Photophysics of Chlorophylls and Carotenoids

Abstract

Chlorophylls and Carotenoids are very important light receptors. Both classes of molecules have a large $\pi$-electron system which is delocalized over many conjugated bonds and is responsible for strong absorption bands in the visible region. In this chapter, we introduce the molecular orbital method for the electronic wavefunction. We apply the free electron model and the Höckel MO method to linear and cyclic polyenes s as model systems and discuss Gouterman’s four orbital model for Porphyrins and Kohler’s simplified CI model for polyenes. Finally, we comment on energy transfer processes involving Chlorophylls and Carotenoids in photosynthesis.

Problems

23.1

Polyene with bond length alternation

Consider a cyclic polyene with 2 N carbon atoms with alternating bond lengths.

The Hückel matrix has the form $H=\left( \begin{array}{ccccc} \alpha &{} \beta &{} &{} &{} \beta '\\ \beta &{} \alpha &{} \beta '\\ &{} \beta ' &{} \ddots &{} \ddots \\ &{} &{} \ddots &{} \alpha &{} \beta \\ \beta ' &{} &{} &{} \beta &{} \alpha \end{array}\right) .$

• (a) Show that the eigenvectors can be written as

  $$ c_{2n}=\mathrm{e}^{\text {i}kn}\quad c_{2n-1}=\mathrm{e}^{\text {i}(kn+\chi )}. $$

• (b) Determine the phase angle $\chi$ and the eigenvalues for $\beta \ne \beta '$.

• (c) We want now to find the eigenvectors of a linear polyene. Therefore we use the real valued functions

  $$ c_{2n}=\sin kn\quad \quad c_{2n-1}=\sin (kn+\chi )\quad n=1\ldots N $$

  with the phase angle as in (b).

We now add two further Carbon atoms with indices 0 and 2N$+$1. The first of these two obviously has no effect since $c_{0}=\sin (0\times k)=0$. For the atom 2N$+$1, we demand that the wavefunction again vanishes which restricts the possible k-values:

$$ 0=c_{2N+1}=\sin ((N+1)k+\chi )=\mathfrak {I}\left( \mathrm{e}^{\text {i}\chi +\text {i}(N+1)k}\right) . $$

For these k-values, the cyclic polyene with 2N$+$2 atoms is equivalent to the linear polyene with 2 N atoms as the off diagonal interaction becomes irrelevant. Show that the k-values obey the equation

$$ \beta \sin ((N+1)k)+\beta '\sin (Nk)=0. $$

(d) Find a similar treatment for a linear polyene with odd number of C-atoms.