# Excited States in Coupled Pigments

If two pigments are coupled via an excitation energy or electron transfer, the rate constant *k _{ET}* can be approximated according to Fermi’s Golden

Rule in first order perturbation theory (Messiah, 1991a,b; Cohen-Tanouji et al., 1999 a,b; Hader, 1999; Schwabl, 2007, 2008):

The rate constant *k _{ET}* which is a probability per time unit (for a transition) can be calculated as the product of the square of the absolute value of the transition matrix element | V

_{12}|

^{2}and the effective density of states (E) of the initial and final states. In photosynthetic complexes the density of states is determined by the vibrational spectrum of the interacting species and the states of their microenvironment (Renger and Schlodder, 2005). The transition matrix element V

_{12}= (

^{V}P

_{2}|

*H*| ^ is

_{WW}the scalar product of the initial state and the final state I'P,,) under the influence of the interacting potential. This potential is represented by the interaction operator *H*_{WW}, e.g. a Coulomb potential *H _{WW} = V_{Coulomb}* as denoted in the appendix in eq. 84. If we investigate the situation where the excited state of one pigment in a coupled trimer is transferred to the coupling partner one can formulate the product states

where Ф, denotes the electronic wave function of the /^{h} electron

in the excited (e) or ground state (g) located at the i^{th} pigment.

Evaluating Fermi’s Golden Rule (eq. 44) with the product states as denoted in eq. 45 delivers matrix elements of the form

denoted Coulomb terms and

denoted exchange terms.

While the Coulomb terms also deliver significant contributions in the weak coupling regime, the exchange terms require a direct overlap of the electronic wavefunctions between the ground state wave functions and the excited state wave functions of both pigments (for details see Renger, 1992).

This direct overlap is only significant if the distance of the interacting fluoropheres is very low. In this case, the electronic wave function is no longer localised at one pigment only but is delocalised among all coupled pigments.

The dominating mechanism for the ET and EET in the short distance regime is called “Dexter transfer”, which denotes a direct electron exchange due to the delocalisation of the electronic probability of presence. For the Dexter transfer the selection rules differ from the selection rules of weakly coupled pigments, including triplet-triplet transfer as proposed for strongly coupled Chl and carotenoid molecules.

In agreement with the exponential decay of the electronic wave functions, Dexter transfer occurs at very small distances typically near to the van der Waals-distance in the order of 0.1-1 nm (Hader, 1999). Concomitantly with the amplitudes of the electronical wave functions, the transfer rate constant decreases exponentially with the distance *r* of the coupled pigments:

where *a* denotes the so called “Dexter coefficient”.

Due to the short distance of the pigments, the excitonic coupling leads to a significant shift of the energetic levels and therefore to a change in the optical spectra. The excitonic coupling leads to excitonic splitting. For excitonically coupled states, the relaxation from the higher excitonic level to the lower excitonic level is typically very fast (<10 ps) and cannot be resolved with time- and wavelength-correlated single photon counting.

In many strong coupled photosynthetic pigments the Dexter transfer is the dominating EET mechanism, i.e. between Chlorophyll molecules in the LHC and in PBS of cyanobacteria and probably also in the PBP antenna of *A.marina* (Theiss et al., 2011). Sometimes it is not fully clear whether the suitable approximation for the dominating EET mechanism is Dexter or Forster. For example in the PBP antenna the transport between different trimers has to bridge distances of about 2 nm which is within both validity areas. It has to be kept in mind that the energy transfer mechanism is neither “Forster” nor “Dexter”, but nothing more than an effect resulting from electromagnetic coupling. Just the approximation conducted is in the Forster or Dexter regime.

At distances above 2 nm the electromagnetic potential of the involved chromophores can be estimated with the model of point-dipole approximation (Renger and Schlodder, 2005). In this case a significant probability for energy transfer can be observed up to a distance of 10 nm

(Lakowicz, 2006). The mechanism of EET using the point-dipole approximation for У_{12} in eq. 44 is called “Forster Resonance Energy Transfer” (FRET).