I am a not a chemistry student but a physics student. Nevertheless, I am quite familiar with molecular orbital theory and similar quantum chemistry concepts. However, I have problems understanding the ‚double bond‘ nomenclature, especially in conjugated molecules.
But let’s start simple: If we have a molecule like $C_2H_4$, there is a double bond between the two C atoms. This double bond is due to the overlap of the p orbitals and the overlap of the $sp_2$ orbitals. Until here, everything is clear for me. $$$$
There are also double bonds between every second carbon atom (structural formular in the top left). In the picture above (b) we see, that all p-orbitals of all carbon atoms overlap, producing four different molecular orbitals. In (a) we also see that there are $sp_2$ bonds between all carbon atoms.
My question is:
Why do we draw double bonds exactly between the two first and the last two C atoms in the structural formula? How does this follow out of the MO-picture (where we have four different $\pi$ type MO)? Why isn’t it possible to draw the double bond between the second and third carbon atom instead of between the first two, or even add an electron and draw a double bond between all carbon atoms (since all p-orbitals overlap)? Does it have something to do with the fact that only for the HOMO-orbital there is a node between the central two carbon atoms (where there is a also single bond drawn).
Pictures are taken from LibreTexts