I'm trying to understand Hartree-Fock well enough to write my own implementation. A point that I seem to be coming to that I'm not sure is correct: the method only produces one-electron orbitals that are linear combinations of the starting orbitals?
That is, if I have $n$ electrons, I start them off in some $n$ orbitals (say, atomic orbitals). Is it correct that the final result will only mix those $n$?
It seems very natural that I might want to include other orbital shapes -- maybe I want the first $m \gg n$ atomic orbitals in my molecule, and let all of those contribute to building lower-energy one-electron orbitals. As best s I understand it, standard HF does not allow for this, because at each step of diagonalizing the Fock matrix it's just using combinations of the existing orbitals -- with no room for the excited states.
Is my understanding correct? Are there other extensions of HF which allow broader mixings?
One possibility that I could imagine would be starting off with $m$ electrons, so that I have $m$ orbitals available. Then I run HF, and at the end I only use the $n$ lowest orbitals. This runs into the problem that I will still be computing Coloumb terms from the $(m-n)$ extra "fake" electrons, which will disturb the shape of my $n$ lowest orbitals. So perhaps the correct approach would be to use $m$ electrons mixing, but when I compute the Fock matrix $F$, I only include $J$ and $K$ terms from the $n$ lowest modes (but letting these affect the energies of all $n$ modes). But that seems messy, because now $F$ isn't symmetric... in the end this seems like a risky path to go down.
Help appreciated, thank you.