Algebraic Determination of Spectral Characteristics of Rovibrational States of Diatomic Molecules. I. Diagram Technique for Determination of Vibrational Dependences of Matrix Elements

TL;DR

This paper introduces an algebraic and diagrammatic method to calculate vibrational matrix elements and their dependence on internuclear distance for diatomic molecules, applicable to various potential models.

Contribution

It develops a novel diagram technique for algebraic calculation of vibrational matrix elements, applicable to arbitrary differentiable functions and potential curves.

Findings

Provides explicit algebraic expressions for vibrational matrix elements.

Develops a diagram technique involving path vectors for calculations.

Applicable to harmonic oscillator and Morse potential models.

Abstract

Explicit algebraic expressions for the expansion of the vibrational matrix elements in series of matrix elements on the wave functions of the ground vibrational state have been obtained for arbitrary sufficiently differentiable functions of the internuclear distance, arbitrary values v and v', and the potential curves whose ladder operators can be constructed. A diagram technique have been developed for it that consists in: 1) the numeration of the matrix elements by points of the 2D diagram with coordinates (l, k), 2) the drawing arrows between points of this diagram corresponding to the action of the annihilation operators on the wave functions; 3) total taking into account of all possible path vectors formed by the continuous sequences of arrows from point (v, v) towards points (0, k). The only requirement is that the action of the operator on the wave functions should give the wave functions of the Schr\"odinger equation with the potential curve having the same parameter values. All necessary data for algebraic calculations of the vibrational dependence of matrix elements for the harmonic oscillator and the Morse potential have been given. Obtained expressions can be used to determine the absolute values and vibrational dependences of various spectroscopic characteristics of both ground and electronically excited states of diatomic molecules.