The Polyene Series
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methyl radical
Now we will explore the electronic properties of the polyenes; methylene, ethylene, butadiene etc. Its useful first to look at the methyl radical CH3·. It has a planar structure with the carbon and three of the four valence in the plane. The hydrogens are in sigma orbitals which are symmetrical about the plane of the bond. The unpaired electron or radical is in a 2pz atomic orbital that is perpendicular to the plane of the molecule. The pi AO for that radical has a lobe above the plane, and a negative lobe below the plane. The probability of the electron is always zero at the nucleus. The lobes in the diagram represent the surface within which there is an 80% probability of finding the electron.
The energy of the π electronic orbital is much higher than the σ orbitals. All of the optical and electrical properties of the polyenes are due to highest occupied molecular orbitals which are the π orbitals.
ethylene
Ethylene is pi conjugated molecule. It can be thought of as two methyl radicals coming together and losing two hydrogens. It is coplanar molecule with a sigma bond between the carbons. Each carbon has a pi electron orbital that overlaps with the other because the carbons are close. In a perfectly planar system the pi electrons will not interact with the sigma bond. But as soon as the molecular becomes non coplanar there can be interaction.
If both pi wavefunctions are the same sign you get a bonding molecular orbital with a higher density between the carbon nuclei (no nodes). If the sign are different you get an antibonding orbital with a node between the nuclei. From a methyl radical like π atomic orbital contribute two pi electrons in to the bonding molecular orbital which will be the highest occupied molecular orbital (HOMO). The antibonding Π * (* denotes unoccupied) corresponds to the lowest unoccupied molecular orbital (LUMO). The energy difference between HOMO and LUMO is 7ev. This is the energy that would have to be supplied by a photon to bring promote an electron to the LUMO level. The visible part of the EM spectrum is between 1.5eV and 3eV. So the first optical transition occurs above the visible spectrum.
butadiene
From the point of view of the π level butadiene corresponds to the interaction between two ethylene sub-units. The pi homo level of one ethylene subunit interacts with the π homo level of the other subunit. There can be a positive combination and a negative combination, one leads to a level that is more stable one that is less stable. The same thing happens for the π* levels. The Pi* LUMO has come down while the π HOMO has increased. As a result the energy difference between HOMO and LUMO (the first optical transition ) has decreased to 5.4 eV.
Considering the wavefunction. The positive combination gives you all bonding for the lowest π bonding orbital. Then mixing two positive with two negative (++--) you fill the HOMO level resulting in a single node. In the π * orbitals the LUMO includes a negative combination (+--+) which results in bonding in the middle with 2 nodes. The fully antibonding molecular orbital has three nodes.
The bonding – antibonding character of the HOMO wavefunction translates to the double-bond / single-bond character of the geometry in the ground state. Higher electron density in the π MO corresponds to the double bond that allows the nuclei get closer (shorter bond length), lower density in single bond corresponds to the node between the antibonding orbitals.
hexatriene
Hexatriene has three ethylene like sub units, 3 occupied levels and 3 unoccuppied pi * levels.