I understand $d_{xy}$ and $d_{yz}$ and those orientations but ones like $d_{z^2}$ are very strange to me.

Can someone please explain that to me?

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    $\begingroup$ Does this help? $\endgroup$ – Klaus-Dieter Warzecha Apr 3 '15 at 16:15
  • $\begingroup$ Welcome to chemistry.SE! Yes. I wanted to add @Klaus's link too. Does that solve your problem? $\endgroup$ – M.A.R. Apr 3 '15 at 16:16
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    $\begingroup$ Generally speaking, and very informally (I'd get shredded if I put this an answer), you want all 3 planes and all 3 axes to have a double dumbbell in them. This is not possible with 5 orbitals, 3 orbitals each put two dumbbells in the planes of coordinate system (xy,yz,xz). The remaining two have 4 dumbbells occupy 3 axes. So 1 of the dumbbells 'folds up' into the donut shape. This is just how the math works out. $\endgroup$ – Lighthart Apr 3 '15 at 22:19
  • $\begingroup$ And even shorter than Lighthart’s description: This is just how the maths works out. $\endgroup$ – Jan Jun 9 '15 at 22:44

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