On the calculation of parameters of tetrahedral splittings in vibrational spectra of $XY_4$ molecules of $T_d$ symmetry: determination of vibrational functions in symmetrized form

Objective: development of a highly effective method of determination of vibrational structure of $XY_4$ molecules of high symmetry, namely, construction of vibrational wave functions, symmetrized in the $T_d$ symmetry group, and determination of analytical expressions for calculation of tetrahedral splittings of the high-excited 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 $XY_4$ molecules of $T_d$ symmetry. The results obtained in this work allow predicting with high accuracy the values of vibrational energies of F1 symmetry states of $XY_4$ ($T_d$) 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.