Interplay of slow bath fluctuations and energy transfer in 2D spectroscopy of the FMO light-harvesting complex: benchmarking of simulation protocols

Recently developed approaches to simulate environment-induced fluctuation effects in two-dimensional (2D) spectroscopy of excitons are compared for the Fenna-Matthews-Olson light-harvesting complex. Fast fluctuations induce population redistribution between exciton energy-levels and raise homogeneous line widths of various peaks in 2D spectra. These effects are easily accounted for in sum-over-eigenstates (SOS) approach and the quasi-particle (QP) scattering approach through relaxation and dephasing rate constants. Slow fluctuations cause correlations of energies at various delay times in 2D photon-echo spectra. These may be calculated either by doing cumulant expansion in SOS techniques or by statistical averaging over static disorder in SOS and QP approaches. We compare the 2D photon-echo signal simulated using two levels of cumulant expansion approaches and two statistical averaging approaches for the same system. These levels differ by the treatment of energy-level correlations at different delay times and give rise to different cross-peak shapes: the cross-peaks retain their original diagonally elongated shapes when correlations are included, while they are more spherically broadened when correlations are neglected. Statistical averaging over disorder give very similar results but requires much higher computational effort. The peak redistribution timescales are very similar for all levels of theory. The spectral signatures at these different levels of theory are compared and simulation cost is estimated. Approaches which do require statistical averaging over disorder are orders of magnitude slower.