Solution of the Roothaan Equations of H2 by Symmetry Arguments FWIW my background is in physics and maths, but I am just starting a chemistry PhD (the last time I took a chemistry class was high school). I have only some background in representation theory, and only an abstract taste, as it was in a maths class (bonus: suggest a reasonable representation/group theory reference with applications to chemistry, preferably a concise treatment).
In Szabo and Ostlund's Modern Quantum Chemistry, section 3.5.2 they solve the ${\rm H}_2$ molecule (in Hartree-Fock theory) using a minimal STO-3G basis, $\phi_1$ and $\phi_2$ (centred on each proton). I believe I understand how to set up the Roothaan equations, guessing the density matrix, and solving self-consistently. However, they go on to say &quot;The canonical molecular orbitals will form a representation of the point group of the molecule.&quot; Um, okay. Why? And what is meant by canonical here? They go on to say they can be labelled by their symmetry $\sigma_u, \sigma_g, \pi_u, \pi_g$, etc. (I know these terms from wikipedia, and it is discussed briefly in another chapter) I don't fully understand why they (can) do this.
Now, I understand we have two molecular orbitals so it makes sense that they choose only the 2 lowest energies. However, from the set of options above, they state without proof that the $\sigma_g$ is the lowest energy, followed by $\sigma_u$ (I also don't really understand where the wave function comes from). How do we conclude this? Is it because the hydrogen atoms are 1.4 Bohr radii away, hence close and hence we expect bonding (not antibonding?), which I understand to be represented by $\sigma_g$? If this is correct, then it's still not rigorous so I would appreciate any more explanation.
After speaking to a coworker and reading some more I think I have a better understanding.
The symmetry of the wave function needs to follow the symmetry of the molecule (unless we are studying a broken symmetry solution, e.g. charge transfer, but that doesn't apply in this case). We are using a minimal basis, and hence we have only two orbitals. From atomic theory, these are s functions. Hence, they cannot form $\pi$ orbitals, and we are left with $\sigma$ orbitals. Between $\sigma_g$ and $\sigma_u$, $\sigma_g$ has the lower energy because it has no nodes, whereas $\sigma_u$ contains a node. From QM we know that this generally implies greater energy, but there's a nice explanation on this here (at the end he also specifically discusses a homonuclear diatomic molecule).
The specifics of how to get the symmetry representation for the molecule and match those with wave functions is still not entirely clear to me, but for this I will read a book on group theory and its chemical applications. It is, however, clear that $\phi_1+\phi_2$ and $\phi_1-\phi_2$ respect $\sigma_g$ and $\sigma_u$ symmetry respectively.