Rotationless Matrix Elements Research Articles

Vibration-rotational and rotational intensities for CO isotopes

Dipole moment matrix elements have been computed for the five most abundant isotopes of CO. The wave functions utilized were obtained from a direct solution of the Schrödinger equation with an accurate RKR potential. The dipole moment function, in the form of a Padé approximant, was chosen to reproduce the experimental measurements near equilibrium, to have the proper united and separated atom limits, and to have the correct long-range asymptotic functional dependence on internuclear separation. Because of the large number of transitions involved, and to facilitate applications, the squares of the dipole moment matrix elements were fitted by a least-squares procedure to polynomials in v and J. Predictions for the 5-0 and 6-0 rotationless matrix elements and Herman-Wallis coefficients are given, and their dependence on the isotopic reduced mass is discussed. For the pure rotational band, v = v′ = 0, explicit Einstein A values and transition frequencies were calculated for the three most abundant isotopes for J up to 55. The corresponding dipole moment matrix elements were also fitted to simple polynomials in m and the dependence of the coefficients on the reduced mass given. The present results incorporate the most accurate and extensive intensity measurements and theoretical dipole moment function data for any heteronuclear diatomic molecule. In view of this, because of the importance of the CO laser, the accuracy of the spectral frequencies, and the ubiquity of the CO molecule, it is reasonable to expect that some of these lines, in particular, in the fundamental band of 12C 16O, can serve as laboratory standards for intensity measurements.

Diagonal vibrational matrix elements for an anharmonic oscillator

General expressions for the diagonal rotationless matrix elements of any power of the internuclear displacement from equilibrium are obtained for 1Σ-state diatomic molecule using an iterative relation. These expressions, including contributions arising from terms of the ninth power in the Dunham potential, are obtained in analytical form. As an application, sixth- and seventh-order contributions to the matrix element 〈0⦶ x 1⦶〉 are calculated for the first eight powers of the displacement.

Variational calculations of vibrational–rotational properties of HF in the X1Σ+and B1Σ+electronic states

The vibrational energy levels for the X1Σ+ and B1Σ+ states of the HF molecule were calculated by the variational method from the corresponding Rydberg–Klein–Rees potentials. The eigenvectors were used to obtain the rotational constants, Bv, Dv and Hv, according to the perturbation theory definitions. The dipole moment function, its rotationless matrix elements and the radiative lifetimes for the vibrational levels corresponding to v= 1, 2, 3 of the ground state were also calculated.

Vibrational matrix elements for diatomic molecules

General expressions for the rotationless matrix elements of the dipole moment are obtained for the transitions υ → υ' (υ ⩽ υ' ⩽ υ + 4), using a quintic internuclear potential, a quartic dipole moment function and third-order perturbation theory.

Moments de transition vibrationnelle des molecules diatomiques. Intensite des raies rovibrationnelles des bandes 0→2 et 0→3 et fonction dipolaire de CO

General expressions for the radial wavefunctions and the rotationless matrix elements of the dipole moment for the transitions 0→0 to 0→4 are obtained using a fifth-power internuclear potential, a quartic dipole moment function, and third-order perturbation theory. Line intensities in the 0→2 and 0→3 bands of CO have been measured for pressures varying from 2 to 5 atm. Using the values of the vibrational transition moments | R 0 2| and | R 0 3| deduced from our measurements and the values of | R 0 0|, | R 0 1| and | R 0 4| previously given, we evaluate the coefficients M 0 to M 4 of the dipole-moment function for CO.

Linienstärken in der 4-0- und 5-0-Rotationsschwingungsbande von Fluorwasserstoff / Line Strengths in the 4-0- and 5-0-Rotation-Vibration-Bands of Hydrogen Fluoride

The strengths of 6 lines in the 4-0- and 5-0-rotation-vibration-bands of hydrogen fluoride have been measured in a multiple reflection cell using a direct measurement method. The absolute values of the rotationless matrix elements of the electric dipole moment have been determined to be |<4|μ(r)|0>| = (3.48 ± 0.13) · 10-22 esu · cm, |<5|μ(r)|0>| = (8.79 ± 0.54) · 10-23 esu · cm.

Dipole moment matrix elements for the 1-0, 2-0, and 3-0 vibration-rotation bands of diatomic molecules

Explicit expressions for the matrix elements of the dipole moment have been derived for the 1-0, 2-0, and 3-0 vibration-rotation bands of diatomic molecules. The derivation is based on a cubic dipole moment function and on radial wave functions obtained using a quintic power series expansion of the internuclear potential and third order perturbation theory. The results are given in the form of a rotationless matrix element multiplied by a rotational or Herman-Wallis factor. The expressions for the Herman-Wallis factors include the leading contributions from each dipole moment coefficient and have a reasonably simple form even in the case of the overtone bands. The rotationless matrix elements are similar to results obtained by Herman and Schuler except that their higher order terms are purposely omitted. The reason for this is discussed as are other approaches to the rotational problem which have incorporated a linear dipole moment function. It is pointed out that the results of the present theory for the case of CO are in very good agreement with the numerical treatment of CO by Young and Eachus. The actual comparison is given in a subsequent paper.