Page 297 - Strategies and Applications in Quantum Chemistry From Molecular Astrophysics to Molecular Engineer

280 P. LAZZERETTI ET AL.

Owing to permutational symmetry of the tensor indices, only

components are distinct for a tensor of rank r appearing in (1). Thus the number of
independent values which completely characterize the various tensors in eq. (1) is 3,
6, 10, 15, 21, ... respectively for Molecular point
symmetry further reduces the number of linearly independent components, see, for
instance, Refs. [4], [5]. For any tensor appearing in (1), denoted in general by
let us rearrange its components as a column vector in cartesian space, i.e.,

If the basis set of unit vectors in cartesian 3-space transforms

under an operation T, then the direct product matrix

can be introduced, so that

If T belongs to a group G and brings the physical system into self-coincidence, then
the array of components will be stable under G, i.e.,

In addition one can always find a transformation leading to a symmetry adapted basis
so that T is brought to the block diagonal form T via the associated similarity
transformation. The matrix can be written as a direct sum