Yves Marrocchi
email: yvesm@crpg.cnrs-nancy.fr

Pierre Beck
email: pierre.beck@univ-grenoble-alpes.fr

The Quest For Water

Keywords: water, planetary bodies, density, circumsolar disk, meteorites

Water played a key role in shaping the Solar System-from the formation of early solids to the processes of planetary and moon formation. The presence of water in molecular clouds influences the initial abundance and distribution of water in the circumsolar disk, which, in turn, affected the water budget of the terrestrial planets and, therefore, their geological activity and habitability. On Earth, surficial and deep-water cycles have largely governed the planet's geodynamical and geochemical evolution. This issue focuses on the past and present distribution of water within the Solar System and how this important molecule affects astrophysical and geological processes.

been distributed among distinct reservoirs: the oceans, the cryosphere (i.e., ice sheets and glaciers), the atmosphere, and terrestrial storage; the latter corresponding to surface and rootzone soil moisture, groundwater, rivers, and lakes, as well as water stored in vegetation. If the oceans represent the main reservoir, with an estimated amount of water of ~1.4 × 10 21 kg, an additional 0.15 ocean masses is stored in other surficial reservoirs [START_REF] Hirschmann | Water, melting, and the deep earth H2O cycle[END_REF]. This visible water is characterized by constant motion, which is referred to as the water or hydrological cycle (FIG. 1). Solar-driven evaporated surficial waters (i.e., oceans, lakes, rivers, streams, land surface and plants) are transferred into the atmosphere under the action of wind, where they experience condensation to form clouds and water droplets, as well as ice crystals (FIG. 1). Hence, atmospheric water comes back to the Earth's surface through precipitation (i.e., rain or snow), replenishing the earthbound parts of the surficial water cycle and percolating downward to form groundwater, which slowly moves through rocks and surface materials to end up in streams, rivers, and lakes, with some water going back directly to the oceans (FIG. 1).

The volume of surficial water and its evolution throughout Earth's history are thus key parameters that control Earth's climate and habitability.

The question of how much water there is on Earth is not trivial to answer, as surficial water represents only a fraction of Earth's total water budget. Although it was discovered in recent decades that a significant quantity of water is stored in the Earth's mantle (which is also the case for other rocky planetary bodies), its precise estimation is still the subject of very active debate. This difficulty lies in the fact that-contrary to refractory elements-volatile elements are lost and fractionated during magma degassing, making the estimation of water abundance in different mantle source regions difficult. Depending on the mantle-derived rocks considered (either those emitted along mid-ocean ridges or at oceanic islands), between 0.1 to 2.5 times the mass of surficial reservoirs could be stored in the mantle [START_REF] Hirschmann | Water, melting, and the deep earth H2O cycle[END_REF]. Another estimate based on the noble gas systematics of the Earth's mantle suggests that the bulk Earth water content could correspond to 10 ± 5 ocean masses [START_REF] Marty | The origins and concentrations of water, carbon, nitrogen and noble gases on Earth[END_REF]. Even by considering the largest estimate of water stored in the mantle, the Earth appears surprisingly dry, as water represents less than 1% of our planet's mass. This peculiar characteristic is thus a key constraint for understanding the origin and evolution of Earth's water.

The amount of water on Earth, and its transfer between the different terrestrial reservoirs, plays a key role in controlling the geological processes and behavior of the solid Earth (FIG. 1).

Deep-Earth water is mainly incorporated as point defects in the silicate minerals of the mantle.

The addition of H into nominally anhydrous minerals profoundly changes their physical properties (i.e., mineral strength, electrical conductivity, diffusion rates of other constituent ions, seismic properties). Water also has a significant influence on mantle rheology and viscosity, as well as the solidus of the mantle and, therefore, the melting regime. Taken together, water greatly affects mantle convection and plate tectonics, which in return influence the influx of water to the mantle through the subduction of wet oceanic lithosphere and back to the surface via mid-ocean ridge, arc, and hotspot volcanism. The ubiquitous presence of water makes this molecule fundamental for both astrophysical, cosmochemical, and geological processes, and highlights the importance of understanding its evolution in time and space throughout the 4.56 billion year history of the Solar System.

WATER IN THE SOLAR SYSTEM: WHERE, HOW, AND HOW MUCH?

Given its significance in planetary and biological evolution, water has been searched for and found throughout the Solar System. Quantifying its abundance and distribution is not easy business, as only a few samples from planetary bodies are available on Earth compared to the stunning diversity of worlds in our Solar System. However, water can be detected using infrared remote sensing techniques at the surface of planetary objects (FIG. 2), because the strong dipole moment of the O-H bond leads to particularly strong absorption in the infrared spectral range.

Water, as well as hydrated or hydroxylated minerals, can be identified and distinguished by analyzing reflected sunlight based on specific absorption features around 1.4, 1.9, 2.2-2.3, and 3 µm (FIG. 2). It can also be observed remotely from the rotational modes of water molecules that can be probed by radio telescopes and, depending on the abundance of water in the line of sight, its isotopic composition may be determined (hydrogen having two stable isotopes: hydrogen H and deuterium D, expressed as the D/H ratio). Water D/H ratios can also be accessed by in situ mass spectrometry measurements onboard orbiters or landers.

Depending on particle density, water ice grains under high vacuum will sublimate at temperatures over 100-150 K. This limit, known as the water snowline (FIG. 1), was effective during the circumsolar disk evolution where its location was a direct function of the stellar luminosity, accretion rate, and gas density. Although its position is generally expected to drift inward during disk evolution (from ~ 3 to 1 AU; astronomical unit AU = Earth-Sun distance = 150 × 10 6 km). The position of the snowline (and its evolution over time) is a fundamental question, because it controls the amount of water available in the asteroidal-and planetaryaccretion regions, a characteristic that, in turn, controls the geodynamic evolution of asteroidal and planetary bodies. The snowline is located near Jupiter's orbit under current Solar System conditions, but specific local conditions at the surface of planetary bodies could disturb this limit and impact the water distribution. Water ice is thus dominantly unstable today at the surfaces of the (atmosphere-free) Moon and Mercury, but has been detected at locations permanently shaded throughout their whole revolution around the Sun. In addition to water ice, -OH-related absorption bands have been also detected at the surface of the Moon at high latitude with a suggested abundance of the order of 100s µg•g -1 of water. Both signatures have been interpreted to reflect an exogenous origin, either from micrometeorites, asteroidal and/or cometary impacts, or by the impact-triggered transformation of solar wind-implanted hydrogen into water molecules.

The density and temperature of the Venusian atmosphere preclude the presence of water-ice deposits on the surface, as well as the presence of most hydrous minerals (but not all). There has been speculation about the amount of water hosted in the interior of Venus, but further evidence through dedicated space missions is needed. In the case of Mars, water has been identified all over the planet, whether in the form of ice, hydrated minerals, and possibly brines (FIG. 2). While the presence of clay minerals in ancient Martian terranes offers a window into past water cycles, there is also an active surface-atmosphere water cycle today. Two massive permanent ice caps have been identified in the north and south poles of the red planet, which provide enough water to fill the Mediterranean Sea. The redistribution of these ice masses in response to Mars' orbital evolution led to "Quaternary" geological processes identified by their morphological fingerprints.

The asteroid belt hosts a menagerie of objects that probably formed in very different locations in the Solar System. To first-order, there is a dichotomy between darker and brighter objects that echoes the dichotomy found in meteorites between ordinary and carbonaceous chondrites (DeMeo and Carry 2014). The silicate-rich asteroids (S-type) do not show strong evidence of hydration on their surface, in line with the general dryness of ordinary chondrites to which they are associated (typically 0.1 wt.% H2O; [START_REF] Vacher | Hydrogen in chondrites: influence of parent body alteration and atmospheric contamination on primordial components[END_REF]). On the contrary, C-type asteroids, which make up roughly half of the mass of the main belt, often show strong evidence of phyllosilicate minerals based on observations of a deep 3-µm band [START_REF] Usui | AKARI/IRC near-infrared asteroid spectroscopic survey: AcuA-spec[END_REF]FIG. 3). The strength of the feature is much higher than has been observed on the Moon, for instance, and suggests the presence of a significant amount of equivalent water (>5 wt.%) on their surfaces.

The meteorite record of aqueous alteration and phyllosilicate production is presented in Vacher and Fujiya (2022 this issue).

The observation that some main-belt asteroids are active, i.e. that they eject a significant amount of dust, is suggestive that, together with OH-bearing minerals, a significant fraction of water ice is present in asteroids, which blurs the boundary with comets. The outer part of the asteroid belt, as well as Jupiter's Trojan asteroids, is dominated by particular types of objects, the P-and D-types, which share strong optical similarities with comets, and can be envisioned as dormant comets [START_REF] Vernazza | Composition of Solar System small bodies[END_REF]. However, unlike C-type asteroids, they generally do not show strong evidence for water ice or hydrated minerals at their surface, although water is probably a major constituent, as indicated by their density and potential D-type-derived meteorites. Their thermal evolution was possibly cold enough to preclude the melting of water and production of phyllosilicate at least on their surface.

Density is the strongest constraint we have today to infer the abundance of water in the interior of small bodies. Accurate density estimates are limited to only a tiny portion of the overall population, but suggest different bulk compositions for S-, C-, and to some extent D-type asteroids (FIG. 4;[START_REF] Vernazza | VLT/SPHERE imaging survey of the largest main-belt asteroids: final results and synthesis[END_REF]. The density of 100-km-sized, S-type asteroids is in the range 3000-4000 kg/m 3 , which is in agreement with measurements on ordinary chondrites found on Earth, and implies a low fraction of water in their interior (FIG. 4). By comparison, the density of C-type asteroids is significantly lower (~1000-2000 kg/m 3 for 100-km objects; FIG. 4), which suggests that they are composed of an important fraction of low-density material (presumably water ice and/or hydrated minerals; FIG. 3). Density estimates for D-type asteroids are rare, but the recently obtained value for 87-Sylvia (1330 ± 70 kg/m 3 ) is slightly lower than that estimated for C-types of similar size (FIG. 4;[START_REF] Vernazza | VLT/SPHERE imaging survey of the largest main-belt asteroids: final results and synthesis[END_REF]. Estimating the bulk water content of asteroids is not straightforward, as it requires constraints or hypotheses on their porosity. Interestingly, density and diameter are correlated for a given class of asteroid, which is interpreted by an increase of porosity in smaller objects, thus allowing the bulk composition of asteroids to be estimated. For instance, the 2060 ± 200 kg/m 3 density of the 420-km-diameter C-type asteroid 10-Hygeia could be explained by a volume mixture of 20% porosity, 55% rock/metal, and 25% water ice.

Comets are, by definition, objects emitting dust as a consequence of volatile sublimation at their surface. Overall, water is the major volatile species within comets, but other volatile species, such as CO or CO2, may play a role in their activity. While the water loss rate from a comet can be determined from the ground, the determination of the water/dust ratio is more complex and requires accurate modeling of the light scattered by the dust particles. Detailed observations in the 3-µm region of the nucleus of comet 67P Churyumov Gerasimenko revealed the general absence of water ice on the surface, and that hydrogen is instead present in surface material in the form of ammonium salt and aliphatic organics (FIG. 3). If the surface seems "dry", the low density of the few cometary nuclei observed by spacecraft (~500-2000 kg/m 3 ; Fig. 4 Optical observations of trans-Neptunian objects (TNOs) and centaurs (objects on unstable orbits between Jupiter and Neptune) have revealed the presence of water ice on their surfaces [START_REF] Barucci | Composition and surface properties of transneptunian objects and centaurs[END_REF], see also Charon in FIG. 2), but not systematically. When present at the surface, water ice appears to be in a crystalline form, rather than amorphous, which is at odds with a very low-temperature origin (<50 K, i.e., directly inherited from the protosolar nebula) and suggests that it may have been thermally processed. The largest known TNOs have wellconstrained densities that are lower than C-type asteroids of similar size (except for 1-Ceres). In the case of the Pluto-Charon system, the determined values are of 1850 and 1700 kg/m 3 , respectively, which translates to 30-40 wt.% of water ice based on a two-component model with a rocky component [START_REF] Mckinnon | Origin of the Pluto-Charon system: constraints from the New Horizons flyby[END_REF]). It should be noted that one of the limitations of the two-component models is that it neglects the presence of carbonaceous compounds, which appear to represent a major constituent of comets, and could then constitute a significant fraction of TNOs.

Looking at the overall picture, if water is present at least locally at the surface of all types of Solar System objects, density estimates suggest that the fraction of volatiles, including water, is higher in objects distant from the Sun. It appears that the fraction of water is homogeneous among C-and D-type asteroids, comets, and TNOs (within an order of magnitude to be conservative), while the inner Solar System is depleted in water compared to these objects.

Understanding this global picture requires a determination of the astrophysical origin of water (and thus what constituents were incorporated into the Solar System) and how and where planetary bodies accreted.

ORIGIN, AMOUNT, AND DISTRIBUTION OF WATER WITHIN THE EARLY SOLAR SYSTEM

The current understanding of the astrophysical origin and pathways of water toward planetary systems is presented in the article by Ceccarelli and Dun in this issue. In the cold and diffuse environment of the interstellar medium (ISM), water may dominantly exist in the form of waterice grains (FIG. 1). In the diffuse ISM, the presence of water ice has only been observed within density-increased molecular clouds where water molecules form through reactions between O and H atoms in the gas phase and/or at the surface of interstellar dust via catalyzed processes (see FIG. 2 in Ceccarelli and Du in this issue). The latter implies an intimate association of water and refractory grains (silicates or carbon-rich grains) at the onset of star formation. Whatever the processes involved, fundamental questions are related to the amount and origin of water present in the starting material from which the Solar System formed (before disk evolution processing) and how its distribution evolved over the 4567-million-year lifetime of the disk. Several approaches can be followed involving more or less straightforward methods. For instance, calculating how elements with solar abundances partition within silicates, carbon-rich grains, sulfides, CO, and H2O allows a water/dust ratio of ~1 to be estimated [START_REF] Min | The thermal structure and the location of the snow line in the protosolar nebula: axisymmetric models with full 3-D radiative transfer[END_REF].

Depending on the amount of carbon incorporated into carbonaceous dust, the dust density estimates range from 1450 to 1800 kg/m 3 . Interestingly, these values are in the range estimated for Pluton and Charon and suggest that their bulk composition may correspond to that of primordial Solar System building blocks (although other large TNOs, like Eris or Haumea, have higher density). In addition, the inferred water/dust content seems much higher than that estimated for comet 67P and for carbonaceous chondrites. The latter show solar abundances for most elements except for the most volatile, which are characterized by water/rock ratios ranging from 0.01 to 0.4 [START_REF] Marrocchi | Origin and abundance of water in carbonaceous asteroids[END_REF]. However, such estimations correspond to final values, and a significant fraction of the initial water may have been degassed during secondary alteration processes of carbonaceous asteroids (Vacher and Fujiya 2022 this issue).

In theory, a more direct approach would be to measure the abundance and H and O isotopic compositions of water trapped in meteorites at both bulk and in situ scales, as they bear invaluable information on the source and abundance of water accreted by asteroids [START_REF] Deloule | Interstellar water in meteorites?[END_REF][START_REF] Alexander | The provenances of asteroids, and their contributions to the volatile inventories of the terrestrial planets[END_REF]). This proves difficult in practice, however, as immediately after reaching the Earth's surface, meteorites are exposed to terrestrial moisture and precipitation that could significantly affect their initial water contents and isotopic characteristics [START_REF] Vacher | Hydrogen in chondrites: influence of parent body alteration and atmospheric contamination on primordial components[END_REF]. Furthermore, secondary minerals formed through fluid circulation within meteoritic parent bodies are intimately mixed with H-bearing organic compounds, thus impeding the accurate estimation of hydrogen isotopic compositions retained in hydrated minerals [START_REF] Piani | A dual origin for water in carbonaceous asteroids revealed by CM chondrites[END_REF]. However, recent studies have overcome these difficulties by developing new analytical procedures, which allow the estimation of the water concentrations and isotopic compositions to be precisely determined in a large set of meteorites [START_REF] Vacher | Hydrogen in chondrites: influence of parent body alteration and atmospheric contamination on primordial components[END_REF][START_REF] Piani | A dual origin for water in carbonaceous asteroids revealed by CM chondrites[END_REF]). If they confirm that the meteorites formed in the outer Solar System are water-rich [START_REF] Mccubbin | Origin and abundances of H2O in the terrestrial planets, Moon, and asteroids[END_REF], it also appears that meteorites formed in the inner Solar System are characterized by higher water abundances than previously thought [START_REF] Vacher | Hydrogen in chondrites: influence of parent body alteration and atmospheric contamination on primordial components[END_REF]. This led to a paradigm shift where the inner Solar System could no longer be considered as completely dry and volatile-poor; these characteristics having a strong influence on the accretion processes within the disk [START_REF] Lichtenberg | Bifurcation of planetary building blocks during Solar System formation[END_REF], as well as on the redox states of the processes that controlled the evolution of the terrestrial planets [START_REF] Gaillard | The diverse planetary ingassing/outgassing paths produced over billions of years of magmatic activity[END_REF]. From an isotopic perspective, it appears that the asteroidal water budget corresponds to a mixture between (i) local water formed via sublimation/recondensation events during the evolution of the disk and (ii) inherited deuterium-enriched interstellar ices from the molecular clouds unaffected by disk processes [START_REF] Alexander | The provenances of asteroids, and their contributions to the volatile inventories of the terrestrial planets[END_REF][START_REF] Piani | A dual origin for water in carbonaceous asteroids revealed by CM chondrites[END_REF]. This could result from protracted interactions with the molecular cloud all along with the disk history [START_REF] Piani | Origin of hydrogen isotopic variations in chondritic water and organics[END_REF]. Interestingly, this model would naturally explain the not-so-dry characteristics of the inner Solar System, as interstellar ices could thus episodically seed the inner Solar System over several million years, especially when chondrule production processes have stopped and the temperature has dropped below the water-ice sublimation temperature [START_REF] Piani | Origin of hydrogen isotopic variations in chondritic water and organics[END_REF].

Taken together, this implies that the water distribution within the early Solar System resulted from complex processes involving local production and molecular cloud legacy, and that the asteroidal water budget depends on both their location and timing of accretion within the disk. This issue of Elements describes several aspects that may have acted in setting the current distribution of water throughout the Solar System. It begins by presenting how water was produced in the molecular cloud from which the Solar System was formed, and how water molecules may have been distributed in the solar protoplanetary disk [START_REF] Ceccarelli | We drink good 4.5-billion-year-old water[END_REF] this issue). At that point, water is present in the vapor phase or as water-ice grains associated with tiny refractory grains. The next step is a big one: growing planetesimals, and then terrestrial planets. The mechanism from which planetesimals can grow and how the dynamical evolution of the early Solar System could have been key in delivering water in the early Solar System is presented in Izidoro and Piani (2022 this issue). Overall, a major constraint on the possible sources of water is the D/H ratio. [START_REF] Peslier | Water in differentiated planets, the Moon and asteroids[END_REF] Charon (McCord et al. 1998;[START_REF] Grundy | Surface compositions across Pluto and Charon[END_REF]. Changes in the shape of the water-ice absorption among these three objects can be seen around 1.6 µm, which are related to temperature. Mars spectra are from the CRISM database. ice on the surface, and the presence of a peculiar signature in the 3-µm region. This spectral signature has been attributed to a mixture of ammonium salts and organic compounds, and is reminiscent of the spectra of (24) Themis.

  ) points toward a significant fraction of interior volatiles. Estimates based on the Rosetta mission are in the range of 10-50 vol% water (Davidsson et al. 2016; Fulle et al. 2019) and an equivalent uncompacted density of approximately 1800-1900 kg/m 3 (McKinnon et al. 2017).

  this issue) present and discuss the variation of water abundance and isotopic composition among various types of inner Solar System materials. We also focus on two different situations where liquid occurred or occurs today in the Solar System. Vacher and Fujiya (2022 this issue) present how carbonaceous chondrites record hydrothermal reactions that occurred just a few million years after the Solar System formed. Finally, many large Solar System objects are now recognized as past or present "ocean worlds", and are a major focus of current Solar System exploration. This is presented in Castillo-Rogez and Kalousová (2022 this issue).Acknowledgments: The authors deeply thank the Elements editorial team for the invitation to guest edit this issue. They also thank Romain Tartèse for his careful review of this article. PB and YM acknowledge support from the Programme National de Planétologie and the Centre National d'Etude Spatiale. PB acknowledges support from the European Research Council (ERC) under the grant SOLARYS ERC-CoG2017-771691.

FIGURE 1 :

 1 FIGURE 1: Schematic representation of astrophysical and terrestrial environments where water plays a key role. Within the circumsolar disk, water-ice grains could have been formed locally or have been inherited from the molecular cloud from which the Sun formed. The distribution ofwater-ice grains within the disk depends on the location relative to the snowline, a limit that corresponds to the distance from the Sun at which the temperature is low enough for water to condense (~150-170 K). Depending on their accretion location, planetary bodies thus can incorporate a significant amount of water that will influence their geodynamic evolution and water cycle, such as on Earth.

FIGURE 2 :

 2 FIGURE 2: Examples of the detection of water ice and hydrated minerals based on infrared reflectance spectroscopy on several Solar System objects. A menagerie of clay minerals has been identified on Mars based on diagnostic absorption related to metal-OH vibration harmonic and combinations, as well as hydrated phases in the form of sulfates (the example of a monohydrated Mg-sulphate is shown), zeolite, or opals. The signature of water ice is also shown as observed on the north polar cap of Mars together with observations of ice on Europa and

FIGURE 3 :

 3 FIGURE 3: Comparison of the reflectance spectra of selected asteroids and a cometary nucleus. The 3-µm region displayed here is particularly sensitive to the presence of water and hydroxylated minerals, because the fundamental OH stretching is encountered in this spectral range. Such observations reveal the presence of phyllosilicates on several dark C-type asteroids (ex: (2) Pallas; Rivkin et al. 2015; Usui et al. 2019). In a few cases only (but including the largest asteroid / minor planet (1) Ceres; De Sanctis et al. (2015)), phyllosilicates appear to bear ammonia in their interlayer space, as evidenced from the presence of an absorption band around 3.06 µm. The asteroid (24) Themis, with an orbit in the outer main belt, shows a different spectrum in the 3-µm region compared to a phyllosilicate-rich object; a spectrum that was initially attributed to the presence of water ice on its surface. Observations of the nucleus of comet 67P by the Rosetta mission revealed the absence of phyllosilicate, the scarcity of water

FIGURE 4 :

 4 FIGURE 4: Density of various types of Solar System objects as a function of diameter (McKinnon et al. 2017; Vernazza et al. 2021). Due to the possible presence of porosity, asteroids, centaurs, and trans-Neptunian objects seem to present density-diameter trends. Still a general decrease of density is present with heliocentric distance that can be explained by an increasing fraction of water ice. The C-type panel (center) also shows the density of D-type asteroid (87) Sylvia (light green) and dwarf planet (1) Ceres (yellow).


domain: math shs

text:
Cécile Ouvrier-Buffet
email: cecile.ouvrier-buffet@u-pec.fr

Exploring students' conceptions of proof in high-school and university: A proposal for collaborations in Europe

Keywords: Proof, conception, survey, high-school and undergraduate students, proof assistants

Although there is a considerable amount of research on proof in mathematics education, very few studies explore students' conceptions of proof at the transition between secondary and tertiary level. Moreover, new didactic issues arise with proof assistants. The aims of this article are twofold: to highlight methods and theoretical backgrounds used in previous research mathematics education in order to identify undergraduate students' conceptions of proof, and to propose a questionnaire (in a large-scale survey perspective) in order to trigger collaborations in Europe on this topic.

Introduction -A rupture between secondary and tertiary education

Proof is epistemologically constitutive of mathematical activity. Fundamental at all levels of schooling and teacher training, proof contributes significantly to the learning processes of mathematical knowledge (e.g. [START_REF] Hanna | The Ongoing Value of Proof[END_REF]. Large-scale surveys at the international level show that proof occupies too marginal a place in education, at all levels of schooling, with insufficient or even inadequate teacher training (e.g. [START_REF] Stylianides | Proving in the Elementary Mathematics Classroom[END_REF]. The rupture between secondary and tertiary education is internationally pointed out by researchers in mathematics education (e.g. [START_REF] Gueudet | Investigating the secondary-tertiary transition[END_REF][START_REF] Selden | Affect, behavioral schemas, and the proving process[END_REF]. In this transition, there is a change in the didactical contract, but also in the relationship to the mathematical concepts and proofs which requires a more complex and formal view of objects and processes of proof with a kind of acculturation to mathematicians' practices (e.g. [START_REF] Dawkins | Values and norms of proof for mathematicians and students[END_REF][START_REF] Selden | Transitions and proof and proving at tertiary level[END_REF]. The students themselves feel like they are struggling, especially with regard to the "ways of thinking for mathematics" (Di [START_REF] Martino | The Mathematical Crisis in Secondary-Tertiary Transition[END_REF] and to the logic and formalism required in the activity of proving and writing proofs (e.g. [START_REF] Selden | Transitions and proof and proving at tertiary level[END_REF][START_REF] Selden | Validations of proofs considered as texts: Can undergraduates tell whether an argument proves a theorem?[END_REF]. Then, internationally, syntheses (those quoted for instance by [START_REF] Stylianides | Proving in the Elementary Mathematics Classroom[END_REF][START_REF] Selden | Transitions and proof and proving at tertiary level[END_REF][START_REF] Gueudet | Transition secondaire-supérieur : Ce que nous apprend la recherche en didactique des mathématiques[END_REF] converge on students' difficulties and their causes: an insufficient knowledge of concepts and mathematical theorems, the abstract mathematical notions raising conceptualisation difficulties, the expectations referring to the expert practices of mathematicians, the lack of meta-knowledge about proof, a difference in institutional culture between secondary and tertiary education, a lack of articulation between semantic, syntactic and pragmatic approaches and, then, difficulties in writing a proof. Early college students (freshmen) use inappropriate reasoning, often based on everyday logic, when faced with mathematical concepts [START_REF] Selden | Validations of proofs considered as texts: Can undergraduates tell whether an argument proves a theorem?[END_REF][START_REF] Selden | Affect, behavioral schemas, and the proving process[END_REF][START_REF] Selden | Transitions and proof and proving at tertiary level[END_REF]. Besides, one does not really know how students validate their proofs, both when they construct them and when they read them (except for the qualitative results of Weber's research). Other factors explaining the "weak performance of students at the secondary and undergraduate levels on proof" exist such as "the disjointedness of reform efforts" (see the still relevant discussion of [START_REF] Harel | Current contributions toward comprehensive perspectives on the learning and teaching of proof[END_REF]. In order to solve the persistent problem of students' difficulties with proof, which seems to be real in all countries, the arrival of proof assistants in didactic research can provide new opportunities for the teaching and the learning of proof in mathematics. I will explain why.

Current issues on proof in mathematics, computer science and didactics

New technologies influence mathematical practices and create new dynamics at the interface with computer science, reflecting the reality of a work done by contemporary mathematicians in proof. [START_REF] Hanna | Proof in mathematics education, 1980-2020: an overview[END_REF] show the need to develop new approaches to proof teaching taking into account proof assistants (henceforth noted PA). PAs (Coq, L∃∀N, Isabelle etc.) used by mathematicians are free open-source software that mechanically verify a proof, making the logic and the formalism visible. These transform the epistemology of proof, the best-known example being the verification of the 4-color theorem. Recently used in teaching with graphical interfaces (D∃∀DUCTION and Edukera for L∃∀N and Coq), they directly question mathematicians and didacticians on how proof processes should be transmitted to teaching. Thus, the question of teaching and learning proofs, problematic at the secondary-tertiary transition, joins current issues in mathematics, computer science and didactics. Isolated case studies show the interest of working at the articulation between formal proofs with PA and paper-and-pencil proofs as well as the contribution of PA for students in terms of their autonomy in controlling and structuring the proofs they produce, and understanding the mathematical statements [START_REF] Kerjean | Utilisation des assistants de preuves pour l'enseignement en L1. Retours d'expériences[END_REF][START_REF] Avigad | Learning logic and proof with an interactive theorem prover[END_REF][START_REF] Thoma | Learning about Proof with the Theorem Prover LEAN: the Abundant Numbers Task[END_REF]. Thus, recent syntheses on the question, based on the hindsight of Technology Enhanced Learning (TELs), and on the nature of the feedback provided by the machine, call for a multidisciplinary structuring of research to analyse the uses and contributions of PAs in the learning of proof [START_REF] Hanna | Proof technology in mathematics research and teaching[END_REF][START_REF] Balacheff | Proof technology and learning in mathematics: common issues and perspectives[END_REF].

Research questions

Bearing in mind the above context, the questions of my current research project coincide with two of those listed by Harel and Fuller (2009, pp. 355-356): What theoretical tools seem suitable to investigate students' conceptions of proof? (Q1). What are students' current conceptions of proof? (Q2). Considering PAs, my third research questions is: How PAs do impact students' conceptions and performances in proof? (Q3). To answer to Q3, in a large-scale study, I will implement PAs with freshmen in three universities in France next year, using a didactical engineering, in the frame of a collaborative national project. My aim is to evaluate how interventions with a PA impact students' conceptions and performances. I will use two devices to answer to Q2: a questionnaire for an initial large-scale study (with open and multiple-choices questions) and interviews with pairs of students to refine the results of the questionnaire. Note that such devices aim at being reusable to evaluate the impact of PAs on students' conceptions and difficulties (before and after the didactical engineering). Designing such a questionnaire is difficult for several reasons: it aims to be large, and it requires synthesising previous results about students' conceptions of proof, while taking into account the underlying theoretical frameworks used about proof. Indeed, the nature of the questions depends on the theoretical frameworks about proof. It leads me to finely consider Q1 with a bibliographical study to identify where and how undergraduate students' conceptions of proof are explored in research. In this paper, firstly, I will present a preliminary bibliographical study (linked with Q1). Secondly, I will explain my theoretical and methodological choices to assess students' proof conceptions. I will exemplify an exploratory questionnaire to identify proof conceptions of high-school and undergraduate students and to conduct pre-and post-diagnosis when specific interventions (with PAs for instance) are implemented (linked with Q2). Thirdly, I will propose a way to explore students' conceptions at the secondary-tertiary transition in a collaborative perspective in Europe.

Bibliographical study

To circumscribe how researchers in mathematics education have investigated the undergraduate students' conceptions about proof, I have conducted a bibliographical study aimed at identifying the nature of the mathematical domains involved, the methods and the theoretical underlying backgrounds used in such surveys. I used only one database, the Education Resources Information Center (ERIC), a free, large, online digital library of education research that is easily accessible to anyone, recognizing that a search in English will not allow us to identify existing research in other languages. My search in ERIC in February 2023 was the following: "(mathematics AND proof AND students AND undergraduate) AND (conception OR survey OR interview OR questionnaire)". I found for "All Publication Type" and "Any Education Level": 63 articles since 2004, 38 since 2014, 17 since 2019, 4 since 2022. 47 are journal articles, 43 are research reports, and the rest are proceedings, conference papers etc. 56 articles are in higher education, 28 in postsecondary education, and 5 in secondary education. For this article, I then focused on "Journal Articles" of the last ten years (since 2014): 30 articles remained (8 are reports or proceedings). I read the 30 articles and classified them with 8 characteristics: name of the journal, year, country of the study, research questions and methods, population analysed in the study, number of participants, theoretical tools and mathematical contents. Regarding my research questions, I only kept articles that deal with a specific population (students in university or at the secondary-tertiary transition) and whose research questions focus on proof and proving processes. The reading of the articles led me to define five exclusion criteria1 and to eliminate 14 articles, leaving a corpus of 16 articles (Table 1). 10 papers of the 16 retained referred to the USA, 1 from Algeria, 2 from Indonesia, 1 from Iran, 1 from South Africa (Lesotho). Europe is poorly represented with only 1 article from Turkey.

Most of the papers used interviews with students, always after a preliminary survey, questionnaire, assessment, or analysis of students' writings or readings proofs. Only one paper conducted a complete large-scale survey with early undergraduate students (Stylianou et al., 2015). The methods and the theoretical background clearly depend on the research questions and on the mathematical contents. Most of the mathematical fields involved in the 16 selected papers are very specific to university (functions and continuity, differentiability, integrability, Cauchy generalized mean value theorem; limits of sequences; infinite series; linear algebra; holomorphic functions; proof by contradiction). Some of the papers explore more transversal domains such as arithmetic, geometry and algebra. The following list of the emblematic theoretical backgrounds used in the selected papers reflects a wide variety of approaches: Tall ( 2008 Bearing in mind my aim to design a large-scale survey questionnaire, I have focused on Stylianou et al. (2015), but also on the theoretical backgrounds used in the other articles (snowball method). When tracing their common references, some authors return regularly: L. Healy and C. Hoyles, G. Harel and L. Sowder, and K. Weber (alone or with others). Moreover, one of the underlying backgrounds shared by Stylianou et al. (2015) and [START_REF] Healy | Justifying and proving in school mathematics[END_REF] is that of Balacheff (1988ab). I have therefore structured my questionnaire using these frameworks and some of the types of questions used in the large-scale study by [START_REF] Healy | Justifying and proving in school mathematics[END_REF] reused in Stylianou et al. (2015) and analysed with the typology of Balacheff (1998ab), itself based on the Lakatosian epistemology. proofs? What are students' global perceptions of proof? My aim is to build a questionnaire which will be usable throughout high-school, college, and pre-service teacher training, mainly with specialist students in mathematics, for several snapshots: to identify students' conceptions and to evaluate the development of these conceptions (also after interventions, for instance with PAs). I have taken into account several criteria: selecting mathematical contents where the obstacle of mastering the involved concepts is minimized; choosing mathematical problems outside of formalism (so that it is not an obstacle) and outside of curricula (to avoid obstacles or ready-made results and processes). So far, the questionnaire is currently being administered in universities in France and Belgium in a pilot testing phase (with freshmen and pre-service secondary teachers training). It is available online (https://hal.science/hal-03987587). Five sections structure the questionnaire: Part I -Identification, enunciation and use of mathematical properties; Part II -Writing proofs; Part III -View of proofs and evaluation of proof; Part IV -General conception of proof; Part V -Respondent information.

Theoretical backgrounds and aspects addressed by the questionnaire

Built upon different theoretical backgrounds and results previously quoted (mainly the significant students' difficulties even with deductive short proofs), my questionnaire explores "basic" components of deductive proofs through the above-described aspects.

In Part I, the students are asked to state a known formal statement (of their choice) and to write a formal statement starting from a representation. They also have to use a given theorem (Thales intercept theorem) in a specific configuration: I want to evaluate their abilities to check the hypotheses of a given theorem by themselves and to specify the instances of this theorem, as well as their abilities with formal statements (to correlate them with students' proof writings in Part II).

Table 2: An example of Part II of the questionnaire (formal proofs to be written by the students)

One defines an imaginary formal system with objects and rules to work with these objects. One wants to prove theorems in this formal system. Our starting points are as follows: Definition: a « TOY » is a sequence of an integer (strictly positive) number of symbols, consisting only of dots and circles, marked with • and O. For example, O•O•• is a TOY. To start building objects, we assume that: A1: O• is valid and A2: •O is valid. Three rules can be used: R1 : For any TOY w and v, if wv and vw are valid then w is valid R2 : For any TOY w and v, if w and v are valid then w•v is valid R3 : For any TOY w and v, if wv• is valid then wO is valid

Prove the following theorems: Thm

A: • is valid; Thm B: ••• is valid; Thm C: ••O is valid.
In Part II, the students are asked to write the three following proofs: a familiar one with several deductive steps in calculus (familiar because learnt at the beginning of high-school in France: "for any a, b positive real numbers, prove that √(a+b)<√a+√b"); a unfamiliar one in arithmetic (but with familiar contents: "prove that the sum of two consecutive odd numbers is divisible by 4"); and a unfamiliar one in a formal, local imaginary axiomatic with non-representational symbols to see how the students make the statements work (see below). The latter is inspired by Monks quoted in Reid and Knipping (2010, p.141). The symbols do not represent anything, and the rules only concern syntax (assessing students' conceptions of this type of proof seems particularly interesting from the perspective of implementing PAs), see Table 2 above.

In Part III, the students are asked to evaluate proofs. The design of the proof proposals follows previous large-scale surveys (Stylianou et al., 2015 with undergraduate students and Healy and Hoyles, 1998 with younger students) and Balacheff (1988ab)'s characterisation of arguments (mainly naïve empiricism, generic argument and intellectual proof). The main difference lies in the choice of the mathematical domains. Four correct mathematical statements are given (both in familiar and unfamiliar fields: arithmetic 2 , geometry 3 , graph theory 4 , combinatorial geometry 5 ). For each of them, there are four proof proposals, presented in a variety of forms and arguments (following Balacheff 1988a's typology), as in Table 3 for the statement in combinatorial geometry. Some of the proof proposals are false or incomplete from a mathematical point of view. The students are asked to tick their answer(s) in multiple choices questions (adjustments of [START_REF] Healy | Justifying and proving in school mathematics[END_REF]Hoyles 1998's andStylianou et al. 2015's questions): Which answer is nearest to what you would have done? Which answer would you choose to explain the "solution" to your classmates? Which answer is the most "thorough" for you? What is/are the correct proof(s) for you? What is/are the incomplete proof(s) in your opinion?

In Part IV, the questions explore the students' global conception of proof (the way they think about proof) with open questions leading to kinds of proofs, functions of proofs, declarative students' tools to write/read/evaluate proofs, and word clouds.

Table 3: An example of Part III -Proofs in combinatorial geometry to be evaluated by the students

Answer A

Let n be the number of vertices of a convex polygon. A vertex can be joined to all vertices except itself and its two neighbours, i.e. (n-3) diagonals per vertex. The same applies to all n vertices, so there are n(n-3) diagonals. But a diagonal joins two vertices X and Y: it is therefore counted twice (once for X and once for Y). The number of diagonals in a polygon with n vertices is therefore necessarily n(n-3)/2. Answer B Suppose that there exists a polynomial function f(n) that calculates the number of the diagonals of a polygon. Suppose it is a first-degree polynomial function of the form: f(n)=an+b where a and b are real numbers. Using special cases (the square, the pentagon, the heptagon) to find values for a and b, I see that there are no values for a and b that work. Therefore, I assume that f(n) is a second-degree polynomial function with the following form: f(n)=an 2 +bn+c where a, b and c are real numbers. Using the same method as above to find the values of a, b and c, I obtain f(n)=n 2 /2-3n/2. Answer C For example, if I take a five-sided polygon: two diagonals start from a vertex. The same applies to each vertex. So, there are 10 diagonals, but I have counted them twice: there are 10 divided by 2, so, there are 5. I conclude that there are n(n-3)/2 diagonals. Answer D There are as many diagonals as edges: this property is true for the 5-sided polygon. Then, check for a polygon with a large number of edges and conclude that the number of diagonals is equal to n (where n is the number of edges).

2 "The product of three consecutive numbers is a multiple of 6." I made adjustments of Healy and Hoyles (1998).

3 "The sum of the angles of a triangle is a straight angle."

4 "The sum of the degrees of the vertices of any (finite) graph is even.". Definitions of graph, vertex and degree are given as well as an example. 5 "Let n be the number of vertices of a convex polygon (n is a strictly positive integer). Calculate the number of its diagonals and prove the formula." I made adjustments of [START_REF] Balacheff | Une étude des processus de preuve en mathématique chez des élèves de collège[END_REF].

Proposals for European collaborations

I contend that all questions of my questionnaire, mathematical contents (familiar for several years or unfamiliar but accessible with discrete concepts) and direct proofs were chosen in order to have a transferable questionnaire, for high-school, university and teacher training. Then, it is a tool to engage discussions to conduct an initial large-scale survey about the development of students' conceptions and difficulties with proof and proving from high-school to university, and also in pre-service teacher training, in Europe. Currently in a pilot testing phase, it is available in open access (in French and English). The questionnaire is designed with mathematical contents that are not specific to university. Only one proof depends on the current French high-school curricula: it can therefore be easily adapted to another country. One can hypothesise that students' conceptions are not very different across Europe. Moreover, the underlying didactic frameworks -shared by our research community -and the chosen mathematical concepts and problems follow those of Balacheff (1988ab) and the design of [START_REF] Healy | Justifying and proving in school mathematics[END_REF] partially redesigned by Stylianou et al. (2015), both in a large-scale study. Besides, a collaborative work may lead researchers to develop additional diagnostic tools to evaluate the impact of the use of specific environments (such as PAs) or interventions on students' conceptions of proof in high-school and university. This is an ambitious goal which will be enriched by working together on -a priori -shared educational problematics.

  )'s three worlds, Harel (1998)'s intellectual need, Mejia-Ramos et al. (2017)'s assessment model, Harel and Sowder (1998)'s proof schemes, Weber (2008)'s classification on proof validation, Balacheff' (1988ab)'s typology of arguments. To date, in the last 10 years, no systematic large-scale diagnostic survey has been conducted in Europe on students' conceptions of proof, particularly at the transition from secondary to tertiary education.

Table 1 : References of the 16 selected papers

 1

5 excluded articles deal with resources (syllabi, workbooks etc.) or teaching methods, task design, etc.; 3 focus on mathematical concepts but not on proof;

focus on mathematicians;

focus on teachers and 1 was off topic.


Andreas Savin

Jacek Karwowski

Second-order adiabatic connection

The adiabatic connection formalism, usually based on the first-order perturbation theory, has been generalized to an arbitrary order. The generalization stems from the observation that the formalism can be derived from a properly arranged Taylor expansion. The second-order theory is developed in detail and applied to the description of two electrons in a parabolic confinement (harmonium). A significant improvement relative to the first-order theory has been obtained. a) Electronic mail: andreas.savin@lct.jussieu.fr b) Electronic mail: jka@umk.pl

INTRODUCTION

The adiabatic connection is used in quantum mechanics to express corrections to models by progressively approaching the system of interest. [START_REF] Güttinger | Das Verhalten von Atomen im magnetischen Drehfeld[END_REF] Usually, this is formally obtained by using for each infinitesimal step the first-order perturbation theory. This paper generalizes the idea of adiabatic connection (as used in quantum mechanics) by applying it at higher orders of perturbation theory [see eq. ( 28)]. Mathematically, it corresponds to using the integral remainder in Taylor's formula. We thus expect improvement by going to higher orders.

The advantage of such an approach originates from using operators that require reduced information about the wave function. In our case, we exploit the shortrange behavior of the wave function. [START_REF] Savin | Models and corrections: Range separation for electronic interaction -Lessons from density functional theory[END_REF] As the short-range behavior has features independent of a specific (electronic) systems, it can be applied "universally", that is, in a system-independent way, in the spirit of density functional theory (DFT).

The approach retains the fundamental role of the adiabatic connection in DFT where it was used not only for explaining what the exchange-correlation density functional should do [START_REF] Gunnarsson | Exchange and correlation in atoms, molecules, and solids by the spin-density-functional formalism[END_REF][START_REF] Harris | The surface energy of a bounded electron gas[END_REF][START_REF] Langreth | The exchange-correlation energy of a metallic surface[END_REF][START_REF] Ziegler | On the calculation of multiplet energies by the Hartree-Fock-Slater method[END_REF] , but also as a guide in constructing density functional approximations (see, e.g., refs. 1 and 2). As in DFT, we need to complement by information provided by the model system. Our approach avoids certain of the limitations present in density functional theory: it is valid for any state, and it needs no fitting to systems such as the uniform electron gas. No use of the Hohenberg-Kohn theorem [START_REF] Hohenberg | Inhomogeneous electron gas[END_REF] , is made. Thus, the method presented here is not restricted to ground states.

Although generally applicable, we illustrate our method by applying it to a system of two electrons in a parabolic confinement (harmonium), as it is sufficient to illustrate the aspects mentioned above.

THEORY

Adiabatic connection

Mathematically, the idea of adiabatic connection relies on the equation

f (b) = f (a) + b a dλ f ′ (λ) (1)
where f ′ (λ) is the first derivative of the function f (λ). In quantum mechanics, the function f (λ) is often associated to an energy. For this, let us consider some model

Hamiltonian, H(µ), and the corresponding Schrödinger equation,

H(µ)Ψ(µ) = E(µ)Ψ(µ). (2)
Note that to simplify the notation, we drop the designations of coordinates and quantum numbers whenever this does not lead to misunderstandings. The quantity of interest is not the energy of the model system, E(µ), but E, the energy produced by the Schrödinger equation of the physical Hamiltonian, H. We write

H(λ, µ) = H(µ) + λ (H -H(µ)) (3)
and

H(λ, µ)Ψ(λ, µ) = E(λ, µ)Ψ(λ, µ) (4)
Note that for λ = 0 we have the model system, and for λ = 1 the physical system.

Furthermore (by first-order perturbation theory, the Hellmann-Feynman theorem),

∂ λ E(λ, µ) = ⟨Ψ(λ, µ)|H -H(µ)|Ψ(λ, µ)⟩ (5)
Applying eq. ( 1), we have an expression for E(λ, µ),

E(λ, µ) = E(µ) + λ 0 dλ ⟨Ψ(λ, µ)|H -H(µ)|Ψ(λ, µ)⟩. (6)
In particular, for λ = 1, we have the correction that added to E(µ) produces the physical energy, E

E(µ) = E -E(µ) (7)
= 1 0 dλ ⟨Ψ(λ, µ)|H -H(µ)|Ψ(λ, µ)⟩ (8)
Eq. ( 8) seems useless, as it requires the knowledge of the wave function for λ ∈ [0, 1].

However, one can exploit eq. ( 8), if the behavior of these wave functions is known for the specific domain probed by the operator H -H(µ). In this paper we consider

H(µ) = T + V + W (µ) ( 9
)
where T is the kinetic energy operator, V = N i=1 v(r i ), and W (µ) = i<j w(|r ir j |, µ) are one-and two-particle potential energy operators. Note that only the twoparticle operator is model-dependent. We choose w(r; µ) as a long-range operator, in order to have

W (µ) = H -H(µ), (10)
a short-range operator to show up in eq. ( 8), and exploit the short-range properties of the wave function. All the numerical examples below are produced with

w(r; µ) = erf(µr) r (11)
w(r; µ) = erfc(µr) r (12)

Second-order adiabatic connection

Eq. ( 1) is only a particular case of the Taylor's expansion with the integral form of the remainder,

f (b) = K-1 k=0 1 k! (b -a) k f (k) (a) + b a dλ 1 (K -1)! (b -λ) K-1 f (K) (λ) (13)
where f (k) is the k th derivative of f ; eq. ( 1) is obtained for K = 1. We consider below the case K = 2,

E = E(µ) + ∂ λ E(λ, µ)| λ=0 + 1 0 dλ (1 -λ) ∂ 2 λ E(λ, µ) (14)
In order to make the distinction between the variants of adiabatic connection, we call the usual one, eq. ( 8) first-order adiabatic connection, AC1, and that given by eq. ( 14) second-order adiabatic connection, AC2. Using higher-order adiabatic connections is possible, but they are not explored in this paper. We would like to stress that eqs. ( 8) and ( 14) are both rigorous, as long as the derivatives exist.

Note that neglecting the integral on the right-hand side of eq. ( 14) gives the firstorder perturbation theory expression, and making the approximation

∂ 2 λ E(λ, µ) ≈ ∂ 2 λ E(λ, µ)| λ=0
gives the second-order perturbation theory expression,

E ≈ E(µ) + ∂ λ E(λ, µ)| λ=0 + 1 2 ∂ 2 λ E(λ, µ) λ=0 (15)
Using ∂ λ E(λ, µ)| λ=0 in eq. ( 14) requires only using eq. ( 5) at λ = 0, that is, the model wave function, Ψ(µ),

∂ λ E(λ, µ) = ⟨Ψ(µ)|W (µ)|Ψ(µ)⟩ (16)
For obtaining the second-derivative in eq. ( 14), we use eq. ( 6),

∂ 2 λ E(λ, µ) = ∂ λ ⟨Ψ(λ, µ)|W (µ)|Ψ(λ, µ)⟩ (17)

Exploiting the short-range behavior of the wave function

In order to avoid the need of knowing Ψ for all λ in eqs. ( 16) and ( 17), we use a short-range operator W , and exploit the "universal" features of the wave function. The behavior of the model wave function approaching the physical system can be analyzed by considering the limit of large µ. [START_REF] Gori-Giorgi | Properties of short-range and long-range correlation energy density functionals from electron-electron coalescence[END_REF] In this limit, the difference between the model and physical wave function is dominated by the behavior at short electron-electron distances (as w(r; µ) differs from the Coulomb potential only in

this domain). For r = |r 1 -r 2 | → 0, Ψ(λ, µ) = ℓ N ℓ φ ℓ (r; λ, µ), ( 18
)
where ℓ is an angular quantum number related to r 1 -r 2 , and N ℓ depends on all variables except r.

Below, we use φ ℓ (r; λ, µ) that is correct to order µ -1 . It satisfies the generalization of Kato's cusp condition to w(r; µ) + λw(r; µ). [START_REF] Kato | On the eigenfunctions of many-particle systems in quantum mechanics[END_REF][START_REF] Kurokawa | General coalescence conditions for the exact wave functions: Higher-order relations for Coulombic and non-Coulombic systems[END_REF][START_REF] Pack | Cusp conditions for molecular wave functions[END_REF][START_REF] Silanes | Cusp conditions for non-Coulombic interactions[END_REF] . Its explicit form is derived in Appendix:

φ ℓ (r; λ, µ) ∝ r ℓ 1 + λ r 2ℓ + 2 + 1 -λ 2ℓ + 2 r erf(µr) + 2ℓ + 2 2ℓ + 1 e -µ 2 r 2 µ √ π (19)
+ Γ(ℓ + 3/2) -Γ(ℓ + 3/2, µ 2 r 2 ) √ π(2ℓ + 1)µ 2ℓ+2 r 2ℓ+1
where the incomplete Γ function is:

Γ(a, z) = ∞ z dt t a-1 e -t
and Γ(a) = Γ(a, 0).

In most quantum-chemical methods one considers the natural singlet and triplet pairing, corresponding to ℓ = 0, and ℓ = 1, respectively. The non-natural singlets and triplets, as they are called by Kutzelnigg and Morgan 12 , correspond to ℓ > 1.

If ℓ > 1 then the centrifugal force keeps electrons apart. As a consequence, for r → 0 the prefactor r ℓ in eq. ( 19) decreases with increasing ℓ. Consequently, the terms with small ℓ dominate in expansion (18). For the system treated in this paper (harmonium), due to separability, the sum is exactly reduced to one term and, in principle, one can study the behavior of the wave function corresponding to an arbitrary ℓ. In the remaining part of this Section the ℓ-dependence is marked explicitly: Ψ(λ, µ) is denoted as Ψ ℓ (λ, µ), and E(λ, µ) -as E ℓ (λ, µ).

The model system

All numerical results presented hereafter are obtained for a system of two electrons in a parabolic confinement (harmonium) with v = 1 2 ω 2 r 2 . If not specified otherwise, ω = 1/2 a.u. The interaction between electrons is generalized to w, eq. ( 11). The radial equation

-∂ 2 r - 2 r ∂ r + ℓ(ℓ + 1) r 2 + 1 4 ω 2 r 2 + w(r; µ) + λw(r; µ) -E ℓ (λ, µ) φ ℓ (r; λ, µ) = 0 (20)
It can be solved to arbitrary accuracy, and this allows us to judge the errors made to approximations. Furthermore, there is no need to take into account the other coordinates in the prefactor N ℓ showing up in eq. ( 18).

Working equations

In first-order adiabatic connection we approximate

∂ λ E ℓ (λ, µ) = ⟨Ψ(λ, µ)|W (µ)|Ψ(λ, µ)⟩ ≈ c ℓ I (1) ℓ (λ, µ), (21)
where c ℓ = N 2 ℓ and

I (1) ℓ (λ, µ) = ∞ 0 dr r 2 |φ ℓ (r; λ, µ)| 2 w(r; µ). (22)
In second-order adiabatic connection we approximate

∂ 2 λ E ℓ (λ, µ) ≈ c ℓ I (2) ℓ (λ, µ) (23)
where

I (2) ℓ (λ, µ) = ∂ λ I (1)
ℓ (λ, µ).

As φ ℓ (r; λ, µ) is explicitly known (in the given limit: r → 0, µ → ∞), the integrals I K ℓ are computed to be (for ℓ = 0 and ℓ = 1),

I (1) ℓ=0 ∝ 1 µ 2 + a 1 + a 2 (1 -λ) µ 3 + a 3 + a 4 (1 -λ) + a 5 (1 -λ) 2 µ 4 (25)
I (1) ℓ=1 ∝ 1 µ 4 + b 1 + b 2 (1 -λ) µ 5 + b 3 + b 4 (1 -λ) + b 5 (1 -λ) 2 µ 6 (26)
where the constants a k and b k are collected in table I.

The proportionality constant, c, is related to the physical system and can be determined by using eqs. ( 21) and (23) for the model system (at λ = 0). We thus get the following working equations: and

E ℓ ≈ E ℓ (µ) + α (1) ℓ (µ) ∂ λ E ℓ (λ, µ)| λ=0 (27)
E ℓ ≈ E ℓ (µ) + ∂ λ E ℓ (λ, µ)| λ=0 + α (2) ℓ (µ) ∂ 2 λ E ℓ (λ, µ) λ=0 (28)
with

α (K) ℓ (µ) = 1 0 dλ (1 -λ) (K-1) I (K) ℓ (λ, µ) I (K) ℓ (λ = 0, µ) (29)
The integrals over λ are trivial, and not shown. Eq. ( 27) corresponds to the firstorder adiabatic connection, while eq. ( 28) corresponds to second-order adiabatic connection. Note that expressions ( 27) and (28) require the same effort as the firstand second-order perturbation theory, respectively, namely computing

∂ λ E ℓ (λ, µ) λ=0 and ∂ 2 λ E ℓ (λ, µ)| λ=0 .
Only the weight of the last term is changed by α (K)

ℓ , K = 1 or 2. Two more remarks on these equations. First, note that by squaring φ ℓ in eq. ( 22) we introduce terms in µ -2 , although a further term to this order may be present in an exact theory, because φ ℓ is constructed to order µ -1 only. Second, higher orders in the adiabatic connection do not improve over perturbation theory with the present form of φ ℓ : in our first-order expression in 1/µ, only terms linear in λ show up. The second derivative is just a constant, and α

(3) ℓ = 1.
Eq. ( 27) can be rewritten as

E ℓ ≈ ⟨Ψ(µ)|H|Ψ(µ)⟩ + α (1) ℓ (µ) -1 ∂ λ E ℓ (λ, µ)| λ=0 ( 30
)
The last term on the r.h.s. appears as approximation for the correlation energy.

One can also write eq. ( 28), as

E ℓ ≈ ⟨Ψ(µ)|H|Ψ(µ)⟩ + E (2) ℓ (µ) + α (2) ℓ (µ) - 1 2 ∂ 2 λ E ℓ (λ, µ) λ=0 ( 31
)
where E

(2)

ℓ (µ) = 1/2 ∂ 2 λ E ℓ (λ, µ)| λ=0
is the second-order energy correction to E ℓ (µ). The last term on the r.h.s. appears as approximation for the remaining correlation energy.

NUMERICAL RESULTS

General considerations

By construction, the approximations leading to the working equations ( 27) and (28) become exact in the limit of the model system approaching the physical system (µ → ∞). However, the cost of performing the calculation of the model system also increases with µ. We are thus interested to use models with small µ, ideally even totally turn off the interaction (µ = 0). The derivation does not tell us how well the approximations used work for small µ. The figures shown in this paper present the errors of the approximation (the difference between the energy obtained using it and that of the physical system) for different models µ. As µ can vary continuously, the plots of the errors show up as curves. We consider µ ∈ [0, 3], as E(µ) approaches anyhow the physical energy E for large µ.

With these approximations, we aim to reach the so-called chemical accuracy, i.e., 1 kcal/mol. [START_REF] Pople | Nobel lecture: Quantum chemical models[END_REF] The region of chemical accuracy is marked in the plots by horizontal dashed lines.

Discussion of the figures

Fig. 1 shows the general trends of the approximations. One can see, that by the choice of using the bare field, v, for all values of µ, cf. eq. ( 9), the error of the model in the limit of small µ is catastrophic (0.5 hartree). First-order perturbation theory leads to the expectation value of the physical Hamiltonian, and thus gives an upper bound to the exact energy. In spite of using a bad wave function, the improvement is important: the error is decreased by an order of magnitude to ≈ 0.06 hartree.

Hartree-Fock, the best value that can be obtained for the non-interacting wave function, is in error by ≈ 0.04 hartree. Second-order perturbation theory further improves the result.

The range of chemical accuracy cannot be distinguished on the scale of the plot shown in fig. 1. In order to discuss the suitability of the results, we have to zoom in (see fig. 2,top). We notice that for the ground state (a singlet state, ℓ = 0), the model does not reach chemical accuracy for the whole range of µ shown in the plot. The full black curve indicates the error of the model, E(µ). Blue curves correspond to first-order: dashed curve for perturbation theory, ⟨Ψ(µ)|H|Ψ(µ)⟩, full curve for first-order adiabatic connection, eq. ( 27). Green curves correspond to second-order: dashed curve for perturbation theory, eq. ( 15), full curve for second-order adiabatic connection, eq. ( 28).

The result of the first-order perturbation theory is improved by using the correction established for µ → ∞, eq. ( 27). Surprisingly, in spite of using an expansion in 1/µ, the error remains negligible even for some range of µ < 1 bohr -1 (to ≈ 0.5 bohr -1 ). However, at smaller µ it worsens dramatically. Going over to the second-order perturbation theory improves over the first-order perturbation theory. Correcting asymptotically, eq. ( 28) improves the result over the whole range of values. However, chemical accuracy is not yet reached for all models.

Let us now apply the method to excited states. The model works much better for first triplet excited state (ℓ = 1), see fig. 2 (middle panel). This improves the results for all approximations.

Considering the first singlet excited state with ℓ = 0, fig. 2 (bottom), we notice that the quality of the model is worse than for the ground state. The corresponding curve does not even show up in the plot. However, the approximations show a behavior that roughly follows that of the ground state. We would like to stress that the corrections for the ground state and for the first excited state with ℓ = 0 use the same factors α (K) ℓ , eq. ( 29). The corrections are, of course, different, as the systemspecific information enters through the model-specific quantities, ∂ K E ℓ (λ, µ)| λ=0 . The full black curve indicates the error of the model, E(µ). Blue curves correspond to first-order: dashed curve for perturbation theory, ⟨Ψ(µ)|H|Ψ(µ)⟩, full curve for first-order adiabatic connection, eq. ( 27). Green curves correspond to second-order: dashed curve for perturbation theory, eq. ( 15), full curve for second-order adiabatic connection, eq. ( 28).

Top panel: ground state; middle panel: first excited state with ℓ = 1; bottom panel: first excited state with ℓ = 0.

CONCLUSION

The integral form of the remainder in Taylor's expansion, eq. ( 13) provides a formula that generalizes the adiabatic connection. We use it to construct approximations to correct energies produced by model Hamiltonians with long range interaction, eqs. ( 27) and (28). They are inspired by approximations used in density functional theory. However, they do not use the Hohenberg-Kohn theorem, and are valid for the ground and excited states. Instead, they are constructed to be valid for the short-range interaction, as one approaches the physical system.

Results are shown only for harmonium. One can notice an improvement with increasing order of the adiabatic connection. No comparison is made with analogous density functional approximations. These can be found in ref. 17.

Of course, one would like to apply the method not only to harmonium. In other systems the sum in eq. ( 18) is extended over all values of ℓ. An explicit treatment of the higher terms is difficult, may be too expensive computationally, and is maybe not needed -as it was alredy discussed, these terms are usually much smaller than the leading one. Thus, techniques such as described in refs. 14, and 20 could be applied.

A strength of the method presented here is the stability of the result for large µ.

Once the stability is lost, it may indicate a worsening of the approximation.

A weakness of the method presented here is that it does not work sufficiently well (within chemical accuracy) for the non-interacting system (µ = 0). This makes the method more expensive. This feature is also present in range-separated density functional methods which possess a striking similarity with the one presented here. One may wonder if the experience gained by constructing density functional approximations cannot be exploited here, too. In particular, a properly constructed effective one-body potential in the model Hamiltonian can reduce the energy error of the physical (interacting) system (see, e.g., figs. 11 and 12 in ref. 17). Alternatively, the method presented here could be used to improve density functional approximations.

Finally, we would like to point out that eq. ( 28) can be seen as a theoretical justification for spin-component-scaled methods

  Constants in eqs. (25) and (26) rounded up to five decimals.

FIG. 1 .

 1 FIG. 1. Errors of different approximations for the two-electron harmonium (ω = 1/2).

FIG. 2 .

 2 FIG. 2. Errors of different approximations for the two-electron harmonium (ω = 1/2).

  [START_REF] Grimme | Spin-component-scaled electron correlation methods[END_REF] .

	r → 0,
	φ ℓ (r; λ, µ) ∝ r ℓ 1 +	µ	1 -λ π(2ℓ + 1) √	+	λ r 2ℓ + 2	.	(A.24)

ACKNOWLEDGEMENT

We dedicate this paper to John P. Perdew who transformed the adiabatic connection to a useful tool not only in understanding DFT, but also for constructing approximations that changed the impact of computational chemistry and physics.

Appendix: A derivation of equation (19) We generalize for an arbitrary ℓ the results obtained for ℓ = 0 in Section III of ref.

3, and in Appendix of ref. 18. We consider the behavior at the limit of large µ of the radial wave function φ(r; λ, µ) which is a solution of the radial Schrödinger equation ( 20)

where

is the radial part of the kinetic energy operator corresponding to a given ℓ,

is the interaction potential, and R(r) is finite at r = 0.

After changing variable r to x = µ r and defining

eq. (A.1) becomes

To solve eq. (A.5) we use the first-order perturbation theory with perturbation parameter 1/µ. We set

where u

ℓ (x; λ) are solutions of the zeroth and the first order perturbation equations:

and

The general solution of the second-order homogeneous differential equation (A.7) is a linear combination of its two independent solutions: f 1 (x) = x ℓ , and

The inhomogeneous equation (A.8) is solved in quadratures:

ℓ (λ) -

where

is the Wronskian of solutions of the homogeneous equation.

According to eqs. (A.3) and (A.9), F (x; λ) = (1 -λ) F (x; 0) + λ F (x; 1). (A.12)

Consequently, from eq. (A.11) we have

ℓ (x; 0) + λ u

ℓ (x; 1). (A.13) Notice that also

where

ℓ (1) depend on neither λ nor µ. The integration constants A (i) ℓ and B (i) ℓ , i = 1, 2, are determined from the requirement that the wave function fulfills the coalescence conditions [START_REF] Kurokawa | General coalescence conditions for the exact wave functions: Higher-order relations for Coulombic and non-Coulombic systems[END_REF][START_REF] Savin | Correcting models with long-range electron interaction using generalized cusp conditions[END_REF] . In particular, neither u

ℓ (x; λ) can be singular at x = 0. The last requirement implies that A (2) = 0.

The evaluation of λ = 1 contribution is straightforward. We have

ℓ (1)

On the other hand, according to eq. ( 26) of ref. 18,

(the Kato's cusp condition 10 ). Therefore, in order to recover the correct behavior of the Coulomb radial function at small x, we have to set B

(1)

ℓ (1) = 0. Finally, according to eqs. (A.6), (A.10), and (A.15),

The second contribution, corresponding to λ = 0, is more difficult to calculate. At the limit of large µ, and for sufficiently large r, φ(r; 0, µ) → φ(r; 1, µ) [START_REF] Gori-Giorgi | Properties of short-range and long-range correlation energy density functionals from electron-electron coalescence[END_REF] To meet this condition we have to set B

ℓ (0) = 0. By fixing

By expanding the right-hand side of eq. (A.18) to a power series of x, we get

Alternatively, eq. (A. 19) can be obtained by the expansion of F (x, 0) and subsequent evaluation of integrals in eq. (A.11). Notice, that the integration constants in both approaches are different.