Comparison of Computational Strategies for the Calculation of the Electronic Coupling in Intermolecular Energy and Electron Transport Processes

Electronic couplings in intermolecular electron and energy transfer processes calculated by six different existing computational techniques are compared to nonorthogonal configuration interaction for fragments (NOCI-F) results. The paper addresses the calculation of the electronic coupling in diketopyrrolopyrol, tetracene, 5,5′-difluoroindigo, and benzene–Cl for hole and electron transport, as well as the local exciton and singlet fission coupling. NOCI-F provides a rigorous computational scheme to calculate these couplings, but its computational cost is rather elevated. The here-considered ab initio Frenkel–Davydov (AIFD), Dimer projection (DIPRO), transition dipole moment coupling, Michl–Smith, effective Hamiltonian, and Mulliken–Hush approaches are computationally less demanding, and the comparison with the NOCI-F results shows that the NOCI-F results in the couplings for hole and electron transport are rather accurately predicted by the more approximate schemes but that the NOCI-F exciton transfer and singlet fission couplings are more difficult to reproduce.

1 Energy expressions and interactions for the Smith-

Michl model
In the following, a represents the HOMO on molecule A, b the HOMO on B, c and d are the LUMO orbitals on molecule A and B, respectively.Furthermore, h ij stands for ⟨i| ĥ|j⟩ with ĥ the one-electron operator and ⟨ij||kl⟩ is the short-hand notation for ⟨ij| 1 r 12 |kl⟩.The overlined orbitals have electrons with β spin, the others represent electrons with α spin.
1. Definition of the electronic states: 2. Diagonal matrix elements of the Hamiltonian      S5.The green curve uses the J AB -values of Table S5, but fixes S AB and the orbital energies to their average values.The other two curves were obtained with varying S AB (red) or varying orbital energies (orange), fixing the other two variables to their average values.0.25 -1.83 0.00 3.42 CAS(10,10) -0.49 1.76 0.00 3.34 CAS(12,12) -0.63 1.75 0.00 3.45 Table S10: TDC for exciton transport (in meV) as function of the intermolecular distance (in Å), with dpp molecule B displaced by ∆x = ∆y = 1.0 Å Table S11: TDC for exciton transport (in meV) as function of the rotation angle of dpp molecule B, displaced by ∆x = ∆y = 1.0 Å, ∆z = 4.5 Å.

Singlet fission coupling
Figure S1: Active orbitals of the CAS(12,12) calculation of the S 0 electronic state of dpp.The orbitals of the smaller active space are similar and are obtained by removing one by one the first and the last active orbital from the previous (larger) active space.

Figure S2 :
Figure S2: DIPRO electron (left) and hole (right) hopping parameter for two dpp molecules as function of the intermolecular distance along the z-axis for different functionals.

Figure S3 :
Figure S3: Effect of J AB , S AB and ϵ A,B on the DIPRO coupling for hole (left) and electron (right) transport between two perfectly stacked parallel dpp molecules (∆z = 3.0 Å) as function of the amount of Fock exchange (α) in the hybrid functional α[ρ HF x ]+(1−α)[ρ B88 x ]+ β[ρ LY P C ] (β = 1).The blue curve represents the γ el if -values listed in TableS5.The green curve uses the J AB -values of TableS5, but fixes S AB and the orbital energies to their average values.The other two curves were obtained with varying S AB (red) or varying orbital energies (orange), fixing the other two variables to their average values.

Figure S4 :
Figure S4: Electronic coupling (in meV) for exciton transport as function of the intermolecular distance of parallel dpp molecules (left) and as of function of the rotation angle of the second dpp molecule (right), which is displaced by ∆x = ∆y = 1.0 Å, ∆z = 4.5 Å.

Figure S5 :
Figure S5: Norm of the largest S 0 S 1 ± S 1 S 0 projections on the lowest eight roots of the CASSCF(8,8) calculation of a dpp dimer as function of the rotation of dpp molecule B, displaced by ∆x = ∆y = 1.0 Å, ∆z = 4.5 Å.

Figure S6 :
Figure S6: NOCI direct singlet fission couplings t 1 (left) and t 2 (right) (in meV) applying different complete active spaces for the fragment wave functions

Figure S7 :
Figure S7: Gallup-Norbeck weights of the S 0 S 1 ± S 1 S 0 dominated MEBFs used to calculate the NOCI total singlet fission couplings t 1 (left) and t 2 (right).

Figure S8 :
Figure S8: NOCI total singlet fission couplings t 1 (left) and t 2 (right) (in meV) applying different complete active spaces for the fragment wave functions

Figure S9 :
Figure S9: Electronic coupling (in meV) for hole transport (left) and electron transport (right) as function of the intermolecular distance of two perfectly stacked parallel tetracene molecules

Figure
Figure S12: Smith-Michl total singlet fission couplings of a tetracene dimer as function of the rotation angle of tetracene B.

Table S1 :
NOCI electronic couplings for hole transport as function of the intermolecular distance (in Å) of two perfectly stacked parallel dpp molecules for different active spaces

Table S3 :
DIPRO electron hopping parameter (in meV) for two dpp molecules as function of the intermolecular distance along the z-axis for different functionals.

Table S4 :
DIPRO hole hopping parameter (in meV) for two dpp molecules as function of the intermolecular distance along the z-axis for different functionals.

Table S6 :
NOCI electronic couplings (in meV) for exciton transport as function of the intermolecular distance (in Å) of two perfectly stacked parallel dpp molecules for different active spaces

Table S7 :
∆E-based electronic couplings (in meV) for exciton transport as function of the intermolecular distance (in Å) of two perfectly stacked parallel dpp molecules for different active spaces

Table S8 :
TDC for exciton transport (in meV) as function of the intermolecular distance (in Å) of two perfectly stacked parallel dpp molecules for different active spaces

Table S9 :
Transition dipole moment of the S 0 → S 1 excitation of a dpp molecule for different active spaces

Table S12 :
Transition dipole moment of the S 0 → S 1 excitation of a dpp molecule for three different ANO-RCC one-electron basis sets using a CAS(12,12)