We apply the Liouville space hierarchical equations of motion method to calculate the linear and two-dimensional (2D) electronic spectra of the Fenna-Matthews-Olson (FMO) protein complex from Chlorobium tepidum, using a widely used model Hamiltonian. The absorption and linear dichroism spectra of the FMO complex, as well as the main features of the 2D spectra are well reproduced. However, comparison with the recent experimental 2D spectra reveals several limitations of the current model: (1) The homogeneous and inhomogeneous broadening seems to be overestimated for the first exciton peak, but may be underestimated for several other exciton peaks. (2) The calculated oscillations of the diagonal and off-diagonal peaks in the 2D spectra are much weaker than the experimental observations, which indicates that an improved model is needed for the excitonic dynamics of the FMO complex.
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