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 Molecular Orbitals in Polyatomic Molecules (Localized Bonds)

Rings

Strained bonds

Molecular Orbitals in Polyatomic Molecules (Delocalized Bonds)

Bond orders

Partial bond order per electron

1.3 butadiene
Here we have sigma bonds, but free rotation is blocked by their incorporation into a ring structure.
This is penicillin, one of many variations of the well-known antibiotic.  
 The five-membered, if a perfect pentagon, would have 108 degree bond angles. The corner atoms of the ring have tetrahedral geometry with 109.5 degree angles between hybrid atomic orbitals allowing effective overlap with neighborinring atomic orbitals.  
 In the four-membered ring, the bond angles are ~90 degrees, causing the overlap between neighboring orbitals to be much less effective (less region of constructive interference) since they spread to larger angles.  
The topic of molecular orbitals is extended to systems in which the orbitals encompass more than just two centers -- atoms -- and are referred to then as delocalized orbitals, in contrast to our previous discussion of localized orbitals (between just two atoms).
A non-essential view of the role that the "coefficients" play in determining bond properties. The coefficients, c, actually emerge when the linear combinations are plugged into the Schrodinger Equation. The bond order that results for a link between A and B is then proportional to the product cAcB and can be positive (bonding), negative (antibonding), or zero (non-bonding). This may be useful to your comfort level, but is not critical.
For two identical atoms coming together, the atomic p orbitals perpendicular to the bond axis form constructive and destructive combinations shown more rigorously here.
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 computer-generated "true" result is shown as well.
The rigorously calculated molecular orbital wave function has coefficients shown here, from which the partial bond order per electron, cicj between linked atoms, may be calculated. Starting from the left carbon, c1*c2 = .37 * .60 = .22. Each electron in this first orbital contributes a pi bond order of .22 to the structure.
Another of the four combinations of the four atomic orbitals is shown here. This is also a bonding molecular orbital.
More rigorous picture and computer generated view of the second delocalized molecular orbital in the butadiene.
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 from the partial bond order per electron that would be given.

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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.
Another property of the molecule that is revealed here is that, because the central bond is not a freely rotating single bond, but rather has partial double bond character restricting rotational motion, there can be two geometrical (cis-trans) isomers.