[ledfrisby]

I know that the dz2 orbital has the same energy as the other d orbitals, and that it has a donut shaped concentration.  What I need to know is why it takes that shape.  

[Demotivator]

The orbitals are real physically, but the interesting thing is that they are derived/predicted  from the mathematics of the wave equation.
There are 5 d orbitals so you would think that there are five solution to the equation. As it turns out , the math actually produces six solutions to Schroedinger equation for l = 2, but there can only be five linearly independent solutions. The standard procedure in differential calculus is to use a linear combination of two dependent functions to produce one independent one. Thus, dz2 looks odd because it is a linear combination of d(z2 - x2) and d(z2 - y2) functions.

[Mitch]

All d-orbitals have 2 nodes the dz2 just happen to have nodes that are cones instead of planes.

[ledfrisby]

The orbitals are real physically, but the interesting thing is that they are derived/predicted  from the mathematics of the wave equation.
There are 5 d orbitals so you would think that there are five solution to the equation. As it turns out , the math actually produces six solutions to Schroedinger equation for l = 2, but there can only be five linearly independent solutions. The standard procedure in differential calculus is to use a linear combination of two dependent functions to produce one independent one. Thus, dz2 looks odd because it is a linear combination of d(z2 - x2) and d(z2 - y2) functions.

Thanks for the speedy reply.  I looked all over the internet, but couldn't find the answer until I came here.