# How can real $d$ orbitals be computed from complex orbitals?

I recently completed MIT's 8.04 quantum mechanics course on edX and have been writing python code to compute hydrogen-like electron orbitals, basically just for fun. My program computes the eigenstates based on $$n$$, $$l$$, and $$m$$ and uses the mayavi 3D graphics package to plot it.

My question concerns computing the $$s$$, $$p$$, $$d$$, and $$f$$ orbitals in real bases by superposition of the (complex) eigenstates. I found how to do this for $$p_x$$ and $$p_y$$ and it works as intended.

$$p_z = p_0 \\ p_x = \frac{1}{\sqrt{2}} \left(p_1 + p_{-1} \right) \\ p_y = \frac{1}{i\sqrt{2}} \left( p_1 - p_{-1} \right)$$

So how do you do something analogous to compute $$d_{z^2}$$, $$d_{xz}$$, $$d_{yz}$$, $$d_{xy}$$, and $$d_{x^2-y^2}$$? From looking at some tables and observing patterns, I'm guessing that I need to do:

$$d_{z^2} = d_0 \\ d_{xz} = \frac{1}{\sqrt{2}} \left(d_1 + d_{-1} \right) \\ d_{yz} = \frac{1}{i\sqrt{2}} \left( d_1 - d_{-1} \right)\\ d_{xy} = \frac{1}{\sqrt{2}} \left(d_2 + d_{-2} \right) \\ d_{x^2 - y^2} = \frac{1}{i\sqrt{2}} \left( d_2 - d_{-2} \right)$$

and similarly for the $$f$$ orbitals with $$m = \pm 1, 2, 3$$.

Thanks for any clarification on this. Nothing important is on the line, but I hope to understand how this works.

## 1 Answer

Simply check the appropriate combinations of spherical harmonics. Indeed the real functions are sums and differences $$Y_{\ell}^m$$ and $$Y_{\ell}^{-m}$$. The real forms are given a little later in the same wiki page.