Application of the free electron model to polyenes

Application of the free electron model to a polyene involves the assumption that the "box" contains the n atoms of the polyene. This is shown for butadiene below. The wavefunction coefficients are derived from the amplitude of the sine function obtained from the solution of the particle in a box. The particle in a box solutions are:

where a is the length of the box and j is the quantum number for a given state. If we imagine that an electron is placed in a box that contains N atoms at positions na/N along the box then the free electron model (FEM) states that the wavefunction coefficients for a polyene will be given by:

 

This is illustrated for butadiene below. The four orbitals shown correspond to the four p orbitals of butadiene.

This is lowest p orbital. It has no nodes. The next highest orbital has one node.

This is the highest occupied orbital as can be seen from Huckel theory as well. Note that both the free electron model and Huckel theory predict that the bonds between atoms 1- 2 and 3 - 4 will be stronger than the bond from 2-3. The end bonds have double bond character whereas the central bond has single bond character. Note the node between atoms 2 and 3 suggesting that the p interaction is anti-bonding and only the s interaction between these two atoms is important. This important prediction is born out for the entire series of polyenes (see the example of hexatriene).

 

The LUMO of butadiene is:

The NLUMO and highest energy p orbital is:

Because there is a node between each atom this orbital is completely p anti-bonding. Using particle in a box energies, the FEM can be used to model the spectral transitions of polyenes.

For a transition from level j to level j + 1 we find that:

According to the Bohr condition

We can evaluate the constants 8mc/h so that l can be expressed in nanometers and a in Angstroms whereupon we obtain:

However, this model does not work particularly well for polyenes themselves. There are several reasons for this.

  1. The geometry of polyenes is not linear, but zig-zag with bond angles of ca. 120 degrees.
  2. The conjugation pattern in the HOMO is an alternating double and single bond pattern and thus the electron is not really "free" as assumed in the FEM.
  3. Interactions among excited states known as configuration interaction may play a role in the transition energies. Configuration interaction is prevalent among polyenes since they tend to have state symmetries that alternate A and B symmetry. By symmetry the A states can mix and the B states can mix. This sometimes changes the ordering of the states.

 Charge transfer states tend to cause the electron to more effectively delocalized along the linear carbon chain. This can be seen by the comparison below.

Note that the charge transfer state changes the pattern of alternation. If we think of the CT state as a resonance structure it tends to cause the electron to be better delocalized along the chain.

Such delocalization is observed in cyanine dyes. For these dyes the FE model works extremely well. Thus, in substituted polyenes and dyes the FE model works better than in polyenes themselves.