What orbitals form sigma bonds?

`sigma` bonds must be made by orbitals that overlap head-on.

POSSIBLE ORBITAL COMBINATIONS TO GENERATE SIGMA MOLECULAR ORBITALS

For simplicity, if we examine only the `s`, `p`, and `d` orbitals, let's suppose that all orbitals we are examining are similar enough in energy to interact.

Let's also suppose that we are ignoring `d`-`d` interactions, since we know those should work (they are the same `l` so it's not so interesting).

Then, orbitals that can capably overlap with each other to form `sigma` bonds include the following linear combinations:

• `s + p_z -> sigma` (bonding)
• `s - p_z -> sigma^"*"` (antibonding)
• `s + d_(z^2) -> sigma` (bonding)
• `s - d_(z^2) -> sigma^"*"` (antibonding)
• `s + d_(x^2-y^2) -> sigma` (bonding)
• `s - d_(x^2-y^2) -> sigma^"*"` (antibonding)
• `p_z + d_(z^2) -> sigma` (bonding)
• `p_z - d_(z^2) -> sigma^"*"` (antibonding)
• `p_x + d_(x^2 - y^2) -> sigma` (bonding)
• `p_x - d_(x^2 - y^2) -> sigma^"*"` (antibonding)
• `p_y + d_(x^2 - y^2) -> sigma` (bonding)
• `p_y - d_(x^2 - y^2) -> sigma^"*"` (antibonding)

where the `z` axis is the internuclear axis (i.e. the axis along which the single bond---which is also a `sigma` bond---is made), and the `x` and `y` axes are where you should expect them to be for the Cartesian coordinate system.

We would also suppose that the antibonding molecular orbitals are unoccupied so that the bond is a standard single bond.

HOW TO DEPICT/IMAGINE THESE ORBITAL OVERLAPS

When you sketch these orbital overlaps:

• All `s` orbitals are spheres. The only way these can change sign is if the whole thing changes sign.
• The `p_z` orbitals can be approximated as dumbbells, regardless of their `n`, without losing the essence of the `sigma` MOs generated (head-on overlap). One lobe is the opposite sign to the other.
• The `d_(z^2)` look almost like `p_z` orbitals, except there is a donut in the middle. You can also approximate these as dumbbells, regardless of their `n`, without losing the essence of the `sigma` MOs generated (head-on overlap). Both lobes are the same sign.
• The `d_(x^2-y^2)` can be approximated as four-leaf clovers, essentially, on the `xy`-plane, with the lobes aligned along the `x` and `y` axes. The opposite lobes along each axis are the same sign. Therefore, they overlap with the `p_x` and `p_y`, which also lie long those axes.

Since they are all aligned along the same axis (`p_z` with `d_(z^2)`, `p_x` with `d_(x^2-y^2)`, and `p_y` with `d_(x^2-y^2)`) AND they are compatible (`s` with `p_z`, `s` with `d_(z^2)`, and `s` with `d_(x^2-y^2)`), they form `sigma` bonding and `sigma^"*"` antibonding MOs. Since we supposed that only the `sigma` bonding MO is occupied, we have a single bond.

(Since `s` orbitals are spheres, it doesn't matter along which axis they bond.)

Of course, there exist `f` orbitals of some sort that can overlap with `s`, `p_z`, `d_(z^2)`, and `d_(x^2-y^2)` in a `sigma` fashion, but that's left up to the really motivated chemist to figure out.