When repeating multi-electron atoms, I often read that closed shells have a symmetric charge density. It was justified by the following addition theorem for spherical harmonics everytime:
which is the sum of the squares of the magnitude of the angular part of the wave functions. Since the radial parts aren't functions of the angles, the whole probability density should be sphericaly symmetric.
But is the probability density of the whole multi-electron wave function really just the sum of the magnitude of the filled orbitals? I mean the wave function for two electrons in two different states (1 and 2) are:
$\Psi(r_1,r_2) \propto \psi_1(r_1) \psi_2(r_2) \pm \psi_2(r_1) \psi_1(r_2)$
so $|\Psi|^2$ is not just a sum over the magnitude of the single states.