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\title{METHOD AND RESULTS OF DETERMINATION IN ANALYTICAL FORM THE
PARAMETERS OF TETRAHEDRAL SPLITTINGS AND BAND CENTERS IN THE XY$_4$
SPHERICAL TOP MOLECULES: $^M$GeH${_4}$ AS AN APPLICATION}

\author{\underline{\FirstName O. N. \LastName ULENIKOV}, \underline{\FirstName E. S. \LastName
BEKHTEREVA}, \underline{\FirstName O. V. \LastName GROMOVA},
\underline{\FirstName A. L. \LastName FOMCHENKO},
\underline{\FirstName P. A. \LastName GLUSHKOV}}
\address{National Research Tomsk Polytechnic University, 30, av. Lenina, 634050 Tomsk,
Russia}

\author{\underline{\FirstName C. \LastName SYDOW} and
  \underline{\FirstName S. \LastName BAUERECKER}}
\address{Institut f\"{u}r Physikalische und Theoretische Chemie, Technische Universit\"{a}t Braunschweig, D - 38106,
Braunschweig, Germany}



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\pagestyle{fancy} \fancyhead[C]{{\footnotesize \textit{HRMS, 28th
Colloquium, Dijon, France,
  Aug. 28 - Sept. 1, 2023}}}
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\category {1,3}
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\maketitle

\index{spherical topmolecules}

The mathematical basis is discusses the for solving an important and
up to the recent time unsolved problem of analytical describing a
complicated tetrahedral structure of the spectra of spherical top
molecules. The main difficulty of the problem until recently was the
impossibility of an analytical presentation of the
quantum--mechanical wave functions of such type molecules. Using the
theory of irreducible tensor sets, we solved this problem and obtain
the necessary results in the most general form for the vibrational
problem (possible vibrational resonance interactions are also taken
into account). Results are obtained for states of arbitrary symmetry
which are united in polyads with a polyad quantum number $P \leq 8$.
As an illustration, band center values are predicted for all bands
of the five isotopologues of ${^{M}}$GeH$_4$ ($M = 70, 72, 73, 74,
76$) up to 10000 cm$^{-1}$.

This work was supported by The Ministry of Science and Higher
Education of the Russian Federation in part of the Science program
(Project FSWW--2023--0003)

\end{document}
