Direction of valence bond and Unsöld's theorem

Is it true that the atomic orbitals of separate atoms already have directionality, which ensures the directionality of the valence bond? Or another words, what is the real spatial distribution of the electron density in the atom, obtained in the one-electron approximation?

Doesn't this contradict Unsöld's theorem that states the square of the total electron wavefunction for a filled or half-filled sub-shell is spherically symmetric?

• I think you might be getting confused by the way those orbitals are drawn. Yes, they have directionality, but they aren't long and narrow like that. For example, a p orbital is more like two big spheres, such that all three p orbitals collectively make a big sphere. That's why if all three of them are half-filled or are all completely filled, the net density is spherical. – Andrew Feb 3 at 16:12
• @Andrew Ok, if we look at the d orbital along z-axix. It differ from y- and x-orbitals. But an atom does not know about any axis, so, what a real angular distribution of electron density in atom with d - orbital? – Sergio Feb 3 at 16:56
• A half-filled subshell means one electron in each of the orbitals, so it doesn't matter that the dz2 orbital looks a bit different from the others. What matters is the sum of all five d orbitals. Look up the equations for the five hydrogen atom d orbital wave functions in spherical coordinates. Square each one to get the electron density. Add all five together and simplify the result. You'll see that all of the angular dependence disappears and the result is only dependent on r, ie it's spherical. – Andrew Feb 3 at 17:22
• @Andrew "You'll see that all of the angular dependence disappears and the result is only dependent on r, ie it's spherical." --- yes, I did it $(2\ell + 1)/4\pi |R(r)|_{nl}^2$. But I'm interested in the direction of valence in this regard. For example, to which end should a two hydrogen atoms join another d-valence orbital atom? – Sergio Feb 3 at 17:28