point-charge dipole model in which there is no vibrational charge flux predicts a simple relation between the permanent moment, M₀, and its derivative with respect to internuclear separation, M₁. This is found to be approximately valid for a surprisingly large number of polar molecules. Next, approximate matrix elements for the fundamental and overtone bands are presented for a linear dipole moment function. Finally, systematic trends for molecules having similar electronic structures are investigated and combined with the approximations discussed above in order to estimate the transition moments for several molecules of astrophysical interest. An extension to the ν₃ fundamental and overtone bands of CH₄ is suggestive that some of these ideas may be applicable to certain polyatomic molecules as well.     

Rotational and vibration-rotational intensities of CS isotopes

Rotational and vibration-rotational intensities (dipole-moment matrix elements and Einstein A coefficients) have been calculated for the four most abundant isotopes of CS for the following sets of rotational and vibrational quantum numbers: J′ − J = ±1 for 0 ≤ J′ ≤ 200, and 0 ≤ v′ − v ≤ 4 for 0 ≤ v′ ≤ 20. The transition moments were calculated using both a series expansion and a Padé approximant representation for the dipole-moment function obtained from ab initio calculations and numerical vibration-rotational wavefunctions computed via a solution of the radial Schrödinger equation with accurate RKR potential-energy functions. Representative numerical results for bandhead positions and dipole-moment matrix elements of vibration-rotational transitions of ¹²C³²S are presented and compared with experimental and theoretical results of other workers. Explicit results are also given for a number of lines in the pure rotational band of CS that are of interest to radio astronomers.     

Fourth-order contributions to the rotational-vibrational matrix elements for diatomic molecules; application to the Fς′v(m) functions for CO

Using the rotational potential function expanded through the quartic term as a perturbation, and considering successively a quartic and a quintic dipole moment function, general expressions are obtained for the vibration-rotation matrix elements corresponding to the transitions vJ→?′J′ with ?′?v+5.The vibration-rotation interaction functions F^?′_v(m) are deduced for the transitions v→?′ (v=0, 10, 20 and?′?v+4) of CO from calculations including third then fourth-order contributions to the matrix elements. The results obtained for the coefficients of the quartic polynomials fitting these functions are compared and their validity is discussed.     

Tensorial development of the rovibronic Hamiltonian and dipole moment operators for XY3Z molecules with a degenerate electronic state: Preliminary application to the CH3O radical

We present a development of the Hamiltonian and transition moment operators of XY₃Z (C_3v) symmetric tops molecules in a degenerate electronic state with the aid of a tensorial formalism developed in a recent paper [A. El Hilali, V. Boudon, M. Loëte, J. Mol. Spectrosc. 239 (2006) 41-50]. Electronic operators are defined from group theory properties. They provide a new approach to build an effective rovibronic Hamiltonian as well as an effective dipole moment operator for rovibronic transition of XY₃Z molecules. This model is studied qualitatively thanks to the tensorial algebra properties. Expressions of the matrix elements are derived for these operators. A first simple application to the ground electronic state of CH₃O is proposed as an illustrative example.     

Shell-model calculations inA=28–32 nuclei

Shell-model calculations for positive-parity levels of A=28–32 nuclei have been performed in a 1d_5/2 2s_1/2 1d_3/2 configuration space. Configuration mixing of at most 250 lowest-lying pure configurations for each (A, J, T) set was taken into account. By a variation of the four effective interaction (MSDI) parameters and three single-particle energies a search was made for the best fit of the calculated energies to the experimental values. The average deviation between theory and experiment for the energies of the 110 fitted levels is 0.44 MeV. By adjusting all the 63 two-body matrix elements (ASDI) and the three single-particle energies, the average deviation of the binding energies was reduced to 0.16 MeV. For the lowest 10–15 levels in all A=28–32 nuclei a one to one correspondence exists between theory and experiment. Electromagnetic transition strengths were calculated both with MSDI and with ASDI wave functions. The ASDI wave functions reproduce the experimental E2 data much better than the MSDI wave functions. For the M1 transition strengths only a minor improvement has been achieved.     

Effective Hamiltonian for degenerate vibrational states in symmetric top molecules

A mixed matrix-operator form of the effective rotational Hamiltonian has been discussed for the degenerate vibrational states of symmetric top molecules. In this scheme, a rotational contact transformation can be applied to the effective Hamiltonian such that the operators of the “2, +2,” “2, −2,” and “2, −1” l-type interactions as well as the operators of the Δk = ±3 and ±4 interactions are eliminated from the first-order terms of the expansion of the rotational Hamiltonian in terms of the small parameter λ. The results have been used to discuss the correlation between various interaction parameters in the effective rotational Hamiltonian for the doubly degenerate fundamental vibrational levels of semirigid symmetric top molecules. For example, for C_3v or D₃ molecules, the parameter of the “2, −1” interaction is correlated with other parameters and cannot be determined separately by fitting the experimental data (unless there are certain accidental resonances between vibrational-rotational levels).     

Adequate representation of the dipole moment function for diatomic molecules and the matrices of vibration-rotation transition moments

An exponential representation of perturbations is used as a basis of the perturbed Morse oscillator approach which is applied, in a matrix form, for calculating the radial matrix elements for diatomic molecules. An analytic procedure is developed to deduce an exponential-power series expansion for the dipole moment function M(r) from experimental spectral intensities. It is shown that for real anharmonic molecules, the series expansion in powers of e^ar (α being the Morse parameter) is an adequate form for representing transition operators, just as the usual series expansion in powers of internuclear distance r is adequate for the case of a harmonic oscillator, and it is equivalent to a series expansion in vibrational wavefunctions. An exponential-power series expansion is derived as well for a model dipole moment function which has a correct long-range dependence and limit. To exemplify the accuracy and efficiency of the technique proposed, the (40 × 40) matrices of vibration-rotation transition moments 〈vJ|M(r)|v′J′〉(v, v′ = 0, 1, …, 39) have been calculated for the ground state of CO. Typical results of these computations are presented (up to v = 35, J = 100, and v′ ? v = 1–4) to illustrate the dependence of vibration-rotation interaction functions on the vibrational and rotational quantum numbers.     

Infrared diode laser spectroscopy of the ν2 fundamental band of SH3+

The high-resolution absorption spectrum of the ν₂ fundamental band of SH₃⁺ has been observed with the magnetic field modulation technique. The ions were generated in a hollow cathode discharge through a mixture of H₂ and H₂S. The molecular constants in the excited vibrational state are determined through a least-squares fit of the data to a standard symmetric top energy level expression. The anomalous sign of the centrifugal distortion constants is explained by the Coriolis interaction between the v₂ = 1 and v₄ = 1 states.     

Probabilities of rovibronic transitions in the I 1Π g − , J 1Δ g − →C 1Π ± u systems of bands of the deuterium molecule

Probabilities of spontaneous rovibronic transitions I ¹Π _g ^? , v′ J′, J ¹Δ _g ^? , v′, J′→C ¹Π _u ^± , v″, J″ of the D₂ molecule (for vibrational and rotational quantum numbers v′=v″=0–3 and J′=1–9, J″=J′±1) have been obtained for the first time. They were determined using (1) the previously proposed nonadiabatic model, which takes into account the electron-rotational interaction of the upper levels; (2) the coefficients of expansion of wave functions of perturbed states in the Born-Oppenheimer basis, which were found from the experimental data on rovibronic terms; and (3) semiempirical b initio data on electronic transition dipole moments of the 3dπ¹Π_g→2pπ¹Π_u and 3dπδ¹Δ_g→2pπ¹Π_u transitions. The dependences of the transition probabilities on J′ for the same bands of both hydrogen isotopomers H₂ and D₂ were found to be identical. They represent monotone functions for R and P branches and functions with a maximum (minimum) for Q branches. The ratios of transition probabilities of different isotopomers for different branches of the same systems of bands and for the same branches of different systems of bands were found to be correlated. The semiempirical values obtained in the paper agree with the experimental values within the limits of the errors of their determination. The nonempirical values of transition probabilities agree with the experiment considerably worse.     

Role of intramolecular interactions in Raman spectra of N2 and O2 molecules

The effect of vibration-rotation interactions and anharmonicities on the polarizabilities matrix elements, Raman scattering cross-sections, and depolarization degrees of N₂ and O₂ molecules for vibrational transitions v→v, v+1, v+2, v+3 have been investigated.     

Electronic and vibrational spectra of MnO2−4 in KBr crystals

Two new features have been observed in the electronic spectrum of KBr crystals doped heavily with MnO^2?₄ ions. The band at 870 nm is assigned to the crystal field transition e → t₂. The band at 600 nm shows a series of vibrational sub-bands at an interval of 740cm^?1 and is ascribed to the coupling between electronic transition and totally symmetric mode of the ion.A line at 830cm^?1, ascribed to totally symmetric mode v₁(A₁), has been observed for the first time in the Raman spectrum. I.R. spectrum of KBr: MnO^2?₄ shows four lines—one due to MnO^2?₄ in Td symmetry and the other three to the split components of v₃(T₂) for MnO^2?₄ in Cs symmetry. I.R. spectrum of KBr: MnO^2?₄: Ca²⁺ shows another s of four lines—one due to MnO^2?₄ in Td symmetry and the other three to the v₃(T₂) mode of the ion in C_2v symmetry. The v₁(A₁) line could not be observed in the i.r. spectra.     

Laser Stark spectroscopy; a case study in the ν2 fundamental band of 14NH3

The ν₂ fundamental band of ¹⁴NH₃ is observed with a laser Stark spectrometer in Doppler-limited resolution. A set of 41 constants including the electric dipole moment and its rotational dependence are determined from 360 Stark resonances observed in the present work and 67 millimeter- and submillimeter-wave inversion and rotation-inversion lines reported in the literature. The rms residual of the fit is 3.1 × 10⁻⁴ cm⁻¹. Field-free transition frequencies of the band in the region 800–1160 cm⁻¹ are calculated from the constants and tabulated together with their estimated uncertainties and relative absorption intensities. Rotational dependence of the electric dipole matrix elements of a C_3v molecule is presented on the basis of the conventional theory of vibration-rotation.     

Calculation of double beta decay rates and the neutrino mass

Predictions for 2v and 0v double beta decay rates are given for all nuclei with A ⩾ 70, for which double beta decay is energetically allowed. These predictions are based on detailed nuclear structure studies of the beta strength distribution and replace earlier estimates basing mostly on phase space considerations. New and more stringent limits on the Majorana neutrino mass are deduced from existing double beta decay experiments. Since the collective effects arising from spin-isospin as well as quadrupole-quadrupole forces are found to lead to a strong reduction of the nuclear matrix elements for two-neutrino double beta decay, but to have only minor influence on the matrix elements M^0v for the neutrinoless decay mode, the smaller limits for m_v result mainly from the fact that the widely used scaling procedure underestimates the 0_v matrix elements. It is further discussed to what extent interference between different neutrinos affects the obtained mass limits.     

Moments of a Single Entry of Circular Orthogonal Ensembles and Weingarten Calculus

Consider a symmetric unitary random matrix V = (v _ij )_{1 ≤ i, j ≤ N} from a circular orthogonal ensemble. In this paper, we study moments of a single entry v _ij . For a diagonal entry v _ii , we give the explicit values of the moments, and for an off-diagonal entry v _ij , we give leading and subleading terms in the asymptotic expansion with respect to a large matrix size N. Our technique is to apply the Weingarten calculus for a Haar-distributed unitary matrix.     

Spectroscopy of XY3Z (C3v) molecules: A tensorial formalism adapted to the O (3) ⊃ C∞v ⊃ C3v group chain

A tensorial formalism adapted to the case of XY₃Z symmetric tops has been developed. We use the O (3) ⊃ C_∞v ⊃ C_3v group chain. All the coupling coefficients and formulas for the computation of the matrix elements are given for this chain. Such relations are also deduced in C_3v group itself.