The polarisability tensor is a symmetric tensor, therefore, a_xy = a_yx , a_xz= a_zx ,a_yz = a_zy
i.e., there are actually six components of polarisability. According to quantum mechanics, the vibration is Raman active if one of these six components of polarisability changes during vibrations, similarly, it is IR active, if one of the three components of the dipole moment (ux, uy, uz) changes during the vibration.

The selection rule for Raman Spectroscopy can be expressed as (a / r) ¹ 0, where r is the distance along the normal co-ordinate i.e., change in polarisability of the molecule w.r.t., vibrational motion must not be zero at the equilibrium position of the normal vibration.
If the plot of polarisability (a) Vs displacement from the equilibrium distance re along the normal coordinate is that represented by fig (a) then the vibration will be Raman active. The vibration will be Raman inactive if the plot is as represented in fig (b)