MITACS Seminar Series on Mathematics of Computer Algebra and Analysis
The Formalism of Polynomials of Quantum Numbers in Problems of Molecular Physics
Konstantin Kazakov, Irkutsk State Technical Univeristy, Irkutsk, Russia.
PREVIEW Vibrational phenomena have always fascinated scientists and engineers. Molecules constitute a vibrational system of an important class that is the subject of our present concern. High-resolution infrared absorption spectra provide information about the distribution of vibration-rotational energy levels and the transition probabilities of real molecules. Spectral lines command physical interest through their interpretation with the aid of physical models, i.e., the relation of frequencies and intensities of spectral lines to molecular motions of various types. As the precision of measurements made with various experimental techniques increases relentlessly, the interpretation of observed spectra becomes correspondingly challenging. This condition stimulates the search for, and development of, innovative methods to investigate vibrational systems for which a conventional description fails. Intuitively, the most natural model of intramolecular motions involves interacting anharmonic oscillations of atomic centers, but this simple physical model lacks a mathematically exact solution. The use of perturbation theory, however, solves the problem. This classical method is simple and clear, but its application is generally limited to the first few orders of theory that any textbook on quantum mechanics describes. The determination of corrections of higher orders becomes complicated through the sheer bulk of the calculations. The calculation of frequencies and intensities of spectral lines with an accuracy defined by experiment hence becomes difficult. A real spectrum of a sample containing even diatomic molecules of a particular chemical compound can comprise lines numbering a few thousands. Despite these difficulties, some success in developing an adequate method of calculation has been achieved, embracing perturbation theory. In the present work, we consider the development of techniques of perturbation theory applied to problems of molecular spectrometry to calculate the frequencies and intensities of vibration-rotational transitions.