## 2 Structure of the Fenna-Matthews-Olson Light-Harvesting Complex
### Interactions within the FMO Complexes and Its Environment  
The folding of the protein will result in the formation of local environments in which the amino acids may be subject to different conditions thus modifying their behaviour. These localized environments influence the optical transition energies of the pigments [1]. These interactions are  
Figure 5: FMO from _Prostheacochloris aestuarii_. (**A**) Complete structure of the protein trimer. (**B**) Close-up to a single monomer. (**C**) The eight numbered chromophores in the monomer exhibiting their localized dipolar momenta. (**D**) A single BChl molecule, one of the eight chromophores in each monomer. Figure produced from Protein Data Bank file 3 EOJ [12]. known as pigment-protein interactions, and their resulting influence on the whole quantum site energies in the FMO complex are schematized in Figure 6 without a detailed quantum description which will be addressed in the development below. The position and orientation of the chromophores is highly related to their excitation energy and their role in the energy transfer path, shifting the absorption energy of the pigments facing the outer antenna towards blue compared to those linked to the reaction center shifted towards red [1]. Determining the site energies of each BChl molecule in the complex protein structure can prove to be very challenging. For this reason, site energies are often treated as parameters that are determined from the fit of optical spectra [13]. Experimental spectroscopy is able to discriminate emissions of excited BChls in short time window intervals, but they are still limited to some few femto-seconds. In addition, such two-dimensional spectroscopy experimental techniques require working at very low temperatures (typically around 77 \({}^{\circ}\)K) thus detecting quantum beats in the BChls. In this sense, spectroscopy has been successful to get, together with theoretical modeling, a concrete quantification of the Hamiltonian corresponding to dipole-dipole interactions and some limited deviations due to the protein scaffolding of the set of BChls inside FMO complex [36]. From a remarkable interest is the novel inclusion of the eighth BChl in the experimental and FMO computer modeling literature [1; 37]. Those Hamiltonians will be discussed in the following subsections and reported in Appendix A. Those main experimentally obtained energies show that the excited BChl 8 and its strongly excitonic coupled BChl 1 are the two main entrance points for excitation energy coming from the baseplate [1]. BChl 6 provides an alternative route for the excitation energy to the reaction center [38]. Thus, for the studies based only on the first seven BChls, authors commonly consider two pathways for the initial excitation energy: BChl 1 and 6. The process of relaxation of the excitation energy is funnelled to BChl 3 along the branches formed by BChl 2 and by BChl 4 to BChl 7 as seen in Figure 6[1]. The latter being completed within 500 fs, while BChl 2 appears to limit the overall relaxation/equilibration time to about 1.5 ps [1]. In any case, the short relaxation times and the large energy gap between the entrance BChl 8 and the sink BChl 3 prevents recombination and leaking [1]. Due to their structural complexity and behaviour, proteins are systems with thousands of degrees of freedom [39]. On the other hand, the functional subsystem (the active site where the BChl cofactors are bound) involves only a few quantum states [39]. The change in the quantum state as the transition from the ground state to the excited electronic state is associated with a change in the electric dipole momentum of the subsystem [39]. The polar residues contained in the protein and its highly polar solvent (water) surroundings result in a strong interaction between the functional subsystem and its environment.