# Why must the ground state of hydrogen molecule be gerade?

As we know, the electronic ground state of hydrogen molecule is $\Sigma_g^+$. My question is, how do we know it from the theoretical perspective?

The nuclear potential in the hamiltonian of hydrogen molecule is of the inversion symmetry, so the electronic wave function of the system must be gerade or ungerade. However, how can we derive that the ground state is of gerade symmetry?