Excited states of ortho-nitrobenzaldehyde as a challenging case for single- and multi-reference electronic structure theory

We present a large set of vertical excitation calculations for the ortho-nitrobenzaldehyde (oNBA) molecule, which exhibits a very challenging excited-state electronic structure like other nitroaromatic compounds. The single-reference methods produce mostly consistent results up to about 5.5 eV. By contrast, the CAS second-order perturbation theory (CASPT2) results depend sensitively on the employed parameters. At the CAS self-consistent field level, the energies of the bright ππ (cid:1) states are strongly overestimated while doubly excited states appear too low and mix with these ππ (cid:1) states. This mixing hampers the CASPT2 step, leading to inconsistent results. Only by increasing the number of states in the state-averaging step to about 40 — to cover all bright ππ (cid:1) states embedded in the double excitations — and employing extended multistate CASPT2 could CASPT2 results consistent with experiment be obtained. We assign the four bands in the molecule's spectrum: The weakest band at 3.7 eV arises from the n NO2 π (cid:1) states, the second one at 4.4 eV from the ππ (cid:1) ( L b ) state, the shoulder at 5.2 eV from the ππ (cid:1) ( L a ) state, and the maximum at 5.7 eV from the ππ (cid:1) ( B a = B b ) states. We also highlight the importance of modern wave function analysis techniques in elucidating the absorption spectrum of challenging molecules.


| INTRODUCTION
Nitro aromatic compounds are an important class of molecules. Several nitrated polycyclic aromatic hydrocarbons are considered important environmental pollutants, 1-4 whose damaging potential is at least partially due to their photochemistry. In turn, some nitro compounds show useful photochemistry, for example ortho-nitrobenzaldehyde (oNBA). oNBA is used as a photoactivated acid (photolabile caged proton), [5][6][7][8] by means of the photo-induced transformation into orthonitrosobenzoic acid. This is one of the oldest reported photochemical reactions 9 and has been intensely studied in the last decades. It is generally accepted that the photochemical mechanism involves first the formation of a ketene intermediate, which later rearranges to the acid. 9,10 For a comprehensive understanding of the photochemistry of nitro aromatic compounds, a precise knowledge of their excited electronic states is indispensable. However, nitro aromatic compounds pose a significant challenge for excited-state electronic structure methods due to, for example, an appreciable multireference character of the ground state, 11 the interaction of several local and charge transfer excitations, the large number of involved orbitals, 12 contributions by doubly excited states, 11 or the presence of twisted intramolecular charge transfer effects. 13,14 For example, the computation of optical spectra of various nitro compounds was reported 15 to require the use of a range-separated hybrid functional to obtain reasonable results for the charge transfer excitations, whereas regular hybrid functionals deliver inaccurate spectra. Another work 11 on nitrobenzene discussed in great detail the quality of the theoretical description of the excited states with a large set of electronic structure methods.
It was shown that a subset of the excited states can only be described accurately with very elaborate methods that properly treat double excitation character, but nonetheless it could not be determined conclusively which electronic structure method provided the most accurate results.
The electronically excited states of oNBA have previously been studied by Leyva et al. 16,17 Experimentally, the absorption spectrum (see Figure S1) of oNBA was reported in cyclohexane 16 and acetonitrile as well as in the vapor phase. 17 The spectrum consists of a series of bands of increasing intensity, whose precise position is modulated by the environment to some extent. The first band in gas phase is a weak shoulder located around 3.7 eV (340 nm), followed by a second band at about 4.4 eV (280 nm). The third, even more intense shoulder is found at about 5.2 eV (240 nm) and the absorption maximum is found at 5.7 eV (220 nm). 17 The same studies also performed a theoretical assignment of this absorption spectrum, 16,17 using time-dependent density functional theory, approximate coupled cluster singles and doubles (CC2), and in particular complete active space self-consistent field (CASSCF) and CAS second-order perturbation theory (CASPT2). The authors proposed the leading electronic configurations that make up the low-lying excited states and general energetic locations and oscillator strengths. However, they also noted that the limitations of active space size and numbers of states lead to an incomplete description of the absorption spectrum at CASPT2 level of theory, whereas the single-reference methods appeared to suffer somewhat from the multireference character of the ground state.
The goal of the present work is to revisit the vertical excitations of oNBA using a larger active space and to investigate the suitability of different electronic structure methods for describing the different parts of the absorption spectrum. On one hand, we employ the singlereference methods CC2, 18 the algebraic diagrammatic construction method to second order for the polarization propagator (ADC(2)), [19][20][21] and equation-of-motion coupled cluster including singles and doubles (EOM-CCSD), 22 as these methods provide an at least qualitative insight into the present single-excitation states in a black-box fashion.
On the other hand, we employ CASPT2 23 with different computation parameters-like number of states, shifts, and multistate (MS) treatment-as this method is supposed to be fully flexible in describing states of single or double excitation character and different needs for electron correlation. The final goal is to estimate the adequacy of the different methods for describing the low-lying electronic states-those that are relevant for the photochemistry of oNBA-as well as the spectroscopically relevant higher electronic states. We also, for the first time to our knowledge, apply modern wave function analysis techniques to the vertical excitations of oNBA, in order to provide a state-of-the-art assignment of the electronic characters that give rise to the observed absorption bands.

| COMPUTATIONAL DETAILS
All vertical excitation calculations were carried out at the ground state minimum geometry optimized at the MP2/cc-pVDZ 24 (18,14) computations, using state averaging over 12 singlet states. In the ground state closed shell configuration, orbitals marked with an asterisk ( * ) are empty.
with Gaussian 16. 25 The Cartesian coordinates can be found in Section S2.
Single-reference excitation calculations were performed with the ADC(2), 19-21 CC2, 18 and EOM-CCSD methods, 22 using the cc-pVTZ basis set. 24 ADC(2) and CC2 calculations were performed with TUR-BOMOLE 7.0, 26 using the resolution-of-identity technique 27 with the default auxiliary basis set. EOM-CCSD calculations were done with Molpro 13.1.2. 28,29 All three calculations are based on Hartree-Fock/ cc-pVTZ orbitals, as shown in Figure 1A. Ground state energies are collected in Table S1 to facilitate reproducibility.
The multireference calculations were carried out using the MS 23,30 and extended multistate (XMS) 31 CASPT2 methods, also with the cc-pVTZ basis. 24 The employed active space considers 18 electrons distributed in 14 orbitals (denoted as CAS (18,14)), which is the union of the three active spaces used in the previous studies by Leyva et al. 16 The active space is shown in Figure 1B for state-averaging over 12 singlet states. It comprises six π=π Ã orbitals of the aromatic ring, three π=π Ã orbitals of the NO 2 group, two lone pair orbitals of the NO 2 group, two π=π Ã orbitals of the carbonyl group, and one lone pair orbital of the carbonyl group. Active spaces calculated with different number of states are shown in Figure S2; For all excitations, we report vertical excitation energies and oscillator strengths (in the length gauge). Besides these basic quantities, we investigated the excitations in detail by means of wave function analysis. The computed single-and multi-reference transitions were analyzed in terms of their transition densities, using TheoDORE 2.0 37 for single-reference methods and the WFA module of OpenMolcas for CASPT2. 38 The following descriptors were computed. First, we computed the norm Ω of the one-electron transition density matrix, which can be interpreted as the fraction of single excitations in the S 0 ! S n transition 39 and thus allows distinguishing single from double/higher excitations. Second, we performed a two-dimensional Löwdin population analysis of the transition density matrix (a charge transfer analysis), 40,41 to identify local and charge-transfer (CT) excitations. For this analysis, the molecule was partitioned into three fragments: the ring (C 6 H 4 ), the nitro group (NO 2 ), and the aldehyde group (CHO). Based on this analysis, we report four descriptors, Ω ring , Ω NO2 , Ω CHO , and CT; the first three quantify local excitations on the respective fragment, the latter quantifies any charge transfer between the fragments. Note that the sum of these four numbers is Ω. Third, we compute the "atomic" CT number, which is computed by treating every atom as an individual fragment and normalizing the obtained CT number by dividing by Ω.
This descriptor is useful because it allows characterizing different kinds of ππ Ã states 42 -in terms of Platt's nomenclature 43 (L b , L a , B b , or B a ) as well as in terms of valence-bond language (ionic "+" or covalent "À" states). [44][45][46] Fourth, we compute the natural transition orbital participation ratio (PR NTO ), 39 which denotes how many natural transition orbitals are needed to describe the excitation, and can be interpreted as a measure for multiconfigurational character. Finally, for all excitations presented in this work, we computed the threedimensional one-electron transition density from the transition density matrices using PySCF. 47 We use plots of the transition density here instead of the more common molecular orbital or natural transition orbital plots, because the transition density allows the direct identification of ππ Ã states in terms of Platt's nomenclature. 43 Note, however, that a transition density plot does not reveal CT character of a state.

| RESULTS AND DISCUSSION
For clarity, in this section we present first the results of the correlated single-reference methods and give a general overview over the excited states that occur in oNBA. Subsequently, we present the CASPT2 results and finally discuss the relative robustness of the methods and the assignment of the absorption spectrum of oNBA.
T A B L E 1 Overview over the settings and employed labels of the CASPT2/cc-pVTZ vertical excitation calculations. Bold-face indicates parameters that were modified with respect to the first line.  Table S6.

| Correlated single-reference methods
For the three lowest excited states, all single-reference methods produce very similar results, so that we will discuss them together. At the employed geometry, the S 1 can be assigned primarily to the excitation of the carbonyl lone pair (n CO π Ã ). The S 2 and S 3 states are excitations from the symmetric and antisymmetric combinations of the two in-plane lone pairs of the nitro group into linear combinations of π Ã orbitals (n À NO2 π Ã and n þ NO2 π Ã ). The transition densities in Figures S3--S5 indicate that these states mix with each other to some minor extent. Based on the excitation descriptors Ω ring , Ω NO2 , and Ω CHO in Tables S3-S5, while S 1 has a sizable CT contribution, all three states are predominantly local excitations. Due to the nonplanarity of oNBA in the ground state, these nπ Ã states-in particular S 2 -acquire small but nonzero oscillator strengths, as shown in Table 2.
The S 4 state is the first predominantly ππ Ã state of oNBA, localized on the ring and at an energy of about 4.7 eV. Inspection of the transition density shows nodes that coincide with the ring carbon atoms (i.e., the transition density is localized on the bonds), hence within the Platt nomenclature 43 this state is denoted as the L b state.
The state has relatively low oscillator strengths-as expected for L b states-that is slightly higher with ADC(2) (0.024) than with the two coupled cluster calculations (0.014-0.016). We find a PR NTO of close to two, indicating that this state is a true multiconfigurational state unlike the nπ Ã states. This is expected for "covalent" L b states 48 based on valence-bond considerations. 46 The atomic CT number is rather high with 0.91, consistent with work on other aromatic systems. 42 Above S 4 , the state characters obtained with the three methods start to diverge. For ADC (2) and CC2, the S 5 is another nπ Ã state, described by an excitation from the carbonyl lone pair to a π Ã orbital delocalized over the ring and the nitro group, according to the charge transfer analysis. This state gains some intensity at ADC(2) and CC2 levels of theory due to slight ππ Ã admixture. It is noteworthy that this state exhibits a relatively small single-excitation contribution Ω of about 85%, indicating relatively high double excitation character (see Tables S3 and S4). This might explain why this n CO π Ã state at EOM-CCSD level is strongly shifted to higher energies (6.15 eV versus 5.14/5.30 eV at ADC(2)/CC2 level) and becomes the S 6 state. The EOM-CCSD S 6 state also more strongly mixes with the ππ Ã state localized on NO 2 , as visible in the transition density plot. The S 6 state at ADC(2) and CC2 level of theory is the first bright ππ Ã state. Energy, oscillator strength, and state character are consistent with the S 5 state as computed at EOM-CCSD level of theory.
The transition density plot identifies this state as the L a state (nodes go through the bonds, that is, transition density localized on the atoms) of the ring. In terms of valence-bond theory, this state is denoted as an ionic "+" state. The state shows lower PR NTO than the L b state, although the differences are significantly smaller than in planar aromatic systems, 42 where PR NTO (and the atomic CT number) can be conveniently used to distinguish the different covalent and ionic ππ Ã states. For the nonplanar, asymmetric oNBA, it seems that these states mix too much with other contributions (e.g., nπ Ã , CT) to enable a direct identification from these descriptors.
The higher electronic states S 7 to S 11 exhibit smaller energy gaps and correspondingly mix to a higher extent, making direct comparisons between the electronic structure methods more difficult. For all three methods, some of these states exhibit very large oscillator strengths, indicating significant ππ Ã character. Based on the wave function descriptors and transition density plots, the characters of these five transitions are most readily identified at the EOM-CCSD T A B L E 2 Results of vertical excitation calculations of oNBA with single-reference methods.
Abbreviation: CT, charge transfer. a See orbitals in Figure 1A.
level of theory, as two linear combinations of the ring ππ Ã (B a and B b ) states, the ππ Ã state localized on NO 2 , and two nπ Ã states with high CT contributions. We remark that in this work, we do not make clear distinctions between the B b state (nodes of the transition density goes through atoms) and the B a state (nodes go through bonds), as their transition densities are too mixed for a clear assignment (compare S 7 and S 8 in Figure S5). Here, the three ππ Ã states show very large oscillator strengths, the two nπ Ã states are rather dark. At CC2 level of theory, the two nπ Ã CT states are shifted to lower energies and mix with the B a and B b states, leaving the NO 2 ππ Ã state relatively unmixed. At ADC(2) level of theory, the states are even more strongly mixed. It seems that the close spacing and significant coupling of states S 7 to S 11 makes it difficult to obtain robust wave function characters.

| Multireference perturbation theory
One possible reason for the difficulty of describing the excited states of oNBA could be a partial multireference character of the ground state, 11 as indicated by the D1 and D2 norms of the single-reference calculations (Table S6). This might not be a severe problem for the equilibrium geometry, but will be very important for other geometries, for example, for the study of the photochemical pathways of oNBA. 49 Thus, and in order to investigate the influence of double excitations, computations at a multireference level of theory are highly desirable for oNBA.
As mentioned above, we carried out MS-CASPT2(18,14)/cc-pVTZ calculations, using the union of the active spaces used previously in the literature. 16 In this earlier work, they employed state-averaging and MS treatment over five or six roots for the different active spaces. In order to accommodate all relevant electronic states in our larger active space, we performed our initial multireference calculations with 12 roots, indicated by the label "M12" for this calculation. Table 3 presents the corresponding results. The results of a wave function analysis and the transition densities are given in Table S7 and Figure S6, respectively.
According to Table 3, the three lowest excitations of oNBA are all of nπ Ã character. The transition densities show that the S 1 is of predominant n CO π Ã character and the S 2 of n À NO2 π Ã character, although some mixing of these two transitions is apparent. The S 3 is primarily of n þ NO2 π Ã , but mixes with the energetically close S 4 that is the ππ Ã (L b ) state. All four states are single excitations (according to Ω) and local excitations (according to the charge transfer numbers) that possess only negligible oscillator strengths that tend to be slightly smaller than in-but still consistent with-the single-reference calculations. The excitation energies also agree reasonably well between single-and multi-reference calculations.
For the excitations from S 5 onward, the agreement of the M12 results with the single-reference results (and the experimental spectrum, see below) deteriorates considerably. The S 5 state at M12 level is predicted to be of ππ Ã (L a ) character and with reasonable energy, but its oscillator strength (0.058) is notably lower than in the singlereference calculations (0.11-0.15). Wave function analysis shows that this state has a significantly lower single-excitation norm Ω (0.53) than S 1 to S 4 (0.72-0.77), indicating a significant double excitation contribution. The remaining computed states of the M12 calculation have a strongly reduced Ω below 0.25, so these states are predominantly doubly excited states, with some small contributions of n CO π Ã , ππ Ã (NO 2 ), and charge transfer. None of the computed states S 6 -S 11 have an oscillator strength of more than 0.015, in stark contrast to the several bright states obtained with the single-reference methods.

| Multireference perturbation theory parameters
In order to investigate the reasons for the bad performance of the MS-CASPT2 calculations in the previous section, we first applied some typical modifications to the CASPT2 settings. Precisely, we increased the imaginary level shift 34 (M12-IM), we set the IPEA shift 35,36 to zero (M12-NO), and we switched from the original MS treatment to XMS treatment 31 (X12). The energies, oscillator strengths, and characters are presented in Table 4. Wave function descriptors are given in Tables S8-S10, and transition densities in Figure S10 compares the energies and state characters (via wave function overlaps 50 ) of the three calculations with M12.
The imaginary level shift 34 is intended to eliminate singularities in the computed energies due to so-called intruder states, which can in principle introduce significant errors in CASPT2 results. However, the M12-IM calculation with large imaginary level shift produces results very similar to the M12 calculation. Both calculations produce qualitatively the same state characters, as in Table S8 and Figures S7 and S10. In particular, the higher excited states are found to be predomi-   Figure 1B.
with the single-reference methods. This shows that the bad performance of the M12 calculation is not due to extensive intruder state problems and cannot be repaired using the imaginary level shift technique.
The IPEA shift is an empirical parameter in CASPT2 that was introduced after identifying systematic errors in CASPT2 energies. 35 The value of this parameter has a significant effect on excitation energies, 36 Figure 1B.
F I G U R E 2 Simulated absorption spectra obtained by Gaussian convolution (full-width at half-maximum of 0.5 eV) of the vertical excitation results presented above. All simulated spectra (filled plots) are normalized with the same normalization constant (such that the ADC(2) spectrum has an absorption of 1 at the maximum) to enable direct comparison of the intensities predicted by the different methods. To better visualize the CASPT2 spectra, we also plot them multiplied by 7 (dashed lines). The experimental spectrum 17 is shown in black. See Figure S11 for plots of individually normalized spectra with an extended wavelength range. and diffuse basis set, and by considering the nuclear vibrational distribution in the ground state. 51 It can be noted that the overall oscillator strength for CC2 is about a third lower than for ADC (2)  Tables S11 and S12, the absence of large oscillator strengths and the presence of several doubly excited states (see also Figure S12) is also found in the underlying SA-CASSCF and SS-CASPT2 calculations.
Thus, it stands to reason that the relevant, high-oscillator strength states are not included in the SA-CASSCF calculation and a larger number of SA-CASSCF roots is necessary to describe them.

| Multireference perturbation theory number of roots
In order to better cover the bright ππ Ã states that are required to describe the absorption spectrum of oNBA accurately, in this section we increase the number of states in the SA-CASSCF and CASPT2 computations. However, a larger number of states in the SA-CASSCF step might reduce the suitability of the orbitals in describing any individual state. 16 Hence, we only increase the number of states by 50% to 18. The results for the resulting M18 and X18 computations are given in Table 5. Corresponding wave function descriptors and transition densities are collected in Tables S13 and S14 as well as in Figures S13 and S14.
Inspection of the M18 calculation quickly reveals that the increase in number of roots did not produce the expected set of bright states-the states above S 5 show very small single excitation character (Ω) and little oscillator strength. In addition, the increase in number of roots seems to have deteriorated the description of the lower-lying states, too. First, in the M18 calculation the n À NO2 π Ã state mixes significantly with the ππ Ã (L a ) state according to the transition density ( Figure S13), gaining some oscillator strength and becoming the lowest state. Likewise, the n þ NO2 π Ã state mixes somewhat with ππ Ã (L b ). The two ππ Ã states (L a and L b ) even mix strongly with each other, as the observed nodes in the transition densities are neither centered on the bonds nor the atoms. Unexpectedly, the main ππ Ã (L a ) contributions are found at around 4.5 eV, about 1 eV lower than with the single-reference methods.
Quite different results are obtained with the X18 calculation. The order of the five lowest states is as expected from the single-reference results: n CO π Ã , n À NO2 π Ã , n þ NO2 π Ã , ππ Ã (L b ), and ππ Ã (L a ). Energies and oscillator strengths resemble very closely the EOM-CCSD results.
In particular, X18 predicts an oscillator strength for the ππ Ã (L a ) state of 0.11, significantly higher than any of the multireference computations discussed so far. Unfortunately, at higher energies (>6 eV) only double excitations are produced, and again bright states are missing.
The results of the M18 and X18 calculations are significantly different (see also Table S15 and Figure S15). In M18, the ππ Ã (L a ) state is clearly overstabilized and mixes extensively with the lower-lying nπ Ã and ππ Ã states. In X18, the different states mix much less than in M18 (compare Figures S13 and S14) and agree qualitatively with the single-reference computations. Nonetheless, both M18 and X18 show that the increase in the number of roots did not notably improve the description of the high-energy part of the absorption spectrum.   Figure 1B.
Considering that the gentle increase in the number of states from 12 to 18 did not alleviate the problems described above, a much larger increase in the number of states-to 40 roots-was performed, based on some preliminary trial calculations. The relevant states of the performed M40 and X40 calculations are presented in are strong mixtures of n À NO2 π Ã and n CO π Ã and some ππ Ã contributions, leading to unexpectedly high oscillator strengths (0.1 in total). The S 3 state at 4.2 eV can be identified as a ππ Ã (B a =B b ) state from the transition density and the very high oscillator strength (0.27). S 4 is the n þ NO2 π Ã . The next four states are all bright (>0.1) ππ Ã states (L b , localized on NO 2 , and B a =B b ). The only state with significant ππ Ã (L b ) character is found at 5.3 eV, much higher than in the other calculations and above the L a state, which is inconsistent with the predictions of Platt. 43,46 All of these states (S 1 to S 8 ) exhibit single-excitation percentages Ω of >70%, and S 9 to S 11 have about 50%. The higher states (S 12 to S 39 ) are predominant doubly or higher excitations (<35%).
The X40 calculation produces results that are significantly more in line with the state ordering expected from the single-reference calculations. The first three states are the n CO π Ã (S 1 ), n À NO2 π Ã (S 2 ), and n þ NO2 π Ã (S 3 ). The transition densities ( Figure S17) show that the n CO π Ã (in S 1 ) only mixes with the n þ NO2 π Ã . On the contrary, S 2 shows admixture with ππ Ã (L a ) and S 3 with ππ Ã (L b ), explaining their sizable oscillator strengths. The S 4 is predominantly ππ Ã (L b ), located only 0.25 eV higher in energy than the S 3 . The next five states are all bright ππ Ã states (localized on NO 2 , L a , B a , B b , and CT contributions). All higher states (S 10 to S 39 ) exhibit <50% single excitation character and are dark. Hence, we obtain the surprising result that in order to describe   Figure 1B.
F I G U R E 3 Simulated absorption spectra obtained by Gaussian convolution (full-width at half-maximum of 0.5 eV) of the vertical excitation results presented above. Both simulated spectra (filled plots) are normalized with the same normalization constant as in Figure 2 (i.e., such that ADC(2) is normalized to 1) to enable direct comparison of the relative intensities predicted by the different methods. The experimental spectrum 17 is shown in black.
the inspection of the oscillator strengths shows that the CASPT2 calculations with 12 and 18 roots are inadequate, only through the analysis of the wave functions and the transition densities were we able to trace back the problem in these calculations.
In order to understand the failure of the different CASPT2 calculations, it is expedient to investigate the underlying CASSCF states.
The energies, oscillator strengths, and wave function descriptors for the SA(40)-CASSCF are given in Table S18, transition densities in Figure S18. The first four states have predominant single-excitation character and can easily be identified as the n CO π Ã , n À NO2 π Ã , n þ NO2 π Ã , and ππ Ã (L b ) states. All  The L a state is mostly covered in the S 10 to S 16 range, whereas the B a =B b states and the ππ Ã (NO 2 ) state are found in the S 17 to S 32 range.
The sum of all Ω values is slightly above 10, so within the 39 computed excitations, only 10 singly excited states can be expectedthese supposedly span the three local nπ Ã states, the five discussed ππ Ã states (L=B a=b and NO 2 ), and some CT states. From these results it is clear that CASSCF calculations with fewer states will only include parts of the L a and B a =B b states.
The finding that the bright ππ Ã states at CASSCF level are too high in energy and buried within doubly excited states can be explained with well-known shortcomings of π-system CASSCF calculations, which are discussed in the literature in much detail for conjugated polyenes and polyacenes, 43,45,46,52 as well as for other molecules. 53,54 Coming from valence-bond theory, the ππ Ã states of such systems are usually classified into ionic "+" and covalent "À" states, where the "+" and "À" signs are related to the involved linear combinations of the different ππ Ã excitations. [44][45][46] It has been shown that within this classification scheme, the covalent "À" states usually have sizable contributions from doubly excited states (and large PR NTO ), but can be described reasonably well with π-system CASSCF. 46,55 An example of these "À" states is the dark ππ Ã (L b ) state that was found as S 4 in most of our results. On the contrary, the ionic "+" states are usually rather pure single excitations. Because they can be described (in the valence-bond picture) as closed-shell zwitterionic states, 56 they polarize the electron density strongly. It can be shown 57 that this leads to a high energetic penalty, formally caused by selfrepulsion of the transition density. A proper inclusion of the dynamic correlation of the σ electrons reduces the self-repulsion, leading to significant energetic stabilization and some reduction in oscillator strength. π-system CASSCF is missing any correlation of the σ electrons and thus severely overestimates the excitation energy of ionic "+" states. A dynamic correlation treatment (e.g., via CASPT2) is required to obtain reasonable energies. 46,52,55 Examples of these "+" transition density plots do not show these in-plane contributions, as they are not plotted from true CASPT2 wave functions, but from CASSCF wave functions remixed by the (X)MS treatment. 30 The covalent "À" states (e.g., L b ) do not exhibit such in-plane contributions in any of the tested methods, as they do not require σ polarization.  In any case, a reasonable CASPT2 treatment is required to properly demix the bright ππ Ã states from the double excitations, as the ionic "+" states are expected to have only little double excitation contributions. 46 Based on the results shown in Table 6 and S16 (especially the Ω values), it appears that regular MS-CASPT2 (M40) did actually disentangle bright states and double excitations, but failed to properly predict the energies of the disentangled states. As shown in Figure S19 (see also Table S19)  In oNBA, the assumption of vanishing off-diagonal elements is not warranted for the dense set of mixed double excitation/bright ππ Ã states, which all interact significantly due to their admixed ππ Ã contributions. This can be seen in Table S20, which lists the full matrix of b H 0 (from the X40 calculation, i.e., with one common b H 0 ), where several off-diagonal elements exceed 2 eV, particularly in the blocks that contain the scattered bright ππ Ã states. The advantage of XMS- The first four states are consistently described with all four methods. The S 1 state is consistently described as being of local n CO π Ã character. It is found at an energy of about 3.5-4 eV and is a completely dark state. The S 2 and S 3 are local n À NO2 π Ã and n þ NO2 π Ã states, energetically located at around 4 eV. These states most likely carry a small oscillator strength due to admixture of ππ Ã contributions, and can be assigned to the first experimental absorption band with maximum at 3.7 eV. 17 The lowest ππ Ã state, the L b state, is rather consistently placed at about 4.5-4.8 eV. Although all methods agree that this is a rather dark ππ Ã (as expected for L b ), its oscillator strength varies slightly (between 0.01 for CC2 and 0.06 for XMS-CASPT2). It is the most likely candidate to assign to the second band of oNBA with maximum at 4.4 eV.
After the lowest four excited states, the state order predicted by the different methods starts to differ. ADC(2) and CC2 predict a lowlying CT n CO π Ã state as the S 5 at an energy of 5.1-5.3 eV, whereas EOM-CCSD and XMS-CASPT2 place this n CO π Ã state at 6-7 eV. It is noteworthy that this state shows a relatively high double excitation character with ADC(2) and CC2, indicating that these methods might not describe it correctly. In any case, this state carries only little oscillator strength, making it difficult to observe experimentally.
The second state of ππ Ã character (S 5 or S 6 ) can be classified as the L a state of the aromatic ring, and is consistently described with an The states that we have discussed above cover the absorption spectrum up to about 6 eV (200 nm), the limit of the experimental absorption spectrum. 17 As there is no experimental reference for higher energies, we refrain from a detailed analysis of any higher states. Based on our single-reference calculations (Tables S3-S5), there might be another band at about 7 eV (175 nm). However, this band is not reproduced even with our most extensive CASPT2 calculation (X40). It is highly likely that at these energies double excitations play a substantial role, making it doubtful that ADC(2), CC2, or EOM-CCSD can adequately predict these states.

| Method assessment and literature comparison
In this last section, we want to briefly discuss the implications of our results for the selection of electronic structure methods for describing oNBA and related molecules. Unsurprisingly, ADC(2) and CC2 produce very similar excitation energies, oscillator strengths, and state characters. ADC(2) tends to give slightly lower energies, although it is difficult to estimate from comparison to experiment whether ADC (2) or CC2 produce more accurate results. Interestingly, despite the fact that the D1 and D2 norms indicate potential problems for ADC(2) and CC2, in general both methods predict plausible excitation energies that are consistent with the experiment and with more elaborate methods. Given that ADC(2) is appreciably cheaper than CC2 and is a Hermitian method-thus produces correct topologies for excited state-excited state conical intersections (e.g., S 1 /S 2 )-we suggest that ADC (2) is the more useful of the two methods for investigations of oNBA and similar compounds, for example, for mapping the excitedstate potential energy surfaces or for nonadiabatic dynamics.
Compared with ADC(2) and CC2, EOM-CCSD produces similar results for the low-lying states, but gives significantly different results for some of the higher states, for example for the second n CO π Ã (due to its higher double excitation character). Hence, for some investigations on oNBA EOM-CCSD might be the more appropriate method than ADC(2) or CC2. However, the high computational cost of EOM-CCSD for energies and analytical gradients make this method challenging to use for potential energy surface exploration tasks.
The presented results have also provided very useful information on the peculiarities of performing CASPT2 calculations for oNBA.
Most importantly, the large active space (CAS (18,14))-which we have chosen to describe both vertical excitation and (in the future) relaxation pathways in a flexible way-seems to permit a large number of double excitations at 6 eV and above, which produces the problems discussed in the previous section. A very large number of stateaveraging roots and XMS treatment are required to recover the bright ππ Ã states that explain the intense absorption bands of the molecule.
Here, we would argue that the benefits of using a large number of roots far outweighs the deterioration of the active space orbitals due to excessive state-averaging. In particular, the results in Table S2 show that the active space orbitals barely change when changing the number of state-averaging roots. Conversely, if only the lowest excited states (n CO π Ã , n À NO2 π Ã and n þ NO2 π Ã , ππ Ã L b ð Þ) need to be described, the requirements for the CASPT2 description are significantly loosened. In this case, already 12 roots and regular MS treatment are sufficient, as these low-lying states are sufficiently well described by SA-CASSCF and thus require only little correction by CASPT2. Given the very high cost of any of the MS-CASPT2 (18,14) calculations, investigations of the excited-state potential energy surfaces benefit from calculations with fewer roots and will need to rely on gradients computed with smaller active spaces/fewer states or with others methods.
Finally, we want to compare our results to the previous vertical excitation computations from Reference 16, at the CC2/TZV(P), TD-B3LYP/6-311G(d,p), and MS-CASPT2 level of theory. Note that these computations employed a different nuclear geometry-the Xray structure 62 -than we employ in the present work (MP2/cc-pVTZ), because it was not possible to reproduce the previous results 16 with the X-ray geometry. The CC2/TZV(P) results 16 are roughly equivalent to our findings with CC2/cc-pVTZ, although the literature data shows slightly lower excitation energies and show a reordering of the lowlying n CO π Ã , n À NO2 π Ã , and n þ NO2 π Ã states. We attribute these differences mostly to the different nuclear geometries used.
The TD-B3LYP results 16 show extensive mixing of nπ Ã and ππ Ã characters according to the reported coefficients and thus the electronic characters cannot be directly compared with our results. However, the excitation energies and oscillator strengths of the low-lying states appears to be comparable to our findings. Interestingly, at higher energies, TD-B3LYP produces a much higher density of states (the S 12 is found at only 5.8 eV) than the other methods in Reference 16 or our work (S 12 only appears at 6.5 eV in ADC(2), 7.4 eV in EOM-CCSD, and 7.2 eV in X40). This indicates that B3LYP underestimates the energies of high-lying states with charge transfer character, as already discussed previously. 15 Finally, the MS-CASPT2 results of Reference 16 loosely resemble our findings. The low-lying n CO π Ã , n À NO2 π Ã , and n þ NO2 π Ã states are reordered, probably due to the change in geometry. These states also exhibit remarkable contributions from charge transfer ππ Ã excitations, which our best CASPT2 calculations do not predict. Reference 16 also reported that with the larger active spaces only few ππ Ã transitions could be computed, and the transitions with large oscillator strength (0.22) required the use of a smaller π-only active space (CAS (12,11)). This is completely in line with our findings here-that large active spaces in oNBA produce many double excitations, which deteriorate the description of the bright ππ Ã transitions. However, here we have shown that the inclusion of a sufficient number of roots and XMS CASPT2 can overcome these problems. We have found no indications that the inclusion of a large number of roots worsens the CASSCF orbitals and description of the low-lying states, as stated in Reference 16.
The information gained in the present work might also be relevant for other, related molecules regarding the peculiarities of computing high-lying states with CASPT2. For example, the prototypical molecule nitrobenzene has gathered quite a lot of attention. Looking at several different multireference perturbation theory calculations from the last decades, 11,12,14,63,64 it can easily be seen that all of them consistently predict a bright ππ Ã state ( 1 A 1 , L a ) at about 5 eV with oscillator strengths of 0.2-0.3, perfectly in line with the experimental absorption at 240 nm. 65 However, at higher energies, the results differ very strongly. A rather early work 63 considered a total of 25 states in their calculations and obtained two more intense bands at 6.2 and 7.5 eV, agreeing with experiment. 65 By contrast, a more recent work 12 only considered 12 states. While they obtained some transitions up to 7.5 eV (including several double excitations), the only bright state is the ππ Ã (L a ) state at 5 eV, with the bright states around 6.2 and 7.5 eV completely absent. It thus appears that the problems encountered in our present work could also be present in calculations of other nitro aromatic compounds.

| CONCLUSIONS
We have presented a large set of vertical excitation calculations for oNBA to investigate the suitability of different electronic structure methods for the description of the excited states. We presented results for the single-reference methods ADC(2), CC2, and EOM-CCSD as well as for multireference MS-and XMS-CASPT2 computations with a large CAS (18,14) active space. For the CASPT2 calculations, we also investigated how the number of included states, the IPEA shift, and the imaginary level shift affect the results.
Judging from consensus of the different electronic structure methods and from comparison to the experimental absorption spectrum, the following electronic states were found. The lowest three excited states are all predominantly of local nπ Ã character, corresponding to excitations out of the carbonyl or nitro lone pair orbitals.
These states are dark, but acquire a small oscillator strength due to the non-planarity of oNBA and thus give rise to the first absorption band at about 3.7 eV. These three low-lying states are rather well behaved, as virtually all methods agree on this picture. At higher energies appear a total of five ππ Ã states, which can be assigned as the ring L b , L a , B b , B a states in Platt's nomenclature and the local ππ Ã state of the nitro group. Only the lowest of these states (the L b state) is consistently described by the employed methods, and produces the second absorption band at around 4.4 eV. The higher states were consistently obtained by the single-reference methods, whereas CASPT2 calculations produced wildly inconsistent results or missing bright states. Reliable results from CASPT2 were only obtained if the number of considered states was increased to about 40 and XMS CASPT2 was employed. Our results indicate that the L a state (and possibly the ππ Ã state of the nitro group) give rise to the third band at 5.2 eV, whereas the B b and B a states are responsible for the fourth band at 5.7 eV. Besides these bright states, at high energies several CT nπ Ã states and a dense set of double excitations were identified.
In general, our work shows clearly that the presence of rather low-energy double excitations in oNBA can make vertical excitation calculations very challenging. In particular, we want to highlight again the finding that we were only able to compute the absorption spectrum of oNBA correctly with CASPT2 if we include around 40 electronic states in the computation-even though only 10 single excitations among the finally obtained XMS-CASPT2 states are present. This indicates that a proper coverage of all desired electronic states in the entire CASSCF/CASPT2 computation of functionalized organic molecules should receive an increased awareness in the future, both in benchmark work 58,[66][67][68] as well as in studies of individual molecules. At least for oNBA, we suggest that the benefits of using a large number of roots-to cover all relevant transitions already at the SA-CASSCF level-far outweighs the potential deterioration of the active space orbitals due to excessive state-averaging. We also suggest that such multireference perturbation theory studies should always be accompanied by black-box single-reference computations and in-depth wave function analysis to ensure that all relevant electronic states are properly characterized and none are missed.