Why is the `SO_3` molecule an electrophile?

Because similar to `BF_3`, which is isoelectronic, `SO_3` has an antibonding MO (labeled `2a_2''`), formed from the sulfur `3p_z` and oxygen `2p_z` *atomic orbitals, that can accept electrons, thereby exhibiting Lewis acid behavior, which is the definition of an electrophile*.

Now, I can't actually find an `"SO"_3` MO diagram online, but I did find some time to derive it from scratch using symmetry-adapted linear combinations, along with the atomic orbital (AO) energies of the `3s` and `3p` AOs of sulfur (as regular AOs) and the `2s` and `2p` AOs of oxygen (as group orbitals; GOs).

I cross-checked it with the isoelectronic `"BF"_3` (Inorganic Chemistry, Miessler, pg. 160), and also used atomic orbital energies from pg. 134. :)

I assume you are unfamiliar with group orbitals, but it is basically a way of treating the interactions of multiple atomic orbitals at the same time.

THE MO DIAGRAM FOR SO3

Here is the MO diagram for `"SO"_3`:

To account for the electrons, we have:

• The `3a_1'` and `2e'` orbitals holding one lone pair for each oxygen (6 electrons total). They are closest in energy to oxygen's atomic orbitals, and so they mainly belong to oxygen.
• The `1a_2'` and `1e''` holding the other lone pair for each oxygen (6 electrons total). These nonbonding orbitals are from the `2p_x` of one oxygen and the `2p_z` of the two other oxygens, which are simply incompatible and cannot overlap with anything from sulfur. They are closest in energy to oxygen's atomic orbitals, and so they mainly belong to oxygen.
• The `4a_1'`, `3e'`, `1a_2''`, and `4e'` orbitals contributing to the `sp^2` double bonds (12 electrons total).

THE LUMO OF A LEWIS ACID IS THE ORBITAL THAT INTERACTS WITH THE HOMO OF A LEWIS BASE

The main thing here is that we can see the `2a_2''` MO is empty and that it is the lowest unoccupied molecular orbital (LUMO).

It is a `\mathbf(pi)` antibonding orbital that corresponds to the interaction between the `3p_z` (`A_2''`) atomic orbital of sulfur and the `2p_z` (`A_2''`) group orbital of the three oxygen atoms.

That `2a_2''` antibonding `pi` MO can accept electrons from an incoming Lewis base, which is why `"SO"_3` is a Lewis acid.

In the context of Organic Chemistry, a Lewis acid is also called an electrophile, or a lover of electrons. `"SO"_3` exhibits electrophilic behavior when a nucleophile (electron donor) tries to make a (non-proton) bond with it.

Since `SO_3` is a good Lewis acid, it is also an electrophile.