ON THE CALCULATION OF PARAMETERS OF TETRAHEDRAL SPLITTINGS IN VIBRATIONAL SPECTRA OF XY4 MOLECULES OF Td SYMMETRY: DETERMINATION OF VIBRATIONAL FUNCTIONS IN SYMMETRIZED FORM

Raspopova Natal'ya Ivanovna, Engineer, sub-department of theoretical and experimental physics, National Research Tomsk Polytechnic University (Tomsk, 30 Lenina avenue), natalia.raspopova@mail.ru
Konov Ivan Aleksandrovich, Engineer, sub-department of theoretical and experimental physics, National Research Tomsk Polytechnic University (Tomsk, 30 Lenina avenue), kiaff1188@mail.ru
Bolotova Irina Batorovna, Postgraduate student, Tomsk State University (Tomsk, 36 Lenina avenue), irbol89@mail.ru
Krivchikova Yuliya Valer'evna, Student, Tomsk State University (Tomsk, 36 Lenina avenue), yuliao_o@mail.ru 

539.194 

Objective: development of a highly effective method of determination of vibrational structure of XY4 molecules of high symmetry, namely, construction of vibrational wave functions, symmetrized in the Td symmetry group, and determination of analytical expressions for calculation of tetrahedral splittings of the highexcited vibrational states using the mentioned symmetrized wave functions. In order to achieve assigned objectives the following tasks were solved: 1. Using properties of symmetry of a molecule, and also results and theorems of the Irreducible Tensorial System Theory the problem of construction of symmetrized wave functions was solved. 2. Development and practical realization of a method of construction of matrix elements of Hamiltonian of a molecule in analytical form using the obtained symmetrized wave functions. 3. Development of algorithms and creation of programs in the MAPLE programming language for determination of expressions describing tetrahedral splittings. The mathematical apparatus of the irreducible tensorial operators theory was used for determination of a Hamiltonian in a symmetrized form and vibrational wave functions of molecules of tetrahedral symmetry. On that basis, using analytical programming language MAPLE and MATHIMATICS, the analytical formulas, allowing to determine the dependence of eigen values of vibrational Hamiltonian of such molecules on the parameters of tetrahedral splittings were obtained. Results: 1. On the basis of the irreducible tensorial operators theory the problem of construction of symmetrized wave functions was solved. 2. The method of construction of matrix elements of Hamiltonian of a molecule in analytical form using the obtained symmetrized wave functions was developed and practically realized. 3. The authors developed an algorithm and created a program on the basis of analytical programming language MAPLE, allowing both analytically and numerically determine the tetrahedral splitting in XY4 molecules of Td symmetry. The results obtained in this work allow predicting with high accuracy the values of vibrational energies of F1 symmetry states of XY4 (Td) molecules up to the states corresponding to eightfold excitation of vibrational quanta, and create a basis for realization of similar calculations of vibrational states of other symmetry. 

molecules of high symmetry, vibrational states, tetrahedral splittings. 

1. Hecht K. J. Mol. Spectrosc. 1960, vol. 5, pp. 355–389.
2. Papousek D., Aliev M. R. Molecular Vibrational Rotational Spectra. Elsevier, Amsterdam, 1982.
3. Fano U., Racah G. Irreducible tensorial sets. New York: Academic Press, 1959.
4. Wigner E. P. Quantum Theory of Angular Momentum. New York: Academic Press, 1965.
5. Varshalovitch D. A., Moskalev A. N., Khersonsky V. K. Quantum Theory of Angular Momentum. Leningrad: Nauka, 1975.
6. Cheglokov A. E., Ulenikov O. N. J. Mol. Spectrosc. 1985, vol. 110, pp. 53–64.
7. Cheglokov A. E., Saveliev V. N., Ulenikov O. N. J. Phys. B-At. Mol. Opt. Phys. 1986, vol. 19, pp. 3687–3693.