Corentin Jouault
email: jouaultc0@gmail.com

of the family Les guêpes pélécinides de l'ambre

Keywords: pelecinid wasps (Hymenoptera, Pelecinidae) support New species, Eopelecinus, Fossil record, Paleobiogeography Nouvelle espèce, Eopelecinus, Registre fossile, Paléobiogéographie

Mid-Cretaceous Burmese amber pelecinid wasps (Hymenoptera, Pelecinidae) support the hypothesis of an Asian origin of the family

Corentin Jouault

Introduction.

Pelecinid wasps Hymenoptera in having a long, thin, pluri-segmented metasoma with elongated, untelescoped segments [START_REF] Goulet | Hymenoptera of the world: An identification guide to families[END_REF]. All the pelecinid females possess a very long (up to 7 cm, in extant taxa) metasoma, used to lay eggs directly on scarab larvae buried in the soil [START_REF] Mason | Structure and movement of the abdomen of female Pelecinus polyturator (Hymenoptera: Pelecinidae)[END_REF][START_REF] Johnson | Revision of the proctotrupoid genus Pelecinus Latreille (Hymenoptera: Pelecinidae)[END_REF].

The Pelecinidae fossil record is rich, dating back to the Jurassic (China, Kazakhstan, and Mongolia) and their highest species diversity occurred during the early Cretaceous (http://fossilworks.org, accessed 10 October 2020). However, their extant record is limited to three species [START_REF] Mccravy | Pelecinid Wasps (Hymenoptera: Pelecinidae)[END_REF]. The numerous Mesozoic genera were widely distributed in Laurasia but are, to date, unknown from all Gondwana deposits. They clearly decreased in diversity after the Cretaceous with only two described Cenozoic species from Baltic amber and Kishenehn Formation [START_REF] Brues | The Parasitic Hymenoptera of the Baltic Amber[END_REF][START_REF] Engel | The fossil pelecinid Pelecinopteron tubuliforme Brues in Baltic amber (Hymenoptera: Pelecinidae)[END_REF][START_REF] Greenwalt | A diminutive pelecinid wasp from the Eocene Kishenehn Formation of northwestern Montana (Hymenoptera: Pelecinidae)[END_REF] and an undescribed specimen from the Eocene Sakhalin amber possibly belonging to the genus Pelecinopteron [START_REF] Kozlov | An Early Cretaceous ichneumon of the family Pelecinidae (Hymenoptera: Pelecinoidea)[END_REF][START_REF] Johnson | The fossil pelecinids Pelecinopteron Brues and Iscopinus Kozlov (Hymenoptera: Pelecinidae)[END_REF].

This peculiar fossil distribution, together with the only phylogenetic analysis proposed for the family, led to hypothesize an Asian origin for the family [START_REF] Shih | Morphology, phylogeny, evolution, and dispersal of pelecinid wasps (Hymenoptera: Proctotrupoidea: Pelecinidae) over 165 million years[END_REF]. However, the recent descriptions of Pelecinidae from the Burmese amber biota [START_REF] Engel | Zoropelecinus zigrasi, a pelecinid wasp in mid-Cretaceous amber from Myanmar (Hymenoptera: Pelecinidae)[END_REF]Guo et al., 2016a;Jouault et al., 2020a) may challenge this assumption. The West Burmese Terrane displays a peculiar geological history [START_REF] Seton | Global continental and ocean basin reconstructions since 200 Ma[END_REF][START_REF] Westerweel | Burma Terrane part of the Trans-Tethyan arc during collision with India according to palaeomagnetic data[END_REF] questioning the fauna origin of its biota. Some taxa described from this amber deposit support Gondwana affinities [START_REF] De Sena Oliveira | Earliest onychophoran in Amber Reveals Gondwanan Migration Patterns[END_REF]Jouault and Nel, 2020;[START_REF] Poinar | Burmese amber: evidence of Gondwanan origin and Cretaceous dispersion[END_REF] while numerous other families and orders suggest an island endemism, first pointed by [START_REF] Zhang | Hymenoptera (wasps, bees and ants) in mid-Cretaceous Burmese amber: a review of the fauna[END_REF] but also by Rasnitsyn and Ohm-Kuhnle (2018). Finally, it is not impossible that taxa with efficient flight abilities might have been able to cross the broad ocean separating West Burmese Terrane from Laurasia before their collision during the Late Cretaceous [START_REF] Gumovsky | Laurasian ancestors and "Gondwanan" descendants of Rotoitidae (Hymenoptera: Chalcidoidea): What a review of Late Cretaceous Baeomorpha revealed[END_REF][START_REF] Martynova | Description of the first sclerogibbid wasp (Hymenoptera: Sclerogibbidae) from Burmese (Myanmar) amber and its phylogenetic significance[END_REF]. The lack of recent exhaustive phylogenetic analysis of extant and fossil pelecinid wasps and of clear apomorphies supporting the extinct Iscopininae, have led [START_REF] Engel | Zoropelecinus zigrasi, a pelecinid wasp in mid-Cretaceous amber from Myanmar (Hymenoptera: Pelecinidae)[END_REF] to synonymize this subfamily with Pelecinidae.

This synonymization has been accepted by recent most publications (Guo et al., 2016a,b;Jouault et al., 2020a) and is followed herein.

In this paper, Eopelecinus marechali sp. nov. is described and figured from a single wellpreserved female. A discussion on the origin of the family is proposed based on the pelecinid fossil record and the geological history of the West Burmese Terrane.

Material and methods.

The amber piece containing the specimen studied herein derives from the deposits of Noije Bum in the Hukawng Valley (26• 29′ N, 96• 35′ E), Kachin State, northern Myanmar (see detailed map in Grimaldi and Ross, 2017: Fig. 2). Radiometric data established an early Cenomanian age (98.79 ± 0.62 Ma) for Kachin amber, based on zircons from volcanic clasts found within the amberbearing sediments [START_REF] Shi | Age constraint on Burmese amber based on U-Pb dating of zircons[END_REF]. Some ammonites found in the amber-bearing bed and within amber corroborates a late Albian/early Cenomanian age [START_REF] Cruickshank | Geology of an amber locality in the Hukawng Valley, northern Myanmar[END_REF][START_REF] Yu | An ammonite trapped in Burmese amber[END_REF].

The amber piece was prepared using a diamond disk and polished with a grinder polisher (Buehler EcoMet 30) by using a very thin silicon carbide sanding paper (grit size = 7000). The holotype of Eopelecinus marechali sp. nov. is housed in the amber collection of the Geological Department and Museum (IGR) of the University of Rennes, France, under the collection number IGR.BU-019. The specimen was examined using a Leica MZ APO stereomicroscope. Photographs have been taken with a Canon 5D Mark II camera, and are digitally stacked photomicrographic composites of several individual focal planes, which were obtained using HeliconFocus 6.7. The figures were composed with Adobe Illustrator CC2019 and Photoshop CC2019 software.

Acronyms for measurements and indices are listed below: HL: maximum head length measured laterally from anteriormost point of head capsule to posteriormost point of vertex; HW: maximum head width measured at eyes level (eyes included); AL: antennal length (length of all antennomeres combined); ED: maximum diameter of eye as measured in lateral view of head to show full surface of eye; PW: pronotum width in dorsal view; ML: mesosomal length measured in lateral view from anterior surface of pronotum to posterior extension of propodeum; WH: mesosoma height (in side view, maximum height measured from lowermost point of mesopleuron (in front of middle coxa) to dorsal edge of mesosoma); MIL: maximal length of first metasomal segment; MIH: maximal height of first metasomal segment; MIIL: maximal length of second metasomal segment; MIIH: maximal height of second metasomal segment; CI: cephalic index (HL/HW × 100); AI: antennal index (AL/HW × 100); OI: ocular index (ED/HL × 100). urn:lsid:zoobank.org:pub:395A96ED-D6B3-4D5B-9704-1B114B48F72F Etymology. The specific epithet is a patronym honoring my friend Arthur Maréchal for his 23rd birthday. The specific epithet is to be treated as a noun in a genitive case.

Diagnosis. Eopelecinus marechali sp. nov. differs from all known Eopelecinus species in having a unique metasomal segment ratio: 1.0; 0.96; 0.87; 0.83; 0.64; 0.52; first segment the longest but only slightly longer than second one; metasomal segments decreasing in size from first to apical one (inmm): 0.78; 0.75; 0.68; 0.65; 0.5; 0.41; vertex strongly concave; propodeum coarsely reticulate; head hemispherical, apical flagellomere nearly twice as long as pre-apical one; small size ca. 4.75 mm; 14 antennomeres. Description. Female; male unknown. Head (Figs. 1 and 2A and B). Hypognathous, hemispherical (CI: 76), wider than long, widest in posterior third in dorsal view; mandibles dentate, oriented ventrally; compound eye large, nearly oval; vertex strongly concave, with three ocelli forming a triangle (hard to define due to the preservation in amber). Antennae 14 antennomeres (Fig. 2B), antenna filiform (probably flattened during fossilization process); scape 0.12 mm, pedicel 0.08 mm, F1 0.06 mm, F2 to F10 all ca. 0.07 mm (preservation does not allow differentiation of lengths), F11 0.08 mm, F12 0.13 mm; scape narrow basally, slightly widened apically, 2.15× longer than wide, slightly shorter than apical flagellomeres; pedicel globular, narrower basally, slightly longer than wide; first flagellomere narrower basally than apex of pedicel, gradually increasing in width apically, more than twice as long as wide; remaining flagellomeres similar to first one except apical one with rounded apex and twice as long as pre-apical flagellomere (laterally deformed due to preservation). Mesosoma (Figs. 1 and 2A) 0.8 mm long, apparently rectangu-lar; pronotum extremely short; mesonotum nearly straight; notauli not clearly visible, metanotal suture conspicuous; mesonotum and propodeum separated by a deep constriction; propodeum coarsely reticulate (Fig. 2A). Fore and mid legs both shorter than hind leg. Fore leg: profemora 0.37 mm long, protatibia: 0.31 mm long, metabasitarsus 0.1 mm long, combined tarsi length ca. 0.28 mm. Mid leg: mesofemora 0.25 mm long, mesotatibia: 0.38 mm long, mesobasitarsus 0.12 mm long, combined tarsi length ca. 0.37 mm. Hing leg: metafemora 0.5 mm long, metatibia: 0.41 mm long, metabasitarsus 0.15 mm long, combined tarsi length ca. 0.45 mm. Tibial spur formula 1-2-2. Tarsal claws unarmed. Arolium present. Forewing (Fig. 2D) 1.5 mm long, with thin costal area; pterostigma short, longer than wide, and broa-dest in its apical third, with posterior margin arched; covered with microsetae. Hind wing 0.8 mm long, without any veins; covered with microsetae.

Metasoma elongated (Figs. 1 and2A andC). First metasomal segment 0.78 mm long, elongated, trapezoidal and broadest slightly before its mid-length; tergite and sternite of first metasomal segment convex laterally and dorso-ventrally, slightly shorter than mesosoma and slightly broader basally than apically, without conspicuous sculpture; other metasomal segments tubuliform, slender and slightly arched medially, shortening and thinning toward metasomal apex, respective length (mm): 0.75; 0.68; 0.65; 0.5; 0.41. Apical-most segment needle-shaped maybe due to apices of ovipositor and ovipositor sheaths exerting slightly from the apex (Fig. 2C).

Color pattern. Colors black to dark brown (as preserved in amber). Legs (assumed)

Discussion

Systematic placement and phylogenetic assumption

The new specimen described herein is undoubtedly assigned to the genus Eopelecinus based on the reduced wing venation (viz. forewing with only veins C and R present), pterostigma short with sides subparallel, pronotum medially short, propodeum reticulate, and metasoma with one broad basal segment and others tubular. Among Eopelecinus species, Eopelecinus marechali sp. nov. can be easily distinguished by the unique metasomal ratio (see differential ratio for all species in Jouault et al., 2020a: tabl. 1), relatively smaller size, antennae 14-segmented, and head with posterior margin conspicuously concave. It (4.75 vs. respectively 20.6, 14.2, 13.3, 18.3, 15.7, 30.8, 8.2, 13.8 mm). E. hodoiporus [START_REF] Zhang | Eight new species of the genus Eopelecinus (Hymenoptera: Proctotrupoidea: Pelecinidae) from the Laiyang Formation, Shandong Province, China[END_REF] is the only other Eopelecinus species possessing a "similar" metasomal ratio but it differs from Eopelecinus marechali sp. nov. in having a rounded head without conspicuous concave vertex (vs. clearly hemispherical with deep posterior concavity in Eopelecinus marechali sp. nov.), and also by its temporal range (Aptian vs. upper Albian-Cenomanian).

According to the phylogenetic analysis of [START_REF] Shih | Morphology, phylogeny, evolution, and dispersal of pelecinid wasps (Hymenoptera: Proctotrupoidea: Pelecinidae) over 165 million years[END_REF], mainly based on wing venation, the genus Allopelecinus was proposed as the putative sister lineage of Eopelecinus; but the numerous genera and species described since this first analysis question these affinities. Additionally, some genera formerly comprised only a reduced number of species while they are today represented by many species with polymorphic characters [START_REF] Shih | Morphology, phylogeny, evolution, and dispersal of pelecinid wasps (Hymenoptera: Proctotrupoidea: Pelecinidae) over 165 million years[END_REF]. [START_REF] Rasnitsyn | The origin and evolution of Lower Hymenoptera. Transactions of the Paleontological Institute[END_REF][START_REF] Rasnitsyn | Origin and evolution of Hymenoptera][END_REF] proposed a theory on a general trend among Hymenoptera consisting in the miniaturization, reduction of venation, enlargement of pterostigma, and reduction of number of flagellomeres, during the evolution of the whole order. This trend has been documented in Aculeata, and non-Aculeata wasps based on fossil specimens (e.g. Jouault et al., 2020b;[START_REF] Li | A new genus and species of Praeaulacidae (Hymenoptera: Evanioidea) from Upper Cretaceous Myanmar amber[END_REF], but it seems to be less obvious for the extant taxa [START_REF] Vargas | Revisited phylogeny of Scleroderminae (Hymenoptera: Bethylidae) reveals a plastic evolutionary history[END_REF]. If we apply Rasnitsyn's hypothesis to Eopelecinus, species of Eopelecinus with 14 antennomeres would have occupied relatively diverging positions in the genus phylogeny of the genus while species with 15 antennomeres would have occupied more inclusive positions. Thereby, Eopelecinus marechali sp. nov.

(with 14 antennomeres) would likely belong to the recently diverged Eopelecinus sub-group including: Eopelecinus marechali sp. nov., E. inopinatus, E. exquisitus, E. giganteus(?), E. hodoiporus, E. huangi, E. laiyangicus, E. mesomicrus, E. pusillus, E. scorpioideus, E. shangyuanensis, E. tumi-dus, and E. vicinus which all possess 14 antennomeres or less. Within this sub-group, it seems that another sub-group, having more recently diverged, emerges on the basis of an antenna with 13 antennomers (E. laiyangicus(?), E. mesomicrus(?), E. pusillus). This assumption fits with the various ages of the different Eopelecinus species: most species with 15 antennomeres are early Cretaceous, while the younger, mid-Cretaceous species had 14 antennomeres. It is therefore likely that Eopelecinus species with 13 antennomeres will be discovered in the Cretaceous deposits. It remains that an analysis of the relationships between the Eopelecinus species is challenging, since most of them only differs in their metasomal ratio and possesses reduced wing venation. It therefore appears that a complete study of the pelecinid phylogeny is necessary, but that falls outside the objective of the present study.

Burmese pelecinid and paleobiogeography

The presence of Eopelecinus in the mid-Cretaceous Burmese amber gives new clues to investigate the paleobiogeography of the genus, on the basis of the geological history of the West Burmese Terrane. During the Lower Jurassic, this Terrane was located near the Australian block in East Gondwana [START_REF] Seton | Global continental and ocean basin reconstructions since 200 Ma[END_REF][START_REF] Van Hinsbergen | Greater India Basin hypothesis and a two-stage Cenozoic collision between India and Asia[END_REF] and separated from East Laurasia (Asian region) by the Meso-Tethys Ocean. Previous studies estimated that the West Burmese Terrane brake away from the Australian block (with Indian block) between the Late Jurassic and the Lower Cretaceous [START_REF] Heine | Reconstructing the lost eastern Tethys ocean basin: convergence history of the SE Asian margin and marine gateways[END_REF][START_REF] Heine | Late Jurassic rifting along the Australian North West Shelf: margin geometry and spreading ridge configuration[END_REF], but more recent studies agree on a separation dated back ca. 125 Ma; the Terrane being an isolated island only after ca. 120 Ma [START_REF] Metcalfe | Allochthonous terrane processes in Southeast Asia[END_REF][START_REF] Metcalfe | Pre-Cretaceous evolution of SE Asian terranes[END_REF][START_REF] Scotese | Atlas of Early Cretaceous Paleogeographic Maps, PALEOMAP Atlas for Arc-GIS[END_REF][START_REF] Westerweel | Burma Terrane part of the Trans-Tethyan arc during collision with India according to palaeomagnetic data[END_REF]. During the mid-Cretaceous, the West Burmese Terrane was isolated geographically and occupied an island position in the Tethys Ocean between Asia and the Indian block, far from both of them [START_REF] Westerweel | Burma Terrane part of the Trans-Tethyan arc during collision with India according to palaeomagnetic data[END_REF], during more than 20 Ma [START_REF] Heine | Reconstructing the lost eastern Tethys ocean basin: convergence history of the SE Asian margin and marine gateways[END_REF][START_REF] Seton | Global continental and ocean basin reconstructions since 200 Ma[END_REF].

On this basis, two evolutionary scenarios are possible to explain the presence of Pelecinidae in the Burmese amber biota (Fig. 3): Burmese pelecinid wasps originated from Laurasia and later migrated to Burmese Terrane (Fig. 3, arrow 2); or originated from Gondwana (Fig. 3, arrow 1). The fossil record and the repartition of the extant Pelecinidae would be in accordance with the first hypothesis because of the numerous genera and species described from Laurasia, while the current fossil record in Gondwana is null (Table 1). The lack of pelecinid wasps described from Gondwana is surprising since numerous Hymenoptera are known from the Crato Formation in Brazil known to be among the best documented Aptian insect Konservat Lagerstätte [START_REF] Martill | Fossils of the Santana and Crato Formations, Brazil. Field guides to fossils[END_REF][START_REF] Martill | Introduction to the Crato Formation[END_REF][START_REF] Milaisey | Santana fossils: An illustrated atlas[END_REF] and is contemporary with the Chinese, Russian and Mongolian deposits. They are also absent from the Turonian deposit of Orapa (Botswana) where more than 2000 insect fossils are known [START_REF] Brothers | Diversity of Hymenoptera and other insects in the Late Cretaceous (Turonian) deposits at Orapa, Botswana: a preliminary review[END_REF], strengthening a Laurasia origin for the family.

The numerous records of pelecinid wasps inhabiting the Northern hemisphere (Laurasia) during the late Jurassic and Cretaceous, when India was located in Southern hemisphere [START_REF] Heine | Reconstructing the lost eastern Tethys ocean basin: convergence history of the SE Asian margin and marine gateways[END_REF][START_REF] Seton | Global continental and ocean basin reconstructions since 200 Ma[END_REF] or latter was an island completely isolated from any continental land masses [START_REF] Westerweel | Burma Terrane part of the Trans-Tethyan arc during collision with India according to palaeomagnetic data[END_REF], would refute a "out-of-India" hypothesis [START_REF] Karanth | Out-of-India Gondwanan origin of some tropical Asian biota[END_REF], in which India was as a raft bringing Gondwanan lineages to Southeast Asia. It also appears very unlikely that Burmese amber pelecinid was a Gondwanan-derived population that had migrated via India since the latter had a hostile paleoclimate during the Cretaceous [START_REF] Spicer | The Cretaceous vegetation and climate of Asia: some insights[END_REF]. Additionally, if a population of Pelecinidae had survived in India, it seems likely that some representatives would be present today in Madagascar and/or Australia since these geological blocks were connected together with West Burmese Terrane before the Gondwana breakup [START_REF] Heine | Reconstructing the lost eastern Tethys ocean basin: convergence history of the SE Asian margin and marine gateways[END_REF][START_REF] Seton | Global continental and ocean basin reconstructions since 200 Ma[END_REF]. It is not the case. The record of Eopelecinus, which is to date the most specious genus of the family, widely distributed in Asia before the Cenomanian, strengthens the possible latter migra-tions from Laurasia to the West Burmese Terrane (Fig. 3). All these geological data agree with the phylogenetic analysis of [START_REF] Shih | Morphology, phylogeny, evolution, and dispersal of pelecinid wasps (Hymenoptera: Proctotrupoidea: Pelecinidae) over 165 million years[END_REF] pointing the Pelecinidae origin from the Asiatic region.

The distribution of extant pelecinid species may be explained by the persistence of Laurasian populations in North America (recorded in Raritan amber, and by extant species; [START_REF] Engel | A Diminutive Pelecinid Wasp in Cretaceous Amber from New Jersey (Hymenoptera: Pelecinidae)[END_REF]) and latter migrations from North America to South America. In fact, before the establishment of the Isthmus of Panama, a series of islands was present between North America and South America during the Miocene [START_REF] O'dea | Formation of the Isthmus of Panama[END_REF] allowing transfer of flying invertebrates between the North America and the South America.

Conclusion.

The highly specialized metasoma of extinct pelecinid wasps are similar to those of the three extant representatives of the family, suggesting similar parasitic behaviors on scarab larvae [START_REF] Bennett | Host location behaviour in Pelecinus polyturator (Hymenoptera: Pelecinidae)[END_REF]. The pelecinid fossil record (particularly that of the genus Eopelecinus), combined with the description of Eopelecinus marechali sp. nov., and the peculiar geological history of the West Burmese Terrane suggest that the origin of the family is likely in Laurasia and particularly in Asia during the Jurassic period. Unlike some other taxa, the shapes and sizes of pelecinids may have played a key role in their lift and favored their transport from continent to the Burmese paleo-island via air currents. Thereby, pelecinid wasps are maybe the testimonies of a late fauna input during mid-Cretaceous period. Future complete phylogenetic analyses, combined with models that estimate ancestral areas with dated phylogenies, current species distribution, and time-stratified paleogeographic matrices may helped to ensure these hypotheses.

Figs

  Figs 1 and 2

Figure 1 :

 1 Figure 1: Eopelecinus marechali sp. nov., holotype IGR.BU-019. Habitus in left-lateral view. Scale

Figure 2 :

 2 Figure 2: Eopelecinus marechali sp. nov., holotype IGR.BU-019. A. Detailed view of mesosoma. B.

Figure 3 :

 3 Figure 3: Maps of West Burmese terrane position at 95 Ma with hypothetic fauna transferts (modi-

  lighter than rest of body. Measurements (in mm): HL 0.38; HW 0.50; AL 1.10; ED 0.22; PW 0.34; ML 0.63; WH 0.37; MIL 0.78; MIH 0.23; MIIL 0.75; MIIH 0.08; MIIIL 0.68; MIIIH 0.07; CI 76; AI 220; OI 57.

  differs from (see. inventor and date in Table 1) E. inopinatus, E. mecometasomatus in having the second metasomal segment shorter than the first. It also differs from E. eucallus, E. fragilis, E. laiyangicus, E. mesomicrus, E. minutus, E. pusillus, E. shan-gyuanensis, E. tumidus, and E. yuanjiawaensis in having the second metasomal segment only slightly shorter than first one (vs. conspicuously shorter). Eopelecinus marechali sp. nov. differs from E. giganteus, E. leptaleus, E. hodoiporus, E. exquisitus, E. scorpioideus, E. huangi, E. similaris, and E. vicinus by its conspicuously smaller size

Table 1 :

 1 Diversity of fossil Pelecinidae.

Diversité des Pelecinidae fossiles.

Acknowledgements

I am grateful to the editor Prof. Didier Néraudeau for managing the manuscript. I sincerely thank the reviewers: Prof. André Nel and Prof. Alexandr. P. Rasnitsyn for their valuable comments on thismanuscript.


MPC-friendly, Snark-friendly and Stark-friendly primitives

Objectives:

• minimize the number of multiplications in large fields

• minimize the size of the polynomial relations representing the execution trace over a finite field.

Examples:

•

= 8 c = 4 RC 1 x RC 2 x RC 3 x A differential distinguisher x 0 x 1 x 2 x 3 x 4 x 5 x 6 x 7 x 8 x 9 x 10 x 11 rc x 3
Original analysis:

best attack with a characteristic over (t + 1) rounds with probability 2q -1 2 .

A better differential:

(0 . . . 0, α, α ) R t-2 → (α, α , 0 . . . 0) R →(α +β, β . . . β, α) R →(β+β . . . β+β , α+β , α +β) For β = -β , we get an iterative differential (0 . . . 0, α, α ) R t
-→ (0, . . . , 0, α + β, α + β) with probability 2q -1

A differential distinguisher

With this t-round differential with proba 2q -1

• A differential characteristic over the 101 rounds with probability 2 -480 for a 732-bit blocksize.

• With structures, we get valid pairs with complexity 2 359 (full permutation) and valid pairs with complexity less than 2 128 for 58 rounds. • With a rebound-like technique, we expect to get valid pairs conforming with the differential over 58 rounds with complexity close to 2 64 (on-going work).

Impossible differentials

Original analysis: best impossible differential over (2t -2) rounds A better impossible differential over (3t -4):

(0, . . . , 0, α)

R 3t-4
(β, 0, . . . , 0)

for any nonzero α = β.

Integral attacks over F q When q = 2 n . For any (affine

) subspace V ⊂ F n 2 with dim V > deg F , x∈V F (x) = 0. Because, for V = b + a 1 , . . . , a v , D a 1 D a 2 . . . D a v F (b) = x∈V F (x)
Not valid in odd characteristic.

Integral attacks over F q

When q = 2 n . For any (affine

) subspace V ⊂ F n 2 with dim V > deg F , x∈V F (x) = 0. Because, for V = b + a 1 , . . . , a v , D a 1 D a 2 . . . D a v F (b) = x∈V F (x)
Not valid in odd characteric.

But for any q

For any exponent k with 0 ≤ k ≤ q -2,

x∈F q x k = 0 General result.
For any F :

F q → F q with deg(F ) ≤ q -2, x∈F q F (x) = 0 .
Less general than the property over F 2 n: For any (affine

) subspace V with dim V > deg F , x∈V F (x) = 0
But for any q

For any exponent k with 0 ≤ k ≤ q -2,

x∈F q x k = 0 General result.
For any F :

F q → F q with deg(F ) ≤ q -2, x∈F q F (x) = 0 .
Less general than the property over F 2 n: For any (affine

) subspace V ⊂ F n 2 with dim V > deg F , x∈V F (x) = 0
α 0 α 1 α t-2 x x ∈ F q x β 1 β t-2 β t-1 z 0 z 1 z t-2 z t-1 x t 0 x t 1 x t t-2 x t t-1 polynomial in x of degree ≤ q -2 Q(x) = t-1 i=1 x t i -(t -2)x t 0
Until the degree does not exceed (q -2)

Input set. X = {(α 0 , . . . , α t-2 , x), x ∈ F q } After (t -1) rounds. X = {(x, β 1 , . . . , β t-1 ), x ∈ F q }
After r rounds, the degree in x of each branch is at most 3 r .

⇒ all branches are balanced if 3 r ≤ q -2.

Adding (t -2) rounds x 0 x 1 x 2 x 3 x 4 x 5 x 6 x 7 x 8 x 9 x 10 x 11 rc x 3
The inputs and outputs of Round satisfy

x i -x i+1 = x -1 i+1 -x -1 i+2 , ∀i ≤ t -3 Over (t -2) rounds, x t-1 0 -x t-1 1 = x 1 t-2 -x 1 t-1
is a polynomial in x of degree ≤ (q -2).

⇒ Distinguisher with complexity q on (2t-3 + log 3 (q -2) ) rounds (59 rounds)

Adding one more round

x 0 x 1 x 2 x 3 x 4 x 5 x 6 x 7 x 8 x 9 x 10 x 11 rc x 3
The inputs and outputs of Round satisfy

x i = x -1 i+1 + (x j -x -1 j+1 ) and x t-1 = x -1 0 ⇒ t-1 i-0 x -1 i -(t -1)x t j = t-1 i=0 x i -(t -1)x j-1 Over (t -1) rounds, t-1 i=0 x 1 i -(t -1)x 1 t-1 = t-1 i=0 x t i -(t -1)x t 0
⇒ Distinguisher with complexity q on (2t-2 + log 3 (q -2) ) rounds (60 rounds)

A few more rounds with two active branches (on-going work).

Zero-sum partition on GMiMC on (3t-3 + 2 log 3 (q -2) ) rounds (109) (x 0 , . . . , x t-1 ) = t-1 i=1 x i -(t -2)x 0 sum to 0 (t -1) rounds ? ? ? ?

? ?

? ?

log 3 (q -2) rounds ? ? ? ? ? ?

? ?

(t -1) rounds ? ? ? ? ? ? ? ?

log 3 (q -2) rounds ? ? ? ? ? ?

? ?

(t -1) rounds ? ? ? ? ? ?

? ?

y 1 0 y 1 1 y 1 t-2 y 1 t-1 α 0 α 1 α t-2 x x ∈ F q polynomial in x of degree ≤ q -2 x β 1 β t-2 β t-1 z 0 z 1 z t-2 z t-1 x t 0 x t 1 x t t-2 x t t-1 polynomial in x of degree ≤ q -2 y t 0 y t 1 y t t-2 y t t-1
(y 0 , . . . , y t-1 ) = t-2 i=0 y i -(t -2)y t-1 sum to 0

Resistance against statistical attacks

Analysed without the partial rounds.

Differential cryptanalysis:

x 3 has differential uniformity 2 over F q .

The best differential characteristic satisfies

EDP ≤ 2 q (t+1)R f /2 → R f = 6 are enough.

Degree of the permutation over F q

Each coordinate is seen as a multivariate polynomial over F q After r rounds:

u=(u 1 ,...,u t ) λ u   t i=1 x u i i   where u i ≤ 3 r
⇒ 39 rounds are enough for Poseidon (40 for Starkad) to reach degree (q -1) in each variable ⇒ log 3 (t) more rounds are enough to get total degree (q -1)t.

Remark: StarkWare challenges with q 2 256 and 96 rounds have degree at most 2 152 in each variable.

Zero-sum partition over F q

State after the last full Sbox layer before the partial rounds.

X = {(α 0 , . . . , α t-2 , x), x ∈ F q }
After 38 rounds forwards.

each coordinate has degree at most (q -2).

≤ q -2 Q(x ) of degree ≤ q -2
Improvement when q = 2 n Each Boolean coordinate is seen as a multivariate polynomial in nt variables over F 2 Degree over F 2 n vs. binary degree. p of binary degree ≤ (n -1)(t -1) q of binary degree ≤ 684

P (x) =

When the MDS matrix has a small order How to propagate a subspace through all partial rounds? Choose V such that all elements in each coset of L(V ) have the same value on the first coordinate.

L(V ) ⊂ H 0 = {(0, x 1 , . . . , x t-1 ), x i ∈ F q } or equivalently V ⊂ M 0 ⊥ .
We can iterate this R P times if

V = H 0 ∩ R P -1 r=0 L r M 0 ⊥ = {0}
This holds if L r = Id for some r ≤ t -2.

When the MDS matrix is an involution

The internal states after each partial layer form a coset of V or of W = L(V ).

Special choice for V .

V = {(xv 0 , . . . , xv t-1 ), x ∈ F q } with v ∈ V.
⇒ The outputs of the partial rounds vary in a coset of {(xw 0 , . . . , xw t-1 ), x ∈ F q } Forward direction.

Each output coordinate is a polynomial in x of degree at most 3 R f /2 ≤ q -2.

⇒ The output coordinates sum to zero. (λx 1/3 + µ) 3 which contains only monomials x 1/3 , x 2/3 , x and a constant term.

Zero

x 2/3 has degree p+1 3 .

⇒ The degree does not increase between the first and second round.

But even by activiting more inputs, we cannot find an integral attack on more than 4 rounds.

  rounds R P partial rounds R f /2 full rounds R f = 8 full rounds and R P = 43 (binary) and R P = 40 (prime)

S 1 3⇒P

 1 -1 : x → x s with s = 2q -Zero-sum for R f = 2 + 4 and R P = 34 (35 for Starkad). (x) of degree at most 3 38

  max{wt(u) : 0 ≤ u < 2 n and λ u = 0} ⇒ The inverse Sbox has binary degree n+1 2 .Several rounds backwards [Boura, C. 13].• Two rounds backwards have binary degree ≤ 684• Three rounds backwards have binary degree ≤ 748Zero-sum partition over F 2 with R f = 3 + 4 and R P = 35 V = {(0, x 1 , . . . , x t-1 ), x i ∈ F 2 n} .

  -sum partition with R f = 2 + 4 and any R P with complexity q

  All candidates follow the same sponge construction with blocksize t and capacity c.

		Parameters of the sponge Performance for 128-bit security Compared performance for these parameters GMiMC with 101 rounds
	M 0 , . . . , M 7 security level log 2 q c t M 8 , . . . , M 15 prime fields are more STARK-friendly than binary fields Best candidate: r	output
	Prime field:	128 bits	64 Variant 128-d: 4 12 variant 128-d
	1. GMiMC		128	2 4 variant 128-a
	2. Rescue 3. Poseidon	128 256 t = 12 and c = 4 over F q 2 12 variant 128-c 1 3 variant 128-b
		π	256	π 1 11 variant 128-e
	Binary field: 1. Vision 2. Starkad	256 bits q =	128 128 2 63 2 61 + 20 × 2 32 + 1 4 8 variant 256-a 4 14 variant 256-b

Cradic [Knudsen Nyberg 92], Misty [Matsui 97] • MiMC [Albrecht et al. 16] SFH contenders StarkWare challenges https://starkware.co/hash-challenge/ Three families of sponges with different permutations • SPN with large blocks: Vision (F 2 n) and Rescue (F p ) [Aly et al. 19] • HadesMiMC permutation: Starkad (F 2 n) and Poseidon (F p ) [Grassi et al. 19] • GMiMC i.e. GMiMC erf over F p [Albrecht et al. 19] Sponge construction

GMiMC

Computing backwards (t -1) rounds ? ? ? ? ? ? ? ?

log 3 (q-2) rounds ? ? ? ? ? ? ? ?

Open question on the complexity of algebraic attacks

Degree of the univariate polynomial of the lexicographical Gröbner basis [Faugère-

Perret].

Complexity for solving the system = D 2 .

Variants aiming at 256-bit security have D 2 170 .

Vision and Rescue

Conclusions

We need to find the right tools for analyzing symmetric primitives over non-binary fields:

• linear attacks and their variants?

• more general integral attacks?

Open question: Does the form of q affect the security? For instance, if p = 2 2 n + 1?


Ivan Petit

Gustavo D Belletti

Théau Debroise

Manuel J Llansola-Portoles

Ivan T Lucas

César Leroy

Christian Bonhomme

Laure Bonhomme-Coury

Dominique Bazin

Michel Daudon

Emmanuel Letavernier

] Jean

Philippe Haymann

Vincent Frochot

Florence Babonneau

Paola Quaino

Vibrational Signatures of Calcium Oxalate Polyhydrates

The vibrational signatures of the calcium oxalate polyhydrates are investigated using a combination of Density Functional Theory-Dispersion corrected, Fourier Transform-Raman and -Infrared (IR) spectroscopies. Most vibrational bands were assigned and the theoretical predictions were compared with in-house and other experimental data, for both, IR and Raman spectroscopies. Such an approach allowed a more accurate analysis of vibrational spectra helping in the completion of the band assignments of the vibrational bands of the mono, di, and tri hydrate calcium oxalate (COM, COD, and COT). Particular attention has been paid to the degree of hydration of COD, the low Raman wavenumbers, and the presence of oxalic acid in natural calcium oxalate polyhydrates. The obtained results are expected to be supportive in the detection of the different polyhydrates in natural samples, such as in kidney stones.

Introduction

Fourier Transform (FT)-Raman and (FT)-Infrared spectroscopies constitute significant tools for bio-medical diagnoses. [1] Two different approaches are at the present investigated to implement these techniques into the regular diagnosis apparatus. The first one is based on statistical analyses of the vibrational spectra. This approach has been used to study the disorder and variations in composition as well as in the crystal structures of deposits in the tissues. [START_REF] Severcan | Vibrational Spectroscopy in Diagnosis and Screening[END_REF] This methodology has been explored for a wide range of pathologies, however only a few are routinely used into applications at the hospital. [3] A major weakness of the implementation of a statistical approach is mainly due to the fact that significant variations in the IR spectrum can be caused by subtle changes in the preparation procedure. [4] Another reason is related to the need for a precise analysis of the shape and the position of absorption bands for these two spectroscopies. [5] At the moment, one of the few applications is to establish medical diagnosis for kidney stones [START_REF] Dao | Infrared and Raman Spectra of Calculi[END_REF] and ectopic calcifications. [START_REF] Lionet | [END_REF] Recently, selected results regarding these research themes, as well as recent experimental developments, such as the ability to perform nanometer scale near-field infrared microscopy [8] provided relevant data in this field. [3] Due to recent technical developments in laboratory apparatus, IR spectroscopy can be implemented at sub-cellular scale in a time frame of seconds to milliseconds. This is of major importance due to the amount of samples to be collected and more than 75,000 kidney stones and 1,100 biopsies have been already characterized at the Necker and Tenon hospitals using the described techniques. The data bank created from those measurements facilitates epidemiologic studies for the clinician to relate urolithiasis to other major public health problems such as type II diabetes. [9] On the contrary, Raman spectroscopy is not routinely used at the hospital for microcrystalline pathologies, mainly because it has some disadvantages compare to IR spectroscopy. For example, in the case of sub-micrometer crystals, the Raman probe, i. e. the excitation laser, can modify the chemistry of the deposit through heat dissipation. In addition, the tissue may display a significant background signal (Raman signature and/ or auto-fluorescence) which can impact on the reliability of the detection of deposits with low Raman scattering cross section. The signal to noise ratio can however be improved through suitable tuning of the excitation laser wavelength from visible to near-IR.

However, Raman shows also major advantages compare to IR spectroscopy. Compositional mapping of samples simply deposited on supports used routinely at the hospital, i. e. glass, can be easily obtained with a sub-micron resolution. Measurements through glass cover slips protecting air sensitive samples are also possible. In addition, in Raman the 50 cm À 1 to 400 cm À 1 spectral range is accessible even down to 10 cm À 1 with enhanced filtering of the laser line, providing precise (unambiguous) chemical signature of the sample under scrutiny.

The use of IR or Raman spectroscopy as diagnostic tools at the hospital ideally presupposes a complete understanding of the origin of the absorption (IR) or scattering (Raman) bands of pathological deposits. This is achievable through the calculation of theoretical ab initio IR and Raman spectra of the different constituting compounds. The complete characterization of the spectra is of utmost importance in the identification for the calcium oxalate compounds present in more than 70% of the kidney stones, and in numerous tissue calcifications.

By computing, at the Density Functional Theory-Dispersion (DFTÀ D) level, the IR and Raman vibrational spectra of the polyhydrates of the calcium oxalates: whewellite (monohydrate: COM), weddellite (dihydrate: COD), and caoxite (trihydrate: COT), we provide data that can be used to identify/quantify the different polyhydrates in natural mixtures such as in biological calcifications or kidney stones.

In this work, the theoretical predictions are compared with the available experimental data, for both, IR and Raman spectroscopies allowing the assignment of the most relevant vibrational bands of calcium oxalate polyhydrates.

Results and discussion

Structural analysis

The unit cell coordinates and parameters of the calcium oxalate polyhydrates were optimized at the PBEÀ D2 or the PBEÀ D3 level. The corresponding parameters are shown in Figure 1 and the unit cell parameters are shown in Table 1.

COM, CaC 2 O 4 ⋅H 2 O (also known as whewellite) is the most abundant calcium oxalate (66%) phase found in kidney stones. [START_REF] Daudon | Épidémiologie actuelle de la lithiase rénale en France[END_REF] This chemical phase has been studied in the past by numerous groups, both experimentally [START_REF] Echigo | [END_REF] and theoretically. [12] The COM phase crystallizes in the monoclinic crystalline system with the space group P2 1 /c. In 1965, Arnott et al. [13] solved the structure of the COM phase, extracted from the stem and leaves of Yucca rupicola, by X-rays. Although containing many impurities, crystals of sufficient size (200 μm) could be selected for a single crystal study, which was not possible with the previous synthetic samples. More recently, several studies have helped to refine the structural parameters of the COM, in particular those from Tazzoli et al., [14] Hochrein et al., [15] and Deganello et al.. [16] These investigations have established COM structures at different temperatures and have shown that there can be a reversible change in space group from P2 1 /c to I2/m at 328 K. Finally, a neutron diffraction study by Daudon et al. [17] on a single crystal of COM allowed to locate the proton positions of the water molecules of the structure. Therefore, this structure has been chosen as the starting geometry for our calculations.

The COD phase is commonly found in kidney stones but is also present as sediment in the Weddell Sea in Antarctica hence its name: weddellite. [18] Its structure has the particularity of presenting zeolitic channels. These channels allow adsorption of large quantities of water molecules that can diffuse 'freely' in the structure. These molecules should not be confused with the 'crystalline' water which is an integral part of the structure. Indeed, the COD phase formula is CaC 2 O 4 ⋅(2 + x) H 2 O, where x represents the amount of water present in the zeolitic channels (x < 1). Even now, the number of water molecules within the structure is subject to discussion. In 1964, Sterling proposed a structure for this phase and addressed the problem of defining the number of water molecules contained in the whole structure. [18][19] He established that the weddellite crystallizes in the tetragonal system with I4/m space group. He also determined the positions and partial occupations of oxygen associated with these water molecules. Tazzoli and Domenegetti [14] later refined the structure and determined an x value of 0.37, which is retained in this study (corresponding to 3 extra water molecules in the COD unit cell). The disorderly nature of the water molecules along the z axis has been demonstrated with the separation of the positions of oxygen atoms in two sites by crystallographic studies. Based on the refinements of the structures of kidney stones by Izatulina et al. [START_REF] Izatulina | Structure, chemistry and crystallization conditions of calcium oxalates -The main components of kidney stones[END_REF] and by Tazzoli and Domenegetti, [14] Izatulina et al. [START_REF] Izatulina | [END_REF] estimated an x value ranging from 0.13 to 0.37.

Additionally, COD is a metastable phase that turns into COM if it is not kept under certain conditions: Conti et al. have followed this transformation by X-ray diffraction and IR [22] suggesting that the presence of this phase in kidney calcification is promoted by the environment in which it evolves.

The COT trihydrate phase also called caoxite is the least common of the three calcium oxalate hydrates. [23] With the chemical formula CaC 2 O 4 ⋅3H 2 O, it presents a triclinic structure with space group P � 1. Its name is derived from the acronym for CAlcium OXalate and Centennial Anniversary Of X-rays and was proposed on the occasion of centenary of X-ray diffraction. [24] Deganello et al. [25] were the first to determine the structure for COT which existence had previously been suspected by Walter-Levy and Laniepce, [26] and Gardner.

First proposed as an intermediary in the formation of forms COM and COD in kidney stones, [23,27] this hypothesis has since been abandoned in favor of that of an amorphous phase of calcium oxalate as a precursor. [28] Although, rarely reported in urinary crystals [5,29] and stones, [30] it is often present in in vitro synthesis. [START_REF] Daudon | Lithiase Urinaire[END_REF] The structure solved by Basso et al. [24a] was used as starting geometry for simulations involving the COT phase.

The calculated parameters for COM, COD and COT show good agreement with the experimental data, as it can be seen in Table 1. PBEÀ D2 approach shows better results in comparison with experimental values, than those obtained using PBEÀ D3. However, the structures of calcium oxalate polyhydrates can be adequately modeled with both methods, with a preference for the former, probably due to the strong ionic character of the crystal (i. e. the presence of Ca 2 + and C 2 O 4 2À ions).

Spectroscopic analysisIR spectra

In Figure 2 we compare the experimental (see experimental details) IR spectra measured at 4 K, with the calculated ones (PBEÀ D2). The spectra are normalized to the most intense peak. The theoretical spectra are obtained by fitting sums of Gaussian functions on the different calculated band intensities.

Whewellite -COM

The IR spectrum can be divided in three different zones of interest for its analysis (See Table 2). First, the "water region" in which 5 bands above 2900 cm À 1, in room temperature experiments and even 6 considering the very broad pic around 3212 cm À 1 in our experiment at very low temperature, whereas only four are calculated. As expected by Shippey, [START_REF] Shippey | [END_REF] and by Petrov et al., [33] the missing line in the calculation is due to an overtone 2δ(HOH), and reinforced by Fermi resonance with one of the neighboring fundamental [33] bands which cannot be calculated in the harmonic approximation.

As expected by Conti et al. [34] propose than the δ(HOH) around 1600 cm À 1 consist on 2 contributions. The first one very broad is associated to the δ(HOH) of the water strongly coupled with the motion of the oxalates molecules. Our calculations confirm this coupling between the water molecule 1 of the structure of Daudon et al. [17] and the oxalates molecules, which resulted in wavenumbers between 1591 cm À 1 and 1623 cm À 1 . This result is in very good agreement with the broad peak observed experimentally around 3212 cm À 1 (supposed to be 2δ(HOH)). The second contributions are a medium weak band proposed by Conti et al. [34] being assigned to δ(HOH) of the more localized water molecule 2 of Daudon et al.. [17] In our calculations, the wavenumbers around 1637 cm À 1 were associated to the δ(HOH) of both water molecules together, and not only of water molecule 2. Nevertheless no implication of the oxalate ion has been observed. This calculated wavenumber fits very well with the main overtone 2δ(HOH) measured around 3268 cm À 1 .

The second and third zones cover respectively the broad ν(CO) region around 1400 cm À 1 , and the low wavenumber region between 1000 and 500 cm À 1 , corresponding to the different COO vibrations (wagging, stretching and bending). The calculated vibrational spectrum is in agreement with experiment from Shippey [START_REF] Shippey | [END_REF] and Petrov et al., [33] except for some small expected band shifts due to the calculation level used. A point raised in the Shippey study concerned the different H-bonds present in whewellite, which were calculated from our optimized whewellite structure. The following H-bond distances were determined: H 2 OÀ Ox along c-axis: 2.638 and 2.713 Å; H 2 O-H 2 O 2.786 Å; H 2 OÀ Ox along b-axis: 2.899 -2.899 Å.

In general, a small but significant redshift compared with the experimental spectrum which is attributed to the use of DFT (PBEÀ D level), which slightly overestimates H-bond interactions, such as those between the oxalate and calcium ions with the neighboring water molecules.

Weddellite -COD

The calculated IR spectrum for weddellite is able to predict, within the experimental error, all bands. The same very small redshift due to the accuracy of the calculation level is observed. The characteristic carbonyl symmetric stretching (exp.: 1589-1578 cm À 1 ) is clearly recovered (theor.: 1689-1539 cm À 1 ), and predicted to be the most intense band, which is in agreement with the experimental spectrum (See Table 3). The zone, between 3200-3700 cm À 1 , corresponds to the OÀ H elongation of the water molecules. The identification of the vibration modes from the calculation presented in Table 3 shows that several vibrational combinations are involved. This is due to the large hydrogen bond network between the water molecules, and between water and oxalate entities. The experimental spectra show a very broad band in this range, due to the presence of mobile zeolitic water in the COD unit cell, in contrast with COM. The mobility of the water molecules in the channels lead to a broader set of conformations and thus a broader range of which vibrations, making it difficult to interpret the part of the spectrum.

In order to investigate possible phase transitions or changes in the degree of hydration, which are related to the x parameter of weddellite, we calculated the IR spectra for x values between 0 to 1 (corresponding to 0 and 8 extra water molecules per unit cell). The spectra are presented in Fig 3 . We show that for 5 water molecules or more per unit cell, intense bands around 2800 cm À 1 appear on the calculated spectra but are not present in the experimental spectrum suggesting that the weddellite does not contain more than 4 zeolitic water molecules per unit cell. This result agrees with the calculated adsorptions energies 50 . For the structures having between 0 and 4 (x = 0 -0.5) water molecules, the IR spectra are quite similar and in good agreement with the experimental one.

Caoxite -COT

The calculated IR spectrum for caoxite presents a good agreement with the experiment (See Table 4). The main difference is the relative intensity between the "water" bands above 2900 cm À 1 and the CO/CC band around 1700 cm À 1 . This inversion in intensity might be due to the dehydration process of the experimental phase. On the basis of the calculated spectra, the assignments for all bands are proposed (See Table 4).

Raman spectra

Raman spectroscopy is a highly reliable technique for the identification of the different Ca oxalate polyhydrates in kidney stones. Previous reports [37] show accurate and reliable Raman spectral identification assignment, e. g. COM can be easily identified from the other ones, if two strong bands appear at ∼ 1489 cm À 1 and ∼ 1462 cm À 1 (in that region, COD and COT show just one band).

In Figure 4 we compare the experimental Raman spectra, with the calculated ones (PBEÀ D3) obtained by fitting sums of Gaussian functions on the different calculated band intensities. The spectra are normalized to the most intense band.

Whewellite -COM

The Raman spectra, similarly to the IR spectra, can be divided in three zones: the "water" region (3600 -3000 cm À 1 ), the CO and CC stretching zone (1650 -1350 cm À 1 ), and the low wavenumbers zone (1000 -10 cm À 1 ).

In Table 5 the theoretical Raman bands for whewellite have been reported, and compared with the experimental results of ref., [34] which are comparable to our experimental Raman data. Five bands corresponding to symmetric and asymmetric OH vibrations of H 2 O molecules are observed in the first zone, in agreement with those reported by Conti et al.. [34] The oxalate vibrations in different planes (the perpendicular ab and bc planes), and a vibration synchronism with the oxalate nearest neighbor occur around 1500 cm À 1 . These bands appear shifted to higher wavenumbers in our calculations. Low wavenumbers zone below 1000 cm À 1 correspond to different OCO vibrations, water vibrations and lattice vibrations where Ca is involved. Around 900 cm À 1 , bending OCO vibrations are observed. Lattice vibrations which involved Ca ions are below 330 cm À 1 , and for an easiest identification CaO 2 Ca rhombus structures have been established (Figure 5).

It is interesting to note that the agreement of the calculated spectrum with the experimental one between 1700 and 400 cm À 1 is particularly good, except for some small expected band shifts and intensities.

Weddellite -COD

The calculated weddellite Raman spectra predict relatively well our experimental results and are in agreement with the reported data, except for some small band shift differences. [35,38] Table 6 shows the results for the calculated spectra and their comparison with the experimental assignation of Frost et al.. [38] The same regions described for COM can be identified. Between 3500-3200 cm À 1 experimental results show only a broad band, but in our theoretical results multiple and narrow peaks of OH stretching are observed because of the nature of the calculation at low temperature. Between 1470-1370 cm À 1 , we observe just one broad peak at 1425 cm À 1 , that can corresponds to the overlap of the strong ν a (CO) band with the ν s (CC) + δ(OCO) vibration bands which is indicated around 1475 cm À 1 in ref.

. [34] Additionally, it is important to note that water vibrational bands are coupled with those of the oxalates and Ca below 900 cm À 1 . For a better description of these vibrations, CaO 2 Ca rhombus structures have been identified and shown in Figure 5.

Caoxite -COT

The Raman spectrum for caoxite exhibits the same three distinguishable zones, which have been described for COM and COD spectra. The assignment of bands in the calculated Raman spectrum and its comparison with previous experimental results [34] are presented in Table 7. Our theoretical results show small shifts in the bands around 1470 cm À 1 with respect to the experimental ones.

In the CO and CC stretching zone between 1000 and 500 cm À 1 , we find the vibration of OCO bending and some vibrations corresponding to water molecules. The zone of low wavenumbers (below 1000 cm À 1 ) presents a band profile similar to COM but with different band positions, which is characteristic for each calcium oxalate polyhydrates. It is interesting to note that water vibrations are important in this region. A detailed description is presented in Table 7, which until now and to the best of our knowledge was not carried out in previous works. For a better analysis, the particular COT structure allows us to define only a CaO 2 Ca rhombus structure, as seen in Figure 6. In addition, the differences of the bands between COT and the other poly hy drates will be discussed below.

In the interesting work of Conti et al. [35] discussing the normal modes of the oxalate ion with average geometry , 2 vibrational modes are still missing. To complete the list of vibrations we attempted to add the missing B 2g mode. It was concluded that this mode might be coupled with the B 1u mode (not observed), showing vibrations wavenumbers for the oxalate ions in one plane (ac plane) of ∼ 838 cm À 1 and ∼ 825 cm À 1 for the other plane (bc plane), which are thus mixed with vibrations of the water molecules.

Identification of the low wavenumber Raman bands

There is not much information about band assignment in the low wavenumber Raman region, because of the existence of both, coupled and lattice vibrations. According to our theoretical results and to shed light on possible deeper characterizations leading to new analysis methodologies, detailed assignments are displayed in Figure 7, and Tables 5, 6 and 7. The pattern of the theoretical spectra do not completely fit the experimental ones, especially the intensities of some bands are not completely well reproduced, which might be caused by the shape of the oxalate particles.

Our results show that around 890 cm À 1 , the COM bands are found at lower wavenumbers than the bands of the other polyhydrates, due to CC stretching and OCO bending. For COD and COT, two easily distinguishable bands are observed, while for COM the bands are strongly overlapping. Previous experimental results [34] are in agreement with ours, although a single band for the three cases was found, which could be due to experimental issues, such temperature and instrumental broadening.

The correct assignment of the Raman bands observed between 600-490 cm À 1 help us to identify the calcium oxalate polyhydrates because this region presents a particular profile for each one. The bands correspond to vibrations in which water, oxalate and Ca ions are involved. The band of high intensity for that zone is found at the following wavenumbers: COM ( ∼ 490 cm À 1 ), COD ( ∼ 498 cm À 1 ) and COT ( ∼ 511 cm À 1 ), which correspond to the experimental bands: [34] 503, 506 and 507 cm À 1 respectively . Interestingly , the trend observes with experimental data is respected through our DFT calculations despite the slightly shifted values.

Another important region is between 300-10 cm À 1 , where other specific water molecules, oxalate and Ca ions are detected. The band around 200 cm À 1 allows to distinguish between COM (∼ 205 cm À 1 ), COD (∼ 210 cm À 1 ) and COT (186 cm À 1 ). That band includes torsional vibration bands of CaO 2 groups coupled with water and OCO torsional vibrations that have a vibration mode that is sy mmetric with the other OCO, which forms the oxalate ion (Figure 8a). In addition, the other OCO torsional vibrations that have an asymmetric vibration with the next one in the ion (Fig. 8 b), coupled with CaO 2 torsional vibrations, are observed as a singular peak at ∼ 139 cm À 1 or 148 cm À 1 for COM and COD, respectively. This assignments can be confirmed and with good agreement with the previous experimental work reported by Conti et al. [34]

Comparison between oxalic acid and Ca oxalate mono and di hydrate

Although, the small probability of the presence of the acidic oxalate form, its spectrum was investigated. Due to its low first pKa around 1.2, it is not expected to be present in the synthetic samples, nevertheless this cannot be said with certainty for natural samples. So, for the sake of comparison we calculated the Raman spectrum of the oxalic acid. It is interesting to note that if oxalic acid would form during the sy nthesis of Ca oxalates, it could be clearly distinguished from its Ca salt counterparts. The theoretical Raman spectrum of oxalic acid dihy drate has been obtained and compared with an experimental one (see Figure 9). Calculated Raman spectrum predicts well the experimental results, and it is in good agreement with previous studies. [39] Only the signal at 1750 cm À 1 (corresponding to the CO stretch band) shifts slightly at higher wavelengths in the theoretical spectrum. This band shows the presence of oxalic acid, since it is absent in the calcium oxalates COM and COD, the most common poly hy drates found in kidney stones. According to these results, it is expected that if oxalic acid is present in natural samples, it can be easily detected and quantified. As expected, in our samples we conclude that no oxalic acid is present.

Conclusions

DFT calculations performed on the calcium oxalate polyhydrate unit cells were used to investigate the vibrational bands measured by Raman and IR spectroscopies. Most bands were assigned and the theoretical predictions were compared when possible with available experimental data, for both, IR and Raman spectroscopies. Such an approach allowed a more accurate analysis of vibrational spectra helping in the completion of the assignments of the vibrational bands of the three calcium oxalate polyhydrates.

The IR spectra of COM and COT were very well reproduced. Due to the presence of zeolitic water molecules in the COD structure, a systematic study on the x parameter in CaC 2 O 4 ⋅(2 + x) H 2 O was undertaken. By calculation of the different spectra for x = 0 -1 we showed that the typical x-value is equal to three, which corresponds to the degree of hydration found in the experimental samples. Moreover, we were able to present several distributions for the water molecules in the channels of COD structure, which motivated us to start to unravel thoroughly the water structure at the molecular level, combin-ing spectroscopic techniques and ab initio calculations. We concluded that the channel available for water adsorption is filled with a maximum of two to three water molecules per unit cell (one channel with two and another with one water molecule).

The Raman spectra for the three calcium oxalate polyhydrates were calculated and completely resolved, which is expected to facilitate the characterization of the calcium oxalate polyhydrates in natural samples.

It has been paid special attention to the low wavenumber Raman active vibrations and the vibrations of oxalic acid. The low vibrational modes in the Raman spectrum are of importance in the identification of the different oxalate polyhydrates, whereas the oxalic acid was studied to investigate its presence in the experimental samples. A theoretical spectrum of oxalic acid was discussed and compared with the calcium polyhydrate forms.

Table 1. Experimental and calculated unit cell parameters for the three calcium oxalate polyhydrates, COM, COD, and COT. Distances in Å, and angles in degrees.

Unit cell parameter Experimental PBEÀ D2 PBEÀ D3

COM [17] COD [START_REF] Echigo | [END_REF] COT 2. PBEÀ D2 calculated whewellite IR wavenumbers corresponding to the different types of vibration (in cm À 1 ): a-antisymmetric, s-symmetric, νstretching, δ-bending, L-libration, R-rocking, W-wagging, and T-twisting.

Next to the wavenumbers the intensity is given by: s-strong, v-very, mmedium, w-weak, b-broad, and sh-shoulder.

Present work Literature [22,32-33,35] Theor

δ(OCO) BC + LÀ R(HOH) δ(OCO) LÀ R(HOH) L-R W (HOH) LÀ R(OCO) a in AC LÀ R(OCO) a in BC LÀ R(HOH) ν(CC) AC ν(CC) BC LÀ W(OCO) BC LÀ W(OCO) BC LÀ W(OCO) AC Ca implicated vibrations
Table 3. PBEÀ D2 calculated IR wavenumbers for the different weddellite vibrational modes (in cm À 1 ): a-antisymmetric, s-symmetric, ν-stretching, δbending, L-libration, R-rocking, W-wagging, and T-twisting. Next to the wavenumbers the intensity is given by: s-strong, v-very, m-medium, wweak, b-broad, and sh-shoulder.

Present work Literature [22,35-36] Theor. band 4. PBEÀ D2 calculated caoxite IR wavenumbers corresponding to the different types of vibration (in cm À 1 ): a-antisymmetric, s-symmetric, νstretching, δ-bending, L-libration, R-rocking, W-wagging, and T-twisting. Next to the wavenumbers the intensity is described as: s-strong, v-very, mmedium, w-weak, b-broad, and sh-shoulder.

Present work

Literature [START_REF] Echigo | [END_REF]22,35] 5. PBEÀ D3 calculated whewellite Raman modes wavenumbers (in cm À 1 ) and assignment with the type of vibration: a-antisymmetric, ssymmetric, ν-stretching, δ-bending, L-libration, R-Rocking, W-wagging, and T-Twisting. The measured mode wavenumbers are reported as: s-strong, vvery, m-medium, w-weak, and b-broad.

Present work

Literature [35] Theor. band the different types of vibration (in cm À 1 ): a-antisymmetric, s-symmetric, νstretching, δ-bending, L-libration, R-rocking, W-wagging, and T-twisting.

Exp. band

Next to the wavenumbers, the intensity is reported as: s-strong, v-very, mmedium, w-weak, and b-broad.

Present work

Literature [35] Theor. band Experimental (gray) and calculated (green) IR spectra for the three calcium oxalate polyhydrates between: 500 and 4000 cm À 1 for the theoretical spectrum and between: 600 et 4000 cm À 1 for the experimental one. From left to right: total spectra, spectra between: 3800-2500 cm À 1 and spectra between: 1900-500 cm À 1 ). 4. Experimental (gray) and calculated (green) Raman spectra for the three calcium oxalate polyhydrates obtained between 4000 and 400 cm À 1 . Three spectra regions are shown when possible for comparison. From left to right: between 3600-3000 cm À 1 , 1700-1300 cm À 1 and 1000-400 cm À 1 .

  ν a,s (OH) ν a (OH) (close to carboxylic group) ν a (OH) (close to water) ν s (OH) ν a (OH) (close to water) ν a (OH) (close to carboxylic group) ν a (OH) (close to water) ν a (OH) (with strong Hbond) ν a (CO) ν s (CO) LÀ T (HOH) L (HOH) ν s (CO) + L(H 2 O) LÀ W(OCO) + L(HOH) δ(OCO) + L(HOH) L(HOH) LÀ R(OCO) + L(HOH) LÀ R(HOH) LÀ W(OCO) + L(OHO) Ca implicated vibrations Table

  ν a (CO) δ(HOH) + ν a (CO) ν s (CO)

  ν s (CC) + δ(OCO) + LÀ W (HOH) W(OCO) + LÀ R(HOH) δ(OCO) + LÀ R(HOH) LÀ R(HOH) + LÀ W(HOH) δ(OCO) + ν s (CC) LÀ R(OCO) + L(HOH) L(HOH) LÀ R(OCO) + L(HOH) Lattice vibration with Ca implicated ν a (CaÀ O) + ν s (CC) + L (HOH) ν a (CaÀ O) on CaO 2 Ca rhombus structure in BC plane 480 + ν s (CC) + L(HOH) ν a (CaÀ O) on CaO 2 Ca rhombus structure in AC plane + ν s (CC) + L(HOH) 301 259 ν a (CO) ν a (CO) -ν a (CO) + ν s (CC) + δ(OCO)s ν s (CC) + δ(OCO) + LÀ W (HOH) ν s (CC) + δ s (OCO) + LÀ W (HOH) water rocking LÀ R(OCO) + L(HOH) ν a (CaÀ O) + ν s (CC) and water vibration L(HOH) LÀ W(CaO 2 ) on CaO 2 Ca rhombus structure + LÀ T (OCO) in AC plane + L (HOH) 253 LÀ W(CaO 2 ) on CaO 2 Ca rhombus structure + LÀ T (OCO) in AC plane + L (HOH) ν s (CaO 2 ) on CaO 2 Ca rhombus structure in AC plane + LÀ R(OCO) + L(HOH)

Figure 1 .

 1 Figure 1. Unit cells of the three calcium oxalate polyhydrates, whewellite (COM), weddellite (COD), and caoxite (COT). The zeolitic water in the COD structure is not shown for clarity.

Figure 2 .

 2 Figure2. Experimental (gray) and calculated (green) IR spectra for the three calcium oxalate polyhydrates between: 500 and 4000 cm À 1 for the theoretical spectrum and between: 600 et 4000 cm À 1 for the experimental one. From left to right: total spectra, spectra between: 3800-2500 cm À 1 and spectra between: 1900-500 cm À 1 ).

Figure 3 .

 3 Figure 3. Calculated COD IR-spectra for unit cells with 0-8 extra water molecules (x = 0 to 1) and different configuration of zeolitic water molecules within the crystal, compared with the experimental IR spectrum at 4 K.

Figure 6 .

 6 Figure 6. Corresponding CaO 2 Ca rhombus structure for COT as discussed in Table7.

Figure

  Figure 4. Experimental (gray) and calculated (green) Raman spectra for the three calcium oxalate polyhydrates obtained between 4000 and 400 cm À 1 . Three spectra regions are shown when possible for comparison. From left to right: between 3600-3000 cm À 1 , 1700-1300 cm À 1 and 1000-400 cm À 1 .

Figure 5 .

 5 Figure 5. The corresponding CaO 2 Ca rhombus structures in AC plane (yellow) and BC plane (green) for (a) COM and (b) COD.

Figure 7 .

 7 Figure 7. Low wavenumber region: comparison between the theoretical Raman spectra of COM, COD and COT obtained between 1000 and 10 cm À 1 .

Figure 9 .

 9 Figure 9. Comparison between the theoretical and experimental Raman spectrum of dihydrate oxalic acid.

Figure 8 .

 8 Figure 8. Oxalate vibrations observed at low wavenumbers region (+ increase of bond length, -decrease of bond length).

Table 6 .

 6 continued

					Table 5. continued
					Present work	Literature [35]
					Theor.	Exp.	Assignment	Assignment
					band	band
					506	522 m	ν(CC) + ν(CaÀ O) in AC
				Assignment	490	503 s	plane ν(CC) + ν(CaÀ O) in BC
						plane
	3610		OH stretching bands	ν(OH)	below	Lattice vibration in which
	-				326	Ca is implicated
	2970				292	ν s (CaO 2 ) on CaO 2 Ca rhom-
	3610	3486 w ν a (OH)			bus structure in AC plane
	-					+ L(HOH)
	3450				256	ν s (CaO 2 ) on CaO 2 Ca rhom-	Elongation/ compression
	3410	3426 m ν a (OH)			bus structure in AC and	of Ca opposite vertices
	-					BC plane	on CaO 2 Ca rhombus
	3390					structure + L(HOH)
	3390	3340 w ν a (OH)		224	ν s (CaO 2 ) on CaO 2 Ca rhom-
	-					bus structure in BC plane
	3360				205	T(CaO 2 ) on CaO 2 Ca rhom-
	3360	3256 w ν s (OH)			bus + L(HOH)
	-
	3260
	3110	3056 w ν s (OH)
	-
	2970
	1645	1629 m ν a (CO)	ν a (CO)
	-
	1585
	1478		ν s (CC) + ν s (CO)
	-
	1380
	1478	1490	ν s (CC) + ν s (CO) on AC	ν s (CC) + ν s (CO)
	-	vs	plane, asym vibr with
	1423	1463	nearest neighb.
		vs
	1423	1396 m	ν s (CC) + ν a (CO) on BC	ν s (CC) + ν s (CO)
	-		plane, asym vibr with
	1380		nearest neighb.
	1040	942 w	δ(OCO) on AC plane +
	-990	b	LÀ W(HOH)
	927 -		ν(CC) + δ(OCO) + LÀ W
	850		(HOH)
	879	896 s	ν(CC) + δ(OCO) on BC	ν(CC) + LÀ W(HOH)
			plane, sym vibr with near-
			est neighb. + LÀ W(HOH)
	874	863 m	ν(CC) + δ(OCO) on AC	ν(CC) + LÀ W(HOH)
			plane, sym vibr with near-
			est neighb. + LÀ W(HOH)
	850 -		LÀ W(OCO)
	804
	839		LÀ W(OCO) on AC plane +
			L(HOH)
	825		LÀ W(OCO) on BC plane
	800-		δ(OCO) + L(HOH)
	645
	785		δ(OCO) in AC plane, asym
			vibr with nearest OCO +
			LÀ R(HOH)
	753		δ(OCO) in BC plane, asym
			vibr with nearest OCO + L
			(HOH)
	740 -		δ(OCO) + L(HOH)
	645
	625 -	597 m	LÀ R(OCO) on AC and BC
	545		plane, asym vibr with
			nearest neighb.
	506 -		ν(CC) + ν(CaÀ O)
	435

Table 6 .

 6 PBEÀ D3 calculated weddellite Raman wavenumbers corresponding to the different types of vibration (in cm À 1 ): a-antisymmetric, ssymmetric, ν-stretching, δ-bending, L-libration, R-rocking, W-wagging, and T-twisting. For some signals, the intensity is reported next to the wavenumbers as: s-strong, v-very, m-medium, w-weak, and b-broad.

	Present work		Literature [38]
	Theor.	Exp.	Assignment	Assignment
	band	band
	3750	3500	OH stretching bands	ν(OH)
	-	-
	3110	3200
		b
	3740		ν a (OH)
	3643		ν s (OH)
	-
	3619
	3565	3467	ν a (OH)	ν(OH)
	-
	3462
	3450		ν s (OH)
	-
	3412
	3407		ν a (OH)
	3393	3266	ν a (OH)	ν(OH)
	-
	3248
	3149		ν a (OH)
	1658		ν a (CO)	ν a (CO)
	-
	1540
	1613	1623	ν a (CO)
		w
	1603	1605	ν a (CO)
	1470	1475	ν s (CC) + ν(CO) + δ(OCO)
	-	vs
	1370	1411
		b
	896	909 s ν s (CC) + δ(OCO) + LÀ W
	883	868
		vw
	829
	753
	725 -
	680
	680 -
	638
	600 -	596
	540	vw, b
	616
	580
	Below
	540
	540 -
	430

Table 7 . continued

 7

	Present work		Literature [35]
	Theor.	Exp.	Assignment	Assignment
	band	band
	247		ν a (CaO 2 ) on CaO 2 Ca rhombus	Elongation/ compres-
			+ LÀ R(OCO) + L(HOH)	sion of Ca opposite
				vertices
	203		LÀ T(CaO 2 ) on CaO 2 Ca rhom-
			bus + LÀ R(OCO) + L(HOH)
	186		LÀ T(CaO 2 ) on CaO 2 Ca rhom-
			bus + LÀ T(OCO) + L(HOH)

Table 7 .

 7 PBEÀ D3 calculated caoxite Raman wavenumbers corresponding to

Acknowledgements The French state funds managed by the ANR within the Investissements d'Avenir programme under reference ANR-11-IDEX-0004-02, and more specifically within the framework of the Cluster of Excellence MATISSE led by Sorbonne Université. HPC resources from GENCI-[CCRT/CINES/IDRIS] (Grant 2016-[x2016082022]

) and the CCRE of Université Pierre et Marie Curie are also acknowledged.

Supporting Information Summary

The computational details including the calculation level and the information on the vibrational spectrum calculations, as well as the experimental details concerning the origin of the samples and the technical details of the experiments are presented in the Supporting Information.

Conflict of Interest

The authors declare no conflict of interest.

Keywords: Calcium oxalate • DFT • IR • polyhydrates • RAMAN


Dr V Escande

Dr C H Lam

Dr P Coish

Prof. Dr P T Anastas
email: paul.anastas@yale.edu

Dr V E Scande

Keywords: aerobic oxidation, cleavage reactions, diols, heterogeneous catalysis, m anganese

Heterogeneous Sodium-Manganese Oxide Catalyzed Aerobic OxidativeCleavage of 1,2-Diols Vincent Escande + ,* Chun Ho Lam + ,Philip Coish, and Paul T. Anastas* Abstract: The aerobic oxidative cleavage of 1,2-diols using aheterogeneous catalyst only based on earth-abundant metals manganese and sodium is reported for the first time.T his reusable catalyst cleaves av ariety of substrates into aldehydes or ketones with high selectivity.T he reaction requires small catalytic loadings and is performed under mild conditions using ambient pressure O 2 or air as the oxidant while producing water as the only by-product. Mechanistic investigations reveal am onodentate,t wo-electron oxidative fragmentation process involving aM n IV species.T he eco-friendly, innocuous catalyst is compatible with aw ide range of functional groups and conditions,making it highly competitive with classical reagents,such as periodic acid or lead tetraacetate,as apreferred method for activated 1,2-diols.

The oxidative cleavage of 1,2-diols to produce aldehydes or ketones is an important transformation in synthetic organic chemistry.Since its discovery in the early 20th century, [1] this reaction has been employed in numerous applications ranging from organic syntheses [2] to analytical studies in biochemistry. [3] Recently,t he oxidative cleavage of 1,2-diols has been employed in biomass valorization for renewable feedstock production. [4] However,d espite human health and environmental concerns,t he reaction conditions continue to use the same archetypal reagents,periodic acid and its salts,and lead tetraacetate.W hile these reagents are known to be selective and efficient, their intrinsic toxicities and safe disposal are non-trivial consequences,q uestioned by sustainability standards. [5] Although several methods have been developed to address the aforementioned challenges using chemically benign oxidant, such as O 2 or hydrogen peroxide, [2a, 6] they either require toxic reagents or high loadings of precious metals for the transformation. Driven by the need for sustainable catalysts,m etal-free routes have been reported recently, [7] as well as catalysts based on more abundant metals, such as complexes of Fe, [8] Mn, [9] Ti, [10] and V. [11] However, in addition to af requent lack of selectivity,acritical drawback persists as most of these catalysts are homogeneous and nonrecyclable.Therefore,development of acost-effective,innocuous,h eterogeneous catalyst that enables ah ighly selective aerobic oxidative cleavage of 1,2-diols is imperative.

Our recent reports using catalysts prepared from Mn-rich plant extracts [12] prompted us to investigate the utilization of aw ell-defined heterogeneous Mn oxide catalyst for the oxidative cleavage of 1,2-diols.T ransition-metal oxides have attracted growing interest for the synthesis of small molecules. They represent amajor class of heterogeneous catalysts used in industry. [START_REF] Aj | Metal oxides:c hemistry and applications, Taylor &F rancis,B oca Raton, FL[END_REF] Among them, manganese oxides are particularly attractive for its abundance and low toxicity. [START_REF] Gerber | [END_REF] These materials exist as avariety of structures,with unique catalytic properties in anumber of reactions. [15] In particular, in layered oxides such as birnessite,e dge-shared MnO 6 units form al attice with exchangeable oxygen molecules.M ns pecies exist in the lattice as the mixed oxidation states Mn II ,M n III and Mn IV . [START_REF] Liu | S uib in Handbook of layered materials[END_REF] We envisaged the coexistence of these different oxidation states and the possibility of redox cycling with O 2 would make Mn layered mixed oxide agood candidate for the aerobic oxidative cleavage of 1,2-diols as ah eterogeneous catalyst. It is worth noting that activated manganese dioxide has been described as an efficient reagent for 1,2-diol oxidation but only with the use of al arge excess of reagent [40:1 ratio of reagent:substrate]. [START_REF] Ohloff | [END_REF] While this amount of reagent has presumably disqualified the method as afeasible method for large-scale reactions,nocatalytic version has been proposed so far.

Herein, we describe an unprecedented protocol for aerobic oxidative cleavage of 1,2-diols into aldehydes or ketones using manganese layered mixed oxide (Mn LMO) as the catalyst (Scheme 1). Thea bsence of toxic solvents or reagents,the use of mild reaction conditions and the ability to cleave av ariety of 1,2-diols within short reaction times by using O 2 or air as oxidants are significant advantages of the method. Most importantly,the catalyst is easily prepared from inexpensive reagents and can be recycled multiple times without losing activity which makes this approach competitive with previous systems.

We prepared Mn LMO under mild conditions,a dapted from ap revious route. [18] Oxidation of Mn 2+ ions from MnCl 2 •4 H 2 Ow ith hydrogen peroxide in presence of sodium hydroxide afforded the corresponding catalyst which was isolated by filtration. Energy-dispersive X-ray (EDX;F igure S1 in the Supporting Information) spectroscopy revealed the oxide nature of the material, and presence of Na along with Mn. Further characterization by inductively coupled plasma atomic emission spectroscopy (ICP-AES;T able S1) and thermogravimetric analysis (TGA;F igure S2) indicated that the material can be formulated as Na 0.274 MnO 2 •6 H 2 O, hereafter referred to as Na-Mn LMO.B ased on this composition, Mn is present with an average oxidation state of approximately + 3.7, [19] which was confirmed by X-ray photoelectron spectroscopy (XPS;F igures S3 and S4). Whereas Na-Mn LMO appeared to be of low crystallinity, its X-ray powder diffraction (XRPD;F igure S5) pattern showed prominent (001) peaks suggesting that Mn oxide layers are uniformly stacked in the birnessite phase with interlayered Na + ions. [20] This is confirmed by Fourier transform infrared (FTIR;F igure S6) spectroscopy which displayed astrong band at 400 cm À1 ,characteristic of layered Mn oxides. [21] We evaluated the performance of our Na-Mn LMO material as ac atalyst for oxidative cleavage using mesohydrobenzoin under the atmospheric pressure of O 2 in amodel reaction. Thee xploratory conditions and results are presented as supplementary material in Table S2. Theo ptimized reaction conditions afforded the desired product, benzaldehyde,i nh igh yield. At 100 8 8Ci n1 -butanol, a9 9% conversion with complete selectivity toward benzaldehyde was obtained in 1h,w ith only 1mol %o fM n( Table 1, entry 1). Ah igh turnover frequency( TOF) of 180 h À1 at 100 8 8Cw as achieved in the initial hour (see Supporting Information). While O 2 gave the shortest reaction times,t he same conversion was obtained in air,in4h(Table 1, entry 2). Control experiments under N 2 confirmed the critical role of oxygen for the reaction (Table 1, entry 3).

Only 5% yield was obtained with commercial MnO 2 and activated MnO 2 ,s upporting the notion that the unique layered structure of Na-Mn LMO was critical to catalytic activity (Table 1, entries 4-5). Interestingly,aNa-birnessite similar to Na-Mn LMO but prepared in less-basic conditions gave al ower conversion (33 %) than Na-Mn LMO (Table 1, entry 6). While this result confirmed that al ayered structure was ak ey parameter for catalysis,i ta lso highlighted the importance of basicity during catalyst preparation.

Theb ase influence on catalyst morphology and activity was assessed by the preparation of catalysts within arange of NaOH concentrations (in excess relative to the amount required for oxidation of Mn 2+ by H 2 O 2 ). Ther esulting catalysts displayed increasing activity with increasing NaOH concentration up to an optimal concentration of 1.8 m (Table S3) while average size of particles as measured by scanning electron microscopy (SEM) decreased to 4.9 mm. However,when the concentration of NaOH reached beyond this limit, average particle size increased dramatically to over 30 mma long with an apparent decline in catalytic activity (Figure 1a).

These observations are consistent with the known influence of basicity on the crystallization of birnessite-materials. [22] Indeed, when we examined aweaker base,tetramethylammonium hydroxide (TMA•OH) and stronger base,cesium hydroxide,w eo bserved an optimal yield with NaOH (Figure 1b). Again, SEM measurements showed that excessively high basicity caused the crystallization of the material, resulting in the formation of large particles (> 30 mm) with low catalytic activity (Figure S7).

Using the most active form of Na-Mn LMO catalyst, avariety of 1,2-diols were readily cleaved under the optimized conditions (Table 2). Although 1mol %ofM nw as sufficient to cleave these substrates,w es elected 10 mol %s ince it provided shorter reaction times.B oth syn and anti as well as symmetrical and asymmetrical benzylic diols were cleaved efficiently (Table 2, entries 1-5). Thea pplicability of the method was illustrated by high, isolated yields (up to 96 %), after simple filtration then evaporation of the solvent (Table 2, entries 1,3). Benzylic or allylic 1,2-diols were the Scheme 1. Previous routes and this work.

Table 1: Oxidative cleavage of meso-hydrobenzoin catalyzed by Na-Mn LMO and other Mn oxides. [a] Entry Catalyst

Atmosphere t [h] Conversion [%] [b] Selectivity [%] [b,c] 1N 6N abirnessite [d] O 2 13 3 > 99

[a] Reaction conditions: meso-hydrobenzoin (42.9 mg, 0.2 mmol), catalyst (1 mol %Mn), 1-butanol (1 mL), gas balloon, 100 8 8C. [b] Conversion and selectivity were determined by GC using dodecaneasinternal standard. Remaining mass balance is recovered starting materials.

[c] Ratio of benzaldehyde yield to conversion as apercentage. [d] Nabirnessite prepared according to Ref. [18].

most reactive,b ut the method was also efficient on less activated benzylic-aliphatic diols (Table 2, entries 5,6). In the particular case of an on-substituted 1,2-diol such as 1f,t he reaction was slower and some degradation occurred after long heating. However,a ddition of ab ase made the oxidative cleavage possible.Ascreening of bases (Figure S8) revealed that the mild base Na 2 CO 3 was most effective.H owever,n o reaction occurred on diols substituted by two aliphatic moieties (Table 2, entries 13,14), showing the catalyst was selective toward diols with aleast one activating substituent. TheNa-Mn LMO catalyst cleaved heterocyclic 1,2-diols with high efficiency (Table 2, entries 8,9), sometimes with only 1mol %o fM n, without any poisoning effect of the heterocyclic atom. Moreover,r eactions were unaffected by the presence of electron-donating and withdrawing aromatic substituents,consistently achieving complete conversion with over 99 %s electivity (Table 2, entries 10-12). Interestingly, atest on (S)-mandelic acid showed that the oxidative cleavage was also possible on this a-hydroxy acid, here again with abeneficial effect of additional Na 2 CO 3 (Table S4).

In addition to substrate surveying, kinetic experiments were devised to gain mechanistic insight. Interestingly,kinetic comparison of meso-a nd (R,R)-hydrobenzoin cleavage revealed the anti isomer, meso-hydrobenzoin, was cleaved faster than the syn isomer, (R,R)-hydrobenzoin (Figure S9). Such observation disapproved the cyclic type Imechanism, [2a] where the opposite trend would be expected. [START_REF] Ohloff | [END_REF]23] Moreover, in ac yclic type Imechanism, the reaction has usually second order (with order one with respect to the substrate and one to the oxidant). [2a] However, our initial rate kinetic study revealed the order with respect to meso-hydrobenzoin to be zero (Figure S10), and displayed af ractional order with respect to the oxidant (Figure S11). We therefore postulate the cleavage proceeded through atype II mechanism wherein an acyclicmonodentate complex intermediate was formed as shown in Scheme 2. In this mechanism, the anti-diol would go b) are the same than for Table 1, entry 1.

Table 2: Aerobic oxidative cleavage of 1,2-diols catalyzed by Na-Mn LMO. [a] Entry Substrate [b] t [h] Product Conv.

[%] [c] Yield [%] [c] 1 [d] 1 99 99

[96] 2 1 93 93

3 [d] 1 15 [e] 53 [e,f ] 7 1 97 97 8 4 94 94

9 [d] 1 99 99 10 1 100 99 11 1 100 99

through an antiperiplanar geometry intermediate (see Scheme 2, A' ')w hich would line up both CÀOs igma orbitals to facilitate the cleavage transition while minimizing the steric hindrance from the phenyl groups. [START_REF] Rubottom | Oxidation in Organic Chemistry, Part D[END_REF] Although less common, the higher reaction rate of anti-diol cleavage compared to its syn isomer has been observed and accounted for with the mechanism described. [25] Based on this monodentate type II mechanism, two scenarios were possible:either ar adical or at wo-electron oxidative fragmentation. [2a] Such mechanistic investigations were previously performed on Ce IV reagents with the same substrate,a nd the two mechanisms were distinguished by addition of acrylamide for radical trapping. [26] Since our experiments showed no difference in the presence or absence of acrylamide (Figure S12), we therefore conclude that the reaction catalyzed by Na-Mn LMO followed atype II mechanism via two-electron oxidative fragmentation as proposed in Scheme 2. This monodentate mechanism also supports the beneficial role of Na 2 CO 3 observed for less-reactive substrates (Table 2, entry 6), which fosters deprotonation of the A' ' intermediate,asmentioned by Criegee et al. for this pathway. [27] Thep roposed catalytic cycle involves Mn IV and Mn II species,with the Mn II species reoxidized by O 2 ,forming H 2 O. Thefractional order of O 2 suggests reoxidation of Mn II was part of the rate-determining step,a nd also indicates as equence of elementary steps,p ossibly involving dissociation processes on the catalyst surface. [START_REF] Houston | Chemical kinetics and reaction dynamics[END_REF] Thus,current effort is devoted to investigate the reoxidation of Mn II into Mn IV ,o n the catalyst surface. [START_REF] Makwana | [END_REF] Such understanding would enable us to design the next generation of catalyst that could yield comparable performance using air, which displays al ower rate than pure O 2 with our current Na-Mn oxide catalyst, as Table 1shows.

As the heterogeneous nature of the catalyst was ak ey attribute,itwas assessed by the removal of the catalyst before the reaction was complete by simple filtration. Analysis of the reaction mixture showed no further conversion after removal of the catalyst (Figure S13). Therecyclability of Na-Mn LMO catalyst was also confirmed. It was isolated, washed with ethanol and dried at 105 8 8Cf or 2h before being reused (Figure S14). Thecatalyst was employed in six successive runs without any loss of activity.

Finally,t of urther assess the utility of Na-Mn LMO catalysis as as ynthetic tool, we examined its utility in multistep,o ne-pot reaction sequences.Aset of diverse domino reactions involving an initial oxidative cleavage of 1,2-diol by Na-Mn LMO were conducted (Scheme 3), and all processes afforded the desired products with high efficiency,indicating the broad applicability of the catalyst.

In conclusion, we have developed an ew method for the aerobic oxidative cleavage of 1,2-diols,with ah eterogeneous catalyst, easily prepared from inexpensive and abundant metals.T his catalyst affords the products in short reaction times,under mild conditions and with facile work-up,and can be reused several times without loss of activity.T he absence of precious metals,t oxic solvents and reagents,the use of O 2 or air in atmospheric conditions with H 2 Oa st he only byproduct, make this catalyst more appealing for the cleavage of activated 1,2-diols to any previously reported systems.T his Scheme 2. Proposed mechanism for the reaction.

Scheme3. Application of Na-Mn LMO-catalyzed aerobic oxidativec leavage of 1,2-diols in domino reactions. Yields and Z/E ratios were determined by 1 HNMR spectroscopy. Entry Substrate [b] t [h] Product Conv.

[%] [c] Yield [%] [c] 13 18 0-14 15 0-

[a] Reaction conditions: substrate (0.2 mmol), Na-Mn LMO (10 mol % Mn), 1-butanol (1 mL), O 2 balloon, 100 8 8C. [b] Unless specified, the starting diols bearing two stereogenic centers were used as variable mixtures of diastereoisomers. [c] Determined by GC using dodecane as internal standard, values in square brackets correspond to isolated yields. Unless specified, remainingmass balance is recovered starting materials. [d] Reaction performed with 1mol %Mnfrom Na-Mn LMO.

[e] Remaining mass balance was unidentified degradationp roducts.

[f]Reaction performed in presence of Na 2 CO 3 (0.4 mmol).

method excels at handling benzylic 1,2-diols,c leaved in excellent yields (92-99 %), with high selectivity (> 99 %). Mechanistic investigations revealed am onodentate,t woelectron oxidative fragmentation, involving Mn IV as the oxidizing species.F inally,w ea pplied this system in several domino reactions,s howing the compatibility of the catalyst with different conditions to demonstrate its value as apowerful synthetic tool.

Figure 1 .

 1 Figure 1. a) Influence of NaOH concentration used for Na-Mn LMO preparation on its catalytic efficiency and on particles size. b) Influence of bases of different strength on catalytic efficiencya nd on particles size. Reaction conditionsf or (a) and (b) are the same than forTable 1, entry 1.

Table 2 :

 2 (Continued)

1 98 98

Acknowledgements

V. E. is grateful for support from French National Center for Scientific Research (CNRS). C.H.L. would like to thank the Donnelley Environmental Foundation for his fellowship support. Dr. Stephen Golledge from University of Oregon and Jinyang Li, from Prof.D r. AndrØ Taylorsg roup at Yale University,are acknowledged for assistance with XPS and TG analysis,r espectively.D r. Sara Hashmi at the Facility for Light Scattering at Yale University is acknowledged for her assistance and instrumentation support.

Conflict of interest

Theauthors declare no conflict of interest.