Classification of doubly excited molecular electronic states

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.

S1. An alternative discussion of W To provide an alternative viewpoint on the 1TDM norm W, we can write |Ψ ! ⟩ as a combination of singly $Ψ ! " % and higher excited $Ψ ! #$… % contributions with respect to |Ψ & ⟩ , i.e.
where the singly excited contribution is defined as and the higher contributions are defined analogously The coefficients in this expansion are not unique, as it is generally possible to reproduce the effect of lower excitations with higher excitation operators. However, following Ref. 1 , we can choose the coefficients to maximize the overlap of $Ψ ! " % with |Ψ ! ⟩ for any normalized set of expansion coefficients '( &! . For this purpose, we write the overlap between |Ψ ! ⟩ and $Ψ ! " % as where In other words, √Ω can be interpreted as the maximum possible overlap of Yf with any possible function that can be constructed via single excitations from Yi, supporting the interpretation of W as a measure of single-excitation character. In practical terms, an W value of zero means that Yf is orthogonal to any function constructed via single excitations from Yi, whereas higher Ω values point to enhanced contributions of single excitations.
Finally, we point out that the above derivation requires that the expansion coefficients '( &! are normalized but Ψ ! " is not generally normalized. Indeed, the norm of Ψ ! " could, in principle, be larger than one, resulting in an W value also larger than one. In practice, values larger than one are not usually observed, but we discuss how to construct such cases in Section S2 of the Supporting Information. In addition, W values slightly larger than one are encountered per construction in TDDFT computations not employing the Tamm-Dancoff approximation. 2 S2. 1TDM norm W larger than 1 Within this section, we exemplify the possibility that the 1TDM norm W is larger than 1 in wavefunction based calculations. The case is usually not encountered in practice but, in principle, such cases can be constructed. Here we consider four different spatial orbitals, f1 to f4. Using these, one can construct two different Slater determinants |11 > 22 > ⟩ and |33 > 44 > ⟩ where either the first two or last two of these are doubly occupied. The wavefunctions of the initial and final states are now constructed as These are normalized and orthogonal. On finds that all off-diagonal 1TDM elements vanish whereas the diagonal 1TDM elements are evaluated as The 1TDM norm W is the sum over the squares of all these elements Thus, indeed, Ω is larger than 1. Similar constructions with more partially occupied orbitals can be employed to obtain even higher values.
Analyzing this case in more detail, we find that the initial and final states both possess the same densities and, thus, we obtain for the promotion and excitation numbers Alternatively, it is possible to compute the fraction of de-excitations according to Ref. 2 .
The excited state is thus classified as being composed of 100% de-excitations highlighting, again, that this is an exotic case.
An W value much larger than one is not usually encountered in practical calculations but appropriate cases can be constructed. In the following we present data on the CO2 4+ cation with one CO distance at 1.16 Å and the other at 2.32 Å. This system possesses four approximately degenerate p-orbitals filled with four electrons, which is intended to reflect the situation sketched above.
Computations were performed in OpenMolcas 3 and wavefunctions were analyzed using the WFA 4 module. An active space containing 4 electrons in the six p/p* orbitals was used. The results are presented in Table S1. Here, the first excited state (S1) is used as a reference for transition and difference density matrix based properties. Crucially, we find that the transition from the S1 state to the degenerate S2 state is indeed characterized by W = 1.988. However, the presented case is clearly far from typical calculations. First, one has to realize that the large W value is only observed for the S1 ® S2 transition and not for any transition involving the ground state. Second, we find that all states possess the same 1DMs, as indicated by vanishing p and h values. As such, we highlight that the meaning of all three descriptors -W, p and h -has to be rethought in the cases of initial states with many open shells.

S3. Generalization of the excitation number
The excitation number between two states described via individual Slater determinants was defined as as described in the main text. 5 Eq. (13) can directly be applied if the initial and final state are both described by single Slater determinant. In the following, we want to discuss how to generalize this to arbitrary wavefunctions. There is no unique way to generalize Eq. (13) and we will evaluate several options. We first realize that the summation in Eq. (13) corresponds to the squared Frobenius norm of the occupied-occupied block of the mixed overlap matrix &!
One furthermore realizes that the mixed overlap matrix is obtained from the MO coefficients and the  6 An additional change has to be made in order to assure that the expression vanishes in the case i=f. For this purpose, we replace the number n in Eq. (17) with a number =!! to assure that the excitation number for the i=f case vanishes. We write Two definitions of =!! were investigated Both definitions vanish for the i=f case, are symmetric with respect to an interchange of i and f and reduce to Eq. (3) if the density matrices derive from a single Slater determinant with doubly occupied orbitals. Results using these definitions are shown in the last two columns of Table S2. Only the combination of Eqs (18) and (20) leads the expected results: singly excited character of the Bu states and partially doubly excited character for 2 1 Ag. We, therefore, use this definition of η in the main text.
Note that this is equivalent to the formula

S6.1. ADC(3) Size Consistency
ADC (3) is size-consistent with respect to the excitation energies, transition moments, and properties of individual states. 7 However, one could extend the definition of size-consistency to multiply excited states and ask if the excitation energy of the doubly excited state of a non-interacting dimer is twice the excitation energy of the singly excited state of the monomer. This cannot be achieved by a method with a fixed number of excitations, such as ADC(3). Indeed, within ADC(3), singly excited states are described in third order but doubly excited states are only in the first order. 7 The lack of this extended type of size consistency explains the difference in excited-state energies and is also seen in the nu,nl (2.46 vs 4.05) and p (1.15 vs 2.00) descriptors, shown in Table 1 of the main text, highlighting that neither doubles appropriately. Indeed, the doubly excited state show values smaller than expected. Since nu,nl and p reflect electron correlation and orbital relaxation effects, lower values than expected mean that the doubly excited state can take advantage of less electron correlation and orbital relaxation compared to the singly excited state.