Lecture #25
  CURMUDGEON GENERAL'S WARNING. These "slides" represent highlights from lecture and are neither complete nor meant to replace lecture. It is advised not to use these as a reliable means to replace missed lecture material. Do so at risk to healthy academic performance in 09-105.
Lecture Outline Molecular Orbitals (delocalized)

Energies via particle-in-a-box

Conjugated systems

Strained bonds

Metallic bond

Here we have a more delocalized system in which the absorbed wavelength has consequently shifted to the visible region of the electromagnetic spectrum and the highest occupied orbital starts out with a larger n quantum number.
We began the look at delocalized molecular orbitals with a discussion of the (energy and geometry) properties of 1,3 butadiene, indicating that we could understand them if the central CC bond had some double bond character.
Yet just previously, we noted a similar looking structure had the flexibility associated with unrestricted rotation about a sigma bond.
The difference, reminding us of what leads to delocalized molecular orbitals, is revealed in looking in more detail at the situation where we do get unrestricted rotation, starting here...
Finally, a look at the atomic orbitals shows that in "rubber" one gets localized pi bonds whereas in the butadiene, delocalized pi bonds arise.
Another major application of the particle-in-a-box model is to describe bonding in metals. We will begin our discussion with a chain of lithium atoms starting with Li2.  
If we next go to Li4, we can note, soon enough, that delocalized molecular orbitals are possible using 2s atomic orbitals.  
 The four atomic 2s orbitals can overlap constructively to give the indicated molecular orbital which is delocalized over the four nuclei (and their inner 1s cores).  
 Shown on the left are the four delocalized combinations, which look somewhat like the delocalized pi orbitals we derived for 1,3 butadiene -- H2C=CH-CH=CH2.

One the right are the four lowest particle-in-a-box density distributions which do a fairly good job of representing the delocalized molecular orbitals on the left.

The diagram contains the 'bond order per electron'. Multiple each such partial bond order by the number of electrons delocalized over the level and add up the contribution from each level...at each bond. Example: 2*(,22)+2*(.22)=.88
 Since the particle-in-a-box does "okay", we can use it to get some information about energy levels associated with delocalized orbitals, information that was missing so far.  
 In continuing to build longer and longer chains of lithium molecules, the levels fill in a simple way, indicated here, with a big jump going to macroscopic amounts (Avogadro's number) of lithiums in one long molecule. This will be the basis of our understanding of the metallic bond.  
 Next to lithium on the Periodic Table is beryllium, but the levels seem to fill to the top. (Bonding and antibonding are equally occupied and there seems to be no metallic bond possible.  
However, remember that there are more levels that could come into play.  
Above the 2s lie the 2p orbitals which contribute another band of levels. The last electrons occupy pi bonding orbitals in the 2p band. There are enough bonding delocalized molecular orbitals for all the valence electrons to be considered delocalized and the system bound as one giant molecule.  
If we consider boron and carbon instead of beryllium, the 2s-2p splitting grows.  
Eventually, as we approach the right side of the periodic table, the 2s-2p are far enough apart that the two bands originating there no longer overlap. There is now an energy gap that most be overcome to access delocalized orbitals spread over the giant molecule. If the energy needed to move an electron into one of these available conducting orbitals is large, we have an electronic insulator, since the electrons are essentially confined to the lower band.