# Why do the $d$ orbitals have such strange symbols?

Why are the five d-orbitals denoted by the symbols $$d_{z^2}, d_{x^2-y^2}, d_{xy}, d_{yz}$$ and $$d_{zx}$$? Does it have to do with the wavefunctions of d-orbitals? The symbols for the f-orbitals are even stranger.

The names of the orbitals tell you how the wavefunction's value depends on orientation. For example, the $$p$$ orbitals are called $$p_x$$, $$p_y$$, and $$p_z$$. That means that for a $$p_z$$ orbital, on each sphere centered at the origin, the value of the wavefunction is proportional to $$z$$. So it has one positive and one negative lobe.
Similarly, the $$d$$ orbital $$d_{xy}$$ has four lobes in the $$xy$$ plane, two positive and two negative. The orbitals $$d_{yz}$$ and $$d_{zx}$$ are similar. The only exception to the pattern is $$d_{z^2}$$, which is an abbreviation of the "proper" name $$d_{z^2 - x^2/2 - y^2/2}$$. The same pattern continues for $$f$$ orbitals, again with some adjustments for the names for brevity.
If you're interested in the formalism, the sets of functions used to name $$d$$-orbitals form a basis for the set of traceless, rank two symmetric tensors. (That's why the name of the $$d_{z^2}$$ orbital has to be adjusted; the trace needs to be taken out. Tracelessness means the average value of the wavefunction on a sphere is zero.) The $$f$$-orbital names similarly are a basis for the set of traceless, rank three symmetric tensors.