Nicolas Brach

Lucas Popek

Mathieu Truong

Claire Laurent

Vincent Bizet

Krishna P Kaliappan

Nicolas Blanchard

Divergent Synthesis of 6-or 7-Aza-Indazoles Via Intramolecular Diels-Alder Cascade of Pyrazines

Pyrazines are reactive 4p partners in intermolecular Diels Alder cycloaddition with exceptionally activated dienophiles or in intermolecular version at elevated temperatures. Herein it is shown that an intramolecular cascade could occur even at room temperature, delivering a collection of 6-or 7-aza-indazoles. An interesting substituent effect of the cycloaddition precursor on the product distribution was uncovered and in situ NMR studies were conducted to gain insights into this unexpected selectivity.

Due to their inherent electron-deficiency, pyrazines can engage in inverse-electron demand Diels-Alder cycloadditions with electron-rich dienophiles, leading to valuable N-heterocycles after a spontaneous retro-Diels-Alder step. [START_REF] Foster | Tandem Inverse-Electron-Demand Hetero-/retro-Diels-Alder Reactions for Aromatic Nitrogen Heterocycle Synthesis[END_REF][START_REF] Boger | Diels-Alder Reactions of Heterocyclic Aza dienes. Scope and Applications[END_REF][START_REF] Ishihara | In Comprehensive Organic Synthesis II[END_REF][START_REF] Duret | Diels-Alder and Formal Diels-Alder Cycloaddition Reactions of Ynamines and Ynamides[END_REF][START_REF] Rinderspacher | Progress in Heterocyclic Chemistry[END_REF] Recent theoretical studies highlighted the crucial role of the two nitrogen atoms (N1 and N4) which contributed to the overall reactivity by lowering the distortion energy and increasing the orbital interaction energy required to reach the transition state of the initial [4+2] cycloaddition. [START_REF] Yang | Diels-Alder Reactivities of Benzene, Pyridine, and Di-, Tri-, and Tetrazines: The Roles of Geometrical Distortions and Orbital Interactions[END_REF] In a seminal report by Neunhoeffer, the intermolecular cycloaddition of 2-methylpyrazine 1a with ynamine 2, an electron-rich dienophile, led to a very low yield of 3-aminopyridines 3a and 3b in a 85:15 ratio, after a month at reflux in acetonitrile (Scheme 1, A). [START_REF] Neunhoeffer | Reaktion von Pyrazinen mit 1-Diäthylamino-propin[END_REF] Two transition states of the initial [4+2] could be invoked to rationalize this product distribution: TS1 which engages the C3C6 portion of 1a and TS2 which engages the C2C5 portion. In addition to this site-selectivity of pyrazine 1a, the unsymmetrical dienophile 2 also presents some a/b-selectivity challenges. Increased reactivity was observed with pyrazine 1b substituted by two esters, as a Diels-Alder cascade with 2 was observed at room temperature, without boron trifluoride. After two weeks, a quantitative yield of three pyridines, 3c, 3d and 3e, was obtained in a 70:15:15 ratio. This product distribution can be attributed to the same classes of transition states, TS3 and TS4, with two divergent retro-Diels-Alder pathways in the case of the former (elimination of 2 different methylcyanoformates). Due to these multiple site-selectivities in the initial intermolecular [4+2] cycloaddition but also in the retro-[4+2] cycloaddition, only a handful of investigations have been reported. [START_REF] Foster | Tandem Inverse-Electron-Demand Hetero-/retro-Diels-Alder Reactions for Aromatic Nitrogen Heterocycle Synthesis[END_REF][START_REF] Boger | Diels-Alder Reactions of Heterocyclic Aza dienes. Scope and Applications[END_REF][START_REF] Ishihara | In Comprehensive Organic Synthesis II[END_REF][START_REF] Duret | Diels-Alder and Formal Diels-Alder Cycloaddition Reactions of Ynamines and Ynamides[END_REF][START_REF] Rinderspacher | Progress in Heterocyclic Chemistry[END_REF][START_REF] Yang | Diels-Alder Reactivities of Benzene, Pyridine, and Di-, Tri-, and Tetrazines: The Roles of Geometrical Distortions and Orbital Interactions[END_REF][START_REF] Neunhoeffer | Reaktion von Pyrazinen mit 1-Diäthylamino-propin[END_REF] In contrast, intramolecular versions have been more developed with insightful studies from van der Plas in the 1980s (Scheme 1, B). [START_REF] Biedrzycki | Intramolecular Inverse Electron Demand Diels-Alder Reactions of Pyrazines[END_REF][START_REF] Haider | Intramolecular Diels-Alder Reactions of Pyrazines With Alkynylphenyl Moieties as Side-Chain Dienophiles[END_REF] Pyrazines 4a-e substituted in the C2-position by an walkynyl side chain undergo a Diels-Alder cascade when heated at 120-210 °C in nitrobenzene. 10 c-Fused pyridines 5 or b-fused pyridines 6 are obtained as the major cycloadducts depending on the nature of the A group in the linker. Interestingly, a small but not negligible solvent effect was observed (t1/2 in p-cymene 1.0, in p-bromotoluene 1.7 and in nitrobenzene 2.9), which led van der Plas to consider that the transition states of the retro-[4+2] might have some ionic character. It was therefore postulated that the nitrogen atom eliminated (N1, light blue) possesses a partial negative charge. As a consequence, substituents A able to stabilize a partial positive charge at C2, such as oxygen/sulfur atoms or N-acetyl, favor TS6 and thus c-fused pyridines 5. On the other hand, strongly electron-withdrawing substituents A, such a SO2, favor TS7 and thus the formation of bfused pyridines 6. Intriguingly, opposite trends have been reported by the same group when A is a methylene group or when the pyrazine is quaternarized by Meerwein reagent. [START_REF] Geurtsen | Intramolecular Diels-Alder Reactions of 2-(Alkynyl)pyrimidines and 2-(Alkynyl)pyridines[END_REF] This brief literature overview demonstrates that pyrazine Diels-Alder cycloadditions present multiple challenges in terms of reactivity and selectivities, challenges that have not found a general solution yet. [START_REF] Chen | For a rare example of intramolecular Diels-Alder cycloaddition between a pyrazine and an alkene[END_REF] Inspired by our recent investigations on Diels-Alder cascade of pyrimidines, [START_REF] Fouler | Activating Pyrimidines by Pre-distortion for the General Synthesis of 7-Aza-indazoles from 2-Hydrazonylpyrimidines via Intramolecular Diels-Alder Reactions[END_REF][START_REF] Brach | Optimized Synthesis of 7-Azaindazole by a Diels-Alder Cascade and Associated Process Safety[END_REF][START_REF] Duret | Inverse Electron-Demand [4 + 2]-Cycloadditions of Ynamides: Access to Novel Pyridine Scaffolds[END_REF][START_REF] Duret | Intramolecular Inverse Electron-Demand [4 + 2] Cycloadditions of Ynamides with Pyrimidines: Scope and Density Functional Theory Insights[END_REF] we report herein the straightforward synthesis of 6-aza-indazoles 8 or 7-aza-indazoles 9 as the major to exclusive cycloadducts via an intramolecular Diels-Alder cascade of pyrazine 7 (Scheme 1, C). Compared to the current state-of-theart, this reaction sequence is characterized by mild reaction conditions and a predictable selectivity depending on the nature of the pyrazine's substitution. In addition, various substituents are tolerated, including halides, alkyls, fluoroalkyls, cycloalkyls, alkenyls, aryls, heteroaryls, alkoxy, protected alcohols and amines. Preparation of the Diels-Alder cascade precursors 7 is efficiently achieved in 2 steps from 2-chloro pyrazines 10 via aromatic nucleophilic substitution with hydrazine hydrate in ethanol (60-70 °C, 18 h) followed by condensation of the intermediate 2-hydrazinyl-pyrazine with ynone 12 under trifluoroacetic acid catalysis (15 mol%) in THF (60 °C, 18 h) (Scheme 2). Overall, 22 different pyrazines 7 were synthesized in 14-78% over 2 steps. We then turned our attention to the reactivity of 7 in Diels-Alder cascades (Scheme 3). Based on our previous activation strategy for pyrimidines, [START_REF] Fouler | Activating Pyrimidines by Pre-distortion for the General Synthesis of 7-Aza-indazoles from 2-Hydrazonylpyrimidines via Intramolecular Diels-Alder Reactions[END_REF][START_REF] Brach | Optimized Synthesis of 7-Azaindazole by a Diels-Alder Cascade and Associated Process Safety[END_REF] we anticipated that acylation of the N7atom from the hydrazone function with trifluoracetic anhydride should trigger a conformational switch of the side chain, bringing the 2-azadiene (the C2C5 moiety of the pyrazine ring) in close proximity to the alkynyl dienophile, thus favoring the initial [4+2] cycloaddition. Indeed, upon treatment of 7a by trifluoroacetic anhydride and 3-pentanone (acting as a trap for the HCN liberated during the retro-[4+2] step, leading to the corresponding cyanohydrin) [START_REF] Martin | Synthesis of Annulated Pyridines by Intramolecular Inverse-Electron-Demand Hetero-Diels-Alder Reaction under Superheated Continuous Flow Conditions[END_REF][START_REF] Zompa | A Holistic Strategy for Cyanide Control and Safety for Pharmaceutical Manufacturing[END_REF] in THF at 60 °C for 4 h, 6-aza-indazole 8a and 7-aza-indazole 9a were produced in a 78:22 ratio (after aqueous hydrolysis). These two heterocycles were easily separated by chromatography, and 8a was isolated in a moderate 43%. The fact that 8a is the major cycloadduct is consistent with previous studies from van der Plas 8-10 and implies an initial [4+2] cycloaddition via TS8. This Diels-Alder cascade has a general character as R 3 can be a hydrogen atom (8b, 33% isolated yield), a cyclopropyl (8c, 85%) or a cyclohexyl (8d, 79%) while keeping R 2 as phenyl. When R 3 is a cyclohexyl, R 2 can be cyclopropyl (8e, 73%), a linear alkyl chain (8f, 61%), a cyclohexenyl (8g, 68%) or a N-methyl pyrazole (8h, 59%). A 2-thienyl substituent (8i, 78%) or an ortho-substituted phenyl (8j, 79%) are also tolerated in the R 2 position. Finally, substituent containing heteroatoms could also be included in the R 2 substituent, such as a N-Boc piperidine (8k, 62%), a N-propyl phthalimide (8l, 80%), a (tert-butyldimethylsilyloxy)ethyl (8m, 53%) or (2-pyrimidinyloxy)ethyl (8n, 59%). In all these 14 examples, the major cycloadduct is the expected 6-aza-indazole 8 as determined from the 1 H NMR analysis of the crude reaction mixture, in ratio ranging from 64:36 to 93:7, independently of the electronics or steric parameters of the R 2 and R 3 substituents of the alkynyl-hydrazone side chain. We next focused on the influence of substituents of the pyrazine on the efficiency and product distribution of this Diels-Alder cascade (Scheme 4). Cycloaddition precursor 7d was selected as a reference substrate leading, as discussed above, to an 8d/9d ratio of 80:20. Interestingly, when TFAA was substituted by Boc2O, an increased ratio was observed between the 6-aza-indazole 8d-Boc and the 7 aza-indazole 9d-Boc (8d-Boc/9d-Boc = 95:5), which might be attributed to a more sterically demanding acyl group. To probe substituent effects, conditions A were selected for the rest of the studies. Thus, substitution of the pyrazine core of 7d in the C3 position was first investigated under standard conditions, with a methyl group (7o), a methoxy (7p), a chlorine (7q) and a trifluoromethyl (7r) (Scheme 3, B). These substituents of opposed steric and electronic requirements all favor the formation of the 6-aza-indazoles 8o-r as the major cycloadduct, via TS9 as the transition state for the retro-[4+2] step. Quite remarkably, when the R 4 substituent is electronwithdrawing (and independently of its mesomeric character), only the 6-aza-indazole could be detected in the crude reaction mixture as can be seen when R 4 is a halide (7q leading to 8q) or a trifluoromethyl group (7r leading to 8r). This convergent effect of the pyrazine C3-substituent on product distribution, enhancing the intrinsic preference of 7 for the 6-aza-indazole cycloadduct 8, was not observed in the case of C6-substituents (Scheme 3, C). Four cycloaddition precursors were prepared with R 5 being a methyl group (7s), a methoxy (7t), a chlorine (7u) and a trifluoromethyl (7v). C6-substitution by an alkyl group led to the 6-aza-indazole 8c as the major cycloadduct (73% isolated yield), the 7-aza-indazole 9s being present only in 22% in the crude reaction mixture. On the other hand, the exclusive formation of 7-aza-indazoles 9t-v was observed for the other substituents, independently of the steric or electronics. This outcome can be rationalized by TS10 for the retro-[4+2] step.

N N H N N N 3 6 N H N N N H N R R R R [R in C3] [R in C6] Diels-Alder cascade N R 3 N R
N N N N N N H N N N 3 6 N H N N N H N CF 3 O R R TS8 8 (from C3-R 7) 9 (from C6-R 7) R R 7 (R
N Cl R 1 N N N H NH 2 R 1 N N N H N R 3 R 2 R 1 11 (9 examples, 45-84%) R 2 R 3
The observed convergency and divergency in product distribution raises the question of an eventual protonation effect of one of the different nitrogen atoms of the pyrazine core. [START_REF] Geurtsen | Intramolecular Diels-Alder Reactions of 2-(Alkynyl)pyrimidines and 2-(Alkynyl)pyridines[END_REF] Indeed, upon trifluoroacetylation of the hydrazone nitrogen atom of 7 with trifluoroacetic anhydride, one equivalent of trifluoroacetic acid is released. The latter might protonate one or several nitrogen atoms of the cycloaddition precursor, in turn affecting the outcome of the retro-Diels-Alder step. Protonation of pyrazines has been reported in a handful of studies, and characteristic [START_REF] Fouler | Activating Pyrimidines by Pre-distortion for the General Synthesis of 7-Aza-indazoles from 2-Hydrazonylpyrimidines via Intramolecular Diels-Alder Reactions[END_REF] C NMR chemical shifts of the C2/C6 and C3/C5 atoms allow for the attribution of the protonated nitrogen. [START_REF] Billes | Quantum-chemical Study on the Successive Protonation of Pyrazine[END_REF][START_REF] Liu | An Investigation of Protonation in 2-Methoxy Pyrazine Using 13 C and 15 N NMR Spectroscopy[END_REF][START_REF] Chupakhin | Application of 1 H, 13 C and 15 N NMR in the chemistry of 1,4diazines[END_REF] A systematic NMR study of the effect of two different acids differing by their H0 acidity function (TFA, -3.03 and 98% H2SO4, -10) [START_REF] Fǎrcaşiu | Evaluation of Acidity of Strong Acid Catalysts I. Derivation of an Acidity Function from Carbon-13 NMR Measurements[END_REF] was conducted on three different model compounds, 7d, 7q and 7u

in the absence or presence of trifluoroacetic anhydride, at 20 and 60 °C in THF-d8. 23 First, a kinetic profile at 20 °C for the cycloaddition cascade of 7d demonstrated that the reaction was highly efficient, an almost complete conversion being obtained after 24h (Scheme 5, A, cf. conversion vs. time curve). The exact same conversion and product distribution was observed when the cycloaddition cascade was conducted in the presence of an organic base (iPr2EtN) for 24h. The influence of acids on 7d, 7q and 7u was then evaluated in the absence of trifluoroacetic anhydride (Scheme 5, B). It appeared that TFA was not able to protonate cycloaddition precursors 7 at 20 °C or 60 °C. This first series of experiments therefore ruled out the protonation of 7 (and hence a fortiori of 13d) with TFA as one of the factors influencing the product distribution when standard cycloaddition conditions are employed (TFAA, 3-pentanone, THF). On the other hand, the addition of H2SO4 to 7q and 7u led to small (for 1 equivalent) to significant (for 4 equivalents) [START_REF] Fouler | Activating Pyrimidines by Pre-distortion for the General Synthesis of 7-Aza-indazoles from 2-Hydrazonylpyrimidines via Intramolecular Diels-Alder Reactions[END_REF] C NMR chemical shifts, indicating that protonation occurs first on the nitrogen atom of the hydrazone, potentially assisted by N1 in the case of 7q (Scheme 5, B, left). [START_REF] García | Hydrolysis Mechanisms for the Acetylpyridinephenylhydrazone Ligand in Sulfuric Acid[END_REF][START_REF] Kalia | Hydrolytic Stability of Hydrazones and Oximes[END_REF][START_REF] Yildiz | A DFT Approach to the Mechanistic Study of Hydrozone Hydrolysis[END_REF] For 7u and in the presence of 4 equivalents of H2SO4, the second protonation occurs at N4 (Scheme 5, B, right). Then, the influence of H2SO4 in the cycloaddition cascade conditions (TFAA, 60 °C) was evaluated for 7q (Scheme 5, C). When it was present at the outset of the reaction, the trifluoroacetylation could not proceed. When it was added after 4 hours of reaction at 60 °C (65% conversion), the presence of H2SO4 immediately led to a plateau and to unidentified products. Although the presence of a strong acid such as H2SO4 could protonate 7 and thus contribute to the electrodeficiency of the pyrazine, it was detrimental to the cycloaddition reaction. Eventually, the impact of the C3-or C6-substitution of the pyrazine core on the cycloadduct distribution might be explained by the nature of the retro-Diels-Alder products. In all cases, the major to exclusive elimination leads to HCN which either spontaneously polymerizes under thermal conditions [START_REF] Ruiz-Bermejo | New Insights into the Characterization of 'Insoluble Black HCN Polymers[END_REF] or is trapped by 3-pentanone to the corresponding cyanohydrine. [START_REF] Martin | Synthesis of Annulated Pyridines by Intramolecular Inverse-Electron-Demand Hetero-Diels-Alder Reaction under Superheated Continuous Flow Conditions[END_REF] The gaseous nature of HCN might thus provide a highly favorable thermodynamic driving force in this Diels-Alder cascade, as is usually observed for 1,2,4-triazines or 1,2,4, 5-tetrazines cycloadditions. [START_REF] Foster | Tandem Inverse-Electron-Demand Hetero-/retro-Diels-Alder Reactions for Aromatic Nitrogen Heterocycle Synthesis[END_REF][START_REF] Boger | Diels-Alder Reactions of Heterocyclic Aza dienes. Scope and Applications[END_REF][START_REF] Ishihara | In Comprehensive Organic Synthesis II[END_REF] In conclusion, we have disclosed that pyrazines substituted by an alkynylhydrazone could undergo intramolecular Diels-Alder cascade even at room temperature. A collection of diversely substituted 6-or 7-aza-indazoles was obtained in moderate to good yields. Interestingly, an influence of the pyrazine substitution on the cycloadduct distribution was observed. Mechanistic investigations have shown that protonation of the cycloaddition precursors with the in situ generated TFA do not account for this observed divergency. Current studies aim at further exploring the selective protonation of pyrazines with strong Brønsted acids as modulators of cycloaddition reactivity.

Corresponding Author

*Nicolas Blanchard -Université de Haute-Alsace, Université de Strasbourg, CNRS, LIMA, UMR 7042, 68000 Mulhouse, France ; orcid.org/ 0000-0002-3097-0548 ; Email : n.blanchard@unistra.fr

Scheme 1 .Scheme 2 .

 12 Scheme 1. Challenges of Pyrazines as dienophiles in Diels-Alder cycloaddition/retro-cycloaddition.

Scheme 3 .

 3 Scheme 3. Scope of Diels-Alder cascade of pyrazines 7. a

  in intermolecular Diels-Alder: challenges in aza-diene selectionNeunhoeffer et al., 1972, ref. 7

  in intramolecular Diels-Alder: influence of the linker = O, S, N retro-[4+2] TS7, A = SO 2 , SO C. This work: enhanced reactivity and divergent synthesis of aza-indazoles

  b (8e/9e = 76:24) c 8g, 68% b (8g/9g = 71:29) c 8f, 61% b (8f/9f = 77:23) c

7

 7

Scheme 4 .

 4 Scheme 4. Convergent and divergent substituent effects. a

Scheme 5 .

 5 Scheme 5. 13 C NMR studies for the protonation of 7. a

■■ 4 ■+0■

 4 Clean reaction leading only to 8d and 9d (80:20) No protonation with TFA ■ Modest to significant 13 C NMR chemical shifts with H 2 SO Δδ are indicated for 1 equiv. a /4 equiv. b 26 a /-1.38 b -0.34 a /-1.84 b +0.11 a /+0.65 b +0.08 a /+0.47 b the cycloaddition cascade at 20 °C and the influence of a base B. 13 C NMR investigations for the protonation of pyrazines 7d, R 1 = R 2 = H 7q, R 1 = Cl, R 2 = H 7u, R 1 = H, R 2 = Cl C. Impact of H 2 SO 4 on the cycloaddition cascade of 7q Conversion vs time (without iPr 2 NEt)

ACKNOWLEDGMENT

The authors thank Didier Le Nouën (LIMA UMR7042 CNRS-Unistra-UHA), Corinne Bailly and Nathalie Gruber (Service de radiocristallographie de la Fédération de Chimie Le Bel FR 2010) who contributed, by their valuable technical and scientific support in NMR and X-ray diffraction, to the achievement of this research project. We thank Michel Grobe (University of Giessen, H. A. Wegner group) for discussions and help in discussing reference 7. We thank the University of Strasbourg for Doctoral Contracts to N. Brach and L. Popek, the University of Upper-Alsace, the Indo-French Centre for the Promotion of Advanced Research -CEFIPRA (project 6105-1) and the CNRS for financial support.


Jiaqi Fang

Oumaïma Bekkouch

Guilhem Zeiser

Yurii Zubchuk

Vincent Bizet

Nicolas Blanchard

Gwilherm Evano

Copper-Catalyzed, Ligand-Controlled N(sp3)-or N(sp)-Selective Arylation of Cyanamides

published or not. The documents may come   L'archive ouverte pluridisciplinaire

ABSTRACT: Cyanamides possess both nucleophilic and electrophilic centers and their arylation reactions are known to proceed at N(sp [START_REF] Prabhath | Recent Advances in Cyanamide Chemistry: Synthesis and Applications[END_REF] ) and C(sp)-sites, leading to N-aryl-cyanamides or amidines. N(sp)-Selectivity has also been reported only in the presence of amines, thus leading to guanidines. Herein, we report a general copper-catalyzed ligand-controlled Chan-Lam-Evans arylation of cyanamides proceeding regioselectively at the N(sp 3 ) or N(sp) atoms and leading to either to N-aryl-cyanamides or dissymmetric carbodiimides. The nature of the ligand, either a bipyridine or a diamine, controls the product distribution and thus offers a divergent entry to useful building blocks from readily available cyanamides.

Cyanamides are versatile building blocks containing three reactive sites: a carbon atom (sp-hybridized) and two nitrogen atoms (sp-and sp 3 -hybridized). 1 This dense array of electrophilic and nucleophilic positions in cyanamides provides a unique reactivity and allows a myriad of ionic, radical, pericyclic and metalcatalyzed transformations. [START_REF] Dhara | Metal-Catalyzed Reactions of Organic Nitriles and Boronic Acids to Access Diverse Functionality[END_REF][START_REF] Prabhath | Recent Advances in Cyanamide Chemistry: Synthesis and Applications[END_REF][START_REF] Bolotin | Metal-Involving Generation of Aminoheterocycles From N-Substituted Cyanamides: Toward Sustainable Chemistry (a Minireview)[END_REF][START_REF] Larraufie | The Cyanamide Moiety, Synthesis and Reactivity[END_REF][START_REF] Nekrasov | Synthesis and Chemical Transformations of Mono-and Disubstituted Cyanamides[END_REF] Moreover, it does also confer to cyanamides the ability to act as efficient ligands for various transition metals. [START_REF] Melekhova | A Novel Family of Homoleptic Copper(I) Complexes Featuring Disubstituted Cyanamides: a Combined Synthetic, Structural, and Theoretical study[END_REF][START_REF] Bokach | Coordination Chemistry of Dialkylcyanamides: Binding Properties, Synthesis of Metal Complexes, and Ligand Reactivity[END_REF][START_REF] Crutchley | Phenylcyanamide Ligands and Their Metal Complexes[END_REF] In line with our interest in copper-catalyzed reactions [START_REF] Evano | Copper-Mediated Coupling Reactions and Their Applications in Natural Products and Designed Biomolecules Synthesis[END_REF][START_REF]Copper-Mediated Cross-Coupling Reactions[END_REF] and transition metal-catalyzed reactions of cyanamides, [START_REF] Dewez | Ring-Closing Metathesis of N-Alkenyl-Cyanamides[END_REF][START_REF] Nitelet | Copper-Catalyzed Alkenylation of Cyanamides[END_REF] we became interested in their N-arylation and more particularly in the control over the chemoselectivity (sp-vs sp 3hybridized nitrogen atom) of the process. Indeed, it has been reported that arylation reactions of cyanamides 1 could proceed with N(sp 3 ), C(sp) or N(sp) -selectivity depending on reaction conditions and aryl partners (Scheme 1A). The N(sp 3 )-arylated product 2 can be obtained selectively from (hetero)aromatic halides by aromatic nucleophilic substitution [START_REF] Pees | Fungicidal 5-Phenyl Substituted 2-(Cyanoamino) Pyrimidines[END_REF][START_REF] Xu | Substituted Benzisothiazole Compound and Application[END_REF][START_REF] Jin | Synthesis of Diverse Boron-Handled N-Heterocycles via Radical Borylative Cyclization of N-Allylcyanamides[END_REF][START_REF] Wang | Intermediate Compound, Carbamazepine and Derivative Thereof as well as Preparation Method of Oxcarbazepine and Derivative Thereof[END_REF] or from diaryliodonium triflates (not shown). [START_REF] Li | Transition-Metal Free N-Arylation of Cyanamides by Diaryliodonium Triflates in Aqueous Media[END_REF][START_REF] Li | Copper-Catalyzed One-Pot Synthesis of Unsymmetrical Arylurea Derivatives via Tandem Reaction of Diaryliodonium Salts with N-Arylcyanamide[END_REF] Transition-metal catalyzed processes have also been disclosed and an efficient palladium-catalyzed (hetero)arylation using (hetero)aryl halides or triflates is shown on Scheme 1B. [START_REF] Stolley | Palladium-Catalyzed Arylation of Cyanamides[END_REF][START_REF] Yang | Copper-Catalyzed Domino Synthesis of Benzimidazo[2,1-b]quinazolin-12(6H)-ones Using Cyanamide as a Building Block[END_REF] On one hand, the use of Pd2dba3 (2.5 mol%), t BuXphos (7.5 mol%) and cesium carbonate in t amyl alcohol at 60-100 °C provides N(sp 3 )-arylated cyanamides 2 in moderate to excellent yields. On the other hand, amidines 3 resulting from a selective C(sp)arylation can be obtained in a palladium trifluoroacetate-catalyzed reaction using 6-methyl-2,2'-bipyridine as a ligand (3 mol%) and trifluoroacetic acid in NMP at 120 °C (Scheme 1B). [START_REF] Rydfjord | Decarboxylative Palladium(II)-Catalyzed Synthesis of Aryl Amidines from Aryl Carboxylic Acids: Development and Mechanistic Investigation[END_REF] In this case, the nucleophilic species arise from the de-carboxylation of benzoic acid derivatives 6 but the corresponding potassium trifluoroborate salts are also competent partners. [START_REF] Sävmarker | Direct Palladium(II)-Catalyzed Synthesis of Arylamidines from Aryltrifluoroborates[END_REF] This C(sp)-selectivity has also been observed in a unique copper-catalyzed cross-coupling between 1 and tolylboronic acid leading to 3 in less than 20% yield. [START_REF] Li | Copper-Catalyzed Oxidative Three-Component Synthesis of N,N′,N″-Trisubstituted Guanidines[END_REF] Finally, the N(sp)selective arylation of cyanamides 1 has only been reported in the presence of an external nucleophile, such as a primary or a secondary amine, thus leading to guanidines 4 (Scheme 1B). [START_REF]In the absence of an external nucleophile, such as an amine, reactions are sluggish and no carbodiimides[END_REF] This transformation can be metal-free with diaryliodonium triflates as arylation reagents, 26 or copper-catalyzed, as exemplified in Scheme 1B, using arylboronic acids. [START_REF] Evano | Copper-Mediated Coupling Reactions and Their Applications in Natural Products and Designed Biomolecules Synthesis[END_REF] Indeed, in the presence of copper(II) chloride (10 mol%), bipyridine (10 mol%), potassium carbonate and oxygen in toluene at 100 °C, guanidines 4 were obtained in good to excellent yields. Depending on the nature of the substituents, further one-pot oxidative copper-catalyzed transformations of 4 can occur and deliver complex heterocycles such as benzimidazoles, quinazolines and their derivatives. [START_REF] Tran | Copper-Catalyzed Domino Three-Component Approach for the Assembly of 2-Aminated Benzimidazoles and Quinazolines[END_REF][START_REF] Rodrigues | Copper-Promoted Tandem Three-Component Access to Quinazolin-4(H)-imines and Benzimidazo[1,2c]quinazolines[END_REF] This brief literature overview demonstrates that copper-catalyzed N(sp 3 )-selective arylation of cyanamides 1 has yet to be discovered. In addition, the fine tuning of the coordination sphere of the active copper species might allow the modulation of the reactive site and redirect the arylation reaction from N(sp 3 ) to N(sp), thereby delivering carbodiimides, central building blocks in organic synthesis. [START_REF] Wang | Carbodiimide-based Synthesis of N-Heterocycles: Moving From Two Classical Reactive Sites to Chemical Bond Breaking/Forming Reaction[END_REF][START_REF] Dunetz | Large-Scale Applications of Amide Coupling Reagents for the Synthesis of Pharmaceuticals[END_REF] In this manuscript, we report our efforts in delineating a unique and divergent N(sp 3 )or N(sp)-selective arylation reaction of cyanamides 1 in a ligand-controlled Chan-Lam-Evans 31-34 reaction (Scheme 1C). Scheme 1. Arylation Reactions of Cyanamides and Control Over the N(sp 3 )-, C(sp)-or N(sp)-Selectivity Optimization of reaction conditions was performed with two distinct cyanamides differing by the nature of the alkyl side chain (butyl for 1a, 1,5-dimethylhexyl for 1b) and phenyl boronic acid 7a (Table 1). [START_REF]Boronic acid proved superior to the corresponding boronate or boroxine, see Table S5 in the Supporting Information[END_REF] Indeed, it was rapidly observed that linear alkyl chains favor the N(sp 3 )-arylation to N-aryl-cyanamide 2 whereas a branched alkyl chain favors the N(sp)-arylation to the corresponding N-arylated dissymmetric carbodiimide 8. It should be noted that the high electrophilicity of carbodiimides 8 leads in most cases to traces of isoureas 9 and 10 arising from the nucleophilic attack of phenol and propan-1ol, respectively, onto the central carbon atom (see below). Initial conditions for the arylation of cyanamide 1a relied on copper(II) acetate (1 equivalent), 4,4'-dimethyl-bipyridine L1 (2 equivalents), t BuOK (3 equivalents) and additives (4Å MS, silica gel 36,37 ) in dichloromethane under an oxygen atmosphere at room temperature for 9 h (Table 1, entry 1). Compound 2a was obtained in 52% (NMR yield) together with approximately 15% of carbodiimide 8a. Catalytic amount of copper acetate (10 mol%) led to a slightly diminished yield (43%, entry 2) which could be increased back to 53% using the more electron rich bipyridine ligand L2 (53%, entry 3). Further structural variations of the ligand (L3-L8, entry 4) were not beneficial. Table 1. Optimization a a Reactions performed using cyanamide 1 (0.3 mmol, 1 equiv), phenylboronic acid (1.5-2.5 equiv), copper acetate (10 mol%), ligand Ln (20 mol%), base (3 equiv), under an O2 atmosphere (1 bar) in CH2Cl2 (0.1 M) at rt for 2 h. b Determined by 1 H NMR using 1,1,2,2-tetrachloroethane as an internal standard. c 120 mg for 0.3 mmol of 1. d 50 mg for 0.3 mmol of 1. e 5 equiv.

Extensive screening of all parameters (including copper(II) sources and loading, bases, solvents, concentration, time) 36 identified cesium carbonate as the preferred base and propan-1ol (5 equivalents) as an essential additive to increase both yields and reproducibility in a shorter reaction time (2 h). Using these N(sp 3 )-selective conditions, 2a was obtained in 71% (NMR yield) together with only 9% of 8a and traces (<5%) of isoureas 9a and 10a (entry 5). Formation of isourea 9a, arising from the nucleophilic attack of 8a by phenol (itself obtained by the copper catalyzed oxidation of 7a under oxygen atmosphere 38 ) could be circumvented by operating under strictly anhydrous reaction conditions. Isourea 10a, arising from the attack of propan-1-ol on 8a, accounts for 5-15% of the reaction mixtures in all cases. It should be noted that these additions of alcohols to carbodiimides have recently been shown to be accelerated by oxygen in copper-catalyzed reactions. [START_REF] Wells | Copper-Catalyzed Addition of Alcohols to Carbodiimides: Oxygen as an Accelerant[END_REF] Thus, a delicate balance must be found with the number of equivalents of propan-1-ol to limit the formation of isoureas 10 while maintaining a good reactivity in the arylation reaction. During the ligand screening phase, it was discovered that diamines such as L9-L12 led to increased amounts of carbodiimide 8b when cyanamide 1b was used as the starting material, all other reaction parameters being identical (entries 6-9). In particular, L10 and L12 led to 8a in 74% and 50% NMR yields, respectively, in a 4.4-5:1 ratio with 2b. Although ligand L10 led to better NMR and isolated yields than L12 (74% vs 50% and 47% vs 28%, respectively), the latter performed better in subsequent studies of the scope of the reaction and was thus selected as the optimal ligand for this N(sp)-selective copper-catalyzed arylation of cyanamides 1. As ligand L12 is also a competent nucleophile in the copper-catalyzed oxidative arylation reaction with phenylboronic acid, ligand L13 was independently prepared [START_REF] Ota | synthesis and evaluation of γ-turn mimetics as LSD1-selective inhibitors[END_REF] and its catalytic activity was evaluated (entry 10). Traces of 2b and 8b were obtained in the standard conditions, demonstrating the superiority of ligand L12. This selectivity switch from N(sp 3 )-to N(sp)-arylation of cyanamides is striking and has not yet been reported to the best of our knowledge and such a divergency holds great interest for the synthetic community by providing valuable nitrogen-containing small molecules that would be otherwise difficult to prepare using a single protocol. [START_REF] Wang | Carbodiimide-based Synthesis of N-Heterocycles: Moving From Two Classical Reactive Sites to Chemical Bond Breaking/Forming Reaction[END_REF] With these optimized conditions in hands for both arylation reactions, we next examined the generality of these two procedures for the N(sp 3 )-and N(sp)-selective arylation of a range of cyanamides 1 with a variety of arylboronic acids 7 (Schemes 2-4), starting with the N(sp 3 )-selective arylation of 1a using ligand L2 with a range of arylboronic acids 7 (Scheme 2). Conversions were found to be good to excellent (80-95%) in all cases and only minor amounts of carbodiimides 8a,c-m were detected in the crude reaction mixtures. Electron-neutral arylboronic acids such as 7a,c,d led to good NMR yields (2a 71%, 2c 61% and 2d 72%) although isolated yields were slightly lower. Scaling up the reaction leading to 2a from 0.3 to 1 mmol scale gave a slightly diminished isolated yield (2a, from 62% to 49%). Electron-rich arylboronic acids 7e-h led to similar results, the N(sp 3 )-arylated cyanamides 2e-h being obtained in 38-65% (NMR yields), the use of a tris(methoxy)arylboronic acid leading to a lower efficiency. Conversion dropped with electron-deficient arylboronic acids substituted by halides (7i-k) or by an ester (7l) and compounds 2i-l were obtained in lower isolated yields (2i, 37%, 2j 26%, 2k 29%, 2l 28%). Besides arylboronic acids, it was found that trans-2-phenylvinylboronic acid 7m was also a competent partner in this transformation, leading to 2m in 43%. Limitations were found with o-cyanophenyl boronic acid and 2-thienyl boronic acid, for which no arylated cyanamides 2 could be detected (Scheme S1). 36 We then turned our attention to the structural variation of starting cyanamide 1, under N(sp 3 )-selective arylation conditions with phenylboronic acid 7a (Scheme 3). Compared to 1a possessing a linear alkyl chain, branched alkyl or aromatic cyanamides 1b,n-y led to diminished NMR and isolated yields, although conversions were generally good to excellent. Minor amounts of dissymmetric carbodiimides 8b,n-y and of the corresponding isoureas could account for this experimental observation. Starting from p-tolyl-cyanamide 1n, the copper-catalyzed phenylation led to 2n in only 27% NMR yield, thus demonstrating the lack of reactivity of this class of cyanamides which is not surprising due to the conjugation of the cyanamide moiety with the tolyl ring. Yields slightly improved with N-benzyl-and N-allyl-cyanamides 1o and 1p, leading to 2o and 2p in 30% and 57% NMR yields, respectively. a-Branched alkyl chains improved yields, as in the case of 2q (72%), although with longer alkyl chains, yields tend to decrease (2r 37%, 2s 42%, 2b 38%). Interestingly, cycloalkyl chains are also tolerated as in the cases of 1t-w which led to the corresponding N(sp 3 )-phenylated cyanamides 2t-w in moderate to good yields. Finally, linear cyanamides 1x and 1y possessing silyloxy-and t butoxycarbamate functionalities (respectively) were evaluated in the arylation reaction with 7a and led to acceptable isolated yields of the corresponding N(sp 3 )-phenylated cyanamides 2x (58%) and 2y (52%). Finally, the scope of the divergent N(sp)-selective arylation was examined with cyanamides 1 possessing a-branched alkyl chains, under conditions relying on ligand L12 (Scheme 4). Dissymmetric aryl,alkyl-carbodiimides 8b,r,z-ah were obtained with moderate to very good in 36-74% NMR yield (36-88%) as opposed to the isolated yields (13-49%). This could logically be traced back to the very high electrophilicity of diimides 8 towards any type of nucleophiles, thus complicating isolation by standard chromatography techniques. However, this single step protocol from cyanamides compares favorably to multi-step sequences required to access dissymmetric cyanamides. [START_REF] Hennessy | Two-Component Coupling of Carbodiimides and Hydrazides Provides Convergent Access to Biologically Active Compounds[END_REF] Cyanamides 1 possessing an acyclic a-branched alkyl chain led to the dissymmetric carbodiimides 8b,r,z-ab in 48-74% NMR yield (8r 68%, 8z 63%, 8aa 71%, 8ab 48% and 8b 50%; 74% with L10). Cycloalkyl side chains performed in a similar manner, delivering carbodiimides 8v-ad in 36-72% NMR yields (8v 38%, 8w 67%, 8ac 72% and 8ad 36%). Finally, we examined four different arylboronic acids while keeping constant cyanamide 1b as starting material. Contrary to the N(sp 3 )-selective conditions, the electronic properties of the arylboronic acids didn't have a dramatic influence on the NMR yields as exemplified with the synthesis of 8ae (60%), 8af (74%), 8ag (70%) and 8ah (58%). The synthesis of carbodiimide 8r is especially striking since it was obtained as a minor product with ligand L2 (Scheme 3), thus highlighting the key role of the ligand in this arylation. Limitation was found with o-cyanophenyl boronic for which no N(sp)-arylated cyanamide 8 could be detected (Scheme S2). 36 Scheme 4. N(sp)-Selective Arylations of Cyanamides a a Reactions performed on 0.3 mmol scale of 1. Yields and conversions determined by 1 H NMR using 1,1,2,2-tetrachloroethane as an internal standard. Isolated yields in parenthesis. b With L10 instead of L12, see Table 1.

A proposed mechanism accounting for the N(sp 3 )-or N(sp)-selective arylation of cyanamides is depicted on Scheme 5, based on previous investigations by Stahl [START_REF] King | Mechanistic Study of Copper-Catalyzed Aerobic Oxidative Coupling of Arylboronic Esters and Methanol: Insights into an Organometallic Oxidase Reaction[END_REF][START_REF] King | Kinetic and Spectroscopic Studies of Aerobic Copper(II)-Catalyzed Methoxylation of Arylboronic Esters and Insights into Aryl Transmetalation to Copper(II)[END_REF] and Watson. [START_REF] West | Mechanistic Development and Recent Applications of the Chan-Lam Amination[END_REF][START_REF] Vantourout | Spectroscopic Studies of the Chan-Lam Amination: A Mechanism-Inspired Solution to Boronic Ester Reactivity[END_REF] In a first step, dimeric copper acetate would undergo denucleation to a mononuclear copper(II) complex A by coordination with the starting cyanamide 1, its more common coordination mode involving its sp-hybridized nitrogen atom. [START_REF] Melekhova | A Novel Family of Homoleptic Copper(I) Complexes Featuring Disubstituted Cyanamides: a Combined Synthetic, Structural, and Theoretical study[END_REF][START_REF] Crutchley | Phenylcyanamide Ligands and Their Metal Complexes[END_REF] Transmetalation with the arylboronic acid would then lead to arylcopper(II) species C via intermediate B, a species that would be in metallotropic equilibrium with D. Disproportionation with a second equivalent of a copper(II) complex would lead to copper(III) species E and F, from which reductive eliminations would deliver the corresponding N(sp)-arylated dissymmetric carbodiimide 8 or the (sp 3 )N-arylated cyanamide 2, respectively. Oxidation with molecular oxygen and coordination of the starting cyanamide 1 would finally close the catalytic cycle. As for the key role of the ligand, the more electron-rich diamine L12 would facilitate the oxidation of copper(II) complexes C and D to the corresponding copper(III) species E and F, a reaction that would be slower with bipyridine ligand L2. Oxidation of C to E would thus be faster than its equilibration to D when using L12 as the ligand, the steric hindrance brought by this diamine ligand, which has in addition a bigger bite angle, moreover disfavoring this equilibration due to a steric clash between this ligand and the cyanamide R substituent in D. Scheme 5. Proposed Mechanism of the Chan-Lam-Evans N(sp 3 )-or N(sp)-Selective Arylation of Cyanamides

In conclusion, we have developed a copper-catalyzed, ligandcontrolled N(sp 3 )-or N(sp)-selective Chan-Lam-Evans arylation of cyanamides, a divergency which has no precedent in the literature. N(sp 3 )-selective arylation was predominant when cyanamides substituted by linear alkyl chains in combination with a bipyridine ligand were used. The use of cyanamides substituted by a-branched alkyl chains in combination with a diamine ligand switched the selectivity to the N(sp)-terminus of the cyanamides, thus allowing the synthesis of dissymmetric aryl,alkyl-carbodiimides. Mechanistic studies shedding light on the origin of this selectivity and their extension to other classes of ambident nucleophiles in Chan-Lam-Evans couplings are underway and will be reported in due course.

Scheme 2 .

 2 Scheme 2. N(sp 3 )-Selective Arylations of Cyanamides: Scope of Arylboronic Acids a

ACKNOWLEDGMENT

We thank the China Scholarship Council for a grant to J.F., the Université libre de Bruxelles (ULB), the University of Strasbourg, the University of Upper-Alsace, the FNRS (PDR T.0160.18) and the CNRS for financial support. The authors thank Dr. Didier Le Nouën, head of the NMR platform of LIMA (UMR7042 CNRS-Unistra-UHA) who contributed, by his valuable technical and scientific support, to the achievement of this research project.

AUTHOR INFORMATION

Corresponding Authors *Nicolas Blanchard -Université de Haute-Alsace, Université de Strasbourg, CNRS, LIMA, UMR 7042, 68000 Mulhouse, France ; orcid.org/ 0000-0002-3097-0548 ; Email: n.blanchard@unistra.fr *Gwilherm Evano -Laboratoire de Chimie Organique, Service de Chimie et PhysicoChimie Organiques, Université libre de Bruxelles (ULB), Avenue F. D. Roosevelt 50, CP160/06, 1050 Brussels, Belgium ; orcid.org/0000-0002-2939-4766 ; Email: Gwilherm.Evano@ulb.be


Mickael Lafontaine
email: mickael.lafontaine@gmail.com

Julie Cloarec-Michaud
email: julie.cloarecmichaud@gmail.com

Kevin Riou
email: kevin.riou@univ-nantes.fr

Yujie Huang
email: yujie.huang@etu.univ-nantes.fr

Kaiwen Dong
email: dongkaiwen@cumt.edu.cn

Patrick Le Callet
email: patrick.lecallet@univ-nantes.fr

Julie Cloarec Michaud

Kinetic particles : from human pose estimation to an immersive and interactive piece of art questionning

the relationship between thought and movement by combining dance, philosophy, numerical arts, and deep learning. We present a novel system that combines a multi-camera setup to capture human movement, state-of-the-art human pose estimation models to automatically analyze this movement, and an immersive 180°p rojection system that projects a dynamic textual content that intuitively responds to the users' behaviors. The demonstration being proposed consists of two parts. Firstly, a professional dancer will utilize the proposed setup to deliver a conference-show. Secondly, the audience will be given the opportunity to experiment and discover the potential of the proposed setup, which has been transformed into an interactive installation. This allows multiple spectators to engage simultaneously with clusters of words and letters extracted from the conference text.

CCS CONCEPTS

INTRODUCTION

An interactive work of art is described by Kluszczyński [START_REF] Kluszczynski | Strategies of interactive art[END_REF] as an event, where an artist doesn't create a final, complete piece of art, but instead produces an area of activity, where the participative behavior of the audience will bring to life the artwork-event. Kluszczyński [START_REF] Kluszczynski | Strategies of interactive art[END_REF] categorizes the audience as participants, performers, executors, or (co)creators of an artwork-event, depending on the interactive art strategy considered, and further details eight interactive art strategies : the strategy of instrument, game, archives, labyrinth, rhizome, system, network, and spectacle. In this work, we focus on the strategy of instrument, where an interface suggests the audience to create a performance by generating visual [START_REF] Kim | Choreographing Digital Water for Interactive Water Performance[END_REF] or audio-visual events [START_REF] Rokeby | Very Nervous System (VNS)[END_REF][START_REF] Sermon | Telematic dreaming[END_REF], based on its behaviors. Digital tools offer extensive ways to explore strategies of instrument, by relying on various sensors to create installations and performances where the activity of the body generates visual and sound universes in real-time [START_REF] Adrien | Hakanaï[END_REF][START_REF] Hanai | Fencing tracking and visualization system[END_REF][START_REF] Manabe | Behind the scenes[END_REF]. Deep learning approaches, including human detection [START_REF] Liao | YOLOv4: YOLOv4: Optimal Speed and Accuracy of Object Detection[END_REF] and human pose estimation [START_REF] Cao | Realtime multiperson 2d pose estimation using part affinity fields[END_REF][START_REF] Cheng | HigherHRNet: Scale-Aware Representation Learning for Bottom-Up Human Pose Estimation[END_REF][START_REF] Jiang | RTMPose: Real-Time Multi-Person Pose Estimation based on MMPose[END_REF], constitute ideal human-art interaction medium, as they allow automatic human gesture analysis, which can be directly used to produce the interactive piece of art. The interactivity offered by such solutions allows for a sensory and poetic augmentation of the gestures of the performer or spectator. Rhizomatiks [START_REF] Hanai | Fencing tracking and visualization system[END_REF] especially showed that coupling such solutions with video projection systems has considerable and innovative interest in the context of a creative conference during which the speaker literally manipulates their concepts and research universe. The video projection system is an important factor that allows the performer or spectators to be immersed, in the literal sense of the term, in these visual universes generated in real-time. With a similar state of mind, we propose a demonstration that combines dance, philosophy, numerical arts and deep learning, and which explores the relationship between thought and movement. Concretely, we propose a multi-camera setup coupled with deep learning approaches that allows to map the movements of several users in the form of a words ballet, that are thereafter projected on the walls surrounding the users. The demonstration is decomposed in two parts, 1) a professional dancer uses the proposed setup to give a conference-show, and 2), the audience is invited to try and explore possibilities of the proposed setup, which is converted to an interactive installation, where several spectators can simultaneously interact with clouds of words and letters taken from the conference text. Note that the number of cameras in our setup can scale with the number of participants, and/or with with the experiment area size, as our software is agnostic to the number of cameras used. Visitors can also add their own text to demonstrate how other researchers can easily re-appropriate this device. Our goal is to present this installation in the 180-degree video projection setup of the XP Lab in order to create a completely immersive experience.

APPROACH 2.1 The artistic project

This work aims at finding connections between thought and movement. It consists of a conference-show combined with an interactive installation that allows the spectator to actively participate in the process and personally and physically experience this linking between speech and action. The user is involved in an immersive universe made up of texts that comes to life according to their movements. The authors of the work give the words a real materiality and autonomy of movement, so that the interaction between them and the users is both intuitive, organic, lively, and choreographic. This artistic performance, both hybrid and transdisciplinary, in the very scope of the lecture and the inherent artistic creation, represents the immutable link between thought and movement, philosophy and dance, words and gestures.

2.1.1

The conference-show time. During the conference, a performer or a duo of performers interact with an immersive textual universe in which they can bring to life the textual content projected on the walls surrounding them. For single performer shows, the performer's movement is typically captured by one 3D camera placed in front of the performer. The human movement is analysed by deep learning approaches, detailed in section 2.2, and propagated to a collection of letters projected around the performer. Fig. 1 illustrates how the letters can be projected in the immersive space. The letters can be structured in different ways, and several behaviors can be attached to them. For instance, in the scenario that will be displayed during the demonstration, the letters can be considered as independent particles controlled by the performer, which interact with each other, with the wall boundaries and are subject to a virtual gravity. These behaviors allow intuitive and choreographic interaction with the performer. In the duo-performer setup, the movement of the two performers is captured by one or more 3D cameras and each user controls different sets of letters, which provides an additional interactive dimension: the performers can interact with each other by the mean of the artistic projection.

Interactive and collaborative installation time.

The idea to combine a whole installation offered to the audience with the showlecture appeared immediately during the creation of the latter with a view of permitting the public to share the live experience. In this installation, several spectators are involved in an immersive video projection where they can simultaneously interact with the textual universes presented during the conference. This installation invites spectators to collaborate both intellectually and choreographically: searching for words, exchanging words, synchronizing movements, etc. This collective participation is undoubtedly innovative given the classical structures offered both in the field of human sciences and the performing arts.

Technical details

The section describes the technical details for the "Particules" scenario, which will run during the conference demonstration. The Figure 2: Illustration of the proposed setup. On the left, the scheme details the pipeline associated to the "Particules" scenario described in section 2.2.1, simplified with 2 projected letters. On the right is a visualization of the real setup, where the circled "camera" corresponds to the camera on the left scheme. Additional cameras, here denoted as "back-cameras" can be added to the setup to tackle occlusions that may occur in multi-user scenarios.

single user setup, associated to the conference-show time is first described. The multi-user setup, associated to the interactive and collaborative installation time is described afterwards.

Single User Scenario.

The proposed interactive setup that will be displayed during the conference is illustrated in Fig. 2. It is composed by (1) a Monocular Human Pose Estimation System, which is a solution to analyse human movement and (2) the "Wrist Velocities to Letter Momentum" module, which is a software that animates textual projections, based on the human movement analysis. Monocular Human Pose Estimation is a long-standing problem [START_REF] Cao | Realtime multiperson 2d pose estimation using part affinity fields[END_REF][START_REF] Chen | Monocular human pose estimation: A survey of deep learning-based methods[END_REF][START_REF] Cheng | HigherHRNet: Scale-Aware Representation Learning for Bottom-Up Human Pose Estimation[END_REF] in computer-vision community, which aims to reconstruct the 2D locations of human body joints in an input image or video. Human Pose Estimation models provide valuable information on the position of the different joints of the user's body, can be further used to recover the user's joints velocities from adjacent frames. The 2D velocities of chosen target joints can further be propagated to the textual content projected around the user. For instance, in the "Particules" scenario, which will be presented during the conference demonstration, the letters of the projected text are considered as independent particles, subject to gravity. By default, the particles are motionless and grounded by this artificial gravity. When a non-zero velocity is detected on one of the user's wrists, a force is applied to the letters, in the direction of the detected velocity. In this "Particules" scenario, both wrists control a respective half of the projected letters. For the demonstration, the ZED M cameras from stereolabs are used, and the "BODY_18" integrated solution for 2D body tracking is used as Monocular Human Pose Estimation System.

2.2.2

Multi-User Scenario. The Multi-User Scenario works similarly to the Single User Scenario, except that independent sets of letters are assigned to the different users. To ensure a temporal smoothness of the movements, the different users are tracked with a unique id to ensure that a given set of letters is always assigned to the same user. In the case of massive amount of users, additional cameras can be added to our setup, to tackle inter-users occlusions. If additional cameras are needed, they are calibrated together using EasyMocap [START_REF]EasyMoCap -Make human motion capture easier[END_REF] calibration tools. The skeletons obtained from the different views are aligned by projecting them in a shared 3D world space using extrinsic camera parameters obtained during the calibration. For each user in the interaction area, the skeleton with the highest prediction confidence is kept, projected to the frontcamera view, and sent to the "Wrist Velocities to Letter Momentum" module.

CONCLUSION AND PERSPECTIVES

This project, through its multidisciplinarity, ranging from philosophy to digital research, including arts and dance, makes it an excellent field for both theoretical and practical experiences. With this project, we hope to bring reflection on our relationship with thought and movement, with the goal of not separating the two in order to better analyze them, but on the contrary, embracing them in their common movement in order to better grasp their own momentum. As future works on the technical part, we wish (1), to migrate the human-pose estimation models used to a multi-view 3D pose estimation approach, to avoid prediction inconsistencies between views and (2), to invest in another area of research that could greatly improve both the immersive and the interactive abilities of the setup. Our goal is create a smart video mapping system that can calculate and adapt the position of projected animations in the 3D space, which would allow projecting content on target objects positioned in the space, including the users' bodies.

  Interactive Art, Digital Art, Immersive Content, Real-time Multi-Person Pose Estimation ACM Reference Format: Mickael Lafontaine, Julie. Cloarec Michaud, Kevin Riou, Yujie Huang, Kaiwen Dong, and Patrick Le Callet. 2023. Kinetic particles : from human pose estimation to an immersive and interactive piece of art questionning thought-movement relationships.. In ACM International Conference on Interactive Media Experiences (IMX '23), June 12-15, 2023, Nantes, France. ACM, New York, NY, USA, 4 pages. https://doi.org/10.1145/3573381.3597228

• Applied computing → Performing arts.

KEYWORDS


Pierre-François Loos

Filippo Lipparini

Denis Jacquemin
email: denis.jacquemin@univ-nantes.fr

Heptazine, Cyclazine, and Related Compounds: Chemically-Accurate Estimates of the Inverted Singlet-Triplet Gap

Molecules that violate Hund's rule and exhibit an inverted gap between the lowest-energy singlet excited state, 𝑆 1 , and the first triplet state, 𝑇 1 , have attracted considerable attention due to their potential applications in optoelectronics, especially in organic light-emitting diodes (OLEDs). Amongst these molecules, the triangularshaped heptazine, and its derivatives, have been in the limelight recently. However, conflicting reports have arisen regarding the relative energies of 𝑆 1 and 𝑇 1 . In the present Letter, we employ highly accurate levels of theory, such as CC3, to not only resolve the debate concerning the sign but also quantify the magnitude of the 𝑆 1 -𝑇 1 gap in heptazine. We also determine the 0-0 energies for both spin symmetries to evaluate the significance of the vertical approximation. In addition, we compute reference 𝑆 1 -𝑇 1 gaps for a series of 10 related molecules. This enables us to benchmark lower-order methods for future applications on larger systems within the same family of compounds. The present contribution can serve as a reliable foundation for the design of larger triangular-shaped molecules with enhanced photophysical properties.

In many technological applications that rely on either lightenergy or energy-light conversions, it is essential to control the energy gap that separates the relevant electronic excited states (ESs). Specifically, the magnitude of the singlet-triplet gap (STG), the energy difference between the lowest-energy singlet excited state, 𝑆 1 , and the lowest-energy triplet state, 𝑇 1 , is of prime importance as key photophysical events typically take place in these states. For example, to design highly fluorescent molecules, a large STG is sought after to minimize the 𝑆 1 → 𝑇 1 intersystem crossing. In contrast, in the pursuit of efficient third-generation organic light-emitting diodes (OLEDs), a vanishingly small STG is required to maximize the reverse intersystem crossing (𝑇 1 → 𝑆 1 ), resulting in the production of more photons.

Despite constant developments, experimental characterizations of ESs remain both costly and challenging. This is why theoretical models, which explore the ES nature and energies are frequently employed hand-in-hand with experimental measurements. Nevertheless, theoretical methods also face challenges in achieving chemically accurate (i.e., error of the order of ±0.05 eV) estimates of ES energies for non-trivial molecular systems. This is in stark contrast with ground-state (GS) properties for which a panel of reliable black-box methods is now readily available. The search for molecular structures having specific, original, and tuneable ES properties are therefore extremely active research lines.

In 2019, Ehrmaier and coworkers published a seminal work demonstrating that heptazine (compound 1 in Figure 1) does not adhere to Hund's rule, with 𝑇 1 being found above 𝑆 1 . [START_REF] Ehrmaier | Singlet-Triplet Inversion in Heptazine and in Polymeric Carbon Nitrides[END_REF] This remarkable and unexpected discovery, supported by both theoretical and experimental analyses (as discussed below), [START_REF] Ehrmaier | Singlet-Triplet Inversion in Heptazine and in Polymeric Carbon Nitrides[END_REF] stimulated many subsequent works. [START_REF] De Silva | Inverted Singlet-Triplet Gaps and Their Relevance to Thermally Activated Delayed Fluorescence[END_REF][START_REF] Ricci | Singlet-Triplet Excited-State Inversion in Heptazine and Related Molecules: Assessment of TD-DFT and ab initio Methods[END_REF][START_REF] Pollice | Organic Molecules with Inverted Gaps Between First Excited Singlet and Triplet States and Appreciable Fluorescence Rates[END_REF][START_REF] Dinkelbach | Large Inverted Singlet-Triplet Energy Gaps Are Not Always Favorable for Triplet Harvesting: Vibronic Coupling Drives the (Reverse) Intersystem Crossing in Heptazine Derivatives[END_REF][START_REF] Ghosh | Origin of the Failure of Density Functional Theories in Predicting Inverted Singlet-Triplet Gaps[END_REF][START_REF] Aizawa | Delayed Fluorescence from Inverted Singlet and Triplet Excited States[END_REF][START_REF] Ricci | Establishing Design Strategies for Emissive Materials With an Inverted Singlet-Triplet Energy Gap (INVEST): a Computational Perspective on how Symmetry Rules the Interplay Between Triplet Harvesting and Light Emission[END_REF][START_REF] Tučková | On the Origin of the Inverted Singlet-Triplet Gap of the 5th Generation Light-Emitting Molecules[END_REF][START_REF] Monino | Connections and Performances of Green's Function Methods for Charged and Neutral Excitations[END_REF][START_REF] Drwal | The Role of Spin Polarization and Dynamic Correlation in Singlet-Triplet Gap Inversion of Heptazine Derivatives[END_REF][START_REF] Blaskovits | Symmetry-Induced Singlet-Triplet Inversions in Non-Alternant Hydrocarbons[END_REF][START_REF] Dreuw | The Inverted Singlet-Triplet Gap: a Vanishing Myth? Front[END_REF] In particular, several groups analyzed in detail the reasons behind this deviation from Hund's rule, [START_REF] Ricci | Singlet-Triplet Excited-State Inversion in Heptazine and Related Molecules: Assessment of TD-DFT and ab initio Methods[END_REF][START_REF] Ricci | Establishing Design Strategies for Emissive Materials With an Inverted Singlet-Triplet Energy Gap (INVEST): a Computational Perspective on how Symmetry Rules the Interplay Between Triplet Harvesting and Light Emission[END_REF][START_REF] Drwal | The Role of Spin Polarization and Dynamic Correlation in Singlet-Triplet Gap Inversion of Heptazine Derivatives[END_REF][START_REF] Blaskovits | Symmetry-Induced Singlet-Triplet Inversions in Non-Alternant Hydrocarbons[END_REF][START_REF] Sandoval-Salinas | Correlation vs. Exchange Competition Drives the Singlet-Triplet Excited-State Inversion in Non-Alternant Hydrocarbons[END_REF] and concurrently, improved substituted molecules have been designed using various modeling approaches. [START_REF] Pollice | Organic Molecules with Inverted Gaps Between First Excited Singlet and Triplet States and Appreciable Fluorescence Rates[END_REF][START_REF] Ricci | Establishing Design Strategies for Emissive Materials With an Inverted Singlet-Triplet Energy Gap (INVEST): a Computational Perspective on how Symmetry Rules the Interplay Between Triplet Harvesting and Light Emission[END_REF][START_REF] Sanz-Rodrigo | Negative Singlet-Triplet Excitation Energy Gap in Triangle-Shaped Molecular Emitters for Efficient Triplet Harvesting[END_REF][START_REF] Sobolewski | Excited-State Singlet-Triplet Inversion in Hexagonal Aromatic and Heteroaromatic Compounds[END_REF] In the original work of Ehrmaier et al., a series of wavefunction calculations performed with the double-𝜁 basis set cc-pVDZ was used to estimate the STG of heptazine (see Table 1). This investigation revealed that time-dependent density-functional theory (TD-DFT) relying on global hybrids -the typical workhorse for ES calculations -failed to deliver qualitatively correct values. [START_REF] Ehrmaier | Singlet-Triplet Inversion in Heptazine and in Polymeric Carbon Nitrides[END_REF] Notably, the most sophisticated levels of theory used in Ref. 1, namely, coupledcluster with singles and doubles (CCSD) and the completeactive-space self-consistent-field method with second-order perturbative correction (CASPT2), provided smaller STG than less advanced second-order approaches ( basis set, def2-TZVP, and reported a STG of -0.139 eV at the 𝑁-electron valence state second-order perturbation theory (NEVPT2) level using a large (12, 12) active space. [START_REF] Ricci | Singlet-Triplet Excited-State Inversion in Heptazine and Related Molecules: Assessment of TD-DFT and ab initio Methods[END_REF] The STG value of Gosh and Bhattacharyya computed with the restricted-active-space method (RASPT2) is similar to the NEVPT2 value (-0.13 eV), whereas similarity-transformed equation-of-motion CCSD (STEOM-CCSD) unexpectedly yields a much larger result (-0.66 eV). [START_REF] Ghosh | Origin of the Failure of Density Functional Theories in Predicting Inverted Singlet-Triplet Gaps[END_REF] A comparison between the multireference values of Refs. 1, 3, and 9 also hints that increasing the size of the basis set from cc-pVDZ to def2-TZVP induces a significant drop of the STG magnitude. In 2022, new experiments were carried out for two substituted heptazines, and a tiny yet negative gap (-0.011 eV) was found for one of them. [START_REF] Aizawa | Delayed Fluorescence from Inverted Singlet and Triplet Excited States[END_REF] Very recently, Dreuw and Hoffmann went a step further by questioning the existence of an inverted STG in heptazine and two closely related molecules (cyclazine, 2, and cyclborane). [START_REF] Dreuw | The Inverted Singlet-Triplet Gap: a Vanishing Myth? Front[END_REF] By systematically ramping up the degree of the algebraic-diagrammatic construction (ADC), going from ADC(2)-s, to ADC(2)-x, and then ADC(3), they observed clear reductions of the STGs, with an ADC(3)/cc-pVTZ gap amounting to a mere -0.07 eV. In addition, they conducted (state-specific) ΔCC calculations based on non-Aufbau reference determinants with a series of increasingly larger basis sets and obtained a positive STG of +0.04 eV with their most refined approach, namely, ΔCCSD(T)/cc-pVTZ. In short, as the level of theory became more accurate, the negative character of the STG in heptazine diminished. Dreuw and Hoffmann, therefore, concluded that the sign (amplitude) of the STG of heptazine was uncertain (notably small), emphasizing the necessity for more accurate calculations [START_REF] Dreuw | The Inverted Singlet-Triplet Gap: a Vanishing Myth? Front[END_REF] This Letter answers their call.

To obtain very accurate estimates of the ES energies of heptazine, we employed CC methodologies including singles, doubles, and iterative triples, along with large basis sets (see computational details section). This approach represents a substantial advancement in comparison to earlier studies. First, we optimized the 𝐷 3ℎ GS structure of heptazine at the CCSD(T)/cc-pVTZ level, ensuring a solid starting point for further calculations. Next, we performed a series of equation-of-motion CC (EOM-CC) calculations with various diffuse-containing basis sets, up to third-order CC (CC3) for the triplet state and even using full triples (CCSDT) for the singlet state (see the SI for the complete set of data). Importantly, we did not consider state-specific approaches, such as ΔSCF and ΔCC, but systematically relied on linear response theory to directly target ES energies.

It is important to note that the so-called GS 𝑇 1 -diagnostic 17 returns a value of 0.02 at the CCSD/aug-cc-pVTZ level for heptazine, indicating the absence of significant multireference character in the GS. Furthermore, our CC3/aug-cc-pVDZ reveals no substantial double excitation character with %𝑇 1 values (which indicates the percentage of single excitation character) of 86.3% and 95.7% for the lowest excited singlet and triplet states, respectively. These observations are consistent with both previous conclusions drawn at the second-order CC level, [START_REF] Tučková | On the Origin of the Inverted Singlet-Triplet Gap of the 5th Generation Light-Emitting Molecules[END_REF] and the negligible difference observed between CC3 and CCSDT for the 𝑆 1 excitation energy (< 0.01 eV, see the SI). There is therefore no clear reason to delve into multiconfigurational methodologies here, as the CC family of systematically improvable methods appears to offer the most efficient path to chemical accuracy.

In the extended quest database, [START_REF] Véril | QUESTDB: a Database of Highly-Accurate Excitation Energies for the Electronic Structure Community[END_REF] it was found that the typical error associated with CC3 is as small as 0.02 eV for such well-behaved ES (having %𝑇 1 > 85%). We are therefore reasonably confident that the data reported in Table 1 stand as the most reliable to date and are likely chemically accurate (error below 0.05 eV). As reported in Table 1, our TBE for the singlet and triplet states are respectively 2.717 and 2.936 eV, resulting in a negative gap of -0.219 eV. It is no surprise that this value is in-between the CC2 and CCSD results obtained with large basis sets. Second-order methods such as ADC(2) and CC2 tend to overestimate the magnitude of the STG, while CCSD and ADC(3) exhibit a tendency to provide an error in the opposite direction. We also note that previous CASPT2 (NEVPT2) significantly underestimate (overestimate) the absolute energies of 𝑆 1 and 𝑇 1 . Finally, it appears that the aug-cc-pVDZ basis set proves sufficiently large to deliver accurate results (at the CC3 level) while the cc-pVDZ basis set produces an excessively negative STG. In any case, the data showcased in Table 1 conclusively affirm that the vertical excitation energy of 𝑇 1 in heptazine is approximately 0.22 eV higher than that of 𝑆 1 .

In Table 2, we provide a similar comparative analysis, juxtaposing our present estimates with prior data from the literature for the singlet and triplet vertical excitation energies of cyclazine (compound 2 in Figure 1). While the two states are notably closer to the GS than in heptazine, the methodological trends outlined in the previous paragraph for heptazine mostly pertain. For example, using a small basis set yields too negative STGs, whereas CCSD and ADC(3) underestimate the gap. For cyclazine, both ADC(2) and CC2 overestimate the excitation energies, but they provide an excellent estimate of the STG, SCS-CC2 being significantly off. Our TBE for the STG in cyclazine is -0.131 eV, which is large enough to affirm the gap inversion in this system. Let us now turn towards the 0-0 energies of heptazine, as this property provides a reliable basis for comparisons with experimental data. [START_REF] Loos | Evaluating 0-0 Energies with Theoretical Tools: a Short Review[END_REF] For obvious computational reasons, we use CCSD/cc-pVDZ, as well as its EOM-CCSD and unrestricted variants (UCCSD) to determine the geometries and vibrational frequencies of 𝑆 0 , 𝑆 1 , and 𝑇 1 , respectively. While the geometries obtained are not as accurate as those computed at the CCSD(T)/cc-pVTZ level (see above and below), it should be stressed that the 0-0 energies are known to be less sensitive to the accuracy of the employed structures than vertical excitation energies. [START_REF] Loos | Chemically Accurate 0-0 Energies with not-so-Accurate Excited State Geometries[END_REF] Besides, the CC3/aug-cc-pVDZ vertical 𝑆 1 excitation energy computed on the CCSD/cc-pVDZ geometry is 2.649 eV, which is quite close to the 2.693 eV result obtained with the CCSD(T)/cc-pVTZ structure.

Starting from the planar 𝐷 3ℎ GS geometry, the EOM-CCSD optimization of 𝑆 1 led to one imaginary frequency associated with the puckering of the central nitrogen atom. The true minimum of the 𝑆 1 structure presents a 𝐶 3𝑣 point group symmetry (see Figure 2). We relied on CC3/aug-cc-pVTZ to compute the adiabatic energy (𝐸 adia = 2.550 eV) for the 𝑆 0 -𝑆 1 transition, which can be corrected by the difference of zero-point vibrational energies between both states (Δ𝐸 ZPVE = -0.044 eV) determined at the (EOM-)CCSD/cc-pVDZ level to obtain a TBE of 2.506 eV for the 0-0 transition between 𝑆 0 and 𝑆 1 . This difference of approximately -0.2 eV from the vertical transition energy reported in Ta-ble 1 aligns with the typical corrections observed for relatively rigid molecules. [START_REF] Goerigk | Computation of Accurate Excitation Energies for Large Organic Molecules with Double-Hybrid Density Functionals[END_REF][START_REF] Jacquemin | A TD-DFT Assessment of Functionals for Optical 0-0 Transitions in Solvated Dyes[END_REF][START_REF] Winter | Benchmarks for 0-0 Transitions of Aromatic Organic Molecules: DFT/B3LYP, ADC(2), CC2, SOS-CC2 and SCS-CC2 Compared to High-resolution Gas-Phase Data[END_REF] On a methodological note, at the CCSD(T)(a) ★ /aug-cc-pVTZ level, we get 𝐸 adia = 2.671 eV (𝐸 0-0 = 2.627 eV), for the 𝑆 0 -𝑆 1 transition, a value 0.12 eV larger than the CC3 result. This data is likely more directly comparable to the results obtained for the triplet state below, for which CCSD with perturbative triples is used to obtain the best estimate.

For the lowest triplet, the 𝐷 3ℎ optimal structure shows two imaginary frequencies at the UCCSD/cc-pVDZ level. The first one leads to a puckered 𝐶 3𝑣 geometry similar to the 𝑆 1 case, and the second one corresponds to an in-plane deformation providing a 𝐶 𝑠 geometry, the 𝐶 3ℎ structure being unstable (Figure 2). Both the 𝐶 3𝑣 and 𝐶 𝑠 structures are genuine minima and have quite similar energies. For the former, the UCCSD(T)/aug-cc-pVTZ value for 𝐸 adia is 3.154 eV and the CCSD-based ZPVE correction is quite large (-0.189 eV) leading to a 0-0 transition at 2.965 eV. For the latter, we get 𝐸 adia = 3.038 eV and 𝐸 0-0 = 3.057 eV, this structure showing an uncommon positive ZPVE correction. In any case, the 0-0 energies of these two triplet structures are clearly larger than the 2.627 eV estimate obtained with a comparable level of theory for the singlet state. In other words, the negative STG obtained from the vertical transitions (-0.219 eV) does not change sign when considering geometry relaxation and zero-point vibrational effects. In fact, it seems to become slightly larger (-0.338 eV). While we acknowledge that the latter value is likely not chemically accurate due to the increased number of approximations made, it is worth noting that previous works also observed that the inverted STG found through vertical calculations persists when considering 0-0 energies. [START_REF] Ehrmaier | Singlet-Triplet Inversion in Heptazine and in Polymeric Carbon Nitrides[END_REF][START_REF] Pollice | Organic Molecules with Inverted Gaps Between First Excited Singlet and Triplet States and Appreciable Fluorescence Rates[END_REF][START_REF] Sobolewski | Are Heptazine-Based Organic Light-Emitting Diode Chromophores Thermally Activated Delayed Fluorescence or Inverted Singlet-Triplet Systems?[END_REF] S0 (D3h) S1 (C3v) T1 (C3v) T1 (Cs) Of course, the state-of-the-art methods discussed above are not readily applicable to the design of novel compounds with enhanced properties, a goal typically achieved through the addition of side substituents or the extension of the conjugated core. [START_REF] Pollice | Organic Molecules with Inverted Gaps Between First Excited Singlet and Triplet States and Appreciable Fluorescence Rates[END_REF][START_REF] Ricci | Establishing Design Strategies for Emissive Materials With an Inverted Singlet-Triplet Energy Gap (INVEST): a Computational Perspective on how Symmetry Rules the Interplay Between Triplet Harvesting and Light Emission[END_REF][START_REF] Sanz-Rodrigo | Negative Singlet-Triplet Excitation Energy Gap in Triangle-Shaped Molecular Emitters for Efficient Triplet Harvesting[END_REF][START_REF] Sobolewski | Excited-State Singlet-Triplet Inversion in Hexagonal Aromatic and Heteroaromatic Compounds[END_REF][START_REF] Sobolewski | Are Heptazine-Based Organic Light-Emitting Diode Chromophores Thermally Activated Delayed Fluorescence or Inverted Singlet-Triplet Systems?[END_REF][START_REF] Pios | Triangular Boron Carbon Nitrides: An Unexplored Family of Chromophores with unique Properties for Photocatalysis and Optoelectronics[END_REF] A wide panel of strategies has been explored in the literature to design new compounds, but the challenge lies in the absence of definitive benchmark values. This lack of a clear reference is evident, for instance, in the varying choices of reference methods made by different research groups when benchmarking double-hybrid functionals, with some using SCS-CC2 [START_REF] Sancho-García | Violation of Hund's rule in molecules: Predicting the excited-state energy inversion by TD-DFT with double-hybrid methods[END_REF] and others opting for CCSD. [START_REF] Alipour | Do any Types of Double-Hybrid Models Render the Correct Order Of Excited State Energies in Inverted Singlet-Triplet Emitters?[END_REF][START_REF] Curtis | Building on the Strengths of a Double-Hybrid Density Functional for Excitation Energies and Inverted Singlet-Triplet Energy Gaps[END_REF] Hence, we decided to perform a benchmark of lower-order approaches for a set of 10 triangulenes, including heptazine, cyclazine, and eight additional molecules displayed in Figure 3, for which we obtained accurate TBE/aug-cc-pVTZ values using the same procedure as in Tables 1 and2. The selection of these molecules was inspired by previous literature. [START_REF] Tučková | On the Origin of the Inverted Singlet-Triplet Gap of the 5th Generation Light-Emitting Molecules[END_REF][START_REF] Drwal | The Role of Spin Polarization and Dynamic Correlation in Singlet-Triplet Gap Inversion of Heptazine Derivatives[END_REF][START_REF] Sancho-García | Violation of Hund's rule in molecules: Predicting the excited-state energy inversion by TD-DFT with double-hybrid methods[END_REF] We underline that the magnitude of the STG varies significantly within this series, as do the absolute singlet and triplet excitation energies (see Table S1 in the SI). The TBEs for the STG range from -0.029 eV (8) to -0.305 eV (10) and thus cover a significant range of possibilities. With these TBE values in hand, we conducted an evaluation of the performance of second-order wave function methods, specifically, CIS(D), ADC(2), CC2, and two of its spin-scaled variants (SOS-CC2 and SCS-CC2), as well as CCSD. We also benchmarked double hybrid functionals within the TD-DFT framework. More specifically, we tested four of them, namely PBE0-2, SOS-PBE-QIDH, SCS-PBE-QIDH, and SOS-RSX-QIDH, as these were previously identified as promising candidates for this particular task in a benchmark study performed by Sancho-Garcia, Adamo, and coworkers. [START_REF] Sancho-García | Violation of Hund's rule in molecules: Predicting the excited-state energy inversion by TD-DFT with double-hybrid methods[END_REF] All results can be found in Tables S2 andS3 of the SI. The results of this benchmark for the STG are displayed in Table 3 in which we report the mean signed error (MSE), the mean absolute error (MAE), and the standard deviation of the errors (SDE) using our TBEs as reference. Equivalent statistical analyses for the singlet and triplet energies themselves can be found in Tables S4 andS5 of the SI. Notably, all tested wave function approaches, except for CCSD, yield a negative MSE, indicating an overestimation of the magnitude of the (negative) STGs. This trend is particularly pronounced for CIS(D), SOS-CC2, and SCS-CC2. In contrast, CCSD tends to provide smaller STG estimates, with no inversion observed for both 7 and 8, for which our TBEs are slightly negative. As one can see by comparing the MSEs and MAEs, these trends are systematic. All wave function approaches, except for CIS(D), provide acceptable SDEs, correctly predicting the chemical trends in the series, which is a satisfying outcome. Among these methods, the most refined scheme, CCSD, produces the smallest SDE, while, as expected, [START_REF] Véril | QUESTDB: a Database of Highly-Accurate Excitation Energies for the Electronic Structure Community[END_REF] the two spin-scaled CC2 models also provide more consistent values than the standard CC2 approach.

Taking into consideration the usual trade-off between absolute and relative accuracies, as well as the computational cost, ADC(2) stands out as the most suitable cost-effective option for assessing the STG in similar systems. It is also noteworthy that ADC(2) delivers satisfactory absolute singlet and triplet energies (see Table S4 and S5 in the SI). Shifting our focus to the TD-DFT results, it is important to highlight that PBE0-2 fails to provide a reasonable triplet energy for the (challenging) compound 6, which very badly affects its otherwise satisfactory statistics. All three QIDH-derived functionals tend to magnify the negative character of the STG and yield relatively large SDEs, equal to or larger than that of CIS(D). Among the double-hybrid functionals tested, SCS-PBE-QIDH proves to be the most reliable, though it offers less consistent results when compared to ADC (2). Finally, we also consider cyclborane, the equivalent of cyclazine 2 but with a central boron atom instead of a nitrogen atom. This compound has been previously studied by Dreuw and Hoffmann. [START_REF] Dreuw | The Inverted Singlet-Triplet Gap: a Vanishing Myth? Front[END_REF] As detailed in Section S3 of the SI, the 𝐷 3ℎ structure presents an inverted STG. Yet this GS geometry is not a genuine minimum (one imaginary frequency at the MP2 level) and there exists a 𝐶 3ℎ GS structure, approximately 1 kcal.mol -1 more stable than the 𝐷 3ℎ conformer at the CCSD(T)/cc-pVTZ level. This 𝐶 3ℎ structure exhibits a clear bond length alternation and conforms to Hund's rule (Table S6). Consequently, this structure was not included in the benchmark discussed above.

In short, this contribution conclusively demonstrates that compounds 1-6, 9, and 10 do indeed exhibit inverted vertical STGs. While 7 and 8 also deviate from Hund's rule according to our TBEs, the amplitudes of their negative STG are likely too small to be definitive. For both the lowest triplet and singlet states of all these compounds, we performed CC3/CCSDT calculations with various diffuse-containing basis sets, as no indications of multireference character or a double-excitation nature were detected. For heptazine, neither the relaxation of the geometry in the excited states nor the inclusion of vibrational corrections alters the conclusion reached in the vertical approximation. In other words, the inverted STG persists when considering 0-0 energies.

When assessing the performance of lower-order methods suitable for computing the properties of substituted or extended triangulenes, using our TBE values, we found that ADC(2) likely strikes the best balance between accuracy, consistency, and cost among the wave function methods. While less consistent than ADC(2), the SCS-PBE-QIDH double-hybrid functional shows promise for TD-DFT calculations. We hope that this Letter can serve as the starting point for the accurate design of systems displaying inverted STGs.

COMPUTATIONAL DETAILS General

All calculations rely on the frozen-core (FC) approximation and, except when noted, the default convergence thresholds and algorithms of a given code. In the text, we dropped the EOM/LR prefix for the CC methods since the two formalisms provide the same transition energies.

Geometries

All GS structures were first optimized at the MP2/6-311G(d,p) level with gaussian 16.A.03, 29 using a 𝑍-matrix enforcing the highest possible point group symmetry. Tight convergence thresholds were applied during these optimizations. Analytical frequency calculations were next performed at the same level of theory, confirming that the structures are true minima. These stable MP2(FC)/6-311G(d,p) geometries were considered as starting point for CCSD(T)/cc-pVTZ analytical optimizations that were performed with cfour. 30,[START_REF] Matthews | Coupled-Cluster Techniques for Computational Chemistry: The CFOUR Program Package[END_REF] For the 0-0 calculations on heptazine, the GS geometry was reoptimized at the CCSD/cc-pVDZ level of theory, and vibrational frequencies were determined as well at this level using gaussian. 29 The structure and vibrational frequencies of the lowest triplet were obtained with the same code at the UCCSD/cc-pVDZ level whereas we applied the corresponding EOM-CCSD/cc-pVDZ level for the lowest singlet ES. Symmetry was lowered when imaginary frequencies were obtained, and the process was restarted until a true minimum was reached.

Transition energies

We used a variety of quantum chemistry software to determine vertical transition energies, using three gaussian basis sets containing both polarization and diffuse functions, namely 6-31+G(d), aug-cc-pVDZ, and aug-cc-pVTZ. CCSD calculations have been performed with dalton [START_REF] Aidas | The Dalton Quantum Chemistry Program System[END_REF] and gaussian. 29 CC3 33 calculations have been performed with dalton [START_REF] Aidas | The Dalton Quantum Chemistry Program System[END_REF] as well as cfour, 30,31 the latter allowing calculations to singlet ES only. CCSDT 34 calculations were also carried out with cfour and were only possible with the most compact basis set. The same code was used for the CCSD(T)(a) ★ 35 and single-point UCCSD(T) calculations with the triple-𝜁 basis set. In cfour, we relied on the QC-SCF algorithm [START_REF] Tommaso Nottoli | General Purpose Quadratic Self-Consistent Field Code with and without Cholesky Decomposition of the Two-Electron Integrals[END_REF] with the correct occupation number set, and a SCF convergence threshold of 10 -9 or 10 -10 au. All CIS(D), 37 ADC(2), [START_REF] Dreuw | The Algebraic Diagrammatic Construction Scheme for the Polarization Propagator for the Calculation of Excited States[END_REF] and CC2 [START_REF] Christiansen | The Second-Order Approximate Coupled Cluster Singles and Doubles Model CC2[END_REF] calculations have been performed with turbomole [START_REF] Balasubramani | Modular Program Suite for ab initio Quantum-Chemical and Condensed-Matter Simulations[END_REF][START_REF]TURBOMOLE V7.3 2018, a development of University of Karlsruhe and Forschungszentrum Karlsruhe GmbH[END_REF] using the aug-cc-pVTZ basis set and applying the RI approach [START_REF] Hättig | CC2 Excitation Energy Calculations on Large Molecules Using the Resolution of the Identity Approximation[END_REF] with the corresponding auxiliary basis. In these calculations, we enforced the default scaling parameters for both SOS-CC2 and SCS-CC2. The TD-DFT calculations based on double hybrids were all performed with orca, [START_REF] Neese | The ORCA Quantum Chemistry Program Package[END_REF] selecting PBE0-2, [START_REF] Chai | Seeking for Reliable Double-Hybrid Density Functionals Without Fitting Parameters: The PBE0-2 Functional[END_REF] SOS-PBE-QIDH, [START_REF] Casanova-Páez | Time-Dependent Long-Range-Corrected Double-Hybrid Density Functionals with Spin-Component and Spin-Opposite Scaling: A Comprehensive Analysis of Singlet-Singlet and Singlet-Triplet Excitation Energies[END_REF] SCS-PBE-QIDH, [START_REF] Casanova-Páez | Time-Dependent Long-Range-Corrected Double-Hybrid Density Functionals with Spin-Component and Spin-Opposite Scaling: A Comprehensive Analysis of Singlet-Singlet and Singlet-Triplet Excitation Energies[END_REF] and SOS-RSX-QIDH. [START_REF] Casanova-Páez | Time-Dependent Long-Range-Corrected Double-Hybrid Density Functionals with Spin-Component and Spin-Opposite Scaling: A Comprehensive Analysis of Singlet-Singlet and Singlet-Triplet Excitation Energies[END_REF] In orca, we set the tightSCF and grid3 options, with the aug-cc-pVTZ and the automatically generated auxiliary basis sets. Note that the Tamm-Dancoff approximation was not enforced in the TD-DFT calculations.

Figure 1 :

 1 Figure 1: Representation of heptazine and cyclazine.

Figure 2 :

 2 Figure 2: Two different views of the CCSD, EOM-CCSD, and UCCSD optimized geometries obtained with the cc-pVDZ basis set.

Figure 3 :

 3 Figure 3: Representation of the eight additional molecules considered in the present benchmark set.

Table

Table 1 : Vertical transition energies to the lowest 𝑆 1 and 𝑇 1 ESs, as well as the corresponding STG of heptazine obtained with various levels of theory. All values are in eV. 𝑎

 1

	Method	𝑆 1	𝑇 1	STG Ref. Year

Table 2 : Vertical transition energies to the lowest 𝑆 1 and 𝑇 1 ESs, as well as corresponding STG of cyclazine obtained with various levels of theory. All values are in eV. 𝑎

 2

	Method	𝑆 1	𝑇 1	STG Ref. Year
	CIS(D)/cc-pVDZ	1.07 1.37 -0.30 2 2019
	ADC(2)/cc-pVDZ	1.04 1.20 -0.16 2 2019
	CCSD/cc-pVDZ	1.09 1.19 -0.10 2 2019
	SCS-CC2/def2-TZVP	1.110 1.334 -0.224 3 2021
	SC-NEVPT2/def2-TZVP	1.224 1.288 -0.044 3 2021
	ADC(3)/cc-pVDZ	0.777 0.869 -0.092 4 2021
	RASPT2/def2-TZVP	0.86 0.89 -0.03 6 2022
	XMC-QDPT2/def2-TZVP			-0.106 9 2022
	CC3/cc-pVDZ	0.98 1.15 -0.17 10 2023
	Mk-MRCCSD(T)/def2-TZVP			-0.18 11 2023
	ADC(3)/cc-pVTZ	0.81 0.87 -0.06 13 2023
	ΔCCSD(T)/aug-cc-pVDZ			+0.015 13 2023
	ΔCCSD(T)/cc-pVTZ			+0.025 13 2023
	ADC(2)/aug-cc-pVTZ	1.001 1.138 -0.137 This work
	CC2/aug-cc-pVTZ	1.051 1.181 -0.130 This work
	CCSD/aug-cc-pVTZ	1.090 1.154 -0.064 This work
	CC3/aug-cc-pVDZ	0.990 1.121 -0.131 This work
	TBE 𝑏	0.979 1.110 -0.131 This work

𝑎

The present calculations are performed on the CCSD(T)/cc-pVTZ GS structure. Note that previous works used (slightly) different geometries. 𝑏 Theoretical best estimate obtained from CC3/aug-cc-pVTZ + [CCSDT/6-31+G(d) -CC3/6-31+G(d)] for 𝑆 1 and CC3/aug-cc-pVDZ + [CCSD/aug-cc-pVTZ -CCSD/aug-cc-pVDZ] for 𝑇 1 .

Table 3 : MSE, MAE, and SDE (in eV) determined for the STG, considering our TBE of Table S1 as reference.

 3 For PBE0-2, 6 is a clear outlier due to a strong orbital mixing in the triplet state (see the SI). Removing it yields MSE, MAE, and SDE of 0.006, 0.046, and 0.056 eV, respectively. Note, however, that removing this challenging compound would improve the statistics of all other double-hybrid functionals.

	Method	MSE	MAE	SDE
	CIS(D)	-0.253	0.253	0.083
	ADC(2)	-0.033	0.033	0.035
	CC2	-0.027	0.027	0.041
	SOS-CC2	-0.159	0.159	0.022
	SCS-CC2	-0.111	0.111	0.021
	CCSD	+0.081	0.081	0.014
	PBE0-2 𝑎	-0.066	0.114	0.235
	SOS-PBE-QIDH	-0.073	0.075	0.084
	SCS-PBE-QIDH	-0.033	0.055	0.085
	SOS-RSX-QIDH	-0.084	0.105	0.121

𝑎

ACKNOWLEDGMENTS

The authors are thankful for the generous allocations of time by the CCIPL/GliCID computational center installed in Nantes. PFL thanks the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (Grant agreement No. 863481) for funding.

SUPPORTING INFORMATION AVAILABLE

Full benchmark results; Molecular orbital plots; Cyclborane results; Cartesian coordinates.