# Why do most character tables include an $(x^2 + y^2)$ term?

Why do most character tables (e.g. $$C_{3h}$$ but not $$C_{2h}$$) include an $$(x^2 + y^2)$$ term?

Is it an abbreviated form for $$d_{z^2}$$, applying only where $$3d$$ (and higher) orbitals might be involved, or just a generic binomial term with no specific relevance?

The only clue I can find is on this page where it lists the full expression for the angular wave function for $$3d_{z^2}$$ as

$$Y_{3d_{z^2}} = √(5/4){2z^2-(x^2 + y^2)}/r^2 × (1/4\pi)^{1/2}$$

where it also comments that "the $$3d_{2z^2-x^2-y^2}$$ orbital is abbreviated to $$3d_{z^2}$$ for simplicity".

Thoughts appreciated.

• Thanks to initial editor of the question, for showing me how to improve super- and sub-scripting – iSeeker Jun 17 at 14:42

The functions on the right hand side of the table are the 'basis functions' for the irreducible representation, which are the rows of character, say $$A_g$$ or $$B_g$$ in C$$_{2h}$$. The basis functions have the same symmetry properties as the atomic orbitals with the same name (but other functions in $$x, x^2, x^3$$ etc and similarly for $$y,z$$, could be used).
To determine where an $$xy$$ function goes, look at a $$d_{xy}$$ orbital shape and subject it to each of the symmetry operations in turn,$$E, C_{2}, i$$ etc depending on point group. (Assume z is the principal axis). Count +1 if the orbital (or function) is unchanged and -1 if not. So in C$$_{2h}$$, a $$C_2$$ operation leaves $$xy$$ unchanged as does $$\sigma_h$$ and $$i$$ so $$xy$$ is a 'basis' for A$$_g$$. The $$xz$$ transforms as B$$_g$$; a $$C_2$$ operation gives -1, as does $$\sigma_h$$, but $$i$$ produces +1.
(note in cases such as $$z^2 -(x^2+y^2)/r^2$$ the $$x^2+y^2$$ part is totally symmetric in the $$xy$$ plane and does not change anything so can be ignored; think of the donut around d$$_{z^2}$$.)