Gabriel Durin

Albane Fontaine

Jean-Claude Berthet

Emmanuel Nicolas

Pierre Thuéry

Thibault Cantat
email: thibault.cantat@cea.fr

Metal-free Catalytic Hydrogenolysis of Silyl Triflates and Halides into Hydrosilanes

The metal-free catalytic hydrogenolysis of silyl triflates and halides (I, Br) to hydrosilanes is unlocked by using arylborane Lewis acids as catalysts. In the presence of a nitrogen base, the catalyst acts as a Frustrated Lewis Pair (FLP) able to split H2 and generate a boron hydride intermediate capable of reducing (pseudo)halosilanes. This metal-free organocatalytic system is competitive with metal-based catalysts and enables the formation of a variety of hydrosilanes at r.t. in high yields (>85 %) under a low pressure of H2 (≤ 10 bar).

Introduction

Hydrosilanes are useful compounds for the production of a variety of organosilicon compounds through hydrosilylation of alkenes or dehydrocoupling reactions. [1] They also promote, in mild conditions, the reduction of functional groups such as esters [2] or amides [3] with high selectivity. In comparison with apolar dihydrogen, the more reducing redox couple E 0 (Si(OEt)4(l)/SiH4(g (-0.51 V vs. NHE) and the polarized and weaker Si-H bond (BDESi-H = 95 kcal.mol -1 < BDEH-H = 104 kcal.mol -1 ) offer some thermodynamic and kinetic advantages [4] relevant for the reduction of oxygenated chemical feedstocks (lignin, plastics and CO2) to chemical fuels and value-added products . [5] Because classical routes for the production of hydrosilanes are energy demanding, [1b, 6] alternative catalytic syntheses that transform Si-X (X = halides, alkoxides) precursors into Si-H groups are appealing. This endeavor has motivated the recent development of catalytic hydrogenolysis routes to hydrosilanes: in the presence of a suitable base, able to facilitate the thermodynamics of the reaction, Si-X bonds (X = OTf, I, Br, Cl) were successfully reduced to Si-H bonds. Yet, the catalysts reported so far by the groups of Shimada, Schneider and Cantat, [7] all use catalytic systems based on molecular complexes of noble metals (Ir, Ru) (Scheme 1, top). To improve the sustainability of this reaction and gain fundamental insights into the generation of hydrosilanes, we have sought to unlock the first metal-free catalytic reduction Si-X into Si-H bonds under H2. Using boranes as catalysts, hydrosilanes (Me3SiH, Et3SiH, Ph3SiH and i Pr2SiH2) were generated in yields up to 91 %, in the presence of an amine base.

Scheme 1. Examples of catalytic routes that convert silyl halides and triflates into hydrosilanes with H2 as hydride source.

Mechanistic investigations derived from the iridium and ruthenium catalyzed hydrogenolysis of Si-X bonds have shown that the generation of a metal hydride from H2 is key to yield a hydrosilane by hydride transfer to the silicon atom. [7c-e] To perform this transformation without a metal, we have sought to use borohydrides as hydrogen transfer reagents. Indeed, borohydrides are known to convert halosilanes into hydrosilanes, as reported by the groups of Klejnot or Nakazawa for instance, who highlighted the stoichiometric reduction of chlorosilanes into hydrosilanes with lithium or sodium borohydride. [8] Moreover, borohydride species can be generated from H2 using Frustrated Lewis Pairs (FLP) chemistry as demonstrated by Stephan and others if an appropriate Lewis base is present. [9] The catalytic hydrogenation of organic functionalities using FLPs has been reported for alkynes, [10] alkenes, [11] ketones, [12] silyl enol ethers, [13] anilines, [14] imines, [15] enamines, [16] amides, [17] aza-Morita-Baylis-Hillman adducts, [18] esters, [19] and recently for the hydrodehalogenation of benzylhalides. [20] In the case of silyl halides and pseudohalides, the presence of a stoichiometric quantity of a nitrogen base is necessary since the reaction is otherwise endergonic. [7c-e] Interestingly, the group of Rieger reported the cleavage of H2 with B/N Frustrated Lewis Pairs (amines and perfluorinated triarylboranes). [21] The group of O'Hare later demonstrated that the generated borohydride is capable of reducing CO2. [22] This result prompted us to consider aryl boranes as catalysts for the hydrogenolysis of Si-X into Si-H bonds, together with a stoichiometric amount of nitrogen bases, effectively forming an FLP system in solution. Using B(C6F5)3 (10 mol%) and TMP (2,2,6,6-tetramethylpiperidine) (1.1 equiv.), only traces of Me3SiH were obtained from Me3SiOTf under 10 bar of H2 at r.t. in CD2Cl2 (Table 1, entry 1). The reactivity of a variety of boranes, i.e. BPh3, B(4-F-C6H4)3, B(2-F-C6H4)3, B(2,6-F2-C6H3)3 and HB(C6F5)2 with distinct Lewis acidities [23] was thus evaluated. The catalytic performance increased slightly with Piers borane [24] HB(C6F5)2 which afforded 21 % NMR yield of Me3SiH (Table 1, entry 2). In contrast, B(2,6-F2-C6H3)3 andB(2-F-C6H4)3 proved to be significantly more reactive, affording Me3SiH in excellent yields (88 and 85 % respectively) (Table 1, entries 3 and4). Compounds BPh3 and its para-F analogue, B(C6F5)3, displayed a low efficiency with a poor Me3SiH yield of 4 % (Table 1, entries 5 and6). With the most reactive FLP system, i.e. the TMP/ B(2,6-F2-C6H3)3 (Table 1, entry 4), a reduction in the yield of Me3SiH was observed either by decreasing the H2 pressure from 10 to 5 bar (88 to 66 % yield, see ESI Section 2.1.6) or the borane catalyst loading from 10 to 5 mol% (67 % yield, see ESI Section 2.1.7). These results represent the first examples of the metal-free hydrogenolysis of (pseudo)halosilanes.

Table 1. Screening of borane catalysts for the hydrogenolysis of Me3SiOTf. [a] Entry Borane

Computed hydricity [25] [kcal.mol -1 ]

Conv. [b] [%] Yield (Select.) [b] [%] (2-F-C6H4)3 48 [c] 96 85 (88) 5 B(4-F-C6H4)3 37 4 4 (99) 6

BPh3 36 5 4 (80) [a] General conditions: Me3SiOTf (0.1 mmol), TMP (0.11 mmol), borane (10 μmol), CD2Cl2 (0.5 mL), H2 (10 bar), 25 °C. [b] Conversions, selectivities, and yields were determined through integration of the Me3Si-signals by 1 H NMR versus an internal standard (1,3,5trimethylbenzene). [c] This value was calculated for this work.

The NMR yields of Me3SiH were plotted against the hydridicity of the different borane catalysts, which corresponds to the energy required to release a hydride ion H -from a putative borohydride [R3BH] -(Figure 1). B-H hydricity values can be obtained from the hydricity scale computed by Heiden et al. for main group hydrides; [25] they quantify the stability of the borohydrides and assist in the evaluation of their reducing power. Figure 1 highlights a volcanotype distribution similar to those resulting from the Sabatier principle for heterogeneous catalysts. [26] It reveals that borohydrides having the lowest hydricities, such as [HBPh3] - (36 kcal.mol -1 ), or the highest, such as [H2B(C6F5)2] -and [HB(C6F5)3] -(61 and 65 kcal.mol -1 ), afford poor yields in hydrosilane. This plot reveals that a balance must be achieved for an efficient catalytic turnover, where the borane is acidic enough to yield a borohydride from the activation of H2, yet still able to transfer a hydride ligand to silicon in a subsequent step.

Figure 1. Volcano plot with NMR yield, obtained from Table 2, plotted against the hydricity [25] of the different arylborane catalysts.

Table 2. Screening of bases for the hydrogenolysis of Me3SiOTf in CD2Cl2. [a] Entry Base pKa [b] Conv. [c] [%] Yield (Select.) [c] [%] 1 2,6-lutidine [a] conditions: Me3SiOTf (0.1 mmol), base (0.11 mmol), B(2,6-F2-C6H3)3 (10 μmol), CD2Cl2 (0.5 mL), H2 (10 bar), 25 °C [b] pKa in MeCN. [27] [c] Conversions, selectivities, and yields were determined by 1 H NMR spectroscopy, through integration of the Me3Si signals versus an internal standard (1,3,5-trimethylbenzene).

The role of the nitrogen base was then evaluated by considering a variety of classical and sterically congested amines, [22,28] i.e. 2,6-lutidine, trialkylamines NEt3 and i Pr2NEt, and piperidines (TMP and PMP = 1,2,2,6,6-pentamethylpiperidine) using the conditions depicted in Table 2. With 2,6-lutidine, the reaction led to the formation of Me3SiH in 15 % yield after 24 h at r.t. (Table 2, entry 1). [START_REF]who demonstrated that when using lutidine and B(C6F5)3, silyl halides are not reduced even under H2[END_REF] Replacing 2,6-lutidine with trialkylamines NEt3 or i Pr2NEt increased the yield in Me3SiH (18 and 55 %, respectively) (Table 2, entries 2 and 3). This yield reached 88 % with the bulky piperidine TMP (Table 2, entry 4 or Table 1, entry 3) but dropped to 24 % with the N-methylated derivative of TMP (1,2,2,6,6-pentamethylpiperidine, PMP), thus underlining a possible influence of the NH group (Table 2, entry 5). A plausible explanation for such a behavior might be the presence of hydrogen bonding N-H•••F interactions between the secondary amine and the fluorine atoms on the triarylborane, reminiscent of the adduct described by Bourissou et al., [30] which pre-organizes the FLP for a more efficient activation of H2 and overall, a more favorable catalytic process. [31] The most productive bases have a pKa greater than 18, while 2,6-lutidine (pKa = 14) afforded the lowest yield in Me3SiH. Nonetheless, as noted previously with transition metal catalysts, [7d] the Brønsted basicity is not the sole parameter influencing the reactivity of the nitrogen base: NEt3 and i Pr2NEt are readily trapped by Me3SiOTf to form the silylated ammonium salts [R3NSiMe3][OTf], which has a decreased reactivity compared to the free silyl triflate. These silylated ammonium salts were, in contrast, not observed with the piperidines for which the hydrogenolysis proceeded in CD2Cl2 with gradual precipitation of the insoluble [TMPH][OTf] or partially soluble [PMPH][OTf] salts (Table 2, entries 4 and 5). When performed with stirring, the reaction proved to be faster in benzene than in dichloromethane, reaching a maximum yield of 91% after 3 h instead of 24 h using the TMP/B(2,6-F2-C6H3)3 pair (ESI, Section 2.1.5).

The generality of this catalytic system using these optimized conditions was evaluated with a series of silyl triflates and halides (Table 3). Similar to Me3SiOTf, Et3SiOTf was fully converted into Et3SiH (90 % yield) with excellent selectivity within 3 h (Table 3, entries 1 and 2). Formation of the bulky Ph3SiH was much slower and required 48 h to reach 88 % yield from Ph3SiOTf (Table 3, entry 3). The double hydrogenolysis of the bis triflate i Pr2Si(OTf)2 was twice as fast as the hydrogenolysis of Ph3SiOTf and provided i Pr2SiH2 in 86% yield (Table 3, entry 4). Attempts to synthesize i Pr2SiH(OTf) by using only 1.1 equivalent of TMP afforded i Pr2SiH2 as the major product (41 %) with low yields of the desired i Pr2SiH(OTf) (9 %) (Table 3, entry 5). Compared to Me3SiOTf, the reduction of Me3SiI and Me3SiBr was slower requiring 7 h and 16 hto reach 88 % and 71 % yields in Me3SiH, respectively (Table 3, entries 6-7).

Finally, only traces of hydrosilane could be detected in the hydrogenolysis of Me3SiCl (Table 3, entry 8). These findings are consistent with the trends obtained using organometallic catalysts for the hydrogenolysis of (pseudo)halosilanes, pointing to a more difficult cleavage of the Si-Cl bond. [7c, 7e] The hydrogenolysis of (pseudo)halosilanes is somewhat more efficient using the borane catalyst compared to the known Ir and Ru catalysts, which require prolonged reaction times and higher temperatures. For example, Me3SiH was obtained in 11 % yield from Me3SiBr using an iridium amido catalyst and i Pr2NEt after 48 h at 60 °C. [7a] In the case of Ru(II) bearing a cooperative ligand, improved performances are reported: Me3SiH was obtained in 85 % yield from Me3SiOTf after 18 h at r.t., with only 1 mol% catalyst loading. These results are close to the 91 % yield obtained after 3 h with 10 mol% loading for our borane-catalyzed reaction (Table 3, entry 1).

To gain insights into the mechanism of the reaction, stoichiometric experiments have been conducted (Scheme 2). In line with the findings of Paradies et al., [32] Table 3. Scope of the reaction. [a] Entry Reagent R4-nSiXn Product R4-nSiHnXn-1

Reaction time [h]

Conv. [b] [%] Yield (Select.) [b] [%] 1 Me3SiOTf Me3SiH [c,d] i Pr2Si(OTf)2 i Pr2SiH2 24 94 86 (91)

5 [d] i Pr2Si(OTf)2 <1 traces [a] General conditions: R4-nSiXn (0.1 mmol), TMP (0.11 mmol), B(2,6-F2-C6H3)3 (10 μmol), C6D6 (0.5 mL), H2 (10 bar), 25 °C [b] Conversions, selectivities, and yields were determined by 1 H NMR spectroscopy, through integration of the R4-nSi signals versus the internal standard 1,3,5-trimethylbenzene. [c] 0.22 mmol of TMP was used. [d] In CD2Cl2. ). This equilibrium might in turn slow down the catalysis, due to quenching of the vacant site of the Lewis acid catalyst. Overall, the rate determining step is the splitting of H2 with an energetic span of 20.2 kcal.mol - 1 (TS1) consistent with the catalytic experimental conditions (10 bar H2, r.t., 3 h for full completion at 10 mol% catalytic loading). A kinetic study based on the time-scale normalization method described by Burés [33] was performed and revealed a partial order of 1 for the base and 0 for the silyl triflate (see ESI, section 2.3). Such values are in agreement with the DFT calculations and the experiments. A complex order (ca. 0.1) for the borane was also found which shows the detrimental influence of the interaction between B(2,6-F2-C6H3)3 and TfO -.

Scheme 3. Computed mechanism for the hydrogenolysis of Me3SiOTf with the FLP (TMP/B(2,6-F2-C6H3)3) (M06-2X/6-311+G(d) level of theory, solvent (benzene) effects are taken into account with SMD).

Capitalizing on this mechanistic knowledge, we next targeted the hydrogenolysis of chlorosilanes, an appealing yet challenging class of substrates. [7] Regardless of the nature of the base, B(2,6-F2-C6H3)3 proved unproductive in the conversion of Me3SiCl to Me3SiH. Nonetheless, as Me3SiCl has a lower reactivity than Me3SiOTf, we have considered a mildly acidic borane (BPh3) in the presence of the strong phosphazene base BTPP (tert-butyliminotri(pyrrolidino)phosphorane) to split H2 and generate a highly reactive borohydride (hydridicity of 36 kcal.mol -1 ) [25] (Scheme 4). When that particular combination was used, 28 % of Me3SiCl was converted after 72 h, forming Me3SiH in 26 % yield. To try to improve this reaction, chloride abstractors were added to the reaction mixture. [7b, 7c] While NaI and Na[BPh4] did not show any influence (25 and 26 % yield respectively), the addition of NaOTf had a positive effect on the reaction yield, and Me3SiH was formed in 52 % yield after 72 h. This particular reaction will be further studied and optimized.

  [HB(2,6-F2-C6H3)3][TMPH] could be generated in 90 % yield by reacting B(2,6-F2-C6H3)3 and TMP (1.1 equiv.) under 10 bar of H2 at r.t. in C6D6. Addition of Me3SiOTf to this solution yielded Me3SiH (93 % yield) within 5 minutes at r.t., together with B(2,6-F2-C6H3)3 and a white deposit of [TMPH][OTf].

Scheme 2 .

 2 Scheme 2. Stoichiometric experiment for the H2 splitting by the FLP B(2,6-F2-C6H3)3/TMP and further hydride transfer into Me3SiOTf.These two reactions suggest a mechanism in two steps depicted in Scheme 3, which is supported by DFT calculations at the M06-2X/6-311+G(d) level of theory (using the SMD model to account for the solvation effects due to benzene). The activation of H2 is exergonic (ΔG = -1.3 kcal.mol -1 ) and proceeds via transition state TS1 (ΔG ‡ = 20.2 kcal.mol -1 ) to yield the ion pair [HB(2,6-F2-C6H3)3][TMPH]. This complex transfers its hydride in a barrierless endergonic step (ΔG = + 1.3 kcal.mol -1 ), affording Me3SiH, free borane B(2,6-F2-C6H3)3, and the salt[TMPH][OTf] as by-product. Since boranes are strong oxophiles, the triflate salt is able to coordinate to the free borane, to form the off-cycle adduct [(TfO)B(2,6-F2-C6H3)3] - (ΔG = + 0.6 kcal.mol -1 ). This equilibrium might in turn slow down the catalysis, due to quenching of the vacant site of the Lewis acid catalyst. Overall, the rate determining step is the splitting of H2 with an energetic span of 20.2 kcal.mol - 1 (TS1) consistent with the catalytic experimental conditions(10 bar H2, r.t., 3 h for full

Acknowledgements

For financial support of this work, we acknowledge CEA, CNRS, the University Paris-Saclay, CINES (HPC Computing time on Occigen, A0100806494) and the European Research Council (ERC Consolidator Grant Agreement no. 818260). T.C. thanks the Fondation Louis D.-Institut de France for its major support.

In summary, we have developed an unprecedented transition metal free catalytic route for the hydrogenolysis of silyl triflates and halides (I, Br), using di-and triarylboranes as catalysts. A delicate balance of the Lewis acidity of the borane catalyst is essential for efficient catalysis. In the presence of the base, the catalyst acts as a B/N FLP and the combination of B(2,6-F2-C6H3)3 and TMP proved the most efficient to afford hydrosilanes in high yields at room temperature and under a low pressure (10 bar H2). Experimental reactions, kinetic data and theoretical calculations support a two-step mechanism with the rate-determining step being the heterolytic splitting of H2. This result paves the way for new catalytic systems for the sustainable synthesis of hydrosilanes, a necessary step towards a closed silicon cycle.

Conflict of interest

The authors declare no conflict of interest.


Dr Federica Solano

Pascale Auban-Senzier

Dr Iwona Olejniczak

Bolesław Barszcz

Tomasz Runka

Pere Alemany

Enric Canadell
email: canadell@icmab.es

Dr Narcis Avarvari
email: narcis.avarvari@univ-angers.fr

Nicolas Zigon
email: nicolas.zigon@univ-angers.fr

E Canadell

Bis(Vinylenedithio)-Tetrathiafulvalene-Based Coordination Networks

Keywords:

Novel coordination polymers embedding electroactive moieties present a high interest in the development of porous conducting materials. While tetrathiafulvalene (TTF) based metal-organic frameworks were reported to yield through-space conducting frameworks, the use of S-enriched scaffolds remains elusive in this field. Herein is reported the employment of bis(vinylenedithio)tetrathiafulvalene (BVDT-TTF) functionalized with pyridine coordinating moieties in coordination polymers. Its combination with various transition metals yielded four isostructural networks, whose conductivity increased upon chemical oxidation with iodine. The oxidation was confirmed in a singlecrystal to single-crystal X-ray diffraction experiment for the Cd(II) complex. Raman spectroscopy measurements and DFT calculations confirmed the oxidation state of the bulk materials, and band structure calculations assessed the ground state as an electronically localized antiferromagnetic state, while the conduction occurs in a 2D manner. These results are shedding light to comprehend how to improve through-space conductivity thanks to sulfur enriched ligands.

INTRODUCTION

Porous conductive materials are appealing for a wide range of applications, in particular as battery or fuel cell electrodes or as supercapacitors. [1][2][3] In that area metal-organic frameworks (MOFs) span a tremendous enthusiasm thanks to their high versatility and various pore size availability. MOFs are crystalline porous materials constructed from metal ions or clusters and organic linkers whose properties are directed by the ligands and metal physical properties and spatial organization. [4,5] Thanks to their crystalline porous structures and to the various chemical compositions accessible, these materials are extremely beneficial to gas storage, controlled drug delivery, optical sensing, heterogeneous catalysis or water harvesting. [6][7][8][9][10] Typically, the vast majority of the networks are formed with organic ligands having a huge HOMO-LUMO gap and/or a poor orbital overlap with the inorganic part, precluding the formation of long-range electronic pathways. Consequently, MOFs usually show poor electrical conduction properties. Conductive MOFs are expected to yield applications in electrochemical sensing [11] , thermoelectrics [12] , fuel cells [2] , and supercapacitors [13] .

It is only lately that electroactive MOFs have emerged as a new class of promising materials, mixing porosity and conductivity. Several mechanisms are reported to bring conductivity in MOFs, such as through-bond, through-space, redox-hopping or guest promoted redox hopping. [14,15] In the "throughspace" approach, and orbital overlap between adjacent organic moieties creates a conduction band within the material. This is particularly true for ligands presenting a tendency to arrange themselves through  stacks. For example, compounds with a large  surface such as naphthalenediimide or triphenylene have been reported to yield conductive MOFs, [16][17][18] or even, recently, metallic ones. [19] In this regard, ligands scaffolds based on tetrathiafulvalenes (TTFs) have been used for a dozen of years in the structure of MOFs to improve their charge mobility and conductivity. [20,21] TTF-tetrabenzoate (TTFTB) represents a landmark in this field, while TTFtetracarboxylic acid provided semiconducting nonporous solids. [22] Dincă et al. demonstrated the high charge mobility of the TTFTB-based networks, and the dependency of the electrical properties towards the cation diameter. [20,23] Using metal-dithiolene in the ligand scaffolds provides an alternative platform to the tetrathiafulvalene and leads to heterobimetallic conductive networks, whose integration in electrodes provided an efficient glucose sensor. [24,25] Pyridine stands as another ubiquitous coordinating moiety for MOF synthesis. They have been also widely used to implement TTF as a redox active ligand in coordination chemistry. [26,27] The TTF-tetrapyridyl ligand was extensively explored for MOFs synthesis and was combined with Fe II to yield spin-crossover materials. [28] The vast coordination chemistry of the pyridyl ligand afforded networks with various topologies, [29] where a strong influence of the counterion was noted. [30] The expansion of the TTF scaffold to a flexible tetrathiopyridyl-TTF provided polymers whose structural properties depended on the conformation adopted by the ligand. [31] Alternative design with an ethylenedithiocapped TTF-bis(thiopyridyl) supplied 1-D coordination polymers where the redox active cores overlapped in an interdigitated fashion, thus enhancing the conduction properties. [32] Tetrathiafulvalene (TTF) and its derivatives have been widely investigated for their impressive electron-donating abilities and for their reversible oxidation to the radical cation (TTF +• ) and dication (TTF 2+ ) forms. [33][34][35] The possibility of obtaining an open-shell within a MOF structure paves the way towards highly conductive materials.

Moreover, extended scaffolds such as bis(ethylenedithio)tetrathiafulvalene (BEDT-TTF) contain four extra sulphur atoms and saturated non planar CH2CH2 group, thus favouring the formation of S•••S multiple side intermolecular interaction in the layers of the radical cations to generate superconducting states at low temperatures for BEDT-TTF based organic conductors. [34,[START_REF] Ishiguro | Organic Superconductors[END_REF] Herein, an unsaturated version of BEDT-TTF, bis(vinylenedithio)tetrathiafulvalene (BVDT-TTF), [START_REF] Kobayashi | [END_REF][38][39][40] functionalized by four pyridines (1, Figure 1) was used as redox active ligand in combination with Cu(II), Zn(II), Cd(II) and Hg(II) to yield four isostructural coordination polymers frameworks as single crystals. After chemical oxidation of the frameworks via iodine doping, pronounced variation in conductivity was revealed in the case of 1-CdCl2 and 1-HgCl2. The oxidation states of the donor molecules in the series of new coordination polymers have been investigated by single-crystal Raman spectroscopy in both the neutral and oxidized state, and band structure calculations have been performed based on the structural data.

RESULTS AND DISCUSSIONS

Synthesis and description of the networks

BVDT-TTF-Py4 1 was prepared according to the literature procedure, [41] starting by a Diels-Alder reaction between 1,2-di(pyridin-4-yl)ethyne [42] and 1,3-dithiolane-2,4,5-trithione. [43] After transchalcogenation with mercury acetate and subsequent phosphite mediated coupling, BVDT-TTF-Py4 was obtained with an overall yield of 60%. Replacement of the ethylene bridge of BEDT-TTF by a vinylene one creates a more rigid structure and avoid the formation of diastereomers, which would be formed upon substitution of the ethylene bridge in BEDT-TTF. [44] The crystalline MOFs were synthesized by slow diffusion at RT during five days: a solution of ligand in DCM (1 mg/mL) was placed in a tube, a buffer solution of DCM/MeOH 1/1 and a solution (1 mg/mL) of metal (CdCl2, HgCl2, ZnCl2) in MeOH were successively layered. The same procedure was used for the synthesis of 1-CuCl2 with the ligand dissolved in CHCl3.

Single crystals suitable for SCXRD were obtained with the four different metals. The four networks were all isostructural (monoclinic P21/c, with medium values for a = 13 Å ; b = 9.6 Å ; c = 16.2 Å ;  = 94°, see SI for details). The structure of 1-ZnCl2 network is presented Figure 2. These four materials all have the same formula 1-MCl2 (M = Cu, Zn, Cd or Hg). The asymmetric unit contains half a ligand and half a metal centre. Additional non-coordinated solvent molecules are also observed in the structures. The metal atoms stand in an octahedral environment, with four pyridyl moieties forming the square base while two chlorine atoms are located in apical positions, with L-M-L angles in the 83 to 96° range (Table 1). The four pyridyl groups around one metal centre come from four different ligands. The metal-ligand distances are reported in Table 1. The longest M-Cl distance of 2.891 Å is observed for 1-CuCl2. The central C=C bonds are in the 1.34 Å range for 1-CuCl2, 1-ZnCl2 and 1-HgCl2, which is in the expected range for a non-oxidized TTF. The distance is slightly longer at 1.36(2)Å for 1-CdCl2, which is probably due to the uncertainty on the bond measurement. Additional Raman experiments (vide infra) did not show a characteristic peak for oxidized BVDT-TTF for pristine 1-CdCl2. The ligands are organized through coordination bonds in 2-D layers (Figure 2b) where metals are separated by 9.5-9.8 Å along the b axis, and 20.8-21.7 Å in the perpendicular direction. The layers are superimposed in a shifted fashion, the central C=C bond of the ligand in a 2-D layer standing below the metal centre of the following layer (Figure 2a and2c). Between two ligands of different layers, the smallest S-S distance is in the 3.21 -3.24 Å range in the four structures. 1-D channels are formed within the packing and are occupied by solvent molecules, representing ca. 30% of the overall crystal volume with pore size of 4.3 x 4.7 Å. The powdered crystals of 1-CuCl2, 1-CdCl2 and 1-HgCl2 preserve their crystallinity, as shown by PXRD after drying in the air at room temperature for 1 day (see SI), but the crystals of 1-ZnCl2 lost their crystalline character and only broad bands were observed by PXRD. Nevertheless, non-grinded single crystals are kept in the air several days without significant loss of diffraction intensity for all the materials.

In order to increase the overall conductivity of the materials, the presence of oxidized BVDT-TTF is necessary. Commonly, crystals are immerged in an iodine solution or exposed to iodine vapours. By exposure of 1-MCl2 dry crystals to I2 vapours overnight, a stringent change of colour from red to dark purple is observed in every case, which is characteristic of the formation of oxidized thiafulvalenes. A loss of crystal quality is visually observed during the oxidation process. Interestingly, this change is reversible and the original colour is regenerated after one day. To our delight, crystals of 1 •+ -CdCl2-I3 - have been formed in a single-crystal to single-crystal fashion and have been characterized by SCXRD despite a loss of data quality due to a decrease of the overall crystallinity. The network is in the C2/m space group, with cell parameters close to the parent 1-CdCl2 cell. A molecule of I3 -has now replaced the solvent molecules within the pores, making the structure densely packed (Figure 3). Due to the lack of precision on the bond length, no characteristic lengthening of the central C=C bond is observed. PXRD were performed on the oxidized materials and present a proper match with the simulated spectra, although broad peaks indicate a possible loss of crystallinity (see SI).

Despite a change of the space group from P21/c to C2/m, the overall structural arrangement of the ligands and metal is preserved between 1-CdCl2 and 1 •+ -CdCl2-I3 -, both networks being practically superimposable. The unit cell parameters are similar +/-2% in length and angles. The S-S interligand bond distances remain in the 3.2-3.7 Å range. The dihedral angles between the pyridyl and vinyl moieties changes from 51-57° to 88° between 1-CdCl2 and 1 •+ -CdCl2-I3 -.

Solid-state cyclic voltammetry

Electrochemical-based cyclic voltammetry (CV) measurements were performed to analyze the redox process of MOF. A custom-designed setup was adopted since the MOFs are not soluble in any solvents.

The approach is based on a solid-state electrochemical configuration described in the experimental part.

In the Table 2 are summarized the oxidation potential and reduction potential of Tetrapyridine BVDT-TTF 1, 1-HgCl2, 1-CdCl2, 1-ZnCl2, 1-CuCl2. Cyclic voltammetry of the ligand (Figure 4) shows a pair of reversible redox waves centred at E1/2 = 0.620(V) in a CH2Cl2 solution. The anodic peak of the ligand in solid state is at a lower voltage compared to those recorded in solution because of a slower diffusion process. It is indeed necessary to consider that the transport of electrons is probably limited to a tight region of the solid particles in contact with the electrode. [45] The broadness of the peak shape is also indicative of the high resistivity of the solid material. The values of the ligand are higher when compared with those of the related BEDT-TTF donor (whose first oxidation potential is E1/2 = 0.49 V). The CV curve of 1-HgCl2 (Figure 5, see SI for 1-CdCl2, 1-ZnCl2, 1-CuCl2) is selected to describe one quasireversible redox process corresponding to the formation of the BVDT-TTF radical cation at 0.74 V. In the case of MOFs, it is expected that the oxidation process will be extended along the entire solid particle thanks to the presence of pores allowing the passage of electrolyte species. Pore size therefore assumes a key role for redox conductivity. The redox properties of the ligand are therefore maintained in the structure of the four networks.

Raman spectroscopy

Raman spectroscopy experiments and theoretical calculations were performed to assess the oxidation state of BVDT-TTF in our compounds in the non-oxidized and oxidized species. The ligand 1 is characterized by strong σ-donating and π-accepting character [41] and the coordination bond between pyridine and metal is responsible for notable cooperative electronic properties of the resulting material.

C=C stretching modes of 1 usually observed in Raman spectra stands in the frequency range 1300 -1700 cm -1 . These modes are known to display a strong dependence of frequency on charge and have been widely used in characterization of charge states in BEDT-TTF-based salts. [46,47] DFT calculations of vibrational modes for both the neutral 1 and the radical cation 1 •+ have been performed in order to estimate the charge localized on 1 in the selected-based coordination polymers. Charge-sensitive vibrations can be used to determine the charge state. The tetrapyridine-substituted BVDT-TTF molecule, based on the fully unsaturated version of BEDT-TTF, [41] is not flat but assume an approximately orthogonal orientation with respect to the main molecular plane. This is clearly seen in Figure 6 that displays comparison of the frontier molecular orbitals in the neutral and the cation. As a consequence of this steric arrangement, molecular dynamics of 1 significantly differs between the neutral and the cation. The charge-sensitive vibration mode labelled C (Figure 7) displays the strongest frequency dependence on charge. In case of the neutral 1, the C mode features in-phase stretching of the three inner double C=C bonds, but the vibration is confined to the central C=C bond for the cation.

The theoretical frequency of C is 1500 cm -1 for the neutral and 1382 cm -1 for the cation, therefore we observe similar frequency shift of the charge-sensitive mode due to oxidation of the donor molecule as the value characteristic for charge-sensitive modes of BEDT-TTF. [46] Figure 8 shows Raman spectra measured at room temperature for selected coordination polymers based on neutral 1, together with the experimental and theoretical Raman spectrum of the neutral 1.

The characteristic vibrational structure in the frequency range of the stretching C=C modes is clearly seen in all experimental spectra, in close agreement with the theoretical spectrum. We assign the strong band centred at 1500 cm -1 in the theoretical spectrum of neutral 1, and in the range 1503-1514 cm -1 in experimental spectra, as the charge-sensitive C mode characteristic for the neutral species (Table 3). The presence of a broad feature at 1410 cm -1 in 1-HgCl2 and at 1414 cm -1 in 1-ZnCl2, that we attribute to C of the cation, suggests that a minute amount of BVDT-TTF(Py)4 •+ radicals is present in these materials. Note that in theoretical spectra, Raman intensity at the frequency of C is two order of magnitude higher in case of the cation than in case of the neutral donor. for the neutral and vC +1 for the cation.

In the theoretical spectrum of BVDT-TTF-Py4 + (Figure 9b) the charge-sensitive vC mode, identified as a signature of the cation, appears as a strong band centred at 1382 cm -1 . In the experimental Raman spectra, this mode is located at 1402 in 1 •+ -CdCl2-I3 -(Table 3). On the other hand, the relatively weak feature centred at 1503 cm -1 is suggestive of remaining neutral BVDT-TTF(Py)4 molecules in the structure (Table 3).

Single-crystal resistivity measurements

Conductivity measurements were performed with a two-contact method on single crystals for compounds 1-CdCl2 and 1-HgCl2 before and after oxidation. 1-CuCl2 and 1-ZnCl2 crystals were too small for connection with the electrodes. For 1-HgCl2 the conductivity goes from 2 -2.5•10 -9 S•cm -1 to 1.2 -2•10 -8 S•cm -1 upon oxidation and for 1-CdCl2 from 7 -9•10 -10 S•cm -1 to 4 -6•10 -9 S•cm -1 (2 crystals measured for each compound). This ca. 8-fold enhancement of the conductivity for both compounds is directly related to the partial oxidation of the ligands, yielding an open-shell structure, and may be partly counter-balanced by the loss of crystallinity observed in X-ray.

Band structure calculations

The electronic delocalization in the present coordination networks occurs essentially via direct through-space S … S contacts within the (BVDT-TTF-Py4) sublattice. As shown in Figure 10a where the short S … S contacts are shown as dotted red lines, the conductive path is confined within ab layers well separated by CdCl2 and the pyridine ligands (and solvent molecules, not shown). These layers (Figure 10b) result from the superposition of two layers as those in Figure 2b. These layers are in fact steplayers (Figure 10d) where every step is made of a series of equivalent donors along b. As schematically shown in Figure 10d, the molecules of successive steps turn around opposite directions. Note that step-layers are in fact the conductive components of many molecular conductors and superconductors. [48] Although the a-and c-directions are interchanged, the same description applies for 1 •+ -CdCl2-I3 -. These step-layers ensure a 2D continuous path of S … S interactions through the overlap mode in Figure 10c and should make the present networks relatively immune to defects. The calculated DFT band structure for 1-CdCl2 is shown in Figure 11a where the energy zero refers to the highest occupied level of the HOMO band. It is the typical band structure of a slightly direct band gap semiconductor with hole type conductivity. The HOMO band is folded along the b*and c*-directions because there are two equivalent molecules in the repeat unit. It exhibits similar dispersion along both the c* (-Z) and b* (-Y) directions, i.e. is a typical 2D band. The total dispersion is ~ 0.3 eV which is not far from that of some of the most studied conducting metal-organic frameworks like Zn2(TTFTB) [H4TTFTB= tetrathiafulvalene tetrabenzoic acid], ~ 0.4 eV. [20] The band gap of the two systems are also quite comparable, 1.47 eV for 1-CdCl2 and 1.75 eV for Zn2(TTFTB). It is interesting to compare the results for the two frameworks even if the first is electronically 2D whereas the second is 1D. The shortest S … S contact in the present compound is 3.217 Å which is considerably shorter than in Zn2(TTFTB), 3.757 Å. It is therefore surprising that the band dispersion is somewhat smaller for 1-CdCl2. Despite intrinsic orbital orientation aspects fixed by the coordination network there is another important aspect worth of notice. In the extended BVDT-TTF donor there are two types of S atoms: those of the inner TTF core and those which are out of this core (noted "i" and "o" respectively in Figure 10c). It is well known for similar ligands like ET [bisethylenedithiatetrathiafulvalene] (see also Figure 6 left) that the participation of the outer S pz orbitals into the HOMO is noticeably smaller than that of the inner S ones (for ET the relation is around 1/3). [48,49] In the case of Zn2(TTFTB) the contacts are of the i … i type but in the present case the 3.217 Å contact is of the o … o type (see Figure 10c) and the two other contacts, 3.546 and 3.721 Å are of the i … o type. Thus, the S … S contacts are less effective than in Zn2(TTFTB) because they all implicate S atoms of the o type but there are more numerous and finally the band dispersion is only a bit smaller. These contacts can be tuned for instance varying the angle (see Figure 10d) or the nature of the transition metal. We believe that using these more extended donors, step-layers with band dispersion comparable to those in the M2(TTFTB) family and other frameworks where the conducting path is mostly 1D can be reached and tuned. In addition, because of the 2D conducting path, the conductivity will be more stable and immune to defects. The calculated DFT band structure for 1 •+ -CdCl2-I3 -assuming double occupation of the levels (i.e., a metallic state) is shown in Figure 11b. Now the HOMO band is half-filled and the Fermi level lies inside the band. The important feature is that the topology of this SOMO band is similar to that of the HOMO band for the non-oxidized donor lattice and the total dispersion is also comparable (264 meV in Figure 11b and 292 meV in Figure 11a. Note the larger energy scale in Figure 11a). Consequently, the nature of the S … S interactions already discussed has not experienced a major disturbance upon the oxidation of the donor lattice. The major change concerns the energy separation between the SOMO and the lower fully empty bands but this does not play any important physical role for 1 •+ -CdCl2-I3 -. Since the band is half-filled and given the stoichiometry, the electronic structure for situations where every donor accommodates a single localized electron must be considered. In Figures 11c and11d are shown the DFT band structures for localized states where all donors have an electron with the same spin i.e. a ferromagnetic state (FM), or where all adjacent donors have opposite spins, i.e. a fully antiferromagnetic state (AF). The metallic state was found to be higher in energy (in meV/formula unit) than both the FM and full AF states: Metallic (0), FM (-23.8) and full AF (-34.0), thus confirming the electronically localized nature of 1 •+ -CdCl2-I3 -. This system should therefore exhibit an antiferromagnetic ground state.

The effect of electronic repulsions is clearly seen comparing the band structures of the metallic (Figure 11b) and FM (Figure 11c) states. The spin up (red) and spin down (blue) bands of the FM state are shifted almost rigidly and have the same shape as those in the metallic state. Because of the electron repulsions, a gap develops separating the filled and empty bands so that the conductivity is activated. However, the gap is of a considerably different nature that that occurring in 1-CdCl2 (Figure 11a) and is dominated by the electron-electron repulsions within the SOMO of the donors, which outweigh the effect of the direct S … S overlap. The bands for the more stable full AF state (Figure 11d) are really the superposition of two bands (one mostly associated with spin up and more localized in the donors of one out of every two steps of the layer while the other, with spin down, is more localized in the donors of the other step). The more favorable electron-electron repulsions in the fully AF state lead to the opening of a very noticeable gap between flatter bands as a result of the spatial confinement of electrons with different spins into different steps of the layer. We have checked that other spin configurations within the step-layer do not lead to energies lower than that of the full AF state by carrying out calculations using a doubled unit cell along the a-and b-directions. Calculations using double cells along the interlayer c-direction led to energy differences of the order of less than 1 K per formula unit, which are within the energy error of the calculations, and thus, we believe that there is no magnetic order between layers. Because of the intercrossed nature of the 2D interactions, it is not likely that the conducting and magnetic properties are seriously disturbed by the presence of a small number of non-oxidized donors.

In summary, the use of extended TTF-based donors such as 1 may lead to interesting new conducting coordination networks mostly because of two features: (i) they favor 2D conducting paths which will make these systems considerably less sensitive to defaults, and (ii) the extended nature of the sulfur-containing fragment of the donor decreases the U intra-site electron repulsion term (i.e. the USOMO) which dominates the activation energy. Further engineering of the conductivity can also be anticipated by controlling the S … S contacts, as noted above, that will play on the overlap component of the hopping.

CONCLUSIONS

In conclusion, four unprecedented isostructural networks have been synthesized by the self-assembly of a BVDT-TTF substituted with four pyridyl moieties and a MCl2 metal centre (M = Cu, Zn, Cd, Hg). The networks present solvent occupied pores and are oxidized post-synthetically by exposition to I2 vapours. For the oxidation of 1-CdCl2 to 1 •+ -CdCl2-I3 -, the reaction occurred in a single crystal to single crystal fashion and was followed by SCXRD. Room temperature Raman study of a series of new coordination polymers based on 1 supports the observed charge of the ligand within the networks. In the frequency range of the stretching C=C modes, a charge-sensitive C mode is identified with the help of theoretical calculations using DFT methods. Raman confirmed the oxidized state of the ligand in the bulk crystals. An 8-fold enhancement of conductivity was measured on single crystals of 1-CdCl2 and 1-HgCl2 after oxidation. Band structure calculations unveiled the 2D nature of the conducting path so that these systems should be unsensitive to defaults. The calculations suggest an antiferromagnetic ground state for the conducting layers but no magnetic order between layers. Cyclic voltammetry performed on bulk powder show that the redox properties of the ligand are maintained for all MOFs. This study demonstrates the relevance of the utilization of TTF based ligands possessing additional sulphur atoms in the design of conductive networks thanks to the formation of 2D conductive pathways. In this regard, it is important to implement the wide variety of electroactive structures found for organic conductors within MOFs scaffolds, which is the main direction of our current researches.

EXPERIMENTAL SECTION

Ligand 1 was synthesized according to reported literature procedures. [41] 1-ZnCl2 synthesis:

Ligand 1 (1 mg, 1.54x10 -3 mmol) was dissolved in 1 ml of DCM and placed in a tube (13 cm height, 1 cm diameter). A buffer solution of DCM/MeOH 1/1 (0.5 mL) and a solution of metal ZnCl2 (2 mg, 0.014 mmol) in MeOH (2 mL) were successively layered. After 5 days, red crystals appeared. The crystals were washed several times with ethanol and DCM. C30H16N4S8 Cl2Zn, 3(CH2Cl2) theor. : C, 36.70; H, 2.05; N, 5.19; S, 23.75. Found: C, 35.05 H, 1.74; N, 5.12; S, 25.13.

1-CuCl2 synthesis:

Ligand 1 (1 mg, 1.54x10 -3 mmol) was dissolved in 1 ml of CHCl3 and placed in a tube (13 cm height, 1 cm diameter). A buffer solution of CHCl3/MeOH 1/1 (0.5 mL) and a solution of metal CuCl2 (2 mg, 0.014 mmol) in MeOH (2 mL) were successively layered. After 5 days, red crystals appeared. The crystals were washed several times with ethanol and CHCl3. C30H16N4S8Cl2Cu, (CHCl3) theor.: C, 39.49; H, 1.82; N, 5.94; S, 27.21. Found: C, 39.58; H, 1.96; N, 5.63; S, 27.43.

1-CdCl2 synthesis:

Ligand 1 (1 mg, 1.54x10 -3 mmol) was dissolved in 1 ml of DCM and placed in a tube (13 cm height, 1 cm diameter). A buffer solution of DCM/MeOH 1/1 (0.5 mL) and a solution of metal CdCl2 (2 mg, 0.010 mmol) in MeOH (2 mL) were successively layered. After 5 days, red crystals appeared. The crystals were washed several times with ethanol and DCM. C30H16N4S8Cl2Cd, 2(CH2Cl2) theor.: C, 36.88; H, 1.93; N, 5.38; S, 24.61. Found: C, 36.40; H, 1.88; N, 5.20; S, 25.67.

1-HgCl2 synthesis:

Ligand 1 (1 mg, 1.54x10 -3 mmol) was dissolved in 1 ml of DCM and placed in a tube (13 cm height, 1 cm diameter). A buffer solution of DCM/MeOH 1/1 (0.5 mL) and a solution of metal HgCl2 (2 mg, 7.36x10 -3 mmol) in MeOH (2 mL) were successively layered. After 5 days, red crystals appeared. The crystals were washed several times with ethanol and DCM. C30H16N4S8Cl2Hg, (CH2Cl2) theor. : C, 35.62; H, 1.74; N, 5.36; S, 24.54. Found: C, 34.53; H, 1.67; N, 5.05; S, 24.70.

General procedure for the post-synthetic oxidation. 10 mg of 1-MCl2 were deposited in a 2 mL vial. It was placed in a larger vial containing I2 (ca. 50 mg), the vial was closed and left overnight at room temperature. A colour change from red to dark purple is observed in every case. The crystals were used as such for the subsequent measurements.

X-Ray structure determinations

Details about data collection and solution refinement are given in Table S1. Data collections were performed on a Rigaku Oxford Diffraction SuperNova diffractometer equipped with an Atlas CCD detector and micro-focus Cu-Kα radiation (λ = 1.54184 Å). The structures were solved by intrinsic phasing and refined on F² by full matrix least-squares techniques with SHELX programs (SHELXT 2018/2 and SHELXL 2018/3) [50,51] using the ShelXle and the Olex2 graphical user interfaces. [52,53] All non-H atoms were refined anisotropically and absorption was corrected by multiscan empirical absorption using spherical harmonics with CrysAlisPro program for 1-CuCl2, 1-ZnCl2, 1-CdCl2 and 1-HgCl2. The H atoms were placed at calculated positions and refined using a riding model. Crystallographic data for the five structures have been deposited with the Cambridge Crystallographic Data Centre, deposition numbers CCDC for 1-CuCl2 : 2203496; 1-ZnCl2 : 2203494 ; 1-CdCl2: 2203493; 1-HgCl2: 2203497; 1-CdCl2-I3 : 2203495.These data can be obtained free of charge from CCDC, 12 Union road, Cambridge CB2 1EZ, UK (e-mail: deposit@ccdc.cam.ac.uk or http://www.ccdc.cam.ac.uk).

Solid state cyclic voltammetry

To measure solid state electrochemical behaviour of MOFs, a cylindrical Teflon cell featuring a hole (0.4 cm diameter) in the bottom was used in a vertical configuration, where a glassy carbon plate as working electrode, WE, was tightened from below. The cell was tightened on a copper plate as the electrical contact. A Pt wire and Ag wire were used as counter electrode (CE) and reference electrode (RE) respectively and placed at the top of the cylinder in contact with the electrolytic solution (0.1M TBAPF6in AN) vs Fc+/Fc. The MOFs were placed over the WE and covered with a Nafion membrane, to prevent direct contact between the solid MOFs and the electrolytic solution.

Computational methods

Theoretical calculations of normal vibrational modes for BVDT-TTF-Py4 isolated molecule were performed with Gaussian 03, [54] using the standard 6-31++G(d,p) basis set and the hybrid density functional (B3LYP). The initial geometry used in calculations was prepared based on the experimental X-ray data. The result of the structure optimization corresponds to energy minima since no imaginary frequencies were found. On the basis of optimized structure, vibrational frequencies as well as Raman scattering activities were calculated. The frequencies computed with a quantum harmonic oscillator approximation tend to be higher than experimental ones, and the scaling factor 0.9614 has been applied. [55]

Band structure calculations

The first-principles calculations for the solid were carried out using a numeric atomic orbitals density functional theory (DFT) approach [56,57] developed for efficient calculations in large systems and implemented in the SIESTA code. [58][59][60][61] We used the generalized gradient approximation (GGA) to DFT and, in particular, the functional of Perdew, Burke, and Ernzerhof. [62] To study the relative energies of states with localized electrons, spin polarized band calculations have been undertaken. All calculations included a Hubbard correction term Ueff = U  J = 6.0 eV for the S 3p states. [63] In previous work [64] we have found that this U term on the chalcogen atoms is needed for appropriately describing the electronic structure of molecular conductors were accurate experimental information on the bandwidth and charge transfer is available. Only the valence electrons are considered in the calculation, with the core being replaced by norm-conserving scalar relativistic pseudopotentials [65] factorized in the Kleinman-Bylander form. [66] We have used a split-valence double- basis set including polarization orbitals with an energy shift of 10 meV for S, C, N, Cl, Cd and H atoms. [67] The energy cutoff of the real space integration mesh was 350 Ry. The Brillouin zone was sampled using grids of (7×7×7) k-points. [68] The experimental crystal structures for (BVDT-TTF-Py4)-CdCl2 and (BVDT-TTF-Py4) •+ -CdCl2-I3

-were used for the computations.

Raman measurements

Crystalline samples selected for Raman measurements were between 0.1 and 0.2 mm in length. Raman spectra in the range 1100 -1700 cm -1 were measured at room temperature using Renishaw inVia Raman microscope equipped with a thermoelectrically (TE)-cooled CCD detector and semiconductor laser emitting light in near-infrared region at 785 nm wavelength. Low-frequency Raman spectra were measured at room temperature in a backward scattering geometry with a LABRAM HR800 Raman spectrometer equipped with a microscope and a He-Ne λ=632.8 nm laser line. The spectral resolution of Raman measurements was better than 2 cm -1 , and the power was reduced to about 0.3 mW to avoid sample overheating.

Conductivity measurements

Electrical conductivity measurements at room temperature were performed in two points on 0.2 to 0.3mm long single crystals. Gold wires were glued with silver paste on both ends of the crystals. We have used a Keithley 486 picoammeter to apply a constant voltage (up to 30V) and measure the current, leading to resistance values of the order of 100G

Figure 1 .

 1 Figure 1. Tetra-pyridine bis(vinylenedithio)tetrathiafulvalene (BVDT-TTF-Py4) 1.

Figure 2 .

 2 Figure 2.Single crystal X-ray structure for 1-ZnCl2. a) Side view of the BVDT-TTF stacking, b) view of a single 2D layer and c) packing (solvent molecules and H atoms have been omitted for the sake of clarity. C: grey ; S : yellow ; N : blue ; Cl : green and Zn : dark blue).

Figure 3 .

 3 Figure 3. Single crystal X-ray structure for 1 •+ -CdCl2-I3 -along the c axis (H atoms have been omitted for the sake of clarity. C: grey ; S : yellow ; N : blue ; Cl : green ; Cd : light yellow and I : purple).

Figure 4 .

 4 Figure 4. CV curves of a) 1 mM ligand in solution in 0.1 M TBAPF6 DCM solution and (b) ligand powder with Nafion. WE/GCE for a) and GCP for b). RE: Ag/AgCl/KClsat,. CE: Pt wire. CV profiles were obtained at 50 mV s -1 scan rate.

Figure 5 .

 5 Figure 5. CV curve of 1-HgCl2 crystal powder coated with Nafion. WE: Glassy carbon plate. RE: Ag/AgCl/KClsat,. CE: Pt wire ; 50 mV s -1 scan rate.

Figure 6 .Figure 7 .

 67 Figure 6. Frontier molecular orbitals: HOMO (Highest Occupied Molecular Orbital) of neutral BVDT-TTF-Py4 (left), and -SOMO (Single Occupied MO) of the BVDT-TTF-Py4 •+ cation (right), calculated using DFT methods.

Figure 8 .

 8 Figure 8. (a) Raman spectra of selected BVDT-TTF(Py)4 coordination polymers measured at room temperature with the 785 nm excitation line; (b) room temperature Raman spectrum of neutral 1 measured with the 785 nm excitation line, together with the respective theoretical Raman calculated at the B3LYP/6-31++G(d,p) level of theory. Note that the spectra in both panels are offset for clarity; the charge-sensitive vC mode is labelled as vC 0

Figure 9 .Table 3 .

 93 Figure 9. (a) Raman spectra of 1 •+ -CdCl2-I3 -measured at room temperature with the 785 nm excitation line. (b) Theoretical Raman spectrum of 1 •+ calculated at the B3LYP/6-31++G(d,p) level of theory. The spectra in panel (a) are offset for clarity; the charge-sensitive vC mode is labelled as vC +1 for the cation.

Figure 10 .

 10 Figure 10. Crystal structure of 1-CdCl2 where the short S … S contacts are shown as red dotted lines; for clarity solvent molecules are not displayed and the pyridine ligands have been omitted in b), c) and d). a) Projection view along the b-direction, b) Overlap mode between pairs of 1 donors where the inner (i) and outer (o) types of S atoms are noted, c) top view of a double layer, and d) schematic lateral view showing that the double layers are step-layers with successive steps built from donors turning in opposite senses.

Figure 11 .

 11 Figure 11.DFT band structure for 1-CdCl2 (a). DFT band structure for the metallic (b), ferromagnetic (c) and fully antiferromagnetic states (d) of 1 •+ -CdCl2-I3 -.  = (0, 0, 0), X = (a*/2, 0, 0), Y = (0, b*/2, 0) and Z = (0, 0, c*/2); M = (0, b*/2, c*/2) in a) but (a*/2, b*/2, 0) in b), c) and d). The dashed line refers to the highest occupied level in all cases. All levels up to the energy zero are doubly filled in a) and b). Spin-up and spin-down bands are shown in red and blue, respectively in (c) and (d). The spin-up and spin-down bands are identical although located in spatially different but equivalent sites in (d) so that they superpose and only the blue bands are visible. Note the larger energy scale in a).

Table 1 .

 1 Selected bond distances and angles for 1-MCl2. i and o correspond to the S atoms located on the inner rings and outer rings of 1

, respectively. Central C=C (Å) M-N distance (Å) M-Cl distance (Å) N-M-N angle (°) N-M-Cl angle (°) Interligand S-S distance (Å) i

							-i	i-o	i-o
	1-CuCl2	1.338(5)	2.062(3) -2.085(3)	2.8907(9)	88.4 -91.7	89.6 -90.3	3.215(1) 3.759(1) 3.713(1)
	1-ZnCl2	1.350(5)	2.193(3) -2.290(3)	2.4466(7)	85.2 -94.8	90.0 -90.4	3.249(1) 3.504(1) 3.707(1)
	1-CdCl2	1.36(2)	2.374(12) -2.443(10)	2.563(3)	84.8 -95.2	89.6 -90.6	3.217(6) 3.556(5) 3.721(6)
	1-HgCl2	1.340(7)	2.437(3) -2.547(3)	2.5286(10)	83.3 -96.6	89.4 -90.6	3.217(1) 3.516(1) 3.738(1)

Table 2 .

 2 Oxidation

		Epa (V)	Epc (V)
	1 in solution	0.665	0.575
	1 in solid state	0.75	0.61
	1-CuCl2 in solid state	0.66	0.40
	1-HgCl2 in solid state	0.74	0.62
	1-CdCl2 in solid state	0.73	0.62
	1-ZnCl2 in solid state	0.77	0.63

potential, Epa, and reduction potential, Epc, of 1, 1-HgCl2, 1-CdCl2, 1-ZnCl2, 1-CuCl2 WE/GCE for a) and GCP for b). RE: Ag/AgCl/KClsat, CE: Pt wire.

Acknowledgements

This work was supported in France by the CNRS and the University of Angers (PhD funding to F.S.). N.A. and I.O. thank the Narodowa Agencja Wymiany Akademickiej -NAWA (Poland, BPN/BFR/2021/1/00001/U/00001) and the French Ministry of Foreign Affairs and the French Ministry of Education and Research (France, PHC Project 48119PG) for financial support through the bilateral Polonium project. The work in Poland was supported within the Statutory Activities of the Institute of Molecular Physics Polish Academy of Sciences and partially supported at Poznan University of Technology by the Research Project of the Polish Ministry of Education and Science 0511/SBAD/2251. Work in Spain was supported by MICIU through Grants PGC2018-096955-B-C44 and PGC2018-093863-B-C22, and Generalitat de Catalunya (2017SGR1506 and 2017SGR1289). E.C. acknowledges support of the Spanish MICIU through the Severo Ochoa FUNFUTURE (CEX2019-000917-S) Excellence Centre distinction and P. A. from the Maria de Maeztu Units of Excellence Program (MDM-2017-0767).

Table of Content

Sulphurous lateral interactions : For electroactive coordination polymers, through space charge transport phenomenon are most often 1D. By increasing the size of the electroactive core, lateral sulphur-sulphur interaction take place and create a 2D conducting pathway. Post-synthetic oxidation is used to increase the materials conductivity.


Hebatalla Elnaggar

Abhishek Nag

Maurits W Haverkort

Mirian Garcia-Fernandez

Andrew Walters

Ru-Pan Wang

Ke-Jin Zhou

Frank De Groot

Affiliations

Magnetic excitations beyond the single-and double-magnons

A photon carrying one unit of angular momentum can change the spin angular momentum of a magnetic system with one unit (Ms = 1) at most. This implies that a two-photon scattering process can manipulate the spin angular momentum of the magnetic system with a maximum of two units. Herein we describe a triple magnon excitation in α-Fe2O3, which contradicts this conventional wisdom that only 1-and 2-magnon excitations are possible in a resonant inelastic x-ray scattering experiment. We observe an excitation at exactly three times the magnon energy, along with additional excitations at four and five times the magnon energy, suggesting quadruple and quintuple magnons as well. Guided by theoretical calculations, we reveal how a two-photons scattering process can create exotic higher-rank magnons and the relevance of these quasiparticles for magnon-based applications.

Introduction

Understanding how to control the spin degree of freedom is a cornerstone for several hot topics of contemporary magnetism research, including ultrafast magnetism and magnonics. The main idea behind magnonics is to use the elementary magnetic excitations (magnons) for information transfer and processing. Magnons are bosonic quasiparticles and are the quanta of magnetic oscillations of systems with periodically ordered magnetic moments [START_REF] Bloch | Zur Theorie des Austauschproblems und der Remanenzerscheinung der Ferromagnetika[END_REF]. A magnon is classically depicted as a phase-coherent precession of microscopic vectors of magnetization in a magnetic medium. When a magnon propagates through a magnetic medium, no electrical charge transport is involved and hence no electrical losses take place. This is the key advantage of using magnons as information carriers. The energy of magnons range typically in the terahertz range (in the order of 1 to 25 THz, i.e., 5 to 100 meV). The magnon frequency has an important impact on the performance of magnonbased devices because the larger the excitation frequency, the faster are the magnons, at least for a fraction of the magnon band. This means that the use of high-frequency (terahertz) magnons could provide a great opportunity for the design of ultrafast devices [START_REF] Zakeri | Terahertz magnonics: Feasibility of using terahertz magnons for information processing[END_REF]. Antiferromagnets represent an appealing playground for the search for new channels of high frequency, long-lived magnons paving the way towards ultrafast magnon-based devices [START_REF] Safin | Excitation of terahertz magnons in antiferromagnetic nanostructures: Theory and experiment[END_REF]4).

Collective excitations such as magnons can be effectively measured using 2p3d resonant inelastic x-ray scattering (RIXS) [START_REF] De Groot | Local spin-flip spectral distribution obtained by resonant x-ray Raman scattering[END_REF]. Here a 2p core electron is resonantly excited from its initial state to the empty 3d orbitals through an electrical dipole transition. This excited state contains a localized core-hole that has a life time of ~100 fs determined by Auger decay. The radiative decay of a 2p core hole brings the system back to the ground state, as well as final states including low-energy excitations. In 1998, it was proposed that 2p3d RIXS can be used to measure magnetic excitations, referred to as spin-flip by de Groot and coworkers [START_REF] De Groot | Local spin-flip spectral distribution obtained by resonant x-ray Raman scattering[END_REF]. Consider a magnetic 3d 9 system where the hole resides in the dz 2 orbital and is spin-up. An incoming photon with the correct energy and polarization can excite a 2p core-electron into the empty dz 2 spin-up hole leading to a 2p 5 3d 10 intermediate state. While the spin-orbit interaction at the 3d shell is in the order of 100 meV, it is ~ 12 eV for the 2p shell: This implies that spin and orbital momenta are mixed in the intermediate state, so neither is a good quantum number and only the total angular momentum is defined. Therefore, the intermediate state can decay to a 3d 9 with a dz 2 spin-down hole final state. This excitation is referred to as a spin-flip excitation because when one compares the initial state with the final state, one finds that the only change between both is the spin projection from up to down and a corresponding counter change in the photon angular momentum. In a magnetic material such as α-Fe2O3, the magnetic excitations are nonlocal collective excitations in the form of magnons. A magnon excitation is commonly interpreted to originate from a local single-site spin-flip RIXS process. This description is widely used because the intermediate state in a 2p3d RIXS experiment is strongly localized due to the 2p core-hole and hence such a local picture is very useful to describe many aspects of the RIXS process. However, the final state excited by RIXS is not necessarily localized and can be a collective excitation such as magnons in a magnetic material. The way to reconcile both the local single-site and collective aspects of RIXS is to realize that the incident photon can be scattered at any equivalent site, leading to a final state that is a superposition of spinflips at equivalent sites. Such a final state carries a nonlocal magnetic excitation and represents the magnon density of states as also confirmed by detailed comparison to inelastic neutron scattering (INS) data [START_REF] Ament | Resonant inelastic x-ray scattering studies of elementary excitations[END_REF]. As a photon in the x-ray regime has a non-negligible linear momentum, one can also measure the dispersion of magnons when spin-flip scattering is allowed in RIXS. This realization has been the main motivation behind the development of high resolution resonant inelastic x-ray scattering (RIXS) beamlines, with the goal to study the spin dynamics of (pseudo)spin S = ½ materials such as cuprates and iridates [START_REF] Dean | Spin excitations in a single La2CuO4 layer[END_REF][START_REF] Peng | Influence of apical oxygen on the extent of inplane exchange interaction in cuprate superconductors[END_REF][START_REF] Kim | Magnetic excitation spectra of Sr2IrO4 probed by resonant inelastic x-ray scattering: Establishing links to cuprate superconductors[END_REF][START_REF] Dean | Ultrafast energy-and momentum-resolved dynamics of magnetic correlations in the photo-doped Mott insulator Sr2IrO4[END_REF].

Spin-half systems represent a special case as only transitions from Ms = -½ to Ms = ½ are allowed on a single atomic site. These magnons propagate a change of one unit of angular momentum and are similar to the magnons observed by other techniques such as INS and Raman scattering (11). We will refer to these magnons as conventional single-magnons. Whereas it is only possible to flip a single spin at a local single magnetic site for cuprates and iridates resulting in a collective singlemagnon excitation in the material, the nickelates can theoretically present single and double spinflip excitations. This is because the Ni 2+ is 3d 8 with S = 1 and hence excitations between Ms = 1, 0, -1 are possible on a single site leading to single-and double-magnons in the extended system. RIXS measurements on NiO has confirmed the presence of single (Ms = 1) and double-magnons Ms = 2) in the system [START_REF] Ghiringhelli | Observation of two nondispersive magnetic excitations in NiO by resonant inelastic soft-X-ray scattering[END_REF][START_REF] Nag | Many-body physics of single and double spin-flip excitations in NiO[END_REF]. We point out that double-magnons are different from bimagnons observed in cuprates. A double-magnon is a Ms = 2 transition, while a bimagnon is composed of two single-magnons, one changing the spin projection with +1 (i.e., Ms = 1) and the other with -1 (i.e., Ms = -1) giving rise to a combined Ms = 0 transition [START_REF] Chaix | Resonant inelastic x-ray scattering studies of magnons and bimagnons in the lightly doped cuprate La(2-x)SrxCuO4[END_REF].

While it is clear that for high spin Ni 2+ ions in NiO possessing two unpaired 3d electrons that only two spins can change their angular momenta (i.e., excitations between Ms = 1, 0, -1), the situation is more complicated for a high spin 3d 5 ion in a magnetic system. In this case, there are conceptually five spins that could be locally reversed resulting in magnons carrying a change of angular momentum of up to 5 units in the extended system (i.e., local spin-flip excitations between Ms = 5/2, 3/2, ½, -½, -3/2, 5/2 leading to a higher-rank magnon final state that is a superposition of higher-rank spin-flips at equivalent sites). This raises the fundamental question: Is it possible to change the spin angular momentum of a system with an amount greater than the change in the xray photon angular momentum of the RIXS experiment?

Here we provide experimental data capable of answering this question by measuring the low-energy magnon spectrum of an α-Fe2O3 single crystal at the ultra-high resolution I21 RIXS setup (E= 32 meV) at Diamond Light Source [START_REF] Zhou | I21: an advanced high-resolution resonant inelastic x-ray scattering beamline at Diamond Light Source[END_REF]. Guided by theory, we show that the crystal lattice acts as a reservoir of angular momentum which provides the extra angular momentum required to excite the higher-rank magnons (i.e., beyond single-and double-magnons). We developed a low-energy effective operator that describes the higher-rank magnons and derived simple selection rules that can be used to predict the best experimental settings for exciting the higher-rank magnons.

Results

The antiferromagnet α-Fe2O3 is an ideal material to initiate this kind of study because the ground state of Fe 3+ has the maximum number of unpaired electrons for the d orbitals providing a platform to test the maximum number of possible single-site spin-flip excitations. Furthermore, the 6 A1 orbital singlet ground state makes a clean case to study solely spin excitations without any orbital contribution.

Figure 1(a) shows the Fe ions in the unit cell of α-Fe2O3. The exchange coupling is dominantly antiferromagnetic with the Néel temperature TN of ∼950 K. In addition to the Néel transition, α-Fe2O3 exhibits another magnetic transition, referred to as the Morin transition (TM of ∼250 K) where below TM it is purely antiferromagnetic. We performed our measurements at 13 K (T < TM) in the collinear antiferromagnetic phase. An exemplary L3 x-ray absorption spectrum (XAS) is shown in Figure 1(b) where we find two main peaks (labelled E1 and E2 where this splitting is due to the crystal field) exhibiting the expected x-ray magnetic linear dichroism signal in line with previous literature [START_REF] Kuiper | X-ray magnetic dichroism of antiferromagnet Fe2O3: The orientation of magnetic moments observed by Fe 2p x-ray absorption spectroscopy[END_REF][START_REF] Miedema | The iron L edges: Fe 2p x-ray absorption and electron energy loss spectroscopy[END_REF][START_REF] Miyawaki | Dzyaloshinskii-Moriya interaction in α-Fe2O3 measured by magnetic circular dichroism in resonant inelastic soft x-ray scattering[END_REF].

The RIXS spectrum measured at E1 is shown in Figure 1(c). The elastic line is observed at zero energy transfer where the final state preserves the initial state spin orientation. A cascade of energy transfer peaks can be seen at 100, 200, 300, 400 and 500 meV. The first energy transfer peak can be assigned to a single-magnon excitation. This agrees well with the observation from INS experiments, where an optical nearly non-dispersing mode is observed at ~100 meV [START_REF] Samuelsen | Inelastic neutron scattering investigation of spin waves and magnetic interactions in Fe2O3[END_REF]. The single-magnon excitation propagates a change of angular momentum of Ms = 1. The second energy transfer peak appears at double the energy of the single-magnon and can be assigned to a double-magnon excitation (Ms = 2) similar to the double-magnon excitation observed in NiO [START_REF] Ghiringhelli | Observation of two nondispersive magnetic excitations in NiO by resonant inelastic soft-X-ray scattering[END_REF][START_REF] Nag | Many-body physics of single and double spin-flip excitations in NiO[END_REF]. The most remarkable feature in our results is the ability of the two-photon RIXS process to excite higher-rank magnons at 3 times, 4 times and potentially at 5 times the energy of a singlemagnon (zoom in Figure 1(d) and its first derivative in (e)). These higher-rank magnons propagate these multiples of angular momentum. We provide the details of the fitting and the full energy loss spectra at two incident energies in the materials and methods in Figures S1, S2 andS3.

We measured the angular dependence of the single-, double-and triple-magnons by rotating the sample in the azimuthal (𝛼) direction to decipher the nature of the transitions involved (see Figure 2). Conceptually, one expects that the angular behavior of the higher-order magnons differs from the single-magnon as the angular momentum selections rules are different (Ms = 1, 2, 3 involves a dipolar, quadrupolar and hexapolar spin-flip process respectively). This is confirmed by comparing the angular behavior in panels (a), (b) and (c) of Figure 2 where the magnitudes of the transitions are reduced approximately with an order of magnitude as we move from single-to double-to triple-magnons, in addition to the change of the angular profile. The angular dependent RIXS of these excitations however, did not show any noticeable dispersion (see Figure S4). The parent single-magnon is an optical mode that shows negligible dispersion according INS measurements which agrees with our measurement [START_REF] Samuelsen | Inelastic neutron scattering investigation of spin waves and magnetic interactions in Fe2O3[END_REF].

Discussion

The implication of our experimental observation is that the two-photon RIXS process can exchange five units of angular momenta with a single magnetic-site. This is an unexpected result because the selection rule for a dipole transition in presence of strong spin-orbital coupling is that the change of the total angular momentum (Mj) is equal to 0 or 1. This means that for a dipole-in (2p →3d transition), dipole-out (3d → 2p transition) 2p3d RIXS process, possible transition should involve Mj = 0, 1, 2 giving rise to only single-and double-magnons. We performed multiplet ligand field theory calculations for Fe 3+ L3-edge RIXS (see Figure 3(a)) which confirms that only singleand double-magnons are expected to be observed and is in line with previous work on NiO [START_REF] Ghiringhelli | Observation of two nondispersive magnetic excitations in NiO by resonant inelastic soft-X-ray scattering[END_REF][START_REF] Nag | Many-body physics of single and double spin-flip excitations in NiO[END_REF]. It is essential to examine the interaction terms of the model Hamiltonian responsible for the singleand double-magnons to find the origin behind the higher-rank magnons in α-Fe2O3.

𝐻 = ∑ 𝑓 𝑘 𝐹 𝑘 + 𝑘 ∑ 𝑔 𝑘 𝐺 𝑘 + ∑ 𝑙 𝑖 . 𝑠 𝑖 + 𝑖 𝐽 𝑒𝑥𝑐ℎ (𝑛. 𝑆) 𝑘 (1)
The model Hamiltonian used for the calculation is given by equation ( 1). The 𝐽 𝑒𝑥𝑐ℎ (𝑛. 𝑆) term is the mean-field super exchange interaction term that determines the energy of the single-magnon. The spin orbit coupling is given by the ∑ 𝑙 𝑖 . 𝑠 𝑖 𝑖 term and is responsible for the spin-flip process by mixing the orbital and spin degrees of freedom and enables the observation of single-magnons as detailed in the work of de Groot et. al. [START_REF] De Groot | Local spin-flip spectral distribution obtained by resonant x-ray Raman scattering[END_REF]. The double-magnon is enabled through the intra-atomic Coulomb exchange given by ∑ 𝑓 𝑘 𝐹 𝑘 + 𝑘 ∑ 𝑔 𝑘 𝐺 𝑘 𝑘 . Fk(fk) and Gk(gk) are the Slater-Condon parameters for the radial (angular operators) part of the direct and exchange Coulomb interactions, respectively. The intra-atomic Coulomb exchange interaction strongly couples the valence and core electrons implying that the spin angular momentum of both the core and valence orbitals are no longer a good quantum number, effectively leading to ΔMs = 0, 1, and 2 excitations.

It is inevitable to conclude that the higher-rank magnons have a different origin compared to the single-and double-magnons reported in previous work [START_REF] Ghiringhelli | Observation of two nondispersive magnetic excitations in NiO by resonant inelastic soft-X-ray scattering[END_REF][START_REF] Nag | Many-body physics of single and double spin-flip excitations in NiO[END_REF][START_REF] Nag | Quadrupolar magnetic excitations in an isotropic spin-1 antiferromagnet[END_REF] and a mechanism that allows the exchange of higher angular momenta is needed. This apparent contradiction can be reconciled by realizing that the angular momenta of electrons only is not a conserved quantum number in real crystals [START_REF] Streib | Difference between angular momentum and pseudoangular momentum[END_REF]. The crystal lattice can exchange angular momentum with the electrons providing the extra angular momentum required for higher-rank magnons involving ΔMs > 2. We performed a full-multiplet ligand field theory calculation for Fe 3+ L3-edge RIXS including the effect of the crystal lattice using an effective octahedral crystal field potential (see Figure 3(b)). In addition to the single-and double-magnons, triple-, quadruple-and quintuple-magnons are now visible confirming that the crystal field potential is the key factor behind the higher-rank excitations as can be seen in details in the linear cuts of the RIXS map in Figure 3(c). Here we stress that the higherrank magnons are generated from a single magnetic-site and the role of the crystal lattice can be considered as a reservoir of angular momentum supplying the extra angular momentum required.

To visualise the role of the crystal lattice, we follow in Figure 4 the fate of an excitation created by the absorption, for example, of a circular right polarized photon. We define the polarization of light as:

|𝑅⟩ = 1 √2 [1, -𝑖, 0], |𝐿⟩ = [1, 𝑖, 0] and |𝑍⟩ = [0,0,1].
The initial state can be represented by the vector shown in Figure 4(a) which includes the 2p and 3d orbitals participating in the RIXS process. The first two numbers shaded in gray represent the occupation of the 2p spin down (red arrow) and 2p spin up orbitals (blue arrow) where we take the spin quantization axis to be the C4 axis of the octahedron. The second two numbers shaded in yellow are the 3d spin down (red arrow) and 3d spin up orbitals (blue arrow) occupation numbers. The projection of the total orbital angular momentum, LZ, is specified in the subscript of the vector to keep track of the orbital angular momentum of the states. This means that the initial state is given by (3,3,5,0)0.

Upon the absorption of the polarized photon, a spin-up electron can be excited from the 2p-1 to the 3d-2 orbital resulting in an intermediate state given by (3,2,5,1)-1 (Figure 4(a)). This intermediate state can decay back elastically by emitting a right polarized photon contributing to the elastic RIXS peak (Figure 4(b1)). Another possible path is shown in Figure 4(b2) where a 3d-2  electron scatters off the crystal field potential to a 3d2  orbital and thereby changes its angular momentum by four units. This extra angular momentum provided by the lattice is the key aspect that makes it possible to excite higher-rank magnons. A cascade of 2p spin-orbit coupling and 2p-3d exchange interaction are required to transfer this orbital angular momentum to spin angular momentum as illustrated in panels (c) to (i) of Figure 4. The first pair of 2p spin-orbit coupling and 2p-3d exchange interaction changes the intermediate state to (3,2,4,2)2. We note that this intermediate state cannot decay to a single-magnon excitation as it cannot reach a LZ = 0 final state through a dipole emission. The second pair of 2p spin-orbit coupling and 2p-3d exchange interaction changes the intermediate state to (3,2,3,3)1. Now this intermediate state can decay to a double-magnon excitation either after the 2p spin-orbit coupling step (2p-1  →3d-2  emitting a left polarized photon -not shown in Figure 4 for visual clarity) or after the exchange interaction step emitting a left polarized photon (Figure 4(g1)). A final 2p spin-orbit coupling step is required to changes the intermediate state to (2,3,3,3)0 which can finally decay to a triple-magnon by emitting a Z polarized photon (Figure 4(h)).

Quadruple and quintuple magnons can be reached by further exchange of angular momentum with the lattice followed by cascades of 2p spin-orbit coupling and 2p-3d exchange interaction. In contrast, when the crystal-field is not considered, only single-and double-magnons can be excited (see the Feynman-diagrams in Figure S5). The transparent Feynman-diagram representation allows us to derive simple selection rules for this example: (i) it is not possible to excite single-magnons using circular right polarized incoming x-rays, (ii) double-magnons can be selectively observed by detecting the left polarized outgoing light. (iii) triple-magnons can be selectively observed by detecting the Z polarized outgoing light. The Z polarized light can be experimentally detected by placing an extra detector in the vertical plane for example. A full RIXS calculation is shown in Figure S6 and confirms the selection rules derived from Figure 4. We developed a low-energy effective RIXS operator that describes low-energy magnetic excitations such as magnons in terms of spin operators based on the work of Haverkort [START_REF] Haverkort | Theory of resonant inelastic x-ray scattering by collective magnetic excitations[END_REF]. The full RIXS cross-section is given by equation ( 2) where the ground state |𝑖⟩ is excited by a photon described by a dipole transition operator 𝑇 𝜖 𝑖 to an intermediate state described by the Hamiltonian (H) and decays to all possible final states |𝑓⟩ emitting a photon described by a dipole transition operator 𝑇 𝜖 𝑜 .

𝑅𝐼𝑋𝑆(𝜔)  ∑ | 〈𝑓|𝑇 𝜖 𝑜 † 1 𝜔 𝑖 -𝐻 + 𝑖Γ/2 𝑇 𝜖 𝑖 |𝑖〉 𝑓 | 2 = ∑ |〈𝑓|𝑅 𝑒𝑓𝑓 |𝑖〉 𝑓 | 2
(2) The effective operator (𝑅 𝑒𝑓𝑓 ) removes the intermediate state from the equation by expanding the intermediate state Hamiltonian in terms of polynomials of spin operators multiplied by x-ray absorption fundamental spectra. The expression of the expansion to the third order is presented in the materials and methods and is summarized in Figure 5(a). The spin flip processes resulting in magnons can be grouped in order of the spin operator rank: linear (shaded in blue), quadratic (shaded in purple) and cubic (shaded in red) operators. The linear spin operators can generate single-magnons while the quadratic spin operators generate single-and double-magnons and produce the majority of the RIXS intensity (see Figure 5(b)). Finally, the cubic spin operators can generate single-, double-and triple-magnons. The main advantage of this expansion is its simple form that allows one to determine general selection rules depending on the polarization of the incoming and outgoing light.

We computed the angular dependence of the single-, double-and triple-magnons based on this expansion in Figure 6. Our calculations capture the essential aspects of the experimental angular dependence where we obtain the correct order of magnitude for the magnons and reproduce the general angular dependence confirming the nature of the higher-rank magnons (compare Figure 2 to Figure 6). Some deviations of the calculated angular dependence from the experiment could be to several factors. On the experimental side, our measurements were performed on a bulk single crystal which is prone to self-absorption and saturation effects. As the RIXS cross-section is a combination of an absorption (photon-in) process and an emission (photon-out) process, two geometrical effects have to be taken into account here: the probing depth is dependent on the x-ray absorption spectroscopy (XAS) cross-section (saturation) and the emitted photons can be reabsorbed (self-absorption). Consequently, the RIXS intensity is distorted in a bulk crystal according to the photon energy and the experimental geometry [START_REF] Wang | Saturation and self-absorption effects in the angledependent 2p3d resonant inelastic X-ray scattering spectra of Co 3+[END_REF]. When the sample is rotated, the probing depth is changed, and the photons emitted at different energies have different escape lengths, hence distorting the angular dependence. It is difficult to correct for these geometrical energy-dependent effects because it is affected by the background absorption (which is the off-resonant contribution from other elements in the sample and in the path of the beam). On the theoretical side, one likely reason for the deviation could be the fact that the Fe sites in hematite have a small trigonal distortion. As a first approximation, the trigonal distortion would not change the ground state of Fe 3+ because the singlet 6 A1 ground state does not split. However, the trigonal distortion would influence the intermediate states, and hence could modify the intensity and consequently the angular dependence [START_REF] Miyawaki | Dzyaloshinskii-Moriya interaction in α-Fe2O3 measured by magnetic circular dichroism in resonant inelastic soft x-ray scattering[END_REF]. Finally, we point out that ligand field-multiplet theory reduces a full solid to a local cluster. This means that any Fe-Fe interactions or intercluster hopping are not considered. We expect that the above approximations can affect the angular dependence [START_REF] De Groot | 2p x-ray absorption spectroscopy of 3d transition metal systems[END_REF].

The comparison between theory and experiment on α-Fe2O3 confirms that the cascade of excitations we observed at triple, quadruple, and quintuple the energy of a single-magnon are higher rankmagnons that propagate these higher multiples of spin angular momentum. From a fundamental point of view, the higher-rank magnons can couple differently with the various degrees of freedom of the system providing a unique platform to investigate magnon interactions. From a technological point of view, these excitations have higher energies than that of single-magnons and hence are potentially more thermally robust. We predict that these higher-rank magnons can also be excited using THz-pulses and can be enhanced using magnonic crystals paving the way towards future magnonic devices.

Methods

Resonant inelastic x-ray scattering measurements

High-resolution (E= 32 meV), Fe L3-edge resonant inelastic x-ray scattering measurements were done at the I21 beamline of the Diamond Light Source, United Kingdom. Linear horizontally (π) or vertically polarized (σ) x-ray beam was used. The angular dependence was measured by rotating the sample about the b-axis (referred to as α rotation) of a polished α-Fe2O3 single crystal cooled down to 13 K. The Morin temperature of Fe2O3 is 220 K. The scattering angle was kept fixed at 150 o . The x-ray absorption spectrum (XAS) shown in Figure 1(b) was measured using total electron yield in the same geometry. The pressure in the experimental chamber was maintained below 5x10 - 10 mbar. The zero-energy transfer position and resolution of the RIXS spectra were determined from subsequent measurements of elastic peaks from an adjacent carbon tape.

Resonant inelastic x-ray scattering data fitting

RIXS data were corrected for self-absorption prior to fitting. The elastic line was fitted with an energy resolution limited Gaussian lineshape (orange shade, Figure 1(c) and SI Figure 1(a)). The phonon peaks close to 43 meV and 150 meV were fitted with asymmetric Lorentzian functions (shown by gray dashed lines) convoluted with the energy resolution. These peaks are clearly visible in RIXS spectra at E2 peak of XAS (see inset of SI Figure 1(b)). The five shaded antisymmetric Lorentzian peaks convoluted with the energy resolution represent the single-(blue), double-(purple), triple-(red), quadruple-(green) and quintuple-(black) magnon excitations (Figure 1c and SI Figure 1(a)). While the energy positions up to the quadruple magnon excitation were kept as a free parameter for fitting, the energy position of the quintuple magnon was fixed to (fitted energy position of the triple magnon)*(5/3). See also SI Figure 2 and SI Figure 3 for the low energy fits to the RIXS data for different α at π and σ polarisations, respectively.

Multiplet ligand-field calculations

The crystal field multiplet model is an effective model Hamiltonian for the description of all charge conserving excitations of ionic transition metal systems. The crystal field multiplet model is valid for the main peaks of 2p x-ray absorption and the low-energy RIXS excitations of ionic transition metal ions, because the 2p3d x-ray absorption and the 3d2p x-ray emission are neutral, selfscreened, transitions, which implies that screening channels such as ligand-metal charge transfer can be approximated by renormalized parameters. We used the quantum many-body program Quanty [START_REF] Haverkort | Multiplet ligand-field theory using Wannier orbitals[END_REF] to simulate Fe XAS and 2p3d RIXS in α-Fe2O3. The Hamiltonian used for the calculations consists of the following terms: (i) Coulomb interaction, (ii) crystal field potential, (iii) spin-orbit coupling, and (iv) effective exchange interaction. The d-d (p-d) multipole part of the Coulomb interaction was scaled to 70% (80%) of the Hartree-Fock values of the Slater integral. The general parameters used for the simulations agree with previous studies of α-Fe2O3 L2,3 edges.

Expression of the effective RIXS operator

The effective operator can be expressed by equation (3) as shown by Haverkort [START_REF] Haverkort | Theory of resonant inelastic x-ray scattering by collective magnetic excitations[END_REF]. Here 𝜖 𝑖𝑛(𝑜𝑢𝑡) is the incoming (outgoing) polarization of the photons. 𝐹 {𝑥,𝑦,𝑧} is the conductivity tensor describing the full magneto-optical response function of the system depending on the local magnetization direction given by {𝑥, 𝑦, 𝑧}.

𝑅 𝑒𝑓𝑓 = -Im[𝜖 𝑖𝑛 * ⋅ 𝐹 {𝑥,𝑦,𝑧} ⋅ 𝜖 𝑜𝑢𝑡 ] (3)
The general form of the conductivity tensor can be expressed as a sum of linear independent spectra multiplied by functions depending on the local magnetization direction as given in equation (4).

Here 𝜃 and 𝜙 define the direction of the local moment with 𝜃 being the polar angle, and 𝜙 being the azimuthal angle. 𝑌 𝑘,𝑚 (𝜃, 𝜙) is a spherical harmonic function and 𝐹 𝑖𝑗 𝑘,𝑚 is the i,j component of the conductivity tensor on a basis of linear polarized light in the coordinate system of the crystal. In symmetries lower than spherical, this expansion on spherical harmonics does not truncate at finite 𝑘 and the angular momentum of the electrons only is not a conserved quantum number in the crystals. We have shown in our previous work that including terms up to 𝑘 = 3 is sufficient to describe Fe 3+ ions in octahedral crystal field [START_REF] Elnaggar | Tensor description of X-ray magnetic dichroism at the Fe L2,3-edges of Fe3O4[END_REF]. The expression is given in equation ( 5) and involves terms up to the 3 rd order in spin leading to single, double, and triple spin flip processes. The RIXS cross-section computed using the three orders of the spin operators.

Fig. 1 .

 1 Fig. 1. Crystal structure and x-ray L3-edge measurements in a (0001) α-Fe2O3 single crystal. (a) Crystal structure of α-Fe2O3 showing only the Fe atoms and the scattering geometry used for all measurements presented in this work. kin(out) are the incident and outgoing wave vectors and the scattering angle (2) was kept fixed at 150°. The incidence angle is α with α = 90° for normal incidence. The antiferromagnetic order is depicted with white arrows showing the orientation of the magnetic moments. (b) Fe L3 XAS measured with  (black) and  (red) polarization. Two main peaks can be identified and are labelled E1 and E2. (c) RIXS spectra measured at E1 (α = 95°,  polarization). The orange shaded Gaussian peak at zero energy transfer corresponds to the elastic peak also having contribution from ΔMS=0 excitation. The five shaded antisymmetric Lorentzian peaks represent the single-(blue), double-(purple), triple-(red), quadrupole-(green) and quintuple-(black) magnon excitations (see Methods for the fitting details). The blue box highlights the spin non-conserving transitions. (d) A zoom on the spin non-conserving transitions. The dots are experimental data, and the black line is the fit. The triple-and quadrupole -magnons can be clearly identified. (e) The first derivative of panel (d)where the signal equals zero at the position of the higher-rank magnons.

Fig. 2 .

 2 Fig. 2. Angular dependence of the magnons measured with  and  polarization. (a) Single-magnon, (b) double-magnon and (c) triple-magnon excitations measured at the incidence energy E1. The angular dependence is measured by rotating the single crystal in the azimuthal direction ( rotation) while the scattering angle was kept fixed at 150°. The error bars shown are least square fitted intensity value errors.

Fig. 6 .

 6 Fig. 6. Computed angular dependence of the magnons measured with  and  polarization using the effective operator expansion. (a) Single-magnon, (b) double-magnon and (c) triple-magnon excitations at the incidence energy E1.

Acknowledgments

We acknowledge the staff of beamline I21 of Diamond Light Source for their help in setting up and running the experiments. We acknowledge the following fundings: Dutch Research Council Rubicon Fellowship (Project No. 019.201EN.010): HE ERC advanced Grant XRAYonACTIVE No. 340279: FdG

Code availability

The code that supports the findings of this study is available online and can be downloaded at: https://www.quanty.org together with a full documentation and instructions to use it. The script file required to reproduce the theoretical figures has been deposited in the zenodo database under accession code at https://zenodo.org/record/7828290.

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Data availability

The data generated and analyzed are included in the paper and its supplementary information and have been all deposited in the zenodo database under accession code at https://zenodo.org/record/7828290. Raw data files will be made available upon request. The authors declare no competing interests.

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