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.