How can you tell if a Gauss-type orbital is an eigenfunction of Hamiltionan $H$? For example:
$$GTO = N z^2 \exp\left(-\alpha r^2\right)$$
I know it is and eigenfunction of $L_z$ and not $L_x$ and $L_y$. But is it not an eigenfunction of the Hamiltonian $H$ and $L^2$? If I know it is one of $H$, I know it will be one of $L^2$ as well since they commute.