Electronic coupling matrix element for electronic excitation energy transfer: S₁S₁S₂+S₀ excited state annihilation

For the singlet-singlet annihilation process S₁S₁S₂S₀, an expression is derived for the electronic coupling matrix element involved in electronic excitation energy transfer (EET) between two equivalent chromophores. Allowance is made for vibronic coupling to occur between the S₁ and S₂ states of the chromophores, for each of which the S₁←S₀ and S₂←S₀ excitations are, respectively, electric-dipole forbidden and allowed. Attention is also given to aspects of the theory for singlet-singlet (π_Dπ_D*) (n_An_A)→(π_Dπ_D) (n_Aπ_A*) EET with vibronic coupling. The theory for S₁S₁→S₂S₀ is illustrated via the results of some model calculations for two CS chromophores. For each chromophore, the calculations treat explicitly the valence-shell pπ atomic orbitals and a sulphur lone-pair atomic orbital. The results of the calculations highlight the nature of the primary contributors to the vibronic coupling component of the electronic EET matrix element, namely Coulombic-type terms. The theory also shows that a Coulombic term is the primary contributor to the vibronic coupling component for singlet-singlet (π_Dπ_D*) (n_An_A)→(π_Dπ_D) (n_Aπ_A*) EET.