Coherent Excitations in Photosynthetic Systems

Abstract

Photosynthetic units of plants and bacteria consist of antenna complexes and reaction centers. Rings of closely coupled chlorophyll chromophores form the light harvesting complexes which transfer the incoming photons very efficiently and rapidly to the reaction center, where the photon energy is used to create an ion pair. In this chapter, we concentrate on the properties of strongly coupled chromophore aggregates. We discuss an exciton model for a strongly coupled dimer including internal charge transfer states and apply it to the special pair of the photosynthetic reaction center. Next, we study circular molecular aggregates, as found in the light harvesting complexes of photosynthesis. We calculate the exciton spectrum including dimerization and apply it to the light harvesting complex LHII. The influence of disorder is discussed including symmetry breaking local perturbations and periodic modulations as well as general diagonal and off-diagonal disorder.

Problems

25.1

Photosynthetic Reaction Center

The “special pair” in the photosynthetic reaction center of Rps.viridis is a dimer of two bacteriochlorophyll molecules whose centers of mass have a distance of 7Å. The transition dipoles of the two molecules include an angle of $139^{o}$.

Calculate energies and intensities of the two dimer bands from a simple exciton model

$$ \mathfrak {\mathcal {H}}=\left( \begin{array}{cc} -\varDelta /2 &{} V\\ V &{} \varDelta /2 \end{array}\right) $$

as a function of the energy difference $\varDelta$ and the interaction V. The Hamiltonian is represented here in a basis spanned by the two localized excited states $|A^{*}B>$ and $|B^{*}A>$.

25.2

Light Harvesting Complex

The circular light harvesting complex of the bacterium Rhodopseudomonas acidophila consists of nine bacteriochlorophyll dimers in a $C_{9}$-symmetric arrangement. The two subunits of a dimer are denoted as $\alpha$ and $\beta$. The exciton Hamiltonian with nearest neighbor and next to nearest neighbor interactions only is (with the index n taken as modulo 9)

$$ \mathcal {H}=\sum _{n=1}^{9}\left\{ E_{\alpha }|n;\alpha> <n;\alpha |+E_{\beta }|n;\beta > <n;\beta |\right. $$

$$ +V_{dim}(|n;\alpha> <n;\beta |+|n;\beta > <n;\alpha |) $$

$$ +V_{\beta \alpha , 1}\left( |n;\alpha> <n-1;\beta |+|n;\beta > <n+1;\alpha |\right) $$

$$ +V_{\alpha \alpha , 1}\left( |n;\alpha> <n+1;\alpha |+|n;\alpha > <n-1;\alpha |\right) $$

$$ +\left. V_{\beta \beta , 1}\left( |n;\beta> <n+1;\beta |+|n;\beta > <n-1;\beta |\right) \right\} . $$

Transform the Hamiltonian to delocalized states

$$ |k;\alpha>=\frac{1}{3}\sum _{n=1}^{9}\mathrm{e}^{\mathrm{i}kn}|n;\alpha>\quad |k;\beta>=\frac{1}{3}\sum _{n=1}^{9}\mathrm{e}^{\mathrm{i}kn}|n;\beta > $$

$$ k=l\, 2\pi /9\quad \quad l=0,\pm 1,\pm 2,\pm 3,\pm 4. $$

(a) Show that states with different k-values do not interact

$$ <k',\alpha (\beta )|\mathcal {H}|k,\alpha (\beta )>=0\quad \text{ if } k\ne k'. $$

(b) Find the matrix elements

$$ H_{\alpha \alpha }(k)=<k;\alpha |\mathcal {H}|k;\alpha>\quad H_{\beta \beta }(k)=<k;\beta |\mathcal {H}|k;\beta > $$

$$ \quad H_{\alpha \beta }(k)=<k;\alpha |\mathcal {H}|k;\beta >. $$

(c) Solve the eigenvalue problem

$$ \left( \begin{array}{cc} H_{\alpha \alpha }(k) &{} H_{\alpha \beta }(k)\\ H_{\alpha \beta }^{*}(k) &{} H_{\beta \beta }(k) \end{array}\right) \left( \begin{array}{c} C_{\alpha }\\ C_{\beta } \end{array}\right) =E_{1,2}(k)\left( \begin{array}{c} C_{\alpha }\\ C_{\beta } \end{array}\right) . $$

(d) The transition dipole moments are given by

$$ \varvec{\mu }_{n,\alpha }=\mu \left( \begin{array}{c} \sin \theta \,\cos (\phi _{\alpha }-\nu +n\phi )\\ \sin \theta \,\sin (\phi _{\alpha }-\nu +n\phi )\\ \cos \theta \end{array}\right) \quad \varvec{\mu }_{n,\beta }=\mu \left( \begin{array}{c} \sin \theta \,\cos (\phi _{\beta }+\nu +n\phi )\\ \sin \theta \,\sin (\phi _{\beta }+\nu +n\phi )\\ \cos \theta \end{array}\right) $$

$\nu =10.3^{o}$, $\phi _{\alpha }=-112.5^{o}$, $\phi _{\beta }=63.2^{o}$, $\theta =84.9^{o}$.

Determine the optically allowed transitions from the ground state and calculate the relative intensities.

25.3

Exchange Narrowing

Consider excitons in a ring of chromophores with uncorrelated diagonal disorder. Show that in lowest order, the distribution function of $E_{k}$ is Gaussian. Hint: write the distribution function as

$$ P(\delta E_{k}=X)=\int d\delta E_{1}d\delta E_{2}\cdots P(\delta E_{1})P(\delta E_{2})\cdots \delta (X-\frac{\sum \delta E_{n}}{N}) $$

and replace the delta function with a Fourier integral.