Giliane P Odin
email: giliane.odin@mnhn.fr

Sylvain Charbonnier
email: sylvain.charbonnier@mnhn.fr

Julien Devillez
email: julien.devillez@edu.mnhn.fr

Günter Schweigert
email: guenter.schweigert@smns-bw.de

On unreported historical specimens of marine arthropods from the Solnhofen and Nusplingen Lithographic Limestones

Keywords: Crustacea, Decapoda, Jurassic, Plattenkalk, Solnhofen, Nusplingen, Germany, collections, conservation Crustacea, Decapoda, Jurassique, Plattenkalk, Solnhofen, Nusplingen, Allemagne, collections, conservation

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INTRODUCTION

The fossil record of marine arthropods delivered by Plattenkalks is particularly diverse and spans several continents (Africa, South America, North America, Europe) and geological era (Cretaceous, Jurassic, Carboniferous). Among these Plattenkalks, the Solnhofen Lithographic Limestones (Bavaria, Germany) are particularly remarkable and deliver beautifully preserved specimens of decapod crustaceans, stomatopods, mysidaceans, and xiphosurans (about 130 taxa; Arratia et al. 2015).

The collection of marine arthropods coming from Solnhofen at the Muséum National d'Histoire Naturelle (MNHN, Paris, France) is the result of 250 years of accumulation, ensuing from exchanges with German institutions, wars pillage, purchases and donations [START_REF] Charbonnier | The marine arthropods from the Solnhofen Lithographic Limestones (Late Jurassic, Germany) in the collections of the Muséum national d'Histoire naturelle[END_REF]. Most of the specimens have been described in a previous paper [START_REF] Charbonnier | The marine arthropods from the Solnhofen Lithographic Limestones (Late Jurassic, Germany) in the collections of the Muséum national d'Histoire naturelle[END_REF]) but here, we report other specimens discovered in the MNHN collections, notably several specimens of the penaeid shrimp Antrimpos undenarius [START_REF] Schweigert | The decapod crustaceans of the Upper Jurassic Solnhofen Limestones: A historical review and some recent discoveries[END_REF] from the d'Orbigny collection. This collection, assembled by the French naturalist Alcide d'Orbigny , was purchased by the MNHN and specimens were inventoried in a catalogue composed of 27 fascicles [START_REF] Lauriat-Rage | La collection d'Invertébrés fossile d'Alcide d'Orbigny et la salle d'Orbigny[END_REF]. In 1898, a dedicated room (called after his name) was allocated in the newly built Gallery of Palaeontology: "la salle d'Orbigny". However, the moving of the invertebrate collections, including arthropods, to another building in 2019-2020 required to map beforehand these collections and has led to the discovery of historical specimens that have not been published before.

MATERIAL AND METHODS

COLLECTION ARRANGEMENT. -As introduced by [START_REF] Charbonnier | The marine arthropods from the Solnhofen Lithographic Limestones (Late Jurassic, Germany) in the collections of the Muséum national d'Histoire naturelle[END_REF], the fossils housed in the National Museum of Natural History (MNHN; Paris, France) are dispersed between several collections and buildings, explaining why new specimens are regularly discovered. Because all the paleontological specimens housed at the MNHN will soon move in a single building, a mapping of the invertebrate collections has been realized.

As a result, unreported historical specimens of marine arthropods from the Solnhofen Lithographic Limestones were discovered in the drawers of the institution and are the subject of this paper. The specimens already described by [START_REF] Charbonnier | The marine arthropods from the Solnhofen Lithographic Limestones (Late Jurassic, Germany) in the collections of the Muséum national d'Histoire naturelle[END_REF] will not be mentioned, unless the fossils are re-attributed to new taxa. FOSSIL LOCALITIES. -Our localities encompass Late Jurassic marine Plattenkalks (i.e. very finely grained limestone precipitated under quiet conditions) of southern Germany (see Audo et al. 2014a: fig 1. for localization) where soft-bodied fossils, including arthropods, were preserved in lagoonal calcareous muds. The diversity of organisms is exceptionally important and the quality of the record is remarkable. Among marine arthropods, taxa such as penaeoidean and caridean shrimps, polychelidan, erymid and glypheid lobsters have notably been recorded (for general reviews see [START_REF] Garassino | The Upper Jurassic Solnhofen decapod crustacean fauna: review of the types from old descriptions. Part I. Infraorders Astacidea, Thalassinidea, and Palinura[END_REF][START_REF] Schweigert | The decapod crustaceans of the Upper Jurassic Solnhofen Limestones: A historical review and some recent discoveries[END_REF][START_REF] Charbonnier | The marine arthropods from the Solnhofen Lithographic Limestones (Late Jurassic, Germany) in the collections of the Muséum national d'Histoire naturelle[END_REF][START_REF] Winkler | Libanocaris annettae nov. sp. (Crustacea: Dendrobranchiata: Penaeidae) from the Upper Jurassic Solnhofen Lithographic Limestones of Eichstätt[END_REF][START_REF] Schweigert | The Upper Jurassic Solnhofen decapod crustacean fauna: Review of the types from old descriptions. Part II. Superfamily Penaeoidea and infraorder Caridea[END_REF].

The Solnhofen Lithographic Limestones, ranging from the late Kimmeridgian to the early Tithonian [START_REF] Schweigert | Ammonite biostratigraphy as a tool for dating Upper Jurassic lithographic limestones from South Germanyfirst results and open questions[END_REF], are located in the southern Franconian Alb (SW Germany).

They include a large set of localities yielding Plattenkalks, such as Eichstätt, Painten, Solnhofen, globally referred to as "Solnhofen" in many old collections [START_REF] Garassino | The Upper Jurassic Solnhofen decapod crustacean fauna: review of the types from old descriptions. Part I. Infraorders Astacidea, Thalassinidea, and Palinura[END_REF]. Used since the late Stone Age as construction stones, fossils were traded by the quarrymen from the 18th century (Barthel et al. 1990: 4-10). These Plattenkalks deliver marine and terrestrial remains, among which jellyfish and feathered dinosaurs (the famous Archaeopteryx) are probably the most remarkable ones. Polychelidan, erymid and glypheoidean lobsters as well as caridean and penaeoidean shrimps account for the majority of Solnhofen crustaceans, the latter being the most abundant (Barthel et al. 1990: 129).

Among xiphosurans, we can cite the Mesolimulus species, which is one of the best-known marine arthropods encountered in Solnhofen (Barthel et al. 1990: 138).

The Nusplingen Lithographic Limestone is a small fossil Lagerstätte, from the late Kimmeridgian, located in the western part of the Swabian Alb (SW Germany). The site includes two close quarries and delivers over 400 taxa of microfossils, land plants, invertebrates, vertebrates and ichnofossils remains [START_REF] Dietl | The Nusplingen Lithographic Limestone -A 'fossil Lagerstaette'of late Kimmeridgian age from the Swabian Alb (Germany)[END_REF][START_REF] Schweigert | The decapod crustaceans of the Upper Jurassic Solnhofen Limestones: A historical review and some recent discoveries[END_REF]. The first fossils from the Nusplingen Lithographic Limestone were reported in the middle of the 19th century [START_REF] Fraas | Beiträge zum obersten weissen Jura in Schwaben[END_REF], and after a commercial use of the limestones, it is now a Protected Excavation Area dedicated to scientific excavation [START_REF] Dietl | The Nusplingen Lithographic Limestone -A 'fossil Lagerstaette'of late Kimmeridgian age from the Swabian Alb (Germany)[END_REF]). Wellpreserved caridean and penaeoidean shrimps and polychelidan lobsters, are typical of the Nusplingen Plattenkalk.

SYSTEMATICS. -The following list of Decapoda and Mysidacea crustaceans and limulids (Xiphosurida) housed in the MNHN is arranged taxonomically. It includes 117 specimens which are all presented Table 1. Three species are known from the Solnhofen Lithographic Limestones [START_REF] Schweigert | The Upper Jurassic Solnhofen decapod crustacean fauna: Review of the types from old descriptions. Part II. Superfamily Penaeoidea and infraorder Caridea[END_REF]. All are present in the palaeontological collections of the MNHN (Fig. 1).

Aeger tipularius (Schlotheim, 1822)

(Fig. 1A-C A. insignis [START_REF] Oppel | Ueber jurassische Crustaceen (Decapoda macrura)[END_REF], and Acanthochirana cordata [START_REF] Münster | Decapoda Macroura. Abbildung und Beschreibung der fossilen langschwänzigen Krebse in den Kalkschiefern von Bayern[END_REF].

COMMENTS. -Our assignation is based on the very short and smooth rostrum. The carapace groove pattern typical of Aegerwith a cervical groove steeply inclined, not joined to the dorsal margin but joined to the antennal groove; an oblique and sinuous branchiocardiac groove crossing all the carapace from the posterior margin to the hepatic region; and a hepatic groove joined anteriorly to the cervical groove and posteriorly to the branchiocardiac groove (see diagnosis in Charbonnier et al. 2017: 36) is particularly visible on specimen A33509 (Fig. 1A-B).

Aeger spinipes (Desmarest, 1817) (Fig. 1D) [START_REF] Charbonnier | The marine arthropods from the Solnhofen Lithographic Limestones (Late Jurassic, Germany) in the collections of the Muséum national d'Histoire naturelle[END_REF] did not mention this species. We add one specimen from Solnhofen, MNHN.GG.2004/57291 (formerly identified as A. tipularius).

DIAGNOSIS
COMMENT. -Our assignation is based on the elongate rostrum bearing one median ventral tooth, particularly visible in UV light (Fig. 1F).

Genus Acanthochirana Strand, 1928 COMMENTS. -Specimen MNHN.F.A33504 (Fig. 3A-C) shows a well-preserved rostrum and a well-marked hepatic spine, while specimen A33519 (Fig. 3G-H) shows evidence of preserved muscle fibres near somites 1 and 2 (see also [START_REF] Briggs | Mineralization of soft-part anatomy and invading microbes in the horseshoe crab Mesolimulus from the Upper Jurassic Lagerstätte of Nusplingen, Germany[END_REF]. For comparison, specimens from the Carnegie Museum of Natural History (CM-33420; from Robalino et al.

2016; Fig. 3D-E) and from the private collection of Roger Frattigiani (Laichingen, Germany; figured in Schweigert 2015: fig. 568 and Fig. 3F) are also illustrated. von Beroldingen probably got them by exchanging fossils with Oscar Fraas (1824-1897) as the two of them knew each other. Indeed, in a letter from 1863, "Mr Béroldingen" mentioned several fossils from the Tertiary of Swabia, including the Jurassic taxon Squatina acanthoderma from Nusplingen, which was described by Oscar Fraas in 1855 [START_REF] Mayer | Württembergische Paläontologen, vorzüglich Liebhaber (Sammler, Förderer, Populisatoren, Präparatoren), die im Catalogus bio-bibliographicus von Lambrecht und Quenstedt (1938) fehlen[END_REF]).

Antrimpos undenarius

Number "13 e " refers to the thirteenth geological stage of d'Orbigny, which is the Oxfordian.

Additional specimens of A. undenarius coming from the SMNS collection are presented in 5A-C, E) is also characteristic of the fossil prawns from Nusplingen. Thus, the preservation in complete dorsal view, as for A49610 (Fig. 4D) and SMNS 66122 (Fig. 5F) is rather rare. Furthermore, the preservation in 3D of the rostrum is particularly remarkable on specimens SMNS 80482 and SMNS 66122 (Fig. 5C,F).

On specimens housed at the MNHN, the diagnostic anatomical characters of the species are poorly preserved and observed alternatively in several specimens (Fig. 4C,E,G). For instance, the rostrum with numerous dorsal teeth (Fig. 4G), the strong hepatic spine, the hepatic ridge, the median lateral processes on s1-s3 are typical of Antrimpos. The large size of the specimens is compatible with A. undenarius.

Infraorder ASTACIDEA Latreille, 1802

Superfamily Erymoidea [START_REF] Van Straelen | Contribution à l'étude des crustacés décapodes de la période jurassique[END_REF] Family Erymidae [START_REF] Van Straelen | Contribution à l'étude des crustacés décapodes de la période jurassique[END_REF] PRELIMINARY REMARKS. -Eryma modestiforme (Schlotheim, 1822) (Fig. 6), Palaeastacus fuciformis (Schlotheim, 1822) (Fig. 7) and Pustulina minuta (Schlotheim, 1822) (Fig. 7) are the three most common erymids found in the Solnhofen Lithographic Limestones. Numerous specimens are almost complete, but the compression often impedes the identification of the carapace grooves, which are important for the systematic of the Erymidae. Instead, we use the general ornamentation and the features of the P1 chelae (shape and ornamentation) to distinguish the erymid lobsters. Indeed, P. fuciformis and P. minuta have P1 chelae with a subrectangular propodus bearing short and wide fingers, while the fingers of E. modestiforme are longer than the propodus and are progressively narrowing to their distal extremity.

Moreover, E. modestiforme is ornamented by small tubercles and depressions, while P. minuta is covered by coarse rounded tubercles and P. fuciformis bears strong spines on its carapace and its P1 propodus.

Genus Eryma Meyer, 1840

TYPE SPECIES.
-Macrourites modestiformis Schlotheim, 1822, by subsequent designation of [START_REF] Glaessner | Crustacea Decapoda[END_REF]. COMMENTS. -Eryma modestiforme and E. veltheimii [START_REF] Münster | Decapoda Macroura. Abbildung und Beschreibung der fossilen langschwänzigen Krebse in den Kalkschiefern von Bayern[END_REF] are the two species encountered in the Solnhofen Lithographic Limestones, the former being the most common.

E. modestiforme is known by numerous specimens, almost complete, stored notably in

European palaeontological collections (e.g. Paris, Berlin, London, Brussels, Basel). Among the MNHN collection, four specimens clearly show the intercalated plate characteristic of the erymid lobsters: MNHN.F.A32408, A33507, A70905, B13463, B13450 (Fig. 6A-D, H; see also Charbonnier & Garassino 2012: fig. 3b). All the specimens exhibit a short P1 propodus with a dense ornamentation, allowing us to ascribe the fossils to the species E. modestiforme (the propodus is more elongate and smooth in E. veltheimii). Moreover, three specimens show additional characters, diagnostic of E. modestiforme: (1) on the specimens MNHN.F.A70905

(Fig. 6C-D) and B13452 (Fig. 6E-F), the carapace groove pattern, in particular the junction between the postcervical and the branchiocardiac grooves [START_REF] Devillez | -Review of the Early Cretaceous erymid lobsters (Crustacea: Decapoda) from the Western Tethys[END_REF][START_REF] Devillez | The genus Eryma Meyer, 1840 (Crustacea: Decapoda: Erymidae): new synonyms, systematic and stratigraphic implications[END_REF]), is easily identifiable; and (2) the morphology of the P1 chelae of the specimen MNHN.F.A32408 (Fig. 6H), shows a subrectangular propodus, thin fingers progressively narrowing to their distal extremity and without teeth on their occlusal margins.

The historical specimen MNHN.F.A32408, figured by Desmarest (1822: pl. 11, fig. 5),

shows the typical mode of preservation and lithology of the Bavarian Lithographic Limestones (Fig. 6H) but the locality of this specimen may be misleading. [START_REF] Desmarest | Les crustacés proprement dits[END_REF] p. 136) refers to "Aichstedt", which is also the locality indicated on the original label, while the locality "Aichtät" is specified on the specimen. In a letter about the Franconian geology to A. Brongniart, L. de Buch (1823) evokes the lithographic limestones exploited at "Eichstedt"

and "Solenhofen", in the Altmuhl valley. However, there is no locality spelled "Eichstedt" next to the Altmühl River (instead, Eichstedt (Altmark) is a city located approx. 80 km west to Berlin) but "Eichstätt". Thus, the designations "Aichtät", "Aichstedt" and "Eichstedt" in the mentioned publications are likely orthographic variations referring to the same locality, nowadays spelled Eichstätt.

Genus Palaeastacus Bell, 1850 TYPE SPECIES. -Astacus sussexiensis Mantell, 1824, by subsequent designation of [START_REF] Glaessner | Crustacea Decapoda[END_REF]. Mecochirus. This view was followed by Schweitzer et al. (2010: 20), who, however, maintained Mecochirus foresti as a separate species. Our examination of the two syntypes did not evidence any significant morphological differences to distinguish two species. Hence, we consider M. foresti as a junior synonym of M. longimanatus.

Specimen MNHN.F.A70929 (Fig. 8C-E) arrived in the MNHN collections in December 1930, through Mr Gravigny, Charles Brongniart's son-in-law. The carapace exhibits the typical cervical groove and the orbital and gastro-orbital carinae. P1 is very elongate and P2 shows a strong and stout propodus as described by [START_REF] Garassino | The Upper Jurassic Solnhofen decapod crustacean fauna: review of the types from old descriptions. Part I. Infraorders Astacidea, Thalassinidea, and Palinura[END_REF].

Infraorder (Fig. 9) STUDIED MATERIAL. [START_REF] Charbonnier | The marine arthropods from the Solnhofen Lithographic Limestones (Late Jurassic, Germany) in the collections of the Muséum national d'Histoire naturelle[END_REF] listed five specimens (Table 1). We do not change their contribution.

COMMENTS. -The specimen MNHN.F.B13436 (Fig. 9A) is preserved in ventral view and exhibits asymmetrical first pereiopods: the right P1 presents the usual aspect whereas the left one is much more reduced. According to [START_REF] Van Straelen | Contribution à l'étude des crustacés décapodes de la période jurassique[END_REF] and Audo (2014), this physiology is not native but rather due to a regenerative heterochely. The crista dentata (serrated inner margin of the third maxilliped ischium) is visible.

Genus Eryon Desmarest, 1817 1).

COMMENTS. -Three of the specimens figured are in ventral view (Fig. 9 C-G), allowing the observation of nicely preserved details such as the mandibles exhibiting sharp triangular teeth of uneven size (specimen MNHN.F.GG2004/8220; Fig. 9D-E). Indeed, the preservation is usually excellent: the antennae, antennules and eyestalks of the specimens MNHN.F.A33545

and GG2004/8218 are clearly visible and they exhibit between eight and ten pereiopods. The dorsally preserved specimen MNHN.F.B13439 (Fig. 9H) is also well-preserved and evidences delicate details on the carapace (branchiocardiac, postcervical and cervical grooves), telson (ponctuations and carinae). The other specimen (A33501; Fig. 9B) present some calcite bundles on the left chela. COMMENTS. -Specimen MNHN.F.A33549 (Fig. 10) presents some pereiopods with terminal dactylus, some pleopods, the tail fan, and the hooked rostrum, which is isolated from the body. The carapace (cephalothoracic shield) is not preserved, neither the pleon. This configuration evokes an exuvia or a predated specimen. Intriguingly, some pereiopods exhibit apodemes as if the muscles were stripped off. COMMENTS. -Specimen MNHN.F.B13432 (Fig. 11A) is poorly preserved and exhibit many calcite bundles (as dumbells and spheres), mostly at the rear of the prosoma. According to [START_REF] Briggs | The role of the calcium carbonate-calcium phosphate switch in the mineralization of soft-bodied fossils[END_REF], among the arthropods of Solnhofen, Mesolimulus is the taxon the most commonly subject to calcite crystals bundles development (we illustrate a specimen of Eryon cuvieri with calcite bundles on Fig. 9B). Such crystals are believed to be formed during the early stage of diagenesis, due to microbial activity. Specimen MNHN.F.B13433 (Fig. 11B) is a dorsal view of a slightly deformed specimen; the prosoma, carapace and telson are beautifully preserved.

Bibliography

  DIAGNOSIS FROM SCHWEIGERT (2001). -Large-sized species of the genus Antrimpos, rostrum with 11 dorsal spines, the posterior one [= hepatical spine] being separated from the others, and one ventral spine located just before the foremost dorsal spine. Long propodite on P3. STUDIED MATERIAL. -Charbonnier & Garassino (2012) did not mention this species. Nine specimens MNHN.F.A06834, A49608, A49609, A49610, A49615, A49616, A49622, A49623 and A49624 from Nusplingen were re-discovered in the Collection d'Orbigny. All these specimens are labelled "Nusplingen, Mr Béroldingen, 13 e " (Fig. 4A). The family von Beroldingen is an old family of nobles originating from Switzerland; Baron Franz Coelestin von Beroldingen (1740-1798) was known as a mineral and fossil collector and after his death, his nephew, Count Joseph Ignace von Beroldingen (1780-?), sold his mineral collection to the London museum (British Museum of Natural History 1904). In our case, "Mr Béroldingen" refers to Paul Ignaz Josef Graf von Beroldingen (1804-1875), another nephew of Baron von Beroldingen, who offered the Antrimpos specimens from Nusplingen to Alcide d'Orbigny.

Fig. 5

 5 Fig. 5 for comparison.

  Fig. 6B, B13452 -Fig. 6E and B13463 -Fig. 6A) and add three new specimens, 1 specimen

  [START_REF] Charbonnier | The marine arthropods from the Solnhofen Lithographic Limestones (Late Jurassic, Germany) in the collections of the Muséum national d'Histoire naturelle[END_REF] mentioned three specimens: two from Solnhofen (MNHN.F.B13449;MNHN.GG.2004/8245) and one from an unknown locality (MNHN.GG.2004/7998-8106). We do not change their attribution.COMMENTS. -Since only one specimen (MNHN.F.B13449) was figured byCharbonnier & Garassinno (2012), we additionally figure the specimen MNHN.GG.2004/8245 (Fig.7A-C), which is particularly well-preserved; notably, the ornamentation is nearly complete.Genus Pustulina Quenstedt, 1857 TYPE SPECIES. -Pustulina suevica Quenstedt, 1857, by subsequent designation ofGlaessnerCharbonnier & Garassino (2012) mentioned three specimens: two from Eichstätt (MNHN.F.B13444, B13445) and one specimen from an unknown locality (MNHN.GG.2004/8101). We do not change their attribution. Family Mecochiridae Van Straelen, 1925 Genus Mecochirus Germar, 1827 TYPE SPECIES. -Macrourites longimanatus Schlotheim, 1820, by subsequent designation of Woods (1927). Mecochirus longimanatus (Schlotheim, 1820) (Fig. 8) STUDIED MATERIAL. -Charbonnier & Garassino (2012) listed 27 specimens including two specimens described as Mecochiria foresti by Secrétan (1968). We add a new specimen from Solnhofen (MNHN.F.A70929). COMMENTS. -We localized the type material of Mecochiria foresti Secrétan, 1968. She listed five syntypes but figured only two (MNHN.F.A33537 -Fig. 8A-B, A33539 -Fig. 8F-G), which are dorsoventrally flattened. According to Förster (1971), these specimens show diagnostic characters of Mecochirus longimanatus, and Mecochiria is a junior synonym of

Superorder

  Elder ungulatus Münster, 1839, by subsequent designation of Oppel (1862). Elder ungulatus Münster, 1839 (Fig. 10) STUDIED MATERIAL. -Charbonnier & Garassino (2012) listed three specimens from Solnhofen (MNHN.F.A33549, B13467; MNHN.GG.2004/8217) but figured none. We do not change their contribution.

FiguresFig. 1 .

 1 Figures

Fig. 5 -

 5 Fig. 5 -Antrimpos undenarius Münster, 1839 from Nusplingen, housed at the SMNS. A, specimen SMNS 80480. B, specimen SMNS 80481. C, specimen SMNS 80482. D, specimen SMNS 80483. E, specimen SMNS 24228. F, specimen SMNS 66122. Scale bars: A-F: 2 cm. Photographs: G. Schweigert.

Fig. 7 -

 7 Fig. 7 -Other Erymoidea from the Late Jurassic Solnhofen Lithographic Limestones (Bavaria, Germany). A-C, specimen MNHN.GG.2004/8245 of Palaeastacus fuciformis (Schlotheim, 1822) from Solnhofen: specimen in natural light (A), specimen in UV light (B) and line drawing (C); D, specimen MNHN.F.B13445 of Pustulina minuta (Schlotheim, 1822) from Solnhofen in UV light. Scale bars: 1 cm, except D: 0.5 cm. Abbreviations: a1: antennule; a2: antenna; e: eye; Mxp3: third maxillipeds; P1-5: periopods 1 to 5. Photographs: L. Cazes. Line drawing: J. Devillez.

Fig. 8 -

 8 Fig. 8 -Mecochirus longimanatus (Schlotheim, 1820). A-B, specimen MNHN.F.33537, A: specimen in natural light, B: specimen in UV light; C, specimen MNHN.F.B13457; D-F, specimen MNHN.F.A70929, D: general view in UV light, E: view of the carapace in natural light and F: line drawing of the carapace; G-H, specimen MNHN.F.A33539, G: specimen in natural light, H: specimen in UV light. Scale bars: 1 cm. Abbreviations: ac = antennal carina, e 1 e = cervical groove, gc = gastro-orbital carina, oc = orbital carina. Photographs: L. Cazes. Line drawing: G.P. Odin.

Fig. 9 -

 9 Fig. 9 -A, Cycleryon propinquus (Schlotheim, 1822): specimen MNHN.F.B13436 from Solnhofen; B-H, Eryon cuvieri Desmarest, 1817: B, specimen MNHN.F.A33501 from Solnhofen; C, specimen MNHN.GG.2004/8218 from Solnhofen area. This specimen was referred to as coming from Pappenheim. Since there are no Solnhofen Lithographic Limestones there, while many traders lived there, this specimen was probably purchased in Pappenheim. D-E, specimen MNHN.GG.2004/8220 from Solnhofen, natural light (D) and UV light (E); F-G, specimen MNHN.F.A33545 from Solnhofen, natural light (F) and UV light (G); H, specimen MNHN.F.B13439 from unknown locality. Scale bars: A, C, F, H: 2 cm; B, D, E, F: 1 cm. Photographs: L. Cazes.

Fig. 10 -

 10 Fig. 10 -Elder ungulatus (Münster, 1839). Specimen MNHN.F.A33549 under UV light. Scale bar: 1 cm. Photograph: L. Cazes.

Table 1 . List of the 117 specimens of marine arthropods coming from the Southern Germany Lithographic Limestones and housed in the MNHN, Paris. Caption: bold: specimens newly described (this work); italic: specimens re-attributed (this work); n: number of specimens; -: not observed.

 1

	Taxa	Charbonnier & Garassino (2012)		This work		Total
	DECAPODA	Collection numbers	n	Collection numbers	n	n
	Aegeridae
	Aeger spinipes (Desmarest, 1817)	MNHN.F.B13441	1	MNHN.F.B13440, B13441, B13442	3	3
	Aeger tipularius (Schlotheim, 1822)	MNHN.F.A33509, B13440, B13442, B13443,	7	MNHN.F.A33509, B13443,	3	3
		R03389, MNHN.GG.2004/57291, 2004/8072		MNHN.GG.2004/8072
	Aeger insignis Oppel, 1862		0	MNHN.GG.2004/57291	1	1
	Acanthochirana longipes (Oppel, 1862)	MNHN.F.A42003	1		0	0
	Acanthochirana cordata (Münster, 1839)		0	MNHN.A33503, A42003, R03389	3	3
	Penaeidae
	Antrimpos speciosus Münster, 1839	MNHN.F.A33504, A33519, A33548, B13465,	6	MNHN.F.A33504, A33519, A33548,	6	6
		R03502, MNHN.GG.2004/8086		B13465, R03502, MNHN.GG.2004/8086
	Antrimpos undenarius Schweigert, 2011		0	MNHN.F.A06834, A49608, A49609,	9	9
				A49610, A49615, A49616, A49622,
				A49623, A49624
	Rauna angusta Münster, 1839	MNHN.F.A42004	1	-	-	1
	Bylgia hexadon Münster, 1839	MNHN.F.A33503	1		0	0
	Albertoppelia kuempeli Schweigert &	MNHN.F.A33521	1	-	-	1
	Garassino, 2004
	Drobna deformis Münster, 1839	MNHN.F.A33531	1	-	-	1
	Caridea
	Hefriga serrata Münster, 1839	MNHN.F.B13464, MNHN.GG.2004/8237	2	-	-	2
	Erymidae
	Eryma modestiforme (Schlotheim, 1822)	MNHN.F.A33507, B13446, B13448, B13450,	8	MNHN.F.A32408, A33507, A70905,	11	11
		B13452, B13463, MNHN.GG.2004/7462,		A70906, B13446, B13448, B13450,
		2004/8078-7471		B13452, B13463,
				MNHN.GG.2004/7462, 2004/8078-7471
	Palaeastacus fuciformis (Schlotheim,	MNHN.F.B13449, MNHN.GG.2004/8245,	3	MNHN.F.B13449,	3	3
	1822)	2004/7998-8106		MNHN.GG.2004/8245, 2004/7998-8106
	Pustulina minuta (Schlotheim, 1822)	MNHN.F.B13444, B13445,	3	MNHN.F.B13444, B13445,	3	3
		MNHN.GG.2004/8101		MNHN.GG.2004/8101
	Glypheidae
	Glyphea pseudoscyllarus (Schlotheim,	MNHN.F.A33506, B13447, B13462,	4	-	-	4

)

  DIAGNOSIS FROM SCHWEIGERT ET AL. (2016). -Very short smooth rostrum; Mxp3 extremely

	long, with movable spines; chelate P1-P3; P1-P2 bearing long movable spines; P3 with spiny
	chelae; achelate P4-P5 and spineless; P3 slightly larger than P1-P2; well-developed pleopods;
	uropodal exopod with rounded diaresis.

STUDIED MATERIAL.

[START_REF] Charbonnier | The marine arthropods from the Solnhofen Lithographic Limestones (Late Jurassic, Germany) in the collections of the Muséum national d'Histoire naturelle[END_REF]

listed seven specimens. Careful examination leads us to re-identify some of them. Three specimens, MNHN.F.A33509 (Solnhofen), B13443 (Eichstätt),

MNHN.GG.2004/8072 (unknown locality)

, are maintained as A. tipularius, while four others are attributed to Aeger spinipes

[START_REF] Desmarest | Crustacés fossiles, in Société de Naturalistes et d'Agriculteurs[END_REF]

,

  FROM SCHWEIGERT ET AL.(2016). -Extremely elongate rostrum, with one medium ventral tooth; Mxp3 extremely long, with movable spines; chelate P1-P3; P1-P2 bearing long movable spines; P3 with smooth chelae; achelate P4-P5 and spineless; P3 slightly larger than P1-P2; well-developed pleopods; uropodal exopod with rounded diaresis. STUDIED MATERIAL.[START_REF] Charbonnier | The marine arthropods from the Solnhofen Lithographic Limestones (Late Jurassic, Germany) in the collections of the Muséum national d'Histoire naturelle[END_REF] listed and figured only one specimen (MNHN.F.B13441, Eichstätt). We add two specimens from Solnhofen, MNHN.F. B13440, B13442 (Fig.1D), formerly identified as A. tipularius.

COMMENT. -Our assignation is based on the extremely elongate rostrum.

Aeger insignis

Oppel, 1862 (Fig. 1E-F) DIAGNOSIS FROM SCHWEIGERT ET AL. (2016). -Short rostrum, with one ventral tooth distally; Mxp3 extremely long, with movable spines; chelate P1-P3; P2 with spiny chelae; P3 with spiny elements and chelae; achelate P4-P5; uropodal exopod with rounded diaresis. STUDIED MATERIAL.

  TYPE SPECIES. -Udora cordataMünster, 1839 (partim), by subsequent designation of The ratio measured on this specimen is of 0.9, implying an assignation to A. cordata. Specimen A33503 (Fig.2A-B), previously attributed to Bylgia hexadon, was misidentified: it clearly shows elongate Mxp3 with moveable spines characteristic of aegerid shrimps. More precisely, the length of Mxp3 fits well with those observed in Acanthochirana. Additionally, this specimen shows a spiny rostral carina prolonged by a smooth rostrum and an epigastric spine, which allow us to ascribe the specimen to Acanthochirana cordata.

	with movable dactyli; cylindric flattened cephalothorax; rostrum dentate; pointed telson and cervical groove rimming the upper segment of hepatic spine; subhorizontal hepatic ridge
	Glaessner(1929). subrounded uropodal exopod. below hepatic spine; s1-s3 with rounded median lateral processes (subtle in s3); Mxp3 short
	and smooth; chelate P1-P3, increasing in length, achelate P4-P5; uropodal exopod with a
	DIAGNOSIS FROM CHARBONNIER ET AL. (2017). -Rostral carina with spines pointing TRANSLATION OF THE EMENDED DIAGNOSIS FROM OPPEL (1862). -Antennules very subrounded diaresis.
	forward; smooth rostrum, spineless; one epigastric spine in the third anterior of dorsal margin; short, of different length, antennae of twice the length of the body, rostrum dentate, spiny
	weak cervical groove, not joined to dorsal margin; oblique branchiocardiac groove joined to Mxp3; chelate P1-P3, increasing in length, and achelate P4-P5; S5 longest; cuticle smooth Antrimpos speciosus Münster, 1839
	hepatic groove at level of strong hepatic spine; short hepatic groove backward directed; except on the dorsal area of the carapace, which is covered with small pits. (Fig. 3)
	cephalic region with postorbital spine; hypertrophied Mxp3 with rows of movable spines; Specimen R03389 (Fig. 2C-D), previously attributed to Aeger tipularius, was also
	branch-like Mxp3 dactylus; chelate P1-P3; achelate P4-P5; uropodal exopod without TRANSLATION OF THE EMENDED DIAGNOSIS FROM VAN STRAELEN (1925). -Rostrum dentate STUDIED MATERIAL. -Charbonnier & Garassino (2012) mentioned 6 specimens:
	diaeresis. -often on both sides; antennules of different length; antennae to twice length of body; chelate 3 specimens from Solnhofen (MNHN.F.A33519, A33548, R03502); 2 specimens from
	P1-P3 and achelate P4-P5. Eichstätt (MNHN.F. A33504, B13465) and 1 specimen from an unknown locality
	EMENDED DIAGNOSIS FROM GLAESSNER (1969). -Antennules very short, antennae to twice
	length of body, rostrum dentate, carapace smooth, thin; length of pereiopods increasing from
	1 to 3; 6 th abdominal somite longest.
	Family Penaeidae Rafinesque, 1815
	EMENDED DIAGNOSIS FROM SCHWEIGERT ET AL. (2016). -Carapace subrectangular, laterally
	Genus Antrimpos Münster, 1839 flattened; rostrum elongate slightly turned upward distally, with nine dorsal teeth and one
	ventral tooth distally; epigastric tooth; Mxp3 short and smooth; uropodal exopod with a
	TYPE SPECIES. -Antrimpos speciosus Münster, 1839, by subsequent designation of subrounded diaresis.
	Glaessner(1929).
	EMENDED DIAGNOSIS (PRESENT WORK). -Carapace subrectangular, laterally flattened;
	TRANSLATION OF THE ORIGINAL DIAGNOSIS FROM MÜNSTER (1839). -Antennules elongate rostrum slightly turned upward distally, with several dorsal teeth and one ventral
	short; antennae very long, with a very long scaphocerite; chelate P1-P3, increasing in length,

Acanthochirana cordata

[START_REF] Münster | Decapoda Macroura. Abbildung und Beschreibung der fossilen langschwänzigen Krebse in den Kalkschiefern von Bayern[END_REF]

(Fig.

2

) STUDIED MATERIAL. -We studied three specimens, MNHN.A33503, R03389

(Solnhofen)

and A42003 (Eichstätt), respectively attributed by

[START_REF] Charbonnier | The marine arthropods from the Solnhofen Lithographic Limestones (Late Jurassic, Germany) in the collections of the Muséum national d'Histoire naturelle[END_REF]

to Bylgia hexadon Münster, 1839, Aeger tipularius, and Acanthochirana longipes

[START_REF] Oppel | Ueber jurassische Crustaceen (Decapoda macrura)[END_REF]

.

COMMENTS.

[START_REF] Charbonnier | The marine arthropods from the Solnhofen Lithographic Limestones (Late Jurassic, Germany) in the collections of the Muséum national d'Histoire naturelle[END_REF]

identified the specimen MNHN.F.A42003 as A. longipes. Careful examination of this specimen leads us to a different identification.

Among the criteria listed in the diagnosis by

[START_REF] Schweigert | The Upper Jurassic Solnhofen decapod crustacean fauna: Review of the types from old descriptions. Part II. Superfamily Penaeoidea and infraorder Caridea[END_REF]

, the number of dorsal rostral teeth and the elongation of Mxp3 are the most important to distinguish A. cordata from A. longipes. Since it was impossible to count the teeth on our specimens, we use the ratio Mxp3 length / cephalothorax length. Based on drawings from Oppel (1862), the ratio is 1 for A. cordata while >>1 for A. longipes. misidentified: it clearly shows elongate Mxp3 with moveable spines characteristic of aegerid shrimps but the length of Mxp3 fits well with those observed in Acanthochirana. Also, the carapace groove pattern is typical of Acanthochirana, showing a weak cervical groove not joined to the cervical margin, an oblique branchiocardiac groove joined to the hepatic groove near the strong hepatic spine and a short hepatic groove backward directed (see diagnosis in

Charbonnier et al. 2017: 32)

. tooth; longitudinal rostral carina; epigastric tooth; postorbital spine; very strong and asymmetrical hepatic spine, with elongate dorsal segment and short ventral one; concave (MNHN.GG.2004/8086). Collections Boué, de Roissy, Hoffstetter, Manchester Museum, d'Orbigny.

  Knebelia schuberti(Meyer, 1836), and Knebelia totoroi[START_REF] Audo | b. -Diversity and palaeoecology of the enigmatic genus Knebelia (Eucrustacea, Decapoda, Eryonidae) from Upper Jurassic Plattenkalks in southern Germany[END_REF] (Audo et al. 2014b).Audo et al. (2014a) mentioned additional species including Soleryon schorri from Wattendorf and Soleryon sp. Macrourites propinquus Schlotheim, 1822, by original designation.

	POLYCHELIDA Scholtz & Richter, 1995
	Family Eryonidae De Haan, 1841
	PRELIMINARY REMARKS. -To date, the following eryonid species have been reported from
	the Solnhofen Lithographic Limestones: Cycleryon propinquus (Schlotheim, 1822),
	Cycleryon orbiculatus (Münster, 1839), Cycleryon elongatus (Münster, 1839), Cycleryon
	wulfi Garassino & Schweigert, 2004, Eryon cuvieri Desmarest, 1817, Knebelia bilobata
	(Münster, 1839), from Zandt.
	Genus Cycleryon Glaessner, 1965
	TYPE SPECIES. -Cycleryon propinquus (Schlotheim, 1822)

  TYPE SPECIES. -Eryon cuvieri Desmarest, 1817, by monotypy.

	Eryon cuvieri Desmarest, 1817
	(Fig. 9)
	STUDIED MATERIAL. -Charbonnier & Garassino (2012) listed 16 specimens including the
	lectotype (Table

  Limulus walchii Desmarest, 1817, by monotypy.

	Genus Mesolimulus Størmer, 1952
	TYPE SPECIES. -Mesolimulus walchii (Desmarest, 1817)
	(Fig. 11)
	STUDIED MATERIAL. -Charbonnier & Garassino (2012) listed four specimens from
	Solnhofen (MNHN.F.A33516, B13432, B13433; MNHN.GG.2004/7473) but figured only
	one specimen (MNHN.F.A33516). We do not change their contribution.
	Class MEROSTOMATA Dana, 1852
	Order XIPHOSURIDA Latreille, 1802
	Family Limulidae Riek & Gill, 1971

Acknowledgements

We thank Lilian Cazes (CR2P, MNHN) for assistance in photographic work. This paper is a contribution from the UMR CNRS 7207 Centre de Recherche en Paléontologie -Paris, and part of the 'PyrEx' project, funded by the DIM Matériaux Anciens et Patrimoniaux. We also would like to thank Denis Audo and Alessandro Garassino for their useful comments on a previous version of the manuscript.

-A monograph of the fossil macrurous Crustacea of England.

Palaeontographical Society of London, London, 122 p.


Tim Gould

Stefano Pittalis

Julien Toulouse

Eli Kraisler

Leeor

Leeor Kronik

Asymptotic behavior of the Hartree-exchange and correlation potentials in ensemble density functional theory

teaching and research institutions in France or abroad, or from public or private research centers.

I. Introduction

Density-functional theory (DFT) [1] is an exact reformulation of the many-body problem, which accesses the total ground-state energy of an electronic system through a variational principle based on the electron density rather than the many-body wave function. DFT is often applied within the popular Kohn-Sham (KS) approach [2], which transforms it into a problem of finding orbital solutions for non-interacting electrons in an effective potential, which is itself a functional of the density. This transforms the difficult many-electron problem into a series of simpler oneelectron problems.

Both quantum mechanics and DFT can be generalised to ensembles of states with different electron numbers, thus achieving a fractional number of electrons. This can be thought of in terms of open systems weakly coupled to the environment, [3] or as an asymptotic limit of a dissociation process. [4,5] Such a generalisation offers insights into subtle quantum mechanical effects that are unavailable from integer systems, including into how well-separated systems behave.

Perdew, Parr, Levy and Balduz [3] (PPLB) showed that the total energy, E, is a piecewise linear function of the number of electrons, N, and that as one crosses an integer, the slope of the E(N) curve may change. The density, and thus electron-ion energy also has this property. It is * t.gould@griffith.edu.au also reflected in DFT, where the sum of the non-interacting kinetic energy functional and the exact Hartree-exchangecorrelation energy functional must also be a piecewise linear function of N. As a consequence of the piecewise linearity of the energy, the energy of a "fractional ion" experiences derivative discontinuities as it passes through an integer. Because this discontinuity must be reflected in the KS system, the KS potential can and usually does experience "jumps" by a uniform constant as the electron number passes through an integer. [3] These derivative discontinuities give rise to surprising yet important phenomena (e.g., Refs. [3][4][5][6][7][8][9][10][11][12][13][14][15][16][17] and references therein). They help to formalise calculations of the ionization potential, the electron affinity, and the fundamental gap.

In applications one must use various density-functional approximations (DFAs), due to the fact that the required exchange-correlation functional is, although well-defined, unknown in general. Therefore there is a strong motivation to determine properties of the exact KS system, so that approximations may be compared against them. Small systems, such as atoms and ions, can be solved within another theory such as quantum Monte-Carlo (QMC) [START_REF] Toulouse | Electron Cor-relation in Molecules -ab initio Gaussian Quantum Chemistry[END_REF] or wavefunction techniques like coupled-cluster theory. [19] Accurate densities and energies can thereby be obtained to serve as benchmarks for testing, and hopefully improving, DFAs. (e.g., Refs. [5,[20][21][22][23][24]) Given the density of a system, one can also find the KS potential v s [n](r) through the numerical procedure of "density inversion", (e.g., Refs. [25][26][27][28][29][30][31][32][33][34][35][36][37]) and thus provide benchmarks for effective potentials, as well as energies.

Density inversion has been applied many times to systems with integer numbers of electrons. Previous work has focused on systems without degeneracies. More rarely, inversion has been applied to systems with broken symmetries or non-integer electron number. [29,[38][39][40][41][42] Only very recently [32] has inversion been applied to systems with degeneracies and fractional electron number -for the case of fractional anions and cations of Li, C, and F.

The fractional charge cases considered by Gould and Toulouse [32] are especially difficult for approximations. They can therefore provide insights into how DFAs fail, and might be improved. However, the question of how the potential resolves into Hartree (H), exchange (x) and correlation (c) components were not addressed in their work. These terms are typically approximated separately in DFAs, making it important for us to understand how they behave separately. Here, we address the question analytically (in the asymptotic limit) and numerically, by studying and reporting Hartree-exchange (Hx) and c components of the KS potential for fractional ions of Li, C, and F.

The manuscript is organized as follows: First, we define the key properties of interest, within the framework of ensemble DFT (EDFT). Second, we derive the main analytic results, specifically that the exact Hx and c potentials can have non-zero asymptotic constants and unusual effective charges in some cases, which cancel when combined. Next, with the objective of quantifying our analytical findings numerically, we describe the numerical method employed and provide results for fractional ions of Li, C, and F. Finally, we conclude.

II. Hartree-exchange and correlation energies in EDFT

EDFT comes in a variety of forms, each dealing with different classes of quantum state ensembles. Here, we are concerned with a form of EDFT which allows us to mix ground states with different electron numbers, as well as to treat high-symmetry densities that are associated with degeneracies. The lowest ensemble energy, E 0 [v ext ], for a given external potential, v ext (r), is found by minimizing an energy density functional over N-representable densities, n(r):

E 0 [v ext ] = min n→N F [n] + dr v ext (r)n(r) , (1)
where the universal, ensemble-generalised functional [3,[43][44][45][46] is

F [n] := min Γ→n Tr[ Γ( T + Ŵ )] ≡ Tr[ ΓI ( T + Ŵ )] . ( 2
)
Here T is the usual kinetic-energy operator, and Ŵ is the Coulomb electron-electron repulsion operator. The minimisation in Eq. ( 2) is carried out over ensemble density matrices Γ formed out of a set of mutually orthonormal states, possibly with different number of electrons. We denote the minimizing ensemble as ΓI , where I indicates it is an interacting state. Note that the quantities of EDFT depend on the weights that specify the ensemble (see below). To indicate this succinctly we use calligraphic letters.

All the ensemble density matrices Γ considered in the min-imisation yield the same density, n(r), i.e. Tr[ Γ n(r)] = n(r)

where n(r) is the density operator. The average electron number, N = drn(r), may be either integer or fractional.

By the convexity conjecture [START_REF] Dreizler | Density functional theory[END_REF], the minimising ensemble density matrix can be written as [3] ΓN

=(M + 1 -N) ΓM + (N -M) ΓM+1 (3)
≡(1 -c) ΓM + c Γ(M+1) , (4)
where M = N is the largest integer less than N, and c = N -M is the fractional part of the electron number. N indicates the smallest integer greater than N, which is M +1 when c > 0, but M when N is integer. We thus use M + 1 when results are the same regardless of choice (as above), but use N when it matters. The density thus becomes,

n N (r) =Tr[ ΓN n(r)] = (M + 1 -N)n M (r) + (N -M)n M+1 (r) . (5)
Traditionally, KS theory can be derived by "switching off" the electron-electron interaction and finding the energy as a functional of the density. Applying this strategy to EDFT gives the two most important energy functionals in the fashion of KS DFT: the kinetic energy functional,

T s [n] := min Γ→n Tr[ Γ T ] = Tr[ Γs T ] = ∑ κ w κ κ s | T |κ s (6)
and the Hartree-exchange-correlation (Hxc) energy functional,

E Hxc [n] :=F [n] -T s [n] . (7)
The trace on the right side of Eq. ( 6) uses the KS noninteracting ensemble density matrix,

Γs = ∑ κ w κ |κ s κ s |, (8)
where |κ s is the κ th KS non-interacting state and w κ is its associated ensemble weight.

The KS states |κ s can be chosen as single Slater determinants (SD) that are constructed from a common set of orbitals, {ϕ iσ (r)}, and defined by means of the occupation factors {θ κ iσ }, where θ κ iσ = 0 or 1. For example, in Li there are four orbitals, {ϕ 1s↑ , ϕ 1s↓ , ϕ 2s↑ , ϕ 2s↓ } which feature in two SD states, with majority up and down electrons. The two SDs are occupied according to 1s 2 2s ↑ (short-hand for {θ 1s↑ = 1, θ 1s↓ = 1, θ 2s↑ = 1, θ 2s↓ = 0}) and 1s 2 2s ↓ , respectively.

The orbitals can be further separated, using ϕ iσ (r) = φ i (r)ξ σ , into spin-independent spatial orbitals, φ i (r), and spin-projectors, ξ σ . The spatial components φ i (r) obey the KS equations

- 1 2 ∇ 2 + v s (r) φ i (r) =ε i φ i (r) . (9)
The electron density is then obtained as

n(r) = ∑ i f i |φ i (r)| 2 , (10)
where f i are the ensemble averaged occupations of the KS orbitals, f i ≡ ∑ κσ w κ θ κ iσ . In the case of Li we get f 1s = 2 and f 2s = 1, as expected.

By construction, the density from Eq. ( 10) must yield the interacting (ensemble averaged) electron densities available from parametrised exact QMC results [32] (see Section IV), for the systems considered. As a consequence of the underlying symmetry of atoms, the multiplicative KS effective potential v s (r) ≡ v s (|r|) is, provably, both spherically symmetric and spin independent.

The next functional to consider is the Hartree-exchange (Hx) energy E Hx . For closed-shell pure states, E Hx is defined unambiguously. Not so for ensembles. In ensembles, the Hartree-exchange energy, and thus the correlation energy, may be defined more flexibly, although, not necessarily more conveniently. In the following discussion we shall adopt a natural and direct generalization of the 'Hx' energy, given by

E Hx [n] := Tr[ Γs Ŵ ] = ∑ κ w κ κ s | Ŵ |κ s , (11)
for the atomic cases we concern ourselves with here; and consistent with previous work. [23,24,32,[START_REF] Gould | [END_REF] Eq. ( 11) may be regarded as a regular expression for the ensemble Hartree-Fock energy which, however, is here evaluated with exact KS orbitals. Correspondingly, the correlation energy can be finally extracted as follows

E c [n] := E Hxc [n] -E Hx [n] ≡ Tr[( T + Ŵ )( ΓI -Γs )] . (12)
While other decompositions are possible, [34,[49][50][51] this particular decomposition comes with formal and practical advantages. [23,24,[START_REF] Gould | [END_REF] It leads to an explicit piecewise linear dependence on N of T s , E Hx , and E c through the ensemble weights (additionally, there is an implicit piecewise non-linear dependence on N, through the orbitals) and, in finite systems, to a derivative discontinuity even when using semilocal DFAs.

In this work we shall show that eqs. ( 11) and ( 12), which we shall call 'Hx' and 'c' when clarity is required, can be used to uniquely define 'Hx' and 'c' potentials. The total Hxc potential is, of course, unique and thus does not need quotes for clarity. We explore formal properties of the associated potentials and thus reveal some surprising properties.

III. Hxc, 'Hx' and 'c' potential for ensembles

In this section we first review known facts concerning the asymptotic behaviour of the Hxc potential, v Hxc (r) beginning with the properties of the KS potential, v s (r). Then we derive our key results for its components, the Hx and c potentials, v Hx (r) and v c (r).

A. Behaviour of the Hxc potential: a brief review

In the traditional derivation of DFT, the KS potential v s (r) is only defined up to a constant, due to the fact that densi-ties could only vary up to a fixed electron number. In EDFT of open systems, we are able to vary both the spatial distribution of the density, and the number of electrons therein. Consequently, the KS potential v s (r) associated with a density n(r) becomes a unique potential (i.e., the additive constant is determined), given by the derivative of the energy with respect to all variations in the density n (see, e.g., Ref. [24,52])

As usual, we may write

v s (r) =v ext (r) + v Hxc (r) (13)
where

v Hxc (r) = δ E Hxc [n] δ n(r) . ( 14
)
Crucially, however, in computing the exchange-correlation potential we have to take the functional derivative of E Hxc [n] with respect to all possible variations of n, namely to account for variations of the density at constant electron number, but also variation of the electron number itself, i.e. we regard v Hxc (r) as an explicit functional of both N and n(r). Then, we mathematically express the Hxc potential as,

v Hxc (r) = ṽHxc (r) + v 0 Hxc (r) = δ E Hxc [n] δ n(r) N + ∂ E Hxc [n] ∂ N n δ N δ n(r) . (15)
In the above expression we distinguish between: a) ṽHxc (r), which is the functional derivative of the Hxc energy at fixed electron number N, and b) v 0 Hxc (r), which equals the partial derivative of the Hxc energy with respect to N, at fixed density (we note that δ N δ n(r) = 1, since N = n(r)dr). By fixed density, we mean that only N is allowed to explicitly vary in Eq. ( 3) and in any related functionals [e.g., n M and n M+1 are held fixed in Eq. ( 5)].

The potential ṽHxc (r) is by itself defined up to an arbitrary constant shift and thus is synonymous to the "naive" xc-potential, i.e. the potential obtained without considering the effect of changing electron number. The term v 0 Hxc (r) requires some further discussion. First, from Eq. ( 15) we realize that this term is actually spatially uniform. Second, we note that for N approaching an integer, the value of ∂ E Hxc /∂ N is generally different when approaching an integer N from the left (N → M -) or from the right (N → M + ), due to the possible discontinuity in the derivative of the energy as a function of N (for non-integer N, left and right limits are the same). As our objective in this work is not directly related to the computation of fundamental gaps, we shall investigate the long-range behavior of the potentials by approaching any given number of electrons always from the left.

With this in mind, we follow Ref. [24] (Supplemental Material, Sec. III) to derive an expression for v 0 Hxc . We account for the fact that, despite E Hxc being formally a functional of n, in practice E Hxc , and particularly its component E Hx (see Sec. B below) are implicit functionals of n, being explicit functionals of the KS orbitals {φ i (r)}. Therefore,

whereas ∂ E Hxc [n] ∂ N n is difficult to obtain, ∂ E Hxc [n] ∂ N {φ i } is avail- able.
To relate the two aforementioned quantities and derive a relation for v 0 Hxc , we express E Hxc as a functional of N and n(r), which is in turn a functional of N and {φ i (r)}:

E Hxc = E Hxc [N, n[N, {φ i (r)}]. Then, ∂ E Hxc [n] ∂ N {φ i } = ∂ E Hxc [n] ∂ N n + dr δ E Hxc δ n(r) N ∂ n(r) ∂ N {φ i } . ( 16
)
Under the integral of the above expression we recognize two terms: the first one is ṽHxc (r). The second term represents the change in the total density that occurs due to the variation of the electron number, while all orbitals are fixed. Consequently,

v 0 Hxc = ∂ E Hxc [n] ∂ N n = ∂ E Hxc [n] ∂ N {φ i } -dr δ E Hxc δ n(r) N ∂ n(r) ∂ N {φ i } . ( 17
)
Recalling that n(r)

= ∑ i f i |φ i (r)| 2 , we readily understand that ∂ n(r) ∂ N {φ i } = ∑ i ∂ f i ∂ N |φ i (r)| 2 .
Due to the aufbau principle, only the occupation factor(s) f h of the highest occupied molecular orbital(s) (HOMO, h) vary (linearly) with N. Therefore, in the absence of degeneracy, ∂ n(r)

∂ N {φ i } = |φ h (r)| 2 .
Our case is more complex, however, as the HOMO is degenerate. Neverertheless, we can obtain

nh (r) ≡ ∂ n(r) ∂ N {φ i } = 1 N h ∑ h |φ h (r)| 2 , ( 18
)
as the average density of the N h -fold degenerate HOMOs, which follows from equi-occupancy of the degenerate HO-MOs (by construction), giving ∂ f h /∂ N = 1/N h for N h degenerate orbitals (furthermore, N h does not change when taking left derivatives). Note that a similar term has been reported in spin-DFT of ensembles. [40]. Therefore, we finally conclude that

v 0 Hxc = ∂ E Hxc [n] ∂ N {φ i } -dr ṽHxc (r) nh (r). (19)
The term v 0 Hxc is a spatial constant. Moreover, it serves to ensure that v Hxc is independent of any additive constant gauge we choose for ṽHxc and is thus unique. Consider two choices: ṽHxc (r) and ṽHxc (r) + C. Neither choice changes the orbitals, density or energy. In the second case, however, Eq. ( 19) has a term -C dr nh (r) = -C, in addition to the value found using ṽHxc (r). Thus, we get ṽHxc (r) + v 0

Hxc or

[ ṽHxc (r) + C] + [v 0 Hxc -C] = ṽHxc (r) + v 0
Hxc , so that the total Hxc potential is independent of our gauge.

We can therefore choose, without loss of generality for the atomic systems we study here, the gauge in which ṽHxc (r) → 0 for |r| → ∞. Then, it may be useful to express

v s (r) = ṽs (r) + v 0 Hxc . (20)
where

ṽs (r) =v ext (r) + ṽHxc (r) , (21)
which also obeys ṽs (r) → 0 for |r| → ∞. Based on this alone, we cannot say if v 0 Hxc = 0, however. To continue the analysis, we seek to find v 0 Hxc . For noninteger N, the density is given by Eq. ( 5), which follows from the convexity conjecture. Thus, the density eventually follows the weaker decaying density of M + 1 electrons, giving n N (r) ∼ (N -M)n M+1 (r) and φ h (r) ∝ n M+1 (r) as r = |r| → ∞. But n M+1 (r) must itself decay with the ionization potential

I N = E M -E M+1 as n M+1 (r) ∼ e -2 √ 2I N r
, and thus we have

-1 2 ∇ 2 φ h (r) ∼ -I N φ h (r) as r → ∞. Since -1 2 ∇ 2 φ h (r) = {ε h -v s (r)}φ h (r)
and lim r→∞ ṽs (r) = 0, we thus see that (ε hv 0 Hxc ) = -I N . Note, this also holds for integer

N = M.
We can then use Janak's theorem [53] on the frontier orbital to obtain ε h = -I N , which yields

v 0 Hxc = 0 , (22)
and therefore lim r→∞ v s (r) = 0. Note that this is an exact behaviour of the Hxc potential [3,54]. It does not necessarily hold true for approximations, which can certainly have a non-zero constant. [13,24] Furthermore, adding this constant can improve the calculation of ionisation potentials and fundamental gaps [55].

Before proceeding further, it is useful to recall that the KS spatial orbitals in spherical atoms can be written as φ i (r) = R nl (r)Y lm (r), where R nl is a real function and Y lm are the real spherical harmonics. Similarly, the corresponding KS orbital energies ε i = ε nl depend only on the quantum numbers n and l.

Another consequence of the spherical symmetry is that the potential cannot reflect any nodal plane structure of the orbitals, which can cause problems in some cases, [56] including systems with 2p orbitals. The reason these nodal planes are not a problem can be shown by noting that any property of the potential cannot depend on the angle. The nodal planes can only occur along the x, y and z axes, for the 2p orbitals. Furthermore, these axes can be rotated without changing the problem. Therefore, the only way the spherical symmetry can be maintained with the nodal structure is if the nodes do not make any impact on the potential.

The asymptotic behavior of v s (r) is v s (r) ∼ -Z ∞,s /r to leading order. Here Z ∞,s is an asymptotic effective charge determined by the asymptotic form of the density. For the exact KS effective potential of integer M electron atomic systems in the ground-state equi-ensemble, we have Z M ∞,s = Z -M + 1. This follows [57,58] from the screening of the nuclear potential v ext (r) = -Z/r by the Hxc potential. From the same convexity condition reasoning given above, it follows that for a fractional N we have v s

[n N ](r) ∼ v s [n M+1 ](r) as r → ∞ and thus Z ∞,s = Z -(M + 1) + 1. Subtracting the external potential term similarly gives Z ∞,Hxc = Z ∞,s -Z = 1 -(M + 1) = -M. In summary, we thus have v s (r) ∼ r→∞ -Z ∞,s r (23) v Hxc (r) ∼ r→∞ -Z ∞,Hxc r , (24)
where

Z ∞,s =Z + 1 -N , ( 25
) Z ∞,Hxc =1 -N . ( 26
)
We remind the reader that N = M for N integer, or M + 1 otherwise.

To avoid risk of confusion, we recall that the well-known derivative discontinuity of the potential emerges only by considering the order of limits in which r is fixed and, then, the total electron number N approaches an integer from above. [42,59] That is to say that lim N→M lim r→∞ v s [n N ](r) = lim r→∞ lim N→M v s [n N ](r). Here, we are considering the order of limits where N is fixed as r → ∞, at which v s [n N ] decays to zero. Moreover, we approach integer numbers of electrons always from the left.

B. Derivation for the 'Hx' and 'c' ensemble potentials: non-trivial asymptotics

Let us decompose the v Hxc (r) potential into its 'Hx' and 'c' components, v Hxc (r) = v Hx (r) + v c (r). Similarly to Eq. ( 15), we start with

v Hx (r) = ṽHx (r) + v 0 Hx , (27)
where ṽHx (r) = δ E Hx [n]/δ n(r)| N and

v 0 Hx = ∂ E Hx [n] ∂ N {φ i } -dr ṽHx (r) nh (r). (28)
We shall proceed to investigate both these terms, first analytically, and later numerically, to show that both are nontrivial in ensemble calculations.

It is useful to note that Eq. ( 11) can be conveniently broken down into its individual Hartree-like and Fock-like contributions as follows [23]

E Hx = 1 2 ∑ i j F S i j + F D i j (ii| j j) -F S i j (i j| ji) , (29)
where

(i j|kl) ≡ drdr |r -r | φ * i (r)φ j (r)φ * k (r )φ l (r ) . ( 30
)
are the usual two-electron repulsion integrals, here evaluated on KS orbitals from EDFT. Note that all the information about the ensemble is encoded in the coefficients F S i j and F D i j -superscript S denotes same-spin terms (↑↑ + ↓↓) and D denotes different-spin terms (↑↓ + ↓↑). After expanding on the orbital occupancy factors, [23] we find

F S i j = ∑ σ f (2) iσ , jσ , F D i j = ∑ σ f (2) iσ , j σ , (31)
here σ means the opposite spin to σ i.e., σ / σ = ↑/↓ or ↓/↑; and, the expressions involve the average pair-occupation factors

f (2) iσ , jσ = ∑ κ w κ θ κ iσ θ κ jσ . (32)
The quantities defined in Eq. ( 31) are the pair-density equivalent of f i = ∑ κσ w κ θ κ iσ . Thus the energy depends on the statistical probability (across the ensemble) of finding orbitals iσ and jσ occupied simultaneously. Note that for closed-shell systems with non-degenerate ground states (w 0 = 1), we recover the 'Hx' energy expressed in terms of the usual Hartree and exchange energies of regular DFT.

From Eq. ( 29), we get

ṽHx (r) = 1 2 ∑ i j (F S i j + F D i j )v (ii| j j) (r) -F S i j v (i j| ji) (r). (33)
where

v (ik| jl) (r) = δ (ik| jl) δ n(r) N , (34)
where these potentials can be determined by numerically solving the optimised-effective-potential (OEP) equations, [60]

dr v (ik| jl) (r ) δ n(r ) δ v s (r) = ∑ I dr δ (ik| jl) δ φ I (r ) δ φ I (r ) δ v s (r) , (35)
where we have assumed real-valued orbitals.

Unlike its general values, the r → ∞ asymptotic behaviour of ṽHx (r) can be found analytically in finite systems. Following Kreibich et al. [START_REF] Kreibich | Density Functional Theory[END_REF] and Della Sala and Görling [56], we obtain

∑ h f h φ h (r) 2 ṽHx (r) ∼ r→∞ ∑ h φ h (r) δ E Hx δ φ h (r) . (36)
Note that both sides depend only on r = |r| after summation over the degenerate HOMOs h. Here, we have exploited again the fact that we have systems with spherical symmetry. Thus the density must be independent of both angle and choice of HOMO, with the latter leading to the sum over h. For the same reason we do not need to worry about nodal planes.

To obtain the asymptote of ṽHx (r) through Eq. ( 36), we use Eq. ( 29) for E Hx , noting that we can easily take derivatives with respect to orbitals. To begin, it is straightforward to show that

δ (ii| j j) δ φ h (r) =δ ih φ i (r) dr |r -r | φ j (r )φ j (r ) + {i ↔ j} (37)
({i ↔ j} means swap labels i and j in the previous expression) and

δ (i j| ji) δ φ h (r) =δ ih φ j (r) dr |r -r | φ i (r )φ j (r ) + {i ↔ j}. ( 38
)
For large r, we can write 1/|r -r | ≈ 1/r and exploit the orthonormality of the orbitals to get

δ (ii| j j) δ φ h (r) ∼ r→∞ (δ ih + δ jh )φ h (r) r , (39)
δ (i j| ji) δ φ h (r) ∼ r→∞ δ i j (δ ih + δ jh )φ h (r) r . (40)
Then, we can use eqs. ( 39) and ( 40) on the right-hand side of Eq. ( 36) to obtain

∑ h φ h (r) δ E Hx δ φ h (r) = 1 2 ∑ h φ h (r) 2 ∑ i j × (F S i j + F D i j ) (δ ih + δ jh ) r -F S i j δ i j (δ ih + δ jh ) r = 1 r ∑ h φ h (r) 2 ∑ j (F S jh + F D jh ) -F S hh . (41)
Finally, we conclude that

ṽHx (r) ∼ r→∞ -Z ∞,Hx r (42)
with the effective charge

Z ∞,Hx = ∑ h F S hh -∑ jh (F S jh + F D jh ) ∑ h f h ≤ 0 (43)
where we have accounted for symmetries. [START_REF]) where only m h varies with h, i.e. the radial term is the same across the degenerate states. By construction, each orbital is occupied equally, giving ∑ h φ 2 h (r)F(r) ∝ R 2 nl (r)F(r) for an arbitrary radial function F(r). This holds for all potential-like properties by symmetry and thus leads to the left and right hand sides of Eq[END_REF] It is easily shown, as done explicitly later, that for an integer number of electrons N = M, this gives Z ∞,Hx = 1 -M, as expected. Not so in general.

Similarly, the correlation potential can be decomposed as

v c (r) = ṽc (r) + v 0 c , (44)
and the asymptotic behaviour of v c (r) is found from eqs ( 22) and ( 24) by simply using v c (r) = v Hxc (r)v Hx (r).

We readily arrive at

Z ∞,c =1 -N -Z ∞,Hx , v 0 c = -v 0 Hx . ( 45
)
The above analysis does not tell us whether or not v 0 Hx = 0, however. In the next section we will be able to determine numerically that, in general, the exact v 0

Hx can be a nonvanishing constant, especially when N is non-integer.

Let us study the effective charge in more detail. first consider a closed-shell system with an even integer M number of electrons, which has only one state with w κ=0 = 1. For this state, θ κ=0 iσ = 1 for i ≤ h = M/2 and zero otherwise. Therefore, f i = ∑ σ θ iσ = 2 for all occupied states i ≤ h. Similarly, F S i j = F D i j = 2. Then, from (43) we obtain

Z M ∞,Hx (pure) = 2 -∑ M/2 j=1 4 2 = 1 -M, (46)
for the potential. This is the usual result [60], i.e. v ext (r) + v Hx (r) ∼ -(Z + 1 -M)/r up to a constant. This means that the outermost electron feels an attractive potential from the Z nuclear charges, balanced by a repulsive force from the remaining M -1 electrons. In such systems we get the "typical" result Z M ∞,c (pure) = 0. Extending our result to ensembles by using Eq. ( 31) lets us rewrite Eq. ( 43) as

Z ∞,Hx = ∑ κσ w κ ∑ h θ κ hσ θ κ hσ -∑ κσ σ w κ ∑ h j θ κ hσ θ κ jσ ∑ h f h =1 - ∑ κhσ w κ θ κ hσ N κ ∑ h f h , ( 47
)
where we used θ κ hσ θ κ hσ = θ κ hσ , ∑ κσ w κ θ κ hσ = f h , and defined N κ ≡ ∑ jσ θ κ jσ to be the number of electrons in state κ. Eq. ( 47) then lets us obtain results for the various cases considered in this work.

Let us continue our analysis by considering the case when all ensemble members have the same number of electrons; i.e. when the electron number is integer, but the ground state is degenerate. In the integer case N κ = M in all members, and we get the simple expression

Z M ∞,Hx (ensemble) =1 - ∑ κhσ w κ θ κ hσ M ∑ h f h = 1 -M ≡ 1 -N , (48)
and Z M ∞,c (ensemble) = 0. Next, consider a fractional Li cation with N = 2 + c electrons. Here we have the states κ ∈ {1s 2 , 1s 2 2s ↑ , 1s 2 2s ↓ } with the weights w κ ∈ {1c, c/2, c/2}. After some work we can show that

Z ∞,Hx (2 < N ≤ 3) = -2 = 1 -(M + 1) ≡ 1 -N , (49)
which is the same as for Z ∞,s , making Z ∞,c (2

< N ≤ 3) = 0.
It may be tempting to think that the result Z N ∞,Hx = 1 -N holds in general. However, let us now consider a system with N = 3 + c electrons. This has states κ ∈ {1s 2 2s ↑ , 1s 2 2s ↓ , 1s 2 2s 2 } with weights w κ ∈ {(1c)/2, (1c)/2, c} which leads, after some work, to

Z ∞,Hx (3 ≤ N ≤ 4) = -2(2N -5) N -2 . (50)
Here, Z ∞,Hx = 1 -N only for N = 3 and 4, i.e. the integers which we previously showed had the expected behaviour. It otherwise has a non-trivial and non-linear dependence on N. Similar arguments for the open p-shell systems considered here give

Z ∞,Hx (5 ≤ N ≤ 6) = -2(3N -13) N -4 , (51)
Z ∞,Hx (6 ≤ N ≤ 7) = -2(4N -19) N -4 , (52)
for fractional ions of C, and

Z ∞,Hx (8 ≤ N ≤ 9) = -4(3N -17) N -4 , (53)
Z ∞,Hx (9 ≤ N ≤ 10) = -4(7N -43) N -4 , (54)
for fractional ions of F. They thus show that this non-trivial screening is common, rather than an exceptional case for Li anions. Note that in these cases the denominator is N -4. This reflects the fact that only the outermost 2p shell plays a role in the asymptotics, with the four core electrons excluded. A similar result can be seen above for Li -c , where the denominator is N -2 reflecting the exclusion of two core electrons. Finally, we observe that eqs. ( 49)- (54) all have the same form, namely Z ∞,Hx = AN-B N-N core , where A and B are constants. This result is not a coincidence. Rather, it follows directly from the fact that all terms in the numerator and denominator of Eq. ( 47) depend linearly on the ensemble weights w κ , and thus linearly on N. The denominator comes from ∑ h f h = N -N core . As discussed above, N core is the number of core electrons (e.g., two for 1s electrons in Li, or four for 1s and 2s electrons in C or F). Thus, given M < N ≤ M + 1 for M integer and M ≥ N core electrons in a degenerate equiensemble ground-state, one obtains,

Z ∞,Hx (N) = (N -1)N core + (M + 1 -2N)M N -N core , (55)
by ensuring that limiting cases are satisfied, i.e.,

Z ∞,Hx (M) = 1 -M and Z ∞,Hx (M + 1) = -M for M > N core , or Z ∞,Hx (M +1) = -M and dZ ∞,Hx (M +1)/dN = 0 for M = N core .
It is easily seen by substitution that Eq. ( 55) covers all previous expressions.

The values for Z ∞,Hx and Z ∞,Hxc are shown in Figure 1 for the systems considered here. Note that Z ∞,Hx is continuous at most (but not all) integer electron numbers. It always takes the "typical" value for N integer. By contrast, ∂ Z ∞,Hx /∂ N is discontinuous at integer electron numbers. As the combined contributions of the Hx and c potentials give Z ∞,Hxc = 1 -N , the non piecewise constant behavior of Z ∞,Hx points, in turn, to a strong non-local functional dependence on the density in the remainder correlation potential. This leads to Z ∞,c = 0 for some non-integer N.

These asymptotic constants are closely related to the idea of freedom from self-interaction in DFT. Thus, prior to concluding this section, let us briefly consider this relation. The result that Z ∞,s = Z + 1 -N can be interpreted as the outermost orbital feeling a nuclear potential that is screened by electrostatic repulsion from the remaining N -1 electrons, i.e., all electrons not in the outermost orbital. This is known as freedom from self-interaction, and is an important quality of the KS potential [START_REF] Barth | [END_REF], albeit one that is rarely met by approximations. The effect of their absence is known as a self-interaction error. In contrast, the correct asymptotic decay of a potential is not always equivalent to freedom from self-interaction, as discussed in detail by Schmidt et al. [64] Typically, we expect the Hx potential to be selfinteraction free, meaning it is responsible for the 1 -N contribution to the screening. This is indeed the case for integer electron number and for Li cations, as shown above. However, this freedom from self interaction is manifestly absent from Hx in the other fractional cases mentioned. It must therefore be compensated for by the c potential.

We conclude that, in general, Z ∞,Hx = Z ∞,Hxc and the ensemble Hx potential has a different long-range behavior compared to the one of the total Hxc potential. Attempts to ensure a correct asymptotic behavior in exchange potential alone based on what is known about the pure-state Hx potential requires a careful revisiting in EDFT.

IV. Numerical results

In this section, we complete our analysis with numerical results obtained for the exact v Hxc (r), v Hx (r), and v c (r). The exact KS potentials v s (r) (and thus their components) were calculated using the densities and methods described by Gould and Toulouse. [32] All our calculations were carried out on logarithmic radial grids with N r = 256 abscissae arranged to accurately capture both core (r → 0) and asymptotic (r → ∞) behaviours. The same grid is used for radial quadrature, derivatives, orbitals, and Green functions (which are used directly and in χ 0 ).

Orbitals {φ k } are calculated by direct diagonalisation of the KS Hamiltonian expressed as an N r × N r matrix. Greens functions are evaluated by summing over all available solutions of the KS equations, i.e. N r orbitals in total from N r radial points. Spatial symmetries allow us to further break the ground state and response problems up via spherical harmonic expansions, in the fashion described in Refs. [23,65].

A. Outline of the procedure

In order to calculate the potential v Hx (r) [and thus, by difference, v c (r)], we need to solve Eq. (35). This equation involves the KS linear-response function to allow direct comparison with the c potential. These plots illustrate, most obviously, the effect of the asymptotic constant on the different potentials. Less obviously they reveal the effect of differing effective charges, since they are constructed to approach zero (or a constant) faster than 1/r for integer electron number cases, but not necessarily for non-integer cases.

χ 0 (r, r ) = δ n(r) δ v s (r ) (56)
and the associated Greens functions

G i (r, r ) = ∑ j,ε j =ε i ρ j (r, r ) (ε i -ε j ) (57)
where ρ i (r, r ) = φ i (r)φ i (r ), such that χ 0 (r, r ) = 2 ∑ i f i G i (r , r)ρ i (r, r ). Therefore, we can rewrite Eq. ( 35), as follows

dr χ 0 (r, r ) ṽHx (r ) = d Hx (r) (58)
where

d Hx (r) = ∑ k dr G k (r, r )φ k (r) δ E Hx δ φ k (r) . (59)
Normally, when one seeks an OEP for exchange, one has access to various minimization procedures. Then, one can use variational techniques or virial theorems to efficiently compute v Hx . [66] Our case, however, is complicated by two features. Firstly, we have a non-integer number of electrons which may or may not cause problems for exact theorems obtained assuming integers. Secondly, rather than obtaining an OEP self-consistently, we instead need to find the functional derivative after "reading" in the potential v s (r) that correspond to some given KS orbitals and orbital energies -those which are obtained from the KS inversion on exact QMC densities. [32] Instead, we approach the problem by basis set expansion, using a good starting guess and additional functions for remaining terms. As our starting guess, we use a KLI approximation, which is known to be correct in the asymptotic limit, and generally good elsewhere. We the follow Yang and Wu [67] and introduce a basis set to represent the difference between the real OEP, and the KLI one. That is, we seek to find values for {c b }, in

ṽHx (r) = ṽHx,KLI (r) + ∑ b c b ξ b (r), (60)
where ξ b (r) are a set of basis functions that depend only on r = |r| due to the spherical symmetry. To determine the coefficients {c b }, we use Eq. ( 60) in Eqs ( 58) and ( 59), then integrate over {ξ b } to give the matrix equation:

c b = ∑ b [A -1 ] bb X b , (61)
where,

A bb = 4πr 2 dr 4πr 2 dr ξ b (r)χ 0 (r, r )ξ b (r ) , (62)
X b = 4πr 2 dr ξ b (r)∆d Hx (r) , (63)
and ∆d Hx (r) = d Hx (r) -4πr 2 dr χ 0 (r, r ) ṽHx,KLI (r ). Again, we take advanatge of spherical symmetry. It remains to describe the basis functions, and how to avoid a spurious numerical constant in the determined potential that appears in some cases. Details are in the appendices.

B. Results

We report results for the exact KS potential and its components for Li, C, and F with fractional numbers of electrons. The neutral atoms are important examples of an electron donor (Li), an electron acceptor (F), and an atom with flexible bonding (C). Moreover, they clearly demonstrate self-interaction errors when treated with semi-local DFAs. [68] In Figure 2 we present components of the potentials for atoms, anions, and cations, at N ∈ {Z -1, Z -0.6, Z, Z + 0.4, Z + 1}, to illustrate the asymptotics of each term. Values for v 0 Hx , and related quantities, are reported in Table I. The numerical results also confirm that, for non-integer electron numbers (c = 0), the potential from the ensemble Hx energy functional defined here does not follow the asymptotic behavior of the Hxc potential given in Eq. ( 26), except in limited cases (e.g., Li (1-c)+ ). Thus, at the Hx level, the outermost orbital spuriously "feels" its own field since Z ∞,Hx = 1 -N , e.g. Z ∞,Hx = -2(2N -5)/(N -2) = 3 for Li c-with 0 < c < 1 and N = 3+c. This means that the Hx potential cannot be regarded as self-interaction free. Its effect is manifested in the asymptotic behaviour of v c , which decays more rapidly than 1/r when v Hx is self-interaction free, and as -Z ∞,c /r otherwise.

We see that the asymptotic constant v 0 Hx is zero for both Li and C at integer electron numbers. But for F -surprisingly -it is zero only in the closed-shell case N = 10 (F -). v 0 Hx is also quite significant in most partial ions, but this is less surprising given the asymptotic self-interaction contribution in those systems. The non-zero values in F likely reflect the fact that the upper "channel" of a shell (i.e., the second electron in an s-shell or 4th-6th electron in a pshell) is more sensitive to ensemble effects than the lower channel, owing to the need to treat electrons of opposite spin within the same ensemble members [meaning F D hh = 0 in the lower channel but F D hh = 0 in the upper channel, e.g. in Eq. (29), where h and h indicate electrons in the frontier orbital]. For this reason, partial anions of Li have a nonzero value and unusual asymptotic potentials but partial cations do not. Consequently, e.g., we expect netural and fully anionic O (not studied here because high-quality reference densities [32,33] are not available) to have a nonzero constant, whereas fully cationic O likely will not.

It is also notable how unvarying the correlation potentials are for cations, even in cases where v c (r) has a long-range Coulombic asymptotic behaviour. This is not entirely surprising, given the dominance of exchange in cations. But it nonetheless bodes well for the ability to approximate v c (r) in such systems. Anions paint a different story. There, correlation is known to be more important, due to the weak binding of the outermost electrons. This is reflected in the greater role played by v c (r) in governing the behaviour of the KS orbitals.

Finally, we note that the findings of this manuscript are also likely to be valid for molecules with degenerate ground-states. A detailed analysis of molecular cases is rather challenging numerically, however, due to the difficulty of obtaining sufficiently accurate reference electron densities [33]. It is therefore outside the scope of this work.

V. Conclusions

In this work, we analytically and numerically investigated exact DFT properties of the 'Hx' and 'c' potentials, for Li, C, and F atoms and fractional ions. Of primary theoretical interest, we revealed two important facts: 1) the exact Hx and c potentials can have equal and opposite asymptotic constants in some cases [Eq. (22) and Eq. ( 28)]; and 2) the effective charge "felt" by the outermost electron can imply a self interaction for the frontier orbitals [Eq. ( 43)], at a fractional number of electrons. This conclusion is contrary to the common wisdom that exact exchange is always one-electron self-interaction free.

The Hx and c potentials of neutral atoms and cations are reasonably well-behaved, with the notable exception (in our tests) of F, which attains asymptotic constants in the Hx and c potentials in its atomic and cationic form, but not in its anion form. The role of the correlation potential in anions is, as might be expected, more complex both for integer and non-integer electron number cases.

Our findings also call for new considerations of how self interaction in the ensemble KS framework may be interpreted. For example, as mentioned above the division into Hx and c we employ here is not the only one. Understanding how similar effects manifest in other definitions may assist further functional development efforts. Similar effects to those found here may also be expected to appear in excited-state EDFT, and this should be investigated.

Figure 1 Z

 1 Figure 1 Z ∞,Hx and Z ∞,Hxc versus N for all the systems (Li, C, and F ions) considered in this work.

Figure 2

 2 Figure 2 Hxc (orange solid line), Hx (navy dashed line), and c (magenta dash-dot line) potentials (with the electron number derivative contribution) for selected ions of Li (left), C (centre), and F (right). The Hx and Hxc potentials are shown asv + Z /r ≡ v + (1 -N )/rto allow direct comparison with the c potential. These plots illustrate, most obviously, the effect of the asymptotic constant on the different potentials. Less obviously they reveal the effect of differing effective charges, since they are constructed to approach zero (or a constant) faster than 1/r for integer electron number cases, but not necessarily for non-integer cases.

Acknowledgements

L.K. is the incumbent of the Aryeh and Mintzi Katzman Professorial Chair and acknowledges support by the Israel Science Foundation.

Table I Asymptotic potential constant v 0

Hx in Ha, and its contributions ∂ E Hx /∂ N and ṽHx,KLI h = dr ṽHx,KLI (r) nh (r) using Eq. (28). We use ṽHx,KLI as a proxy for ṽHx to avoid spurious constants (see Appendix B). Errors from the KLI approximation are expected to be just a few mHa. For the potential basis functions {ξ b (r)} we begin with the orbital-derived functions

System

that comprise terms that enter into the KLI potential.

Here n k (r) = ∑ m |φ n k l k m (r)| 2 and ρ kk (r) = δ l k l k ∑ m φ n k l k m (r)φ n k l k m (r) are shell averaged densities and mixed densities, respectively. We expect terms of these forms to contribute significantly to the Thus, e.g., we get ξ 1 (r) = n 1s (r)/n(r), ξ 2 (r) = dr ρ 1s1s (r )/|r -r |, ξ 3 (r) = dr ρ 1s2s (r )/|r-r | and ξ 4 (r) = dr ρ 2s2s (r )/|r-r | for Li and its ions.

The orbital-derived basis in Eq. (A1) is then augmented by a set of Gaussian functions e -ζ k r 2 for k ∈ {1, . . . , N Aug } to capture details missed by the orbital-derived functions. In our calculations we use 16 augmenting functions. Thus, e.g., ξ 5 (r) = e -ζ 1 r 2 , . . . , ξ 20 (r) = e -ζ 16 r 2 for Li and its ions. During testing, we varied these functions to investigate the stability of various integrated quantities to changes in the basis -these tests revealed the issue discussed in the next section.

All together, these basis functions let us capture the important properties of the potential, including manifestations of shell structure. Results near the nucleus (r Z -1/3 ) are imperfect, however. Fortunately they are also unimportant given the dominance of the nuclear potential -Z/r over v Hxc (r) for small r. We thus exclude them from most of our plots in Sec. IV

B. Avoiding spurious numerical constant terms

In addition to the general numerical considerations given above, we highlight an important detail in our analysis of the potentials ṽHx (r), namely how to avoid a spurious numerical constant.

To begin with, let us consider the mathematical details of the problem. It is clear from the form of the KS Hamiltonian that the electron density of any quantum system is unchanged by a constant shift in its effective potential. Thus the density linear-response function has an eigenfunction with eigenvalue 0. Its inverse is thus ill-defined in general, and constant terms must thus be excluded from χ -1 0 for meaningful numerical results.

In the case of an exact inversion, this is not a problem. However, in our calculations we expand χ -1 0 on a finite basis. In some cases this basis truncation leads to an effective pseudo-constant shift in the potential, when compared to the asymptotically decaying and otherwise similar KLI potential v Hx,KLI (r). That is, we find v Hx (r) =v Hx,KLI (r) + ∆v Hx (r) +Ch(r) (B1)

where ∆v Hx (r) is the true difference between the potentials, that decays to zero faster than 1/r. The additional term involves a constant C and pseudostep function h(r) obeying h(r < r c ) ≈ 1 and h(r > r c ) ≈ 0. When this error is present, the density at the cutoff radius r c is typically n(r c ) ≈ 10 -12 , meaning that drn(r)h(r) ≈ N and drn i (r)h(r) ≈ 1 for all i. This contribution Ch(r) to the potential is almost certainly an artifact of numerics rather than a real effect. This is made especially apparent by varying the basis set, which leads to large changes to ṽHx h in the anomalous cases, and virtually no changes to ṽHx h in cases when it is close in value to ṽHx,KLI h . Unfortunately, the presence of Ch(r) can make analysis difficult in cases where we requires ṽHx (r), since it introduces an uncontrolled (and physically meaningless) "gauge". To ensure reasonable results for all systems we thus add a correction

to our potentials to avoid the spurious pseudo-constant, i.e. C determined above cancels Ch(r) in the region where it causes problems. In cases where we need ṽHx h = dr nh (r) ṽHx (r) this is equivalent to replacing ṽHx (r) by ṽHx,KLI (r) inside the average.

The effect of the correction C can be tested on "good" cases where no shift is apparent. These case suggest that ṽHx h differs from ṽHx,KLI h by at most a few mHa, more than an order of magnitude less than correlation energies. We further note that in calculations where we use v Hx (r) = ṽHx (r) + v 0 Hx , the pseudo-constant does not make any difference at all to final results as C it is cancelled out through the termdrn h (r) ṽHx (r) in v 0 Hx (at least in the region r < r c of interest).

We note that it might be possible to obtain v 0 Hx more directly, by using relationships between Kohn-Sham and Hartree-Fock quantities. [58,69] However, this requires further scrutiny in the case of using the exact (as done here), rather than OEP (as done previously), orbitals in v Hx ; and a proper ensemble generalization of Hartree-Fock quantities in several cases. Due to these outstanding theoretical issues, this strategy is not pursued in this work.