Atomic orbitals I just studied atomic orbitals in a theoretical QM class, and I'm left with several questions, that are probably more questions in quantum chemistry:


*

*Many orbitals seem to have a preferred axis - how is that axis
"chosen"? Does it move constantly, or is it actually a fixed
direction in space? Why is there an axis at all when the problem is
spherically symmetric? Does the shape of the orbitals have any
bearing on chemistry? 

*We can see lobes in preferred directions - do
these actually play a role in chemical reactions, or maybe in
crystallography for the arrangement of crystals? 

*If I understand
correctly, electrons are actually represented by superpositions of
spherical harmonics - with an arbitrary superposition, the
probability density for the electron could be anything - can we
actually find the coefficients of the superposition an electron
actually is in?

*Lastly, there is an orbital which has a high probability of the electron being at the center of the nucleus (I don't remember which one) - is that anything interesting?

 A: Regarding your first two points:
The symmetry axis of an orbital is free for a free atom. If it's bound to some other atom through one of these one-dimensionally elongated orbitals, the orientation of one orbital is fixed.
If you take e.g. carbon, silicon or germanium, you have one s orbital and three p orbitals, which are oriented perpendicular to each other. Because they are so close in energy, they hybridize and therefore lead to a an sp3 hybrid. In a simple picture, the electrons repel each other as far as possible, which leads to bonding in a tetrahedral configuration with bond angles of 109.4 °. Therefore the atomic orbitals influence the crystal structure or symmetry.
Your third point is honestly unclear to me.
I'm not sure, what you mean by the fourth one. The "center of mass" has different spacings with respect to the core. But there should not be an orbital, which has its highest probability located at the core. This would mean that electrons overlap with the core, which is not energetically favorable.
A: $$
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I'll try to answer the last two.

with an arbitrary superposition, the probability density for the electron could be anything - can we actually find the coefficients of the superposition an electron actually is in?

I'm a little confused about what you mean here. If we are given $\ket{\psi}$ as a combination of, say, $\ket{n \ell m}$ states, we can use that to calculate which spherical harmonics are present, since we know (for hydrogenic atoms) how to decompose $\psi(\vec{x}) = R_{n \ell} (r) Y_{\ell m} ( \Omega)$. Sure, the probability density "could be" anything, but if you want something really weird you might have to engineer it. (Look into recent work on Rydberg atoms for some fun superpositions.) On the other hand, do you mean, given a real electron in an atom, is there a way to directly measure the probability distribution? That's a much harder experimental question.

Lastly, there is an orbital which has a high probability of the electron being at the center of the nucleus (I don't remember which one) - is that anything interesting?

Actually yes. It's the $s$ ($\ell = 0$) orbital, because any angular momentum will keep you pushed out of the nucleus. This becomes important in hyperfine splitting, where the electron interacts with the nuclear magnetic moment, and this is obviously much easier if you spend time close to the nucleus, so the effect is larger.
