Amal Aït Brahim

Rabah Tighilt Ferhat

Gilles Zurfluh

Incremental extraction of a NoSQL database model using an MDA-based process

published or not. The documents may come

-In recent years, the need to use NoSQL systems to store and exploit big data has been steadily increasing. Most of these systems are characterized by the property "schema less" which means absence of the data model when creating a database. This property brings an undeniable flexibility by allowing the evolution of the model during the exploitation of the base. However, the expression of queries requires a precise knowledge of this model. In this paper, we propose an incremental process to extract the model while operating the document-oriented NoSQL database. To do this, we use the Model Driven Architecture (MDA) that provides a formal framework for automatic model transformation. From the insert, delete and update queries executed on the database, we propose formal transformation rules with QVT to generate the physical model of the NoSQL database. An experimentation of the extraction process was performed on a medical application.


Alexander T Tesfaye

Frederic Dumur

Didier Gigmes

Sébastien Maria

Laure Monconduit

Thierry Djenizian
email: thierry.djenizian@emse.fr

Frédéric Dumur

Superior Electrochemical Performance of Thin-Film Thermoplastic Elastomer-Coated SnSb as an Anode for Li-ion Batteries

ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

There is great interest in developing rechargeable Li-ion batteries (LIBs) with high energy density and long cycle life for a wide variety of applications such as portable electronic devices, electric and hybrid vehicles, and grid-scale energy storage systems [START_REF] Armand | Building better batteries[END_REF][START_REF] Etacheri | Challenges in the development of advanced Li-ion batteries: a review[END_REF][START_REF] Nitta | Li-ion battery materials: present and future[END_REF][START_REF] Ellis | Three-dimensional self-supported metal oxides for advanced energy storage[END_REF] . Currently, commercialized LIBs use carbonaceous materials as the negative electrode, however, the technology reaches a limit because of the low specific capacity (372 mAh g -1 ) corresponding with the formation of LiC6 [START_REF] Kamali | Review on carbon and silicon based materials as anode materials for lithium ion batteries[END_REF] . Antimony and tin have long been considered as an attractive replacement for graphite electrode in LIB systems due to their superior lithium storage capacities of 660 and 994 mAh g -1 , respectively [START_REF] Kamali | Review on carbon and silicon based materials as anode materials for lithium ion batteries[END_REF][START_REF] Zhang | A review of the electrochemical performance of alloy anodes for lithium-ion batteries[END_REF][START_REF] Wang | Nanospheres of a new intermetallic FeSn5 phase: synthesis, magnetic properties and anode performance in Li-ion batteries[END_REF][START_REF] Antitomaso | Ultra-fast dry microwave preparation of SnSb used as negative electrode material for Li-ion batteries[END_REF] .Unfortunately, the electrochemical performance of these materials fades during cycling because of the electrical contact loss induced by large volume variations combined with the continuous growth of a solid electrolyte interphase (SEI) layer at the surface hindering the electron transfer [START_REF] Winter | Electrochemical lithiation of tin and tin-based intermetallics and composites[END_REF] . The use of multi-phase active materials, such as SnSb, instead of pure metals (Sn or Sb) has been used to limit the large volume change by reacting with Li + at different stages during the charge and discharge process [START_REF] Li | Nano-alloy anode for lithium ion batteries[END_REF][START_REF] Wachtler | Tin and Tin-based intermetallics as new anode materials for Lithium-ion cells[END_REF] . SnSb reacts reversibly with Li + by an alloying/dealloying mechanism according to equation ( 1) and (2) [START_REF] Winter | Electrochemical lithiation of tin and tin-based intermetallics and composites[END_REF][START_REF] Wachtler | Tin and Tin-based intermetallics as new anode materials for Lithium-ion cells[END_REF][START_REF] Wilhelm | Significant electrochemical performance improvement of TiSnSb as anode material for Li-ion batteries with composite electrode formulation and the use of VC and FEC electrolyte additives[END_REF][START_REF] Simonin | SnSb micron-sized particles for Li-ion batteries[END_REF] .

(1) (2) Unfortunately, the alloying/dealloying reaction is still accompanied by a volume change which results in the loss of electrical contact with the active material, Sn agglomeration, and the formation of SEI film on the electrode surface leading to poor electrochemical performance [START_REF] Chen | Recent progress in advanced materials for lithium ion batteries[END_REF][15][START_REF] Tesfaye | The electrochemical behavior of SnSb as an anode for Li-ion batteries studied by electrochemical impedance spectroscopy and electron microscopy[END_REF] . Much effort has been devoted to improving the cycling performance, which includes chemical modification of the electrolyte by adding various additives like vinylene carbonate (VC) and fluoroethylene carbonate (FEC) to create a thin stable protective [START_REF] Wilhelm | Significant electrochemical performance improvement of TiSnSb as anode material for Li-ion batteries with composite electrode formulation and the use of VC and FEC electrolyte additives[END_REF][START_REF] Zhang | A review on electrolyte additives for lithium-ion batteries[END_REF] . It has also been reported that the electrochemical properties of Si and TiO2 can be improved by coating the electrode surface with inorganic or organic compounds, such as Ni, Al2O3, self-healing polymer, and polyaniline [START_REF] Wang | Self-healing chemistry enables the stable operation of silicon microparticle anodes for high-energy lithium-ion batteries[END_REF][START_REF] Sopha | ALD Al2O3-Coated TiO2 Nanotube layers as anodes for Lithium-ion batteries[END_REF][START_REF] Wu | Stable Li-ion battery anodes by in-situ polymerization of conducting hydrogel to conformally coat silicon nanoparticles[END_REF] . In the present work, the latter approach is investigated by coating micron-sized SnSb with a new thermoplastic elastomer (TPE) thin layer. Typically, TPEs are nanostructured materials based on block (or graft) copolymers with short hard blocks and long soft blocks. The glassy nanodomains dispersed in the rubbery matrix ensure rubber elasticity thanks to reversible physical crosslinking and give mechanical resistance to the TPEs. In order to prevent mechanical damage, an elastomer showing high strain property and good adhesion to the surface of the active material has been designed. The synthesis of a block copolymer poly(styrene-b-2-hydroxyethyl acrylate) so-called PS-b-PHEA is studied using polystyrene (PS) as the hard block and PHEA as the soft block, which is rubbery at room temperature and in addition known for its biocompatibility and lack of toxicity. Moreover, the block copolymer is expected to have a good affinity for the electrode thanks to hydrogen bonds formed between the hydroxyl groups present as pending functions on PHEA molecules and terminated groups at the surface of SnSb particles. In this work, the synthesis of the targeted block copolymer PS-b-PHEA is investigated by nitroxide mediated polymerization (NMP), which is a simple method of controlled radical polymerization (CRP) allowing the tuning of the mechanical properties of the copolymer through precise control of the molar mass and dispersity of both the hard and soft blocks [START_REF] Nicolas | Nitroxide-mediated polymerization[END_REF] . Compared to other synthesis methods, NMP is an unimolecular technique involving only one alkoxyamine initiator with no additional controller and no purification step [START_REF] Wang | Controlled/" living" radical polymerization atom transfer radical polymerization in the presence of transition-metal complexes[END_REF][START_REF] Chiefari | Living free-radical polymerization by reversible additionfragmentation chain transfer: the RAFT process[END_REF] . Recently, PS-b-PHEA could be synthetized by different CRP methods including NMP with TEMPO-and TIPNObased nitroxide [START_REF] Greesh | Preparation of poly (styrene-b-2hydroxyethyl acrylate) block copolymer using reverse iodine transfer polymerization[END_REF][25][START_REF] Choi | Composite polymer electrolyte membranes comprising triblock copolymer and heteropolyacid for fuel cell applications[END_REF][START_REF] Cao | Synthesis and characterization of well-defined block copolymers containing pendant, self-complementary quadruple hydrogen bonding sites[END_REF][START_REF] Yin | Nitroxidemediated homo-and block copolymerization of styrene and multifunctional acryland methacryl derivatives[END_REF] . But to the best of our knowledge, only short PHEA blocks not exceeding 50 wt. % of the copolymer have been reported limiting the deformation property.

Sn Sb Li e Li SnSb +  + + - + 3 3 3 4 . 4 0    + + - + x Sn Li xe xLi Sn x layer
For the first time, we report that PS-b-PHEA carrying long PHEA blocks can be obtained by NMP controlled with SG1 nitroxide. We also show that a PS-b-PHEA thin layer deposited onto micron-sized SnSb can stabilize the electrode/electrolyte interface, leading to significant electrochemical performance.

Results and Discussion

Material Synthesis

Figure 1a describes the synthesis of PS-b-PHEA. In the first step, Figure 1a (i and ii), polystyrene PS-SG1 with low molecular weight and dispersity (Mn = 6500 g mol -1 and Đ = 1.13) was synthetized in bulk thanks to the presence of MAMA-SG1 as an initiator/ controlling agent at 120 °C. Then, a reversible termination reaction is induced where propagating macro-radicals react reversibly with stable nitroxides SG1 to give macroalkoxyamine dormant species, Figure 1a (ii). This equilibrium is fast, allowing the controlled addition of monomers, and shifted toward the dormant species, diminishing the radical concentration and thus the probability of a bimolecular termination reaction. Figure 1a (iii) shows the use of PS-SG1 as alkoxyamine macroinitiator to perform the controlled polymerization of HEA in DMF at 120 °C. A small amount of free nitroxide SG1 was also initially added, since it is known to better control the polymerization of acrylates [START_REF] Benoit | Kinetics and mechanism of controlled free-radical polymerization of styrene and n-butyl acrylate in the presence of an acyclic β-phosphonylated nitroxide[END_REF] . Indeed, acrylates have a high propagation rate constant and the complete monomers conversion can occur before enough nitroxides are generated to establish the control equilibrium.

The study of kinetics was carried out to show that the NMP polymerization of HEA from PS-SG1 occurred in a well-controlled manner. As shown in Figure 2a,ln([M]o/[M]) vs. time is linear suggesting that the concentration of propagating radicals remains constant in the course of the polymerization, indicating that a sufficient amount of free nitroxide was initially added, and therefore the undesired bimolecular termination reactions should be negligible. Indeed, as shown in Figure 2b, the average molar mass of the copolymer increased linearly with the conversion and the molar mass distributions stay low with dispersity values below 1.5 until 65% conversion. This result is good considering that the polymerization occurs from a PS macroinitiator and that a high molar mass was targeted, i.e. 70,000 g mol -1 at 100% conversion. The desired PS-b-PHEA copolymer revealed a PS weight fraction of 10% (determined by 1 H NMR), which represents the suitable proportion for producing organized PS nanospheres in the PHEA matrix.

The average number molar mass Mn obtained by size exclusion chromatography (SEC) was 65,200 g mol -1 , which is higher than the value of 55,000 g mol -1 calculated from the conversion of 82% measured by 1 H NMR, but SEC molar masses are measured relative to PMMA standards. The copolymer exhibits a dispersity Đ of 1.91, which is quite large but the conversion and the PHEA molar mass were high.

The chemical structure and composition of the block copolymer was investigated by [START_REF] Armand | Building better batteries[END_REF] H NMR spectroscopy and FTIR spectroscopy. Figure 3a shows the 1 H NMR spectrum of the PS-SG1 macroinitiator. Signals a and (b, c) at 7.00 and 6.6 ppm, respectively, are attributed to the phenyl rings of PS. Signals d and e at 1.8 and 1.4 ppm are ascribed to the aliphatic (-CH) and (-CH2) protons of the PS chain, respectively. The 1 H NMR spectrum obtained for PS-b-PHEA is shown in Figure 3b. The signals at 6.50 -7.30 ppm are related to the protons of the aromatic styrene groups (a, b, c). The strong signals at 3.55 and 4.05 ppm are attributed to the methylene (-CH2) protons of the hydroxyethyl group (d and e) and the signal at 4.8 ppm corresponds to the -OH of hydroxyethyl group. The presence of peaks belonging to styrene and HEA are consistent with the literature data reported for the copolymerization of PS-b-PHEA [START_REF] Greesh | Preparation of poly (styrene-b-2hydroxyethyl acrylate) block copolymer using reverse iodine transfer polymerization[END_REF][START_REF] Choi | Composite polymer electrolyte membranes comprising triblock copolymer and heteropolyacid for fuel cell applications[END_REF] .

FTIR spectroscopy analysis of PS and PS-b-PHEA are shown in Figure 4. In the FTIR spectrum of PS the aromatic (C=C stretching) band produced three peaks at 1602, 1492, and 1451 cm -1 , the aromatic (C-H) bands was observed at 740 cm -1 , and aliphatic band (C-H stretching) appeared at 3024 cm -1 . The FTIR spectrum of PS-b-PHEA exhibits new stretching bands at 3432, 1714, and 1160 cm -1 , which are assigned to (-OH), -C=O, and C-O of PHEA, respectively [START_REF] Greesh | Preparation of poly (styrene-b-2hydroxyethyl acrylate) block copolymer using reverse iodine transfer polymerization[END_REF][START_REF] Vargün | Polymerization of 2-hydroxyethyl acrylate in bulk and solution by chemical initiator and by ATRP method[END_REF] . These FTIR spectroscopy results are clear evidence of the copolymerization of styrene and HEA by NMP. cycle for SnSb electrodes before and after PS-b-PHEA coating, respectively. In agreement with previous studies, the electrochemical reactivity for both electrodes is confirmed by the presence of five peaks during the first cycle. The broad peak around 0.8 V vs. Li/Li + corresponds to the alloying of Li + with SnSb to form Li3Sb and Sn and the formation of the SEI film at the electrode surface [START_REF] Yang | Sub-Microcrystalline Sn and Sn-SnSb powders as lithium storage materials for Lithium-ion batteries[END_REF][START_REF] Fernández-Madrigal | X-ray diffraction, 7Li MAS NMR spectroscopy, and 119Sn Mössbauer spectroscopy study of SnSb-based electrode materials[END_REF][33] . The two peaks at 0.5 V and 0.6 V vs. Li/Li + are attributed to the lithiation of Sn to form LixSn. Further alloying of Sn occurs between 0.3 -0.1 V vs. Li/Li + 34-36 . Five distinct peaks were obtained during the first anodic scan. The peaks at 0.45 V, 0.7 V, 0.75 V, and 0.8 V vs. Li/Li + are assigned to the delithiation of LixSn alloy to form Sn. The peak at 1 V vs. Li/Li + is ascribed to the dealloying of Sb to form again SnSb [START_REF] Antitomaso | SnSb electrodes for Li-ion batteries: the electrochemical mechanism and capacity fading origins elucidated by using operando techniques[END_REF] .

Electrochemical Characterization

For both electrodes, the anodic/cathodic peaks are reversibly obtained for the first five cycles.

After the 5 th cycle, the peak intensity corresponding to Li + reaction with Sb is diminishing for the as-prepared SnSb electrode (Figure 5a). This is attributed to the pulverization of the active material and the loss of electrical contact between active particles, conductive additives and the current collector as the result of a large volume change during the alloying/dealloying mechanism in addition to the formation and growth of the SEI film. In contrast, reversible alloying/dealloying peaks for PS-b-PHEA-coated SnSb were observed up to 40 cycles (Figure 5b). Moreover, the characteristic of the initial CV for PS-b-PHEA-coated SnSb is different from the subsequent CVs as oxidation/reduction peaks tend to enlarge. This effect can be explained by the creation of new active surfaces during the successive alloying/dealloying reactions. The stability of the redox peaks is attributed to the elastomer polymer film at the surface of SnSb. Indeed, highly stretchable PS-b-PHEA is able to bear the mechanical stress caused by the large volume variations and thus maintains good quality at the electrode/electrolyte interface and the inter-grains electronic percolation pathways during successive charging and discharging of the battery as depicted in Figure 1b. The elastomeric property of PS-b-PHEA was confirmed by a tensile stress test (see Supplementary Fig. S2 online). The polymer can be easily stretched up to 200% and reveals a very low value Young's modulus of 0.01 MPa, which is typical for TPEs with a tensile strength of 0.8 MPa [START_REF] Puskas | The effect of hard and soft segment composition and molecular architecture on the morphology and mechanical properties of polystyrene-polyisobutylene thermoplastic elastomeric block copolymers[END_REF][START_REF] Dufour | Polar three-arm star block copolymer thermoplastic elastomers based on polyacrylonitrile[END_REF][START_REF] Nasiri | Enhanced mechanical and adhesion properties in sustainable triblock copolymers via non-covalent interactions[END_REF] . Additionally, the mechanical test also shows that PS-b-PHEA can be elongated up to 400% without rupture.

The electrochemical performance of PS-b-PHEA-coated SnSb was evaluated through the examination of the charge/discharge profiles obtained by galvanostatic cycling tests. Figure 6a shows the discharge capacity vs. cycle number for the as-prepared SnSb and the polymercoated SnSb cycled at C/10 (i.e. 0.0827 A g -1 ). It is clearly apparent that the PS-b-PHEAcoated SnSb has superior cyclability than the non-coated SnSb with a reversible capacity of 720 mAh g -1 obtained, whereas only 356 mAh g -1 was retained after 50 cycles for the asprepared SnSb electrode. The remarkable stable capacity is attributed to the presence of PS-b-PHEA film conferring to SnSb enhanced mechanical and chemical properties. Actually, the copolymer is able to limit the formation of fracture as well as the continuous growth of the SEI. The beneficial effect of the polymer layer is also evidenced at faster kinetics (1C) over 100 cycles (Figure 6b). Indeed, the polymer-protected electrode is able to maintain a capacity of 308 mAh g -1 , whereas the as-prepared SnSb retains only 93 mAh g -1 .

To study the operational stability of PS-b-PHEA-coated SnSb, the cycle life performance has been studied at multiple C-rates (Figure 6c). A stable gravimetric capacity of 800 mAh g -1 , 368 mAh g -1 , and 170 mAh g -1 were obtained at C/10 (0.0827 Ah g -1 ), C/2 (0.4135 Ah g -1 ), and 2C (1.654 Ah g -1 ) rates, respectively. When the rates was subsequently restored to C/2 and C/10, capacity values of 344 mAh g -1 and 770 mAh g -1 were obtained respectively. The excellent rate capability is again attributed to the PS-b-PHEA film, which enables the Li + transfer while ensuring the mechanical stability of the electrode.

Coulombic efficiency (CE) and capacity retention are important criteria for evaluating the electrochemical performance of electrodes for LIBs [START_REF] Tesfaye | The electrochemical behavior of SnSb as an anode for Li-ion batteries studied by electrochemical impedance spectroscopy and electron microscopy[END_REF][START_REF] Wang | Self-healing chemistry enables the stable operation of silicon microparticle anodes for high-energy lithium-ion batteries[END_REF][START_REF] Tarascon | Issues and challenges facing rechargeable lithium batteries[END_REF][START_REF] Tesfaye | Porous silicon nanotube arrays as anode material for Li-ion batteries[END_REF] . The CE obtained for PS-b-PHEA-coated SnSb at the first cycle was 87% and reached more than 98% while discharge capacity retention of more than 95% was obtained after 50 cycles, for the as-prepared SnSb a CE of 92% and 90% obtained after the first and 50 cycles (Figure 6d). These values likely indicate relatively more stable SEI formation on the surface of the polymer-covered electrode even after long term cycling.

To provide further evidence for the positive contribution of the elastomer film on the electrochemical performances, SEM analysis was carried out before and after cycling tests. 7c). This is due to the repeated volume change, which results in the pulverization of the SnSb particles, and thus poor cycling stability (Figure 6a). For comparison, the surface of PS-b-PHEA-coated SnSb electrode is almost unchanged (Figure 7d). It can be simply noted the apparition of a few small cracks and lumps. Thus, no significant modification of the surface nor delamination were observed thanks to the highly rubbery nature of the polymer coating, which mitigates the cyclic volume variations.

Conclusion

In this work, enhanced electrochemical performance of SnSb can be achieved by coating the surface with a new TPE. Highly rubbery PS-b-PHEA was synthesized by the NMP method using a specific polystyrene macroinitiator allowing the presence of long hydroxyethyl acrylate blocks. The excellent cycle life performance and rate capability of PS-b-PHEAcoated SnSb obtained is attributed to the high failure strain characteristics of the elastomer film. The copolymer can strongly mitigate the volume expansion/contraction during Li + alloying/dealloying reactions with SnSb and the SEI formation. As a result, the SnSb electrode is chemically and mechanically stable beyond 100 cycles even at fast kinetics.

Moreover, the use of the PS-b-PHEA coating can extend to other high capacity materials like silicon and conversion electrodes that suffer from large volume changes during cycling.

Methods

Material Synthesis and Instrumentation

Tetrahydrofurane (THF), N,N-Dimethylformamide (DMF), Methanol, 2-hydroxyethyl acrylate (HEA) and styrene (S) were purchased from Sigma-Aldrich and were used as

received. (N-tert-butyl-N-(1′-diethylphosphono-2,2′-dimethylpropyl)-O-(2-carboxyl-prop-2- yl) (MAMA-SG1
) alkoxyamine initiator (registered as trademark BlocBuilder by Arkema) and N-tert-butyl-N-(1'-diethylphosphono-2,2'-dimethylpropyl) (SG1, 85%) free nitroxide were provided by Arkema and were used as received.

The chemical structure and composition of the block copolymer was investigated by proton nuclear magnetic resonance spectroscopy ( 1 H NMR) and Fourier Transform Infrared (FTIR) spectroscopy. NMR experiments were performed with a Bruker Avance III 400 MHz Nanobay spectrometer. [START_REF] Armand | Building better batteries[END_REF] H NMR spectra were recorded at a frequency of 400 MHz with a 11.3 μs 30° pulse, a repetition time of 4 s and 16 scans. NMR chemical shifts were reported in standard format as values in ppm relative to deuterated solvents. Infrared analysis was carried out using a Perkin Elmer Spectrum 2 equipped with a single reflection diamond module (ATR). IR spectrum was recorded in the 400 -4000 cm -1 range, at 4 cm -1 resolution.

Synthesis of Macroinitiator, PS-SG1

MAMA-SG1 (0.768 g, 25.6 mmol) and styrene (20 g, 192 mmol, 95 eq.) were placed in a three-neck round-bottom flask equipped with a reflux condenser and a magnetic stir bar. The mixture was purged for 30 min with argon to remove oxygen. The mixture was immersed in an oil bath and heated to 120 °C (in the solution) with a 30 min ramp temperature. After 2 h, cooling down the mixture in an ice bath stopped the polymerization. The crude product was diluted in THF (20 mL) and then poured into a large excess of cold ethanol. The solid was collected by filtration, re-dissolved in THF, and re-precipitated in cold methanol. Then, the precipitate was dried under vacuum (5 × 10 -2 mbar) after filtration. A Mn of 6500 g mol -1 with a Đ of 1.13 were determined by size exclusion chromatography (SEC) carried out in THF at 70 °C relative to PS standards (see Supplementary Fig. S1 online).

Synthesis of Poly(styrene-b-2-hydroxyethyl acrylate) Block Copolymer, PS-b-PHEA

PS-SG1 macroinitiator (0.4418 g, 0.068 mmol) and SG1 free nitroxide (1.98 mg, 0.00674 mmol, 0.1 eq.), dissolved in DMF (4 g), were placed in a three-neck round-bottom flask equipped with a reflux condenser and a magnetic stir bar. Then HEA (4.002 g, 34.5 mmol, 507 eq.) were added and the solution was purged for 30 min with argon to remove oxygen.

The mixture was immersed in an oil bath and heated to 120 °C (in the solution) with a 20 min temperature ramp. Aliquots were withdrawn periodically for monitoring conversions by [START_REF] Armand | Building better batteries[END_REF] H NMR and molar masses by SEC. After 90 min, cooling down the mixture in an ice bath stopped the polymerization. The final conversion was 82%. The crude product was diluted in THF (15 mL) and then poured into a large excess of cold diethyl ether. The precipitated viscous polymer stuck on the bottom of the beaker and was collected by decanting the solvent in the fridge, followed by washing 3 times with diethyl ether and drying under reduced pressure. The solid was re-dissolved in DMF and then re-precipitated in cold diethyl ether.

Finally, the solvent was decanted in the fridge, followed by washing 3 times with diethyl ether, and drying under reduced pressure, yielding the desired copolymer with Mn = 65,200 g mol -1 and Đ = 1.91 (determined by SEC in DMF at 70 °C relative to PMMA standards, see Supplementary Fig. S1 online).

Electrode Fabrication

PS-b-PHEA was dissolved in DMF (1.5 mL) and then mixed with Carbon black SP (Alfa Aesar) and Li-salt bis(trifluoromethane)sulfonimide-LiTFSi (Sigma Aldrich) with a mass ratio of 70:15:15 using magnetic stirrer. The homogenous suspension was then drop-casted onto Teflon and dried overnight at 80 0 C to form a conductive elastomer thin film.

Crystallized SnSb powder up to 1 g was obtained using a very simple dry microwave route under one minute and a half [START_REF] Antitomaso | Ultra-fast dry microwave preparation of SnSb used as negative electrode material for Li-ion batteries[END_REF] . The SnSb electrode was fabricated using a slurry cast method.

Figure 5 a

 5 Figure 5 a and b shows curves obtained by cyclic voltammetry (CV) recorded up to the 40 th

Figure 7 a

 7 Figure 7 a and b show the SEM top-view images of the as-prepared SnSb and PS-b-PHEA-

First,

  SnSb powder was mixed with polyvinylidene fluoride (PVDF) as a binding agent, and Carbon black SP as the conducting agent with a mass ratio of 70:15:15 in vibrational ball milling for 25 min at 20 Hz. Then, the slurry was tape casted on a Cu current collector at room temperature for 24 h and then further dried at 110 0 C for 12 h under vacuum. Then, the SnSb electrode was heated to 90 0 C on a hot plate. The conductive composite copolymer film was then melted at 100 0 C and coated on the SnSb electrode with a sharp blade. The electrodes were degassed in vacuum at room temperature overnight and transferred to an argon glove box for the battery assembly. Standard two-electrode Swagelok cells were assembled in a glove box filled with high purity argon. The half-cells consisted of PS-b-PHEA-coated SnSb as the working electrode and a battery grade Li foil as the counter electrode. The two electrodes were separated by a Whatman glass microfiber soaked in the liquid electrolyte: 1M LiPF6 in ethylene carbonate, propylene carbonate, and dimethyl carbonate (1:1:3 volume ratio) with 1 vol.% VC and 5vol.% FEC. The electrochemical characterization was performed by cyclic voltammetry (CV) and chronopotentiometry using a VMP3 potentiostat (Bio Logic, France). The CV was performed in the potential window of 0.01 -1.75 V vs. Li/Li + at a scan rate of 0.1 mV s -1 .The galvanostatic tests were done at various C-rates, where C/n means the battery is fully charged or discharged up to its total storage capacity in n hours (for this work 1C = 0.827 A g -1 ).The morphology of PS-b-PHEA-coated SnSb before and after cycling experiments was examined by a field-emission scanning electron microscope (FE-SEM Ultra-55 Carl Zeiss).

Figure 1 .

 1 Figure 1. (a) The synthesis mechanism of PS-b-PHEA using NMP: (i) Initiation of radicals,

Figure 2 .

 2 Figure 2. (a) Variation of ln[Mo/M] vs. time and (b) average number molar mass Mn and

Figure 3 . 1 H

 31 Figure 3. 1 H NMR spectrum of (a) PS-SG1 macroinitiator in CDCl3 and (b) PS-b-PHEA in

Figure 4 .

 4 Figure 4. FTIR spectra of PS and PS-b-PHEA.

Figure 5 .

 5 Figure 5. Cyclic voltammograms of (a) as-prepared SnSb and (b) PS-b-PHEA-coated SnSb recorded at a scan rate of 0.1 mV s -1 in the potential window of 0.01 V -1.75 V vs. Li/Li + .

Figure 6 .

 6 Figure 6. Cycling stability study showing the variation of the discharge capacity versus the

Figure 7 .

 7 Figure 7. SEM images of (a) as-prepared SnSb and (b) PS-b-PHEA-coated SnSb before

Acknowledgements

This work is financially supported by ALISTORE-ERI. We thank Philippe Antitomaso for providing the SnSb powders.

Author Contributions

Additional Information Supplementary Information

Competing Interests: The authors declare no competing interests as well as no financial competing interest.


B G Giraud

S Karataglidis

Algebraic density functionals

Keywords: numbers: 21.60.De, 31.15.A-, 71.15.-m

A systematic strategy for the calculation of density functionals (DFs) consists in coding informations about the density and the energy into polynomials of the degrees of freedom of wave functions. DFs and Kohn-Sham potentials (KSPs) are then obtained by standard elimination procedures of such degrees of freedom between the polynomials. Numerical examples illustrate the formalism.

Existence theorems [1] for DFs do not provide directly constructive algorithms. Fortunately, the Kohn-Sham (KS) method [2] spares the construction of a "kinetic functional" and reduces energy and density calculations to the tuning of a local potential, v KS (r). Hence, a considerable amount of work has been dedicated to detailed estimates of electronic correlation energies and the corresponding KSPs, see for instance [3][4][5]. Many authors were also concerned with representability and stability questions, see for instance [6] and, for calculations in subspaces, see [7] and [8]. For cases where the mapping between potential and density shows singularities, see [9]. For reviews of the rich multiplicity of derivations of DFs and KS solutions and their properties, we refer to [START_REF] Dreizler | Density Functional Theory[END_REF] and [START_REF] Parr | [END_REF], and, for nuclear physics, to [12].

Local or quasi local approximations use the continuous infinity of values ρ(r), ∀r, as the parameters of the problem. However, whether for atoms, molecules or nuclei, a finite number of parameters is enough to describe physical situations. For instance, Woods-Saxon nuclear profiles notoriously make good approximations, depending only on a handful of parameters, and it is easy to add a few parameters describing, for example, long tails and/or moderate oscillations of the density. (High frequency oscillations are unlikely, for they might cost large excitation energies.) We can stress here, in particular, the one-dimensional nature of the radial density functional (RDF) theory [13], valid for nuclei and/or atoms, isolated, described by rotationally-invariant Hamiltonians; the constrained density minimization of energy [START_REF] Levy | Proc. Natl. Acad. Sci[END_REF] returns isotropic densities, with radial profiles, ρ(r), 0 r < ∞. The number of parameters to describe a nuclear density, therefore, can be restricted to maybe ∼ 10 at most; situations with ∼ 20 parameters are a luxury. For molecules, shapes are much more numerous, but a finite, while large number of parameters, truncating a list of multipoles for instance, still makes a reasonable frame. Practical DFs, therefore, can boil down to functions of a finite number of parameters. Functional variations can then be replaced by simple derivatives.

This Letter shows how information about both the density and the energy can be recast into polynomials. This allows elimination of part of the parameters. Further polynomial manipulations locate energy extrema. Only density parameters are left. The same method gives KSPs. Finally we offer a discussion and conclusion.

Consider a basis of n orthonormalized, single-particle states, ϕ α (rστ ), where spin and isospin labels στ will be understood. The orthonormalized Slater determinants φ i made out of the ϕ α 's for N fermions make a finite subspace, of some dimension N , in which eigenstates of the physical Hamiltonian H can be approximated by configuration mixings, Ψ =

N i=1 (C i + iC ′ i )φ i .
Here C i and C ′ i are the real and imaginary parts, respectively, of the mixing coefficients, but, in practice, with real matrix elements, H ij = φ i |H|φ j , of the Hamiltonian H, the imaginary parts C ′ i vanish. Both the energy η and the normalization are quadratic functions of such coefficients,

η = N i,j=1 C i H ij C j , N i=1 C 2 i = 1. ( 1
)
Let a † r and a r be the usual creation and annihilation operators at position r. Tabulate the matrix elements φ j a † r a r φ j . The density corresponding to Ψ is, again, quadratic with respect to the C i 's,

ρ(r) = ij C i φ j a † r a r φ j C j , (2)
and any parameter that is linear with respect to moments of the density is also a quadratic function of the C i 's.

Let {S ν (r)}, ν = 1, . . . , ∞, be a complete orthonormal set of "vanishing average" functions. Namely, the two sets of conditions, dr S ν (r) = 0, ∀ν, and, drS µ (r)S ν (r) = δ µν , ∀µν, are satisfied. Such sets are easy to find; in the case of one-dimensional problems, including radial ones, they can be implemented by means of orthogonal polynomials [15,16] and a generalization to more dimensions is easy. Then subtract from ρ some reference density, ρ 0 , obtained by some approximation relevant for the N fermions. The difference, ∆ρ = ρρ 0 , is of a vanishing average, since, by definition, both ρ and ρ 0 integrate out to N . Then the Fourier coefficients,

∆ ν = dr S ν (r)∆ρ(r), (3)
define ρ, as ρ = ρ 0 + ∞ n=1 ∆ ν S ν . As already stated, this expansion of ρ can be truncated. at some realistic order N ′ , lower than the number of independent parameters

C i . The ∆ ν 's are quadratic in the C i 's, ∆ ν = ij C i dr S ν (r) φ j a † r a r φ j C j -ρ 0ν . ( 4
)
Note the auxiliary numbers, ρ 0ν = dr S ν (r) ρ 0 (r).

It is then trivial to use the N ′ density constraints, Eqs. ( 4), and the normalization in Eqs. (1), to eliminate, for instance, the last (N ′ + 1) coefficients C i .

This leaves a polynomial relation, R(η, ∆ 1 , . . . , ∆ N ′ , C 1 , . . . , C N -N ′ -1 ) = 0, between the energy, the density parameters, and the remaining coefficients C i . Finally, the energy must be minimized with respect to such remaining coefficients, via still polynomial conditions, ∂R/∂C i = 0, i = 1, . . . , N -N ′ -1. This gives a polynomial relation, E(η, ∆ 1 , . . . , ∆ N ′ ) = 0, between the energy and the density parameters. This polynomial E is our "algebraic" DF. It accounts for all contributions to the energy, both without and with correlations, for only matrix elements of the full H are used.

The procedure can be further simplified in the following way. Let H be the matrix representing the Hamiltonian on an orthonormal basis for a suitable subspace of wave functions, and, similarly, let, for instance, D 1 , D 2 be the matrices representing two constraints selected to parametrize the density, such as, for instance, two among the parameters (∆ ν + ρ 0ν ). Set the equation, polynomial in all three variables ε, λ 1 , λ 2 ,

P (ε, λ 1 , λ 2 ) ≡ det (H -λ 1 D 1 -λ 2 D 2 -ε) = 0. (5)
Here ε is the free energy, lowest eigenvalue of (Hλ 1 D 1λ 2 D 2 ), and the λ's are Lagrange multipliers. It is well known that ∂ε/∂λ i = -D i , i = 1, 2, where D i ≡ D i is the expectation value of the corresponding constraint. From Eq. ( 5) such partial derivatives read, ∂ε/∂λ i = -(∂P/∂λ i )/(∂P/∂ε), i = 1, 2, hence two more polynomial relations are obtained,

Q i (D i , ε, λ 1 , λ 2 ) ≡ (∂P/∂ε) D i -(∂P/∂λ i ) = 0. (6)
Replace in Eqs. (5,6) the free energy by its value, ε = ηλ 1 D 1λ 2 D 2 , in terms of the energy, η ≡ H and the constraints, D 1 , D 2 . This creates three polynomials in terms of η, D 1 , D 2 , λ 1 , λ 2 , out of which λ 1 , λ 2 can be eliminated, for a final polynomial equation, E(η, D 1 , D 2 ) = 0. This easy Legendre transform generates our "algebraic DF". A generalization to any number of quadratic constraints is trivial. Such algebraic DFs are not open formulae of the form, η = F (D 1 , . . . , D N ′ ), but they provide roots for η at any realistic degree of numerical accuracy. Incidentally, they may also give excited energies and/or spurious ones, a well known property [17] of DFs.

For an illustrative toy model, we consider two fermions only and set the one-body part of H as,

K = -d 2 /(2dr 2 1 ) -d 2 /(2dr 2 2 ) + (r 2 1 + r 2
2 )/2, the sum of two harmonic oscillators, and its two-body part as a translation invariant, separable potential, defined in coordinate representation by,

r 1 r 2 |V | r ′ 1 r ′ 2 = -V 0 δ [(r 1 + r 2 -r ′ 1 -r ′ 2 )/2] × e -[(r2-r1) 2 +(r ′ 2 -r ′ 1 ) 2 ]/4 (r 2 -r 1 )(r ′ 2 -r ′ 1 )/ √ 2π. (7)
Then, given the first 4 wave functions, ϕ 0 , . . . , ϕ 3 , of the one-dimensional harmonic oscillator, we create, to prepare a configuration mixing, a basis of 4 negative parity Slater determinants. These read, in a transparent notation, {ϕ 0 , ϕ 1 }, {ϕ 0 , ϕ 3 }, {ϕ 2 , ϕ 1 }, {ϕ 2 , ϕ 3 }. We set V 0 = 3 for a numerical test. To constrain H, we choose the second moment operator, r 2 1 + r 2 2 . The matrices representing H and the constraint in the toy subspace read,

H =     -1 0 0 0 0 7/4 3 √ 3/4 0 0 3 √ 3/4 13/4 0 0 0 0 45/8     , (8)
and

D =     2 3/2 1/2 0 3/2 4 0 1/2 1/2 0 4 3/2 0 1/2 3/2 6     . (9)
The equations which correspond to Eqs. ( 5),( 6) read,

P toy (ε, λ) = -360+154ε+344ε 2 -154ε 3 +16ε 4 +1464λ + 1692ελ -1636ε 2 λ + 256ε 3 λ + 725λ 2 -5140ελ 2 + 1408ε 2 λ 2 -4192λ 3 + 3072ελ 3 + 2064λ 4 = 0, Q toy (D, ε, λ) = -1464-1692ε+1636ε 2 -256ε 3 -1450λ+ 10280ελ -2816ε 2 λ + 12576λ 2 -9216ελ 2 -8256λ 3 + (154 + 688ε -462ε 2 + 64ε 3 + 1692λ -3272ελ+ 768ε 2 λ -5140λ 2 + 2816ελ 2 + 3072λ 3 )D = 0. ( 10
)
Finally, the substitution, ε = η -λD, followed by the elimination of λ, generates the desired polynomial equation, E toy (η, D) = 0. (This polynomial E toy is of order 12 in both η and D and is a little cumbersome for a publication here. It is available to the interested reader.)

We show in Fig. 1 the contour line, E toy (η, D) = 0. The ground state is found at the lowest point of the oval envelope, with coordinates, D = 2, η = -1. The highest and lowest eigenvalues of H are, 45/8 and -1, and those of D are, 4± 4 + √ 15, namely ∼ 6.81 and ∼ 1.19. This is confirmed by the extremal points, up, down, right and left, of the oval. The inside pattern refers to excited states. The concavity of the lowest part of the envelope and convexity of its highest part are transparent properties of the theory. They generalize for any dimension of the subspace and any number of constraints; we tested this generalization with further toy models. Moreover, when, via embedded subspaces, the dimension N of the matrices, H, D i , grows while H and the constraints are kept the same, a growth of the envelope is found and the bottom of the envelope converges towards a limit, as expected. This gives numerical estimates for an extrapolation of this concave part towards its limit for N → ∞.

Such concavities should also occur in DF theories with a continuous infinity of constraints. But they are often difficult to verify, and are, therefore, overlooked, although they are an important test of soundness.

A byproduct of the procedure consists of a polynomial relating the potential energy to the constraints. Set the Hamiltonian as, H = h + V , with V = -V 0 V, where V 0 is an interaction strength and V gives all details of interaction shapes. Nothing prevents one from considering V 0 as a Lagrange multiplier and obtain, via the polynomial method pushed one step further, a polynomial, F ( h , V , D 1 , . . . , D N ′ ), linking h to the expectation values of V and the constraints. A standard result of this

Legendre transform is, ∂ h /∂ V = V 0 , i.e., G(V 0 , h , V , D 1 , . . . , D N ′ ) ≡ (∂F /∂ h ) V 0 -∂F /∂ V = 0. (11)
Replace, in F and G, the quantity h by η + V V 0 . Then eliminate η and V 0 between E and such modified F and G. This links V , hence V , to the D i . It must be stressed here that now V should not be minimized with respect to the D i ; rather, those D i values to be used are those that minimize the total energy η.

A similar argument provides the kinetic energy, or any other part of η, in the same context of total energy constrained minimization. Such results are of interest for a detailed analysis of corrections induced by correlations.

The direct approach resulting from Eqs. ( 5) and ( 6) bypasses the KS approach. For the sake of completeness, however, we now show how this theory can handle determinants and also calculate a KSP. Consider a basis of n single-particle states, ϕ α , α = 1, . . . , n, a Slater determinant Φ made of N orthonormal orbitals, ψ γ = n α=1 c γα ϕ α , and a Hamiltonian with its one-body and two-body parts, H = K +V , assuming real matrix elements,

K αβ = ϕ α |K|ϕ β and V αβγδ = ϕ α ϕ β |V |ϕ γ ϕ δ .
The energy of Φ becomes quartic in the orbital coefficients, c γα , because of V , and even needs order 6 if threebody forces are introduced, but the N (N +1)/2 orthonormalization constraints and the density remain quadratic. Obviously, a few parameters constraining the density of Φ, or its difference from some ρ 0 , can again be chosen as quadratic in the coefficients c γα . Elementary eliminations then yield a polynomial relation between Slater energy and density parameters.

The following toy model, in which the number of free parameters reduces to N = 2 and we choose that of density constraints as N ′ = 1, illustrates the strategy. From the first 4 wave functions, ϕ 0 , . . . , ϕ 3 , of the onedimensional harmonic oscillator, set a Slater determinant Φ made of one positive and one negative parity orbitals,

ψ + = tϕ 0 + uϕ 2 , ψ -= vϕ 1 + wϕ 3 . ( 12
)
One constraint is spared by such orbital parities, which ensure orthogonality. Normalization constraints can also be spared if they are ensured by a "trigonometric" form of the components,

t = (1-a 2 )/(1+a 2 ), u = 2a/(1+a 2 ), v = (1 -b 2 )/(1 + b 2 ), w = 2b/(1 + b 2 )
, with both parameters, a, b, real numbers. The density is a Gaussian modulated by a polynomial,

ρ(r) = π -1 2 e -r 2 a 6 r 6 + a 4 r 4 + a 2 r 2 + a 0 , (13)
with two independent coefficients only, because of the two parameters only, a, b, for Φ. One of the relations between a 6 , . . . , a 0 is linear, since the integral,

∞ -∞ dr ρ(r) = 15 8 a 6 + 3 4 a 4 + 1 2 a 2 + a 0 = 2, ( 14
)
must equate to the particle number. The other comes from the condition that gives the density of Φ,

|ψ + | 2 +|ψ -| 2 = π -1 2 e -r 2 (a 6 r 6 +a 4 r 4 +a 2 r 2 +a 0 ). ( 15
)
Insert Eqs. (12) into Eq. ( 15) and take advantage of the harmonic oscillator basis states. The density constraint, Φ ⇒ ρ, then means 4 conditions in terms of t, u, v, w,

4 w 2 /3 = a 6 , t 2 - √ 2 t u + u 2 /2 = a 0 , 2 u 2 + 4 2/3 v w -4 w 2 = a 4 , 2 √ 2 t u -2 u 2 + 2 v 2 -2 √ 6 v w + 3 w 2 = a 2 . ( 16
)
We can use these, Eqs. ( 16), rather then Eqs. ( 4), for our argument. In terms of a, b, these Eqs. ( 16) read,

a 6 = 16 b 2 3 (1 + b 2 ) 2 , a 0 = 1 -2 √ 2a + 2 √ 2a 3 + a 4 (1 + a 2 ) 2 , 3 a 4 (1 + a 2 ) 2 (1 + b 2 ) 2 = 8 (3 a 2 + √ 6b + 2 √ 6 a 2 b + √ 6 a 4 b -6 b 2 -6 a 2 b 2 -6 a 4 b 2 - √ 6 b 3 -2 √ 6 a 2 b 3 - √ 6 a 4 b 3 + 3 a 2 b 4 ), a 2 (1 + a 2 ) 2 (1 + b 2 ) 2 = 2 (1 + 2 √ 2 a -2 a 2 -2 √ 2 a 3 + a 4 -2 √ 6b -4 √ 6 a 2 b -2 √ 6 a 4 b + 4 b 2 + 4 √ 2 ab 2 - 4 √ 2 a 3 b 2 + 4 a 4 b 2 + 2 √ 6 b 3 + 4 √ 6 a 2 b 3 + 2 √ 6 a 4 b 3 + b 4 + 2 √ 2 ab 4 -2 a 2 b 4 -2 √ 2 a 3 b 4 + a 4 b 4 ). ( 17
)
For the sake of simplicity, we select a 6 and a 0 as primary, independent parameters of ρ and eliminate a, b between those of Eqs. ( 17) that give a 6 , a 0 , a 4 . The result, 256 -1024a 0 + 1536a 2 0 -1024a 3 0 + 256a 4 0 -768a 4 + 1792a 0 a 4 -1280a 2 0 a 4 + 256a 14) to link a 4 , a 2 to a 6 , a 0 . Incidentally, Eqs. (17) show that 0 ≤ 3a 6 ≤ 4 and 0 ≤ 2a 0 ≤ 3.

The same toy Hamiltonian H as was used to generate Fig. 1 induces the Slater energy,

η ≡ Φ|H|Φ = (t 2 + 5u 2 + 3v 2 + 7w 2 )/2 -V 0 /8 × 2 (4 t 2 + u 2 ) v 2 -4 √ 3 t u v w + (6 t 2 + u 2 ) w 2 = [ 2 + 12a 2 + 2a 4 + 12b 2 + 40a 2 b 2 + 12a 4 b 2 + 2b 4 + 12a 2 b 4 + 2a 4 b 4 -V 0 (1 -a 2 + a 4 -2 √ 3ab + 2 √ 3a 3 b + b 2 - 2a 2 b 2 + a 4 b 2 + 2 √ 3ab 3 -2 √ 3a 3 b 3 + b 4 -a 2 b 4 + a 4 b 4 ) ] / [(1 + a 2 ) (1 + b 2 )] 2 . ( 19
)
Given H, the DF is defined by the constrained minimization [START_REF] Levy | Proc. Natl. Acad. Sci[END_REF], F [ρ] = Min Φ⇒ρ Φ |H| Φ , where the constraint, Φ ⇒ ρ, will now be interpreted as just a constraint Φ ⇒ a 6 . We motivate this choice of the maximum degree coefficient by at least two reasons, namely, i) it is an interesting degree of freedom, since it can be interpreted as a "halo driving" parameter, ii) it will actually turn out that the ground state corresponds to a 6 = 0, (hence, no halo!), this value 0 interestingly sitting on an edge of the convex domain of densities; variational calculus at edges of domains are notoriously challenging. We can, therefore, eliminate b between Eq. ( 19) and the first among Eqs. (17). This implements the constraint, Φ ⇒ a 6 , in a precursor situation before energy minimization with respect this constrained Φ, whose last free parameter is a. This "precursor" energy is given by, P(η, a 6 , a) = 1024 + 12288a 2 + 38912a 4 + 12288a 6 + 1024a 8 + 1536a 6 + 12288a 2 a 6 + 21504a 4 a 6 + 12288a 6 a 6 + 1536a 8 a 6 + 576a 2 6 + 2304a 2 a 2 6 + 3456a 4 a 2 6 + 2304a 6 a 2 6 + 576a 8 a 2 6 -1024η -8192a 2 η -14336a 4 η -8192a 6 η-

1024a 8 η -768a 6 η -3072a 2 a 6 η -4608a 4 a 6 η- 3072a 6 a 6 η -768a 8 a 6 η + 256η 2 + 1024a 2 η 2 + 1536a 4 η 2 + 1024a 6 η 2 + 256a 8 η 2 -1024V 0 - 5120a 2 V 0 + 4096a 4 V 0 -5120a 6 V 0 -1024a 8 V 0 - 576a 6 V 0 + 384a 2 a 6 V 0 + 384a 4 a 6 V 0 + 384a 6 a 6 V 0 - 576a 8 a 6 V 0 + 144a 2 6 V 0 + 288a 2 a 2 6 V 0 + 288a 4 a 2 6 V 0 + 288a 6 a 2 6 V 0 + 144a 8 a 2 6 V 0 + 512ηV 0 + 512a 2 ηV 0 + 512a 6 ηV 0 + 512a 8 ηV 0 -96a 6 ηV 0 -192a 2 a 6 ηV 0 - 192a 4 a 6 ηV 0 -192a 6 a 6 ηV 0 -96a 8 a 6 ηV 0 + 256V 2 0 - 512a 2 V 2 0 + 768a 4 V 2 0 -512a 6 V 2 0 + 256a 8 V 2 0 - 96a 6 V 2 0 -480a 2 a 6 V 2 0 + 960a 4 a 6 V 2 0 -480a 6 a 6 V 2 0 - 96a 8 a 6 V 2 0 + 9a 2 6 V 2 0 + 432a 2 a 2 6 V 2 0 -846a 4 a 2 6 V 2 0 + 432a 6 a 2 6 V 2 0 + 9a 8 a 2 6 V 2 0 = 0. ( 20
)
This is now combined with the energy minimization, ∂P/∂a = 0, with respect to a, thus eliminating a,

Q(η, a 6 ) = (32 + 24a 6 -16η -16V 0 + 3a 6 V 0 ) × (128 + 48a 6 -32η -8V 0 + 3a 6 V 0 ) (4096+ 4608a 6 + 1152a 2 6 -3072η -1536a 6 η + 512η 2 - 2304V 0 -480a 6 V 0 + 216a 2 6 V 0 + 640ηV 0 - 144a 6 ηV 0 + 128V 2 0 -144a 6 V 2 0 + 63a 2 6 V 2 0 ) = 0. ( 21
)
This polynomial Q, Eq. ( 21), is an algebraic DF for the Slater Φ. In turn, with a final minimization, ∂Q/∂a 6 = 0, "with respect to the density", actually here w.r.t. the a 6

parameter, the polynomial equation for η reads,

S(η) = (η + V 0 -2) (4η + V 0 -16) (8η + V 0 - 48) (4η + 3V 0 -16) (64 + 36V 0 -2ηV 0 + V 2 0 )× (-1024 -1152V 0 + 64ηV 0 + 316V 2 0 -348ηV 2 0 + 47η 2 V 2 0 -264V 3 0 + 52ηV 3 0 + 5V 4 0 ) = 0. (22)
An elimination of η between the same conditions, Q = 0 and ∂Q/∂a 6 = 0, yields the condition for a 6 , a 6 (3a 6 -4) (3a

6 V 0 -24V 0 -64) (524288+ 491520V 0 -49152a 6 V 0 + 151552V 2 0 + 46080a 6 V 2 0 - 54144a 2 6 V 2 0 + 18432V 3 0 + 7680a 6 V 3 0 -10152a 2 6 V 3 0 + 3024a 6 V 4 0 -2961a 2 6 V 4 0 ) = 0. ( 23
)
For the numerical illustrations that follow, set V 0 = 3. Then Eq. ( 21) becomes, (16 -

33a 6 + 16η) (104 + 57a 6 -32η) (-1664 + 1872a 6 + 2367a 2 6 -1152η -1968a 6 η + 512η 2 ) = 0. ( 24
)
For V 0 = 3 the lowest root of Eq. ( 22) is, η = -2.98623.

But it is soon recognized as spurious, because, inserted into Eq. ( 21), it returns absurd, negative only values of a 6 . This is confirmed by a detailed consideration, in the only allowed domain, 0 ≤ a 6 ≤ 4/3, of the solution branches yielded by Eq. ( 24), namely η = -1 + 33a 6 /16, η = 13/4 + 57a 6 /12, η = (72 + 123a 6 ± 18496 + 2736a 6 -3807a 2 6 )/64. These are shown in Fig. 2, and clearly validate the second lowest root, η = -1, of Eq. ( 22), together with that root, a 6 = 0, of Eq. ( 23), hence b = 0.

It is then trivial to take advantage of Eq. ( 19) by inserting the results, η = -1, b = 0, and obtain, a = 0, hence a 0 = 1, then a 4 = 0 and a 2 = 2. The optimal density is, therefore, ρ = (1 + 2r 2 ) e -r 2 / √ π. Notice, incidentally, that we have five equations at our disposal, namely Eq. ( 19) and Eqs. (17) to directly relate η and the a i 's via an elimination of a, b, via polynomial conditions of the form T (η, a 6 , a 4 ) = 0, and U(η, a 6 , a 2 ) = 0, for instance. We verified that the same set, {a 0 = 1, a 2 = 2, a 4 = 0, a 6 = 0}, results from such a direct use of the values, V 0 = 3, η = -1.

While usually many wave functions can give the same density, this toy model allows the wave function to be identified. It is that Slater determinant Φ gs made of ϕ 0 and ϕ 1 . This Φ gs is, obviously, the first two-fermion eigenstate of our K, a sum of two harmonic oscillators, with eigenvalue, 2 = 1/2 + 3/2. The same Φ gs is also an eigenstate of V, Eq. ( 7), since in the following Jacobi coordinates, R = (r 1 + r 2 )/2, r 21 = r 2r 1 , the wave function of Φ gs reads, ∝ exp(-R 2 ) r 21 exp(-r 2 21 /4), while the representation of V is,

Rr 21 |V |R ′ r ′ 21 ∝ δ(R -R ′ ) r 21 r ′ 21 exp[-(r 2 21 + r ′2 21 )/4],
showing an obvious projector on that relative motion expressed by Φ gs . The corresponding eigenvalue is, -V 0 , hence our result, η = -1, when V 0 = 3. We took great care to verify that the same results are obtained if, instead of a 6 , we use other choices for "density constraint", such as the parameter a 0 , or a moment such as the second one, a 0 /2 + 3a 2 /4 + 15a 4 /8 + 105a 6 /16, or the local value ρ(θ) at some testing point r = θ. Such rearrangements of information with respect to the wave function parameters may be of some interest for questions of physics or numerical convenience, but do not change the nature of the algebra nor the the final results. It can be noted here that what is important for the method is that the energy and the constraints be polynomials of the parameters. The fact that, in the toy model, the density is described by a polynomial of r is not essential. It only makes the algebra slightly simpler. With wave functions more complicated than harmonic oscillator ones, any choice of moments, or local values of ρ, still makes eligible constraints.

An issue which will arise in all future models using this polynomial method is that the final minimization of η must be performed within a convex domain of densities: what conditions must the coefficients a i , or those other selected parameters (moments, local values, etc), satisfy to maintain ρ positive? This question was recently [18] solved by means of the Sturm criterion, for a general class of positive functions having positive Fourier transforms. The criterion gives the number of real roots of a polynomial, and can be used to ensure that a polynomial has no real roots. As seen in the toy model, the detailed structure of the calculation can be a guide to define the physically acceptable domain of parameters, see the bounds found for a 6 and a 0 . For more subtle questions about the topology of acceptable functional spaces of densities and trial functions, we refer to [19], but will conjecture, with-out proof, that here with traditional functions (harmonic, Coulomb) and their configuration mixings, the positivity of ρ should be sufficient.

There is also the question of spurious solutions. The elimination of that spurious solution found in the Slater toy model turned out to be trivial. For more complicated systems, spurious solutions [7,8] might certainly pop up, but an analysis for their detection remains easy. In particular, for other toy models that we tested, spurious solutions were found to induce values of physical parameters out of their allowed range, and/or even complex values while only real ones are acceptable. We can insist that the final, polynomial equation for the energy, S(η) = 0, can only create a finite number of candidate solution branches to be investigated.

This concludes our toy model as a demonstration of a handling of determinants in this algebraic approach. But we can still take advantage of it for a study of the "kinetic Kohn-Sham functional". First notice that the "harmonic energy", Φ|K|Φ , and the kinetic energy of Φ differ by only an explicit functional of the density, namely half of its second moment, dr r 2 ρ(r). The search for a functional for Φ|K|Φ , therefore, is a problem equivalent to that for the kinetic energy. Set now V 0 = 0 in Eqs. (19). The same program of elimination that was used for a full energy functional now returns a simpler, and very transparent, form of Eq. ( 22), (η -6)(η -4) 2 (η -2) = 0. The corresponding version of Eq. ( 21), (2η -4 -3a 6 ) 2 (2η -8 -3a 6 ) 2 = 0, gives η = 2 and η = 4 if a 6 = 0. This means determinants made of {ϕ 0 , ϕ 1 } and {ϕ 2 , ϕ 1 }, respectively. For a 6 = 4/3, at the other edge of the domain, the harmonic energies are η = 4 and η = 6, with determinants {ϕ 0 , ϕ 3 } and {ϕ 2 , ϕ 3 }, respectively.

After this proof that the method is basically the same for determinants as for configuration mixings, we can stress that configuration mixings have the technical advantage that the energy is quadratic only and permits the short cut described at the stage of Eqs (5,6).

A constructive derivation of KSPs is available. For instance, truncate some single particle basis and let P be the projector upon the resulting, finite dimensional subspace for a system of N fermions, with their Hamiltonian H, or rather now, PHP. Given the kinetic energy operator T, choose a local potential w 0 (r), hence a one-body operator W 0 = N i=1 w 0 (r i ), hence a one-body Hamiltonian H 0 = T + W 0 , so that the ground state of PH 0 P, a Slater determinant Φ 0 , be non degenerate and providing an approximate density ρ 0 for the system. For any density ρ in the subspace, the integral, ∆ρ, of the difference, ∆ρ = ρρ 0 , vanishes as already stated. (Here and in the following, the integral sign, , means r d-2 dr depending on the d-dimensional problem under consideration.) Expand, as already discussed, ∆ρ in a basis of orthonormal functions S β (r), "constrained by vanishing averages" [15,16], ∆ρ(r) = ∞ β=1 b β S β (r). Truncate the expansion at some suitable order N ′ . Again, given a de-terminant Φ with the parameters c α nℓm of its orbitals, or given a correlated state, Ψ = q C q Φ q , the constraints, Φ ⇒ b β or Ψ ⇒ b β , are polynomials of the parameters. Given H 0 , the polynomial method returns a polynomial K(κ, b 1 , . . . , b N ′ ) for a reference functional, such that the lowest root of the equation, K = 0, represents the constrained minimum, κ ′ = Min Φ⇒b1,...,b N ′ Φ|H 0 |Φ , for the determinants in the subspace. In the same way, given the full H, the method gives a polynomial E(η, b 1 , . . . , b N ′ ), the lowest η root of which is the constrained minimum, η ′ = Min Ψ⇒b1,...,b N ′ Ψ|H|Ψ , for correlated states in the subspace. Then it is trivial to derive from K and E a polynomial, Ω(ω; b 1 , . . . , b N ′ ), for the difference, ω = ηκ. The diagonalization of PHP then reads,

∂κ ∂b β + ∂ω ∂b β = 0, β = 1, . . . , N ′ . ( 25
)
With the ratio, v β = -(∂Ω/∂b β )/(∂Ω/∂ω), representing ∂ω/∂b β , define the one-body, local potential, 25), read as the diagonalization for a determinant Φ with the same density ρ as that of the eigenstate Ψ of PHP. The potential, P(w 0 + v ∆ )P, is a KSP valid for the subspace, up to the convergence of the truncation with N ′ terms. This polynomial method most often uses a very non local parametrization of ρ, that deviates from the quasilocal tradition of the field. In every case, our unconventional parametrization of ρ creates a new zoology of DFs. Nothing of this zoology is known to us, but its interest is obvious, since manipulations of polynomials and properties of their roots, including bounds, are basic subjects. Moreover, extrapolations of polynomials, and criticism of such extrapolations, are easy. The number of available, exactly solvable models is huge. It is limited only by computational power. For nuclei or atoms, the models will be "radial" [13], somewhat simple. For nuclear physics, our ultimate goal will be to see whether particle number can be used as a constraint, to generate a mass formula. For electrons in molecules or extended systems (metals, thin layers, etc.), however, a necessary algebra of functions of 2 or 3 variables will burden the models. Anyhow, one can always test whether our polynomials from "smaller" models may remain good approximations for "larger" ones, if, for instance, scaling properties can be established. Asymptotic properties of a sequence of "DF polynomials" might guide towards derivations of more traditional DFs. In particular, the polynomial models allow comparisons between the KS and the true kinetic energies of correlated systems. They also provide explicit terms for those correlation energies due to interactions.

v ∆ (r) = N ′ β=1 v β S β (r). Let Φ be the ground state of P H 0 + N i=1 v ∆ (r i ) P. Notice that Φ|PS β P|Φ = Φ|S β |Φ . Then the energy E of Φ has derivatives, ∂E/∂v β = (∆ρ + ρ 0 ) S β = b β + b β0 , (26)
In conclusion, this algebraic method simplifies density functional theory into energy minimization under finite numbers of constraints, under very elementary manipulations of polynomials. It retains all essential information about the density and all components of the energy. In a forthcoming paper, we shall investigate a more realistic problem than the toy models used for this Letter.

SK acknowledges support from the National Research Foundation of South Africa and thanks CEA/Saclay for hospitality during part of this work. BG thanks Rhodes University and the University of Johannesburg for their hospitality during part of this work.

FIG. 1 :

 1 FIG.1: Contour Etoy(η, D) = 0 for the configuration mixing model with 4 × 4 matrices, as described in the text.

5 ΗFIG. 2 :

 52 FIG.2: Trajectories η(a6), see Eq. (21), for the toy model for a Slater determinant, as described in the text.

  because of the orthonormality of the S β 's. The numbers, b β0 = ρ 0 S β , are easily pretabulated. The quantities, v β and (b β + b β0 ), are Legendre conjugates, and, moreover, ∂/∂(b β + b β0 ) = ∂/∂b β . The conditions, Eqs. (


B G Giraud

P Moussa

Schrödinger equations for the square root density of an eigenmixture and the square root of an eigendensity spin matrix

, a consequence of the rotational invariance of the nuclear Hamiltonian; when nuclear ground states (GSs) have a finite spin, the RDFT uses eigenmixture density operators to simplify predictions of GS energies into one-dimensional, radial calculations. We also study Schrödinger equations governing spin eigendensity matrices.

The theorem of Levy, Perdew and Sahni [1] may be described as follows: i) let H A be a Hamiltonian for A identical particles, with individual mass m,

H A = A i=1 [-h 2 ∆ ri /(2m) + u( r i )] + A i>j=1 v( r i , r j ), (1)
ii) consider a GS eigenfunction of H A , ψ( r 1 , σ 1 , r 2 , σ 2 , ..., r A , σ A ), where σ i denotes the spin state of the particle with space coordinates r i , iii) use a trace of |ψ ψ| upon all space coordinates but the last one, and upon all spins, to define the density,

ρ( r) = A σ1...σA d r 1 d r 2 ...d r A-1 |ψ( r 1 , σ 1 , r 2 , σ 2 , ..., r A-1 , σ A-1 , r, σ A )| 2 , (2)
iv) then there exists a local potential v ef f ( r) so that,

[-h 2 ∆ r /(2m) + v ef f ( r)] ρ( r) = (E A -E A-1 ) ρ( r) , (3) 1
where the eigenvalue is the difference of the GS energy E A of the A-particle system and that, E A-1 , of the (A -1)-particle one. Can this theorem be generalized for densities derived from eigenoperators, D ∝ N n=1 w n |ψ n ψ n |, corresponding to cases where H has several (N > 1) degenerate GSs ψ n ? The degeneracy situation is of a wide interest in nuclear physics for doubly odd nuclei, the GSs of which often have a finite spin, and, if only because of Kramer's degeneracy, for odd nuclei. In particular, because of the rotational invariance of the nuclear Hamiltonian, the density operator of interest for the RDFT [2] reads, D = M |ψ JM ψ JM |/(2J +1), where J and M are the usual angular momentum numbers of a degenerate magnetic multiplet of GSs ψ JM . Actually, more generally, it will easily be seen that the argument which follows holds for a degenerate multiplet of excited states as well.

This paper proves the generalization, by closely following the argument used for one eigenstate only [1]. Furthermore, there is no need to assume identical particles. No symmetry or antisymmetry assumption for eigenfunctions is needed. Let p i and σ i be the momentum and spin operators for the ith particle, at position r i . Single out the A-th particle, with its degrees of freedom labelled r and σ rather than r A and σ A . For a theorem of maximal generality, with distinct masses, one-body and two-body potentials, our Hamiltonian may become,

H A = H A-1 + V A + h A , with H A-1 = A-1 i=1 [-h 2 ∆ ri /(2m i ) + u i ( r i , p i , σ i )] + A-1 i>j=1 v ij ( r i , p i , σ i , r j , p j , σ j ) , V A = A-1 j=1 v Aj ( r, r j , p j , σ j ) , h A = -h 2 ∆ r /(2m A ) + u A ( r) . (4)
The potentials acting upon the first (A -1) particles may be non local and spin dependent, but, for a technical reason which will soon become obvious, those potentials acting upon the A-th particle in V A and h A must be strictly local and independent of the A-th spin. For notational simplicity, we choose units so that h2 /(2m A ) = 1 from now on. As in the one eigenstate case [1] we select situations where there exists a representation in which, simultaneously, the Hermitian Hamiltonian H A and all the eigenfunctions ψ( r 1 , σ 1 , ..., r A , σ A ) under consideration are real. This reality condition does not seem to be restrictive, in view of time reversal invariance.

Let E A be a degenerate eigenvalue of H A . The degeneracy multiplicity being larger than 1, select N ≥ 2 of the corresponding eigenfunctions ψ n , orthonormalized. Their set may be either complete or incomplete in the eigensubspace. The density operators,

D = N n=1 |ψ n w n ψ n | , N n=1 w n = 1 , (5)
with otherwise arbitrary, positive weights w n , are normalized to unity, Tr D = 1, in the A-body space. They are eigenoperators of H A , namely

H A D = E A D.
The partial trace of a D upon the first (A -1) coordinates and all A spins,

τ ( r) = N n=1 w n σ1...σA-1σ d r 1 ...d r A-1 [ψ n ( r 1 , σ 1 , ..., r A-1 , σ A-1 , r, σ)] 2 , ( 6
)
defines a "density" τ , normalized so that d r τ ( r) = 1. Let now φ n r σ be, in the space of the first (A -1) particles, an auxiliary wave function defined by,

φ n r σ ( r 1 , σ 1 , ..., r A-1 , σ A-1 ) = ψ n ( r 1 , σ 1 , ..., r A-1 , σ A-1 , r, σ)/ τ ( r) . (7)
Note that this auxiliary wave function depends on the choice of the weights w n . Now the density operator in the space of the first (A -1) particles,

D ′ r = nσ |φ n r σ w n φ n r σ |, is normalized, Tr ′ D ′ r = 1
, where the symbol Tr ′ means integration upon the first (A -1) coordinates and sum upon the first (A -1) spins. Since this normalization of D ′ r in the (A -1)-particle space does not depend on r, two trivial consequences read, ∇ r Tr ′ D ′ r = 0 and ∆ r Tr ′ D ′ r = 0. More explicitly, this gives,

n σ σ1...σA-1 d r 1 ...d r A-1 w n φ n r σ ( r 1 , σ 1 , ..., r A-1 , σ A-1 ) × ∇ r φ n r σ ( r 1 , σ 1 , ..., r A-1 , σ A-1 ) = 0 , (8) and n σ σ1...σA-1 d r 1 ...d r A-1 w n { [∇ r φ n r σ ( r 1 , σ 1 , ..., r A-1 , σ A-1 )] 2 + φ n r σ ( r 1 , σ 1 , ..., r A-1 , σ A-1 ) ∆ r φ n r σ ( r 1 , σ 1 , ..., r A-1 , σ A-1 ) } = 0 .(9)

Then one can rewrite the eigenstate property, (H

A -E A ) ψ n = 0, into, (H A-1 + V A + h A -E A ) √ τ φ n r σ = 0 . (10)
This also reads,

√ τ (H A-1 + V A + u A -E A ) φ n r σ -(∆ r √ τ ) φ n r σ = 2 (∇ r √ τ ) • (∇ r φ n r σ ) + √ τ (∆ r φ n r σ ) . (11)
The right-hand side (rhs) of Eq. ( 11) occurs because the Laplacian, ∆ r , present in h A , acts also upon the parameter, r, of φ n r σ . This is where the local, multiplicative nature of u A and v Aj in the last particle space is used and avoids the occurrence of further terms, that would induce a somewhat unwieldy theory. Define, for any integrand Ψ n rσ , the following expectation value in the first (A -1)-particle space,

Ψ n r σ = σ1...σA-1 d r 1 ...d r A-1 Ψ n r σ (r 1 , σ 1 , ..., r A-1 , σ A-1 ) . ( 12
)
Multiply Eq. ( 11) by φ n r σ and integrate out the first (A -1) coordinates and spins, to obtain,

φ 2 n r σ [ E exc nσ ( r) + E A-1 + U nσ ( r) + u A ( r) -E A -∆ r ] τ ( r) = 2 φ n r σ (∇ r φ n r σ ) • [∇ r τ ( r)] + φ n r σ (∆ r φ n r σ ) τ ( r) , (13)
where

E exc nσ ( r) is defined from, φ 2 n r σ [E exc nσ ( r) + E A-1 ] = σ1...σA-1 d r 1 ...d r A-1 φ n r σ (r 1 , σ 1 , ..., r A-1 , σ A-1 ) [H A-1 φ n r σ ] (r 1 , σ 1 , ..., r A-1 , σ A-1 ) , (14)
and U nσ ( r) results from,

φ 2 n r σ U nσ ( r) = A-1 j=1 σ1...σA-1 d r 1 ...d r A-1 φ n r σ (r 1 , σ 1 , ..., r A-1 , σ A-1 )v Aj ( r, r j , p j , σ j )φ n r σ (r 1 , σ 1 , ..., r A-1 , σ A-1 ).( 15
)
The square norm of φ n r σ in the (A -1)-particle space results from Eq. ( 12) with Ψ = φ 2 . In Eq. ( 14) the expectation value of H A-1 is explicited as the sum of the GS energy E A-1 of H A-1 and a positive, excitation energy E exc nσ ( r). From Eq. ( 15), the Hartree nature of the potential U nσ ( r) is transparent.

Keeping in mind that, ∀ r, the density operator D ′ is normalized to unity, namely, that nσ w n φ 2 n r σ = 1, multiply Eq. ( 13) by w n and perform the sum upon n and σ. This gives,

[ U exc ( r) + E A-1 + U Hrt ( r) + u A ( r) -E A -∆ r ] √ τ = nσ w n [ 2 φ n r σ (∇ r φ n r σ ) • (∇ r √ τ ) + φ n r σ (∆ r φ n r σ ) √ τ ] ,( 16
)
where the "mixture excitation potential",

U exc ( r) = nσ w n φ 2 n r σ E exc nσ ( r) , (17)
is local and positive and the "mixture Hartree-like potential",

U Hrt ( r) = nσ w n φ 2 n r σ U nσ ( r) , (18)
is also local. Because of the frequent dominance of attractive terms in v Aj , it may show more negative than positive signs. Then notice that, because of Eqs. (8), and Eq. ( 12) with Ψ = φ ∇φ, the sum in the rhs of Eq. ( 16), nσ w n φ n r σ (∇ r φ n r σ ) , vanishes. Note also, from Eqs. (9), and Eq. ( 12) with Ψ = φ ∆φ, that, again for the rhs of Eq. ( 16), the following equality holds,

- nσ w n φ n r σ (∆ r φ n r σ ) = nσ w n (∇ r φ n r σ ) • (∇ r φ n r σ ) , (19)
where, again, the symbol denotes the trace Tr ′ , an integration upon the first (A -1) coordinates together with summation upon their spins. The rhs of this equation, Eq. ( 19), defines a positive, local potential,

U kin ( r) = nσ w n (∇ r φ n r σ ) • (∇ r φ n r σ ) . ( 20
)
Finally, according to Eq. ( 16), the sum of local potentials, U ef f = U exc + U Hrt + U kin + u A , drives a Schrödinger equation for √ τ ,

[-∆ r + U ef f ( r)] τ ( r) = (E A -E A-1 ) τ ( r) . ( 21
)
This is the expected generalization of the LPS theorem. Note that, if the (A-1) particles are not identical, then E A-1 , the GS energy of H A-1 , means here the mathematical, absolute lower bound of the operator in all subspaces of arbitrary symmetry or lack of symmetry. In practice, however, most cases imply symmetries in the (A-1)-space, and E A-1 means the ground state energy under such symmetries.

For nuclear physics, this generalization can be used in two ways: i) The first one consists in considering hypernuclei or mesonic nuclei, where the A-th particle is indeed distinct. Theoretical calculations with local interactions for the distinct particle may be attempted while non local and/or spin dependent interactions for the A-1 nucleons are useful, if not mandatory. Then, obviously, the density τ refers to the hyperon or the meson and, given the neutron and proton respective numbers N and Z, wave functions ψ n and φ n r, σ belong to both N -and Z-antisymmetric subspaces. The energy E A-1 is the GS energy of nucleus {N, Z}, a fermionic GS energy, rather than the absolute lower bound of the mathematical operator H A-1 in all subspaces.

ii) The second one consists in setting all A particles to be nucleons, at the cost of restricting theoretical models to local interactions. Such models are not without interest indeed, although interactions which are spin dependent are certainly more realistic. The antisymmetric properties of the functions ψ n are requested in both N-and Z-spaces. If the singled out, A-th coordinate is a neutron one, the density τ defined by Eq. ( 6) is the usual neutron density, divided by N ; the functions φ n r σ are antisymmetric in the (N -1)-neutron space and the Z-proton space; the energy E A-1 now means the fermionic GS of nucleus {N -1, Z}, not that absolute, mathematical lower bound of operator H N -1,Z . Conversely, if the A-th coordinate is a proton one, then, mutatis mutandis, τ is the usual proton density, divided by Z, and E A-1 is the GS energy of nucleus {N, Z -1}.

In both cases, the Hamiltonians to be used are scalars under the rotation group, and, therefore [2], the density operators D considered by the RDFT are also scalars. Hence, all calculations defining U ef f reduce to calculations with a radial variable r only.

We shall now extend our previous results to the case where we allow spin dependence for all interactions, a most useful feature if all A particles are nucleons. Polarized eigenmixtures are also interesting and need also be considered. Hence, a generalisation of our approach, which uses the "spin-density matrix" (SDM) formalism [3] [4], is in order. The Hamiltonian may become,

A i=1 [-h 2 ∆ ri /(2m i ) + u i ( r i , σ i )] + A i>j=1 v ij ( r i , σ i , r j , σ j ) . ( 22
)
It allows subtle differences between neutrons and protons, besides the Coulomb interactions between protons. More explicitly, there can be two distinct one-body potentials, u n , u p , namely one for neutrons and one for protons, but within the neutron space the function u n ( r i , σ i ) obviously will not read u ni ( r i , σ i ). Similarly in the proton space, the Hamiltonian contains terms u p ( r i , σ i ) rather than u pi ( r i , σ i . The same subtlety allows terms v pp ( r i , σ i , r j , σ j ), v pn ( r i , σ i , r j , σ j ), v np ( r i , σ i , r j , σ j ) and v pp ( r i , σ i , r j , σ j ), rather than v ppij ( r i , σ i , r j , σ j ), ... etc. (Of course, v pn = v np .) But non localities of potentials and interactions, in the sense of explicit dependences upon momenta p i , remain absent.

Then the A-th particle is again singled out, with degrees of freedom again labelled r and σ, and the Hamiltonian is split as a sum, K A-1 + W A + k A , somewhat similar to the split described by Eqs. (4). For simplicity, we shall use short notations, K, W and k, rather than K A-1 , W A and k A . With two spin states, σ = ±1/2, for the A-th nucleon, we represent eigenstates ψ n as column vectors, ψ n = ψ n+ ψ n- , and operators as matrices such as,

W = W ++ W +- W -+ W -- , k = k ++ k +- k -+ k -- and ū = u A++ u A+- u A-+ u A-- . The matrix, K = K 0 0 K , is a
scalar in spin space, since K does not act upon the A-th particle. The spin density matrix, ρ n , results from an integration and spin sum over the (A -1)-space of the matrix, ψ n × ψ T n , where the superscript T denotes transposition,

ρ n ( r) = ψ n+ ψ n- × [ ψ n+ ψ n-] = (ψ n+ ) 2 ψ n+ ψ n- ψ n-ψ n+ (ψ n-) 2 . ( 23
)
It depends on the last coordinate, r, and its matrix elements are labelled by two values, {σ, σ ′ }, of the last spin. For an eigenmixture one defines, obviously, θ( r) = n w n ρ n ( r), and the trace in the last spin space, [θ ++ ( r) + θ --( r)], is that density, τ ( r), defined by Eq. (6). The SDM, θ, is symmetric and positive semidefinite, ∀ r. Except for marginal situations, it is also invertible, in which case there exists a unique inverse square root, also symmetric and positive. Define, therefore, a column vector φ n r of states in the (A -1)-space according to,

φ n r = θ -1 2 ( r) ψ n . (24)
Then the following property,

n w n φ n r × φ T n r = θ - 1 2 ( r) n w n ψ n × ψ T n θ -1 2 (

 12 r) = 1 , (25) holds ∀ r. Here 1 denotes the identity matrix. Hence, the following gradient and Laplacian properties also hold,n w n ∇ r φ n r + φ n r - × [ φ n r + φ n r -] + n w n φ n r + φ n r - × ( ∇ r [ φ n r + φ n r -] ) = 0 , ∀ r ,

and

The eigenvector property, K + W + k -E A 1 ψ n = 0, also reads,

Right-multiply Eq. ( 28) by the row vector, φ T n r , integrate and sum over the (A -1)-space, weigh the result by w n and sum upon n. Because of Eq. ( 25), the weighted sum of averages over the (A -1)-space simplifies into,

and

The rhs of Eq. ( 29) can be simplified, but less than that of Eq. ( 16). Indeed Eq. ( 26) does not imply that the coefficient of ∇ r θ 1 2 ( r) vanishes. In fact this coefficient is,

and Eq. ( 26) shows that the matrix on the rhs is antisymmetric. In turn, from Eq. ( 27), the "Laplacian induced coefficient" in the rhs of Eq. ( 29) may be listed as,

In the rhs of Eq. ( 33), the similarity of its first term with potential U kin , Eq. ( 19), is transparent. Also transparent is the antisymmetry of the second term.

With the definitions,

and

the Schrödinger equation for the square root of the spin density matrix reads,

This concludes our generalizations of the LPS theorem. On the one hand, see Eq. ( 21), we obtained for the square root density of an eigenmixture a Schrödinger equation, most similar to the LPS equation. On the other hand, at the cost of a slightly less simple result, we also obtained, see Eq. ( 37), an LPS-like equation that drives the square root of the spin density matrix.

This opens a completely new zoology of local, effective potentials, of which very little is known, but the interest of which is obvious, since they drive a reasonably easily measurable observable, the square root of an eigenmixture density, which depends on one degree of freedom r only. A connection of such potentials with optical potentials, or rather their real parts, is likely, but yet remains an open problem.