Exploring Non-Condon Effects in a Covalent Tetracene Dimer: How Important Are Vibrations in Determining the Electronic Coupling for Singlet Fission?

Abstract

Singlet fission (SF) offers opportunities for wavelength-selective processing of solar photons with an end goal of achieving higher efficiency inexpensive photovoltaic or solar-fuels-producing devices. In order to evaluate new molecular design strategies and for theoretical exploration of dynamics, it is important to put in place tools for efficient calculation of the electronic coupling between single-exciton reactant and multiexciton product states. For maximum utility, the couplings should be calculated at multiple nuclear geometries (rather than assumed constant everywhere, i.e., the Condon approximation) and we must be able to evaluate couplings for covalently linked multichromophore systems. With these requirements in mind, here we discuss the simplest methodology possible for rapid calculation of diabatic one-electron coupling matrix elements-based on Boys localization and rediagonalization of molecular orbitals. We focus on a covalent species called BT1 that juxtaposes two tetracene units in a partially cofacial geometry via a norbornyl bridge. In BT1, at the equilibrium C2v structure, the "nonhorizontal" couplings between HOMOs and LUMOs (t(HL) and t(LH)) vanish by symmetry. We then explore the impact of molecular vibrations through the calculation of t(AB) coupling gradients along 183 normal modes of motion. Rules are established for the types of motions (irreducible representations in the C2v point group) that turn on tHL and tLH values as well as for the patterns that emerge in constructive versus destructive interference of pathways to the SF product. For the best modes, calculated electronic coupling magnitudes for SF (at root-mean-squared deviation in position at 298 K), are within a factor of 2 of that seen for noncovalent tetracene dimers relevant to the molecular crystal. An overall "effective" electronic coupling is also given, based on the Stuchebrukhov formalism for non-Condon electron transfer rates.

Solution-Phase Singlet Fission in a Structurally Well-Defined Norbornyl-Bridged Tetracene Dimer

“…For the estimated BT1 reorganization energy mentioned above (λ tot ≈ 0.5 eV) we found an effective coupling for SF (V eff ) on the order of 5 meV. 48 In light of the experimental data presented herein, this now appears to be a modest overestimate by ∼1 order of magnitude. Application of Marcus theory with V eff = 5 meV, λ tot = 0.5 eV, and ΔE SF = 0.052 eV would suggest a time constant for forming the 1 TT of 630 ps, ∼100 times faster than our observation of 70 ns.…”

“…In theoretical work we showed that diabatic electronic coupling for conversion from a dimer (AB) state S 1 S 0 or S 0 S 1 (or a superposition of these) to 1 TT will be zero as long as the molecule maintains a symmetry plane passing through both chromophores as it does in the C 2v ground state. 48 The effect is due to orbital symmetry (we explored this with a frontier orbital basis set) and is true for both direct and charge-transfer mediated pathways. Diabatic coupling for SF can emerge through vibrations that break this plane of symmetry; however, one might intuit that rates of S 1 to 1 TT conversion should be small in BT1 compared to systems where such coupling is nonzero at the static geometry of the ground state.…”

“…For nearest-neighbor noncovalent Tc dimers within the crystal, diabatic couplings were calculated by us that reach 7.3 meV. 48 By comparison in BT1, this coupling is formally zero due to orbital symmetry properties as long as the dimer maintains a plane of symmetry passing through both tetracene chromophores, as it does in its C 2v -symmetric ground state. Vibronic coupling can emerge for the S 1 → 1 TT photoreaction, but it requires that vibrations break the aforementioned symmetry plane.…”

“…48 With these ideas in mind, it is worthwhile to look back at our previous theoretical work that focused on calculation of diabatic couplings 48 and ask how those results do in predicting time constants for production of 1 TT. First, for noncovalent Tc dimers in a geometry relevant to polycrystalline films, application of Marcus theory suggests τ fiss = 33 ps using our calculated diabatic coupling (V = 7.3 meV) 48 along with parameter values taken from the literature: ΔE SF ≈ 0.1 eV 14,75 and λ tot = 0.18 eV. 16 This time scale is of the right order of magnitude to what is observed (31 to 125 ps depending on polycrystalline grain size).…”

Orbital Optimization in the Active Space Decomposition Model

“…14 The resulting model Hamiltonians have been used to benchmark approximate methods in more recent studies. 16 However, our ASD method was not applicable to covalently linked chromophores in previous works. This was mainly due to the difficulty in finding appropriate ASD subspaces.…”