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.

Thursday July 14th, 2010, at 3:30pm in K9509.


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.