Symmetry of ground state electron configuration Take a molecule, whose atoms have a symmetry $S\in \text{O}(3)$ (i.e. S permutes the atoms). 
$S$ also acts on the wavefuncion $\psi(x_1,...,x_n)$ of the $n$ electrons in the molecule, by its action on the $x_i$ (maybe the most natural action also permutes the $x_i$, I'm not sure).
Is it true that the electron configuration $\psi_0$ of least energy (the ground state one) is invariant under $S$? If not, when is it/is it not?
 A: Generally, it need not be the case that the electronic ground state of any given molecule will be invariant under a symmetry transformation $S$ which preserves the overall configuration of the nuclear positions. This can be shown to be the case if the ground state is non-degenerate, but if you have a degenerate ground-state manifold, then the generic case is for the symmetry transformations to swap between the different states within that degenerate manifold.
For an example where this happens, consider the nitric oxide molecule NO, for which the ground state includes one unpaired electron in a pi orbital, which has a degeneracy of 2, i.e. it has a $\pi_x$ and a $\pi_y$ orbitals that congregate electron density in the $x$ (resp. $y$) directions orthogonally to the molecular axis (along $z$). As such, a 90° rotation about the molecular axis will not respect the molecular ground state, but it will change the $\pi_x$ state to the $\pi_y$ state, and vice versa.
More generally, if you have an arbitrary molecule, then one of the first parts of the analysis is to understand its symmetry group $G\leq \rm SO(3)$, and then to understand its representations, using the formal machinery of representation theory within quantum mechanics, as it is the representations of $G$ that will encode the different ways that the symmetry group can act within each degenerate eigenspace. Most molecular physics and quantum chemistry textbooks (or at least, those beyond the pure introductory presentations) devote large sections to this material.
