Lecture #23
  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 Quantum Theory of the Chemical Bond

Molecular Orbitals in Polyatomic Molecules

Molecular orbitals from combinations of hybrid atomic orbitals

The rigid double bond

The freely rotating single sigma bond

Strained bonds

A restricted rotation double bond can undergo a "photochemical excitation" to an electronic state in which the pi electrons revert to their atomic orbitals owing to the cancelation of bonding vs antibonding contributions. In this excited state, the CC bond can now rotate briefly, while the excited molecule lasts.
Using a four-carbon system whose Lewis structure indicates has two double bonds and one single bond, we will show the molecular orbitals describing the pi bonding are delocalized.
Each carbon atom has a sigma framework involving sp2 orbitals leaving a pure 2p atomic orbital from which the pi molecular orbital will arise.
One of the linear combinations of the four atomic 2p orbitals is entirely constructive -- in phase over all four atoms -- and its resulting delocalized bonding molecular orbital allows any electrons it contains to be "spread" over the entire structure.
The contribution of an overlap region to bond order may be quantified with a simple relationship involving the weighting coefficients "c" from the combining atomic orbitals. For homonuclear diatomics, all the c's are equal. For heteronuclear diatomic molecules, in the bonding molecular orbital, c1 << c2 if c2 is the weighting factor of the more electronegative species.  
 This shows the equal contributions from 2p orbitals in a homonuclear diatomic molecule. The actual weighting coefficients are given (and will be provided, their derivation being beyond what we need). The lower pi bonding combination gives a "partial bond order per electron" of 0.50 meaning two electrons in this state (spin up and spin down) would give a bond order of 1.00 to the system. The antibonding partial bond order is negative and exactly cancels the effect of an electron in its bonding counterpart.  
 For the 4 overlapping 2p orbitals in 1,3 butadiene, the weighting coefficients in the lowest, bonding pi orbital are shown and these give rise to the "bond order per electron" indicated between the orbitals. The general trend in amplitudes (contributions) is shown by the dashed line.  
Another of the four combinations of the four atomic orbitals is shown here. This is also a bonding molecular orbital.
The actual, quantitative "partial bond orders per electron" and the weighting coefficients on the next linear combination are given here (along with the general trend as a dashed line).
A look at the original Lewis structure reminds us that there are four pi electrons -- two from each of the "double" part of the double bonds -- to be distributed into the pi molecular orbitals.
Placing one of the pairs of pi electrons into the lowest orbitals allows us to calculate the bond order due to these electrons at each carbon-carbon link. We introduce the idea of a partial bond order per electron associated with each interatomic link for each molecular orbital. Here, that partial bond order per electron is half the result we got for the pair of electrons or one-sixth for each CC link in pi orbital #1.The bond order -- one-half the average number of electrons per bond -- may be obtained from a given bond's partial bond order per electron by multiplying by the number of electrons in the orbital. Vice versa, if you knew how the electron(s) in a given molecular orbital were distributed over the various links, you could get the partial bond order per electron as we did for the lowest pi-orbital.
The remaining (second) pair of pi electrons goes into the next state. We again calculate the bond order at each carbon-carbon link due to these two electrons. The partial bond order per electron is one-quarter for each of the outer links and zero for the center CC link.
The complete electron configuration for the four pi electrons.
The complete bond order for each carbon-carbon link with contributions considered from the sigma and pi molecular orbitals summed.
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.
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.