Lecture #24
Much not in text; some in Section 14.5
  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)

1,3 butadiene example

Excited states

Benzene

Ozone

Particle-in-a-box energies
The difference in the molecule's structure when molecular orbital theory is used can be accommodated with resonance Lewis structures that are not what we call "preferred" structures, but which nevertheless affect the overall appearance of the molecule. The bonds are neither pure double bonds nor pure single bonds.
Next, we'll consider structural changes in the molecule when it is in an excited state.
Interpretation of the change follows in an uncomplicated manner.
Way back, when we were finishing up Lewis structures, we used benzene as the prime example of where resonance entered into play. In the molecular orbital view, the resonance structures arise from place electrons into delocalized orbitals constructed from various combinations of atomic p orbitals. Note that steric number three at each carbon gives us a trio of sp2 hybrid atomic orbitals and a pure 2p atomic orbital perpendicular to their plane.  
The six 2p atomic orbitals are in position to overlap in various ways, the most energetically stable being constructive interference everywhere. This provides a picture of the first pi molecular orbital.  
A second combination is shown here in which a couple of nodes are implied where the orbital amplitude switches from + to -.  
The highest energy, least stable, molecular orbital comes from the combination of atomic orbitals that has the most destructive interference. Wave amplitude between neighboring carbons is depleted in this combination which is antibonding everywhere.  
The six combinations, some of which were not illustrated, have energies indicated in this diagram. The construction of the diagram is beyond this course, but its use is straightforward since a total of six electrons must be placed...as shown.  
The ozone molecule's Lewis structure shows that even the preferred structure in this case indicates resonance is required for a complete description of the molecule's valence electrons. Note that a pair of electrons involved in bonding oxygens is delocalized. But also note that a "non-bonding", lone pair is also resonating between both end sites in the Lewis structure.
Of the 18 valence electrons in ozone, two pair are involved in resonance. The remaining 14 electrons constitute the sigma bonded framework shown here as coming from sp2 hybrids at each oxygen (since each is in an electronic geometry involving three groups).
There is a pure 2p atomic orbital at each oxygen perpendicular to the plane of the molecule. These lead, via linear combinations, to three delocalized pi molecular orbitals. The one involving constructive interference leading to a bonding molecular orbital is illustrated at the right.
Here is the second linear combination.
And the third, the antibonding delocalized molecular orbital corresponding to the bonding one. You should be able to figure out the "partial bond order per electron" for an electron in each of these three orbitals.
The pi molecular orbital energy diagram for ozone into which are distributed four valence electrons. Two are in the bonding orbital and yield a bond order of 0.7, which when added to the sigma bond corresponds to a bond order of 1.7 between each of the oxygens. The remaining pair is distributed at each end, corresponding to the negative charge simultaneously at each end oxygen in the Lewis structure.
 Changing the electron configuration of non-
bonding orbitals does not affect the structure significcantly.		  
Another triatomic molecule is carbon dioxide. The text presentation of this is awkward, so it's completely re-constructed here following the arguments that should seem familiar. Here's the underlying framework with the hybrids on the end oxygens lying in two planes perpendicular to each other. This arrangement will lead, in the next slide, to the ready formation of two double bonds.
The pure p orbitals perpendicular to the hybrids overlap effectively in pairs -- not all three at once -- as shown here. Two pure double bonds arise in this manner. The placement of all 16 valence electrons is indicated.
The particle-in-a-box model can now be profitably revisited. Showing the first five waves and their energies. There are, of course, an infinite number of such possibilites corresponding to shorter and shorter wavelengths and higher and higher energies.
The shape of the n=1 particle-in-a-box wave function approximates the appearance of the ground state pi wave function for the delocalized molecular orbital in 1,3 butadiene for instance.
The shape of the n=2 particle-in-a-box wave function approximates the appearance of the p2 orbital in a delocalized system.
The shape of the n=3 particle-in-a-box wave function approximates the appearance of the p*3 delocalized orbital.
The shape of the n=4 particle-in-a-box wave function approximates the appearance of the p*4 delocalized orbital.
There are four electrons "free" to delocalize in the 1,3 butadiene system H2C=CH-CH-CH2
The average carbon-carbon bondlength (C=C and C-C) is 142 pm and allows calculation of the wavelength of the absorbed light by using the particle-in-a-box to estimate the energy levels. The ultraviolet wavelength is 210 nm. We allow the box to extend beyond the outer nuclei by another one-half bond at each end.