Knowledge Bank

The adaptation of the ``tetrahedral tensorial formalism"" to symmetric top molecules whose equilibrium configuration belongs to the $C3v$ point group is presented. Tensorial algebra and computer programs have been set up consistently with previous works about tetrahedral $XY4$ molecules. In particular, the systematic introduction of any interaction term of any order is one major advantage when studying intricate systems of vibrational interacting bands. Making full use of group theory, all tensor operators can be generated systematically: those involved in the transformed Hamiltonian as well as those taking part in the transformed Dipole Moment or Contact Transformation expansions. This systemacy appears especially well suited to numerical computation: for instance, the matrix elements of the rotation-vibration Hamiltonian are calculated by using one unique formula. Even if, as far as simple problems (such as isolated band analyses) are concerned, the tensorial formulation is more complex than classical ones, both the mathematical and computer treatments are easily made and in principle can directly fit any complex interacting system, leaving the spectroscopist with physical consideration only. We show some similarities and differences of this extension with respect to the tetrahedral formalism. Correspondences with classical treatments, such as relations between classical and tensorial spectroscopic coefficients, are also given. A prospective application to the analysis of the lowest polyads of the $CHD3$ molecule is discussed in some detail.