(a) (b)
Figure 7. Evolution of the state populations of a seven-state system, modeling one unit of the FMO pigment–protein complex. Simulations are performed at two temperatures: a) 77 K, chosen to model a cryogenic temperature and limit decoherence, and b) 300 K, chosen to model physiological temperature.
We also illustrate the use of our HEOM implementation in modeling of exciton transfer in the FMO system, a pigment–protein complex found in green sulfur bacteria.15 The complex consists of several identical units that each contain seven bacteriochlorophyll molecules. Each of these molecules can be treated as an individual state (site). The cite energies and couplings (in \(cm^{-1}\)) are given by a 7 x 7 Hamiltonian:15
\(H=\par \begin{pmatrix}410&-87.7&5.5&-5.9&6.7&-13.7&-9.9\\ &530&30.8&8.2&0.7&11..8&4.3\\ &&210&-53.2&-2.2&-9.6&6.0\\ &&&320&-70.7&-17.0&-63.3\\ &&&&480&81.1&-1.3\\ &&&&&630&39.7\\ &&&&&&440\\ \end{pmatrix}\) (18)
The calculations are conducted with the parameters: KK = 1 (two Matsubara frequencies), LL = 5 (maximal hierarchy level), γ = 0.02 fs-1 (decoherence rate), and η = 35 cm-1 (reorganization energy). The 1 ps trajectory with the integration timestep of 1 fs requires an order of thirteen minutes running with 1 thread, when run on Intel(R) Xeon(R) CPU E5-2620 v3 @ 2.40GHz. The resultant population dynamics (Figure 7) for both the cryogenic temperature (panel a) and physiological temperature (panel b) matches those reported earlier by Ishizaki and Fleming.15

3.2.2. Calculation of the absorption line shapes.

An example snippet to run the absorption line shape calculations is shown in Figure 8. It is structured similarly to the one for the density matrix evolution (Figure 4), but has a number of exceptions. First of all, the Hamiltonian is defined differently – the ground electronic state is not coupled to any of the excited states, but the two states are coupled to each other via \(H_{12}=H_{21}=-J\). Such a Hamiltonian corresponds to the excitonic dimer, as for instance defined by Shi et al.25
In addition, we define the transition dipole moment operator, which couples the ground and excited states and is given by\(\mu=\sum_{n=1}^{2}\left(\left.\ |n\right\rangle\left\langle 0|\right.\ +\left.\ |0\right\rangle\left\langle n|\right.\ \right)\). The matrix representation of this operator is given in lines 12-15 of the snippet in Figure 8. The initial conditions in the absorption line shapes calculations are also different from those in the bare population dynamics. In this case, although the system starts in its electronic ground state,\(\rho_{\text{gs}}=\left.\ |0\right\rangle\left\langle 0|\right.\ \), the transition dipole moment operator evolves this initial density matrix to a different state, in which coherences between the ground and each of the excited states are initialized,\(\rho_{\mathbf{0}}\left(t\right)=\mu\rho_{\text{gs}}\). This setup is done in lines 18-19 of the snippet.