Question

show why am even/odd rule is necessary for improper rotation operations

Answer 1

In geometrical point of view, an improper rotation, also called roto-reflection, rotary reflection,or roto inversion is, depending on context, a linear transformation or affine transformation which is the combination of a rotation about an axis and a reflection in a plane perpendicular to that axis. An improper rotation of an object thus produces a rotation of its mirror image. The axis is called the rotation-reflection axis. This is called an n-fold improper rotation if the angle of rotation is 360°/n. The notation S_ndenotes the symmetry group generated by an n-fold improper rotation (not to be confused with the same notation for symmetric groups). The notation n is used for n-fold roto inversion, i.e., rotation by an angle of rotation of 360°/n with inversion. The Coxeter notation for S_2n is [2n⁺,2⁺], and orbifold notation is n×, order 2n. The direct subgroup, index 2, is C_n, [n]⁺, (nn), order n, as the roto reflection generator applied twice. S_2n for odd n contain inversion, with S₂ = C_i is the group generated by inversion. S_2n contain indirect isometries but not inversion for even n. In general, if odd p is a divisor of n, then S_2n/p is a subgroup of S_2n. For example, S₄ is a subgroup of S₁₂.

The symmetry of a vibrational level is the direct product of the symmetries of all of the contributing modes - vibrational ground states are always totally symmetric. Since the direct product of any non-degenerate irrep with itself is totally symmetric, overtones involving an even number of quanta will be totally symmetric. Levels involving an odd number of quanta of a vibration will have the symmetry of the mode. The symmetries of overtone levels of degenerate vibrations are not so easily ascertained and will not be discussed here. However, an excellent discussion is available. This is the reason even/odd rule is necessary for improper rotation operations.