Let's have an atomic carbon with the following electron configuration:
$$ 1s^2 2s^2 2p^2 $$
One of it's levels is ${}^1S_0$, which is corresponding with the following state:
$$ \begin{align} \left| S=0, M_S=0 \right> &= \frac{1}{\sqrt{2}}\Big( \alpha(1)\beta(2) - \alpha(2)\beta(1) \Big)\\ \left| L=0, M_L=0 \right> &= \frac{1}{\sqrt{3}}\Big( \left| p^+(1)p^-(2) \right> - \left| p^0(1)p^0(2) \right> + \left| p^-(1)p^+(2) \right> \Big)\\ \left| S=0, M_S=0, L=0, M_L=0 \right> &= \frac{1}{\sqrt{6}} \Big( \left| p^+(1)\alpha(1)p^-(2)\beta(2) \right>- \left| p^+(1)\beta(1)p^-(2)\alpha(2) \right> \\&- \left| p^0(1)\alpha(1)p^0(2)\beta(2) \right> + \left| p^0(1)\beta(1)p^0(2)\alpha(2) \right> \\&+ \left| p^-(1)\alpha(1)p^+(2)\beta(2) \right> - \left| p^-(1)\beta(1)p^+(2)\alpha(2) \right> \Big) \end{align} $$
Now I know, that ${}^1S_0$ state is supposed to be totally symmetric, i.e. $A_g$ in the $D_{2h}$ point group. But is it possible to derive it "rigorously" from the wavefunction?