why not dx^2 or dx^2-z^2 or dz^2-y^2 orbitals exist ?

Question

is it because of choice of axes is made by us or is it related to wave function?

Answer 1

There are many solutions of the Schrodinger equation for the set of d-orbitals. The only catch is that they have to have the right angular momentum and fit into a set of five orthogonal orbitals. The set that is commonly used is established by convention. You could easily make an orbital set based around a dx^2 orbital (you'd have xz,yz,xy and z^2-y^2 in there too), but people would look at you funny when you start mentioning orbitals they don't use.

Answer 2

The orbital d are at the number five . You find them by solving Schrôdinger's equation . orbitals d correspond to the subshell l=2 and the only solutions are dxy, dyz , dzx , dz^2 and dx^2-y^2.