Why not just use the x,y,z as basis functions instead of linear/quadratic functions of these unit vectors?

Question

I am going through Applications of Group Theory to the Physics of Solids by M. S. Dresselhaus.

In chapter 4 he states

This is with respect to the P(3) group.

I understand basis functions must transform like the given representation e.g for the identity representation $\Gamma_1$ all operators leave any unit function unchanged hence you could use any of $x,y,z,z^2$ separately as basis functions because they are left unchanged in this representation.

For a 2D representation $\Gamma_2$ often functions like $(xz,yz)$ and $(xy,x^2-y^2)$ are also stated as basis functions. Why not simply use $(x,y)$ for the 2D case?

Do we list all other basic functions in character tables for completeness? Or is it because in certain cases basis functions like $(xy,x^2-y^2)$ may be more useful than $(x,y)$ [I am thinking that the former basis function has the symmetry of a d orbital according to Harris and Bertolucci hence it may be more useful to consider in the context of looking at atoms/molecules]

Answer 1

In a single word, the answer is d-orbitals, as you guessed!

It's very frequently useful to determine how specific orbitals on an atom transform, because this in turn determines which orbitals it can overlap with to form bonds / molecular orbitals.

Each d subshell has five d-orbitals, and often these are written as $\mathrm{d}_{xy}$, $\mathrm{d}_{yz}$, $\mathrm{d}_{xz}$, $\mathrm{d}_{z^2}$, and $\mathrm{d}_{x^2 - y^2}$. These transform in the same way as the respective Cartesian products in the subscripts, so for example, the $\mathrm{d}_{xy}$ orbital transforms in the same way as $xy$. (More technically, the mathematical form of the $\mathrm{d}_{xy}$ orbital is $xy$ multiplied by some spherically symmetric term, which doesn't affect the symmetry.)

In fact there are also some other uses for the quadratic terms, including Raman spectroscopy (vibrational modes that transform the same way as quadratic terms are Raman active), but I imagine that d-orbitals are the most common use, and one that most chemistry students will encounter when studying transition metal chemistry.

(The $x, y, z$ linear functions are useful for lots of things (determining symmetries of translational modes, etc.), but they're also there because of the p-orbitals, which transform in the same way as $(x, y, z)$.)