Generalized Master Equation with Non-Markovian Multichromophoric Förster Resonance Energy Transfer for Modular Exciton Densities

A generalized master equation (GME) governing quantum evolution of modular exciton density (MED) is derived for large scale light harvesting systems composed of weakly interacting modules of multiple chromophores. The GME-MED offers a practical framework to incorporate real time coherent quantum dynamics calculations of small length scales into dynamics over large length scales, and also provides a non-Markovian generalization and rigorous derivation of the Pauli master equation employing multichromophoric Förster resonance energy transfer rates. A test of the GME-MED for four sites of the Fenna-Matthews-Olson complex demonstrates how coherent dynamics of excitonic populations over coupled chromophores can be accurately described by transitions between subgroups (modules) of delocalized excitons. Application of the GME-MED to the exciton dynamics between a pair of light harvesting complexes in purple bacteria demonstrates its promise as a computationally efficient tool to investigate large scale exciton dynamics in complex environments.

Figure 1

Schematic of a modular system. Arrows represent transition dipoles of each chromophore, dotted and dashed lines their electronic couplings (within and between complexes, respectively), and wavy lines exciton-bath couplings. The gray region represents MED.

Figure 2

(a) Decomposition of the first four BChls of FMO complex into two modules. The parameters defining $H_n^e$ and $H_c$ (all in ${\mathrm{cm}}^{-1}$) are $E_{1_1}=12\hspace{0.17em}400$; $E_{2_1}=12\hspace{0.17em}520$; $E_{1_2}=12\hspace{0.17em}200$; $E_{2_2}=12\hspace{0.17em}310$; ${\mathrm{\Delta }}_1=-87$; ${\mathrm{\Delta }}_2=-53$; $J_{1_1{1}_2}=5$; $J_{1_1{2}_2}=-5$; $J_{2_1{1}_2}=30$; $J_{2_1{2}_2}=8$. (b) Time-dependent populations of module 1 calculated with HEOM and with two different approximations for GME-MED. Insets show HEOM populations at each BChl. In all figures, numbers within parentheses represent the site of initial excitation.

Figure 3

Exciton dynamics between donor (D) and acceptor (A) B850 units separated by the center-to-center distance of 80 Å. (a) Time-dependent exciton population at the donor B850 unit. (b) Plot of effective forward rate $k_{\text{eff}}$ (see text and [23]) from donor to acceptor with temperature. The black solid line is based on GME-MED-1 and the red dashed line its Markovian limit rate, which corresponds to the PME with MC-FRET rates.