Anton A Ivanov

Clément Falaise

Alexandra A Shmakova

Nathalie Leclerc

Stéphane Cordier

Yann Molard

Yuri V Mironov

Michael A Shestopalov

Pavel A Abramov

Maxim N Sokolov

Mohamed Haouas

Emmanuel Cadot

Cyclodextrin-Assisted Hierarchical Aggregation of Dawson-type Polyoxometalate in the Presence of {Re6Se8} Based Clusters

CLUSPOM salt' was obtained from aqueous solution of the cationic cluster and the polyoxometalate. In this solid, the arrangement between the polymetallic building blocks is mainly governed by long-range coulombic interactions. In presence of γ-CD, the Dawson anion and the cationic cluster are assembled differently, forming a hierarchical supramolecular solid, K2[{Re6Se8}(H2O)6]2{[P2W18O62]@2γ-CD}•42H2O, where the organic macrocycle acts as a ditopic linker between the inorganic building blocks. In such an edifice, the short-range molecular recognition dominates the long-range coulombic interactions leading to a specific 3D organization. Interestingly, the assembling of anionic [P2W18O62] 6-with the anionic rhenium cluster [{Re6Se8}(CN)6] 4-is also achieved with γ-CD despite the repulsive forces between the nano-sized anions. The resulting solid, K10{[{Re6Se8}(CN)6]@2γ-CD}[P2W18O62]•33H2O, is built from 1:2 inclusion complexes {[{Re6Se8}(CN)6]@2γ-CD} 4-linked by a POM unit interacting with the exterior wall of the organic macrocycle. Multinuclear NMR and small-angle X-ray scattering (SAXS) investigations support supramolecular pre-organization in aqueous solution prior to crystallization.

INTRODUCTION

Discrete polymetallic entities exhibit prodigious structural diversity, chemical composition and properties, making them particularly appealing as preformed building units for tailoring innovative materials. [START_REF] Long | Polyoxometalates: Building Blocks for Functional Nanoscale Systems[END_REF][2][START_REF] Molard | Clustomesogens: Liquid Crystalline Hybrid Nanomaterials Containing Functional Metal Nanoclusters[END_REF][4][START_REF] Cadot | A Building Block Strategy to Access Sulfur-Functionalized Polyoxometalate Based Systems Using {Mo2S2O2} and {Mo3S4} as Constitutional Units, Linkers or Templates[END_REF] The crystal engineering methodology taking benefits from the supramolecular recognition processes between complementary nano-sized entities attracts considerable attention due to its capacity to create synergies relevant for applications in optic, catalysis, separation, gas adsorption, etc. [START_REF] Pinkard | Molecular Clusters: Nanoscale Building Blocks for Solid-State Materials[END_REF][7][START_REF] Roy | Nanoscale Atoms in Solid-State Chemistry[END_REF][START_REF] Xiao | Luminescence Responsive Charge Transfer Intercluster Crystals[END_REF][START_REF] Uchida | Frontiers and Progress in Cation-Uptake and Exchange Chemistry of Polyoxometalate-Based Compounds[END_REF][START_REF] Turkiewicz | Assembling Hierarchical Cluster Solids with Atomic Precision[END_REF] These supramolecular mixed polymetallic-scaffolds based solids are generally built from large cationic and anionic complexes, and therefore can be viewed as salts of complexes wherein the anionic partners are spatially organized to minimize coulombic repulsions, as observed in many examples. [12][START_REF] Gruber | Salt-like Structures of Oligomeric Gold Complexes and Polyoxometalates[END_REF][START_REF] Yoshinari | A 1:1 Intercluster Compound Consisting of +6 and -6 Charged Rh III 4 Zn II 4 Octanuclear Cations and Anions with Aminothiolate Ligands[END_REF][START_REF] Falaise | From Aqueous Speciation to Supramolecular Assembly in Alkaline Earth-Uranyl Polyoxometalates[END_REF][START_REF] Kawamoto | Amphiphilic Guest Sorption of K2[Cr3O(OOCC2H5)6(H2O)[END_REF][START_REF] Uchida | A Breathing Ionic Crystal Displaying Selective Binding of Small Alcohols and Nitriles: K3[Cr3O(OOCH)6(H2O)3][α-SiW12O40]⋅16 H2O[END_REF][START_REF] Liu | The Transformation of Polyoxometalates in the Formation of Intercluster Compound [Ag41(α-SiW10O37)[END_REF] However, the structural organization obeys sometimes to more intricate rules. For instance, aggregates built from equally charged polymetallic entities are observed in few materials. [START_REF] Gruber | Structure-Directing Forces in Intercluster Compounds of Cationic [Ag14(CCtBu)12Cl] + Building Blocks and Polyoxometalates: Long-Range versus Short-Range Bonding Interactions[END_REF][START_REF] Gruber | Packing Requirements and Short-Range Interactions as Structure-Directing Forces in the Intercluster Compounds Based on Silver Clusters[END_REF][START_REF] Schulz-Dobrick | Structure-Directing Effects in the Supramolecular Intercluster Compound[END_REF] This underlines that short-range supramolecular interactions, such as hydrogen bonds or van der Waals contacts must be taken into account as additional forces that drive the assembly of the preformed inorganic units in the solid-state. Therefore, mastering the conglomerate of collective weak forces during the aggregation processes could offer novel opportunities in the crystal engineering of mixed cluster solids such as, for instance, i) using neutral polymetallic entities, [START_REF] Turkiewicz | Assembling Hierarchical Cluster Solids with Atomic Precision[END_REF] ii) assembling cationic (or anionic) building blocks together or even better iii) programming the assembly into predicted architectures.

Among the library of well-defined metal-containing aggregates, polyoxometalates (POMs), that correspond to anionic metal-oxo clusters of mainly group VI elements in their higher oxidation states (e.g. Mo 6+ and W 6+ ), exhibit striking redox 4 and catalytic properties [START_REF] Wang | Recent Advances in Polyoxometalate-Catalyzed Reactions[END_REF] , and therefore, are relevant components to design advanced materials. Consequently, POMs are widely used in the preparation of large anion-cation materials, and have been combined with Al(III) polycations, 12,[START_REF] Son | New Ionic Crystals of Oppositely Charged Cluster Ions and Their Characterization[END_REF][24][START_REF] Son | Polymorphism in Intercluster Salt System: Two Crystal Structures of[END_REF][START_REF] Mizuno | Control of Polymorphisms and Functions in All-Inorganic Ionic Crystals Based on Polyaluminum Hydroxide and Polyoxometalates[END_REF] trinuclear Ru(III) cations, [START_REF] Ibrahim | Large Cation-Anion Materials Based on Trinuclear Ruthenium(III) Salts of Keggin and Wells-Dawson Anions Having Water-Filled Channels[END_REF] 3d-metal based macrocations [START_REF] Kawamoto | Amphiphilic Guest Sorption of K2[Cr3O(OOCC2H5)6(H2O)[END_REF][START_REF] Uchida | A Breathing Ionic Crystal Displaying Selective Binding of Small Alcohols and Nitriles: K3[Cr3O(OOCH)6(H2O)3][α-SiW12O40]⋅16 H2O[END_REF][START_REF] Uchida | Zeotype Ionic Crystal of Cs5[Cr3O(OOCH)6(H2O)3][α-CoW12O40[END_REF][START_REF] Jiang | Channel-Selective Independent Sorption and Collection of Hydrophilic and Hydrophobic Molecules by Cs2[Cr3O(OOCC2H5)[END_REF][START_REF] Tagami | Size-Selective Sorption of Small Organic Molecules in One-Dimensional Channels of an Ionic Crystalline Organic-Inorganic Hybrid Compound Stabilized by π-π Interactions[END_REF][START_REF] Uchida | Zeotype Organic-Inorganic Ionic Crystals: Facile Cation Exchange and Controllable Sorption Properties[END_REF][START_REF] Eguchi | Inverse and High CO2/C2H2 Sorption Selectivity in Flexible Organic-Inorganic Ionic Crystals[END_REF][START_REF] Seino | Reduction-Induced Highly Selective Uptake of Cesium Ions by an Ionic Crystal Based on Silicododecamolybdate[END_REF][START_REF] Hitose | Rapid Uptake/Release of Cs+ in Isostructural Redox-Active Porous Ionic Crystals with Large-Molecular-Size and Easily Reducible Dawson-Type Polyoxometalates as Building Blocks[END_REF] , gold [START_REF] Gruber | Salt-like Structures of Oligomeric Gold Complexes and Polyoxometalates[END_REF][START_REF] Schulz-Dobrick | Structure-Directing Effects in the Supramolecular Intercluster Compound[END_REF][START_REF] Schulz Dobrick | Supramolecular Intercluster Compounds Consisting of Gold Clusters and Keggin Anions[END_REF][START_REF] Schulz Dobrick | Characterization of Gold Clusters by Crystallization with Polyoxometalates: The Intercluster Compounds[END_REF][START_REF] Schulz Dobrick | Synthesis and Characterization of Intercluster Compounds Consisting of Various Gold Clusters and Differently Charged Keggin Anions[END_REF][START_REF] Nomiya | Intercluster Compound between a Tetrakis{triphenylphosphinegold(I)}oxonium Cation and a Keggin Polyoxometalate (POM): Formation during the Course of Carboxylate Elimination of a Monomeric Triphenylphosphinegold(I) Carboxylate in the Presence of POMs[END_REF][START_REF] Yoshida | Two Types of Tetranuclear Phosphanegold(I) Cations as Dimers of Dinuclear Units, [{(Au{P(p-RPh)3})2(μ-OH)}2] 2+ (R = Me, F), Synthesized by Polyoxometalate-Mediated Clusterization[END_REF][START_REF] Yoshida | Various Oxygen-Centered Phosphanegold(I) Cluster Cations Formed by Polyoxometalate (POM)-Mediated Clusterization: Effects of POMs and Phosphanes[END_REF] and silver [START_REF] Xiao | Luminescence Responsive Charge Transfer Intercluster Crystals[END_REF][START_REF] Gruber | Structure-Directing Forces in Intercluster Compounds of Cationic [Ag14(CCtBu)12Cl] + Building Blocks and Polyoxometalates: Long-Range versus Short-Range Bonding Interactions[END_REF][START_REF] Gruber | Packing Requirements and Short-Range Interactions as Structure-Directing Forces in the Intercluster Compounds Based on Silver Clusters[END_REF] clusters. The structural organization of these solids results from a sophisticated balance between long-range and short-range bonding interactions, making any predictions of the crystalline arrangement highly challenging.

Gamma-cyclodextrin (γ-CD) is a water-soluble macrocyclic polysaccharide constituted of 8 glucopyranose units, exhibiting a rim lined by the eight hydroxymethylene groups (primary face) and a larger rim lined by sixteen hydroxyl groups (secondary face). Each γ-CD face is able to interact with polyoxometalates or other ionic polynuclear entities in aqueous solution leading to highly stable supramolecular adducts. [START_REF] Moussawi | Cationic Cluster, and γ-Cyclodextrin: From Primary Interactions to Supramolecular Hybrid Materials[END_REF][START_REF] Wu | Complexation of Polyoxometalates with Cyclodextrins[END_REF][START_REF] Assaf | Water Structure Recovery in Chaotropic Anion Recognition: High-Affinity Binding of Dodecaborate Clusters to γ-Cyclodextrin[END_REF][START_REF] Falaise | Probing Dynamic Library of Metal-Oxo Building Blocks with γ-Cyclodextrin[END_REF][START_REF] Ivanov | Host-Guest Binding Hierarchy within Redox-and Luminescence-Responsive Supramolecular Self-Assembly Based on Chalcogenide Clusters and γ-Cyclodextrin[END_REF][START_REF] Ivanov | Tuning the Chaotropic Effect as an Assembly Motif through One-Electron Transfer in a Rhenium Cluster[END_REF][START_REF] Ivanov | Size-Exclusion Mechanism Driving Host-Guest Interactions between Octahedral Rhenium Clusters and Cyclodextrins[END_REF][START_REF] Moussawi | Nonconventional Three-Component Hierarchical Host-Guest Assembly Based on Mo-Blue Ring-Shaped Giant Anion, γ-Cyclodextrin, and Dawson-Type Polyoxometalate[END_REF] The main driving force that governs this recognition process has been identified as resulting from a water structure recovery process induced by the chaotropic nature of the solvated macro-ions. [START_REF] Assaf | The Chaotropic Effect as an Assembly Motif in Chemistry[END_REF] We reported recently that γ-CD can act as a ditopic supramolecular linker to associate in a predictive way the Dawson-type POM [P 2 W 18 O 62 ] 6-and the tantalum octahedral cluster [{Ta 6 Br 12 }(H 2 O) 6 ] 2+ to give the first example of a CLUSPOM compound, obtainable either as hydrogel or highly ordered crystal. [START_REF] Moussawi | Cationic Cluster, and γ-Cyclodextrin: From Primary Interactions to Supramolecular Hybrid Materials[END_REF] This auspicious result opened the possibility to combine an electron-rich cluster (CLUS; as donor) with an electron-poor polyoxometalate (POM; as acceptor), and encouraged us to explore further the crystal engineering of CLUSPOM solids using other metal atom cluster types like the face-capped cluster-based units bearing different charge and carrying different physical properties than [{Ta 6 Br 12 }(H 2 O) 6 ] 2+ . Herein, we report on the crystal engineering of novel supramolecular CLUSPOM solids obtained from aqueous mixtures containing γ-CD, Dawson anion [P 2 W 18 O 62 ] 6-and face-capped [{Re 6 Se 8 }L 6 ] m+/n-based cluster units (Figure 1). It is worth noting that, the cluster core {Re 6 Se 8 } can be derived either as the aqua cationic ([{Re 6 Se 8 }(H 2 O) 6 ] 2+ ) or as the cyano anionic ([{Re 6 Se 8 }(CN) 6 ] 4-) building units. This allows investigations upon the role of attractive and repulsive long-range coulombic forces on the CLUSPOM architectures involving γ-CD.

Moreover the [{Re 6 Se 8 }L 6 ] m+/n-units exhibit the best luminescent properties among the rhenium chalcogenide cluster series making them particularly attractive to prepare innovative multifunctional solids. [START_REF] Cordier | Advances in the Engineering of Near Infrared Emitting Liquid Crystals and Copolymers, Extended Porous Frameworks, Theranostic Tools and Molecular Junctions Using Tailored Re6 Cluster Building Blocks[END_REF] Prior to evaluate the role of the γ-CD in the assembling process, we studied the co-crystallization of [P 2 W 18 O 62 ] 6-(noted {P 2 W 18 }) and [{Re 6 Se 8 }(H 2 O) 6 ] 2+ (noted {Re 6 ca }).

This chemical system leads to the isolation of an original CLUSPOM material where the polymetallic units are packed following the NiAs structural arrangement. Then, we show that the presence of γ-CD, modifies the crystal formation process producing a hierarchical organization wherein two cationic clusters sandwiched a POM unit through two distinct γ-CDs acting as ditopic linker. Using the anionic [{Re 6 Se 8 }(CN) 6 ] 4-species (noted {Re 6 an }) instead of the cationic derivative, another supramolecular CLUSPOM solid was also isolated, highlighting that γ-CD is even able to maintain inter-cluster supramolecular interactions despite non favorable repulsive anion-anion coulombic forces. Finally, we investigate the pre-aggregation processes occurring in aqueous media by multinuclear 1D NMR, DOSY NMR and small-angle x-ray scattering (SAXS) techniques.

EXPERIMENTAL SECTION

Synthesis

The compounds described herein were synthesized from aqueous mixtures of γ-CD H, 2.4; N, 0.9. EDS shows Re : Se : K : P : W ratio = 6 : 7.9 : 9.8 : 2.1 : 18.0. TGA revealed a weight loss of about 6.0% from 50 to 150 °C (the calculated weight loss of 33 H2O is 6.0%).

Single Crystal X-ray Diffraction

Crystals of compounds {Re6 ca }{P2W18}, {Re6 ca }@CD, {Re6 ca }2{P2W18}@2CD and {Re6 an }@2CD{P2W18}, were selected under polarizing optical microscope and glued in paratone oil to prevent any loss of crystallization water. X-ray intensity data were collected at low temperature (T = 200(2) K) on Bruker D8 VENTURE diffractometer equipped with a PHOTON 100 CMOS bidimensional detector using a high brilliance IμS microfocus X-ray MoKα monochromatized radiation (λ = 0.71073 Å). Data reduction was accomplished using SAINT V7.53a. The substantial redundancy in data allowed a semi-empirical absorption correction (SADABS V2.10) to be applied, on the basis of multiple measurements of equivalent reflections.

Using Olex2, [START_REF] Dolomanov | OLEX2: A Complete Structure Solution, Refinement and Analysis Program[END_REF] the structure was solved with the ShelXT 55 structure solution program using Intrinsic Phasing and refined with the ShelXL 56 refinement package using Least Squares minimization. The remaining non-hydrogen atoms were located from Fourier differences and were refined with anisotropic thermal parameters. Positions of the hydrogen atoms of γ-CD were refined in the calculated positions. In the compound {Re6 an }@2CD{P2W18}, numerous free water molecules (H2O) as well as the counter cations (K + ) located inside the voids are disordered.

Thereby the contribution of solvent-electron density was removed using the SQUEEZE routine in PLATON, producing a set of solvent-free diffraction intensities. A disorder of the rhenium clusters on two positions (occupancy 50% / 50%) is observed in {Re6 an }@2CD{P2W18} (details are given in the Supporting Information, see Figure S2). Crystallographic data for single-crystal Xray diffraction studies are summarized in Tables S1-4 and the bond lengths of the inorganic building blocks are reported in Tables S5-11. CIF files can be obtained free of charge from the Cambridge Crystallographic Data Centre via https://www.ccdc.cam.ac.uk. Deposit number:

{Re6 ca }{P2W18} : 1995746, {Re6 ca }@CD : 1995747, {Re6 ca }2{P2W18}@2CD : 1995749, and {Re6 an }@2CD{P2W18} : 1995748.

NMR studies.

All solution NMR spectra were measured in D2O at 27 °C. 1 H, [START_REF] Uchida | Zeotype Organic-Inorganic Ionic Crystals: Facile Cation Exchange and Controllable Sorption Properties[END_REF] P and 77 Se NMR spectra were obtained on a Bruker Avance 400 spectrometer at Larmor frequencies of 400.1, 162.0, and 77.3

MHz respectively, while 183 W NMR on a Bruker Avance 500 spectrometer at Larmor frequency of 20.8 MHz. Translational diffusion measurements were performed using Bruker's "ledbpgs2s" stimulated echo DOSY pulse sequence including bipolar and spoil gradients. Apparent diffusion coefficients were obtained using an adapted algorithm based on the inverse Laplace transform and maximum entropy. Chemical shifts are reported relative to standards: Me4Si (CDCl3, 1 wt. %) for 1 H, H3PO4 for 31 P, Me2Se for 77 Se, and Na2WO4 (D2O, 1 M) for 183 W. [START_REF] Harris | NMR Nomenclature. Nuclear Spin Properties and Conventions for Chemical Shifts (IUPAC Recommendations 2001)[END_REF] Small Angle X-ray Scattering.

SAXS experiments were performed at the SWING beamline of the SOLEIL synchrotron (Saint-Aubin, France). Measurements were carried out using a fixed energy of 12 keV. Samples were inserted in glass capillaries (Ø = 1.5 mm), the exposition time was fixed to 1000 ms and scattering patterns were recorded on EIGER-4M (Dectris) camera. Simulated scattering curves of polymetallic unit as in their inclusion complexes with CD were generated using SolX, [START_REF] Zhang | Probing Protein Fine Structures by Wide Angle Solution X-Ray Scattering[END_REF] utilizing structural files (xyz).

Infrared spectroscopy.

Fourier Transform Infrared (FT-IR) spectra were recorded on a 6700 FT-IR Nicolet spectrophotometer, using diamond ATR technique. The spectra were recorded on non-diluted compounds in the range 400-4000 cm -1 . The IR spectra are provided in Supporting Information (see Figure S5-6).

Energy-Dispersive X-ray Spectroscopy.

EDS measurements were performed using a SEM-FEG (Scanning Electron Microscope enhanced by a Field Emission Gun) equipment (JSM 7001-F, Jeol). The measurements were acquired with a SDD XMax 50 mm 2 detector and the Aztec (Oxford) system working at 15 kV and 10 mm distance. The quantification (P, W, Se, Re, K, Na) is realized with the standard library provided by the constructor using Lα lines.

Thermogravimetric analysis (TGA).

Analysis was performed on Mettler Toledo TGA/DSC 1, STARe System apparatus under oxygen flow (50 mL min -1 ) at a heating rate of 5°C min -1 up to 600°C.

RESULTS AND DISCUSSION

Formation of 'CLUSPOM salt'

Before any investigation in the presence of γ-CD, we attempted the co-crystallization of Dawson type POM with cationic aqua complex [{Re6Se8}(H2O)6] 2+ . Evaporation of an equimolar mixture of [P2W18O62] 6-and [{Re6Se8}(H2O)6] 2+ in weakly acidic condition yields large yellow hexagonal crystals of K4[{Re6Se8}(H2O)6][P2W18O62]•25H2O ({Re6 ca }{P2W18}). X-ray structural analysis reveals a 'CLUSPOM salt' crystallizing in the centrosymmetric space group P63/m.

Briefly, this compound consists of an assembly of Dawson type anion and Re6 cluster cation generating the nickeline structure type (NiAs) often observed in transition-metal chalcogenides and pnictogenides. [START_REF] Alsén | Röntgenographische Untersuchung Der Kristallstrukturen von Magnetkies, Breithauptit, Pentlandit, Millerit Und Verwandten Verbindungen[END_REF] In such an arrangement, the gravity center of the [P2W18O62] 6-ion lies on the As site while the [{Re6Se8}(H2O)6] 2+ cation is located on the Ni site (Figure 2). Structural analysis reveals that the geometrical parameters of the polymetallic units correspond to those usually observed in their salts. The packing of the polymetallic units can be described as a hexagonal packing of the [{Re6Se8}(H2O)6] 2+ cation, forming symmetrically equivalent layer along the sixfold axis. The [P2W18O62] 6-units are located between these layers of octahedral clusters and occupy alternately one half of the trigonal prismatic sites. Consequently, the polyoxometalate unit is surrounded by six cationic clusters in the corners of a trigonal prism, while the cluster is surrounded by six Dawson anions forming an octahedron. The distance between the gravity center of cationic and anionic unit is about 10.7 Å and the shortest distance between these two molecular units is 3.01 Å corresponding to a very weak hydrogen bond between the terminal aqua ligand of the rhenium cluster and the terminal oxygen atom of the polyoxometalate (Re-

OH2•••O t =W).
Herein, the contribution of the short-range interaction between the surfaces of the building blocks appears limited. Therefore, we conclude that the 3D organization is mainly driven by long-range coulombic interactions between large [P2W18O62] 6-anions and [{Re6Se8}(H2O)6] 2+ cations and by the intrinsic high symmetry of the polymetallic building blocks, which is D3h and Oh for [P2W18O62] 6-and [{Re6Se8}(H2O)6] 2+ , respectively. The charge balance of {Re6 ca }{P2W18} is compensated by four potassium cations (confirmed by EDS analysis).

Crystallographic data refinement indicates that disordered K + ions with 33% occupancy (more details in the Supporting Information), are surrounded by free water molecules, terminal aqua ligands of the rhenium cluster or oxygen atoms of the POM unit.

Binary systems involving γ-CD

Predicting the self-assembly process between CD and two distinct building blocks involves first to identify the rules governing the binary systems in the presence of γ-CD . [START_REF] Moussawi | Cationic Cluster, and γ-Cyclodextrin: From Primary Interactions to Supramolecular Hybrid Materials[END_REF] In our previous reports on the binary aqueous systems [P2W18O62] 6-/γ-CD and [{Re6Se8}(CN)6] 4-/γ-CD, we have shown that γ-CD binds both these polymetallic units with similar association constant (about 10 3 M -1 ), but interacts with different rims (Figure 3). [START_REF] Moussawi | Cationic Cluster, and γ-Cyclodextrin: From Primary Interactions to Supramolecular Hybrid Materials[END_REF][START_REF] Ivanov | Host-Guest Binding Hierarchy within Redox-and Luminescence-Responsive Supramolecular Self-Assembly Based on Chalcogenide Clusters and γ-Cyclodextrin[END_REF] The Dawson anion assembles through the smaller γ-CD face giving variable stoichiometry in the solid-state (from 1 to 3 CDs per POM unit) (Figure 3b). [START_REF] Moussawi | Cationic Cluster, and γ-Cyclodextrin: From Primary Interactions to Supramolecular Hybrid Materials[END_REF][START_REF] Moussawi | Nonconventional Three-Component Hierarchical Host-Guest Assembly Based on Mo-Blue Ring-Shaped Giant Anion, γ-Cyclodextrin, and Dawson-Type Polyoxometalate[END_REF] Oppositely, the [{Re6Se8}(CN)6] 4-anion interplays with γ-CD through the larger rim of the macrocycle, forming 1:2 inclusion complex (Figure 3c). [START_REF] Ivanov | Host-Guest Binding Hierarchy within Redox-and Luminescence-Responsive Supramolecular Self-Assembly Based on Chalcogenide Clusters and γ-Cyclodextrin[END_REF] The binary system [{Re6Se8}(H2O)6] 2+ /γ-CD has been also investigated, revealing very weak binding constant according to 1 H NMR data, however no structural data was reported. [START_REF] Abramov | Supramolecular Adduct of γ-Cyclodextrin and [{Re6Q8}(H2O)6] 2+ (Q=S, Se)[END_REF] The stability of inclusion complexes involving {Re6Se8} clusters and γ-CD strongly depends on both the chaotropicity of the cluster and the host-guest size-matching. [START_REF] Ivanov | Tuning the Chaotropic Effect as an Assembly Motif through One-Electron Transfer in a Rhenium Cluster[END_REF][START_REF] Ivanov | Size-Exclusion Mechanism Driving Host-Guest Interactions between Octahedral Rhenium Clusters and Cyclodextrins[END_REF] The cationic cluster (V = 350 Å 3 ) is probably too small to fit well with γ-CD cavity, in contrast to the bigger anionic cluster (V = 420 Å 3 ). To fill the lack of knowledge about the arrangement of [{Re6Se8}(H2O)6] 2+ and γ-CD in the solid-state, we successfully crystallized, by evaporation from a solution with both components, the 1:1 supramolecular adduct {[{Re6Se8}(H2O)6]@ γ-CD}(NO3)2•10H2O, notated as {Re6 ca }@CD. This compound is isostructural to that previously reported with the sulphur containing cluster [{Re6S8}(H2O)6] 2+ . [START_REF] Abramov | Supramolecular Adduct of γ-Cyclodextrin and [{Re6Q8}(H2O)6] 2+ (Q=S, Se)[END_REF] Complete structural description is included in supplementary information (Fig. S3 and S4). In short, the crystal structure consists of a 1:1 hostguest complex, where the cationic cluster is entrapped in the centre of the larger rim of the tori host (Figure 3d). The closest host-guest distance (about 3.2 Å) is observed between four water molecules of the cluster unit and four hydroxo groups located on the secondary face of the macrocyclic host.

Ternary systems involving the cationic cluster

The results obtained on binary systems give fundamental insights to construct, in a predictive way, novel CLUSPOM materials based on the combination of the three components namely the γ-CD, the cationic octahedral cluster, and the Dawson-type POM. Actually, because [P2W18O62] 6- and [{Re6Se8}(H2O)6] 2+ interact with the different γ-CD faces, the macrocycle is expected to act as a ditopic supramolecular linker wherein the secondary face nests the cluster and the POM is inserted in the primary face. (Figure 3b). In addition, short contacts between H6 protons of γ-CD and "equatorial" terminal oxygen atoms of the POM (dC-H•••O=W ≈ 2.7 Å) reinforce the supramolecular assembly. In this supramolecular adduct {[P2W18O62]@2γ-CD} 6-, the γ-CDs are perpendicular to the three-fold axis of the POMs and its cavity embeds the two trimeric units {W3O13} of the Dawson anion. This feature shows some analogy with the inclusion complex prepared with the Keggin type polyoxometalate [PMo12O40] 3-in which the trimeric metalate unit {Mo3O13} are also enclosed into the host cavities. [START_REF] Wu | Complexation of Polyoxometalates with Cyclodextrins[END_REF] Interestingly, the disposition of γ-CD around the Dawson unit in {Re6 ca }2{P2W18}@2CD differs from those previously observed in absence of rhenium clusters, in which the supramolecular complexation occurs on the larger side of POM. [START_REF] Moussawi | Cationic Cluster, and γ-Cyclodextrin: From Primary Interactions to Supramolecular Hybrid Materials[END_REF] This observation highlights the versatility of the binding between [P2W18O62] 6-and CD.

Each remaining γ-CD faces of {[P2W18O62]@2γ-CD} 6-nests a cationic [{Re6Se8}(H2O)6] 2+ cluster (Figure 4a). This inorganic entity deviates strongly from the center of the macrocycle contrarily to the structural motif observed without the POM (see structural description of {Re6 ca }@CD in SI). The structural analysis indicates hydrogen bonding between one apical water molecule of the octahedral cluster and a hydroxyl group of the secondary face (dO-H•••H2 O-Re ≈ 2.7 Å) (Figure 4b). In addition, short contacts are also depicted between the inner selenide ligand and two hydrogen atoms of the γ-CD cavity (labeled H3). Hence, γ-CD acts as a ditopic supramolecular ligands, allowing the hierarchical association of cationic Re6 clusters and Dawson anion which are distanced by 3.2 Å, like in the CLUSPOM salt {Re6 ca }{P2W18}.

The 3D organization is rather challenging to depict. As we have previously mentioned, two crystallographically independent {[{Re6Se8}(H2O)6]2•[P2W18O62]@2γ-CD} 2-motifs constitute the unit cell (Figure 5a). These supramolecular three-component assemblies are associated together through a set of weak interactions (hydrogen bonds and short contacts) involving the inorganic units from one motif and the cyclodextrin from the other (Figure 5c). In brief, hydrogen bonds between the terminal water molecules and hydroxo group of primary γ-CD face (dO-H•••H2O-Re ≈ 2.7 Å), as well as various short contacts between oxygen atoms of the POM and hydrogen atoms located on the external wall of γ-CD (dC-H•••O-W ≈ 2.5-2.8 Å) contribute to the 3D cohesion. Moreover, the supramolecular architecture is reinforced by hydrogen bonds between hydroxo groups (secondary face) of the two distinct cyclodextrins belonging to crystallographically equivalent bamboo fragments (Figure 5b). It results a packing of nano-sized rods aligned along [-111]

(Figure S1). The electroneutrality of {Re6 ca }2{P2W18}@2CD is ensured by two potassium cations per {[{Re6Se8}(H2O)6]2•[P2W18O62]@2γ-CD} 2-assembly (confirmed by EDS analysis). In a previous study dealing with the CLUSPOM [P2W18O62] 6-/γ-CD/[{Ta6Br12}(H2O)6] 2+ , [START_REF] Moussawi | Cationic Cluster, and γ-Cyclodextrin: From Primary Interactions to Supramolecular Hybrid Materials[END_REF] we isolated an infinite chain showing a periodic -cluster-macrocycle-POM-macrocycle-sequence where each {[P2W18O62]@2γ-CD} 6-supramolecular motif links two others by the tantalum cluster closely embedded into two γ-CDs from distinct {[P2W18O62]@2γ-CD} 6-assemblies. Oppositely, the molecular nature of {Re6 ca }2{P2W18}@2CD assemblies would indicate that the cationic {Re6Se8} cluster acts as a "stopper". This difference is explained from the distinct supramolecular behavior of clusters in presence of γ-CD. In fact, investigations in solution showed that γ-CD exhibits very high affinity to edge-bridged [{Ta6Br12}(H2O)6] 2+ , generating a strong 1:2 inclusion complex, whereas γ-CD showed a very weak affinity to [{Re6Se8}(H2O)6] 2+ leading to a fragile 1:1 adduct. [START_REF] Moussawi | Cationic Cluster, and γ-Cyclodextrin: From Primary Interactions to Supramolecular Hybrid Materials[END_REF][START_REF] Abramov | Supramolecular Adduct of γ-Cyclodextrin and [{Re6Q8}(H2O)6] 2+ (Q=S, Se)[END_REF] These two examples highlight that weak supramolecular forces can control the aggregation between inorganic building blocks having opposite charges into the hierarchical CLUSPOM architecture which can be anticipated by considering the properties of the binary systems i. e. γ-CD and polymetallic units.

Ternary systems involving the anionic cluster

The promising results of using cyclodextrin to organize the ionic packing of preformed polynuclear inorganic entities in the solid-state encourage us to push the limits of this approach by assembling two different anionic building blocks, such as [{Re6Se8}(CN)6] 4-and [P2W18O62] 6-. In such systems long-range electrostatic repulsion and short-range supramolecular forces should compete within the aggregation process while cyclodextrin would act as a POM-cluster connector. From this three components system, we successfully isolated a crystalline solid containing both anionic {Re6Se8}-based cluster and the polyoxometalate unit. This compound notated {Re6 an }@2CD{P2W18} was isolated by evaporation of an aqueous mixture of [{Re6Se8}(CN)6] 4-,

[P2W18O62] 6-and γ-CD in the molar ratio cluster:POM:γ-CD = 1:1:2. This CLUSPOM solid crystallizes in F222 space group. The unit cell volume is quite large (V ≈ 108020 Å 3 ) and the asymmetric unit consists of one polyoxometalate unit and two inclusion complexes {[{Re6Se8}(CN)6]@2γ-CD} 4-centred at the Wyckoff position 8j (0, 1/4, 1/4) and 8h (1/4, 1/4, 0)

(Figure 6). The inclusion complexes {[{Re6Se8}(CN)6]@2γ-CD} 4-exhibit the same structural arrangement as those observed in absence of the polyoxometalate. [START_REF] Ivanov | Host-Guest Binding Hierarchy within Redox-and Luminescence-Responsive Supramolecular Self-Assembly Based on Chalcogenide Clusters and γ-Cyclodextrin[END_REF] In brief, the octahedral interacts with oxygen atoms of four Dawson anions in a square geometry through a set of short contacts and hydrogen bonds (Figure 6). The adduct {[{Re6Se8}(CN)6]@2γ-CD} 4-located at the Wyckoff position 8j shows a different arrangement of the neighbouring polyoxometalate units.

Actually, this inclusion complex is located within a square prism defined by eight Dawson anions which interact through weak bonds with the external surface of the γ-CD (Figure 6). In such a structural organization, each Dawson anion is located at the centre of trigonal-prism where the shortest distance between the two inorganic units is about 5.5 Å. Ten remaining K + cations per polyoxometalate unit (confirmed by EDS analysis) are required for charge balancing.

They are distributed with water molecules within the large voids (36% of cell volume) generated by the crystal packing.

It should be worth mentioning that in {Re6 an }@2CD{P2W18}, the [P2W18O62] 6-does not form any inclusion complex with γ-CD and interacts preferably only with its outer surface. Thus, the situation observed here differs from those observed with the supramolecular CLUSPOM edifices built with cationic clusters such as [{Ta6Br12}(H2O)6] 2+ or [{Re6Se8}(H2O)6] 2+ . Therefore, we conclude that these differences arise from equilibrated balance between the different attractive forces that operate within the aggregation process. They result from the short-range supramolecular interactions which govern the nature of the host-guest arrangement, and from long-range coulombic repulsions that increase the shortest distances between the anionic POM and cluster species. Nevertheless, using γ-CD permits isolation of supramolecular CLUSPOM solids built from distinct anionic building blocks. To the best of our knowledge, only gigantic ring-shaped POM species were previously identified as able to aggregate with anionic polymetallic units. [START_REF] Moussawi | Nonconventional Three-Component Hierarchical Host-Guest Assembly Based on Mo-Blue Ring-Shaped Giant Anion, γ-Cyclodextrin, and Dawson-Type Polyoxometalate[END_REF][START_REF] Miras | Unveiling the Transient Template in the Self-Assembly of a Molecular Oxide Nanowheel[END_REF][START_REF] Fang | Mo24Fe12 Macrocycles: Anion Templation with Large Polyoxometalate Guests[END_REF][START_REF] Xuan | Self-Templating and In Situ Assembly of a Cubic Cluster-of-Clusters Architecture Based on a Mo24Fe12 Inorganic Macrocycle[END_REF][START_REF] Xuan | Stereoselective Assembly of Gigantic Chiral Molybdenum Blue Wheels Using Lanthanide Ions and Amino Acids[END_REF] Characterizing the solution phase by multi-nuclear NMR and Small-Angle Xray Scattering

As aqueous mixtures of {Re6Se8} based clusters, taken either in anionic or cationic form, γ-CD, and Dawson type polyoxometalate promote three-component systems in the solid-state, it could be postulated that pre-aggregation process should take place in solution. In such context, multinuclear NMR ( 1 H, 31 P, 77 Se, 183 W) experiments were first performed to describe in details the local interactions between the three components. For a complete insight about possible aggregation processes, we carried out DOSY NMR and small-angle x-ray scattering (SAXS), which is a powerful technique to observe self-assembly involving inorganic polynuclear entities in solution. [START_REF] Nyman | Small-Angle X-Ray Scattering to Determine Solution Speciation of Metal-Oxo Clusters[END_REF] [P2W18O62] 6-/γ-CD/[{Re6Se8}(H2O)6] 2+ system.

For this system, the following cluster:γ-CD:POM molar ratios 1:2:1, 0:2:1, 1:2:0 were studied by NMR spectroscopy, as well as each component separately ([POM] and

[cluster] = 10 mM ; [γ-CD] = 20 mM). Note that increasing the cluster amount up to 2 equivalents (2:2:1), leads rapidly to the precipitation of {Re 6 ca } 2 {P 2 W 18 }@2CD.

According to 1 H NMR spectra, γ-CD signals are strongly influenced by POM units (strong shift of H6). However the γ-CD signal is almost not affected by the cationic cluster (Figure 7a). Similarly, the addition of cationic clusters to a solution of POM and γ-CD provokes only tiny changes, characterized by a small chemical shift of H6 and H3 (0.02 ppm) and a weak broadening of proton signals. The 31 P and 183 W NMR spectra (Figures S7 andS8), show the signatures of the Dawson POM [P 2 W 18 O 62 ] 6-. While 31 P NMR exhibited only very small changes presumably because phosphorus nuclei are protected inside the internal POM cavity, 183 W NMR spectra are mostly affected by the presence of the CD and the effect of cluster appears insignificant. From the cluster side, the 77 Se spectra revealed very weak interaction with the γ-CD as ca. 1 ppm downfield shift is observed, whereas the POM does not affect significantly the cluster spectrum (Figure S9). Thus, in this system, the interactions of POM with γ-CD seem to be predominant and only a very weak effect of the cluster is observed. At first sight, this suggests the formation of [[P 2 W 18 O 62 ]@xγ-CD} 6-assembly as main species. The NMR conclusions are supported by SAXS investigations performed in similar condition (same concentration of reactants). Figure 7b shows X-ray scattering curves of a cluster and polyoxometalate containing solution with or without γ-CD. It should be mentioned that nonaggregated cyclodextrin is almost silent in SAXS analysis. For both scattering curves, a plateau is observed in low q region, indicating the absence of inter-particle interferences which means that no large aggregates are formed. The SAXS curves were fitted in the Guinier region to extract the average Rg value (radius of gyration, the root-mean-square distance of all the electrons from the gravity centre of the particles). In absence of γ-CD, the Rg value is about 4 Å; this is in agreement with a solution containing Dawson type anions (theoretical Rg = 4.5 Å) and cationic {Re6Se8} based clusters (theoretical Rg = 2.5 Å). Interestingly, addition of γ-CD increases the Rg value (4.8 Å), meaning an expansion of the particle size as expected from the binding of POM unit by γ-CD. Another significant difference between the SAXS curves, concerns the relative I(0) value which is about 0.86 and 1.23 in absence and presence of γ-CD, respectively, involving a I(0)CD/I(0)noCD ratio of 1.43. This value should be compared to the theoretical I(0) ratio determined for the following mixtures: i) 1 free cluster + 1 free γ-CD + 1 POM@1γ-CD (1.2), ii) 1 free cluster + POM@2γ-CD (1.4) and iii) POM@2γ-CD@cluster (2.2) (details are provided in the Supporting Information, see Table S12 and Figure S14).

Consequently, SAXS and NMR investigations conclude that the solution is mainly dominated by supramolecular aggregates involving mainly the POM unit and γ-CD macrocycle. Such a supramolecular assembly is observed in the crystal structure of {Re6 ca }2{P2W18}@2CD, in which the pre-associated anionic motif {[P2W18O62]@2γ-CD} 6-is decorated by two cationic clusters.

[P 2 W 18 O 62 ] 6-/γ-CD/[{Re 6 Se 8 }(CN) 6 ] 4-system.

As previously, this study of [P2W18O62] 6-/γ-CD/[{Re6Se8}(CN)6] 4-system was performed using fixed γ-CD concentration of 20 mM. Thus, according to 1 Н NMR spectra (Figure 8a), the signals of cyclodextrin protons are influenced by both inorganic components and balanced between both [{Re6Se8}(CN)6] 4-/γ-CD and [P2W18O62] 6-/γ-CD systems. This suggests that the γ-CD-based inclusion complexes involving POM or {Re6Se8} cluster exist simultaneously in solution. Such a result would be consistent with previous studies showing that the stability constants determined by ITC fall in the same range of magnitude (K = 2⋅10 3 -3⋅10 3 mol.L -1 ) for both adducts. [START_REF] Moussawi | Cationic Cluster, and γ-Cyclodextrin: From Primary Interactions to Supramolecular Hybrid Materials[END_REF][START_REF] Ivanov | Host-Guest Binding Hierarchy within Redox-and Luminescence-Responsive Supramolecular Self-Assembly Based on Chalcogenide Clusters and γ-Cyclodextrin[END_REF] Formation of higher aggregates resulting from multi-components cluster-macrocycle-POM based adducts is excluded according to DOSY NMR (see below). Thus, the observed 1 H NMR spectrum of the three-component mixture should correspond to a situation of a competition between POM and cluster entities to bind γ-CD. The 31 P and 183 W, and the 77 Se NMR spectra (Figures S10-12) are also consistent with significant interactions between inorganic clusters (POM and rhenium cluster) only in presence of the γ-CD in solutions.

As observed by NMR, SAXS technique clearly evidences strong interactions of

[{Re6Se8}(CN)6] 4-with the γ-CD (Figure 8b). In fact, adding CD to aqueous mixture of anionic clusters drastically modifies the scattering curve (increase of scattering intensity and apparition of an oscillation located q = 0.4 Å -1 ). The Rg values extracted from the fitting of experimental data are 2.2 and 5.3 Å in absence and in presence of γ-CD, respectively. The experimental Rg of the γ-CD containing solution corresponds to an intermediate value between Rg calculated for 1:1 and 1:2 inclusion complexes (4.8 vs 5.7 Å for 1:1 and 1:2, respectively, see Figure S15). SAXS investigation reveals that the solution behaviour of the [P 2 W 18 O 62 ] 6-/γ-CD/[{Re 6 Se 8 }(CN) 6 ] 4-system differs from the system with the cationic cluster. In fact, SAXS curves exhibit a broad coulombic peaks, characterized by a decrease of the intensity at low q values (Figure 8c), indicating a structuring arising from cluster-cluster repulsions in solution. Comparison of the log(q)-log(I(q)) scattering curves of the rhenium cluster solution with (green curve) and without γ-CD (grey curve). c) Comparison of the log(q)-log(I(q)) scattering curves of the [P2W18O62] 6-/[{Re6Se8}(CN)6] 4-system (blue curve) with the [P2W18O62] 6-/γ-CD/[{Re6Se8}(CN)6] 4-system (red curve). All solutions were equilibrated for 24 h prior to NMR or SAXS measurements. This could be expected since the solution contains only anionic units. Although inter-particle interferences prevent accurate interpretation of the I(0) increase with the γ-CD addition, we assume that this indicates mainly the formation of inclusion complexes. This is also supported by Guinier fitting of the SAXS curves showing that adding γ-CD induces only a slight Rg increase (Rg no( γ-CD) = 4.3 Å vs Rg (γ-CD) = 5 Å). The SAXS observations coupled with NMR data allow us to conclude that the aqueous [P 2 W 18 O 62 ] 6-/γ-CD/[{Re 6 Se 8 }(CN) 6 ] 4-system is dominated by inclusion complexes, such as {[{Re6Se8}(CN)6]@xγ-CD} 4-and {[P 2 W 18 O 62 ]@xγ-CD} 6-, rather than three-component supramolecular aggregates mostly cancelled by repulsive coulombic forces. Interestingly, the evaporation of such aqueous mixtures led to the formation of the crystalline compound {Re6 an }@2CD{P2W18} that exhibits exclusively inclusion complexes built from the anionic cluster, {[{Re6Se8}(CN)6]@2γ-CD} 4-.

The POM unit interacts with the external wall of the CDs that encapsulated the anionic cluster. This situation, which differs from those observed in solution, can be due to the minimization of lattice energy in the solid-state (e.g. limiting the long-range coulombic repulsion).

Diffusion measurements of γ-CD by DOSY NMR. DOSY NMR spectroscopy could provide additional information about interaction of the inorganic clusters with the γ-CD and their aggregation in solution. Table 1 summarizes the measured diffusion coefficients of γ-CD in the same solutions studied by SAXS shown in Figures 7 and8

CONCLUSION

Herein, we report the crystal engineering of CLUSPOM compounds resulting from the supramolecular aggregation of an electron-poor polyoxometalate and an electron-rich rhenium cluster. Using a "classical" synthetic approach based on long-range coulombic interactions between a macro-anion ([P2W18O62] 6-) and a macro-cation ([{Re 6 Se 8 }(H 2 O) 6 ] 2+ ), a 'CLUSPOM salt' was successfully isolated from aqueous mixtures without supramolecular structure-directing agent. On the other hand, using an innovative strategy consisting in cyclodextrin-driven assembling, a hierarchical CLUSPOM architecture was designed in a predictive way. In addition, we reported that presence of γ-CD provides "counter-intuitive" association containing both

ASSOCIATED CONTENT Supporting Information

The Supporting Information is available free of charge on the ACS Publications website.

Crystal data, FT-IR spectra of {Re6 ca }{P2W18}, {Re6 ca }2{P2W18}@2CD and {Re6 an }@2CD{P2W18}. 77 Se and 183 W NMR spectra. Simulated SAXS curves.

Figure 1 .

 1 Figure 1. (a) Structural representation of the molecular components used for designing supramolecular inter-cluster compounds. The inorganic building blocks are Dawson-type polyoxometalate [P2W18O62] 6-and octahedral rhenium clusters [{Re6Se8}L6] m+/n-with L = H2O or CN - built from a {Re6Se8} core formed by six Re III atoms linked through metal-metal bonds. The supramolecular connector is the γ-CD resulting from the condensation of 8 glucopyranose units. Colour code: purple polyhedron: WO6 unit, grey polyhedron = PO4 unit, green sphere = Re, orange sphere = Se, grey sphere = apical ligand (CN -or H2O).

Figure 2 .

 2 Figure 2. a) Illustration of the structural arrangement derived from NiAs structure where the Dawson anion and the cationic octahedral cluster lie on the As 3-and Ni 3+ positions, respectively. b) Projections of the crystal structure {Re6 ca }{P2W18} along the 001 (left) and 100 (right) directions. The water molecules and free potassium atoms are omitted for clarity.

Figure 3 .

 3 Figure 3. a) Selective binding of the γ-CD (yellow) with the metallic units, especially the Dawson anion (purple oval) and Re6 based clusters (green sphere). b) Schematic illustration of the different supramolecular adduct previously observed in the solid-state for [P2W18O62] 6-(b), [Re6Se8}(CN)6] 4-(c) and [{Re6Se8}(H2O)6] 2+ (d).

Figure 4 .

 4 Figure 4. a) View of the three-component supramolecular adduct showing a central Dawsontype polyoxometalate decorated by two γ-CDs interacting with the primary face while the secondary face of these macrocyclic host interact with two cationic rhenium clusters. The shortest distance between the inorganic entities is about 3.2 Å. b) Top plan views of the strongest interactions between the cyclodextrin and the polymetallic units (left with the cluster; right with the POM).Mixing cationic rhenium cluster (obtained by dissolving of K4[{Re6Se8}(OH)6]•8H2O in an acidic medium) in 2 M sulphuric acid with an aqueous solution containing γ-CD, in the molar ratios of {Re6 ca }:{P2W18}:γ-CD: = 2:1:2 (fast precipitation of tiny crystals) and 1:1:2 (formation of high quality crystals) leads to crystallization {Re6 ca }2{P2W18}@2CD. SCXRD analysis indicates that the unit cell (space group P1) contains two Dawson type anions, four cationic clusters and four γ-CDs. This supramolecular hierarchically built CLUSPOM solid can be viewed as a packing of two crystallographically independent bamboo-shaped fragments built from the following cluster-macrocycle-POM-macrocycle-cluster sequence where γ-CD acts as ditopic linker (Figure4a). Within this motif, POMs and clusters interplay with the same binding mode

Figure 5 .

 5 Figure 5. a) Projection of the unit cell along a axis showing the two crystallographically independent bamboo fragments observed in {Re6 ca }2{P2W18}@2CD. The water molecules and free potassium atoms are omitted for clarity. b) Each independent bamboo fragments are linked to each other by a set of hydrogen bonds between two adjacent γ-CDs, generating zig-zag chain along b axis. c) View of the supramolecular interactions (hydrogen bonds and short dC-H•••O-W

Figure 6 .

 6 Figure 6. View of the structural arrangement in the compound {Re6 an }@2CD{P2W18} built from [P2W18O62] 6-and inclusion complexes {[{Re6Se8}(CN)6]@2γ-CD} 4-located at the Wyckoff position 8j and 8h. The water molecules and free potassium atoms are omitted for clarity. The green dashed lines indicate short contacts and hydrogen bonds between Dawson-type POM and γ-CD.

  cluster disordered over two positions (50%/50%), is closely embedded into two γ-CDs facing their wider rim. The equatorial cyano groups interact through hydrogen bonds with the hydroxo groups of the secondary face, giving N•••O distances in the 2.8-3.0 Å range. The exterior surface of the supramolecular adduct {[{Re6Se8}(CN)6]@2γ-CD} 4-centred at the Wyckoff position 8h

Figure 7 .

 7 Figure 7. a) 1 Н NMR spectra of γ-CD ([γ-CD] = 20 mM) in acidified D2O for different mixtures containing [{Re6Se8}(H2O)6] 2+ or [P2W18O62] 6-. b) Comparison of the log(q)-log(I(q)) scattering curves of the [P2W18O62] 6-/[{Re6Se8}(H2O)6] 2+ system (blue curve) with the [P2W18O62] 6-/γ-CD/[{Re6Se8}(H2O)6] 2+ system (red curve). The concentration of [{Re6Se8}(H2O)6] 2+ and [P2W18O62] 6- is 10 mM. SAXS and NMR studies have been carried out in diluted sulfuric acid (1M). All solutions were equilibrated for 24 h prior to NMR or SAXS measurements.

Figure 8 a) 1 Н

 81 Figure 8 a) 1 Н NMR spectra of γ-CD ([γ-CD] = 20 mM) in D2O in presence and absence of [{Re6Se8}(CN)6] 4-and [P2W18O62] 6-. b)Comparison of the log(q)-log(I(q)) scattering curves of the rhenium cluster solution with (green curve) and without γ-CD (grey curve). c) Comparison of the log(q)-log(I(q)) scattering curves of the [P2W18O62] 6-/[{Re6Se8}(CN)6] 4-system (blue curve) with the [P2W18O62] 6-/γ-CD/[{Re6Se8}(CN)6] 4-system (red curve). All solutions were equilibrated for 24 h prior to NMR or SAXS measurements.

  Figures7 and 8. The diffusion rate D drops from 235 to 187 µm 2 /s when 0.5 equivalent of

Table 1 . 7 a

 17 Diffusion coefficients from 1 H DOSY NMR of γ-CD ([CD] = 20 mM) in D2O in presence and absence of [{Re6Se8}(CN)6] 4-, [{Re6Se8}(H2O)6] 2+ and [P2W18O62] 6-. Solution composition (molar ratio) a D (µm 2 /s) Solutions equilibrated for 24 h.

[P2W18O62] 6 -

 6 and [{Re 6 Se 8 }(CN) 6 ] 4-anions into a supramolecular edifice. These results evidence that cyclodextrin behaves as efficient supramolecular connector to assemble together polymetallic unit in the solid-state. Solution investigations by complementary techniques (multinuclear and DOSY NMR, and SAXS) revealed however that aggregation of the molecular components results from processes in competition including short-range supramolecular recognition and long-range coulombic attraction/repulsion forces. Interestingly, although the CLUSPOM compounds prepared in this study are built from luminescent rhenium clusters, all these solids are non-emissive when irradiated with UV light (λ = 365 nm), suggesting an efficient quenching process in the presence of the electron-acceptor POM. Further work on cyclodextrin-assisted aggregation of functional polymetallic units, namely the CLUSPOM compounds, is underway.

{Re6Se8}(H2O)6][P2W18O62]•25H2O ({Re6 ca }{P2W18}) K4

	(C 48 H 80 O 40 , K TGA revealed a weight loss of about 8.2 % from 25 to 250 °C (the calculated weight loss of 31 TCI, 99%), K 6 [P 2 W 18 O 62 ]•19H 2 O,
	H2O is 8.17 %).
	{[{Re6Se8}

4

[{Re 6 Se 8 }(CN) 6 ], and K4[{Re6Se8}(OH)6]•8H2O. Dawson-type polyoxometalate and {Re 6 Se 8 } based clusters were synthesized as described in literature.

[START_REF] Mbomekalle | High Yield and Reagent-Saving Synthesis of Pure α-K6P2W18O62•14H2O[END_REF][START_REF] Naumov | Synthesis and Crystal Structure of K4[Re6Se8(CN)6]•3.5H2O[END_REF][START_REF] Yarovoi | Octahedral Hexahydroxo Rhenium Cluster Complexes [Re6Q8(OH)6] 4[END_REF]

K4[[{Re6Se8}(OH)6]•8H2O (50 mg, 0.028 mmol) was dissolved in 2 mL of 2 M H2SO4. In parallel K6[P2W18O62]•19H2O (140 mg, 0.028 mmol) were dissolved in 2 mL of water. Then, the resulting solutions were mixed and left to stay on the air. After 5 days, hexagonal crystals of {Re6 ca }{P2W18} were formed and separated from the solution, washed with cold water and dried in air. Yield 0.13 g (70 %) based on rhenium cluster. EDS shows Re : Se : K : P : W ratio = 6.8 : 7.8: 3.8 : 18.

(H2O)6]@γ-CD}(NO3)2•10H2O ({Re6 ca }@CD)

	0.704 g (0.54 mmol) of solid γ-CD was added directly to this solution under stirring. After
	dissolution of the macrocyclic cavitand the reaction mixture was allowed to slowly evaporate in
	air to leave a very small volume of a concentrated solution. When the solution was concentrated,
	the product started to decompose, therefore, the complex was isolated only as several crystals.
	Yield: a few crystals. EDS shows Re : Se ratio = 6.0 : 8.2 and confirms the absence of K.
	K2[{Re6Se8}
	K4[{Re6Se8}(OH)6]•8H2O (0.584 g ;0.27 mmol) was dissolved in 20 mL of water, after that 0.5 M
	HNO3 was added drop wise under stirring until pH reached 1.5 producing a clear yellow solution.

(H2O)6]2{[P2W18O62]@2γ-CD}•42H2O ({Re6 ca }2{P2W18}@2CD)

	K4[{Re6Se8}(OH)6]•8H2O (0.088 g, 0.04 mmol) was dissolved in 2 mL of 2 М H2SO4.
	K6[P2W18O62]•19H2O (0.1 g, 0.02 mmol) and γ-CD (0.057 g, 0.04 mmol) were dissolved in 1.5 mL
	of water. The resulting solutions were mixed and left to stay on the air. After 12 hours,
	crystalline compound {Re6 ca }2{P2W18}@2CD were formed and separated from the solution, washed
	with cold water and dried in air. Yield 0.164 g (70 %) based on cluster complex. Anal. Calcd for
	C96H268K2O196P2Re12Se16W18: C, 9.99; H, 2.34. Found: C, 10.0; H, 2.4. EDS shows Re : Se : K : P : W
	ratio = 12 : 16.0 : 2.2 : 2.1 : 18.0. TGA revealed a weight loss of about 6.4% from 50 to 110 °C
	(the calculated weight loss of 42 H2O is 6.4%).

K10{[{Re6Se8}(CN)6]@2γ-CD}[P2W18O62]•33H2O ({Re6 an }@2CD{P2W18}).

	K4[{Re6Se8}(CN)6] (0.129 g, 0.062 mmol) was dissolved in 1 mL of water. K6[P2W18O62]•19H2O
	(0.3 g, 0.062 mmol) and γ-CD (0.174 g, 0.124 mmol) were dissolved in 3 mL of water. The
	resulting solutions were mixed and left to stay on the air. During the slow evaporation of the
	solution, crystals

of {Re6 an }@2CD{P2W18} were formed after one day, which were separated from

	the solution, washed with cold water and dried in air. Yield 0.454 g (74 %) based on cluster
	complex. Anal. Calcd for C102H226K10N6O175P2Re6Se8W18: C, 12.41; H, 2.31; N, 0.85. Found: C, 12.6;

Accession Codes: CCDC 1995746-1995749 contain the supplementary crystallographic data for this paper. These data can be obtained free of charge viawww.ccdc.cam.ac.uk/data_request/cif.

ACKNOWLEDGMENT

We thank Javier Perez for assistance in using beamline SWING. We acknowledge SOLEIL Synchrotron for providing beamtime and synchrotron radiation facilities. Authors gratefully acknowledge financial support from IRP-CNRS CLUSPOM, CNRS MOMENTUM and LabEx CHARMMMAT (grant number ANR-11-LBX-0039). This work was also supported by i) University of Versailles Saint Quentin, ii) CNRS, iii) Région Ile de France through DIM Nano K. NIIC thanks the Ministry of Science and Education of the Russian Federation. A.A. Ivanov and A. A. Shmakova thank Embassy of France in Russia for the Vernadsky scholarship for postgraduate students.

ABSTRACT:

The association of metallic clusters (CLUS) and polyoxometalates (POM) into hierarchical architectures is achieved using γ-cyclodextrin (γ-CD) as supramolecular connector. The new self-assembled systems, so-called CLUSPOM, are formed from Dawson-type polyoxometalate

Notes

The authors declare no competing financial interest.

For Table of Contents Use Only.

This study evidences that γ-cyclodextrin, a natural macrocyclic polysaccharide, behaves as efficient supramolecular connector to assemble an electron-poor polyoxometalate ([P2W18O62] 6-) and electron-rich rhenium cluster ([{Re6Se8}(H2O)6] 2+ or [{Re6Se8}(CN)6] 4-) in the solid-state.

However, solution investigations revealed that aggregation of the preformed molecular components results from processes in competition including short-range supramolecular recognition or long-range coulombic attraction/repulsion forces.


Achilleas Lazarides

Arnab Das

Roderich Moessner

Fate of many-body localization under periodic driving

We study many-body localised quantum systems subject to periodic driving. We find that the presence of a mobility edge anywhere in the spectrum is enough to lead to delocalisation for any driving strength and frequency. By contrast, for a fully localised many-body system, a delocalisation transition occurs at a finite driving frequency. We present numerical studies on a system of interacting one-dimensional bosons and the quantum random energy model, as well as simple physical pictures accounting for those results.

Introduction-The study of disorder and localisation has a long and productive history, beginning with the seminal work of Anderson [1]. More recently, the effects of disorder on interacting systems have been considered under the heading of many-body localisation (MBL) [1,2], in part motivated by fundamental questions relating to thermalisation in closed quantum systems.

At the same time, significant theoretical effort has been devoted to understanding thermalisation in periodically driven systems. There has been work recently on the long-time behaviour of both integrable [4,5] and nonintegrable [2,3,8] (with Ref. [8] also studying locally driven MBL systems). For clean systems or MBL systems in their delocalised phase, it has been found that driving leads to a state equivalent to a fully-mixed state, satisfying a special case of the Eigenstate Thermalisation Hypothesis (ETH, see [2,3,[8][9][10][11][12]). Local periodic driving of MBL systems in their localised phase, on the other hand, has been argued not to have any global effects [8].

In this work, we study the effects of global periodic Table I: Effect of driving frequency in the presence and absence of a mobility edge driving, and find that there exists a regime where MBL survives. We identify two mechanisms by which periodic driving might destroy MBL, depending on the existence or otherwise of a mobility edge. The first, rather robust, mechanism is the mixing of undriven eigenstates from everywhere in the spectrum by the driving; if there is a mobility edge, this results in delocalisation of all states of the effective Hamiltonian. The second mechanism is more subtle and involves strong mixing of states [2] which cause a delocalisation transition at finite frequency. Our findings are summarised in Table I.

� � � � � /� � � � �� � � � � � �� �/� ω � /�
In what follows, we begin by studying the case of no mobility edge. We introduce and numerically solve a system described by a local non-integrable Hamiltonian. After establishing the existence of the aforementioned critical frequency using level statistics, we demonstrate that ETH is (is not) satisfied below (above) this frequency, and present a physical picture explaining this phenomenon. We then move to the case where a mobility edge exists. As a case study, we use the Quantum Random Energy Model (QREM) which has recently been shown to display a mobility edge. A direct numerical solution confirms that driving delocalises the entire spectrum, consistent with an intuitive argument we sketch. Finally, we point out open questions.

We shall concentrate throughout on systems described by Hamiltonians of the form

H(t) = H 0 + H D (t), (1)
so that their time evolution is described by an effective Hamiltonian H ef f ( ) for each instant during the period T , defined by Left (right): Driving with a low (high) frequency (see Fig. 1) results in the probability distributions which does (does not) develop a central peak upon increasing system size, signalling delocalisation [2]. Data is disorder averaged over 10 4 (100) realisations for L = 14 (L = 16). Inset: Level statistics parameter versus inverse system size in the localised (bottom, red) delocalised (top, blue) phases. The parameter η = ´ds sP (s) with P (s) the probability distribution of the level statistics [1,3], taking the value η P/CU E in the localised/delocalised regime. Data averaged over 1000 disorder realisations for L = 8, 10, 12, 100 realisations for L = 14.

exp (-iH ef f ( ) T ) = T exp -i ˆ +T dt H(t
• • • • • • • • • • • • • • • • • • • ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ 0.0 0.2 0.4 0.6 0.8 0 2 4 6 n j=3 P(n j=3 ) ω/J=4, w/J=16 • • • • • • • • • • • • • • • • • • • • ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ • L=16, static ■ L=14 ◆ L=16 0.0 0.2 0.4 0.6 0.8 1 2 3 n j=3 P(n j=3 ) ω/J=2, w/J=7 • • • • • • • • • ω/J=.1, w/J=7 • ω/J=8, w/J=9
Without loss of generality we set = 0 (see Ref. [4]). The eigenvalues of H ef f ( ), called the quasienergies, are independent of and effectively play the role of energy eigenvalues.

We now define what we mean by localised and delocalised phases. In a localised phase, the (quasi-)energy level statistics do not display level repulsion, and the expectation values of operators in the eigenstates of the (effective) Hamiltonian fluctuate wildly from eigenstate to eigenstate. In a delocalised phase, the opposite is true: the levels repel each other, and the expectation values of physical, local operators in nearby energy or quasienergy states are similar. Other definitions are possible and, in general, equivalent (see eg Ref. [13]). The connection between the eigenstates of H ef f and the applicability of ETH was elucidated in Ref. [2]. The framework developed there turns out to be natural for discussing the case of a system which, in the absence of driving, is in the MBL phase.

No mobility edge: Local Model-Let us introduce a model of interacting hard-core bosons described by a driven, local Hamiltonian (Eq. 1) with

H 0 = H hop + 2 r=1 V r L-1 i=1 n i n i+r + L i=1 U i n i (3)
where

H hop = -1 2 J L-1 i=1 b † i b i+1 + b † i+1 b i + hc
is a hopping operator, the b are hard-core bosonic operators, U i an on-site random potential uniformly distributed between -w and +w and H D (t) a time-periodic hopping term

H D (t) = δ δ(t)H hop (4)
with δ a dimensionless constant, δ(t) = -1(+1) in the first (second) half of each period T = 2π/ω . Via Jordan-Wigner transformations this model is related to a fermionic interacting system as well as to a spin-1/2 chain. Throughout this work we will concentrate on the specific case V 1 /J = V 2 /J = 1, although our qualitative conclusions are not sensitive to this.

To locate the transition in the undriven model we use the standard technique [1] involving finite-size scaling of the level statistics (see inset of Fig. 6 and Supplemental Material). At half-filling there thus appears to be a transition at a disorder amplitude w u c /J (≈ 6 for our interaction parameters V 1 /J = V 2 /J = 1) [30].

We now drive this system δ = 0. The level statistics of the quasienergies of H ef f (Eq. 2), show level repulsion in the clean limit [3] but are found to cross freely (indicating localisation) in the MBL regime if driven locally, as reported in Ref. [8]. Here, we show that globally periodically driving the system in the MBL regime delocalises the system if the driving frequency is below a (system size independent) critical value. We argue that this is a consequence of the structure of the effective Hamiltonian for an MBL system [14][15][16][17][18].

As established above, the undriven system is in the delocalised phase for disorder amplitude w < w u c ; driving at this disorder is qualitatively similar to driving any nonintegrable system [2], a case that has been studied in Ref. [2]. We have indeed confirmed quasienergy level repulsion for w < w u c . To study the MBL regime, w > w u c , we switch on periodic driving (Eq. 4) with amplitude δ/J = 0.1 (our results do not change qualitatively for different δ provided the system is large enough that the local level spacing is less than δ). We directly calculate H ef f and its level statistics. As our central result we find that for each disorder amplitude there exists a driving frequency ω c (w) above which the system remains in the localised phase under driving (see Supplemental Material), while for ω < ω c (w) the system delocalises. This frequency is plotted in Fig. 1 as a function of disorder amplitude w, while examples of the level statistics results are shown in the inset of Fig. 6. We expect ω c (w) to diverge as w approaches w u c from above. Having established a transition via the level statistics, we now show in addition that the phases above (below) ω c do (do not) satisfy the form of ETH discussed in Ref. [2], further reinforcing our interpretation of ω c as a "delocalisation frequency". We consider a localised undriven system and provide in Fig. 6 direct evidence for the fullymixed nature of the eigenstates of H ef f for slow -but not for fast -driving. The quantity under consideration is the probability distribution for the eigenstate expectation values (EEVs) [2] of the density operator. Driving faster than the delocalisation frequency (right panel) yields little change in the probability distribution. By contrast, driving slowly (left panel), a central peak is seen to de-

-ξ c 0 ξ c 2ξ c ϵ 0 ϵ 0 -ω ϵ 0 ω + ϵ 0 2 ω + ϵ 0 n Figure 3
: Schematic illustration of sub systems and their energy levels in an MBL system. The horizontal axis indexes the conserved quantity (increasing n corresponds to increasing spatial index i); the vertical indexes energy. ξc is some localisation length, setting the typical spatial size of the subsystems. A periodic coupling of the subsystems with frequency ω couples the red state in the middle block to both the purple and the green levels in the two neighbouring blocks, while a time-independent coupling would only couple it to the green ones. The width of the purple and green strips is set by the amplitude of the driving. Critically, the limit of ω greater than the typical subsystem bandwidth is indistinguishable from a time-independent driving. By contrast, the limit of ω → 0 collapses the local spectra, wiping out the effect of disorder.

velop with increasing system size, corresponding to the EEVs all being equal and given by n j=3 = 0.5. This is the fully-mixed result for our system at half-filling, corresponding to delocalisation [2].

In passing, let us remark that, since the EEVs of the instantaneous Hamiltonian show the same behaviour as in Fig. 6 (data not shown), our localised phase is not unlike the localisation in energy space discussed in Ref. [19], even though the underlying physics is quite different.

Physical picture-We now relate our numerical findings to a physical picture valid for weak driving. In the MBL phase and in the absence of driving, the system is effectively integrable in that there exist extensively many local integrals of motion [14][15][16][17][18]. The system may thus be thought of as a set of local subsystems, of finite spatial extend, therefore of finite energetic bandwidth as schematically shown in Fig. 3. As a result, if the driving frequency is larger than the typical local subsystem bandwidth, the system cannot absorb energy from the driving and does not react. Therefore, driving with a frequency much higher than the typical local bandwidth cannot destroy MBL. In contrast low-frequency driving may be understood by viewing our driving protocol as a series of quenches: as MBL systems eventually reach a steady-state after an instantaneous quench [21,22], periodic driving with the protocol we use can be thought of as a series of non-adiabatic perturbations. It is quite natural then to expect this to cause the system to spread in energy space, delocalising it.

Let us elaborate this pair of arguments, beginning with high-frequency driving.

High-frequency driving-The most general form of H M BL consistent with known phenomenology such as vanishing of the conductivity at all energies is

H G M BL = n H ( ) n + m<n H ( ) n V ( ) m,n H ( ) m . . . (5)
with the H ( )

n Hamiltonians for local subsystems (with local spatial support) and n a spatial index indicating the site about which the subsystem is centred [14][15][16][17][18]. Due to its locality, each H ( ) n has a local spectrum of some typical, finite width set by the disorder amplitude and other system details and independent of the other blocks (see Fig. 3, where the spectra for three H ( )

n are sketched schematically).
Driving H G M BL with a sum of local terms such as in Eq. 1 couples each H ( ) n to its neighbours [START_REF]conserved quantities, driving couples blocks at all instances by exponentially weakly[END_REF] via terms allowing energy and matter transfer. Consider a single energy level for n = 0 (middle block, Fig. 3), indicated by the red line in the middle block. A timeindependent coupling between the blocks couples it to the green blocks on each side, while a periodic coupling with frequency ω couples it to both the green and purple blocks by virtue of folding the energy spectrum into the ω-periodic quasienergies. Crucially, for ω larger than the typical width of the blocks, folding the local spectra has no effect [20] and a weak coupling does not delocalise the system, as it acts similarly to a time-independent perturbation [32]. In other words, the system can react to the driving by absorbing energy quanta ω only if there exist levels separated by this energy. In the presence of MBL the typical local bandwidth sets the maximum driving frequency to which the system can react. [START_REF]A simple example of driving faster than this is given in the Supplemental Material[END_REF] Low frequency-In the limit of low-frequency driving disorder is effectively suppressed and the delocalised phase is always reached.

This phenomenon is best understood in the time domain as follows. Consider time evolving with Hamiltonian H 1(2) for the first (second) half of the period. This series of nonadiabatic changes to the system generically results in a broadening of the energy distribution, provided that the half-period T /2 is longer than the characteristic relaxation time [21,22]. Typically, this eventually leads to a fully-mixed state occupying the entire Hilbert space equiprobably.

There are two central ingredients to this argument. The first is that the relaxation time does not diverge with system size so that the half-period T /2 can be longer. The existence of a dephasing timescale independent of system size [21,22] ensures that this is the case. The second is that repeatedly dephasing in the two different eigenstate bases does lead to energy delocalisation. Since H 1,2 are both MBL Hamiltonians, the eigenstates of one are in general localised in terms of the eigenstates of the other. Nevertheless, repeated cycles of dephasing to alternating bases do indeed eventually lead to a fully-mixed state, as is shown in the Supplemental Material.

A mobility edge: The QREM as a case study-We now turn to the case in which a mobility edge is present in the undriven spectrum. Our central result is based on the observation [2] that a periodic perturbation acting on a system couples each undriven state to states spread uniformly throughout the spectrum of H 0 . As a result, if part of the spectrum corresponds to delocalised eigenstates then all eigenstates of H ef f will necessarily be delocalised. We numerically confirm this by studying the Quantum Random Energy Model (QREM), recently studied in Ref. [23] where it was shown to have a mobility edge. This model is described in Ref. [23]: it is defined for N Ising spins with the Hamiltonian H = E σ z j -Γ j σ x j , where E is a random operator diagonal in the σ z basis (that is, it assigns a random energy to each spin configuration) and Γ a transverse field. Extensivity of the many-body spectrum is satisfied if the random energies are drawn from a distribution

P (E) = 1 √ πN exp -E 2 /N .
The diagnostic of localisation we use is the participation ratio (PR), defined for the state |ψ as φ = n | n| ψ | 4 with n enumerating Fock states. φ approaches unity for a state localised on a single Fock state and 2 -N for one fully delocalised in Fock space. The leftmost panel shows in Fig. 4 shows the average φ versus energy (scaled with system size) of the 256 eigenstates of an undriven N = 8 system averaged over 1000 disorder realisations, demonstrating the existence of a mobility edge.

Next, we drive the system by modulating Γ (t) = Γ 0 1 + δ δ (t) , δ (t) = +1(-1) for the first (second) half of the period with an amplitude δ = 0.2 and frequency ω = 2π/T = 0.1. The PR of the eigenstates of H ef f are shown in the second panel of Fig. 4. As expected, periodic driving causes delocalisation of the entire spectrum so long as part of the undriven spectrum at the same Γ 0 is delocalised.

Outlook-We have shown that many-body systems can remain many-body localised, with Poissonian level statistics, when they are subjected to slow driving. On the other hand, for fast driving or in the presence of a mobility edge, delocalisation will occur, with driving inducing level repulsion.

This "classification" of the behaviour of MBL systems under driving immediately raises further questions. What are the timescales involved in reaching the longtime state we have discussed, how do they depend on the driving amplitude and frequency, and how do they differ between the localised and the delocalised limit? What is the precise difference between local and global driving as far as both the long-time state and the approach to it are concerned? More broadly, we have concentrated on systems with a bounded local spectrum. What happens if it is unbounded, as in the cases of a continuum system or of a lattice boson system? What if we bring the system in contact with a heat bath?

We believe that the dual out-of-equilibrium situationdriving and MBL -is only beginning to be explored and will prove to be fertile ground for future research. man and particularly T. Scheler for discussions as well as D. Abanin, A. Chandran and L. D'Alessio for exchanges in the course of this project.

Note added: After the completion of this work, two related works [25,26] have appeared. Each of these takes a somewhat different perspective but they all establish phenomenologies essentially consistent with the one we report.

I. DETERMINATION OF THE TRANSITION

To accurately locate the localisation-delocalisation transition for the undriven system, we first study the level statistics of the eigenvalues of H 0 , as for example in Ref. [1].

That is, after obtaining the (quasi-)energies n , we calculate the following ratio involving adjacent level spacings δ n = n -n+1 : r n = min (δ n , δ n-1 ) / max (δ n , δ n-1 ). The mean η = ´1 0 dr rP (r) distinguishes between Wigner-Dyson-type and Poisson statistics. We calculate η for a sequence of system sizes and extrapolate the limit of η as L → ∞.

To obtain the frequency ω above which delocalisation sets in for a driven system, we again calculate the mean of the distribution function of the quasienergy statistics η = ´1 0 dr P (r) as a function of disorder amplitude w, averaged over 10000 disorder realisations and for several system sizes. Typical results are shown in Fig. 5. The transition is located at the crossing point of the lines for different system sizes: if increasing system size results in larger η then we conclude that the system is delocalised, since η = η CU E for a delocalised system and η = η P for a localised system with η P < η CU E . Here η CU E is the value for the CUE ensemble [3].

To ensure that our results are applicable to the thermodynamic limit we need to take a frequency low enough so that the width of the energy spectrum of the undriven Hamiltonian is larger than the driving frequency ω. The main practical problem is the following: with decreasing disorder amplitude w and for fixed system size, the value ω c increases while the energetic width of the DOS decreases (see Sec. IV). Since ω must be small compared to the width in order for our extrapolation to the thermodynamic limit to be meaningful, the ω c for values of the disorder close to w c are inaccessible for the system sizes available to us. The width of the DOS is indicated in Fig. 5 by vertical lines; the crossing point of the curves cannot lie to the right of this line, since otherwise the finite size of the system would be important (and thus the results would not be reliable in the thermodynamic limit).

Fig. 5 reveals the following features: for w/J ≤ 6 (where the undriven system is delocalised) the lines for succesive, increasing L do not cross for values of ω below the bandwidth, indicating that the thermodynamic limit is delocalised, as expected. For w/J > 6, there is a clear crossing point, which indicates the position of the transition. The crossing value of ω determined by this method is plotted as a function of w/J in Fig. 1 in the main text.

II. EIGENSTATE EXPECTATION VALUES

As discussed in Ref. [2] and the main text (Fig. 2 of the main text), periodically-driven ergodic (or delocalised) systems develop a peak in the probability distribution of the eigenstate expectation values (EEVs). Fig. 6 shows explicit examples of the EEVs in the case of slow driving (system remains localised, left panel) and fast driving (system delocalises, right panel).

III. SINGLE-PARTICLE LOCALISATION LENGTH

In Fig. 7 we show the single-particle localisation length as a function of the eigenstate energy for the noninteracting Hamiltonian

H 0 = - 1 2 J L-1 i=1 b † i b i+1 + b † i+1 b i + hc + L i=1 U i n i (6)
with U i an on-site random potential uniformly distributed between -w and +w and we take w/J = 5. This plot demonstrates that, for the values of w/J where the interacting Hamiltonian of the main text is in the MBL phase, the single-particle localisation length is well below the accessible system sizes. This ensures that the finiteness of this length is not a source of finite-size effects.

IV. WIDTH OF SPECTRUM OF LOCAL MODEL

The density of states (DOS) for a large class of systems with bounded local Hilbert spaces, including the type we study, is Gaussian:

d ( ) = N -1 exp -( -0 ) 2 /2 2 w [4]
. An example for a particular disorder realisation of our HCB model is shown in Fig. 8, while a plot of the fitted width w as a function of disorder amplitude w is shown in Fig. 9. Ref. [4] shows that, in the absence of disorder, w /J ∝ L, while Fig. 9 suggests that in the presence of strong disorder w /J ∝ L 1/2 . This may be understood via the central limit theorem: for strong disorder, the system's eigenvalues are approximately sums of uniformly distributed random numbers (the random potential at each site), and the probability distribution of a sum of L uniformly distributed random numbers approaches a normal distribution with width L 1/2 . In any case, in Fig. 5 we use the actual values of w /J obtained by fitting and averaging over a number of realisations.

V. DRIVEN TWO-LEVEL SYSTEM

To illustrate how the energy bandwidth of a system sets a natural frequency above which driving has little effect we consider a model with finite bandwidth, namely The driving amplitude is δ/J = 0.1 w/J, ω/J, and each point represents an average over 10000 disorder realisations. The dashed vertical lines indicate half the width of the energy spectrum; for ω greater than this, our results cannot be extrapolated to the thermodynamic limit (see IV). The colours correspond to system sizes L = 8, 10, 12 from bottom to top for the smallest ω. The values ηGOE and ηP correspond to the presence and absence of level repulsion, respectively, which in turn correspond to localised and delocalised phases. The dotted vertical lines correspond to the typical spectral width of the system, for frequencies above which our results cannot be used to infer the thermodynamic limit.

• • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ � � � η� ��� ���� ��� η��� ω/� �/�=� • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ◆◆ ◆◆◆ ◆◆ ◆◆◆ ◆◆◆◆◆ ◆◆◆◆ ◆◆◆◆◆◆ ◆◆◆ ◆◆◆◆◆◆◆◆◆◆◆◆◆ � � � � η� ��� ���� ��� η��� ω/� �/�=� • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ◆ ◆ ◆ ◆ ◆◆ ◆◆ ◆ ◆◆◆◆◆ ◆◆◆◆◆◆ ◆◆◆ ◆◆ ◆◆◆◆ ◆◆◆◆◆ ◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆ � � � � η� ��� ���� ��� η��� ω/� �/�=� ••• •• •• •••• •••••••• •••••••••••••••••••••••••• ••••••••••••• ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ◆ ◆ ◆ ◆ ◆◆ ◆◆ ◆◆ ◆◆◆ ◆◆◆ ◆◆◆◆◆ ◆◆ ◆◆◆◆◆◆ ◆◆◆◆◆◆◆ ◆◆◆◆◆ ◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆ � � � � � η� ��� ���� ��� η��� ω/� �/�=� •••• ••• •••• •••••••• ••••• •••••••••••• ••••••••••••••••••••••••• ••••• ■ ■■ ■ ■■ ■■ ■■■■ ■■■■■■■■■■■■■ ■■ ■ ■■■■■■■■■ ■■■■■■■■■■■■■■■■■■■■■■■■■■■■■ ◆ ◆ ◆ ◆◆ ◆◆ ◆◆◆◆ ◆◆◆◆ ◆◆◆◆◆◆ ◆◆ ◆◆◆◆ ◆◆◆◆◆◆◆◆◆ ◆◆◆◆◆◆◆◆ ◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆ ◆◆ � � � � � � η� ��� ���� ��� η��� ω/� �/�=� ••• ••• ••••• ••••••••••••••••••••••••••••• •••••••••••••••••••••••••••••••••• ■ ■■ ■■ ■■■■ ■■■ ■■■■■■■■■■■■■ ■■■■■■■■■■■■■■■■■■■■■ ■■■■■■■■■■■■■■■■■■■■■■■■■■■■ ◆ ◆ ◆ ◆ ◆ ◆◆ ◆◆ ◆◆◆◆ ◆◆◆◆◆ ◆◆◆◆◆◆◆ ◆◆◆◆◆◆ ◆◆◆◆◆◆◆◆◆ ◆◆◆◆◆◆◆◆◆ ◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆ ◆◆ � � � � � � � η� ��� ���� ��� η��� ω/� �/�=�
Figure 6: Plots of eigenstate expectation values (EEV) of the density at a single arbitrarily chosen site in all the eigenstates of H ef f for a system with w/J = 8.0, size L = 18 for a Hilbert space dimension of DH = 48620 and driving amplitude δ/J = 0.1. For driving frequency above the blue line in Fig. 1 of the main text, ω/J = 8.0 (left), the EEVs fluctuate wildly between different eigenstates of H ef f . In contrast, for a driving frequency below the blue line, ω/J = 0.1 (right), there is markedly less eigenstate-to-eigenstate variation, consistent with all states being fully mixed. This is the expected behaviour of the EEVs for clean (therefore delocalised) driven systems (see Ref. [2]). In the undriven system the EEVs appear qualitatively similar to those in the left panel.

the two-level system (TLS) [5], driven with a small amplitude: This has a threshold frequency above which it does not react to the driving. The simplest way to see this is to directly calculate

H T LS = h δ cos (ωt) δ cos (ωt) -h . ( 7
) • • • • • • • • • • •• •• •• •• ••• •••••••• •• •• • •• • •• • •• • • • • • • • -4 -2 0
exp (-iH ef f ( ) T ) = T exp -i ˆ +T H T LS (τ ) dτ
which describes the time evolution over one period. If tem. Fig. 10 shows the Frobenius norm H ef f -H 0 T LS where H 0 T LS is the undriven TLS Hamiltonian (Eq. 7 with δ = 0). This figure shows that the norm vanishes as a power law of ω/h.

H ef f ≈ h 0 0 -h = T -1 ´T 0 dt H T LS (
• • • • • • • • • • • • • ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ • �=� ■ �=�� ◆ �=�� � � � � �� �� �� � � � � � � � � �/� ε � ��� �/�  ������ ����� �� ��� ��� ������ ν=�/�
The situation for (for example) a single particle hopping on a lattice (whether to nearest-neighbour sites or with some exponentially small amplitude to hop to any distance) is completely identical.

The result described here is valid for weak driving amplitude; different physics might emerge in other limits [6].

VI. DELOCALISATION VIA DEPHASING

We show that an initial state which is an eigenstate of the Hamiltonian H 1 and driven by alternating between H 1 and H 2 (as in the protocol described in the main text for the local disordered model) spreads out to fill the entire Hilbert space if it dephases between each change of Hamiltonian. Specifically, we construct the density matrix ρ (0), diagonalise it in the eigenbasis of H 1 and then discard off-diagonal elements. It is then rotated to the eigenbasis of H 2 and, again, the off-diagonal terms are discarded. This procedure constitutes one period. Discarding the off-diagonal states is supposed to model dephasing, and is similar in spirit to the Boltzmann Stosszahlansatz. ranges from 1 to 1/D 2 H during the process, with D H the dimensions of Hilbert space), with c α the projection of the state onto the eigenstates of H 1 , as a function of period for an initial state that as an eigenstate. The time evolution is carried out using the driving protocol described in the main text for the local model (Eqs. 3 and 4 in the article), with the additional operation of dephasing carried out by hand (that is, off-diagonal elements in the energy basis are discarded by hand). The paramerers used are indicated in the caption.

The conclusion to be drawn from Fig. ?? is that dephasing clearly leads to the system spreading out in energy space. While not unexpected in general (a series of sudden perturbations of the system will, in general, increase its energy), this calculation confirms that the conclusion remains valid in the case of MBL Hamiltonians.

VII. THE QUANTUM RANDOM ENERGY MODEL

Consider the set of 2 N Fock states for N spin-1/2 spins defined by fixing all the σ z

[ i] for i = 1 . . . N , labelling them by |n with n = 1 . . . 2 N . These form a complete basis and may be thought of as the (2 N ) vertices of an Ndimensional hypercube. To each edge/Fock state assign an energy at random, drawn from a distribution

P (E) = 1 √ πN exp -E 2 /N
(ensuring extensivity of the energies, bandwidth etc). [ i] is the Laplacian on the hypercube so that Γ plays the role of a hopping amplitude. Overall, the Hamiltonian is the Anderson problem with hopping Γ and on an N -dimensional hypercube. Formally, the Hamiltonian is given by Ĥ = -Γ ∇2 + V with V the random potential. This model is not local in real space, but it is local on the hypercube, ie, the state space of a spin-1/2 model.

Figure 1 :

 1 Figure1: Plot of driving frequency ωc below which the system delocalises as a function of disorder amplitude w. The shaded areas correspond to delocalisation. The red dots are obtained from finite-size studies of the level statistics of the system. The disorder amplitude wc is the value below which the undriven system is delocalised in the absence of driving. The blue line is a guide to the eye.

Figure 2 :

 2 Figure 2: Main plot: Probability distribution of the eigenstate expectation values (EEVs) of the density at site j = 3.Left (right): Driving with a low (high) frequency (see Fig.1) results in the probability distributions which does (does not) develop a central peak upon increasing system size, signalling delocalisation[2]. Data is disorder averaged over 10 4 (100) realisations for L = 14 (L = 16). Inset: Level statistics parameter versus inverse system size in the localised (bottom, red) delocalised (top, blue) phases. The parameter η = ´ds sP (s) with P (s) the probability distribution of the level statistics[1, 3], taking the value η P/CU E in the localised/delocalised regime. Data averaged over 1000 disorder realisations for L = 8, 10, 12, 100 realisations for L = 14.

5 Figure 4 :

 54 Figure 4: Driving the QREM. The top left figure shows the participation ratio φ for the eigenstates of the undriven model, showing a mobile region (blue) surrounded by a localised region (red).Driving with frequency ω/J = 0.1 and amplitude δ/J = 0.2 (top right) causes all states at a given Γ0 to become as delocalised as the least localised state at that Γ0 in the undriven model. This is also shown in the bottom panel which shows φ for Γ0 = 0.01, 0.1, 0.5 (red, blue and green line, from top to bottom) in the absence (presence) of driving with darker (lighter) colour. The driven points always lie below the undriven points for the corresponding Γ0. This is due to the strong mixing of all undriven eigenstates by the driving. All data in this figure is for 8 spins and averaged over 1000 disorder realisations.

Figure 5 :

 5 Figure 5: Level statistics for various disorder amplitudes w/J as a function of driving frequency ω. The driving amplitude is δ/J = 0.1w/J, ω/J, and each point represents an average over 10000 disorder realisations. The dashed vertical lines indicate half the width of the energy spectrum; for ω greater than this, our results cannot be extrapolated to the thermodynamic limit (see IV). The colours correspond to system sizes L = 8, 10, 12 from bottom to top for the smallest ω. The values ηGOE and ηP correspond to the presence and absence of level repulsion, respectively, which in turn correspond to localised and delocalised phases. The dotted vertical lines correspond to the typical spectral width of the system, for frequencies above which our results cannot be used to infer the thermodynamic limit.

  loc. length vs eigenstate energy for w/J=5

Figure 7 :

 7 Figure 7: Single-particle localisation length for a noninteracting Anderson problem with parameters similar to the typical values used in our main text. The localisation length is well below most of the system sizes we have used.

Figure 8 :

 8 Figure 8: Comparison between actual density of states and Gaussian fit.

•

Figure 9 :

 9 Figure 9: Width of DOS as a function of disorder amplitude for two system sizes L = 10, 12. This was obtained by explicitly calculating the DOS for 100,400 realisations of the disorder for L = 12, 10, respectively.

||Figure 10 :

 10 Figure10: Approach of H ef f to undriven HT LS with increasing frequency. This is the limiting form for large h/δ (weak driving); in practice, even h/δ = 1 gives a result indistinguishable from this.

Figure ?? shows a plot of ln |c α | 2 (

 2 the logarithimic scale is necesary as |c α | 2

Figure 11 :

 11 Figure 11: Stroboscopic demonstration of the spreading of an initial energy-localised state driven via alternating between two Hamiltonians and forcing dephasing by hand (see Eqs. 3 and 4 in the main text for the Hamiltonians). The parameters used are system size L = 10 at half-filling, driving amplitude δ/J = 0.2, disorder amplitude w/J = 8 (inside the localised phase) and interactions V1/J = V2/J = 1. The system delocalises in energy even though both Hamiltonians are localised as a direct result of dephasing.

Acknowledgements-We thank J. Bardarson, M. Haque, V. Khemani, J. Kjall, V. Oganesyan, S. Sondhi, O. Tiele-