Extracting the Excitonic Hamiltonian of the Fenna-Matthews-Olson Complex Using Three-Dimensional Third-Order Electronic Spectroscopy  Dugan Hayes, Gregory S. Engel  Biophysical Journal  Volume 100, Issue 8, Pages 2043-2052 (April 2011) DOI: 10.1016/j.bpj.2010.12.3747 Copyright © 2011 Biophysical Society Terms and Conditions

Figure 1 Third-order response pathways represented graphically using double-sided Feynman diagrams. Time progresses upward along the axis shown to the left on which the intervals τ, T, and t are indicated. The left and right vertical lines represent the ket and bra of the density operator, respectively, and the curved arrows indicate interactions with a radiation field. The ground state of the system is given by |g〉, and different single-exciton states are given by |ea〉 and |eb〉. These two diagrams represent pathways that contribute to the signal in the ks = –k1 + k2 + k3 phase-matching direction. Diagram A depicts a rephasing pathway in which the coherence and rephasing frequencies are not equal, yielding a crosspeak. Diagram B depicts a nonrephasing pathway in which these frequencies are equal, yielding a peak along the main diagonal. In both cases, the system is in a zero-quantum coherence during T, resulting in a periodic beating of the corresponding signal. Biophysical Journal 2011 100, 2043-2052DOI: (10.1016/j.bpj.2010.12.3747) Copyright © 2011 Biophysical Society Terms and Conditions

Figure 2 (a) 3D third-order electronic spectrum of FMO. Features in the spectrum are presented in both color and opacity scales to permit visualization of the entire data set. (b) 2D spectrum extracted from the 3D spectrum by cutting through the solid at a fixed waiting frequency. As expected, the spectrum shows features broadened along the diagonal that appear above and below the diagonal. These features correspond to positions in the spectrum where the absolute value of the difference between the coherence frequency (x axis) and the rephasing frequency (y axis) is equal to the fixed waiting frequency. (c) 2D spectrum taken at a fixed rephasing frequency. A series of spectra of the latter type corresponding to each exciton are shown in Fig. 3. Biophysical Journal 2011 100, 2043-2052DOI: (10.1016/j.bpj.2010.12.3747) Copyright © 2011 Biophysical Society Terms and Conditions

Figure 3 (a–g) 2D spectra taken from the 3D spectrum at fixed values of rephasing frequency corresponding to each of the seven excitons. Each spectrum is normalized to its respective maximum. The guidelines overlaid on the spectra correspond to positions at which the absolute value of the difference between the coherence frequency (y axis) and the fixed rephasing frequency is equal to the waiting frequency (x axis). A total of 19 crosspeaks appear along these lines. In panel h, points along the main diagonal of the nonrephasing data set are plotted against waiting frequency. The dotted lines correspond to the diagonal frequencies of the seven excitons. The peaks in this spectrum provide additional redundancy for calculating the transition energies of the system. The same slices taken from a simple 3D Gaussian dressed stick model are presented in panels i–p for comparison. Biophysical Journal 2011 100, 2043-2052DOI: (10.1016/j.bpj.2010.12.3747) Copyright © 2011 Biophysical Society Terms and Conditions

Figure 4 Experimental (solid blue line) and calculated (dashed red line) linear absorption spectra of FMO. Biophysical Journal 2011 100, 2043-2052DOI: (10.1016/j.bpj.2010.12.3747) Copyright © 2011 Biophysical Society Terms and Conditions