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.
 A: 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.
