Dipole Moment Function of Diatomic Molecules

Abstract

The connection between the dipole moment function (dipole moment as a function of internuclear distance), M(u), and its matrix elements has been studied for the one-dimensional model of a diatomic molecule. It is shown that M(u) is completely determined by a single row or column of the matrix 〈v | M(u) | v′〉, but that the diagonal matrix elements do not completely determine M(u). Attention is given to the problem of approximating M(u) when a finite number of matrix elements are known. The usual method (called by us the ``polynomial approximation'') is compared with a new approximation scheme which we call the ``wave function approximation.'' These methods are equivalent for a harmonic oscillator, but not in general. Estimates of the errors of the two methods are obtained in the case where the anharmonicity of the potential is small. The ``wave-function expansion'' of M(u) which is developed and used to carry out the calculations is a new way of relating matrix elements to M(u) which makes it much easier to calculate the dipole moment function from its matrix elements. This method has been used to obtain explicit formulas relating the dipole moment function to its matrix elements for the harmonic oscillator and for the Morse oscillator.