Vibrational Intensities in Diatomic Infrared Transitions. The Vibrational Matrix Elements for CO

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Abstract
Second‐order perturbation theory has been employed to obtain general formulas for the (rotationless) vibrational matrix elements of diatomic molecules in infrared transitions. The mechanical and electrical anharmonicities have been taken into account by expanding the potential function in a power series through quartic terms and the electric dipole moment through cubic terms in the internuclear separation. The formulas obtained in this manner are shown to be quite accurate for the fundamental and the first two overtones. These expressions were then employed to evaluate the vibrational matrix elements for CO (1Σ+) up to v′=4 using the experimental integrated absorption coefficients as auxiliary data for the determination of the electric dipole moment expansion coefficients. A comparison of the vibrational matrix elements for CO involving the use of perturbed harmonic oscillator wave functions with those obtained using Morse wave functions with both linear and quadratic expansion of the electric dipole moment indicates that the Morse treatment is applicable through higher vibrational quantum numbers and higher transitions than the perturbation treatment. The comparison furthermore shows that the simple Morse expression with a linear expansion of the electric dipole moment yields vibrational matrix elements which would appear to be sufficiently accurate for many purposes. The anharmonic oscillator wave functions obtained would also prove useful in the evaluation of vibrational matrix elements in electronic transitions if the array for the harmonic oscillator overlap integrals were extended through higher vibrational quantum numbers.