  For the theoretical screening of Singlet Fission (SF) rates in molecular
aggregates, commonly dimer model systems are employed. However, there is
experimental evidence, that the SF process proceeds from the $^{1}$(TT) state
via an triplet-triplet energy transfer process to a further intermediate: a
$^{1}$(T...T) state with two non-adjacent, spin-correlated triplets, which
cannot be captured by the dimer models. In this work, we extend Michl's
diabatic frontier orbital model to trimer systems, for which we automatically
generate the diabatic two and three-center couplings using symbolic algebra. We
apply this method to study the packing dependence of the $^{1}$(T...T)
formation in the perylene diimide (PDI) trimer stack. We find that efficient
triplet-triplet energy transfer is facilitated by structural motifs for which
also significant excimer character can be observed. Furthermore, the coupling
shows a local maximum for the structural motif that has been assigned to
efficient $^{1}$(TT) population. Employing second order perturbation theory, we
study the interference of the individual electronic pathways that arise in the
PDI trimer system, allowing us to reproduce the packing dependence of the SF
rates derived from the Redfield simulations in the work by Mirjani et. al.
(Phys. Chem. C 2014, 118, 26, 14192-14199).
