  Coherent energy transfer in pigment-protein complexes has been studied by
mapping the quantum network to a kinetic network. This gives an analytic way to
find parameter values for optimal transfer efficiency. In the case of the
Fenna-Matthews-Olson (FMO) complex, the comparison of quantum and kinetic
network evolution shows that dephasing-assisted energy transfer is driven by
the two-site coherent interaction, and not system-wide coherence. Using the
Schur complement, we find a new kinetic network that gives a closer
approximation to the quantum network by including all multi-site coherence
contributions. Our new network approximation can be expanded as a series with
contributions representing different numbers of coherently interacting sites.
  For both kinetic networks we study the system relaxation time, the time it
takes for the excitation to spread throughout the complex. We make
mathematically rigorous estimates of the relaxation time when comparing kinetic
and quantum network. Numerical simulations comparing the coherent model and the
two kinetic network models, confirm our bounds, and show that the relative
error of the new kinetic network approximation is several orders of magnitude
smaller.
  Keywords: exciton transfer, quantum efficiency, kinetic networks, FMO,
coherent energy transfer, quantum networks, Schur complement.
