Linear Optical Spectra

In the following, a short survey of the theory of optical spectra is given. For a detailed discussion, we refer to refs (3, 18, 30, and 31). The theory is based on a standard Hamiltonian H_ppc for the pigment–protein complex that describes the pigments as coupled two-level systems interacting with vibrational degrees of freedom of the pigments and the protein.³ It is composed of three parts, H_ppc = H_ex + H_ex–vib + H_vib. The exciton part H_ex includes the site energies E_m of the pigments, defined as the optical transition energies at the equilibrium position of nuclei in the electronic ground state, and the excitation energy transfer couplings V_mn. The exciton–vibrational part H_ex–vib describes the modulation of site energies and excitonic couplings by the vibrations. In the spirit of a NMA,¹⁷ it is assumed that the site energies and excitonic couplings depend linearly on the displacements of the vibrational coordinates from their equilibrium values. The vibrational part H_vib describes uncoupled harmonic oscillators.

For the calculation of optical spectra, the Hamiltonian of the PPC is transformed to the basis of delocalized exciton states |M⟩ which are given as linear combinations of local excited states |m⟩.

The exciton coefficients c_m^(M) and excitation energies , are obtained from the solution of the eigenvalue problem of the exciton Hamiltonian H_ex. The absolute square of the exciton coefficient c_m^(M) describes the probability that the mth pigment is excited when the PPC is in the Mth exciton state. The exciton–vibrational Hamiltonian H_ex–vib contains diagonal elements, which lead to vibrational sidebands and off-diagonal elements, which cause lifetime broadening of the optical lines.¹⁸