Question

If I have two atoms that come in with one half-filled orbital each, do they make a `sigma` bond?

Answer 1

Well, yeah. If you have two atoms each with a singly-occupied atomic orbital, if they come together and have ideal orbital overlap to form molecular orbitals, the number of electrons is conserved and the number of orbitals is conserved.

If each atom comes in with one electron in an atomic orbital, then the molecule must have a total of two electrons contributed in total from those two atoms (and two molecular orbitals result, one bonding and one antibonding).

(Here, the bonding orbital is the `sigma_(1s)`, and the antibonding, `sigma_(1s)^"*"`.)

If two electrons, one in each atomic orbital, start off unpaired, they become paired (if and only if the degeneracy`""^([1])` of the orbital is `1`!), as they are expected to follow the Aufbau principle and fill all orbitals of one energy level before going on to the next energy level.

And in fact, for the first covalent bond formed in any molecule, one in-phase and one out-of-phase `bbsigma` (sigma) overlap (head-on overlap) occur, which forms two molecular orbitals each with a degeneracy of `1` as seen here in general for any `s` orbital overlap:

So, by the definition of degeneracy, there is only one sigma molecular orbital in its own energy level, and thus the two electrons have to go in the same `sigma` molecular orbital before going into the `sigma^"*"`.

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Footnote:

`""^([1])` Degeneracy is just the number of orbitals with the same energy. A degeneracy of `2` means there are two orbitals that non-coincidentally have the same exact energy. `sigma` orbitals always have a degeneracy of `1`, so that if there are two electrons that are to fill the particular energy level, they will pair up.