Jianyu Wei

Peter L Rodríguez-Kessler

Jean-François Halet

Samia Kahlal

Jean- Yves Saillard
email: jean-yves.saillard@univ-rennes1.fr

Alvaro Muñoz-Castro
email: alvaro.munoz@uautonoma.cl

On Heteronuclear Isoelectronic Alternatives to Au

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Introduction

The fastly developing chemistry of gold nanoclusters (AuNCs) is currently of huge topical interest owing to their many potential interdisciplinary applications, [START_REF] Levi-Kalisman | Synthesis and Characterization of Au102 (p-MBA) 44 Nanoparticles[END_REF][START_REF] Tsukuda | Aerobic Oxidations Catalyzed by Colloidal Nanogold[END_REF][START_REF] Zhao | State of the Art in Gold Nanoparticle Synthesis[END_REF][START_REF] Wuithschick | Turkevich in New Robes: Key Questions Answered for the Most Common Gold Nanoparticle Synthesis[END_REF][START_REF] Jin | Atomically Precise Colloidal Metal Nanoclusters and Nanoparticles: Fundamentals and Opportunities[END_REF][START_REF] Weerawardene | Origin of Photoluminescence of Ag 25 (SR) 18 -Nanoparticles: Ligand and Doping Effect[END_REF][START_REF] Zhao | Endohedrally Doped Cage Clusters[END_REF][START_REF] Luo | Reactivity of Metal Clusters[END_REF][START_REF] Mingos | Gold Clusters, Colloids and Nanoparticles I[END_REF][START_REF] Mingos | Colloids and Nanoparticles II[END_REF][START_REF] Gao | Halogen Effects on the Electronic and Optical Properties of Au 13 Nanoclusters[END_REF][START_REF] Kamei | Generation of Small Gold Clusters with Unique Geometries through Cluster-to-Cluster Transformations: Octanuclear Clusters with Edge-Sharing Gold Tetrahedron Motifs[END_REF][START_REF] Zhu | Catalysis Opportunities of Atomically Precise Gold Nanoclusters[END_REF][START_REF] Kwak | Selective Determination of Dopamine Using Quantum-Sized Gold Nanoparticles Protected with Charge Selective Ligands[END_REF][START_REF] Sakai | Photovoltaic Properties of Glutathione-Protected Gold Clusters Adsorbed on TiO(2) Electrodes[END_REF][START_REF] Wu | Well-Defined Nanoclusters as Fluorescent Nanosensors: A Case Study on Au(25) (SG)(18)[END_REF][START_REF] Murray | Nanoelectrochemistry: Metal Nanoparticles, Nanoelectrodes, and Nanopores[END_REF][START_REF] Galloway | Biotemplated Magnetic Nanoparticle Arrays[END_REF][START_REF] Du | Atomically Precise Noble Metal Nanoclusters as Efficient Catalysts: A Bridge between Structure and Properties[END_REF][START_REF] Ebina | Two-, and Three-Dimensional Self-Assembly of Atomically Precise Metal Nanoclusters[END_REF] Numerous theoretical and experimental research efforts have contributed to the rationalization of their structure and the viability of novel noble-metal nanostructures. 21- 38 The most recent developments of this field encompass the design of doped AuNCs, 39- 48 an efficient strategy to tune the properties of the parent homonuclear clusters. 24,25,44,49- 57 AuNCs consist of atomically precise metallic cores further passivated by a protecting shell of ligand, to which in some cases are added additional Au(I) centers. 58- 62 The Au 13 icosahedron is one of their most recognizable motifs as earlier characterized in the very first phosphine derivatives [Au 13 (PMe 2 Ph) 10 Cl 2 ] 3+ by Mingos and coworkers [START_REF] Briant | Synthesis and X-Ray Structural Characterization of the Centred Icosahedral Gold Cluster Compound [Au 13 (PMe 2 Ph) 10 Cl 2 ](PF 6 ) 3 ; the Realization of a Theoretical Prediction[END_REF] and later in [Au 13 (dppe) 5 Cl 2 ] 3+ (1), [START_REF] Zhang | Diphosphine-Induced Chiral Propeller Arrangement of Gold Nanoclusters for Singlet Oxygen Photogeneration[END_REF][START_REF] Shichibu | HCl-Induced Nuclearity Convergence in Diphosphine-Protected Ultrasmall Gold Clu ter : A Novel Synthetic Route to "Magic-Number" Au 13 Clusters[END_REF] precluding the most recent extensive research on [Au 25 (SR) 18 ] -, [START_REF] Ding | Water-Soluble Au13 Clusters Protected by Binary Thiolates: Structural Accommodation and the Use for Chemosensing[END_REF][START_REF] Sheong | An [Au 13 ] 5+ Approach to the Study of Gold Nanoclusters[END_REF][68][START_REF] Kang | Au 25 (SR) 18 : The Captain of the Great Nanocluster Ship[END_REF][START_REF] Zhu | Reversible Switching of Magnetism in Thiolate-Protected Au25 Superatoms[END_REF] which have been extended to other related clusters. [START_REF] Tsukuda | Protected Metal Clusters: From Fundamentals to Applications[END_REF][START_REF] Gam | Coinage Metal Superatomic Cores: Insights into Their Intrinsic Stability and Optical Properties from Relativistic DFT Calculations[END_REF][START_REF] Saillard | Structure and Bonding Patterns in Large Molecular Ligated Metal Clusters[END_REF][START_REF] Fehlner | Molecular Clusters. A Bridge to Solid State Chemistry[END_REF][START_REF] Häkkinen | Atomic and Electronic Structure of Gold Clusters: Understanding Flakes, Cages and Superatoms from Simple Concepts[END_REF][START_REF] Mäkinen | Modeling Thiolate-Protected Gold Clusters with Density-Functional Tight-Binding[END_REF] In the last decade compound 1 has been further explored revealing its interesting optical and chiroptical properties as investigated separately by Konishi [START_REF] Zhang | Diphosphine-Induced Chiral Propeller Arrangement of Gold Nanoclusters for Singlet Oxygen Photogeneration[END_REF] and Li, [START_REF] Zhang | Diphosphine-Induced Chiral Propeller Arrangement of Gold Nanoclusters for Singlet Oxygen Photogeneration[END_REF][START_REF] Shichibu | HCl-Induced Nuclearity Convergence in Diphosphine-Protected Ultrasmall Gold Clu ter : A Novel Synthetic Route to "Magic-Number" Au 13 Clusters[END_REF] showing an enhanced behavior compared to other phosphine-protected clusters. [START_REF] Wen | Crystal Structure, Electrochemical and Optical Properties of [Au 9 (PPh 3 ) 8 ](NO 3 ) 3[END_REF][START_REF] Woehrle | Ligand Exchange Reactions Yield Subnanometer, Thiol-Stabilized Gold Particles with Defined Optical Transitions[END_REF][START_REF] Schmid | Au 55 [P(C 6 H 5 )3] 12 Cl 6 -Ein Goldcluster Ungewöhnlicher Größe[END_REF] In addition, the high quantum yield of the lowest triplet state (T 1 ) leads to promising singlet-oxygen ( 1 O 2 ) photogeneration capabilities in comparison to organic dyes such as by anthracene, as well as related Au 25 (SR) 18 and Au 38 (SR) 24 clusters . [START_REF] Ashen-Garry | Singlet Oxygen Generation by Cyclometalated Complexes and Applications[END_REF][START_REF] Derosa | Photosensitized Singlet Oxygen and Its Applications[END_REF] Within the superatom model, [START_REF] Tsukuda | Protected Metal Clusters: From Fundamentals to Applications[END_REF][START_REF] Walter | A Unified View of Ligand-Protected Gold Clusters as Superatom Complexes[END_REF][START_REF] Häkkinen | Atomic and Electronic Structure of Gold Clusters: Understanding Flakes, Cages and Superatoms from Simple Concepts[END_REF] 1 is described as a 8-cluster-electron (8-ce) species. Indeed, only the electrons significantly involved in metal-metal bonding, that is the 6s(Au) valence electrons of the [Au 13 ] 5+ core, are included in the superatom ce count. The 5d(Au) electrons are not considered for they have been recognized a long time ago to play a negligible role. [START_REF] Mingos | Molecular-Orbital Calculations on Cluster Compounds of Gold[END_REF] The highest occupied orbitals of 1 correspond to the 1P shell [START_REF] Walter | A Unified View of Ligand-Protected Gold Clusters as Superatom Complexes[END_REF][START_REF] Häkkinen | Electronic Structure: Shell Structure and the Superatom Concept[END_REF][START_REF] Lopez-Acevedo | Structure and Bonding in the Ubiquitous Icosahedral Metallic Gold Cluster Au 144 (SR) 60[END_REF] and the lowest unoccupied levels to the 1D manifold. The 1P→1D electronic transitions account for the character of the low-energy UV/Vis spectrum and the related luminescent properties. In this context, the exploration of 18-ce related counterparts allows further understanding of the role of electron counts in the optical properties, based on heteroatom doping of the cluster-core, [START_REF] Ghosh | Atomic-Level Doping of Metal Clusters[END_REF][START_REF] Kurashige | Atomic-Level Understanding of the Effect of Heteroatom Doping of the Cocatalyst on Water-Splitting Activity in AuPd or AuPt Alloy Cluster-Loaded BaLa 4 Ti 4 O 15[END_REF][START_REF] Kumar | Gold Nanoclusters as Electrocatalysts: Size, Ligands, Heteroatom Doping, and Charge Dependences[END_REF][START_REF] Puls | A Novel Concept for the Synthe i of Multiply Doped Gold Clu ter [(M@AunM′m)Lk] Q+[END_REF][START_REF] Negishi | Effect of Copper Doping on Electronic Structure, Geometric Structure, and Stability of Thiolate-Protected Au 25 Nanoclusters[END_REF][START_REF] Negishi | Palladium Doping of Magic Gold Cluster Au 38 (SC 2 H 4 Ph) 24 : Formation of Pd 2 Au 36 (SC 2 H 4 Ph) 24 with Higher Stability than Au 38 (SC 2 H<sub>4</Su[END_REF] as earlier theoretically proposed and experimentally demonstrated by the bare W@Au 12 cluster by Pyykkö and Li, respectively. [START_REF] Pyykkö | Icosahedral WAu12: A Predicted Closed-Shell Species, Stabilized by Aurophilic Attraction and Relativity and in Accord with the 18-Electron Rule[END_REF][START_REF] Li | Experimental Observation and Confirmation of Icosahedral W@Au 12 and Mo@Au 12 Molecules[END_REF] A proper candidate is given by doping the parent cluster 1 by Os resulting in a neutral 18-ce [OsAu 12 (dppe) 5 Cl 2 ] (2) species (assuming that Os lies at the cluster center, see below). It is important at this stage to note that although [Au 13 (dppe) 5 Cl 2 ] 3+ (1) and 18-ce (2) are isoelectronic in terms of their total and valence number of electrons, within the superatom model, we suggest considering them as respectively 8-ce and 18ce species. This difference originates in the fact that the fully occupied 5d(Au) orbitals are low-lying and contracted, whereas the 5d(Os) ones are higher in energy and more diffuse. Therefore, contrarily to the former, the latter can participate to cluster bonding. This is why we include the 5d electrons of Os into the ce count of (1), whereas in (2) we do not include that of the encapsulated Au. Likewise, the naked W@Au 12 cluster is also commonly considered as a 18-ce species . [START_REF] Pyykkö | Icosahedral WAu12: A Predicted Closed-Shell Species, Stabilized by Aurophilic Attraction and Relativity and in Accord with the 18-Electron Rule[END_REF][START_REF] Li | Experimental Observation and Confirmation of Icosahedral W@Au 12 and Mo@Au 12 Molecules[END_REF] We thus consider 2 as having the 1S 2 1P 6 1D 10 configuration, with 1D HOMOs (or near HOMOs) of large 5d(Os) nature, giving rise to optical properties different from that of 1, whose configuration is 1S 2 1P 6 .

These different situations are sketched in Scheme 1, where the superatomic electronic structures of M@Au 12 (M = Au, Os) cores are showns on the basis of the interaction of M with its M 12 icosahedral cage. [Os@Au 12 ] 2+ case is similar to that of [W@Au 12 ]. [START_REF] Mingos | Molecular-Orbital Calculations on Cluster Compounds of Gold[END_REF][START_REF] Häkkinen | Electronic Structure: Shell Structure and the Superatom Concept[END_REF] Obviously, the two situations crudely illustrated in Scheme 1 should be considered as two limit cases and there should be a gradual transition between them, i.e.

when going from W to Au. Thus, the description of a [M@Au 12 ] n+ (M = W, Au; n = 0, +5) cluster core as a 18-ce rather than a 8-ce species (or conversely) is somewhat formal. From the relativistic density functional (DFT) calculations described below, we suggest placing the formal 18-ce/8-ce borderline on the right side of Os. In this investigation, the stability and bonding patterns of the hypothetical cluster 2 are explored and its electronic, optical, chiroptical and potential luminescent properties are analysed, with systematic comparison with its homonuclear parent [START_REF] Hirai | Understanding Doping Effects on Electronic Structures of Gold Superatoms: A Case Study of Diphosphine-Protected M@Au 12 (M = Au, Pt, Ir)[END_REF] thus supporting the fact that the hypothetical title Os species is feasible.

Computational Details

Calculations were carried within the Density Functional Theory 92 (DFT) formalism using the ADF2019 code. [START_REF]Amsterdam Density Functional (ADF) Code[END_REF] Scalar relativistic corrections via the ZORA Hamiltonian were considered [START_REF] Van Lenthe | Relativistic Total Energy Using Regular Approximations[END_REF] together with the generalized gradient approximation (GGA) Perdew-Burke-Ernzerhof (PBE) exchange-correlation (xc) functional, [START_REF] Perdew | Generalized Gradient Approximation for the Exchange-Correlation Hole of a Many-Electron System[END_REF][START_REF] Perdew | Generalized Gradient Approximation Made Simple[END_REF] owing to its reliable performance at reasonable computational cost on AuNCs. [START_REF] Lopez-Acevedo | Structure and Bonding in the Ubiquitous Icosahedral Metallic Gold Cluster Au 144 (SR) 60[END_REF][START_REF] Akola | On the Structure of Thiolate-Protected Au 25[END_REF][START_REF] Jiang | The Expanding Universe of Thiolated Gold Nanoclusters and Beyond[END_REF][START_REF] Akola | Thiolate-Protected Au 25 Superatoms as Building Blocks: Dimers and Crystals[END_REF][100][START_REF] Pyykkö | Icosahedral WAu12: A Predicted Closed-Shell Species, Stabilized by Aurophilic Attraction and Relativity and in Accord with the 18-Electron Rule[END_REF] Dispersion forces were considered via the empirical pairwise corrections of Grimme (DFT-D3). 101 Basis sets of triple- Slater quality, plus two polarization functions (STO-TZ2P) were employed for valence electrons. The frozen core approximation was applied to the [1s 2 -4f 14 ] shells for Au and Os, [1s 2 -4p 6 ] Ru, [1s 2 -3p 6 ] Fe, [1s 2 ] for C, and [1s 2 -2s 2 ] for P and Cl, leaving the remaining electrons to be treated variationally.

Ground and excited states geometry optimizations were performed without any symmetry constraint, via the analytical energy gradient method implemented by Versluis and Ziegler, 102 with energy convergence criteria set at 10 -4 Hartree, gradient convergence at 10 -3 Hartree/Å and radial convergence of 10 -2 Å. Electronic excitation energies were calculated via time-dependent-DFT (TD-DFT) considering the van Leeuwen-Baerends (LB94) xc-functional as in previous similar studies. [START_REF] Alkan | Relativistic DFT Investigation of Electronic Structure Effects Arising from Doping the Au 25 Nanocluster with Transition Metals[END_REF]100,103 The current approach was tested on the description of dual photoluminescence properties of a ligand-supported hexanuclear Au(I) framework, with good agreement to experimental emission and absorption energies. 104

Results and Discussion

The question of the location of heteroatoms in doped group 11 1). [START_REF] Zhang | Diphosphine-Induced Chiral Propeller Arrangement of Gold Nanoclusters for Singlet Oxygen Photogeneration[END_REF] The calculated M (center) - Relative energies are given in brackets (kcal.mol -1 ). Os atom is in blue. The electronic structures of 1 and 2 are illustrated in Figure 2. The occupied 1D shell of 2 is intercalated between the 1P x,y HOMOs and 1P z superatomic orbitals (shell splitting allowed by symmetry). In average, the 1D shell of 2 is composed of 52% 5d(Os) character mixed in a bonding way with 21% 6s/6p(Au), and 27% from protecting ligands. Such an occupied 5d(M (center) ) shell does also exist in 1. However, these low-lying combinations are fully non-bonding, with no significant 6s/6p(Au (ico) )

Au (Cl) distance in 2 (2.
admixture and thus cannot be considered as the superatomic 1D shell. Rather, the true 1D orbitals of 1 are its five lowest vacant ones, which are of dominant 6s/6p(Au) composition, with Au (center) -Au (ico) bonding character (Figure 2). On the other hand, the lowest vacant levels of 2 are not of superatomic character, but *-ligand centered orbitals.

The singlet state of 2 is largely favored in comparison to the triplet and quintet states (by 53.2 kcal.mol -1 and 91.9 kcal.mol -1 , respectively). Consistently, its HOMO-LUMO is large (1.53 eV) and similar to that of 1 (1.96 eV). The [Ru@Au 12 (dppe) 5 Cl 2 ] and [Fe@Au 12 (dppe) 5 Cl 2 ] relatives have also a substantial preference for isomer I (+88.7 and +128.4 kcal.mol -1 from the next isomer, respectively, see Supporting Information).

They also have significant HOMO-LUMO gaps (1.52 and 1.09 eV, respectively) and show similar frontier orbitals. In order to evaluate the variation of the interaction between the dppe ligands and the rest of the molecule, we computed the overall interaction energy (ΔE int ) between the [(dppe) 5 ] phosphine shell and the [M@Au 12 Cl 2 ] n+ (M = Au, n = 3; M = Os, n = 0) fragment. It amounts to -160.7 kcal.mol -1 for 1 and -96.5 kcal.mol -1 for 2 (per dppe unit). Values similar to 2 were found for the M = Ru and Fe relatives, i.e., -96.0 and -91.7 kcal.mol -1 , respectively. The origin of the difference in interaction energy ΔE int between the homonuclear Au cluter and its heteronuclear congeners can be interpreted within the Energy Decomposition Analysis (EDA) framework given by Ziegler and Rauk, 106-108 according to:

ΔE int = ΔE Pauli + ΔE elstat + ΔE orb + ΔE disp
leading to different chemically meaningful quantities accounting for the destabilizing character of the interaction given by the repulsion between occupied orbitals (∆E Pauli ), and for the stabilizing electrostatic, covalent and dispersion (charge transfer) characters, ΔE elstat, ΔE orb and ΔE disp ). The character of the interaction is mainly electrostatic, similar to other phosphine protected gold clusters. 109 The EDA analysis shows that the stabilizing ligand-core interaction, involving one dppe ligand from [Au 13 Cl 2 (dppe) 5 ] 3+ to [Os@Au 12 (dppe) 5 Cl 2 ], varies mostly by the decrease in the orbital interaction term (ΔE orb ) from -190.4 to -146.6 kcal.mol -1 , retaining a smaller variation from the electrostatic character of the interaction (-336.9 to -343.5 kcal.mol -1 , respectively), which accounts for the variation in the ligand-core interaction. In the Ru and Fe counterparts, a similar variation is found with ΔE orb amounting to -144.0 and -144.8 kcal.mol -1 , respectively. The proposed 18-ce phosphineprotected clusters are thus structurally related to the parent [Au 13 Cl 2 (dppe) 5 ] 3+ , with weaker ligand-core interaction energy indicating the smaller role of the protecting ligands to achieve stable species with sizable HOMO-LUMO gap, owing to the extra 1D 10 electrons in the cluster core of the former. Analysis of the overall interaction energy of the [M@Au 12 ] n+ metallic core with the whole protecting ligands shell leads to a similar trend (Supporting Information). Optical properties were also addressed in order to investigate the expected differences between 1 (8-ce) and 2 (18-ce). The experimental UV/Vis spectrum of 1 (Figure 3) features a moderate shoulder between 620 and 520 nm, followed by a more intense signal at 488 nm. [START_REF] Zhang | Diphosphine-Induced Chiral Propeller Arrangement of Gold Nanoclusters for Singlet Oxygen Photogeneration[END_REF] The calculated spectrum of 1 exhibits peaks at 590 and 473 nm accounting nicely for the experimental observation, 110 with 1P x,y → *-ligand and 1P z → 1D transition character, respectively. In the case of 2, several peaks appear in the calculated spectrum (Figure 3). That of lowest energy (801 nm) is associated with a 1P x,y → *-ligand transition. It is substantially red-shifted with respect to the parent cluster 1 (590 nm). The next peak manifold is composed of five signals at 672, 642, 603, 577, and 541 nm, accounting for different 1D → *-ligand transitions, which are not found in the 8-ce parent 1. Another peak at 499 nm is found of mixed 1D → *ligand and 1P z → *-ligand character, resembling the peak of 1 at 473 nm (488 nm experimentally [START_REF] Zhang | Diphosphine-Induced Chiral Propeller Arrangement of Gold Nanoclusters for Singlet Oxygen Photogeneration[END_REF] ). Thus, the existence of an occupied 1D shell in 2 gives rise to a manifold peak pattern expected from the UV/Vis spectrum characterized mainly by metal → *-ligand transitions, which are desired for photoinduced electron injection over surfaces from anchored gold nanoclusters. 111 A quite similar simulated UV/Vis behavior is obtained for the Ru relative, with the first peak at 796 nm of 1P x,y → *-ligand nature followed by a 671-542 nm manifold of 1D→*-ligand character with mixed 1D → *-ligand and 1P z → *-ligand transition at 490 nm. In the case of the Fe species, the simulated spectrum exhibits a weak lowenergy peak at 796 nm (1P x,y → *-ligand) followed by a 663-526 nm peak manifold of 1D → *-ligand character, with the appearance of the related 1P z → *-ligand transition at 450 nm. The stereochemical constraints of the dppe ligands in 1 induce a substantial distortion of the icosahedral Au 13 core [START_REF] Briant | Synthesis and X-Ray Structural Characterization of the Centred Icosahedral Gold Cluster Compound [Au 13 (PMe 2 Ph) 10 Cl 2 ](PF 6 ) 3 ; the Realization of a Theoretical Prediction[END_REF][START_REF] Zhang | Diphosphine-Induced Chiral Propeller Arrangement of Gold Nanoclusters for Singlet Oxygen Photogeneration[END_REF][START_REF] Shichibu | HCl-Induced Nuclearity Convergence in Diphosphine-Protected Ultrasmall Gold Clu ter : A Novel Synthetic Route to "Magic-Number" Au 13 Clusters[END_REF] and their helical orientation induces some chirality, resulting in two possible enantiomers insphering circular dichroism (CD)

properties. The CD spectra were computed for the right-handed enantiomers (Figure 4).

That of [Au 13 Cl 2 (dppe) 5 ] 3+ (1) reproduces well its experimental CD spectrum, [START_REF] Zhang | Diphosphine-Induced Chiral Propeller Arrangement of Gold Nanoclusters for Singlet Oxygen Photogeneration[END_REF] with a positive peak at 560 nm (563 nm experimental) and a negative peak at 462 nm (472 nm experimental). That of [Os@Au 12 (dppe) 5 Cl 2 ] (2) shows similar amplitudes but different shape. Its peak at 801 nm appears as a positive band, followed by another positive band peaking at 642 nm. The next peak manifold shows a negative and positive band, resulting in a pattern fingerprint different than for 1. A pattern quite similar to that of 2 is found for its Ru and Fe counterparts, with the exception of a negative band around 800 nm for the Fe species. A comparison of the strength of the CD signal, suggest a slightly more pronunced for heterometallic species, which can be of interest for further studies. The remarkable photoluminescent properties reported for 1 [START_REF] Zhang | Diphosphine-Induced Chiral Propeller Arrangement of Gold Nanoclusters for Singlet Oxygen Photogeneration[END_REF][START_REF] Shichibu | HCl-Induced Nuclearity Convergence in Diphosphine-Protected Ultrasmall Gold Clu ter : A Novel Synthetic Route to "Magic-Number" Au 13 Clusters[END_REF] are ascribed to a T 1 →S 0 decay [START_REF] Zhang | Diphosphine-Induced Chiral Propeller Arrangement of Gold Nanoclusters for Singlet Oxygen Photogeneration[END_REF] process, providing interesting singlet-oxygen sensitization applications.

For a comparison, they were also evaluated for the 18-ce cluster counterparts and are discussed here. The emission wavelength of 1 was found experimentally at 766 nm (1.61 eV) by Konishi and Shichibu. [START_REF] Shichibu | HCl-Induced Nuclearity Convergence in Diphosphine-Protected Ultrasmall Gold Clu ter : A Novel Synthetic Route to "Magic-Number" Au 13 Clusters[END_REF] Computations indicate a transition between the related T 1 →S 0 states at the T 1 geometry at 809 nm (1.53 eV), 110 which corresponds to a 1D → 1P decay. The excited state affords an increase of the axial AuCl-AuCl distance from 5.096 (S 0 ) to 5.184 Å (T 1 ). For [Os@Au 12 (dppe) 5 Cl 2 (2) the computed emission wavelength is red-shifted to the NIR at 1045 nm (1.19 eV). This time the T 1 states retain an axial ClAu-AuCl distance (5.18 Å) similar to that in S 0 ), but affords a particular equatorial PAu-AuP change from 5.22 (S 0 ) to 5.43 Å (T 1 ), in contrast to the structural rearreagnments at the excited state from 1.

The calculated emission energies of the Ru and Fe counterparts are similar, amounting to 1076 nm (1.15 eV) and 1210 nm (1.02 eV), respectively. In the case of Ru, the T 1 states exhibits also a similar axial ClAu-AuCl distance (5.180 Å ) than its S 0 ground-state, (5.217 Å) but affords a particular equatorial PAu-AuP change from 5.245 (S 0 ) to 5.536 Å (T 1 ). In the case of Fe, the S 0 and T 1 states retain similar ClAu-Au-Cl distance (4.967 and 4.886 Å, respectively), but show a decrease of the PAu-AuP distance from S 0 (5.248 Å) to T 1 (5.164 Å).

Analysis of related Hg-doped species given by [Hg@Au 12 Cl 2 (dppe) 5 ] 4+ species, leads to the preference for isomer III, i.e. Hg located at the M-Cl site, suggesting the possible extension to other doped species at different sites of the MAu 12 core, besides the central position (Supporting information).

We would like at this stage to come back to the fact that 1 and 2 are both isoelectronic and have different superatom electron counts (18-ce and 8-ce, respectively). They have the same total number of cluster valence electrons (162 including the ligand lone pairs and all the 5d electrons). This electron count satisfy the rule established by Mingos in the 1980s for spherical high-nuclearity gold phosphine clusters (12xn s + 18, where n s = number of surface gold atoms). 112,113 On the other hand, 1 and 2 are 18-ce and 8-ce superatoms, respectively. This difference originates from the fact that the 5d(M (center) ) orbitals (and electrons) participate to the bonding in the former but do not participate significantly in the later (keeping the total electron number constant), the jellium core potential which controls the superatomic electron configuration becomes shallower at the central position. [START_REF] Hirai | Understanding Doping Effects on Electronic Structures of Gold Superatoms: A Case Study of Diphosphine-Protected M@Au 12 (M = Au, Pt, Ir)[END_REF] Although different, this rationalizing approach is equivalent to ours, but would be less useful for interpreting optical properties.

Finally, It we would like to mention that the isoelectronic substitution of one Au atom in [Au 13 Cl 2 (dppe) 5 ] 3+ by Hg + substantially disfavors configuration I (Figure 1).

Indeed, the even lesser availability of the 5d(Hg) orbitals, as compared to 5d(Au), for for bonding with the Au 12 cage, leads to an energetic preference by 27 kcal/mol for isomer III, i.e. Hg located at the M-Cl site, suggesting the possible extension to other doped species at different sites of the MAu 12 core, besides the central position (Supporting information).

Scheme 1 .

 1 Scheme 1. Interaction of a formally d 10 encapsulated metal atom M with an icosahedral [Au 12 ] 4+ icosahedral cage. Left: M = gold; right: M = osmium. Note that the

  596 Å) is slightly enlarged in comparison to those in 1 (from 2.548 to 2.596 Å). Similar Au-Cl separations (from 2.201 to 2.288 Å) are computed for 1 and 2, denoting that the Au 12 Cl 2 cage remains similar upon osmium doping.

Figure 1 .

 1 Figure 1. Relaxed structures of the three isomers I-III of [OsAu 12 Cl 2 (dppe) 5 ] (2).

Figure 2 .

 2 Figure 2. Electronic structure of [Au 13 Cl 2 (dppe) 5 ] 3+ (1), [Os@Au 12 Cl 2 (dppe) 5 ] (2), [Ru@Au 12 Cl 2 (dppe) 5 ] (3) and [Fe@Au 12 Cl 2 (dppe) 5 ] (4), showing 1P and 1D superatomic shells.

Figure 3 .

 3 Figure 3. Calculated UV/Vis absorption spectra of [Au 13 Cl 2 (dppe) 5 ] 3+ , and [M@Au 12 Cl 2 (dppe) 5 ] (M = Os, Ru, Fe). The experimental spectrum of [Au 13 Cl 2 (dppe) 5 ] 3+ 64 is also shown for comparison.

Figure 4 .

 4 Figure 4. Calculated circular dichroism spectra for right-handed isomers of [Au 13 Cl 2 (dppe) 5 ] 3+ , and [M@Au 12 Cl 2 (dppe) 5 ] (M = Os, Ru, Fe). A broadening of 0.2 eV was employed. Values centered between ±500 × 10 -40 esu 2 .cm 2 . The experimental spectrum of [Au 13 Cl 2 (dppe) 5 ] 3+ 64 is also shown for comparison.

(Scheme 1 )

 1 . As discussed in the introduction, there should be a continuum with intermediate situations when going from Os to Au, with the 10 5d(M (center) ) electrons participating in an intermediate fashion to the bonding. The very recently published compounds [IrAu 12 (dppe) 5 Cl 2 ] + and [PtAu 12 (dppe) 5 Cl 2 ] 2+ are now elegantly completing the continuum series. 91 They were described by Tsukuda and coworkers as 8-ce species, but differing from [Au 13 (dppe) 5 Cl 2 ] 3+ in that when doping the central position by Pt or Ir

  IrAu 12 (dppe) 5 Cl 2 ] + and [PtAu 12 (dppe) 5 Cl 2 ] 2+ were published by Tsukuda and coworkers,

	1. The isoelectronic
	clusters [MAu 12 (dppe) 5 Cl 2 ] (M = Fe, Ru) are also discussed. As we were working on
	these hypothetical species, the isolation and full characterization of [

  metal clusters has been first addressed in 1989.105 In the case of 1, doping it by formally replacing one Au atom by Os (keeping constant the total electron number) results in three possible configurations for 2. One with Os on the central site (isomer I), another one with Os bonded to a dppe ligand (isomer II) and the third one when bonded to chlorine (isomer III). The resulting optimized structures (Figure1) indicate a large preference for isomer I, being largely favored by +99.1 and +101.0 kcal.mol -1 with respect to isomers II and III, respectively. This is consistent with the fact that only isomer I can be described as a superatom with 18-ce count (see above). This observation agrees with the recently published structures of the closely related compound [IrAu 12 (dppe) 5 Cl 2 ] + .

[START_REF] Hirai | Understanding Doping Effects on Electronic Structures of Gold Superatoms: A Case Study of Diphosphine-Protected M@Au 12 (M = Au, Pt, Ir)[END_REF]

Thus, hereafter, we refer our results to isomer I for 2. The relaxed structures of 1 and 2 exhibit similar M (center)-Au (ligand) distances of 2.622 and 2.612 Å, respectively, which compares well to experimental data available (2.552 Å) for 1 (Table

Table 1 .

 1 Structural

		Exp Au a	Au (1)	Os (2)	Ru	Fe
	M (center) -Au (ligand)	2.552	2.622	2.612	2.623	2.624
	M (center) -Au (Cl)	2.486	2.548	2.596	2.608	2.483
	Au-Cl	2.169	2.201	2.288	2.307	2.331

parameters for [M@Au 12 (dppe)Cl 2 ] n+ species (M = Au, n = 3; M = Os, Ru, Fe, n = 0).

a Experimental data from ref.

[START_REF] Zhang | Diphosphine-Induced Chiral Propeller Arrangement of Gold Nanoclusters for Singlet Oxygen Photogeneration[END_REF]

Table 2 .

 2 Energy decomposition analysis between the [(dppe) 5 ] phosphine shell and the [M@Au 12 Cl 2 ] n+ (M = Au, n = 3; M = Os, Ru, Fe , n = 0). Values in kcal.mol -1 .

		1		2		3		4
		Au		Os		Ru		Fe
	ΔE Pauli	390.2		416.5		400.5		389.1
	ΔE elstat	-336.9	61.1%	-343.1	66.9%	-329.5	66.4%	-311.9 64.9%
	ΔE orb	-190.4	34.6%	-146.6	28.6%	-144.0	29.0%	-144.8 30.1%
	ΔE disp	-23.6	4.3%	-23.3	4.5%	-23.0	4.6%	-24.1	5.0%
	ΔE int	-160.7		-96.5		-96.0		-91.7

Acknowledgments

The authors are grateful to the Chilean-French ECOS-CONYCYT program (project C18E04 and ECOS180004) and to the French-Chilean International Associated Laboratory for "Multifunctional Molecule and Material " (LIA-CNRS N°1027). The GENCI (Grand Equipment National de Calcul Intensif) is acknowledged for HCP support (project a0010807367). J.W. thanks the China Scholarship Council for a Ph.D. scholarship. A.M.-C. acknowledges support from Fondecyt 1180683. P.L.R.-K acknowledges to FONDECYT 3190329 grant.

Concluding remarks

Here we predict the stabilization of 18-ce superatoms based on the central doping of the M@Au 12 Cl 2 core, showing characteristic patterns for both UV/Vis and CD spectra. The optical properties are dominated by core-to-ligand charge transfer in contrast to the 8ce cluster 1, showing 1D → *-ligand and 1P → *-ligand character transitions. Such differences are also noted in the calculated CD spectra with a peak manifold between 950 and 650 nm, not observed for the parent Au 13 cluster. The emission properties derived from the T 1 →S 1 decay exhibit a red-shift from 809 nm calculated for 1 to 1045 nm for [Os@Au 12 (dppe) 5 Cl 2 ], which involves an equatorial distortion of the Au 12 Cl 2 core for the emissive excited state, rather than the axial distortion observed for 1. The proposed neutral 18-ce [M@Au 12 (dppe) 5 Cl 2 ] (M = Os, Ru, Fe) clusters exhibit weaker ligand-core interaction energy owing to a decrease in the ligand-core bonding interaction due to the extra 1D 10 shell, denoting the smaller role of the protecting ligands to achieve stable species. The resulting sizable HOMO-LUMO gaps computed for these species indicate substantial stability. As already explored extensively for MAu 24 (SR) 18 clusters, similar structural features are found for 8-ce and 18-ce species. This suggests further exploration of different doped species derived from the ligandprotected Au 12 Cl 2 core that would allow to tuning molecular properties, and further designing novel building blocks for nanostructured materials under the superatom approach.

AUTHOR INFORMATION

Corresponding Authors

Jean-Yves Saillard* jean-yves.saillard@univ-rennes1.fr Alvaro Muñoz-Castro* alvaro.munoz@uautonoma.cl

Authors

Jianyu Weiorcid.org/0000-0001-8540-2233 P. L. Rodríguez-Kesslerorcid.org/0000-0003-2373-7729 ean-ranco i Haletorcid.org/0000-0002-2315-4200 Samia Kahlal -orcid.org/0000-0003-1121-0292 Jean-Yves Saillard orcid.org/0000-0003-4469-7922 Alvaro Muñoz-Castro orcid.org/0000-0001-5949-9449

Author Contributions

The manuscript was written through the contributions of all authors.


Hao-Hai Wang

Jianyu Wei

Fahime Bigdeli

Farzaneh Rouhani

Hai-Feng Su

Ling-Xiao Wang

Samia Kahlal

Jean-François Halet

Jean-Yves Saillard

Ali Morsali

Kuan-Guan Liu

Monocarboxylate-Protected Two-Electron Superatomic Silver Nanoclusters with High Photothermal Conversion Performance

The first series of monocarboxylate-protected superatomic silver nanoclusters was synthesized and fully characterized by Xray diffraction, Fourier-transform infrared spectroscopy (FT-IR), X-ray photoelectron spectroscopy (XPS) and electrospray ionization mass spectrometry (ESI-MS). Specifically, compounds [Ag16(L)8(9-AnCO2)12] 2+ (L = Ph3P (I), (4-ClPh)3P (II), (2-furyl)3P (III), and Ph3As (IV)) were prepared by using solvent-thermal method under alkaline condition. These clusters exhibit a similar unprecedented structure containing a [Ag8@Ag8] 6+ metal kernel, of which the 2-electron superatomic [Ag8] 6+ inner core describes a flattened and puckered hexagonal bipyramid of S6 symmetry. Density functional theory calculations provide a rationalization of the structure and stability of these 2-electron superatoms. Results indicate that the superatomic 2 electrons occupy a superatomic molecular orbital 1S that has a substantial localization on the top and bottom vertices of the bipyramid. The π systems of the anthracenyl groups, as well as the 1S HOMO, are importantly involved in the optical and photothermal behavior of the clusters. The four characterized nanoclusters show high photo-thermal conversion performance of sun light. These results show that the unprecedented use of mono-carboxylates in the stabilization of Ag nanoclusters is possible, opening the door for the introduction of various functional groups on their cluster surface.

Introduction

High-nuclearity atom-precise silver nanoclusters (NCs) are currently the subject of intense interest due to their fascinating structures and specific properties [1][2][3][4][5][6][7] . Factors such as electronic shell closing, coordination saturation, nature of surface organic ligands, atom packing, and steric hindrance greatly influence the structures, properties, and stability of such NCs 8,9 . Their structure consists of a metallic kernel, usually exhibiting a core/shell structure, passivated by a protecting ligand shell. Their closed-shell stability is attributed to their magic number of bonding electrons providing from the 5s(Ag) atomic orbitals (AOs). This can be rationalized within the superatom concept 10,11 based on the spherical jellium model. 12 It is noteworthy that the ligands are not only acting as stabilizers of the highly reactive cluster metal kernel, but they also significantly influence the properties of the clusters. [13][14][15][16] Silver NCs protected by ligands containing nitrogen and oxygen donor atoms are attracting intense interest due to their unique properties with various potential applications. 8,[17][18][19][20][21] However, according to the hard-soft-acid-base (HSAB) theory, [22][23][24] preparing a silver cluster with hard base-type ligands such as oxygen-donor ligands is a challenging synthetic process, owing to the rather soft nature of the silver Lewis acid. A series of solutions such as inorganic O-donors (metal oxo or polyoxo) has been developed for the synthesis of these types of O-donors ligand-protected clusters. 25,26 Using organic O-donors, we recently opened the synthetic route to perfluoroglutarate-protected NCs, 27 and Zang and coworkers developed a series of carboxylate-protected clusters confined in Tiorganic cages 28 . Despite these developments, the use of simple organic O-donor ligands such as mono-carboxylates has not been reported yet. Designing easy methods for the synthesis of monocarboxylate-protected NCs would render possible the access to a series of species that would benefit from the advantages of carboxylate functional groups, such as availability, affordability and, overall, the huge variety of carboxylate functional groups. 29 Herein, the synthesis and characterization of a novel series of phosphine-and arsine-mediated monocarboxylate-protected superatomic silver NCs are described. These new molecular models show high photothermal response. It is noteworthy that photothermal materials based on silver NCs have been shown to be good candidates to efficiently convert light into heat by increasing the solar energy absorption in the high-energy region 30 . Thus, such systems constitute good models for understanding the photothermal conversion mechanisms at the molecular level 5,6 . In our compounds, a bulky monocarboxylic acid (9-anthracene carboxylic acid) as a protecting agent was used. The choice of such carboxylate ligands originates from their multi-ring aromatic π-system that has been widely used as UV absorbers 31,32 , photosensitizers 33,34 or transmitters 20 for their sensitization and great absorption of the short wavelength irradiation 21 to improve their ability in optical absorption. Moreover, the presence of free electrons in the core of these superatomic species can promote a bathochromic shift in the absorption of long wavelength irradiation, 35,36 favoring photothermal efficiency.

Experimental Section

Synthesis of the nanoclusters

The general method used for the synthesis of the I-IV NCs is described in Scheme 1. 3 mmol) was added to a solution of AgOTf (0.077 g, 0.3 mmol) in methanol (6 mL) corrections 47 were employed in order to take into account dispersion effects. Because of computational limitations, natural atomic orbital (NAO) populations were calculated with the Gaussian16 package 48 via its NBO3.1 module, [START_REF] Glendening | NBO 6.0[END_REF] by using the ADF optimized geometries with a Def2-SVP basis [START_REF] Weigend | [END_REF] set and BP86 functional. In order to reduce computational efforts, the Gaussian16 package was also used for calculating the UV-Vis optical transitions by time-dependent DFT (TD-DFT) at the Def2-SVP/PBE0 51 level of theory.

Synthesis of [Ag16

Results and Discussion

Structure Description

The clusters I-IV•(OH)2 were prepared by using a similar solventthermal method, as reported in our previous work, under the alkaline condition. 19 TEA was used as both a base source and a (weak) reductant. 52,53 The structures of clusters I-IV were characterized by single-crystal X-ray diffraction (Table S1). Clusters I and IV crystallize in the cubic Pa3 ̅ space group, whereas clusters II and III crystallize in the trigonal 𝑅3 ̅ group. Based on these results, the general formula of I-IV is: [Ag16(L)8(9-AnCO2)12] 2+ (L = Ph3P (I), (4-ClPh)3P (II), (2-furyl)3P (III), Ph3As (IV)). The presence of two hydroxide anions in the formula could originate from the use of imperfectly dried solvent. The dicationic nature of the clusters [Ag16(L)8(9-AnCO2)12] 2+ was also confirmed by ESI-MS (see below). In the crystal, they all adopt the same structure of perfect S6 symmetry with a unique Ag8@Ag8 coreshell arrangement (Figure 1). The eight inner silver atoms (purple) describe a flattened hexagonal bipyramid, with the hexagonal base somewhat chair-distorted. The S6 symmetry axis passes through the top and bottom vertices of the bipyramid (Figures 1 andS1), which are in close contact (~2.7 Å) with each other. All but one of the bipyramid edge lengths are lower than 2.9 Å (Table S2), indicating some degree of covalency. Alternatively, the Ag8 inner core can also be seen as forming a parallelepiped or compressed rhombohedron. Each of the eight inner silver atoms is bonded to three carboxylate oxygen atoms in a pyramidal fashion. The eight outer Ag atoms (green) lie on the top of each inner Ag atoms, but they are not in contact with them or with any other metal center (Ag…Ag > 3.4 Å). They are only bonded to ligands. The six outer silver atoms surrounding the puckered hexagonal base of the Ag8 bipyramid are bonded to one L ligand and two carboxylate oxygen atoms in a nearly planar tri-coordination mode. The two other outer silver atoms that lie on the S6 axis are bonded to one L ligand and three oxygen atoms in a tetrahedral fashion. Based on the silver-ligand interactions above described, two kinds of new surface motifs (Figure 2a and2c) can be distinguished. The first one (Figure 2a) consists of a [(L)Ag(9-AnCO2)]6 "hexamer" surrounding the hexagonal base of the Ag8 bipyramid and composed of the six tri-coordinated outer silver atoms, six phosphines or arsines, and six μ4-η 2 trans,η 2 trans anthracene-9carboxylates. The second one (Figure 2c) is a [(L)Ag(9-AnCO2)3] triangular "tie" located at the top and bottom of the Ag8 bipyramid and composed of one tetrahedrally coordinated outer silver atom, one phosphine or arsine, and three μ4-η 2 trans,η 2 cis anthracene-9carboxylates.

The carboxylate ligands being monoanionic, one is left with a formally [Ag16] 14+ core in [Ag16(L)8(9-AnCO2)12] 2+ . Owing to the fact that the eight outer Ag atoms are isolated from any of their congeners, one is led to consider them as in a +I oxidation state, making locally stable planar-tri-coordinated 16-electron (for six of them) and tetrahedrally coordinated 18-electron (for two of them) metal centers, respectively. Thus, one is left with an [Ag8] 6+ core, which, with two delocalized (so-called free) electrons originating from the two formally occupied 5s(Ag) atomic orbitals (AOs), can be viewed as a closed-shell 2-electron superatom of 1S 2 configuration. 54 Note that a 2-electron bicapped octahedral (or rhombohedral) [Ag8] 6+ NC core is also known, 19 as well as a tetracapped tetrahedral [Ag8] 6+ NC core. 28 However, a new [Ag8] 6+ polyhedral architecture is spotted here for the first time. Its mixed-valent nature is consistent with the fact that most of the Ag…Ag contacts (Table S2) are shorter than that 'argentophilic' interactions (2.8 Å), 55 similarly as in many other mixed-valent silver NCs 18,56,57 . The FT-IR spectra, as well as the recorded and simulated powder XRD patterns of these NCs are consistent with their single-crystal X-ray structure (Figures S2 andS3). It should also be minded that a related hexagonal bipyramidal structure has been shown to be adopted by a series of 2-electron [Au2Cu6L2(SR6)] (L = phosphine) clusters. 58,59 However, this structure differs from ours by the planarity of the Cu6 hexagonal base (the Au2Cu6 core is of D3h symmetry) and by the very different coordination modes of the ligands. To finish this structural analysis of the cluster core, one should notice that there is a continuous pathway which connects the above-mentioned D6h hexagonal bipyramid to the D3d bicapped octahedral arrangement adopted by the [Ag8] 6+ core of [Ag8(pfga)6] 6-(pfga = perfluoroglutatate) 19 . It turns out that the [Ag8] 6+ inner core of clusters I-IV adopts one of the intermediate structures of S6 symmetry that connect these two geometrical limits. under ultrasonication. 42 μL of TEA (0.3 mmol) and 0.023 g (0.1 mmol) of (2-furyl)3P were added to the resulting muddy solution, successively, which was still muddy. Then 2 mL of DMF were added. A cloudy solution was obtained after ultrasound treatment. The reaction mixture was sealed and kept at 70 ℃ for 20 h. After cooling to room temperature, yellow block crystals of III•(OH) 2 ). Absorption corrections were applied by using the program CrysAlis (multi-scan). The structures were solved by direct methods, and all non-hydrogen atoms were refined anisotropically by least-squares on F 2 using the SHELXTL program 1-6, 30, 37-39 . The hydrogen atoms of the organic moieties were generated geometrically.

Photothermal Experiments

A Xenon lamp (300 W) was used as the light source to simulate the solar irradiation during the photo-thermal conversion experiments (by using an AM 1.5 G optical filter, with light intensity 0.1 W•cm -2 ). The sample was placed in a 2.0×2.0 cm 2 glass sample cell right under the light source (30 cm). The sample surface temperature changes with time recorded by FLIR infrared thermography. The test conditions were performed under ambient temperature (29 ℃) and humidity (30 %). The photothermal conversion efficiency was calculated according to a previous method. 40

Other Analyses and Characterization

Microanalyses of C, H, and N were carried out with a CE instruments EA 1110 elemental analyzer. IR spectra were recorded from KBr pellets in the range 4000-400 cm -1 with a Nicolet AVATAR FT-IR330 spectrometer. Emission spectra were measured on a Hitachi F-7000 spectrometer. UV/Vis absorption spectra were measured on a SHIMADZU UV2550 spectrophotometer. Luminescence, lifetime and quantum yield (solid state) were measured on an Edinburgh FLS980 apparatus. ESI-MS was performed on an Agilent Technologies ESI-TOF-MS spectrometer. Qualitative analysis was carried out using GCMS-QP2010 Plus with EI ion source, EI-MS standard database, and an RTX-5 quartz capillary column. X-ray photoelectron spectroscopy (XPS) studies were performed on PHI Quantum-2000 XPS. The sample was put under UHV to reach the 10 -8 Pa range. The nonmonochromatized Al Kα source was used at 10 kV and 10 mA. All binding energies were calibrated using the C (1s) carbon peak (284.6 eV), which was applied as an internal standard. High resolution narrow-scan spectra were recorded with the electron pass energy of 50 eV and takeoff angle of 55° to achieve the maximum spectral resolution.

Computational Details

Geometry optimizations were carried out within the density functional theory (DFT) formalism using the ADF2018 code. 41,42 Scalar relativistic corrections via the ZORA Hamiltonian 43 were included together with the Becke-Perdew (BP86) functional. 44,45 A triple-ξ Slater basis set plus a polarization function (TZP) was used. 46 The frozen core approximation was applied to the [1s 2 -4p 6 ] shells for Ag, [1s 2 -2p 6 ] for P, and [1s 2 ] for C and O, leaving the remaining electrons to be treated variationally. Grimme's DFT-D3 empirical

Mass Spectrometry and X-ray Photoelectron Spectroscopy Analysis

The charge state and chemical composition of cluster IV were determined by positive-ion ESI-MS. Its spectrum in CH3OH solution is shown in Figure 3 S4). Overall, the ESI-MS spectra confirmed M= [Ag16(Ph3As)8(9-AnCO2)12] as a dication. The high-resolution XPS spectrum further confirmed the mixed-valent nature of the clusters (Figure S5). The Ag 3d5/2 XPS peak was deconvoluted into two peaks with the almost one-to-seven ratio of Ag 0 to Ag + (Figures 4 andS5). 27

Optical Properties

For further understanding the light absorption mechanism of NCs I-IV, 60 their optical features were investigated. Their solid-state UV-Vis spectra show strong and multiple absorption bands in the 200-500 nm range, suggesting good absorption upon short wavelength irradiation (Figure S6a). Their optical HOMO-LUMO gaps were estimated from the onset of these spectra (Figure S6b) (I: 2.34 eV, II: 2.56 eV, III: 2.33 eV, IV: 2.40 eV). They indicate negligible effect of the phosphine (or arsine) nature. Their UV-vis absorption spectra in DMF show two distinct peaks at 3.28 eV (α; 378 nm, εmax = 5.5×10 4 L⋅mol -1 cm -1 ) and 4.71 eV (β; 263 nm, εmax = 9.2×10 4 L⋅mol -1 cm -1 ) (Figure 5) that are consistent with the simulated spectra obtained from time-dependent density functional theory (TD-DFT) calculations (see below). The photoluminescence properties of clusters I-IV were also investigated in the solid state. At room temperature, the emission (fluorescence) spectra show a major peak around 420 nm (Figure 6). At 77 K, the peak around 420 nm is still present, but in the case of compounds II-IV a more intense peak appears around 300 nm (Figure 7). It is noteworthy that the free ligand in its acidic form 9-AnCOOH fluoresces at 466 nm at room temperature in CH2Cl2 solution.

Photo-thermal Conversion Properties

Considering the efficient UV-vis absorption properties of clusters I-IV, with the conjugated aromatic anthracene group over the ligand sphere, potential photothermal conversion properties are expected for these compounds. 61 Indeed, the carboxylate ligands, which contain multiple aromatic rings, can strongly absorb light, and on the other hand, the silver core with unique electrical properties and special heat capacity is well suited for converting light into heat 30 . To accurately evaluate the photothermal conversion performance of the four NCs, the cluster powder was fixed on a piece of quartz glass to form a uniform film. The photothermal conversion properties of the four NCs were conducted with simulated sunlight absorption. The temperature of all the NC samples increased rapidly under 1-sun irradiation (0.1 W•cm -2 , 420-780 nm), as confirmed by an IR camera. As shown in Figure 8 ℃, respectively. Then the NC samples reached a quasi-steady-state, over the next 40 min, the temperature increasing by only 0.5~2.9 ℃ (Figure 8). To confirm the role of the components in photothermal effect, a set of control experiments, including the background (quartz glass), the metal precursors (AgOTf) and the 9-Anthracene carboxylic acid ligand, were performed (Figure S8). After 1 h irradiation, the temperatures of the background, AgOTf and 9-AnCOOH ligand increased only by 18.8 o C, 20.5 o C and 22.4 o C, indicating that the observed high photothermal performance is mainly derived from the photothermal effect of the NCs. The photothermal conversion efficiency of these NCs can be calculated from their cooling curves 62 (See Figure S9), the obtained conversion efficiencies for I-IV are 41.1 %, 35.8 %, 40.7 % and 33.6 %, respectively. In addition, the I-IV samples could be re-used for four photothermal cycles without any inactivation, suggesting excellent stability and photothermal recyclability (Figure S10). Moreover, the XRD patterns of these cluster samples after four photothermal cycles confirms their structural stability (Figure S11), except for cluster II, which partially decomposes to Ag2O after four photothermal cycles (Figure S11). Owing to the existence of multiple π-delocalized anthracene groups on the cluster surface and that of free electrons within the [Ag8] 6+ core, an efficient intramolecular charge transfer is expected to enhance light absorption 38,63,64 , followed by a ligand-to-metal charge transfer (LMCT) which quenches the photon re-emission process, thus favouring the photothermal conversion. This lightinduced LMCT is confirmed by DFT calculations (see below). Compared to other silver-based photothermal materials, clusters I-IV show the best temperature increment under 1-sun (0.1 W•cm 2 ) by far. 65 Our results suggest that Ag clusters can be efficient candidates in various photothermal applications, for instance photothermal anti-counterfeiting 66 , photothermal water evaporation 67 , solar thermoelectric production [68][69][70][71][72] , etc., strongly encouraging further exploration of the Ag NC chemistry in this direction.

Computational Analyses

DFT calculations at the ZORA-BP86/TZP/D3(BJ) level were performed on cluster I (see Computational Details). The ideal symmetry of I is S6, but its fully optimized geometry was found to be slightly distorted (Ci). Relevant computed data are provided in Table 1 and its Kohn-Sham MO diagram is shown in Figure 9a. The large computed HOMO-LUMO gap (1.70 eV) is consistent with cluster stability. The MO diagram also indicates that no other closed-shell electron counts are possible for such a structure, confirming the dicationic nature of I. The optimized Ag-Ag distances within the Ag8 inner core, ranging from 2.698 Å to 2.994 Å, are consistent with their experimental X-ray counterparts. The HOMO of I shows a significant localization on the inner Ag8 core (with 5s/5p participation) and can be identified as the superatomic 1S orbital, with a sizable participation of the ligand (An) shell. The six lowest unoccupied molecular orbitals show major ligand (π*(An)) character. The LUMO+6 and LUMO+7, which contain large 5s/5p(Ag) character, can be easily identified as the superatomic 1Px and 1Py orbitals. Owing to the oblate (flattened) nature of the cluster, the 1Pz orbital is destabilized at substantially higher energy and is not shown in this diagram. Thus, I can be considered as a 2electron superatom with 1S 2 configuration. The preferred localization of the 1S HOMO on the Ag8 inner core and the computed NAO charges (Table 1) confirm the {[Ag8] 6+ }@{(Ag + )8} charge partitioning suggested in the structural analysis above. The nonspherical nature of the [Ag8] 6+ core is not uncommon for 2-electron superatoms, 27,28,[73][74][75][76][77][78][79][80][81][82] providing it is compact enough to give the full in-phase 1S orbital a sufficiently bonding character. This strong anisotropy of the S6 inner core induces a neat differentiation of its two types of symmetry-equivalent vertices, which is reflected by their different charges and configurations (Table 1). Obviously, the two free electrons are substantially more localized on the Agcap atoms (top and bottom of the bipyramid) rather than on those constituting the chair-distorted hexagonal base of the bipyramid (Aghex). A similar result was computed for the bare (unligated) [Ag8] 6+ fragment. Stretching to the limit our formal charge partitioning, the [Ag8] 6+ inner core of I could be described as a neutral (Agcap)2 dimer surrounded by a crown of six Aghex + ions. This extreme view is however at least in part contradicted by the computed Aghex-Aghex Wiberg bond index (Table 1), which, when compared to its Agcap-Agcap counterpart, is consistent with non-negligible (although weak) covalency, thus indicating some delocalization of the free electrons on the Aghex centers. In any case, the marked difference between Agcap and Aghex is consistent with the fact the high-resolution XPS spectra of I-IV can be resolved into two peaks, tentatively assigned to Ag(0) and Ag(I) in an approximate one-to-seven ratio, respectively (see above). We suggest that these experiments could not distinguish the six Aghex core atoms, which bear a minor part of the free electrons, from the outer Ag(I) ones.

It is of note that calculations carried out with simplified phosphine and/or carboxylic ligands yielded geometries that are different from those found for the clusters. Therefore, the specific architecture of clusters I-IV is the result of a compromise between ligand steric hindrance, metal/ligand charge equilibrium and metal-metal bonding. This compromise results in a cluster bearing a dicationic charge, which in turn seeks for two hydroxide counterions from water traces in the solvent. The TD-DFT-simulated UV-vis spectrum of I (Figure 9b) is in a good agreement with its experimental counterpart (Figure 5). The experimental weaker absorption band at 460 nm and the four absorption bands between 350 nm and 400 nm are well reproduced, with a reasonably small red shift in the high energy region (one peak at 448 nm and two peaks at 413 nm and 396 nm). A detailed frontier orbital analysis in Figure 9c shows that the transitions associated with the absorption band at 448 nm are from the ligand-type (mainly π(An)) HOMO-4 orbital to the mixed metal/ligand-type LUMO and to the two superatomic 1P levels (LUMO+6, LUMO+7), thus of LMCT major character. The more intense bands are associated with several transitions that are of major LLCT character (anthracenyl π→π* essentially).

Owing to the particularly large size of the cluster, it was not possible for us to calculate by TD-DFT the fluorescence emission wavelength corresponding to the transition from the first excited singlet state to the ground state. On the other hand, such calculations were performed on the free ligand 9-AnCOOH, for which the value computed at the same level of theory is 477 nm, a value close to the room-temperature experimental counterpart (466 nm, see above) and not much different from those recorded for clusters I-IV (~ 420nm), suggesting that the emissive behavior of the clusters is ligand (π*→π) in nature. Another computed emission with large oscillator strength was computed at 274 nm. It is close to that observed at low temperature for clusters II-IV (see above). One can therefore conclude that the weak emissive behavior of clusters I-IV is to be associated with that of their An substituents. This is consistent with the hypothesis that the photothermal behavior consists in a ligand (π*(An)) to metal (1S HOMO) non-radiative deactivation.

Conclusion

In summary, the first monocarboxylate-protected superatomic silver NCs [Ag16(L)8(9-AnCO2)12] 2+ (L = phosphine or arsine) were isolated and fully characterized for the first time. These 2-electron heteroleptic species feature an [Ag8@Ag8] 14+ core-shell structure, with a [Ag8] 6+ superatomic inner core, adopting an unprecedented structure that describes a distorted hexagonal bipyramidal structure of S6 symmetry, whereas the outer Ag8 shell is made of the unique "Ag(PR3)-AnCOO-Ag(PR3)" staple motifs. DFT calculations indicate that the two free electrons are preponderantly located on the top and bottom atoms of the bipyramid, approaching a dimeric Ag2 state. The NC surface also exhibits unprecedented structural features, formed by the original arrangement of eight outer Ag + , monocarboxylates and phosphines/arsines. DFT calculations show also that these distinctive architectures are due, not only to their electron count, but also to the nature and steric hindrance of the carboxylate ligands, which is very rare so far. The investigated NCs show excellent photothermal conversion ability, so far overpassing other silver-based photothermal materials. TD-DFT calculations suggest that this striking property is associated with the existence of multiple π-delocalized anthracene groups on the cluster surface and that of two free electrons within the [Ag8] 6+ core. Because of the wide availability and variability of monocarboxylic acids, this work could pave the way for the synthesis of new monocarboxylate-protected coinage metal NCs with larger metal kernels and with various functionalized carboxylates for the design of NCs with desired properties.

(Ph3P) 8 ( 9 - 1

 891 Scheme 1 The synthetic routes of crystals I-IV•(OH)2. think this is related to the different solubilities of the phosphines in the reaction system. Anal. Calcd for NC II•(OH)2: C324H206O26P4Cl24Ag16: C, 53.93; H, 2.88. Found: C, 54.23; H, 3.92.Synthesis of {Ag16[(2-furyl)3P]8(9-AnCO2)12}(OH)2 (III•(OH)2): 9-Anthracene carboxylic acid (0.067 g, 0.3 mmol) was added to a solution of AgOTf (0.077 g, 0.3 mmol) in methanol (6 mL) corrections47 were employed in order to take into account dispersion effects. Because of computational limitations, natural atomic orbital (NAO) populations were calculated with the Gaussian16 package 48 via its NBO3.1 module,[START_REF] Glendening | NBO 6.0[END_REF] by using the ADF optimized geometries with a Def2-SVP basis[START_REF] Weigend | [END_REF] set and BP86 functional. In order to reduce computational efforts, the Gaussian16 package was also used for calculating the UV-Vis optical transitions by time-dependent DFT (TD-DFT) at the Def2-SVP/PBE0 51 level of theory.

2. 2 X

 2 -ray Structure Analysis X-ray-quality crystals of the clusters I-IV•(OH)2 were directly obtained by solvothermal method, and X-ray crystallographic data were recorded at 154 K for I•(OH)2, 159 K for II•(OH)2, 293 K for III•(OH)2, 193 K for IV•(OH)2 by mounting a single-crystal of clusters I•(OH)2 (0.21 ×0.15 ×0.07 mm 3 ), II•(OH)2 (0.07 ×0.06 ×0.06 mm 3 ), III•(OH)2 (0.06 ×0.05 ×0.04 mm 3 ), and IV•(OH)2 (0.22 ×0.17 ×0.09 mm 3 ), respectively. Intensity data of the NCs were collected on a Bruker SMART APEX CCD diffractometer (Mo Kα source for I•(OH)2, II•(OH)2, IV•(OH)2; Cu Kα source for III•(OH)2

Fig. 1

 1 Fig. 1 The full structure (a), side view (b) and top view (c) of [Ag16(L)8(9-AnCO2)12] 2+ [L = Ph3P (I); (4-ClPh)3P (II); (2-furyl)3P (III); Ph3As (IV)]. Color code: grey ball, grey stick, red stick and orange stick represent Ag, C, O and P (or As), respectively. H atoms are omitted for clarity.

Fig. 2

 2 Fig. 2 Two different binding motifs of the monocarboxylates: (a) the [(L)Ag(9-AnCO2)]6 hexamer, (b) the schematic representation of the surface motifs: and (c) the triangular "tie" [(L)Ag(9-AnCO2)3]. Color code: green, surface Ag; purple, inner Ag.

  . It has three dominant peaks (A, B and C). The prominent peak A centred at m/z = 3472.31 is ascribed to [M + H2O + 3CH3OH] 2+ (calculated m/z = 3472.31) with M = [Ag16(Ph3As)8(9-AnCO2)12]. Peak B centred at m/z = 3424.21 and peak C centred at m/z = 3307.33 are assigned to the [M + H2O] 2+ (calculated m/z = 3424.28) and [M -2Ag 0 ] 2+ (calculated m/z = 3307.40) formulae, respectively (Figure

Fig. 3

 3 Fig. 3 Positive-ion mode ESI-MS of the crystal IV of M = [Ag16(Ph3As)8(9-AnCO2)12] dissolved in CH3OH. Inset: Zoom-in ESI-MS of the experimental (black line) and simulated (red line) isotopic patterns for species A = [M+H2O+3CH3OH] 2+ at 3472.31.

Fig. 4 Fig. 5

 45 Fig. 4 XPS spectra of nanocluster I.

Fig. 6 Fig. 7

 67 Fig. 6 Photoluminescence analysis of nanoclusters I-IV in the solid state at room temperature.

3 ℃

 3 , when the light was turned on, the temperature of I-IV rises rapidly to a maximum of 81.6 ℃, 72.3 ℃, 70.4 ℃, 70.in 20 min (Figure S7), increasing by 58.5 ℃, 37.5 ℃, 38.6 ℃, 38.6

Fig. 8

 8 Fig. 8 Temperature variation curves of nanoclusters I-IV.

Fig. 9

 9 Fig. 9 (a) Kohn-Sham MO diagram of I . The blue and red levels correspond to the superatomic 1S and 1P levels, respectively. (b) The simulated UV-vis spectra of I obtained from TD-DFT calculations. (c) Frontier MO composition and electronic transitions associated with the main absorption bands.

  Ph3As were added to the resulting muddy solution, successively, which was still muddy. Then 2 mL of DMF were added. A cloudy solution was obtained after ultrasound treatment. The reaction mixture was sealed and kept at 70 ℃ for 20 h. After cooling to room temperature, yellow block crystals of IV•(OH)2 were obtained. Anal.

		were
	obtained. Anal. Calcd for NC III•(OH)2: C276H178O56P8Ag16: C, 52.09; H,
	2.82. Found: C, 52.38; H, 2.98.
	Synthesis of	[Ag16(Ph3As)8(9-AnCO2)12](OH)2 (IV•(OH)2): 9-
	Anthracene carboxylic acid (0.067 g, 0.3 mmol) was added to a
	solution of AgOTf (0.077 g, 0.3 mmol) in methanol (6 mL) under
	ultrasonication. 42 μL of TEA (0.3 mmol) and 0.031 g (0.1 mmol) of

Calcd for NC IV•(OH)2: C324H230O26As8Ag16: C, 56.69; H, 3.38. Found: C, 56.74; H, 3.66

Table 1

 1 Relevant computed data for I (averaged to S6). X-ray-measured distances are reported for comparison.

			DFT	X-ray
		Agcap-Agcap (inner core)	2.708 [0.331]	2.753
	Atomic distances (Å) and WBIs a	Agcap-Aghexa (inner core) Aghexa-Aghexa (inner core) Aginner…Agouter	3.092 [0.156] 2.895 [0.161] 2.698 [0.111] 3.520 [0.067]	3.131 2.852 2.711 3.598
		Ag-O	2.417 [0.112]	2.400
		Ag-P	2.323 [0.394]	2.355
	HOMO-LUMO gap		1.70 (eV)
		2  Agcap (inner core)	-0.09
	NAO charges and electronic configurations	6  Aghexa (inner core) 8  Agouter (av.)	4d 9.87 5s 0.37 5p 0.85 +0.35 4d 9.90 5s 0.19 5p 0.57 +0.60
			4d 9.86 5s 0.30 5p 0.24

a In brackets.

Acknowledgements

The experimental work was supported by the National Natural Science Foundation of China (Grant No. 92161106), the Natural Science Foundation of Ningxia (Grant No. 2022AAC05017), the Foundation of State Key Laboratory of Physical Chemistry of Solid Surfaces (Grant No. 202013) and Tarbiat Modares University. We are extremely grateful to Wenyan Dan of Tongji University for his help with the crystal structure determination. The Grand Equipment National de Calcul Intensif is acknowledged for HCP support (Project a0010807367).

Notes and references

Author Contributions

The manuscript was written through contributions of all authors.

Conflicts of interest

The authors declare no competing financial interest.


Ali Khalesi
email: ali.khalesi@eurecom.fr

Petros Elia
email: petros.elia@eurecom.fr

Multi-User Linearly Separable Computation: A Coding Theoretic Approach

Keywords: Distributed computation, Linearly separable function, Coding theory

In this work, we investigate the problem of multiuser linearly separable function computation, where N servers help compute the desired functions (jobs) of K users. In this setting each desired function can be written as a linear combination of up to L (generally non-linear) sub-functions. Each server computes some of the sub-tasks, and communicates a linear combination of its computed outputs (files) in a singleshot to some of the users, then each user linearly combines its received data in order to recover its desired function. We explore the range of the optimal computation cost via establishing a novel relationship between our problem, syndrome decoding and covering codes. The work reveals that in the limit of large N , the optimal computation cost -in the form of the maximum fraction of all servers that must compute any subfunction -is lower bounded as γ ≥ H -1 q ( log q (L) N

), for any fixed log q (L)/N . The result reveals the role of the computational rate log q (L)/N , which cannot exceed what one might call the computational capacity Hq(γ) of the system.

I. INTRODUCTION

There is a recent surge in the use of distributed computing system such as MapReduce [START_REF] Dean | Mapreduce: simplified data processing on large clusters[END_REF], Spark [START_REF] Zaharia | Spark: Cluster computing with working sets[END_REF] for processing nonlinear and computationally hard functions. This surge has been further intensified by recent developments relating to training large-scale machine learning algorithms such as deep neural networks with high data complexity (cf. [START_REF] Verbraeken | A survey on distributed machine learning[END_REF]). In many such applications, the main goal is to utilize distributed parallel processing techniques to offload computations to a group of distributed servers in order to reduce the computation time. Such approaches can be found in various settings such as in [START_REF] Abadi | {TensorFlow}: A system for {Large-Scale} machine learning[END_REF].

This parallelization among distributed workers and servers presents new challenges in terms of accuracy, scalability, latency and straggler mitigation, privacy and security, communication and computation complexity [START_REF] Verbraeken | A survey on distributed machine learning[END_REF], [START_REF] Ulukus | Private retrieval, computing and learning: Recent progress and future challenges[END_REF]. To this end, various information-and coding-theoretic works, have sought to design algorithms that alleviate the above problems, as well as establish the fundamental performance limits in various scenarios. Such works can be found in the context of computational accuracy [START_REF] Jahani-Nezhad | Codedsketch: A coding scheme for distributed computation of approximated matrix multiplication[END_REF]- [START_REF] Wan | Cache-aided matrix multiplication retrieval[END_REF], latency and straggler mitigation [START_REF] Raviv | Gradient coding from cyclic mds codes and expander graphs[END_REF]- [START_REF] Tandon | Gradient coding: Avoiding stragglers in distributed learning[END_REF], scalability [START_REF] Li | Nearoptimal straggler mitigation for distributed gradient methods[END_REF], [START_REF] Soleymani | Analog lagrange coded computing[END_REF], security and privacy [START_REF] Soleymani | Distributed multi-user secret sharing[END_REF]- [START_REF] Wan | On secure distributed linearly separable computation[END_REF], as well as in the context of communication and computation complexity [START_REF] Li | Compressed coded distributed computing[END_REF]- [START_REF] Wan | On the tradeoff between computation and communication costs for distributed linearly separable computation[END_REF]. For a detailed survey of various such works, the reader is encouraged to see [START_REF] Ng | A survey of coded distributed computing[END_REF], [START_REF] Li | Coded computing: Mitigating fundamental bottlenecks in large-scale distributed computing and machine learning[END_REF].

In this paper we adapted a data parallel, centralized topology [START_REF] Verbraeken | A survey on distributed machine learning[END_REF], where there exist a master or dealer node who acts as a total trusted authority and manages N worker\server nodes, in the presence of L datasets\sub-tasks. Our setting also considers K agents\users, where each such user sends its demand to the master node. In particular, our setting here considers a master node that manages N server nodes that must contribute in a distributed manner to the computation of the desired function by each of the K different users. Under the linearly-separable assumption (cf. [START_REF] Wan | Distributed linearly separable computation[END_REF]), we consider that each user k ∈ {1, 2, . . . , K} asks for a linearly-separable function that takes as input up to L sub-functions.

To derive bounds to the computational costs required (at the servers) to complete the computation for all users, we here establish a novel relationship between our computing problem, linear codes, syndrome decoding, and the here-introduced class of partial-covering codes.

In this paper, Section II introduces the system model, Section III formulates the problem, Section IV makes the connection to coding theory, Section V presents the main algebraic converse, and finally Section VI concludes the paper.

Notations: For some n ∈ N, we define [n] {1, 2, . . . , n}. For some matrix (or vector) X, we use ω(X) to denote the corresponding hamming weight. We will denote the finite field GF(q) as F. For some vector x ∈ F n and some code C ⊆ F n , we will use d(x, C) to represent the hamming distance of x to the nearest codeword in C. We will dedicate the use of letter ρ to correspond to a code's normalized covering radius, while we will use ρ(C) when we want to emphasize on the normalized covering radius of a specific code C ∈ F n . For some matrix H, we will use C H to represent the linear code whose parity-check matrix is H. Similarly, when we write H C , we refer to the parity-check matrix of a linear code C. For some k ≤ n, k, n ∈ N, we will also use C(k, n) to represent a linear code of message length k and codeword length n. We will use V q (n, ρ) to represent the volume of a Hamming ball in F n with radius ρn. For 0 ≤ x ≤ 1 -1 q , x ∈ R, we will use the notation H q (x)

x log q (q -1)

-x log q (x) -(1 -x) log q (1 -x)
to represent the q-ary entropy function. Finally we will use sup(x ⊺ ) to represent the support of some vector x ⊺ ∈ F n , i.e., to represent the set of indices of non-zero elements of that vector.

II. PROBLEM STATEMENT

In this section, we explain the Multi-user Linearly Separable Computation setting (cf. Fig. 1) which consists of K users, N servers, and a master node that coordinates servers and users 1 .

In this problem, user k ∈ [K] demands a function

F k (D 1 , . . . , D L ) of L independent datasets D ℓ , ℓ ∈ [L]
, where this function takes the general linearly-separable form

F k (D 1 , D 2 , . . . , D L ) (1)
f k,1 f 1 (D 1 ) + f k,2 f 2 (D 2 ) + . . . + f k,L f L (D L ) (2) = f k,1 W 1 + f k,2 W 2 + . . . + f k,L W L k ∈ [K], (3)
where in the above,

W ℓ = f ℓ (D ℓ ) ∈ F, ℓ ∈ [L]
is a so-called 'file' output, and where

f k,ℓ ∈ F, k ∈ [K], ℓ ∈ [L]
are the linear combination coefficients which belong, together with the entries of W ℓ , in some finite field F.

A. Phases

The model involves three phases, with the first being the demand phase, then the assignment and computation phase and then the transmission and decoding phase.

In the demand phase, each user k ∈ [K] sends the information of its desired function F k (.) to the master node, who then deduces the linear decomposition of this function according to [START_REF] Verbraeken | A survey on distributed machine learning[END_REF]. Then based on these K desired functions, during the assignment and computation phase, the master assigns some of the datasets to each server, who then proceeds to calculate the corresponding files W ℓ = f ℓ (D ℓ ) for their locally available datasets. Based on this assignment, each dataset D ℓ will be placed at all the servers in some chosen server set W ℓ .

In the assignment and computation phase, the master in accordance with the desired task functions assigns some datasets to each server, and then each server n ∈ [N ] computes all the computationally hard files from the locally-available datasets. To this end, we define the set of indices of the servers that dataset ℓ ∈ [L] has been assigned to as

W ℓ ⊂ [N ].
During the transmission phase, a linear combination of the locally available output files at the server is transmitted, where each server n ∈ [N ] transmits

z n ℓ∈[L] e n,ℓ W ℓ n ∈ [N ], (4)
in a single shot (slot) over a broadcast channel to a subset of users in

T n ∈ [K].
In the above, we can easily see that the encoding coefficients e n,ℓ , which are indeed determined by the master node, satisfy e n,ℓ = 0, ∀(n, ℓ)

∈ [N ] × [L], n / ∈ W ℓ .
1 Our setting incorporates the underlying assumption that the tasks performed at the servers substantially outweigh in computational complexity the basic linear operations that are performed at the different users and also we have assumed that each server node is connected to all of the users through a broadcast channel.

F k (.), each component subfunction W ℓ = f ℓ (D ℓ
) is evaluated at each server in W ℓ . Each server n transmits a linear combination zn (of the locally available files) to all users in Tn. This combination is defined by the coefficients e n,ℓ . Finally, to decode, each user k ∈ [K] linearly combines (based on decoding vectors d k ), its received signal from all the servers it is connected to. Decoding must produce for each user its desired function

F k (D 1 , . . . , D L ).
Finally during the decoding part, each user k linearly combines the received signals as follows

F ′ k n∈[N ] d k,n z n , (5)
for some decoding coefficients d k,n ∈ F, n ∈ [N ] determined again by the master node. Naturally d k,n = 0, ∀k / ∈ T n . Decoding is successful when

F ′ k = F k for all k ∈ [K].

B. Computation Cost

Remembering that |W ℓ | indicates the number of servers that compute a subfunction

W ℓ = f ℓ (D ℓ ), ℓ ∈ [L], our normalized computational cost metric takes the form γ max l∈[L] |W l | N , (6)
to represent the maximum fraction of all servers that must compute any subfunction. We wish to establish a lower bound on this computational cost.

III. PROBLEM FORMULATION

To formulate the problem we first define the below vectors,

w [W 1 , W 2 , . . . , W L ] ⊺ , (7)
e n [e n,1 , e n,2 , . . . , e n,L ] ⊺ , n

∈ [N ], (8)
E [e 1 , e 2 , . . . , e N ] ⊺ , (9)
d k [d k,1 , d k,2 , . . . , d k,N ] ⊺ , k ∈ [K], (10)
z [z 1 , z 2 , . . . , z N ] ⊺ , (11)
f k [f k,1 , f k,2 , . . . , f k,L ] ⊺ , k ∈ [K], (12)
f [F 1 , F 2 , . . . , F K ] ⊺ , (13)
f ′ [F ′ 1 , F ′ 2 , . . . , F ′ K ] ⊺ , (14)
F [f 1 , f 2 , . . . , f K ] ⊺ . (15)
Note that from (3), we have that

f = [f 1 , f 2 , . . . , f K ] ⊺ w, (16)
and then from (4) we conclude that

z = Ew = [e 1 , e 2 , . . . , e N ] ⊺ w, (17)
which indicates what each server transmits. This transmission is defined by the so-called encoding matrix E. Now from [START_REF] Ulukus | Private retrieval, computing and learning: Recent progress and future challenges[END_REF] we see that each user, after decoding, receives

F ′ k = d T k z, (18)
which in turn defines the set of all decoded signals

f ′ = [d 1 , d 2 , . . . , d K ] ⊺ z (19)
across all the users. This decoding is itself fully defined by the so-calleddecoding matrix

D [d 1 , d 2 , . . . , d K ] ⊺ ∈ F K×N . ( 20
)
Decoding is successful and correct if and only if

f = f ′ (21)
for any D ℓ , ℓ ∈ [L]. Directly from ( 16), ( 17), ( 19) into ( 21), we see that correctness holds -for any D ℓ , ℓ ∈ [L], i.e., for any w ∈ F L×1 -if and only if 2

DE = F. (22)
Noting that W l = sup(E(:, {l}) ⊺ ) and that |W l | = ω(E(: , {l})), we have that

max l∈[L] ω(E(:, l)) = max l∈[L] |W l | (23)
which simply tells us that our computational cost γ from (6) takes the form

γ = 1 N max l∈[L]
ω(E(:, l)).

We here provide a simple example to help clarify the setting and the notations.

A. Simple Example

As described in Figure 2, we consider the example of a system with a master node, N = 8 servers, K = 4 users, L = 6 datasets, and a field of size q = 7.

Let us assume that the users ask the following functions:

F 1 = 2f 1 (D 1 ) + 4f 2 (D 2 ) + 4f 3 (D 3 ) + 5f 4 (D 4 ) + 5f 5 (D 5 ), F 2 = 3f 1 (D 1 ) + 4f 2 (D 2 ) + 5f 3 (D 3 ) + 2f 4 (D 4 ) + 6f 5 (D 5 ) + 6f 6 (D 6 ), F 3 = 2f 1 (D 1 ) + 4f 2 (D 2 ) + 6f 3 (D 3 ) + 5f 4 (D 4 ) + 2f 5 (D 5 ), F 4 = 3f 1 (D 1 ) + 5f 2 (D 2 ) + 2f 4 (D 4 ) + 3f 5 (D 5 ) + f 6 (D 6 )
2 Since the master node does not know about W ℓ , ℓ ∈ where designing the scheme, to make sure that [START_REF] Li | Compressed coded distributed computing[END_REF] where F k , k ∈ [START_REF] Abadi | {TensorFlow}: A system for {Large-Scale} machine learning[END_REF] is defined in (3) (after which, the corresponding f k , w are respectively defined in ( 7) and ( 12)). Consequently from [START_REF] Soleymani | Analog lagrange coded computing[END_REF], our demand matrix takes the form

F =   2
In the assignment phase, the master allocates D 1 , D 2 , . . . , D 6 to the 8 servers according to

W 1 = {1, 2, 3, 5, 8}, W 2 = {1, 2, 3, 4, 6, 7}, (26)
W 3 = {1, 2, 3}, W 4 = {1, 4, 5, 7} (27)
so that for example dataset 3 resides at servers {1, 2, 3}, and where server 2 is assigned datasets D 1 , D 2 , D 3 , D 5 and thus has to compute

W 1 = f 1 (D 1 ), W 2 = f 2 (D 2 ), W 3 = f 3 (D 3 ), W 5 = f 5 (D 5 ).
A quick inspection shows that the normalized computation cost (cf. ( 6)) is equal to

γ = max l∈[6] |W l | 8 = 6/8. (29)
After computing their designated output files, each server n transmits z n as follows

z 1 = 2W 1 + 6W 2 + 3W 3 + W 4 + 2W 5 , (30)
z 2 = 4W 1 + 5W 2 + 2W 3 + 3W 5 , (31)
z 3 = W 1 + 2W 2 + W 3 + 2W 6 , (32)
z 4 = W 2 + 2W 4 + 4W 5 + W 6 , (33)
z 5 = 2W 1 + W 4 + 3W 5 + 2W 6 , (34)
z 6 = 2W 2 + 5W 5 + 3W 6 , (35)
z 7 = W 2 + 2W 4 + 4W 6 , (36)
z 8 = 2W 1 + 4W 5 + 5W 6 , (37)
corresponding to an encoding matrix (cf. ( 17)) of the form

E =        
2 6 3 1 2 0 4 5 2 0 3 0 1 2 1 0 0 2 0 1 0 2 4 1 2 0 0 1 3 2 0 2 0 0 5 3 0 1 0 2 0 4 2 0 0 0 4 5

        . ( 38
)
We can quickly verify (cf. ( 29)) that indeed max l∈ [START_REF] Jahani-Nezhad | Codedsketch: A coding scheme for distributed computation of approximated matrix multiplication[END_REF] ω(E(: , l))/8 = 6/8 = γ. Subsequently, the master asks each server n to send its generated z n to its designated receiving users, such that for each server, these user sets are:

T 1 = {2, 4}, T 2 = {1, 3}, T 3 = {3}, T 4 = {1, 2, 3, 4}, (39) T 5 = {1, 2, 3, 4}, T 6 = {1, 2}, T 7 = {1, 4}, T 8 = {4}, ( 40
)
where for example server 2 will transmit z 2 to users 1 and 3.

To decode, each user k ∈ [START_REF] Abadi | {TensorFlow}: A system for {Large-Scale} machine learning[END_REF] computes the linear combination

F ′ k as F ′ 1 = 2z 2 + 3z 4 + 4z 5 + 2z 6 + z 7 , (41)
F ′ 2 = 4z 1 + 2z 4 + z 5 + 3z 6 , (42)
F ′ 3 = 4z 2 + 5z 3 + 2z 4 + z 5 , (43)
F ′ 4 = 4z 1 + 2z 3 + z 4 + 2z 5 + 4z 7 + 5z 8 , (44)
A quick verification 3 reveals the correctness of decoding and that indeed F ′ k = F k for all k = 1, 2, 3, 4. For example, for the first user, we see that

F ′ 1 = 2z 2 + 3z 4 + 4z 5 + 2z 6 + z 7 = 2(4W 1 + 5W 2 + 2W 3 + 3W 5 ) + 3(W 2 + 2W 4 + 4W 5 + W 6 ) + 4(2W 1 + W 4 + 3W 5 + 2W 6 ) + 2(2W 2 + 5W 5 + 3W 6 ) + (W 2 + 2W 4 + 4W 6 ) = 2W 1 + 4W 2 + 4W 3 + 5W 4 + 5W 5 + 0W 6
which indeed matches F 1 . In this example, each user recovers their desired function, with a corresponding normalized computational cost γ = 3/4 . This has just been an example to illustrate the setting. The effort to find a solution with reduced computation cost, follows in the section below.

IV. A CODING THEORETIC APPROACH

This section establishes a conceptual bridge between our problem and coding theory.

Our first aim is to decompose F into F = DE under a constrained computation cost, i.e., under a sparsity constraint on E. For E ℓ E(:, l) and F ℓ F(:, l) denoting the lth column of D and E respectively, we can rewrite our decomposition as

DE ℓ = F ℓ , ℓ ∈ [L]. (46)
3 Let us recall that each decoded symbol takes the form

F ′ k = d ⊺ k z where d ⊺ k is the kth row of D, and where z = [z 1 z 2 • • • z N ] T .
If we viewed D ∈ F K×N as a parity check matrix of a code C ⊂ F N , referred to as H C , then we could view E ℓ ∈ F N as an arbitrary error pattern, and F ℓ ∈ F K as the corresponding syndrome. Since we wish to sparsify E ℓ , we are interested in E ℓ being the minimum-weight coset leader whose syndrome is F ℓ . This is simply the output of the minimum-distance syndrome decoder. To get a first handle on the weights of E ℓ , we can refer to the theory of covering codes which bounds the weights of coset leaders.

To derive our results for any L ≤ q K4 , we will first need to explore certain coding-theoretic properties, and for this reason we will transition to the traditional coding-theoretic notation which speaks of an n-length code C of rate k/n, where for us n = N and k = N -K. The parity check matrix H C ∈ F (n-k)×n will generally be associated to our decoding matrix D ∈ F K×N , the received (or error) vectors x ∈ F n will be associated to the encoding vectors E ℓ ∈ F N , and its syndrome s x ∈ F n-k (or just s depending on the occasion) will be associated to F ℓ ∈ F K . As suggested before, when we write C D (or C H ) we will refer to the code whose parity check matrix is D (or H).

V. RESULTS

As a first step, we extend the concept of covering codes to the following class of codes,

Definition 1. Let ρ ∈ (0, 1]. We say that a set X ⊆ F n is ρ-covered by a code C ⊆ F n iff d(x, C) ≤ ρn, ∀x ∈ X ( 47
)
in which case, we say that C is a (ρ, X )-partial covering code.

Naturally when X = F n , the code is simply the traditional covering code.

We are now able to link partial covering codes to our distributed computing problem via the following thheorem, . This e min is the coset leader associated to syndrome F(:, l).

Since though DE = F, we also have that DE(:, l) = F(:, l). Since E(:, l) and e min are in the same coset (of the same syndrome F(:, l)), and e min is the minimum-weight coset leader, we can conclude that ω(E(:, l)) ≥ e min . Thus the assumption that ω(e min ) > ρn implies that ω(E(:, l)) > ρn which contradicts the complexity requirement that ω(E(: , l)) ≤ ρn from ( 23) and ( 6) and. Thus if D does not correspond to a partial covering code that covers X F,D , the complexity constraint is violated.

On the other hand, recalling that C D is a partial covering code for X , means that for any x ∈ X then d(x, C D ) ≤ ρn. For the same x ∈ X , let c min be again its closest codeword, and let e min = xc min , where again by definition (of the partial covering code), ω(e min ) ≤ ρn. Since, like before, De min = F(:, l) for some l ∈ [L], then we simply set E(:, l) = e min whose weight is sufficiently low to guarantee the complexity constraint. We recall that for each F(:, l), this coset leader E(:, l) = e min can be found using the minimumdistance syndrome decoder.

We can now proceed with the characterisation of the converse.

Theorem 2. For the distributed linearly separable problem with K users, N servers and any number L of sub-functions, the optimal computation cost is lower bounded as

γ ≥ H -1 q ( log q (L) N ). (49)
Proof. Let γ = ρ. To prove our theorem, we first need to prove that a code

C(k, n) ⊆ F n q can ρn-cover a set X ⊆ F n q of size |X | = q k L, only if log q (L) ≤ log q (V q (n, ρ)). (50)
To prove (50) we simply apply the union bound after noting that each of the q k codewords can only ρn-cover V q (n, ρ) vectors. This implies that

Lq k ≤ V q (n, ρ)q k , (51)
which in turn directly yields (50) after applying the logarithm on both sides of the inequality.

To complete the proof, we first assign the above set X to be the X in Theorem 1, where now X ⊇ X F,D (cf. (48)). We know from above that |X | = Lq k . We can also see that |X F,D | = Lq k because, by definition (cf. (48)), this set X F,D is the reverse image -to the ambient space F n -of the all the syndromes corresponding to F (i.e., the reverse image of all the columns of F). Thus now we know that X = X F,D . Then by substituting N = n, K = n -k, we see that log q (L) ≤ log q (V q (N, ρ)). By applying the well known (asymptotically tight) bound q N Hq(ρ)-o(N ) ≤ V q (N, ρ) ≤ q N Hq(ρ) , we can conclude that, in the limit of large L, log q (L) ≤ N H q (ρ), thus proving that H -1 q ( log q (L) N

) ≤ ρ.

Remark 1. We see that the lower bound is only dependent on log q (L) N

and if L = q K , the result reduces to sphere-covering bound, which has been already known in the literature.

VI. CONCLUSION

We have explored the computational cost of the multiuser linearly separable function computation setting, which is a broad setting that captures several distributed computing problems such as the distributed gradient coding problem [START_REF] Tandon | Gradient coding: Avoiding stragglers in distributed learning[END_REF], the distributed linear transform problem [START_REF] Dutta | Short-dot: Computing large linear transforms distributedly using coded short dot products[END_REF], and the distributed matrix multiplication and the distributed multivariate polynomial computation problems [START_REF] Yu | Straggler mitigation in distributed matrix multiplication: Fundamental limits and optimal coding[END_REF], [START_REF] Jia | Cross subspace alignment codes for coded distributed batch computation[END_REF], among others. The work established a novel relationship between our problem and coding theory, and provided an algebraic converse that reveals that the normalized computational cost -in the form of the maximum fraction of all servers that must compute any subfunction -is lower bounded as γ ≥ H -1 q ( log q (L) N

) in the limit of large L and fixed log q (L)/N .

In the journal version of this work which is available online now on https://arxiv.org/abs/2206.11119, we have provided achievable schemes for the same setting and in detail insights on this problem.

  This work was supported by the European Research Council (ERC) through the EU Horizon 2020 Research and Innovation Program under Grant 725929 (Project DUALITY)

Fig. 1 .

 1 Fig.1. The K-user, N -server Linearly Separable Computation setting. After each user informs the master of its desired function F k (.), each component subfunction W ℓ = f ℓ (D ℓ ) is evaluated at each server in W ℓ . Each server n transmits a linear combination zn (of the locally available files) to all users in Tn. This combination is defined by the coefficients e n,ℓ . Finally, to decode, each user k ∈ [K] linearly combines (based on decoding vectors d k ), its received signal from all the servers it is connected to. Decoding must produce for each user its desired function F k (D 1 , . . . , D L ).

W 5 =

 5 {1, 2, 4, 5, 6, 8}, W 6 = {3, 4, 5, 6, 7, 8}

Theorem 1 .

 1 A solution to the multi-user linearly separable problem DE = F with complexity γ = 1 N max ℓ∈[L]ω(E(:, l))exists if and only if D is the parity check matrix to a (γ, X )partial covering code C D for some existing set X ⊇ X F,D , where X F,D is defined as,X F,D {x ∈ F N |Dx = F(:, l), for some ℓ ∈ [L]}. (48)With such D in place, each E(:, l) is the output of the minimum-distance syndrome decoder of C D for syndrome F(:, l).Proof. To first prove that the complexity constraint indeed requires D to correspond to a partial covering code that covers X , let us assume that D does not have this property, and thus there exists an x ∈ X such that d(x, C D ) > ρn.Let c min be the closest codeword to x in the sense that d(x, c min ) = d(x, C D ). Now let e min = xc min and note, directly from the above assumption, that ω(e min ) > ρn. Naturally Dx = D(e min + c min ) = De min by virtue of the fact that D is the parity check matrix of C D . Since x ∈ X , we know that ∃ ℓ ∈ [L] : Dx = F(:, l) which directly means that ∃ ℓ ∈ [L] : De min = F(:, l)

There are at most L = q K possible distinct columns for F. We naturally assume the worst case where the columns of F are different.


Adeline Kerner

Régine Vignes-Lebbe

 We analyze genera discrimination focusing on the different sections (graph 1).  ~44% not discriminated in transverse section.  ~90 % discriminated with a transverse and a longitudinal sections.  >95% discriminated with an oblique section.

 A single oblique section is the most efficient for discrimination: all structures are visible but their shapes are difficult to recognize.  Traditionally, it is advisable to use a transverse AND a longitudinal section: discrimination is lower efficient but characters interpretation is easier.  Practically, longitudinal sections are difficult to find.  New recommendation to prepare samples:

When a longitudinal section is not seen, redirect the transverse section to make an oblique.



  Fossils are examined through oriented thin sections.  Transverse sections are easily found contrary to longitudinal ones.  What is the influence of the thin section orientation during the identification process?  Archaeocyathan knowledge base comprises the 307 valid genera described with 120 descriptors.  Checkbase (Xper² tool) compares all couples of genera (fig.2A&C)  Then each genus is assigned to a single category: not discriminated, may be discriminated, discriminated (fig.2B)

Graph 1 :Figure

 1 Figure 1: Homepage of archaeocyathan website