Daniele Fazzi

Mario Barbatti

Walter Thiel

Unveiling the Role of Hot Charge-Transfer States in Molecular Aggregates via Nonadiabatic Dynamics

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INTRODUCTION

Excitons are the primary outcome of light-matter interactions. Their formation and relaxation play a key role in a variety of areas, including chemical reaction dynamics 1 , ultrafast electron-nuclei processes in organic and inorganic materials [2][3][4] , dynamics of semiconductors and quantum dots 5 , and light harvesting and energy transfer in photosynthetic systems 6 . In particular, ultrafast exciton dynamics govern the energy flow and the charge generation/recombination mechanisms in organic -electron conjugated materials and interfaces 7 .

In the last two decades, our understanding of ultrafast phenomena ruling exciton dynamics and charge generation processes in organic materials has significantly improved. This progress took place in the wake of scientific and industrial developments in photovoltaics, electronics, spintronics, and thermoelectrics, with strong focus on soft materials [8][9][10][11][12][13][14][15] .

However, several features of exciton and charge generation dynamics at the molecular scale are still obscure. The usage of fairly similar concepts such as frontier orbitals and interfacial states is still common in the literature. Even the role of one of the primary process, the high-energy (hot) exciton relaxation, is still under debate [16][17][18][19][20] in the organic and hybrid photovoltaics communities: it is unclear whether hot excitons would favour an instantaneous electron-hole charge separation, thus increasing the overall charge generation efficiency, or alternatively lead to unproductive nonradiative decay toward low-lying electron-hole bound states.

Experimental evidence of fast (~30-100 fs) electron transfer processes and polaron formation from photoinduced hot excitons has recently been reported for thiophene-based molecules 21 , acenes 22 , diverse polymers 23 , and polymer/fullerene interfaces 17,18,21,[24][25][26] . This ultrafast population of hot CT states competes with nonradiative mechanisms within the manifold of excited states 27 . Internal relaxation can bring hot excitons to the bottom of the excited-state band, thus reducing the lifetime of hot CTs and allowing for nonradiative pathways and recombination processes that are detrimental for efficient charge generation 28 .

An appropriate theoretical description and modelling of ultrafast exciton relaxation in organic aggregates is urgently needed 29,30 . Pioneering recent studies by Rossky et al. 31,32 , Tamura et al. 33,34 , Tretiak et al. 4,13,35 , and Prezhdo et al. 10,36,37 describe the nonadiabatic excited-state dynamics of mediumsized systems based on different approximations. Limited to model Hamiltonians, single-determinant Kohn-Sham treatments or low-dimensional 1-D systems, these simulations have contributed to the understanding of the dynamic processes, but without addressing in detail the nature of the electronic states accessed during the relaxation process. This knowledge gap is tackled in our present work.

To understand ultrafast exciton relaxation phenomena and the population of hot CT states from an atomistic and time-dependent viewpoint, we have performed full-dimensional nonadiabatic excitedstate dynamics simulations based on linear-response time-dependent density functional theory (TDDFT) 29,38 for the bi-thiophene stacking dimer (Figure 1), a molecular aggregate representative of a large class of solid-state organic -conjugated materials. Our aim is to understand how hot excitons relax and whether we can control their lifetime. For this purpose, we use nonadiabatic excited-state dynamics to understand and predict i) the excited-state relaxation time scales for high-energy excitations; ii) the interplay between inter-and intra-molecular nonadiabatic relaxations; iii) the population of CT states; (iv) exciton delocalization/localization processes; and v) the nuclear vibrations governing exciton dynamics in thiophene-based molecular aggregates.

Our working hypothesis is that productive charge separation and transfer processes in organic electronics and photovoltaics depend on the population and lifetime of CT states. However, such CT states are usually not among the lowest states in the manifold of excited states in organic homo-and hetero-junctions 39 . If the lifetime of high-lying (hot) excited states is extended, the time the dimer spends in CT states rises, which increases the odds of productive charge separation. In this spirit, we propose a new goal for the design of functionalized molecular aggregates: if we can modify the excitedstate electronic structure by introducing energy gaps between high-and low-energy bands, this will disfavour unproductive ultrafast exciton relaxation and increase the residence time of the photoexcitation on productive hot CTs (see Scheme 1). The main assumption underlying this hypothetical scenario is that charge separation is triggered by a charge transfer process between two monomers. However, it might also be possible that charge transfer states extended over several monomers act as triggers for (long-range) charge separation 40 . We do not address this possibility here, but only point out that in principle it can be studied using the same methodology.

COMPUTATIONAL METHODS

Ground-and excited-state properties of the bi-thiophene dimer were investigated with DFT and TDDFT. The ground-state and the first three excited-state equilibrium geometries were optimized at the B97XD level, using the 6-31G*, 6-311+G*, and aug-ccpVTZ basis sets. This approach, which incorporates long-range and dispersion corrections, has been previously shown to provide semiquantitatively correct predictions for this type of systems 39 . Static calculations were done both in vacuum and in a dielectric using the conductor polarisable continuum model (CPCM). Ground-state geometries and vertical electronic transitions were also evaluated using the algebraic diagrammatic construction to second order (ADC(2)) 41 , combined with SVP and def2-TZVP basis sets. Further details are given in the Supplementary Information (Supplementary Note 1) including a more thorough description of the computational procedures and additional numerical results.

Absorption cross sections were computed with the nuclear ensemble method based on harmonicoscillator Wigner distributions. These calculations employed the B97XD functional and the 6-31G* and 6-311+G* basis sets, both in vacuum and in a dielectric continuum (see Supplementary Data). 150 points were considered for spectrum generation.

Nonadiabatic excited-state dynamics were done with TDDFT using the fewest-switching surface hopping (FSSH) method as implemented in NEWTON-X interfaced with Gaussian 09 42 . Dynamics were run at the TD-B97XD/6-31G* level, using a microcanonical ensemble. The time step for integration of the classical equations was 0.5 fs. The integration of the quantum equations was done with 0.025 fs steps using interpolated properties between classical steps. Time-dependent coefficients were corrected for decoherence effects as described in ref. 43 .

The large density of states in the dimers leads to numerous state crossings. In particular, the occurrence of trivial crossings between non-interacting or weakly interacting states has been recognized as potentially challenging for FSSH. In such cases, conventional FSSH becomes costly, requiring very small time steps 44,45 . This problem has been tackled by different strategies to track down the diabatic nature of the crossings, including the local diabatization method 46,47 , the self-consistent FSSH 45 , the flexible surface hopping techninque 48 , the global flux surface hopping 49 , and the Min Cost algorithm 44 .

In the present work, nonadiabatic couplings between excited states were computed from time-dependent wavefunction overlap integrals as proposed by Hammes-Schiffer and Tully 50 based on wavefunctions built through Casida's Ansatz [START_REF] Casida | In Recent advances in density functional methods[END_REF] . This approach has been extensively used for computating nonadiabatic couplings in dynamics simulations [START_REF] Barbatti | [END_REF][53][54][55] and has been found to be very stable even in the presence of trivial crossings 47 .

Due to the limitations of the linear-response TDDFT and Kohn-Sham DFT methods in providing reliable electronic states near intersections with the ground state, nonadiabatic transitions between the first excited state and the ground state were not computed. Trajectories were run for a maximum of 300 fs or until a crossing with the ground state (within 0.15 eV) was reached. In such cases, the last time step was taken to indicate internal conversion to the ground state. Under this assumption, the occupation of the ground state and the correction to the S1 occupation were computed at each time step as reported in ref. 56 .

The excited states were classified as localized, delocalized, or charge-transfer using the method introduced and discussed in refs. 39,57 . In brief, the molecular aggregate is split in two units A and B (in this case, the two monomers); then the contribution of each molecular orbital to each unit is computed with a Mulliken partition. With this information, the amount of localization ( I A P  ) and of charge transfer ( I A P  ) for each electronic state I is computed for unit A based on a configuration-interaction wavefunction approximation, using linear-response coefficients. Finally, the densities

RESULTS AND DISCUSSION

Absorption spectrum

The computed TDDFT vertical excitations and absorption cross sections for the bi-thiophene monomer and the van-der-Waals (vdW) dimer are reported in Figure 1. Just like in other noncovalent organic dimers, 58 the ground-state structure of bi-thiophene dimer is asymmetric causing a split of the low-lying excited states (S1 AB , S2 AB ). Analysis of the dimer excited states shows S1 AB and S2 AB to be localized within each monomer. Delocalized excited states are predicted at higher energies, and excited states with CT character are not present below 6.5 eV in the vertical spectrum. The dimer absorption cross section gives rise to two bands, one at low energy (LE, ~4.5 eV) and another at high energy (HE, ~5-6 eV). A manifold of excited states contributes to the HE band, and among them S 8 has non-negligible oscillator strength (f = 0.13). At distorted ensemble geometries, the bright state may also occur as S9.

High-energy nonadiabatic dynamics

TDDFT nonadiabatic excited-state dynamics was initiated in the HE band, restricting the excitation to the 5.5 ± 0.5 eV spectral window (shaded area in Figure 1). A total of 30 trajectories, considering 10 coupled singlet electronic states in each case, were initiated in the S9 state and propagated up to 300 fs or until the S1 AB /S0 AB gap became smaller than 0.15 eV. Six trajectories reached the S1 AB /S0 AB crossing during the simulations, while 24 had still large S1 AB /S0 AB energy gaps at 300 fs. These findings can be compared to very recent TDDFT 56 and ADC(2) 59 nonadiabatic excited-state dynamics performed on an isolated bi-thiophene monomer. In this case, HE-band excitation leads to over 80% occupation of S0 within 300 fs, while for the dimer the fraction is only 19%. Therefore, the ultrafast nonradiative decay through S1/S0 crossing is significantly delayed in the vdW dimer compared to the monomer.

To better understand the exciton dynamics and the transfer processes in the dimer, two exemplary trajectories are illustrated in Figure 3. Traj A is representative of those remaining in the excited-state manifold during the entire simulation, while traj B is an example of those few that reach the S1/S0 crossing before 300 fs.

In traj A, the photoexcited dimer decays from a high-energy excited state (S9 AB ) to the low-lying S2 AB and S1 AB states in 150 fs. Thereafter, the dynamics continues in the LE regime and does not reach the ground state. In traj B, the photoexcited system relaxes from S9 AB to S1 AB in ~30 fs and arrives at the S1 AB /S0 AB intersection seam in about 100 fs.

To gain insight into the different deactivation mechanisms and character of the excited states involved in the dynamics, we have monitored in each time step the electronic-density difference between the occupied excited state and the ground state for these two trajectories. This information is also plotted in Figure 3. Three types of electronic states are observed during the dynamics: localized, delocalized, and CT states.

In traj A, a localized state is initially populated (Figure 3). Within ~50 fs, this high-energy exciton evolves into a hot CT state, which features a net electron-hole separation between the two monomers.

This CT state persists up to ~150 fs, remaining populated for 100 fs. After 150 fs, the low-lying states are populated and energy transfer occurs between the two bi-thiophene units, leading to an oscillation between localized and delocalized states, with no further sign of a CT state.

In traj B, the dynamics starts in a delocalized state. The inter-molecular exciton localizes within 30 fs (Figure 3) and remains localized until the crossing to the ground state takes place. The radiationless deactivation mechanism to the ground state resembles that occurring in the bi-thiophene monomer: a ring-opening process. 56

Role of the ultrafast nuclear oscillations in the hot exciton dynamics

The time-dependent evolution of the excited states is governed by ultrafast nuclear oscillations, namely: i) CS vibrations (Figure 4), ii) ring puckerings, and iii) dihedral rotations (SCCS). The time scales for processes i and ii are shorter than for iii (50-100 fs vs. 250-300 fs), making CS bond elongations and ring puckerings the major nuclear motions contributing to nonadiabatic relaxation. Here we discuss the role of CS vibrations, being the most effective deactivation channel, while ring puckerings and dihedral rotations are addressed in Supplementary Note 3.

For traj A, in the early stages of the relaxation process (t ≤ 150 fs), the CS stretchings control the ultrafast nonadiabatic transitions amongst high-lying excited states, bringing the dimer from the initial state (S9 AB ) to the hot CT state. Once the population is transferred from the hot CT to the LE band (S3 AB → S2 AB , t ~150 fs), exciton localization (intra-molecular) and delocalization (inter-molecular) processes occur (Figure 4). In the LE band, exciton localization tends to take place whenever the amplitude of a CS vibration is larger (smaller) than a quantity we call the average Bond Length Amplitude Oscillation (<BLAO>, grey horizontal lines in Figure 4). It is defined as: <BLAO> = <BL> ± , where <BL> is the average CS bond length, as computed at the equilibrium ground-state geometry of the dimer (S0 AB state), and  is the average CS bond length fluctuation around <BL>, as extracted from a 300 fs S1 excited-state dynamics run of a single bi-thiophene molecule 56 . <BLAO> can be interpreted as an This process leads to exciton localization and subsequent radiationless deactivation through the S1 AB /S0 AB crossing (Figure 3). The mechanism is similar to the S1 → S0 nonradiative ring-opening pathway found for the isolated bi-thiophene monomer in Ref 56 . The analysis of all trajectories reaching the maximum simulation time (without returning to the ground state) shows the same exciton dynamics as observed for traj A: i) an initial decay (~20-50 fs) from high-lying excitons to hot CT states; ii) survival and evolution of the hot CT states for ~50-100 fs; iii) transfer to the low-lying excited states (S2 AB , S1 AB ) originating from the symmetry breaking; and iv) oscillations between localized and delocalized states in the LE regime, governed by CS vibrations.

The ultrafast population of hot CT states, as computed from our TDDFT nonadiabatic dynamics, matches well with experimental studies a variety of organic materials 4,18,[21][22][23]32 , which report formation of hot CTs within ~30-60 fs and further ultrafast photoinduced polaron generation.

We have already mentioned that the fraction of trajectories reaching the intersection with the ground state is much smaller in the dimer than in the monomer. The nonradiative pathways toward the ground state are controlled by CS bond stretchings, which destabilize the S0 state and stabilize S1 56,60 .

Therefore, to understand the reason for the fewer radiationless processes in the dimer, we compare the excited-state potential energy profiles along the CS stretching and the average CS fluctuations in the monomer and in the dimer (Figure 5). As shown in Figure 5a, the crossing between the S1 AB state and the ground state is reached via an activated process in the dimer. From the S1 AB minimum, a potential energy barrier of ~1 eV needs to be overcome to reach this crossing. In the case of the bi-thiophene monomer, the process is barrierless instead. The barrier in the S1 AB profile of the dimer impedes radiationless transitions to the ground state, which explains the slower return to the S0 AB state, compared to the monomer.

The source of this strong difference between the reaction paths in the monomer and the dimer along the CS stretching coordinate is due to the energetics of the lowest * state in the region near the S1 minimum (R(CS) < 2 Å). As expected, in the monomer, the CS stretching has little effect on the energetics of this state. In the dimer, however, the CS stretching strongly destabilizes the lowest * state because it breaks the intermolecular interactions that had stabilized the * in first place. As result, while in the monomer the path to the crossing to the * state (which is strongly stabilized along this coordinate and gives rise to the intersection with S0) is almost barrierless, in the dimer a large barrier connects the * to the * state. Thus, the - stacking interactions change the shape of the excitedstate potential energy surfaces and thus ultimately govern the exciton dynamics, 61 as also indirectly observed via Raman spectroscopy in the case of oligothiophene aggregates 62 . Additional information about the reaction path along the CS stretching is provided in the Supplementary Note 3 (Points 4 and

5).

As shown in Figure 5b, the mean CS bond length variations are smaller in the dimer (1.7 Å < R(CS) < 1.85 Å) than in the monomer (1.7 Å < R(CS) ≤ 2.1 Å), which lowers the chances for deactivation by the ring-opening mechanism. Thus, the S1 AB potential energy barrier and the hindering of the CS oscillations in the dimer have the overall effect of increasing the lifetime of the low-energy excited states (e.g. S2 AB and S1 AB ) and thus lowering the odds of radiationless deactivation to the ground state.

The CS bond stretchings play another fundamental role even before the S1 AB state is populated: they are ultimately responsible for driving the exciton relaxation through the HE band and on to the LE band. This is illustrated in the potential energy profiles from rigid scans shown in Figure 5c. These profiles were computed for two cases: i) elongation of one CS bond (Error! Reference source not found.c, left) from its equilibrium value (~1.72 Å) and ii) concomitant stretching of two inner CS bonds belonging to different units (Figure 5c, right). Both motions destabilize the HE states and lead to state crossings that may act as funnels for exciton relaxation through the HE band and for the population of S2 AB in the LE band.

The stabilization of S3 AB leading to a state crossing with S2 AB is of special interest. As discussed before (traj A), the S3 AB → S2 AB relaxation represents the population transfer from a hot CT state, belonging to the HE band, to a localized state belonging to the LE band. The faster the S3 AB → S2 AB relaxation occurs, the shorter is the lifetime of hot CT states; and the lower is the probability of charge separation. As can be seen from Figure 5c, the stretching of the CS bonds governs the evolution of the S3 AB -S2 AB energy gap (S3 AB -S2 AB ), and thus the deactivation from HE to LE bands. Although we cannot exclude that S2 may have CT character at some specific geometries, the data analysis shows that non-CT regions of the S2 surface dominate the dynamics.

This gives rise to the following question: Is there a way, by making use of design rules, to slow down the hot exciton deactivation processes, thus reducing the HE→LE relaxation and increasing the lifetime of high-energy states, which serve as a gateway to CT states?

Design rules for long-lived hot CT states

A possible answer to this question is suggested by additional TDDFT nonadiabatic dynamics simulations on the bi-thiophene dimer, in which the Cartesian coordinates of the sulphur atoms and the nearest bonded carbons are frozen, as indicated in Figure 6 (where only one bi-thiophene molecule is shown for clarity). In these constrained simulations (in which the C-H bonds and the C-C bonds in the thiophene ring are still free to oscillate), there are still ultrafast nonadiabatic transitions within the HE band, but those from the HE to the LE band are completely suppressed (Figure 6), as expected because of the artificial constraint on the CS distances. This supports our hypothesis that CS stretching vibrations are associated to the ultrafast relaxation within the manifold of excited states.

The exciton transfer from HE to LE is ruled by the energy gap (S3 AB -S2 AB ). In Figure 6 we compare the gaps obtained from the constrained dynamics (red line) and from all unconstrained trajectories reaching the maximum simulation time (grey lines). For the constrained dynamics, the computed gaps oscillate between 0.55 and 0.20 eV, while they vary over a much larger range in the unconstrained case, often reaching values as small as 0.01 eV. By constraining the CS bond lengths, the S3 AB → S2 AB crossing is evidently inhibited, with a concomitant increase in the lifetime of the highenergy excited states in the HE band. The simple intuitive design rule emerging from these restricted-dynamics simulations is that the lifetime of hot excitons 24,25 should be increased by finding ways to reduce the amplitude of the CS vibrations. Considering that the quantum oscillator amplitude is given by (ħ/)  , where  is the reduced mass and  the angular frequency, this could be achieved by increasing either  or , for example by introducing into the molecular backbone isoelectronic heavy atoms (e.g. Se vs. S, or Si vs.

C as recently reported 63 , increase in ), or fused thiophene rings which flatten the structure and enhance the local C-S electron density (e.g. thienoacene units, increase in ).

We note that these proposed chemical modifications are motivated only by qualitative arguments and that we do not yet have any computational data to support them. Moreover, they focus exclusively on the control of the nuclear vibrations, without considering their impact on the electronic structure. In Figure 7 (left side) we schematically sketch and summarize our findings. For (typical) thiophene-based aggregates, the initial high-energy exciton evolves via nonadiabatic transitions to hot CT states that are populated in tens of femtoseconds. Hot CT states last for ~100 fs, and then decay to low-lying excited states. At the bottom of the excited-state LE band, the exciton oscillates between localized and delocalized states without ever returning to a CT state. For a hypothetical functionalized aggregate where the HE and LE bands are kept energetically apart (Figure 7 right side), the exciton is trapped in the HE band and does not populate the LE band (on an ultrafast time scale), which would increase the probability to populate CT states.

The readjustment of the electronic structure due to repopulation of excited states favoring specific relaxation channels resembles the so-called shishiodoshi transfer mechanism, which denotes a concerted unidirectional electronic and vibrational energy transfer through a chain of excited states with descending energy; this process is enhanced by vibronic relaxations 7 .

The ultrafast decay via hot CT states, observed in our full-dimensional TDDFT nonadiabatic dynamics, has important consequences for thiophene-based homo-and heterogeneous organic interfaces used for electronic and photovoltaic applications. The possibility to populate in few femtoseconds a high-energy CT state that may live over longer times opens the way for ultrafast electron transfer and charge separation, before relaxation to the bottom of the interfacial excited states can happen.

CONCLUSION

Full-dimensional nonadiabatic excited-state dynamics based on linear-response TDDFT allow us to gain insight on the photophysics of high-energy (hot) excitons at organic/organic interfaces. Ultrafast exciton deactivation processes were investigated for a bi-thiophene dimer, considered as representative of a large class of molecular aggregates, homogeneous and heterogeneous organic interfaces. The time scales and relaxation processes emerging from our TDDFT simulations nicely agree with data from state-of-the-art ultrafast spectroscopy on small thiophene-based conjugated molecules, acenes, homoand co-polymers, and even on polymer/fullerene interfaces 22,25,26,32,[64][65][66][67][68] . Our findings are of general relevance for the interpretation and understanding of ultrafast excitonic processes in functional conjugated molecular aggregates, and for the design of new efficient solid-state photoactive materials.

In the early stages of the relaxation process (~20-50 fs), ultrafast nonadiabatic transitions occur within the manifold of high-energy excited states, populating hot CT states. Hot CTs live for ~100 fs and then decay to low-energy states, namely S2 AB and S1 AB . Thereafter, the system oscillates between localized and delocalized states, in a process governed by CS vibrations. Already at 300 fs, 80% of the occupation is transferred to the bottom of the excited-state manifold, from where nonradiative decay processes through S1/S0 crossings occur, but much less frequently than in the monomer case 56,59 . This can be traced back to an energy barrier between the S1 minimum and the S1/S0 crossing, which impedes non-radiative S1/S0 deactivation in the vdW dimer.

Nonadiabatic within the manifold of the excited states are fostered by CS vibrations.

CS bond elongations stabilize the excited states, thus promoting fast depopulation of hot CT states and reducing the energy gap between HE and LE bands (via pronounced stabilization of the S3 state).

Increasing the lifetime of hot excitons is a possible way to enhance the occupation of the CT states, which may lead to polaron formation in molecular aggregates. We explored this possibility by running constrained nonadiabatic excited-state dynamics, in which the promoting CS vibrations were suppressed by freezing the corresponding coordinates. This prevented the ultrafast population of lowlying excited states (Figure 6), without inhibiting nonadiabatic transitions within the high-energy excited-state manifold. The resulting increase in the lifetime of hot excitons improves the odds of populating long-lived hot CT states (Figure 7).

Based on these findings, we conclude that the search for ways to extend hot-exciton lifetimes may be a productive goal in the design of molecular aggregates for photovoltaics. For example, focusing on the mechanics of the nuclear frame, we anticipate that replacement of Se for S (or maybe Si for C) in the molecular backbone may mechanically reduce the amplitude of the relevant promoting vibrations and that introduction of fused thiophene rings (e.g. thienoacene units) 69 may lead to a higher electron density in the CS bonds and strengthen them; these measures may increase the lifetime of hot excitons and open an energy gap between the HE and LE bands, thus impeding nonadiabatic HE-LE transitions (with localization and recombination). However, we note that these measures will also impact the electronic structure, in ways that we have not yet evaluated.

Independently of the success of such heuristic approaches, we emphasize that the bi-thiophene dimer investigated in this work should be understood as a prototype for photoinduced charge-separation in vdW organic dimers. Although it is gratifying that our simulations agree with the measured photo-

Scheme 1 .

 1 Scheme 1. Left side: colour code for excited-state classification and sketch of a molecular dimer. Right

  compared to predefined thresholds to classify the states.

Figure 1 .

 1 Figure 1. Top: DFT (B97XD/6-31G*) molecular structures for the bi-thiophene monomer and dimer (lateral and side views). Bottom left: TDDFT (B97XD/6-31G*) vertical transition energies for the bi-thiophene monomer and dimer, and excited-state classification. Bottom right: absorption cross

Figure 2

 2 Figure2shows the occupations of the states in the HE band (sum over S3 AB to S9 AB ) and the LE band (sum over S1 AB and S2 AB ) and also of the S0 AB ground state as a function of time. The HE band

Figure 2 .

 2 Figure 2. TDDFT (B97XD/6-31G*) nonadiabatic excited-state dynamics initiated by populating a high-energy excited state (S9 AB ). Left: state occupations vs. time. Right: sketch of ultrafast relaxation within the manifold of high-energy excited states (HE band) toward the low-energy states (LE band, S1 AB and S2 AB ) and of nonradiative relaxation to the ground state.

Figure 3 .

 3 Figure 3. Two exemplary trajectories: left sidetraj A, t = 0 -300 fs; right sidetraj B, t = 0 -100 fs. Top: snapshots of the electronic-density difference between the occupied excited state and the ground state for typical localized (loc), delocalized (deloc), or CT states; orange/green regions indicate loss/gain of electron density upon excitation. Bottom: time evolution of the excited-state energies (TD-B97XD/6-31G*) with indication of the prevailing state character. Also shown is the molecular structure close to the S1 AB /S0 AB intersection seam featuring the typical thiophene ring opening.

  oscillation interval that allows us to decide whether exciton localization occurs (CS vibrations beyond <BLAO> interval) or not (CS vibrations within <BLAO> interval), once the LE band is populated. At the B97XD/6-31G* level, <BL> = 1.73 Å and  = ± 0.1 Å.In traj A, following the population of the LE band (~150 fs), exciton localization occurs in two time windows, 150 ≤ t 1 ≤ 200 fs and t 2 ≥ 250 fs (Figure4-left), when the amplitude of some CS vibrations is larger than <BLAO>. For traj B, the CS bonds tend to oscillate between 1.6 and 2.0 Å (similar to traj A); however, at around 30 fs one CS bond on molecular site B starts to strongly stretch (> 2 Å), soon exceeding the upper limit of <BLAO> and continuing until it cleaves (Figure4-right).

Figure 4 .

 4 Figure 4. Variation of the CS distances in the bi-thiophene vdW dimer (molecular sites A and B) and classification of the occupied excited states during the nonadiabatic dynamics. Left: traj A (t = 0 -300 fs); right: traj B (t = 0 -100 fs). Grey lines: <BLAO> (see text).

Figure 5 .

 5 Figure 5. Panel a): TDDFT (B97XD/6-31G*) relaxed potential energy profiles of LE states (S2 AB , S1 AB ) and ground state (S0 AB ) for the bi-thiophene dimer (left side). The profiles are computed by optimizing the S1 AB state as a function of the CS distance (in increments of 0.02 Å) starting from the S1 AB equilibrium structure; analogous curves are shown for the monomer at the right; dashed lines are

Figure 6 .

 6 Figure 6. Top, left: TD-B97XD/6-31G* high-energy nonadiabatic excited state dynamics of a bi-thiophene dimer, for a trajectory with frozen Cartesian coordinates of the carbon and sulfur atoms (highlighted with black dots in the molecular structure). Top, right: sketch of the excited states involved in the deactivation process. Bottom: evolution of the gaps (S3 AB -S2 AB ) in all trajectories reaching the maximum simulation time in the unconstrained dynamics (grey lines), in the constrained trajectories with frozen coordinates (red line), and in trajectory A (black, see text).

Figure 7 .

 7 Figure 7. Schematic representation and classification of the excited states in an organic aggregate at different time steps during the excited-state dynamics. Left side: typical thiophene-based aggregate, with a sketch of the computed ultrafast deactivation via hot CT states. Right side: functionalized aggregate, with a sketch of the processes that tend to increase the lifetime of hot excitons (thus favouring CT processes) and to keep a high (≥ 0.2 eV) and constant energy gap (S3 AB -S2 AB ) (thus disfavouring exciton transfer to the LE regime).

ACKNOWLEDGMENTS D.F. acknowledges the Alexander von Humboldt foundation for a postdoctoral research fellowship. M.B. thanks the support from the A*MIDEX grant (n° ANR-11-IDEX-0001-02) and from the project Equip@Meso (ANR-10-EQPX-29-01), both funded by the French Government «Investissements d'Avenir» program.

induced charge transfer time constants in organic crystals and semiconducting polymers, we believe that the most interesting aspect of our study is the demonstration of how the first-principles characterization of the time evolution of the electronic structure may lead to new insights on molecular design. Rather than expecting larger complexes to share the same electronic features as our small prototype, we believe that the goal should be the following: given a certain organic crystal, we should identify and locate the CT states within the manifold of the electronically excited states relevant to ultrafast relaxation, and then try to devise ways to increase their lifetime by isolating them at the bottom of the electronic band. We believe that this approach is a true step forward in relation to the common analysis based on HOMO-LUMO gaps and vertical excitation spectra. Future work will need to focus on the investigation of singlet-state dynamics in larger molecular aggregates and at donor/acceptor interfaces 32 and on singlet-triplet exciton dynamics 70 . This will require nonadiabatic excited-state dynamics simulations with theoretical methods, such as linear response TDDFT, that are able to reasonably describe the excited-state electronic structure 39,71 of medium-large systems.

ASSOCIATED CONTENT

Supporting Information. Supplementary Note 1: ground state optimized geometries, excited state (singlet and triplet) vertical transition energies, double excitations, and absorption spectra. Supplementary Note 2: details on the computational methods and fitting procedure for the HE and LE state population. Supplementary Note 3: time-dependent oscillations of internal coordinates (ring puckering, C-C bond length and bithiophene dihedral angle oscillations); electron densities evaluated along traj A, traj B and trajectory with frozen degrees of freedom; energy of the triplet states along the dynamics; and S1 relaxed potential energy profiles for different values of the inter-molecular dimer distance. Supplementary Note 4: optimized Cartesian coordinates and absolute energies.

AUTHOR INFORMATION

Corresponding Authors fazzi@kofo.mpg.de; mario.barbatti@univ-amu.fr; thiel@kofo.mpg.de

Author Contributions

All authors contributed to analyse the data and write the manuscript.

FUNDING SOURCES

No competing financial interests have been declared.


Ljiljana Stojanović
email: stojanovic@kofo.mpg.de

Gessenildo Pereira Rodrigues

Saadullah G Aziz

Rifaat H Hilal
email: rhilal@kau.edu.sa

Mario Barbatti
email: mario.barbatti@univ-amu.fr

Photochemistry of methyl hypobromite (CH 3 OBr): excited states and photoabsorption spectrum

Keywords: methyl hypobromite, excited states, photoabsorption spectrum

The singlet and triplet excited states of CH 3 OBr with the excitation energies up to ~9.5 eV are studied using multi-reference configuration interaction with singles and doubles method (MRCI-SD) and several single-reference methods, including time-dependent density functional theory (TD-DFT), coupled-cluster (linear-response CC2 and equation-of-motion CCSD), and algebraic diagrammatic construction (ADC(2)). Among the single-reference methods, CCSD gives vertical excitation energies and oscillator strengths comparable to the MRCI-SD values for the majority of excited states. The absorption cross section in the gas phase in the region between 2 and 8.5 eV was simulated with CCSD using the nuclear ensemble approach. The computed spectrum predicts two intense absorption bands.

The first band, peaked at ~7.0 eV, is induced by Rydberg excitation. The second band has a strong overlap between a broad σσ* transition and three Rydberg transitions, resulting in two peaks at 7.7 and 7.9 eV. The spectrum also features a low-intensity band peaked at ~4.5 eV due to n* excitation. The intensity of this band is influenced by spin-orbit coupling effects. We analyzed the dissociation pathways along O-Br and C-O coordinates by computing rigid potential energy curves of the ground and the lowest-lying singlet and triplet excited states, and discussed the possible dissociation products.

Due to the specific electronic structure of the excited states, characterized by multireference, double excitations, and Rydberg states occurring in the low-energy region, their correct description along dissociation coordinates is feasible only with MRCI-SD.

INTRODUCTION

The discovery of the stratospheric ozone depletion in the late 1970s has brought a lot of attention to the photochemistry of chlorine and bromine organic compounds, as they have been noticed to play a significant role in stratospheric and tropospheric ozone depletion events. For instance, bromine compounds released from the Earth's surface reach the stratosphere, where they undergo photodissociation producing highly reactive species, such as atomic bromine (Br) or bromine monoxide (BrO). These reactive species interfere with natural ozone cycle, taking part as catalysts in the main chemical reactions in which ozone is depleted. [1][2][3] Even though bromine compounds are much less abundant than chlorine ones, it is estimated that they are more efficient in ozone destruction, causing significant fraction of the overall ozone loss. [4][5][6] In addition to its importance for ozone depletion processes, bromine radicals can also induce DNA damage. 7,8 According to one of the proposed mechanisms, bromine radicals, which are usually formed in a tissue by reduction of bromate ions (BrO 3 -) by glutathione and Cystein (GSH/Cys), can induce one-electron oxidation of guanine. In this way cation radical of guanine are formed, and it is further subjected to a reaction with a water molecule and oxidation, yielding oxygenated derivatives of guanine. 7 Methyl hypobromite (CH 3 OBr), as a potential source of atomic bromine and bromine monoxide, is a molecule of interest for both ozone depletion and DNA damage. It is also the simplest prototype for organic hypobromites investigations. In spite of that, only a small number of theoretical and experimental studies have been devoted to elucidation of its electronic structure, absorption spectrum, and photochemical mechanisms.

In the first theoretical study on CH 3 OBr, Guha and Francisco 9 investigated the geometry, vibrational spectrum, and energetics of CH 3 OBr in the ground state using density functional theory (DFT), and computed vertical excitation energies to the six lowest-lying singlet excited states using configuration interaction with singles (CIS). In their next computational study, 10 these authors reported vertical excitation energies to the five lowest-lying singlet and five lowest-lying triplet states computed with multi-reference configuration interaction with singles and doubles (MRCI-SD), as well as potential energy curves for these states along the O-Br coordinate computed with multi-configurational self-consistent field (MCSCF).

Later, Papayannis et al. 11 proposed six possible mechanisms for CH 3 OBr decomposition. Based on unrestricted Møller-Plesset perturbation theory to the second order (UMP2), they optimized geometries and computed vibrational frequencies for reactants, transition structures, and products involved in the proposed decomposition channels. Finally, they also computed G2MP2 potential energy surfaces and used Rice-Ramsperger-Kassel-Marcus (RRKM) theory to estimate the microcanonical rate constants for each mechanism; concluding that the most important decomposition pathways are O-Br bond cleavage and 1,2-elimination pathway leading to HCHO and HBr.

To our knowledge, in the only existing UV-VIS spectroscopic study of CH 3 OBr, Benter et al. 12 reported the gas-phase absorption spectrum restricted to the UVA region, between 230 and 400 nm, observing a week band with a peak at ~280 nm (~4.4 eV).

This series of physical-chemical studies have been of fundamental importance to our current understanding of the photochemistry of organic hypobromites. This investigation field, however, has been dormant for the last ten years and revisiting the photochemical properties of these compounds in terms of new methods and higher computational levels is especially timely. Some of us have already taken the first steps in this direction. In a recent computational study, Aziz et al. 13 performed high-level coupled cluster (CCSD(T)), MP2, and DFT studies on the geometric features of tautomeric CH 3 OBr/CH 3 BrO/BrCH 2 OH and HCOBr(cis)/HCOBr(trans)/BrCHO systems. The latter tautomeric system is derived from the decomposition products of the former. Vibrational frequencies were computed for all species and the topology of the electron density was analyzed using quantum theory of atoms in molecules (QTAIM).

In the present study, we turn to the excited-state properties, and investigate singlet and triplet excited states of CH 3 OBr with the excitation energies up to ~9.5 eV, employing different correlated multi-reference and single-reference approaches. We compare the results obtained with singlereference methods and analyze their applicability for description of the excited states in the Franck-Condon region and along the O-Br dissociation coordinate. Since the CCSD describes correctly the vertical excitation energies of the molecule (except for few specific states, which will be discussed later), we employ this method together with the nuclear ensemble approach for the simulation of the UV photoabsorption cross section and action spectrum in the gas phase up to 8.5 eV. Furthermore, we study dissociation pathways of the molecule along O-Br and C-O bonds by computing the rigid potential energy curves of the ground and the low-lying excited states at MRCI-SD level. This benchmark investigation not only provides significantly improved information about electronic structure and photochemistry of CH 3 OBr, but it also reveals several challenges that further computational theoretical investigations must face for adequately describing the excited states of organic hypobromites. These challenges include low-energy double-excitations, bright valence states very sensitive to active spaces definition, innumerous very diffuse Rydberg states, and strong correlations out of the Franck-Condon region.

COMPUTATIONAL DETAILS

The ground state geometry of CH 3 OBr molecule was optimized at the Møller-Plesset perturbation theory at the second order [14][15][16][17] (MP2) with aug-cc-pVTZ basis set 18,19 . The C s symmetry, theoretically confirmed in earlier studies 9,11 , was imposed in the geometry optimization and in all further computations (except those for the absorption cross section). The optimized geometry of the molecule is given in the Supporting Information, Section S1. The molecule has trans (gauche) conformation. It is oriented such that Br, O, C, and one of the hydrogen atoms (H1) lie in the xy-plane, which coincides with the mirror plane, whereas the other two hydrogen atoms (H2 and H3) lie on the opposite sides of the xy-plane (Figure 1). The oxygen atom has a lone electron pair in a 2p z orbital (out-of-plane), whereas two in-plane 2p orbitals take part in σ(O-Br) and σ(C-O) bonds. The bromine atom has two lone electron pairs, one in a 4p z orbital (out-of-plane) and another one in an in-plane 4p orbital (a linear combination of 4p x and 4p y orbital). The other in-plane 4p orbital is involved in the σ(O-Br) bond.

We performed state-averaged multi-configurational self-consistent field (SA-MCSCF) 20,21 computations of the ground and excited singlet and triplet electronic states. Separately one from another, fourteen singlet and fourteen triplet states (7A' and 7A'') were computed at the SA-MCSCF level, averaging all states with equal weights. The diagonalization of the Hamiltonian was performed in the basis of configuration state functions (CSFs).

After a set of preliminary SA-MCSCF computations of potential energy curves of the ground and excited states along O-Br coordinate, we concluded that the CSFs formed by excitations of 12 active electrons within the active space composed of 11 orbitals [SA-MCSCF (12,11)] are necessary to describe the excited states of interest. The state-averaged molecular orbitals include seven valence and four Rydberg orbitals (within C s group the occupied orbitals are 5a'', 17a', 18a', 6a'', and 19a', and the virtual orbitals are 20a'', 21a'', 8a'', 22a', and 23a'). The σ(C-H) orbital, two linear combinations of σ(C-O) and σ(O-Br) orbitals, the symmetric and antisymmetric linear combination of 4p z (Br) and 2p z (O) orbitals (denoted further as n' and n''), the in-plane 4p orbital of Br (denoted as n Br ), and the σ*(O-Br) orbital comprise valence orbitals; and the Rydberg orbitals are mainly represented by 5s (Br), 5p xy (Br), 5p' xy (Br), 5p z (Br) orbitals (5p xy and 5p' xy are two linear combinations of bromine 5p x and 5p y orbitals).

A graphical representation of molecular orbitals in the active space and their compositions are given in the Supporting Information, Figure S1 and Table S1.

Analyzing the compositions of the electronic MCSCF wave functions, we have found that the CSFs obtained by double excitations to Rydberg orbitals do not contribute significantly to the composition of the excited states' wave functions for the studied O-Br stretching geometries. For this reason, we restricted the maximum number of excitations to these orbitals to one, reducing significantly computational time.

The state-averaged MCSCF wave functions were used as reference functions in the subsequent internally-contracted multi-reference configuration interaction with singles and doubles (MRCI-SD) computations [22][23][24][25] . Within MRCI-SD computations, the closed-shell orbitals were doubly occupied in all CSFs and they were not correlated through single and double excitations. The contribution of higher excitations corrections were taken into account by generalized Davidson's method (designated as MRCI-SD+Q). MCSCF and MRCI-SD computations were performed with MOLPRO 2012.1 code. [START_REF] Werner | version 2012.1, a package of ab initio programs[END_REF] We have computed spin-orbit matrix elements between fourteen singlet ((1-7) 1 A′ and (1-7) 1 A′′) and fourteen triplet states ((1-7) 3 A′ and (1-7) 3 A′′) employing state-interacting method implemented in MOLPRO code. The method implies the diagonalization of the sum of electronic and Breit-Pauli (BP) operator ( BP el H H  ) in the basis composed of MRCI-SD electronic wave functions. The transition dipole moments between the ground state and the obtained spin-orbit coupled excited states are computed.

Besides the multi-reference approach, we have tested the performance of single-reference methods, namely the full time-dependent density functional theory (TD-DFT) and Tamm-Dancoff approximation to the TD-DFT (TDA-DFT), three coupled cluster methods, and an algebraic diagrammatic construction method in the description of the excited states of CH 3 OBr.

Linear-response (LR) TD-DFT and TDA-DFT computations were performed employing the long-range corrected density functionals CAM-B3LYP 27 and ωB97XD 28 , which are proven 29,30 to be more suitable for modeling electron excitations to delocalized orbitals, compared to hybrid functionals.

These functionals are constructed to improve the long-distance behavior of the non-Coulombic part of the exchange functional. Moreover, the ωB97XD functional also includes dispersion correction, which may be important for description of the molecule in regions in which photodissociation occurs.

Among the coupled cluster methods, linear-response coupled cluster with approximated doubles and resolution-of-identity approximation (CC2) [31][32][33] , equation-of-motion (EOM) coupled cluster singles and doubles (CCSD) [34][35][36][37] , and completely renormalized equation of motion coupled cluster singles and doubles with noniterative triples correction (CR-EOM-CCSD(T)) [38][39][40] computations were performed.

Furthermore, computations were carried out with the linear-response algebraic diagrammatic construction method to approximated second order (ADC(2)) 41,42 , using MP2 energies as the reference ground state. The CC2 and ADC(2) excited-state energies were computed with TURBOMOLE 43,[START_REF]TURBOMOLE V6.6 2014, a development of University of Karlsruhe and Forschungszentrum Karlsruhe GmbH[END_REF] .

TD-DFT, TDA-DFT, and CCSD computations as well as the geometry optimization at MP2 level were performed with Gaussian 09 software. [START_REF] Frisch | Gaussian 09, Revision D.01[END_REF] CCSD(T) computations were performed with GAMESS code. [START_REF] Schmidt | [END_REF][START_REF] Gordon | Theory and Applications of Computational Chemistry[END_REF] Dunning's generally contracted aug-cc-pVTZ basis set was chosen for description of H, C, and O atoms, whereas in the case of bromine atom, the aug-cc-pVTZ basis set was augmented by a set of diffuse s, p, d, and f functions added in an even-tempered manner. The exponents for the additional diffuse functions are computed in the same way as in the case of Dunning's d-aug-cc-pVTZ set [START_REF] Woon | [END_REF] , which is developed for the elements of the first three rows of the periodic table. These exponents form a geometric progression with the exponents of the two most diffuse functions of the series, i.e. they are computed as αβ, where α is the exponent of the most diffuse function in the corresponding series of the aug-cc-pVTZ set and β is the ratio of the two most diffuse functions of the set.

The mentioned methods were used to compute the vertical excitation energies, oscillator strengths, and the O-Br potential energy curves of the ground and excited singlet and triplet states. C-O potential energy curves were also computed with MRCI-SD with the reference space discussed later.

Besides that, the spatial extents (the expectation value of the operator 2 r in the electronic wave function (

2 2 2 2 z y x r   
)) of each of the studied electronic states were also computed using MCSCF and MRCI-SD wave functions at the ground state equilibrium geometry.

Finally, the UV photoabsorption cross section of the molecule was simulated using the nuclear ensemble approach 49 as implemented in the Newton-X code 50,[START_REF] Barbatti | NEWTON-X: a package for Newtonian dynamics close to the crossing seam[END_REF] . 500 different nuclear geometries were sampled using a harmonic oscillator Wigner distribution (at 0 K) computed from harmonic normal modes in the ground state. Vertical excitation energies and oscillator strengths into eight excited singlet states were computed with EOM-CCSD method for each geometry in the ensemble. This information was used to compute the photoabsorption cross section, which was convoluted with normalized Lorentzian functions with a phenomenological width of 0.05 eV.

RESULTS AND DISCUSSION

In first three subsections, we present and discuss the vertical spectrum computed with different classes of methods. The focus of these sections is more technical, emphasizing the suitability of each method for dealing with this molecule. In the subsequent subsection, a more spectroscopic discussion is done.

The last two sections deal with the photodissociation problem.

Vertical excitations with MCSCF and MRCI-SD

We present in Table 1 the results for the vertical excitation energies, oscillator strengths (for singlet states), main configuration weights, and 2 r values obtained at MCSCF and MRCI-SD levels, as well as vertical excitation energies obtained employing Davidson's correction for the singlet excited states of the CH 3 OBr molecule. The results for the triplet states are presented in Table 2. Only the leading configurations with weights larger than 15% are shown and we simplify the states' assignment by not distinguishing between the different orbitals of the same type (for instance, both non-bonding n' and n'' orbitals are designated as n orbitals). A more detailed description of the excited states' wave functions in terms of configuration weights is given in the Supporting Information (Tables S2 andS3).

According to the MRCI-SD results, several intense electronic transitions are expected in the far-UV region, from 7 to 8 eV (Table 1). They correspond to the transitions to the ns Rydberg state (3 1 A''),

to the Rydberg states with mixed ns and np characters (4 1 A' and 5 1 A'), and to the σσ* state (6 1 A').

Most of the excited states are dominated by one configuration, for instance, the first several valence excited states (as 1 1 A'', 2 1 A'', 1 3 A', 2 3 A', and 1 3 A'') and the highest Rydberg states (as 7 1 A', 6 1 A'', 7 3 A', 7 3 A''). On the other hand, other states (for instance, 4 1 A' and 5 1 A') have multiconfigurational character. The mixing of valence and Rydberg configurations occurs in the case of several excited states on MCSCF level, but the inclusion of additional correlation through MRCI-SD leads to a pure valence (for instance, 6 1 A') or pure Rydberg character (7 1 A').

One of the most interesting features of the computed excited states is the occurrence of doublyexcited states even in a low-energy region, such as the 3 1 A' and 5 1 A'' in the singlet manifold and the 5 3 A', and 4 3 A'' in the triplet. These states are dominated by various valence n 2 σ* 2 configurations, including all possible combinations of double excitations from n' and n'' orbitals to σ*(O-Br) orbital.

Moreover, the wave functions of these states also include up to the 15% of mixed valence-Rydberg doubly-excited configurations, n 2 σ*p and n 2 σ*s (see Table S2 and S3 in the Supporting Information).

A large number of the remaining excited states also have up to 10% of double excitations. The existence of the excited states with significant contributions from double excitations in a low-energy region is not exclusive of this molecule and has also been predicted for other organic conjugated systems [START_REF] Starcke | [END_REF]53 .

On a more technical note, the effect of the dynamical correlation on the vertical excitation energies and wave functions of excited states can be elucidated comparing the results obtained with MCSCF and MRCI methods. The nature of the several excited singlet states changes upon inclusion of dynamical correlation at MRCI level. For instance, the 3 The calculated MRCI-SD correlation energies, defined approximately by the differences between MRCI-SD and MCSCF energies ( -), are given in Table S2 and Table S3 of the Supporting Information. The MRCI-SD correlation energies of the ground and valence excited states are larger than those of the states with mixed valence-Rydberg and Rydberg character. The valence excited states are more stabilized by additional dynamic correlation obtained within MRCI-SD than the Rydberg excited states, because in their case the larger number of electrons is correlated comparing to the Rydberg ones. The inclusion of additional electronic correlation through the MRCI-SD reduces the vertical excitation energies of valence excited states (for instance, 2 1 A' and 1 1 A''), and significant raises (from ~0.04 to ~0.88 eV) the vertical excitation energies of the excited states with Rydberg character, compared to the MCSCF results. Similarly, in the case of triplet states, additional correlation included by MRCI-SD mostly decreases the vertical excitation energies of the valence excited states and doubly-excited states, and increases the excitation energies of Rydberg states.

The extensivity correction included through the MRCI-SD+Q level does not significantly influence the excitation energies. The root-mean-square deviation (RMSD) between MRCI-SD+Q and MRCI-SD energies is 0.1 eV.

We show in the next sections that the MRCI-SD method predicts singlet and triplet * energies that are 0.5 to 1.0 eV larger than those predicted by all other tested methods. It also predicts an oscillator strength for the singlet  excitation that is about half of that given by the other methods.

This deviation in the * estimate can be noticed in the comparison between the MRCI-SD results and the single-reference methods in Figure 2-top. The good agreement between coupled-cluster, ADC, and TD-DFT methods for these quantities indicates that MRCI-SD may be providing inferior results for the * state. Unfortunately, there are no available experimental results for comparison. Although we have not investigated this feature in detail, we have evidences (see next section) that the usage of a reduced active space, instead of the full valence space (which would be too computationally demanding), may be the reason underlying this problem.

Apart from the * mismatch, the multireference character of the MRCI-SD, built upon active and reference spaces systematically deduced from a series of preliminary MCSCF computations, supports using the MRCI-SD results as the reference ones for the comparison with results of singlereference methods.

Spin-orbit coupling

As a relatively heavy atom, Br may induce spin-orbit coupling (SOC) effects in the electronic states. To assess them, we have computed the couplings between fourteen singlet ((1-7) 1 A', (1-7) 1 A'') and fourteen triplet states ((1-7) 3 A', (1-7) 3 A'') at MRCI-SD level. A summary of the spin-orbit effects on the singlet states after coupling to triplet states is provided in Table 3. Full information for all states is given in Table S4 of the Supporting Information.

The inclusion of spin-orbit coupling lifts the degeneracy of the triplet states; the initial fourteen triplet states result in 42 spin-orbit coupled states, where the energy separation between states arising from the same triplet state is ~10 -3 -10 -2 eV (Table S4). The spin-orbit interaction affects the vertical excitation energies of the majority of singlet states by less than 0.25 eV, except in the case of the highest states in the considered domain, whose energy shifts are very pronounced, up to 1.1 eV for the 6 1 A' state and 1.6 eV for 7 1 A' state. (The spin-orbit shifts of the triplet states follow similar trend.)

These very large deviations for the high states, however, are most likely artifacts of the method and we will not draw any physical-chemical conclusion out of them.

Considering the values of oscillator strengths, the distribution of transition intensities among states is not changed significantly upon inclusion of spin-orbit effects. Transitions to the triplet state components have negligible oscillator strength values, except in the case of m = -1 component of 5 3 A'' state, for which the relatively large oscillator strength (f = 0.0110, Table S4) is induced by the singlettriplet interaction with the close-lying 4 1 A'' state. The oscillator strengths for transitions to the singlet states are not significantly changed, except in the two anomalous cases of the 6 1 A' and 7 1 A' states. The effects of spin-orbit coupling on the photoabsorption spectrum of CH 3 OBr will be further discussed in the "Photoabsorption spectrum simulation" Section.

Vertical excitations with TD-DFT

Since single-reference methods are widely used for describing excited states in general, we have tested their applicability for describing the excited states of CH 3 OBr. We start by the popular LR TD-DFT, even though the occurrence of low-lying double excitations revealed by MRCI-SD preannounces the limitations of the method.

The results of the vertical excitation energies, oscillator strengths (for singlet states), and approximate configuration weights obtained with TD-DFT are shown for singlet excited states in Table 4 and for triplet excited states in Table 5. In the cases where it is not possible to quantify the As expected, the most striking discrepancy between TD-DFT and MRCI-SD results is the inability of TD-DFT method to reproduce doubly-excited states (7 1 A', 5 1 A'', 5 3 A', and 6 3 A'' according to MRCI-SD results). Within the linear-response approximation, the susceptibility of a real interacting system is calculated from the susceptibility of its non-interacting Kohn-Sham counterpart and Hartreeexchange-correlation kernel 54 . The susceptibility of the Kohn-Sham system has poles only at single excitations, and in the adiabatic approximation (A) the Hartree-exchange-correlation kernel is frequency independent. Therefore, within LR ATD-DFT the susceptibility of the interacting system has poles only at single excitations and it is not able to reproduce the excited states with double excitation character 55 .

In general, the vertical excitation energies and oscillator strengths of most of valence and lowlying Rydberg states obtained with TD-DFT are in a good agreement with MRCI-SD results. However, the vertical excitation energies of the highest Rydberg states (6 1 A', 4 1 A'', 5 1 A'', and 6 1 A'') are significantly underestimated. Even though range separated functionals are constructed to give better description of delocalized states (as the Rydberg ones) by improving the asymptotic behavior of the exchange-correlation potential 56 , they do not describe well these high-lying Rydberg states.

TD-DFT vertical excitation energies of σσ* states are significantly smaller than the corresponding MRCI-SD values (1.00 eV for 1 σσ* state and 1.09 eV for 3 σσ* state). TD-DFT also predicts significantly larger value of the oscillator strength for the transition to this state, compared to MRCI-SD value. As we already mentioned, in this case, we favor the TD-DFT results over those from MRCI-SD. Even though the σσ* character of this state is predicted by both TD-DFT and MRCI-SD, there is an important difference in compositions of the corresponding MRCI-SD and TD-DFT σ and σ* orbitals, which helps to understand the discrepancies in excitation energies and oscillator strengths.

Within MRCI-SD, the σ orbitals are linear combinations of σ(O-Br) and σ(C-O) orbitals, whereas in the case of TD-DFT, the corresponding Kohn-Sham σ orbitals are more delocalized and have more complex compositions. This indicates that the active and reference spaces chosen for the MRCI-SD computations might not encompass all the orbitals relevant for description of σ orbitals, leading to the overestimation of the vertical excitation energies of σσ* states. Overall, ωB97XD performs slightly better than CAM-B3LYP. Excluding high Rydberg 5 1 A'' and 6 1 A'' states (according to TD-DFT assignation), the RMSD of the vertical excitation energies for the singlet states between TD-ωB97XD and MRCI-SD is 0.36 eV, whereas between TD-CAM-B3LYP and MRCI-SD it is 0.42 eV.

The vertical excitation energies of triplet states obtained with TD-DFT have significantly larger deviations from corresponding MRCI-SD results compared to the singlet states. The RMSD of the One of the reasons for discrepancy between ADC(2)/CC2 and MRCI-SD vertical excitation energies for the states with partial doubly-excited character is that, within the ADC(2) and CC2, double excitations are treated approximately. In CC2, the doubles equations taken from CCSD are approximated by truncation to the first order of the fluctuation potential. 59 In the case of the ADC(2), it can be shown that its results are equivalent to those of the symmetrized Jacobian of the configuration interaction singles with iterative variant of the doubles correction (CIS(D ∞ )) [START_REF] Hättig | Advances in Quantum Chemistry[END_REF] . Since CIS(D ∞ ) method is hierarchically lower than CC2, the approximations applied to the description of the double excitations within ADC(2) should be even stronger than in CC2.

Another pitfall that we have identified is that pure Rydberg 4 1 A'', 5 1 A'', and 6 1 A'' states are mainly described as mixed valence-Rydberg states with significant contribution of nσ* configurations (~20%) within ADC(2) and CC2, leading to a large deviation of their vertical excitation energies compared to the MRCI-SD values. As in the case of TD-DFT, the vertical excitation energies of triplet states have even larger deviations from their MRCI-SD counterparts.

CCSD brings significant improvement over ADC (2) and CC2 in the description of the excited states in the Franck-Condon region. The vertical excitation energies of the majority of the singlet states are in excellent agreement with MRCI-SD results. The mean deviation of the vertical excitation energies for the first nine CCSD singlet excited states from the corresponding MRCI-SD values is only 0.12 eV. The treatment of the states with partial doubly-excited character is improved compared to the CC2 method. However, as the other single-reference methods, CCSD is not able to predict the existence of pure doubly-excited states. This happens because while CCSD is correct to second order for single-electron excitations, it is correct only to the first order for double-electron excitations 59 .

The vertical excitation energy of the σσ* state is also lower than the corresponding MRCI-SD value by ~0.5 eV. The computed CCSD oscillator strength value (0.1467) is much closer to the corresponding TD-DFT (0.1321) and CC2 (0.1360) values, than to the MRCI-SD value (0.0636).

The vertical excitation energies of the highest Rydberg 5 1 A'' and 6 1 A'' states (according to the CCSD assignment) are significantly underestimated compared to the MRCI-SD values.

A number of benchmark studies on organic molecules (for instance, on the set of organic chromophores [START_REF] Moore | [END_REF] , DNA nucleobases [62][63][64] ) have shown that the EOM-CCSD(T) method gives correct descriptions of both localized and delocalized excited states. Comparing the vertical excitation energies of the singlet excited states obtained with CR-EOM-CCSD(T) and EOM-CCSD methods (Table 6), we conclude that triplet corrections do not affect significantly vertical excitation energies, except in the cases of the highest Rydberg 7 1 A' and 7 1 A'' states, where it increases vertical excitation energies for 0.13 and 0.19 eV, respectively. The RMSD between CCSD and CCSD(T) energies is only 0.07 eV.

Photoabsorption spectrum simulation

Due to the high computational cost, which requires single point calculations of the vertical excitation energies and oscillator strengths for 500 molecular geometries distributed around the equilibrium geometry, we could not employ the MRCI-SD method in the photoabsorption cross section simulation.

Instead, the simulation was done at CCSD level without including the spin-orbit coupling effects, which performs well near the Franck-Condon region. As already explained, the doubly-excited singlet n 2 σ* 2 states (which correspond to the 3 1 A' and 5 1 A'' states within C s symmetry) could not be reproduced by this method. Because the oscillator strengths for excitations into these states are negligible, their absence among the CCSD excited states does not influence significantly the simulated spectrum.

The photoabsorption cross section of CH 3 OBr simulated at CCSD level in the region from 2.5 to 8.5 eV is shown in Figure 3, including the contributions from the transitions to nine excited states ((2-5) 1 A' and (1-4) 1 A''). We did not include the highest Rydberg states (5 1 A'' and 6 1 A''), whose vertical excitation energies are underestimated, in the spectrum simulation.

The absorption spectrum has two high-intensity bands. The first intense band is peaked at ~7.0 eV and it is mostly induced by n-s (+n-p) excitation (transition to 3 1 A'' state according to CCSD results). The intense transition to the σσ* state gives a very broad contribution extending from 7 to 7.7

eV. The features at 7.35 and 7.5 are due to this transition, although they are most likely artifacts caused by statistical noise in the Monte-Carlo integration. The second intense band is mostly induced by an overlap of four transitions. The peak at 7.7 also has a large contribution from *. The peak at ~7.9 eV is due three Rydberg states with mixed ns-np character (they are the 4 1 A', 5 1 A', and 5 1 A'' CCSD states within the C s group).

Furthermore, in the low-energy region, the spectrum features a very low-intensity band with a peak at ~4.6 eV (it is shown in the inset of the Figure 3). Comparing the peak positions with the vertical excitation energies, one could conclude that the low-intensity band at ~4.6 eV is due to the nσ* excitation (transition to 2 1 A' state).

We compared the obtained results in low-energy region with the experimental UV/VIS absorption spectrum 12 of CH 3 OBr in the range from 400 to 230 nm (corresponding to ~3.1 to ~5.4 eV).

They are shown together in the inset of the Figure 3. The experimental results reveal a very weak absorption band at 280 nm (4.43 eV) and 0.0016 Å 2 .molecule -1 (52 M -1 .cm -1 ). The position and width, and intensity of the low-intensity peak associated with nσ* transition obtained in the simulation agree very well with the experimental results, but the intensity of the simulated peak (0.0035 Å 2 .molecule -1 ) is overestimated by approximately a factor two compared to the experimental intensity. We will discuss the reason for this overestimation later in this section.

Under normal terrestrial irradiance, photoactivation of CH 3 OBr should be restricted to the n* band at 4.6 eV. It is possible, however, that in extraterrestrial environments excitation of high-energy bands also contributes to the photoactivation. In fact, the action spectrum for extraterrestrial irradiance, computed as the product of the extraterrestrial solar irradiance [START_REF]Standard Solar Constant and Zero Air Mass Solar Spectral Irradiance Tables[END_REF] and the simulated absorption cross 20 section, shows that the ns+np band at 7 eV strongly contributes to the photoactivation too (Figure 3bottom).

We analyze the effect of spin-orbit coupling on photoabsorption spectrum comparing the MRCI-SD vertical excitation energies and oscillator strengths of the eight excited states included in spectrum simulation in the cases with and without spin-orbit coupling (Table 3).

Mostly, the spin-orbit interaction induces the change of vertical excitation energies by less than ~0.2 eV. The spin-orbit interaction slightly increases the vertical excitation energy of 2 1 A' state (~0.15 eV) and decreases its oscillator strength by approximately factor two. Thus, although the effect of the spin-orbit coupling on the position of the band at 4.6 eV (induced by the transition to 2 1 A' state) is minor, it may substantially affect its intensity, reducing it by approximately factor two. Thus, the overestimation of the simulated intensity compared to the experimental intensity mentioned above can be attributed the lack of spin-orbit coupling effects in the CCSD photoabsorption spectrum simulations.

The first intense absorption band (peaked at ~7 eV) is not importantly affected by spin-orbit coupling, since the vertical excitation energy and oscillator strengths for the transition to 3 1 A'' state are not changed significantly upon inclusion of spin-orbit effects. Together with the reduction of the band intensity at 4.6 eV, we may expect a change of the relative importance of the two bands in the action spectrum shown in Figure 3-bottom, with lower intensity at 4.6 eV relative to the intensity at 7 eV.

Dissociation through O-Br and C-O cleavages

We investigated the mechanism of CH 3 OBr decomposition that occurs through O-Br bond dissociation, by computing MRCI-SD stretching curves for the ground state and the thirteen lowest-lying singlet and triplet excited states ((1-4) 1 A', (1-4) 3 A', (1-3) 1 A'', and (1-3) 3 A'') of CH 3 OBr.

As in the case of CH 3 OCl (Ref. [START_REF] Li | [END_REF] ), most of the computed excited states of CH 3 OBr are strongly repulsive along the O-Br coordinate, indicating that O-Br bond cleavage is one of the favorable mechanisms, and agreeing with the conclusions of Papayannis et al. 11 based on chemical kinetics modeling.

The ground state and the first eleven excited states (2 1 A', 3 1 A', (1-3) 1 A'', (1-3) 3 A', (1-3) 3 A'') converge to the same dissociation limit, yielding the CH 3 O radical in the ground 1 2 E state and Br atom in the ground 2 P state. The photodissociation in the higher 4 1 A' and 4 3 A' states yields CH 3 O radical in the excited 1 2 A 1 state and Br atom in the ground 2 P state (Figure 4). The computed excited states do not cross along the O-Br coordinate. Some of the pairs of singlet and triplet excited states (like 2 1 A' and 2 3 A'', and 2 1 A'' and 2 3 A') are quasi-degenerate along the O-Br coordinate.

Since among the suggested photodissociation mechanisms of CH 3 OBr, the cleavage of the C-O bond has been also proposed 11 , we also examined the behavior of the excited singlet and triplet states along C-O coordinate. The active space that has been chosen to describe the energies of excited states along the O-Br bond was not sufficient in the case of the C-O stretching. Therefore, we use somewhat larger active space composed of 13 active orbitals with 14 active electrons. Even at this level of theory, we could describe only the ground and the seven lowest-lying singlet and triplet states.

The rigid potential energy curves of five states (1 1 A', 2 1 A', 1 1 A'', 1 3 A', 1 3 A''), computed with MRCI-SD method, are shown in Figure 4. The ground and the first three excited states (1 1 A'', 1 3 A', 1 3 A'') exhibit the same dissociation limit producing CH 3 radical in the ground 1 2 A'' state and BrO radical in the ground 1 2 Π state. In the dissociation limit, the higher 2 1 A' state produces CH 3 radical in the ground 1 2 A'' and BrO radical in the 1 2 Σ -state.

Contrary to the potential energy curves along the O-Br coordinate, the lowest-lying excited states show minima along the C-O coordinate, indicating that in the case of photodissociation mechanism that occurs through C-O bond cleavage, a potential energy barrier has to be overcome (the potential energy curves of CH 3 OCl show similar behavior [START_REF] Li | [END_REF] ). This fact implies that the C-O bond cleavage is a less probable mechanism compared to the O-Br cleavage.

Taking together these dissociation curves and the results for the absorption and action spectra, under terrestrial solar irradiation, where only the 4.6 eV band contributes to CH 3 OBr photoactivation, we may expect CH 3 O radicals to be formed exclusively in the 1 2 E state. Above the ozone layer, however, under extraterrestrial irradiance, the action spectrum shows that the 7 eV band should also contribute to photoactivation. In this case, we may expect CH 3 O radicals to be formed in both 1 2 E and 1 2 A 1 states. In any case, atomic Br should be released exclusively in the 2 P state.

Methods benchmark: O-Br cleavage

In the next subsections, we discuss the suitability of the applied single-reference methods for the description of excited states along O-Br stretching coordinate. MRCI-SD is taken as the reference method in all of the following comparative analysis. The potential energy curves of the first seven singlet and seven triplet excited states computed with MRCI-SD are shown in Figure 4. We restricted our computations to fourteen electronic states, because of root-flipping problems occurring in the SA-MCSCF computations of the potential energy curves, when higher electronic states intrude in state averaging.

O-Br dissociation with TD-DFT

The results for the potential energy curves computed with TD-ωB97XD method are shown in Figure 5 (the TD-CAM-B3LYP results are given in Figure S2 of the Supporting Information). Not surprisingly, TD-DFT does not describe correctly most of excited states in their dissociation limit (for example, 3 1 A', 3 1 A'', 3 3 A', and 3 3 A''), and the obtained potential energy curves exhibit incorrect topography. For example, TD-DFT predicts an energy barrier for the dissociation in 4 1 A' and 4 3 A' states, states that are purely repulsive along O-Br coordinate according to MRCI-SD. TD-DFT also predicts the existence of several intersections in the singlet and triplet manifolds (for example, between 1 3 A' and 1 3 A'' at 2.2 Å, Figure 5-bottom-left), which do not appear in MRCI-SD computations.

It is well-known 67,68 that the description of excited states along dissociation coordinates poses a serious challenge to TD-DFT. These problems arise because the states gain more multi-reference character as the dissociation limit is approached and LR TD-DFT is not suited to deal with that. In fact, even the ground state along dissociation coordinate is already a challenge for DFT. The correlation effects in DFT are included through the xc-functional. In the cases of strong static correlations, such as the dissociation of bonds when near degeneracy effects arise, hybrid xc-functionals usually underestimate electronic correlation energy 69 . Since the description of the excited states within TD-DFT relies on description of the ground state, most of the excited states are not described correctly.

We analyzed the compositions of MRCI-SD electronic wave functions of the ground and excited singlet electronic states along the O-Br coordinate. An interesting feature is that the contribution of the doubly-excited CSFs to the ground state at the final point of the curve becomes significant (it reaches ~14%). Obviously, the electronic ground state becomes strongly correlated along O-Br coordinate, with doubly-excited configurations being of non-negligible importance, leading to the failure of the method.

These problems with strong correlation can be in principle alleviated by spin-unrestricted TD-DFT computations 70 . With this in mind, we computed the potential energy curves of the studied electronic states with TD-UDFT (ωB97XD) starting from broken-symmetry orbital guesses. As the potential energy curves for the singlet and triplet states show (Figure S3 of the Supporting Information), the additional electronic correlation recovered through spin-unrestricted approach does not significantly improve the description of the studied states.

O-Br dissociation with ADC(2), CC2, and CCSD

The corresponding potential energy curves computed with CC2 method are shown in Figure 5, whereas the results of ADC(2) and CCSD are shown in Figure S4 and Figure S5 of the Supporting Information.

ADC(2), CC2, and CCSD, as single-reference methods, also experience problems with description of bond dissociations. Like TD-DFT, they all predict incorrect asymptotic behavior of several excited states and their potential energy curves exhibit incorrect topographies. This also includes the prediction of intersections along the stretching of the O-Br coordinate, which do not occur in the MRCI-SD results.

This behavior can also be explained by inability of these methods to describe correctly correlations appearing in the ground state along the dissociation curve. ADC(2) method is constructed upon MP perturbation theory, and thus it is expected that it can yield reasonable results only if the electronic ground state is described reasonably at the MP2 level, which excludes cases when ground state has pronounced multi-reference character. Likewise, the ability of CC2 and CCSD to describe the excited states relies on the performance of these methods for the description of the ground state.

The conclusions that ADC(2) and CC2 are not suitable for the computation of excited states along O-Br dissociation are supported by the D 1 diagnostic 71 . The D 1 values along the O-Br coordinate computed for MP2 and CC2 ground states are compared with their recommended limits in the Figure S6 in the Supporting Information. D 1 becomes very large along the O-Br coordinate (an order of magnitude larger than the recommended limit), which indicates that the ground state is not described well on MP2/CC2 levels.

CONCLUSIONS

In this study, the electronic structure of the CH 3 OBr molecule is investigated by correlated singlereference and multi-reference methods. The excited singlet and triplet electronic states are computed at the equilibrium geometry of the ground state and along O-Br dissociation coordinate employing multireference configuration interaction with singles and doubles method, time-dependent density functional theory with range-separated functionals, algebraic-diagramatic-construction method, and coupled cluster methods. Spin-orbit coupled states are also computed at the multi-reference configuration level.

Most of the excited electronic states of CH 3 OBr exhibit complex electronic structure, characterized by a mixing of several configurations. Moreover, doubly-excited configurations contribute significantly to the wave functions of several excited states even in a low-energy region. For these reasons, the correct description of the excited states in Franck-Condon region and beyond is a true challenge for the majority of the single-reference methods.

Among applied single-reference methods, CCSD describes the best most of the states dominated by single excitations in Franck-Condon region, with an exception of the highest Rydberg states lying above 8.5 eV (5 1 A'' and 6 1 A'' according to the MRCI-SD assignation). ADC(2) and CC2 methods describe correctly the lowest-lying valence states, but underestimate the vertical excitation of the majority of the Rydberg states. TD-DFT, tested with two range-separated functionals, describes well most of the valence and the low-lying Rydberg excited states, but underestimates the excitation energies of the highest Rydberg states. All single-reference methods have failed to describe states with pure doubly-excited character. All these methods also provide very similar excitation energies and oscillator strengths for the bright *, whose values are significantly different from those predicted by MRCI-SD. We believe that in this case MRCI-SD is providing the poorer results.

We simulated the UV photoabsorption cross section simulation with the nuclear ensemble approach, using vertical excitation energies and oscillator strengths into nine excited states computed with CCSD at the geometries sampled from a Wigner distribution. The inability of CCSD to reproduce the doubly-excited 3 1 A' state does not significantly affect the simulated spectrum, since the predicted oscillator strength for the transition to this state is negligible. The computed photoabsorption cross section of the molecule is characterized by two intense bands in the region from 6.5-8. Not surprisingly, the aforementioned pitfalls of single-reference methods to describe the vertical excitations become even more pronounced in the description of the potential energy curves of the excited states along the O-Br stretching coordinate. All methods provided very poor results compared to those at MRCI-SD. The reason for their inadequacy is the strong electronic correlations occurring in the ground electronic state along the stretching. The inapplicability of single-reference methods in this case has been confirmed earlier for diverse other systems 72,73 . For the correct description of the excited states in wide geometry range, a multi-reference method as the MRCI-SD seems to be mandatory.

ACKNOWLEGEMENTS

  contributions of individual configurations due to significant mixing of orbitals of different types (typically σ* and Rydberg s and p), only qualitative contributions are given. Moreover, we compare the vertical excitation energies and oscillator strengths for the transitions to excited states obtained by different methods in Figure 2 (top), and we show the correlation between the reference MRCI-SD vertical excitation energies of the singlet states and vertical excitation energies obtained using singlereference methods in Figure 2 (bottom).

  5 eV, and a lowintensity band with a peak at 4.6 eV. The position and width of peak at 4.6 eV is in excellent agreement with the available experimental spectral data, but its intensity is overestimated by a factor two, due to spin-orbit coupling effects. The simulated action spectrum for extraterrestrial solar irradiance indicates while near Earth surface photoactivation of CH 3 OBr should be restricted to the 4.6 eV band, in the upper atmosphere the 7.0 eV bands should also significantly contribute to photoactivation. Additionally, we investigated the dissociation pathways in the low-lying excited states along the O-Br and C-O bond cleavages using MRCI-SD method. The studied excited states are repulsive along the O-Br coordinate, implying that dissociation of O-Br bond occurs without energy barrier, and that decomposition after excitation should occur fast along this coordinate. On the other hand, most of the electronic states have minima along C-O coordinate, implying that this mechanism for dissociation is less probable compared to O-Br cleavage, because certain energy barrier to dissociation has to be surpassed to dissociate C-O bond. The analysis of dissociation curves together with the action spectrum indicates that while under terrestrial solar irradiation CH 3 O radicals should be formed in a single state upon Br elimination, above the ozone layer this radical may be formed in two distinct electronic states.

This

  Project was funded by the King Abdulaziz City for Science and Technology (KACST) under grant number 11-ENV 1995-03. The authors, therefore, acknowledge with thanks KACST for support for scientific research. Also, the authors appreciate the kind cooperation provided by the Deanship of Scientific Research (DSR), King Abdulaziz University. GPR acknowledges the Brazilian agencies CAPES and CNPq (Project 232180/2013-0), and the German agency DAAD for financial support. MB thanks the support of the A*MIDEX grant (n° ANR-11-IDEX-0001-02) funded by the French Government « Investissements d'Avenir » program.

Figure 1 .

 1 Figure 1.Molecular structure of CH 3 OBr alongside the atom labels.

Figure 2 .

 2 Figure 2.Comparison of the vertical excitation energies and oscillator strengths of the excited

Figure 3 .

 3 Figure 3. Top: The simulated UV photoabsorption cross section of CH 3 OBr. The first ionization

Figure 4 .

 4 Figure 4. The potential energy curves of the excited singlet and triplet states along O-Br (top) and

Figure 5 .

 5 Figure 5. The potential energy curves of the excited singlet (top) and triplet (bottom) states along

Table 2

 2 SA-MCSCF, MRCI-SD, and MRCI-SD+Q results for vertical excitation energies (E v in eV), spatial extent ( 2 r in a 0 2 ), and main configuration weights of the excited triplet electronic states.

			SA-MCSCF				MRCI-SD			MRCI-SD+Q
	State	E v	weights	r	2	E v	weights	r	2	E v
	1 3 A' 4.46 0.85 nσ*	65.8 4.11 0.85 nσ*	66.2	3.97
	2 3 A' 6.13 0.89 σσ*	64.7 6.35 0.85 σσ*	64.8	6.37
	3 3 A' 7.15	0.93 np	113.6 7.85	0.86 np	104.2	7.98
	4 3 A' 7.17	0.90 np	124.7 7.91	0.84 np	130.3	8.05
	5 3 A' 7.94	0.93 np	133.1 8.69 0.79 n 2 σ* 2 69.2	8.72
	6 3 A' 8.43	0.85 ns	125.8 8.70	0.87 np	130.5	8.81
	7 3 A' 8.57 0.83 n 2 σ* 2 70.3 9.02	0.83 ns	131.2	9.09
	1 3 A'' 3.30 0.88 nσ*	65.5 3.11 0.85 nσ*	65.5	3.02
	2 3 A'' 6.10 0.78 nσ*	67.3 5.89 0.79 nσ*	66.0	5.76
	3 3 A'' 6.37	0.99 np	104.8 6.99	0.82 np	103.0	7.11
	4 3 A'' 7.19	0.90 np	132.7 7.75 0.68 n 2 σ* 2 76.7	7.73
	5 3 A'' 7.51	0.83 ns	125.6 7.88	0.74 np	124.0	7.94
	6 3 A'' 7.88 0.76 n 2 σ* 2 72.6 8.07	0.78 ns	129.7	8.11
	7 3 A'' 8.07	0.93 np	132.4 8.85	0.88 np	132.0	9.01

Table 3

 3 Vertical excitation energies (eV) and oscillator strengths for pure singlet states and for singlet states spin-orbit coupled to triplets. The same information for triplet states coupled to singlets is provided in the Supporting Information, TableS4.

		Without SOC	With SOC
	State	E v	f	E v SOC	f
	1A'	0.00	-	0.00	-
	1A''	3.71	0.0000	3.73	0.0000
	2A'	4.77	0.0012	4.93	0.0005
	2A''	6.25	0.0012	6.23	0.0009
	3A''	7.04	0.0439	7.05	0.0432
	3A'	7.40	0.0003	7.45	0.0001
	4A'	7.88	0.0301	7.94	0.0075
	4A''	7.81	0.0000	8.01	0.0000
	5A'	7.95	0.0435	8.11	0.0433
	5A''	7.90	0.0004	8.13	0.0032
	6A''	8.82	0.0052	9.00	0.0014
	6A'	8.14	0.0636	9.25	0.0100
	7A''	9.55	0.0113	9.60	0.0178
	7A'	8.77	0.0007	10.35	0.1020

Erreur ! Source du renvoi introuvable.

vertical excitation energies for the triplet states obtained with ωB97XD functional and with MRCI-SD is 0.56 eV and the corresponding value for CAM-B3LYP functional is 0.61 eV.

The results of the TDA-DFT computations with CAM-B3LYP and ωB97XD are compared to the full TD-DFT results in Tables S5 andS6 in the Supporting Information. Even though it has been claimed that Tamm-Dancoff approximation to the full TD-DFT method could improve the description of triplet states, 57,58 the obtained TDA-DFT vertical excitation energies of both singlet and triplet excited states are very close to their TD-DFT counterparts. For both functionals, the RMSD is only 0.05 eV for the singlet manifold and 0.12 eV for the triplet.

Vertical excitations with ADC(2), CC2, CCSD, CCSD(T)

The vertical excitation energies, oscillator strengths (for singlet states), and approximate configuration weights obtained with ADC(2), CC2, and CCSD are shown for singlet excited states in Table 6, and for triplet excited states in Table 7. Like for TD-DFT, wherever it is not possible to quantify the contributions of individual configurations due to the mixing of orbitals of different types, only the qualitative contributions are given. ADC(2) and CC2 give very similar results for the vertical excitation energies and oscillator strengths of the studied excited states (the largest difference between corresponding vertical excitation energies obtained with ADC(2) and CC2 is ~0.04 eV). They describe well the several low-lying valence states with single excitation character (2 1 A', 1 1 A'', and 2 1 A''). Just like TD-DFT, they also predict vertical excitation energy for the 1 σσ* state smaller (by ~0.6 eV) than that given by MRCI-SD.

The remaining states, which are mostly of Rydberg character and have partial doubly-excited character, are poorly described by both ADC(2) and CC2. Their vertical excitation energies and oscillator strengths remarkably deviate from the corresponding MRCI-SD values (see Figure 2). Furthermore, both ADC(2) and CC2 are not able to reproduce purely doubly-excited states.

TABLES

TABLE OF CONTENTS

From photoabsorption to photodissociation, from MRCI to TDDFT; a comprehensive benchmark on the photochemistry of methyl hypobromite (CH3OBr) is provided.