Electronic states with partial or complete doubly excited character play a crucial role in many areas, such as singlet fission and non-linear optical spectroscopy. Although doubly excited states have been studied in polyenes and related systems for many years, the assignment as singly vs. doubly excited, even in the simplest case of butadiene, has sparked controversies. So far, no well-defined framework for classifying doubly excited states has been developed, and even more, there is not even a well-accepted definition of doubly excited character as such. Here, we present a solution: a physically motivated definition of doubly excited character based on operator expectation values and density matrices, which works independently of the underlying orbital representation, avoiding ambiguities that have plagued earlier studies. Furthermore, we propose a classification scheme to differentiate three cases: (i) two single excitations occurring within two independent pairs of orbitals leaving four open shells (DOS), (ii) the promotion of both electrons to the same orbital, producing a closed-shell determinant (DCS), and (iii) a mixture of singly and doubly excited configurations not aligning with either one of the previous cases (Dmix). We highlight their differences in underlying energy terms and explain their signatures in practical computations. The three cases are illustrated through various high-level computational methods using dimers for DOS, polyenes for Dmix, and cyclobutane and tetrazine for DCS. The conversion between DOS and DCS is investigated using a well-known photochemical reaction, the photodimerization of ethylene. This work provides a deeper understanding of doubly excited states and may guide more rigorous discussions toward improving their computational description while also giving insight into their fundamental photophysics.
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