(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.