Lecture #24
  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 Delocalized molecular orbitals


Carbon dioxide

Energies via particle-in-a-box

Light-induced transitions

Particle-in-a-box and delocalized molecular orbitals

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.
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 0.5, which when added to the sigma bond corresponds to a bond order of 1.5 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.
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.