Triplet state photochemistry and the three-state crossing of acetophenone within time-dependent density-functional theory.

Triplet state photochemistry and the three-state crossing of acetophenone within timedependent density-functional theory Miquel Huix-Rotllant and Nicolas Ferré Citation: The Journal of Chemical Physics 140, 134305 (2014); doi: 10.1063/1.4869802 View online: http://dx.doi.org/10.1063/1.4869802 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/140/13?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Communication: Orbital instabilities and triplet states from time-dependent density functional theory and longrange corrected functionals J. Chem. Phys. 135, 151103 (2011); 10.1063/1.3656734 Efficient exact-exchange time-dependent density-functional theory methods and their relation to time-dependent Hartree–Fock J. Chem. Phys. 134, 034120 (2011); 10.1063/1.3517312 Mixed time-dependent density-functional theory/classical trajectory surface hopping study of oxirane photochemistry J. Chem. Phys. 129, 124108 (2008); 10.1063/1.2978380 Troubleshooting time-dependent density-functional theory for photochemical applications: Oxirane J. Chem. Phys. 127, 164111 (2007); 10.1063/1.2786997 A global investigation of excited state surfaces within time-dependent density-functional response theory J. Chem. Phys. 120, 1674 (2004); 10.1063/1.1635798

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THE JOURNAL OF CHEMICAL PHYSICS 140, 134305 (2014)

Triplet state photochemistry and the three-state crossing of acetophenone within time-dependent density-functional theory Miquel Huix-Rotllanta) and Nicolas Ferréb) Institut de Chimie Radicalaire (UMR-7273), Aix-Marseille Université, CNRS, 13397 Marseille Cedex 20, France

(Received 14 February 2014; accepted 18 March 2014; published online 4 April 2014) Even though time-dependent density-functional theory (TDDFT) works generally well for describing excited states energies and properties in the Franck-Condon region, it can dramatically fail in predicting photochemistry, notably when electronic state crossings occur. Here, we assess the ability of TDDFT to describe the photochemistry of an important class of triplet sensitizers, namely, aromatic ketones. We take acetophenone as a test molecule, for which accurate ab initio results exist in the literature. Triplet acetophenone is generated thanks to an exotic three-state crossing involving one singlet and two triplets states (i.e., a simultaneous intersystem crossing and triplet conical intersection), thus being a stringent test for approximate TDDFT. We show that most exchange-correlation functionals can only give a semi-qualitative picture of the overall photochemistry, in which the threestate crossing is rather represented as a triplet conical intersection separated from the intersystem crossing. The best result overall is given by the double hybrid functional mPW2PLYP, which is even able to reproduce quantitatively the three-state crossing region. We rationalize this results by noting that double hybrid functionals include a larger portion of double excitation character to the excited states. © 2014 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4869802] I. INTRODUCTION

Time-dependent density functional theory (TDDFT) has gained an overwhelming popularity and has become a routine tool as cost-effective computational approach for predicting vertical or adiabatic electronic transitions of large molecular and solid systems.1–3 Such a success has eventually encouraged the application of TDDFT to photochemical problems. A theoretical treatment of photochemistry with TDDFT is challenging though, since one needs a balanced and accurate treatment of electron correlation. Present approximations to exact TDDFT are only able to capture some portions of the correlation energy (frequently referred as the lack of double excitation character in the TDDFT excited states4, 5 ). As such, most approximate TDDFT models lack state-specific correlation effects. This is an important drawback when approximate TDDFT is used to explore the potential energy surface (PES) of excited states, especially when electronic state crossings occur. Frequently, the crossing regions are crudely represented or even absent when approximate functionals are used.6 Here, we assess the ability of TDDFT in the adiabatic approximation for describing the photochemistry of aromatic ketones. We employ acetophenone for the present test, since (i) experimental data and quantum chemical benchmarks results are available, and (ii) several singlet and triplet states of different electronic structures and their crossings are involved in its photochemistry. Triplet acetophenone is generated thanks to the simultaneous crossing of one singlet and two triplet excited states, the so-called three-state a) Electronic mail: [email protected] b) Electronic mail: [email protected]

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intersection.7 These crossings involve excited states only, thus avoiding the wrong dimensionality problem and the noninteracting v-representability problem in regions of degeneracy between the ground and excited states.6, 8 The description of a three-state crossing is challenging for approximate TDDFT. In the present study, we highlight the possible pitfalls of approximate TDDFT by testing several flavors of exchange-correlation (xc) functionals, in order to set up the optimal way to describe the photochemistry of aromatic ketones. Triplet states of aromatic ketones have attracted considerable experimental interest due to their photo-reducing activity in the presence of hydrogen atom donors,9 and their capacity to transfer the excitation energy in the presence of triplet quenchers.10 In both cases, the nature of the lowest-lying excited triplet states determines the photo-physical activity of the ketone. Many experimental studies have tried to characterize the nature of the lowest triplet state in aromatic ketones either by phosphorescence, electron spin resonance,11 laser flash photolysis,12–14 and ultrafast electron diffraction.15 These ketones exhibit two close-lying states, a 3 (nπ ∗ ) state, in which a lone-pair electron of the carbonyl oxygen is delocalized in the π conjugated system, and the 3 (π π ∗ ) state which is a typical transition of aromatic compounds. The energy gap between these two triplet states, which determines the photochemistry of the ketone, strongly depends on the type of ketone, the substituents, the polarity of the solvent, and the initial absorbing state.16 The photochemistry of aromatic ketones is difficult to predict by a priori arguments. Even, experiments are often not concluding enough about the nature of the lowest lying triplet states, due to the presence of a energetically close

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lying manifold of states. Therefore, the theoretical characterization of such molecules becomes essential. In the literature, there are few examples on the modeling of the photochemistry of aromatic ketones. Mostly, these studies use multi-reference methods such as complete active-space selfconsistent field (CASSCF), which generally need large active spaces and expensive post-CASSCF treatments for an accurate description of the electronic correlation effects. The CASSCF study of phenalenone by Segado and Reguero,17 at the burden of present computational capacities, exemplifies the necessity of much cheaper computational methods, such as TDDFT, which can describe the photochemistry of aromatic ketones. Due to the complex electronic structure of aromatic ketones, involving singlet and triplet states, only few studies have employed TDDFT.15, 18–20 To date, TDDFT has not been validated for describing triplet state photochemistry, especially when singlet-triplet or triplet crossings occur. Triplets are frequently less accurate than singlets in approximate TDDFT. Additionally, they have a stronger dependence on the choice of xc functional, and are more affected from ground state instabilities.21, 22 In the present work, we perform a critical assessment of TDDFT to describe the photochemistry leading to triplet acetophenone by comparing it with highly-accurate multireference results for this system.7 The whole photochemistry can be described by considering three singlet states 1 (nπ ∗ ), 11 (π π ∗ ), and 21 (π π ∗ ), and two triplet states 3 (nπ ∗ ) and 3 (π π ∗ ). Here, we focus on the description of the lowest singlet and triplet states, which offers the most efficient paths to the triplet manifold.7 This paper is organized as follows: Section II contains the technical details of the computations and the strategy to locate approximate conical intersections. Section III contains the results and discussion, and has been divided in the assessment of geometric parameters (Sec. III A), the spectra of energies (Sec. III B), and the three-state crossing (Sec. III C). We end up with some conclusions in Sec. IV.

II. COMPUTATIONAL DETAILS

Full response calculations will be referred by the name of the xc functional. Addition of the TDA label indicates the use of the Tamm-Dancoff approximation. For the present assessment, we have selected six different flavors of DFT xc functionals, namely, a hybrid functional (PBE0), a range-separated hybrid functional (CAMB3LYP), a range-separated hybrid meta-GGA (M11), a hybrid meta-GGA (M06-2X), a hybrid meta-GGA with screened exchange (MN12SX), and a double hybrid functional (mPW2PLYP).23–32 The choice has been guided by the will to have one representative functional for each rung in the Jacob’s ladder of DFT functionals.33 The underlying assumption is that sophistication of the xc functionals (hybrid, meta-GGA, range-separation, etc.) might have a larger impact on the description of electron correlation (and hence on the photophysics) than different parameterizations within a given rung.

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All TDDFT calculations have been carried out with except for mPW2PLYP calculations, which have been performed with ORCA.35 Analytic gradients have been used for all calculations, except for the mPW2PLYP/TDA, for which numerical gradients have been employed using a two-step formula and a step size of 0.01 a.u. The geometry of all minimum-energy structures discussed here can be found in the supplementary material.36 Approximate conical intersections have been searched with TDDFT using the penalty function approach of Levine, Coe, and Martinez as implemented in the CIO PT code.37 These are not real conical intersections, but rather minimumenergy crossing points. The optimized penalty function is given by GAUSSIAN 09 34

= fI J (R)

+ EJ (R) EI (R) 2 − EI (R) 2 EJ (R) + λ , − EI (R) +α EJ (R)

(1)

is the adiwhere I and J are indexes of electronic states, EI (R) abatic electronic energy, R is the geometry coordinates, and λ and α are parameters set to 30 and 0.02, respectively, and updated during the optimization as implemented in the CIO PT code.37 Multi-reference CASSCF and second-order extended multi-configuration quasi-degenerate perturbation theory (XMCQDPT2) calculations have been performed with the FIREFLY code.38, 39 The active space comprises 10 electrons in 9 orbitals (8 π orbitals and the lone pair orbital of oxygen). State Average CASSCF calculations have been performed over three roots for singlet states and over two roots for triplet states. XMCQDPT2 calculations have been performed in a space of 10 states for singlets and 8 states for triplets. All XMCQDPT2 calculations have employed an intruder state avoidance shift of 0.02 a.u.40 Unless otherwise stated, all calculations have been performed with a mixed triple zeta basis set, consisting of a cc-pVTZ basis set for hydrogen and carbon centers and augcc-pVTZ for oxygen centers.41, 42

III. RESULTS AND DISCUSSION A. Minimum energy structures

In this section, we present a detailed comparison of TDDFT and XMCQDPT2 geometries (the latter taken from Ref. 7). Among all the geometrical degrees of freedom, we will focus on the C–O bond length and the phenyl ring C–C distances (see Fig. 1 for the numbering of carbon atoms). The C–O bond length strongly varies within each electronic state and can be used as the main photochemical coordinate that explains the photochemical population of the triplet manifold.7 The phenyl ring distortion from aromaticity is an important test for assessing the performances of TDDFT, which is known to fail in the description of bond-length alternation in the excited states of some poly-conjugated molecules.43


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FIG. 1. Structure of acetophenone with numbering of the carbon atoms on top of each center.

1. The C–O bond length

Correlation graphs comparing the TDDFT and XMCQDPT2 C–O distances are depicted in Figure 2. This bond length is highly sensible to the amount of correlation energy included in the theoretical method7 and therefore, TDDFT is only expected to reproduce this distance approximately. Indeed, all tested xc functionals underestimate the C–O distance for all the considered electronic state minima. Only MN12SX and M06-2X overestimate the distance for the triplet state 3 (π π ∗ ), but this is probably fortuitous, as the corresponding TDA distances are underestimated. For mPW2PLYP/TDA and M11, the 21 (π π ∗ ) state minimum was not found, because the minimum of this state rests in the intersection region with the 11 (π π ∗ ) state. Overall, the deviations with respect to XMCQDPT2 are numerically small (the maximum deviation is around 0.06 Å). All functionals except mPW2PLYP/TDA (discussed separately) give C–O bond lengths with an uneven accuracy. While the 1 (π π ∗ ), and the triplets 3 (π π ∗ ) and 3 (nπ ∗ ) show similar deviations from XMCQDPT2, larger differences are observed for 1 (nπ ∗ ) and 21 (π π ∗ ). The less accurate description of the latter state can be expected, since high-energy states are normally of less quality in TDDFT due to a too strong mixing with the continuum states.44 For the other state, it might be attributed to an inaccurate description of electron correlation. Indeed, most of the tested xc functionals do not account for state-specific correlation, since this would require an explicit functional dependence on the nature of the electronic state transition or the spin multiplicity. The double hybrid mPW2PLYP functional gives the best accuracy for the C–O bond length, featuring an almost constant and tiny deviation from XMCQDPT2. These results stress the importance of including a portion of secondorder Møller-Plesset (MP2) correlation to describe the excited states. From this result, we may infer that MP2 correlation includes to some extent a larger portion of double excitation character, which seems necessary to describe the specific correlation effects of electronic states of different nature and multiplicity.

FIG. 2. Correlation graph of the C–O bond length (in Å) between DFT and TDDFT (top), TDDFT/TDA (bottom), and XMCQDPT2 for the relevant state energy minima. The ground state [1 (π π )] calculated with DFT is common for both TDDFT and TDDFT/TDA, and thus is only shown once in the figure. Vertical lines mark the C–O distance at the corresponding XMCQDPT2 minimum energy structure. XMCQDPT2 structures are taken from Ref. 7.

2. The phenyl C–C bond lengths

The distortion of the C–O bond length is expected to affect the conjugation of the whole molecule. Consequently, the C–C bond lengths of the phenyl ring reflects important structural changes that should be well described for applying TDDFT to aromatic ketones. In Figure 3, we report the differences between TDDFT (left graph) or TDDFT/TDA (right graph) with respect to XMCQDPT2 C–C bond lengths for each of the state energy minima. The C–C bond lengths are in general shorter than XMCQDPT2 ones. On average, errors are numerically smaller (about 0.01–0.02 Å) than in the case of the C–O distance. The ground 1 (π π ) state C–C bond lengths are uniformly well represented by all functionals, giving the closest agreement with XMCQDPT2. Concerning the excited states, the deviations affecting the different C–C bond lengths are slightly


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FIG. 3. Acetophenone phenyl C–C bond lengths: DFT or TDDFT deviations with respect to XMCQDPT27 for each of the electronic state minima. For each excited state, we report TDDFT (left) and TDDFT/TDA (right) results. For the nomenclature of the carbon atoms, see Fig. 1.

xc-dependent. Actually, their amplitudes show a more pronounced dependence of the location of the C–C bond in the ring. The application of the TDA does not introduce any overall significant improvement, although the distances are more uniformly represented with respect to XMCQDPT2 than in full response. This is clearly seen for the 21 (π π ∗ ) state for example. All the six xc functionals give similar trends. Accordingly, the deviations with respect to XMCQDPT2 must be ascribed to the TDDFT approximations rather than to the design of the xc functionals. It is noteworthy that TDDFT C–C bond lengths are of better quality for (nπ ∗ ) states than for (π π ∗ ) states. mPW2PLYP/TDA, which was shown to get the most accurate C–O distance, does not perform much better than the other xc functionals for the phenyl C–C bond lengths. Among the largest deviations, CAM-B3LYP and M11 functionals have troubles in describing the compression of the C4 – C5 and C7 –C8 bonds in the 3 (π π ∗ ) state.

B. Theoretical spectroscopy

The established accuracy of TDDFT prompts for an investigation of its reliability in the acetophenone spectroscopy (vertical and adiabatic) first. Besides the usual study in the Franck-Condon region, we also investigate the electronic transitions at each selected singlet and triplet minimum energy geometry.

1. Vertical transitions

The low energy absorption spectrum of acetophenone is characterized by three transitions: an almost dark 1 (nπ ∗ ) state, a weakly absorbing 11 (π π ∗ ) state, and a bright 21 (π π ∗ ) state. The vertical transitions are experimentally found at 3.38, 4.22, and 5.17 eV, respectively (see Ref. 15 and references therein), while XMCQDPT2 simulations give vertical excitations at 3.67, 4.46, and 5.07 eV, respectively.7 The experimental values for the 1 (nπ ∗ ) state and the 11 (π π ∗ ) state cannot be directly resolved from the absorption spectrum, due to their low intensity. These are obtained from sensitized phosphorescence and optoacoustinc spectroscopy, respectively, while the brightest peak can be resolved directly from the absorption spectrum.15 Therefore, the accuracy of the experimental values is uneven. Additionally, we did not calculate zero-point energies in our TDDFT calculations, and therefore, it is more appropriate to compare TDDFT with XMCQDPT2. In Fig. 4, we compare the simulated absorption spectra computed with TDDFT, TDDFT/TDA, and XMCQDPT2. For the sake of comparison of the shapes of the spectra, all peaks have been normalized so that the 21 (π π ∗ ) state has an oscillator strength of 1. For the present discussion, the transition energies and the oscillator strengths have been also collected in Table I. As a general observation, all functionals seems to blue-shift the position of the main peak by more than 0.4 eV except MN12SX and PBE0, for which it is limited to


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4 4.5 5 5.5 6

FIG. 4. (a)–(f) Simulated absorption spectra at TDDFT (blue solid), TDDFT/TDA (blue dashed), and XMCQDPT2 (red). Vertical excitation energies are in eV. The vibrational structure has been approximated by centering gaussian functions at the vertical excitations with a broadening of 0.25 eV. For the sake of comparison, oscillator strengths have been normalized, so that the maximum peak corresponds to one. For the numerical values, see Table I.

0.2 eV. This overestimation of the excitation energy is even larger when using the TDA. The vertical excitation energy to the 11 (π π ∗ ) state is also blue-shifted by all functionals, with deviations as large as 0.7 eV (the minimum one, 0.47 eV, being already significantly large). None of the functionals can clearly reproduce the step in the spectrum due to the 11 (π π ∗ ), since the gap with the 21 (π π ∗ ) state is in general too small and the transition to the 11 (π π ∗ ) state remains hidden in the theoretical spectrum. Only the mPW2PLYP/TDA spectrum and the PBE0/TDA one to a lesser extent let appear a small hump. However, the TDDFT oscillator strengths are one order of magnitude smaller than the XMCQDPT2 ones (see Table I), whatever the xc functional used. Given the systematic TDDFT blue-shift of the energies, such a discrepancy in the oscillator strength values must be related to much smaller TDDFT transition dipole moments computed at the minimum-energy structures of each the acetophenone excited states. Finally, the vertical transition to the 1 (nπ ∗ ) state is characterized by an almost negligible oscillator strength, in good agreement with XMCQDPT2. The corresponding excitation energy is again overestimated by all functionals, with deviations comprised between 0.13 and 0.41 eV. TDDFT/TDA tends to produce larger blue-shifts than TDDFT. From the tested xc functionals, PBE0 gives the best agreement with the reference XMCQDPT2 spectrum, with deviations of only 0.15 eV for the vertical transitions to 1 (nπ ∗ ) and 21 (π π ∗ ) states and a larger 0.47 eV for the

TABLE I. Vertical and adiabatic transition energies (eV); oscillator strengths (in parenthesis). TDDFT/TDA values are in italics.

XMCQDPT2 CAM-B3LYP M06-2X M11 MN12SX PBE0 mPW2PLYP

XMCQDPT2 CAM-B3LYP M06-2X M11 MN12SX PBE0 mPW2PLYP

1 (nπ ∗ )

11 (π π ∗ )

3.67 (0.003) 4.02 (0.000) 4.05 (0.000) 3.84 (0.000) 3.92 (0.000) 3.80 (0.000) 3.89 (0.000) 4.06 (0.000) 4.08 (0.000) 3.81 (0.000) 3.84 (0.000) 3.91 (0.000)

4.46 (0.225) 5.12 (0.018) 5.20 (0.016) 5.21 (0.017) 5.29 (0.015) 5.33 (0.018) 5.42 (0.015) 4.99 (0.020) 5.05 (0.019) 4.93 (0.019) 4.99 (0.018) 4.99 (0.019)

1 (nπ ∗ )

11 (π π ∗ ) 4.32 4.94 5.04 5.02 5.12 5.14 5.26 4.76 4.85 4.72 4.81 4.80

3.66 3.72 3.75 3.58 3.66 3.57 3.66 3.74 3.76 3.46 3.49 3.46

(a) Vertical excitations 21 (π π ∗ ) 5.10 (1.450) 5.52 (0.225) 5.74(0.272) 5.61 (0.238) 5.85 (0.285) 5.70 (0.224) 5.95 (0.269) 5.32 (0.210) 5.53 (0.256) 5.35 (0.237) 5.57 (0.291) 5.65 (0.225) (b) Adiabatic excitations 21 (π π ∗ ) 4.72 5.22 5.50 5.30 5.59 5.67 5.07 5.35 5.11 5.38

3 (nπ ∗ )

3 (π π ∗ )

3.55 3.46 3.52 3.44 3.50 3.39 3.43 3.77 3.83 3.25 3.31 3.76

3.66 3.33 3.78 3.87 4.09 3.53 3.96 3.74 3.98 3.30 3.68 3.60

3 (nπ ∗ )

3 (π π ∗ )

3.32 3.16 3.23 3.16 3.23 3.16 3.21 3.68 3.51 2.91 2.98 3.47

3.34 2.85 3.34 3.30 3.58 3.04 3.50 3.24 3.52 2.87 3.27 3.15

Triplet gap 0.11 −0.11 0.26 0.43 0.59 0.14 0.53 −0.03 0.15 0.05 0.37 −0.14 Triplet gap 0.02 −0.31 0.11 0.14 0.35 −0.12 0.29 −0.44 0.01 −0.04 0.29 −0.32


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11 (π π ∗ ) state. Note that PBE0 and MN12SX feature an additional peak around 6.5 eV which is absent in the XMCQDPT2 spectrum. This transition corresponds to a higher energy singlet state, largely overstabilized by TDDFT. The application of the TDA solves this problem, at the expense of increasing the deviations with respect to XMCQDPT2 for the rest of the states. Now considering the triplet state minima, the TDDFT relative energies are quantitatively in better agreement with XMCQDPT2 ones than in the case of the singlet states. The maximum deviation is 0.43 eV for the 3 (π π ∗ ) state using M06-2X/TDA (see Table I). We observe a strong dependence of the triplet excitation energy with the xc functional, thus confirming the previous observation of Jacquemin et al.22 Moreover, an incorrect ordering of the triplets is observed for CAM-B3LYP, MN12SX, and mPW2PLYP/TDA, where we observe a more stable 3 (π π ∗ ) state already at the FranckCondon region (reported as negative triplet gaps in Table I.) This is largely solved by the TDA except in the case of mPW2PLYP/TDA. Interestingly, the TDA systematically enlarges the triplet gap for all functionals, mainly due to a deeper impact on the 3 (π π ∗ ) state energy. Finally, the comparison of the xc functional performances show that CAM-B3LYP/TDA is the best choice among the tested functionals for describing the triplet state energies.

2. Adiabatic transitions

Besides the usual discussion regarding the relevance of vertical versus adiabatic computed transitions from/to the excited states,45 a comparison of the TDDFT adiabatic excitation energies with the XMCQDPT2 ones may also indicate error cumulation at the minimum energy structures. Moreover, the differences between vertical and adiabatic excitation energies can be used as an indication of the slope of the initial photochemical paths in the singlet excited states. Assuming the cancellation of zero-point energies, the corresponding 0-0 transition energies are reported in Table I. At the XMCQDPT2 level of theory, the adiabatic transition energy to the 1 (nπ ∗ ) state is essentially equal to the corresponding vertical excitation. This feature is not reproduced by any of the tested xc functional. However, the TDDFT adiabatic energies are in semi-quantitative agreement with the XMCQDPT2 one (maximum deviation equal to 0.2 eV), thus indicating a good energetic description of the 1 (nπ ∗ ) minimum. The adiabatic transition to the 11 (π π ∗ ) state is 0.14 eV smaller than the vertical excitation at the XMCQDPT2 level. This stabilization is quantitatively well reproduced by all xc functionals. However, both the vertical and the adiabatic excitation energies remain largely overestimated with respect to XMCQDPT2 (as large as 0.96 eV for M11/TDA). A similar trend is observed for the 21 (π π ∗ ) state (maximum deviation: 0.95 eV for M11/TDA again). These last results indicate that the energetic positioning of these two singlet states are not sufficiently accurate using TDDFT. As far as triplets are concerned, some of the tested xc functionals give an incorrect ordering. In the Franck-Condon

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region, CAM-B3LYP, MN12SX, and mPW2PLYP/TDA give a more stable 3 (π π ∗ ) state than 3 (nπ ∗ ) states. This is also observed for the corresponding minima, for which M11 also gives an overstabilization of the 3 (π π ∗ ) minimum. This is largely solved by the TDA.

C. Three-state crossing

The three-state crossing between the 1 (nπ ∗ ), the 3 (π π ∗ ), and the 3 (nπ ∗ ) states is the main route to the generation of triplet acetophenone. It consists of an efficient intersystem crossing between the 1 (nπ ∗ ) and 3 (π π ∗ ) states, and a conical intersection between the 3 (π π ∗ ) and 3 (nπ ∗ ) triplets. At the XMCQDPT2 level of theory, both crossings are found at almost the same geometry.7 In the present section, we aim at exploring the three-state intersection region with TDDFT. In a first step, we determine the triplet minimum energy crossing geometry at each level of theory using the penalty function approach of Levine, Coe, and Martinez,37 while in a second step, we explore the PESs around the triplet conical intersection.

1. Triplet intersection geometry

The principal bond lengths of the minimum energy crossing structures are reported in Fig. 5. At the XMCQDPT2 level, the triplet minimum energy crossing geometry is characterized by an elongation of the C–O bond and by the compression of the C1 –C3 , C4 –C5 , and C7 –C8 distances with respect to the ground state structure. The C–O distance is largely underestimated by most of the tested xc functionals. The resulting deviations can be as large as 0.2 Å. Interestingly, mPW2PLYP/TDA gives instead a noticeably smaller 0.023 Å difference, in line with the results of Sec. III A. The large deviations of most xc functionals are expected, given the relatively large energy gap between the triplet energy profiles, at variance with the quasi-degeneracy obtained with XMCQDPT2 and mPW2PLYP/TDA (see Figure S1 of the supplementary material.36 ) The TDDFT C–C bond lengths and the alternation of formal double and single bonds always significantly deviate from the XMCQDPT2 ones. As a matter of fact, the C1 –C3 distances are comprised between 1.445 Å and 1.522 Å, always longer than the 1.369 Å XMCQDPT2 value. The TDA only slightly corrects this behavior, or even make the situation worse (PBE0). Concerning the phenyl ring, CAM-B3LYP results in inverted bond lengths, C4 –C5 and C7 –C8 being longer than the other bonds. If the application of the TDA helps to improve the geometry of the triplet crossing, the resulting phenyl ring now looks quasi-aromatic with bond lengths equal to 1.397 ± 0.015 Å. The same quasi-aromaticity of the phenyl moiety is also found for full-response TDDFT using the Minnesota or the PBE0 functionals. In that case, the use of TDA further improves the geometries, in qualitative agreement with the XMCQDPT2 ones. Similar to our previous observations, mPW2PLYP/TDA gives an overall accuracy of the C–C distances similar to the rest of the functionals.


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From these results, we can conclude that most functionals favor a minimum-energy crossing geometry biased towards the 3 (π π ∗ ) structure, with a C–O distance with a too strong double bond character and the C1 –C3 with a too strong single bond character. This is in contradiction with XMCQDPT2, for which these two bonds have a similar length. The XMCQDPT2 distances could only be represented by the mPW2PLYP functional, thus indicating that correlation effects coming from double excitations (such as those included by the MP2 correction) are necessary to describe the changes of conjugation in the excited state. 2. Potential energy surfaces of the three-state crossing

The accuracy of the crossing geometries is complemented by the comparison of the PESs in the vicinity of the intersection. Accordingly, we have determined the energy profiles of the 3 (nπ ∗ ), 3 (π π ∗ ), and 1 (nπ ∗ ) states along the unscaled gradient difference (UGD) direction defined using the two triplet states. This 2D cut of the crossing region is appropriate to observe the main topological features of this intersection, as the UGD vector can be exactly calculated in TDDFT. The principal components of the UGD vectors are reproduced in Fig. 5. At the XMCQDPT2 level of theory, they mainly involve modifications of the C–O, C3 –C4 , and C6 –C7 bonds. This is well reproduced by all xc functionals except for CAM-B3LYP and mPW2PLYP/TDA, which erroneously involves the C4 –C5 and C7 –C8 bonds. In general, the TDA-TDDFT UGD vectors are in good agreement with the XMCQDPT2 ones. The PESs around the crossing geometry are plotted in Fig. 6. The geometries along the x coordinate have been generated as CI + i · 0.01 · ∂ (EJ − EI ) , i = R (2) R ∂R

FIG. 5. (a)–(g) Structure of the 3 (nπ ∗ ) / 3 (π π ∗ ) minimum energy crossing point. For each structure, the unscaled gradient difference direction is shown. XMCQDPT2 geometry is taken from Ref. 7.

CI is the minimum energy crosswhere i is the x coordinate, R ing geometry, and EJ/I are the adiabatic energies of states 3 (π π ∗ ) and 3 (nπ ∗ ), respectively. Note that the triplet crossing is set to the 0 UGD coordinate arbitrarily. Note that the XMCQDPT2 crossing appears at 1 UGD coordinate. This is because the minimum-energy crossing structure were obtained in Ref. 7 using a 6-31G* basis set, while the energies have been calculated with the mixed cc-pVTZ/aug-cc-pVTZ used here. The XMCQDPT2 energy profiles clearly evidence the three-state crossing region, since the 3 (π π ∗ ) state crosses the 3 (nπ ∗ ) and 3 (π π ∗ ) states at almost the same geometry. Either using TDDFT or TDDFT/TDA, most xc functionals feature indeed a triplet intersection and a nearby intersystem crossing, although the latter is found at a significantly different position along the UGD displacement vector. On the contrary, mPW2PLYP/TDA predicts a three-state crossing very clearly. This is because mPW2PLYP/TDA, unlike other functionals, is able to reproduce the quasi-degeneracy of the 3 (nπ ∗ ) and 1 (nπ ∗ ), which is the ultimate reason of the existence of a three-state crossing.7 All other xc functionals predict a too large gap between the 1 (nπ ∗ ) and 3 (nπ ∗ ) states (see


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Figure S1 in the supplementary material.36 ) Still, even if the three-state crossing is not apparent with most functionals, we observe a pathway that may lead to the population of the triplet manifold for most xc functionals. In other words, using TDDFT in this region would probably result in a less efficient 1 (nπ ∗ ) to 3 (nπ ∗ ) transfer of population that may ultimately modify the triplet generation quantum yield. In general, most functionals predict a triplet intersection between 0.5 and 1 eV higher than XMCQDPT2. The intersystem crossing is in general 0.5 eV higher than the triplet crossing, while XMCQDPT2 gives an intersystem crossing isoenergetic to the triplet conical intersection. Only mPW2PLYP/TDA is able to reproduce accurately the energetics of the intersection. However, the same level of theory shows an unusual issue: the singlet state is energetically lower than the corresponding (nπ ∗ ) triplet. This is most probably due to a double counting of correlation due to the perturbative correction on the diagonal elements of the TDDFT matrix (see Ref. 32.) Another source of error, which breaks the degeneracy between the intersystem crossing and the triplet conical intersection, is the underestimation of the gap between the triplet 3 (π π ∗ ) state and the higher 23 (π π ∗ ) one, resulting in a nonnegligible coupling between these states. This is clearly seen for CAM-B3LYP in Fig. 6, in which we observe an avoided crossing between the (π π ∗ ) triplet states. For the remaining TDDFT calculations, a weaker coupling is observed. Nevertheless, we observe strong topological modifications of the 3 (π π ∗ ) energy profiles, certainly due to this unrealistic coupling. The mPW2PLYP/TDA gap between the two higher triplets is well reproduced, although some interaction seems to lack among them, and so the 3 (π π ∗ ) state has a steeper increase of energy at positive UGD vector at variance with the corresponding XMCQDPT2 surface. IV. CONCLUSIONS

FIG. 6. (a)–(g) Potential energy profiles of 1 (nπ ∗ ) (blue), 3 (nπ ∗ ) (blue dashed), 3 (π π ∗ ) (red dashed), and 23 (π π ∗ ) (green dashed) states near the triplet crossing. The origin corresponds to the triplet crossing minimumenergy structure. Other points have been computed using extrapolated structures along the UGD direction, using a displacement step of 0.01 a.u. (see Eq. (2) in the text.) The XMCQDPT2 crossing geometry (taken from Ref. 7) where optimized using a 6-31+G* basis set, while all energies and the gradient difference vectors have been computed with cc-pVTZ/aug-cc-pVTZ basis set. All the profiles are shifted so that the minimum of the 1 (π π ) state is 0. All energies are in eV.

We have performed an exhaustive assessment of linearresponse TDDFT to describe the triplet population of acetophenone. Six flavors of xc functionals have been tested. Moreover, we have analyzed the differences between full response TDDFT and TDDFT in the Tamm-Dancoff approximation. In agreement with experimental results and high-level quantum chemical calculations, we have shown that TDDFT predicts a pathway from the absorbing singlet state to the triplet manifold, due to the existence of an intersystem crossing between the 3 (π π ∗ ) and 1 (nπ ∗ ) states. From all the tested functionals, the double hybrid functional mPW2PLYP gives by far the most accurate description of the photochemistry of acetophenone. This functional accurately describes the main geometric and energetic parameters, giving the best overall qualitative and semi-quantitative agreement with multi-reference results. More importantly, this is the only functional that features a three-state crossing topologically similar to XMCQDPT2. All other tested functionals give instead an intersystem crossing relatively far from the triplet intersection. The main source of error is the unbalanced description of the electron correlation for states of different transition


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character or multiplicity. While the energies of the triplet states are underestimated, the singlet states are overestimated. Additionally, different precisions are obtained for states with (nπ ∗ ) and (π π ∗ ) character. Therefore, the state crossings occur at geometries significantly different from the reference ab initio ones.46 The good performance of mPW2PLYP is explained by the fact that MP2 corrections include a larger portion of double excitation correlation in the excited states. Regarding more specifically the triplet surfaces, we have observed wrong topologies probably due to unphysical interactions with higher-energy triplet states. This is especially important for the CAM-B3LYP functional. Interestingly, the TDA largely solves this issue, since triplet states are energetically better described and coupling with higher states is weakened. Therefore, we expect that the TDA give an overall more accurate description of the triplet state surfaces needed in photochemical applications of acetophenone. In conclusion, mPW2PLYP/TDA is the up-to-now most suitable way to study quantum-chemically the photochemistry of acetophenone. We therefore recommend this functional for future studies of aromatic ketones with TDDFT. The present results also prompt for the development of improved double hybrid functionals for use with TDDFT. A deeper study of the MP2 perturbative corrections is necessary to avoid the double counting of electron correlation. ACKNOWLEDGMENTS

We thank the Agence Nationale de la Recherche for funding through the IMPACT project ANR-11-BS08-0016. We also thank Fabien Archambault for technical support and the granted access to the HPC resources of Aix-Marseille Université financed by the project Equip@Meso (ANR-10-EQPX29-01) of the program “Investissements d’Avenir” supervised by the Agence Nationale pour la Recherche. 1 E.

Runge and E. Gross, Phys. Rev. Lett. 52, 997 (1984). E. Casida, in Recent Advances in Density Functional Methods, Part I, edited by D. P. Chong (World Scientific, Singapore, 1995), p. 155. 3 A. D. Laurent and D. Jacquemin, Int. J. Quantum Chem. 113, 2019 (2013). 4 N. T. Maitra, F. Zhang, R. J. Cave, and K. Burke, J. Chem. Phys. 120, 5932 (2004). 5 M. Huix-Rotllant, A. Ipatov, A. Rubio, and M. E. Casida, Chem. Phys. 391, 120 (2011). 6 B. G. Levine, C. Ko, J. Quenneville, and T. J. Martinez, Mol. Phys. 104, 1039 (2006). 7 M. Huix-Rotllant, D. Siri, and N. Ferré, Phys. Chem. Chem. Phys. 15, 19293 (2013). 8 M. E. Casida, B. Natarajan, and T. Deutsch, in Fundamentals of TimeDependent Density Functional Theory, Lecture Notes in Physics Vol. 837, edited by M. A. Marques, N. T. Maitra, F. M. Nogueira, E. Gross, and A. Rubio (Springer, Berlin, 2012), pp. 279–299. 9 A. V. Buettner and J. Dedinas, J. Phys. Chem. 75, 187 (1971). 2 M.

J. Chem. Phys. 140, 134305 (2014) 10 W.

M. Moore and M. Ketchum, J. Am. Chem. Soc. 84, 1368 (1962). A. Lamola, J. Chem. Phys. 47, 4810 (1967). 12 A. C. Bhasikuttan, A. K. Singh, D. K. Palit, A. V. Sapre, and J. P. Mittal, J. Phys. Chem. A 102, 3470 (1998). 13 A. K. Singh, A. C. Bhasikuttan, D. K. Palit, and J. P. Mittal, J. Phys. Chem. A 104, 7002 (2000). 14 S. Yabumoto, S. Shigeto, Y. Lee, and H. Hamaguchi, Angew. Chem. Int. Ed. 49, 9201 (2010). 15 J. S. Feenstra, S. T. Park, and A. H. Zeweil, J. Chem. Phys. 123, 221104 (2005). 16 H. Lutz, M. C. Duval, E. Breheret, and L. Lindqvist, J. Phys. Chem. 76, 821 (1972). 17 M. Segado and M. Reguero, Phys. Chem. Chem. Phys. 13, 4138 (2011). 18 Y.-W. Wang, H.-Y. He, and W.-H. Fang, J. Mol. Struct.: THEOCHEM 634, 281 (2003). 19 H.-Y. Xiao, Y.-J. Liu, and W.-H. Fang, J. Mol. Struct.: THEOCHEM 802, 99 (2007). 20 Q. Ou and J. E. Subotnik, J. Phys. Chem. C 117, 19839 (2013). 21 M. J. G. Peach, M. J. Williamson, and D. J. Tozer, J. Chem. Theor. Comput. 7, 3578 (2011). 22 D. Jacquemin, V. Wathelet, E. A. Perpète, and C. Adamo, J. Chem. Theor. Comput. 5, 2420 (2009). 23 C. Adamo and V. Barone, J. Chem. Phys. 110, 6158 (1999). 24 A. D. Becke, J. Chem. Phys. 98, 5648 (1993). 25 S. H. Vosko, L. Wilk, and M. Nusair, Can. J. Phys. 58, 1200 (1980). 26 T. Yanai, D. P. Tew, and N. C. Handy, Chem. Phys. Lett. 393, 51 (2004). 27 R. Peverati and D. G. Truhlar, J. Phys. Chem. Lett. 2, 2810 (2011). 28 Y. Zhao and D. G. Truhlar, Theor. Chem. Acc. 120, 215 (2008). 29 Y. Zhao and D. G. Truhlar, J. Chem. Phys. 125, 194101 (2006). 30 R. Peverati and D. G. Truhlar, Phys. Chem. Chem. Phys. 14, 16187 (2012). 31 T. Schwabe and S. Grimme, Phys. Chem. Chem. Phys. 8, 4398 (2006). 32 S. Grimme and F. Neese, J. Chem. Phys. 127, 154116 (2007). 33 J. P. Perdew and K. Schmidt, in Density Functional Theory and Its Applications to Materials, edited by V. Van Doren, C. Van Alsenoy, and P. Geerlings (American Institute of Physics, 2001), Vol. 577, pp. 1–20. 34 M. J. Frisch, G. W. Trucks, H. B. Schlegel et al., GAUSSIAN 09, Revision D.01, Gaussian, Inc., Wallingford, CT, 2009. 35 F. Neese, “ORCA–an ab initio, Density Functional and Semiempirical Program Package, Version 2.9.1,” Max-Planck Institut für Bioanorganische Chemie, Mülheim an der Ruhr, Germany. 36 See supplementary material at http://dx.doi.org/10.1063/1.4869802 for the Cartesian coordinates of the minimum-energy structures, minimum-energy crossing points, and unscaled gradient different vectors for all the tested exchange-correlation functionals and the spectrum of excited states along the C–O coordinate. 37 B. G. Levine, J. D. Coe, and T. J. Martinez, J. Phys. Chem. B 112, 405 (2008). 38 A. A. Granovsky, J. Chem. Phys. 134, 214113 (2011). 39 A. A. Granovsky, Firefly version 7.1.G, http://classic.chem.msu.su/gran/ firefly/index.html. 40 H. A. Witek, Y.-K. Choe, J. P. Finley, and K. Hirao, J. Comput. Chem. 23, 957 (2002). 41 T. H. J. Dunning, J. Chem. Phys. 90, 1007 (1989). 42 R. A. Kendall, J. T. H. Dunning, and R. J. Harrison, J. Chem. Phys. 96, 6796 (1992). 43 M. Wanko, M. Garavelli, F. Bernardi, T. A. Niehaus, T. Frauenheim, and M. Elstner, J. Chem. Phys. 120, 1674 (2004). 44 M. E. Casida, K. C. Casida, and D. R. Salahub, Int. J. Quantum Chem. 70, 933 (1998). 45 A. Chantzis, A. D. Laurent, C. Adamo, and D. Jacquemin, J. Chem. Theor. Comput. 9, 4517 (2013). 46 M. Casida and M. Huix-Rotllant, Ann. Rev. Phys. Chem. 63, 287 (2012). 11 A.


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