Unified model for singlet fission within a non-conjugated covalent pentacene dimer

When molecular dimers, crystalline films or molecular aggregates absorb a photon to produce a singlet exciton, spin-allowed singlet fission may produce two triplet excitons that can be used to generate two electron–hole pairs, leading to a predicted ∼50% enhancement in maximum solar cell performance. The singlet fission mechanism is still not well understood. Here we report on the use of time-resolved optical and electron paramagnetic resonance spectroscopy to probe singlet fission in a pentacene dimer linked by a non-conjugated spacer. We observe the key intermediates in the singlet fission process, including the formation and decay of a quintet state that precedes formation of the pentacene triplet excitons. Using these combined data, we develop a single kinetic model that describes the data over seven temporal orders of magnitude both at room and cryogenic temperatures.

and fits to the data. c) Species-associated spectra using the kinetic model given in Fig. 3. d) Population kinetics for each species. The total number of T1 states per S1 created is twice the population of the T1 species indicated in d difference spectra for NC obtained at 105 K in butyronitrile. b) Single wavelength kinetics and fits to the data. c) Species-associated spectra using the kinetic model given in Fig. 3. d) Population kinetics for each species. The total number of T1 states per S1 created is twice the population of the T1 species indicated in d). Multiplying the species-associated spectra in c) by the corresponding species populations in d) yields the complete A vs. time and wavelength data set.   The nsTA data for NC in liquid benzonitrile at 295 K were first fit using a sum of four exponentials convoluted with a Gaussian instrument response to obtain the effective rate constants for each state in the unified kinetic model ( Supplementary Fig. 16). The nsTA data were then globally fit using the unified kinetic model established by the TA and TREPR measurements of NC in solid butyronitrile at 105 K which required the same number of effective rate constants. The total deactivation of each state in the unified kinetic model was fixed to the corresponding effective rate constants obtained above, while the individual rate constants were allowed to vary. The fit was deemed satisfactory when (i) the species-associated spectra of

Supplementary Methods 4  Computational Details.
We have investigated different conformers of NC using density functional theory.
Specifically, we have found the global minimum for the NC system using different methods.
This resulted in three different conformers NC1, NC2, and NC3 shown in Supplementary Fig.   17. At the M06-2X/6-31G(d) level of theory, 6, 7 the conformer NC2 is more stable than conformers NC1 (which is found to be a saddle point with a small imaginary frequency of -6. To gain insight into the SF mechanism, we have investigated the diabatic electronic states relevant for the process and their interstate couplings. 8 These diabatic electronic states can be obtained by a suitable unitary transformation of the adiabatic electronic states. Among the different methods available, in this work we have employed the four-fold-way diabatization method of Truhlar and co-workers. 9,10 In the following, first we will briefly describe the procedure of the four-fold-way and then we will present our results. The four-fold-way is based on the configurational uniformity concept proposed by Ruedenberg and co-workers. 11,12 In this work, the adiabatic wavefunctions used in the diabatization process have been built using all configuration state functions (CSFs) whose coefficients are more than 0.20 in any of the adiabatic electronic states considered. The adiabatic (canonical) molecular orbitals of the active space that define the CSFs are then rotated such that they satisfy the so called three-fold density criterion and MORMO conditions to form diabatic molecular orbitals (DMOs). 9,10 The adiabatic states are then re-expressed by CSFs defined by DMOs and are transformed to diabatic states by applying a unitary transformation.
To quantify the contribution of the direct and mediated mechanism to the overall SF process we have calculated the effective coupling, , of | 1 (S1S0)> to | 1 (T1T1)> (or | 1 (S0S1)> to | 1 (T1T1)>): 16 it is assumed that the mixing of the CT states with | 1 (T1T1)> and | 1 (S1S0)> | 1 (S0S1)>) can be described perturbatively. The contributions of the direct and mediated (via CT states) mechanisms are accounted for by the first and second term on the right hand side, respectively.
To substantiate the hypothesis of the involvement of a quintet state in the singlet fission mechanism, we have calculated the energy of the lowest-lying quintet state of | 5 (T1T1)> type and shown that this state is nearly degenerate with | 1 (T1T1)>. This indicates that a mixed singletquintet spin state is formed in the singlet fission process. Specifically, for a system of two coupled triplet states, without external magnetic field and neglecting for simplicity spin-orbit coupling, the spin part of the full Hamiltonian ( ) originates from the spin-spin exchange and magnetic dipole-dipole interaction between the electrons of the two triplet centers. This can be expressed as 18,19 = + where , and are the zero-field and electron-electron interactions, respectively that have the general form = ℎ = ℎ 1 2 where ℎ is Planck´s constant, Si is the spin at chromophore i, D is the zero-field splitting tensor and J is the exchange coupling tensor. Diagonalization of the full Hamiltonian (i.e. the sum of the electrostatic and magnetic terms) generates nine eigenfunctions that are linear combinations of singlet (| ⟩), triplet (| ⟩), and quintet (| ⟩) spin states. 8,20 When no external magnetic field is present, the spin basis set can be built up from all pair combinations of zero field triplet spin functions of each chromophoric site. 8,[18][19][20] Employing this basis, and assuming the energies of | 1 (T1T1)> and | 5 (T1T1)> to be nearly degenerate, it can be shown 19