Constance Lecourt

Raquel Utrera Melero

Jana Schiller

Florent Moutier

Vincent Dorcet

Guillaume Calvez

Corentin Poidevin

Karine Costuas

Manfred Scheer

Christophe Lescop

Phase dependence and mechanical and thermal ductility of the luminescence properties of tetranuclear Cu( i ) metallacycle assemblies stabilized by ditopic organo

Introduction

An ever-rising attention is devoted to the study of luminescent Cu(I)-based derivatives as new photo-functional molecular materials for emitting devices or/and luminescent sensors. Such derivatives offer particularly appealing perspectives for future developments, not only due of the relatively high abundance and low cost of copper offering a competitive alternative to the well-known phosphorescent compounds based on expensive heavy metal complexes, but also regarding the various photophysical properties these Cu(I) complexes exhibit. Indeed, emission covering the full visible light spectrum have been reported for such compounds, combined with a large diversity of radiative relaxation processes, including very efficient Thermally Activated Delayed Fluorescence (TADF) processes. [START_REF] Yersin | Highly Efficient OLEDs: Materials Based on Thermally Activated Delayed Fluorescence[END_REF][START_REF] Czerwieniec | Cu(I) complexes -Thermally activated delayed fluorescence. Photophysical approach and material design[END_REF] In such complexes presenting TADF properties, spatially well-separated HOMO and LUMO orbitals induce small energy gap between the lowest energy singlet-triplet states. This enables a thermal population of the lowest energy excited singlet state (S 1 ) from the lowest energy triplet excited state (T 1 ), via a reverse intersystem crossing (RISC). Consequently, efficient radiative relaxation processes from this thermally populated S 1 state to the ground state S 0 can take place allowing exalted solid-state luminescence properties

Scheme 1 Molecular structure of the derivatives 1, 49 2, 50 3 51 and 4. [START_REF] Galimova | Luminescent Cu 4 I 4cubane clusters based on N-methyl-5,10-dihydrophenarsazines[END_REF][START_REF] Galimova | Cu 4 I 4 -cubane clusters based on 10-(aryl)phenoxarsines and their luminescence[END_REF] close to room temperature (RT). Importantly, in Cu(I)polynuclear systems, non-covalent cuprophilic interactions characterized by short intermetallic distances (d(Cu-Cu) bellow ca. 2.9 Å) [START_REF] Harisomayajula | Cuprophilic Interactions in and between Molecular Entities[END_REF] can be found in the ground state and/or in the excited state relaxed geometries. [START_REF] Sayed Moussa | A solid state highly emissive Cu(I) metallacycle: promotion of cuprophilic interactions at the excited states[END_REF][START_REF] Bassoli | Phosphorescence enhancement by close metal-metal interaction in T1 excited state in a dinuclear copper(I) complex[END_REF] Such interactions induce an enhancement of the spin-orbit coupling (SOC) constants compared to the one observed with non-interacting Cu(I) ions for which intrinsic individual SOC is generally moderate. Consequently, the radiative rate constants for the T 1 state are increased, promoting efficient RISC to the S 1 state but also a fast and efficient phosphorescent relaxation process to the ground state (S 0 ). 48 In order to prepare emissive Cu(I) assemblies bearing larger SOC constant value and likely improved radiative relaxation rates and luminescence properties, one approach could consist in the introduction of heavy donor elements bearing high individual SOC values in the coordination sphere of the Cu(I) ions. Interestingly, a general synthetic strategy toward highly luminescent Cu(I)-based systems relies on the use of aromatic organophosphorus ligands that confer a remarkable stability to Cu(I) complexes but also, very frequently, generate electronic configuration promoting emergence of TADF properties. In a marked contrast, the use of analogous heavier pnictogenbased ligands, in particular arsines, remains particularly scarce while such ligands could be advantageous over lighter pnictogen donors due to higher intrinsic SOC (such as ξ = 1202 cm -1 for As atom while ξ = 230 cm -1 for P atom). [START_REF] Montalti | Handbook of photochemistry[END_REF] Thus, Artem'ev et al. described recently the preparation and the photophysical studies of a series of complexes 1 (Scheme 1) in which CuX (X = Cl, Br, I) sub-units are stabilized by the bis(2pyridyl)phenylarsine ligand. [START_REF] Artem'ev | Pyridylarsine-based Cu(I) complexes showing TADF mixed with fast phosphorescence: a speeding-up emission rate using arsine ligands[END_REF] It was reported that these derivatives present short-lived luminescence at RT assigned to a combination of fast phosphorescence and TADF processes. Interestingly, these dinuclear Cu(I)-arsine complexes displayed significantly shorter decay times than their phosphine analogous, which was tentatively assigned to a consequence of the higher SOC strength carried by the arsenic atoms versus phosphorus atoms. in 2020 Naka et al. reported the study of 'Cu 2 I 2 '-based bimetallic phosphorescent complexes 2 (Scheme 1) stabilized by the triphenylarsine ligand and showing mechanochromic luminescence properties. [START_REF] Kobayashi | Stimuli-Responsive Emission of Dinuclear Rhombic Copper(I) Iodide Complexes Having Triphenylarsine and N-Heteroaromatic Co-Ligands[END_REF] In 2021, this group also reported a comparative study of the effect of a arsenic ligand (bis(diphenylarsino)methane) versus its phosphorus-based congener (bis(diphenylphosphino)methane, dppm) on the formation of inter-and intramolecular metallophilic interactions in Au(I)-Cu(I) ionic-pair luminescent derivatives 3 (Scheme 1). [START_REF] Kobayashi | Homo-and hetero-metallophilicity-driven synthesis of highly emissive and stimuli-responsive Au(I)-Cu(I) double salts[END_REF] In These studies triggered our interest for exploring such potential impact of the replacement of phosphorus donor atoms by arsenic donor atoms in ligands coordinated to Cu(I) metal centers on the alteration of photophysical properties of the resulting complexes. Herein, we report the preparation of a tetrametallic Cu(I) metallacycle A (Scheme 2) stabilized by the ''mixed'' arsenic-phosphorus bridging ligand [(diphenylarsino)methyl]diphenylphosphane L As,P (Scheme 2a). This tetrametallic assembly is obtained considering an adaptation of the coordination-driven supramolecular synthetic approach. Its synthesis, characterization, X-ray single crystal molecular structure and solid-state luminescence properties are reported and compared to its analogous tetrametallic compound B (Schemes 2b) previously published by our group, based on the diphosphine dppm ligand. 32 Density functional theory (DFT) calculations reveal that the SOC of the heavier arsenic element does not show a direct influence on the photophysical properties in A compared to the derivative B. However, the establishment of structureproperty relationships reveals an impact of the L As,P ligand on the crystalline packing of A compared to the organization observed in the crystals of the lighter congener species B. This raises the question of the relative impacts of the SOC values on the luminescence efficiencies of these derivatives versus the extended supramolecular organizations in the crystalline solidstates. In line with this point, luminescent thermochromism and mechanochromism are observed and analyzed for both A and B assemblies, that moreover also display in the solid-state at high temperature an original irreversible thermal crystalline transition that impacts their photophysical properties. All in all, this study highlights the subtle and competitive effects that can rule out the photophysics of Cu(I) polymetallic assemblies based on organo-pnictogen ligands and reveals their stimulisensitive behaviours, allowing the facile preparation of different phases bearing contrasted efficient luminescence properties.

Results and discussion

Preparation and characterization of the tetrametallic metallacycle A

The high flexibility presented by Cu(I) metal complexes was in general initially considered as being very detrimental in the perspective of designing highly luminescent Cu(I) complexes. Indeed, it allows easy geometrical re-organization processes to be activated in the excited states, connected with fast nonradiative decay rates and concomitant small RT emission quantum yields. Such unfavorable situation was firstly avoided by coordinating Cu(I) ion with bulky chelating ligands. [START_REF] Czerwieniec | Cu(I) complexes -Thermally activated delayed fluorescence. Photophysical approach and material design[END_REF] As a result, little freedom for distortions in the exited state is imposed to the Cu(I) ion's coordination sphere and RT photoluminescence quantum yields as high as the unity can be measured in particular in systems where TADF relaxation processes take place. In an alternative approach, in 2016 Yam and Lescop described the selective preparation of the sterically constrained highly emissive tetrametallic Cu(I) metallacycle B (Scheme 2b) from the self-assembly reaction of the preassembled bimetallic precursor [Cu 2 ( 2 -dppm) 2 ] 2+ with cyanide connecting ligands. [START_REF] Sayed Moussa | A solid state highly emissive Cu(I) metallacycle: promotion of cuprophilic interactions at the excited states[END_REF] Within this tetrametallic assembly bearing multiple chelates and metallacycle subunits, the Cu(I) ions' coordination sphere can undergo only reduced structural re-organization alteration in the excited states at the time scale of the photoexcitation events, due to the high energetic cost of such process over the whole constrained polycyclic architecture. Consequently, non-radiative deactivation pathways are disfavored which promotes emergence of enhanced solid-state photophysical properties in these selfassembled derivatives, also likely favored by the cuprophilic interactions observed within this assembly (d(Cu-Cu) = 2.866(9) Å in the [Cu 2 ( 2 -dppm) 2 ] 2+ subunits). Indeed, the derivative B reveals to be an efficient TADF luminophore, emitting in the solid state at RT upon photoexcitation an eyeperceived blue light with a remarkable emission quantum yield of 72 %. This result motivated us to replace the chelating dppm ligand by its L As,P analog ligand and to explore whether similar selfassembling could take upon its combination with Cu(I) metal centers and cyanide ligands. The synthesis of the L As,P chelating ligand was conducted based on an adaptation of a previously reported procedure, [START_REF] Dube | α-Cationic Arsines: Synthesis, Structure, Reactivity, and Applications[END_REF][START_REF]For experimental details, spectroscopic, X-ray diffraction data and computational studies views see ESI[END_REF] H} solution NMR spectrum of derivative A shows a singlet at -9.45 ppm (Fig. S1). This signal is deshielded compared to the noncoordinated L As,P ligand ( = -21.6 ppm, Fig. S1), suggesting that a coordination occurred with the Cu(I) ions. This chemical shift is consistent with the two broad singlets (-8.5 ppm and -10.5 ppm) observed in the RT 31 P{

1 H} spectrum of the metallacycle B, supporting a similar molecular structure. [START_REF] Sayed Moussa | A solid state highly emissive Cu(I) metallacycle: promotion of cuprophilic interactions at the excited states[END_REF] The presence of PF 6 ˉ counter-anions is confirmed by a septet at -144.4 ppm in the 31 P{ 1 H} spectrum (and by the typical (PF 6 ˉ) vibrational band at 834 cmˉ1 in the infrared spectrum (Fig. S4).

The RT 1 H solution NMR spectrum revealed the substitution of the acetonitrile ligands as no peak assigned to this fragment is detected (Fig. S1). This is in accordance with the solid-state infrared spectrum of the derivative A in which the typical signature of the (C≡N) stretching mode of the acetonitrile ligand [START_REF] Kritchenkov | Efficient one-pot green synthesis of tetrakis(acetonitrile)copper(i) complex in aqueous media[END_REF] coordinated to a Cu(I) metal center at 2311 cm -1 is absent while a (C≡N) vibrational band at 2118 cmˉ1 characteristic of a Cu(I)-C≡N-Cu(I) fragment is observed (Fig. 1), at a similar energy than reported for the analogous metallacycle B.

32

Crystal structure of the tetrametallic metallacycle A

The molecular structure of the derivative A was determined by single crystal X-ray diffraction study at 150 K. [START_REF]For experimental details, spectroscopic, X-ray diffraction data and computational studies views see ESI[END_REF] The ORTEP diagram is represented in Fig. S10 and all the coordination sphere parameters are reported in Table S1

-S3. A [Cu 4 (µ 2 - L As,P ) 4 (CN) 2 ](PF 6
) 2 ] assembly is revealed, having a tetrametallic scaffold similar to that of the compound B (Fig. 2). However, while derivative B crystallizes in the P2 1 /n space group of the monoclinic system, [START_REF] Sayed Moussa | A solid state highly emissive Cu(I) metallacycle: promotion of cuprophilic interactions at the excited states[END_REF] the assembly A crystallizes in the P-1 space group of the triclinic crystal system (Table 1). 2+ fragment are crystallographically different and are denoted Cu(1) and Cu(2), Cu(1) being located at vertices of the acute angles of the metallacycle scaffold and Cu(2) at vertices of the obtuse ones. The four Cu(I) centers forming the metallacycle lie in the same plane. The carbon and nitrogen atoms of the cyanide bridging ligand are found to be disordered on two atomic positions for which equal occupancies were assigned. A similar treatment was applied for the arsenic and phosphorus atoms of the L As,P ligand since they are determined being equally distributed on both atomic positions corresponding to the donor atoms of the assembling L As,P ligands coordinated to the metal centers. Each Cu(I) ion is coordinated to two donor atoms (phosphorus and/or arsenic atoms) from two distinct L As,P ligands and to one CNˉ ligand, resulting in a trigonal planar geometry characterized by six possible coordination sphere environments (P 2 (N or C) 1 , As 2 (N or C) 1 or P 1 As 1 (N or C) 1 ). For the sake of simplicity in the description of the metric parameters of the inorganic metallacyclic 'Cu 4 (CN) 2 ' backbone of the assembly A and by analogy with the conclusions that were obtained from geometry optimization calculations based on DFT considering the possible arrangements of the carbon and nitrogen atoms of the cyano units in the molecular scaffold of the derivative B, [START_REF] Sayed Moussa | A solid state highly emissive Cu(I) metallacycle: promotion of cuprophilic interactions at the excited states[END_REF] it will be considered that the nitrogen atoms of the cyano linkers are coordinated on the Cu(2) 'obtuse' metal center. Finally, in the solid-state crystal packing, metallacycles A share -CH and - interactions with neighboring assemblies involving the phenyl rings of the L As,P ligands. Moreover these discrete scaffolds are separated by the PF 6 ˉ counter-anions and included CH 2 Cl 2 molecules that occupy the cavities lying between the tetrametallic units A. In this triclinic P-1 packing, all metallacycles are orientated parallel ( Fig. 3). This is in a marked contrast with the monoclinic P2 1 /n packing observed in the crystals of the congener derivative B in which two markedly different orientations of the metallacycle are alternating ( Fig. 3).

Photophysical properties

The RT solid-state absorption spectrum of derivative A shows a broad band in the UV region centered at 291 nm and a shoulder at 350 nm, typical of * transition centered on the aromatic moieties of the L As,P ligands (Fig. S13). The excitation spectrum at room temperature shows a large excitation domain from 300 nm to 365 nm with an intensity maximum centered at 356 nm (Fig. S15).

Fig. 4 Temperature dependent (80-300 K) normalized emission spectra of derivatives A, Ag and A' upon excitation at 356 nm, 372 nm and 365 nm, respectively, and of derivatives B, [START_REF] Sayed Moussa | A solid state highly emissive Cu(I) metallacycle: promotion of cuprophilic interactions at the excited states[END_REF] Bg and B' upon excitation at 330 nm, 374 nm and 325 nm, respectively.

Upon excitation at 356 nm at 300 K, a polycrystalline batch of compound A exhibits a cyan eye-perceived solid-state luminescence (Fig. S12), characterized by a single broad unstructured emission band (FWHM = 3713 cmˉ1, Fig. 4) centered at 456 nm together with biexponential decay times of  1 = 38 µs (58% relative rate) and  2 = 16 µs (42% relative rate) and an emission quantum yield (EQY) of 84 %. Upon cooling from 300 K to 80 K, the emission spectra of the molecular assembly A show a gradual and almost linear redshift ( em(300K-80K) = 0.176 eV (1420 cmˉ1), Fig. 4) associated with an increasing signal intensity (Fig. S16). At 80 K, an intense greenish eye-perceived emission is observed characterized by a large unstructured emission band (FWHM = 2344 cmˉ1) centered at 483 nm, while the emission lifetimes  1 (65%) and  2 (35%)increases up to 224 µs and 111 µs, respectively. The temperature dependent shift of the solid-state emission spectra of A is completely reversible and the cyan luminescence observed at 300 K is recovered once the sample is warmed up. The detailed study of the temperature dependence of the excited state emission lifetime of A (Fig. S22) reveals biexponential radiative decay all along the temperature range from 300 K to 80 K. The thermalvariation profiles of  1 (long component) and  2 (short component) are reproducibly irregular. Likely, such a behavior can be assigned to the variety of the molecular conformations that can be generated due to the relative different localizations of the nitrogen and carbon atoms of the cyano ligands and/or of the phosphorus and the arsenic atoms of the L As,P ligands in the tetrametallic metalacyclic backbone (affording six different possible coordination spheres for each metal center). For all of these possible conformers of compound A, different radiative relaxation emission processes probably coming from various excited emissive states (see quantum chemical calculation section) might be implied at each temperature. This renders tricky and questionable a fine assignment of the origin of the luminescence properties of compound A. However, all in all, the gross photophysical properties recorded for compound A probably emerge from an admixed between pure phosphorescence and TADF processes, this assumption being also supported by the TADF process highlighted as dominant radiative relaxation pathway in the case of the parent compound B. The photophysical properties of the analogous compound B were previously published [START_REF] Sayed Moussa | A solid state highly emissive Cu(I) metallacycle: promotion of cuprophilic interactions at the excited states[END_REF] and compare mostly to the ones characterized in this study for the derivative A. Indeed, metallacycle B shows a clear blue eye-perceived luminescence under excitation at 320 nm, centered at 449 nm at room temperature with biexponential decay times of 40 and 10 µs and an EQY of 72 % (Table 2). Upon cooling from 300 K to 80 K, a gradual redshift of the whole spectrum (Fig. 4) is also observed affording an emission maximum at 475 nm, together with an increase of the biexponential decay times reaching 200 µs and 40 µs at 80 K. The temperature-dependent emission of the derivative B was assigned to a TADF mechanism. Such similarities highlighting only minor differences in the photophysics of derivatives A and B are quite puzzling, regarding previous results published in the literature, comparing phosphorus-based and arsenic-based related compounds that mostly reported different luminescence properties. [START_REF] Artem'ev | Pyridylarsine-based Cu(I) complexes showing TADF mixed with fast phosphorescence: a speeding-up emission rate using arsine ligands[END_REF][START_REF] Kobayashi | Stimuli-Responsive Emission of Dinuclear Rhombic Copper(I) Iodide Complexes Having Triphenylarsine and N-Heteroaromatic Co-Ligands[END_REF][START_REF] Kobayashi | Homo-and hetero-metallophilicity-driven synthesis of highly emissive and stimuli-responsive Au(I)-Cu(I) double salts[END_REF][START_REF] Galimova | Luminescent Cu 4 I 4cubane clusters based on N-methyl-5,10-dihydrophenarsazines[END_REF] Regarding these unexpected results, theoretical molecular calculations were conducted to get insights into the influence of the substitution of one phosphorus atom by one arsenic atom on the molecular structures, the electronic properties and the photophysical properties of the derivatives A and B.

Quantum chemical calculations

Quantum chemical calculations have been performed employing DFT and time-dependent DFT (TD-DFT). The computational details are provided in the ESI. Full (A opt-full (S 0 ), B opt-full (S 0 )) and partially constrained (A opt-const (S 0 ), B opt-const (S 0 )) geometry optimizations in vacuum of compounds A and B in their ground state and full optimizations of their first singlet and triplet excited states (S 1 , T 1 ) were performed (Tables S4-S6, Fig. S31). Electronic excitations (absorption properties; Table S7 and Fig. S32-S33) and excited states properties (emission; Table S8 and Fig. S34) were further simulated. Considering the possible conformers resulting from the different relative orientations of the cyano and the L As,P ligands in the coordination spheres of the Cu(I) ions, the positioning of the carbon atoms versus the nitrogen atoms in the cyano units, and of the phosphorus and arsenic atoms in the L As,P pnictogen ligands have been studied. The most stable arrangement corresponds to the configuration shown in the Fig. 2d with a centrosymmetric architecture having the nitrogen atoms of the cyano ligands coordinated on the Cu(I) obtuse metal center of the metalacyclic 'Cu 4 (CN) 2 ' backbone [START_REF] Sayed Moussa | A solid state highly emissive Cu(I) metallacycle: promotion of cuprophilic interactions at the excited states[END_REF] and a phosphorus and an arsenic atom coordinated to each metal centers. This arrangement was kept for the rest of the computational study.

Regarding the A opt-full (S 0 ) and B opt-full (S 0 ) geometries, the gross metric data compare well with X-ray diffraction structures with deviations of less than 1.4 % on the bond distances. However, the metallacycle of the fully optimized structure A opt-full (S 0 ) is not strictly planar with about 6° of deviation from planarity whereas the B opt-full (S 0 ) metallacycle is quasi planar. Moreover, these fully-optimized geometries show shorter Cu⋯Cu distances than the X-ray structures (Table S5). These discrepancies strongly indicate that experimental metric data of the internal 'Cu 4 (CN) 2 ' metalacyclic backbones are influenced by the intermolecular environment. Indeed, previous investigations conducted on related luminescent 'Cu 6 ' and 'Cu 8 Pd 1 ' assemblies have revealed that the crystal packing greatly influences the solid-state molecular conformations observed in single-crystals and therefore the metric data and the solid-state photophysical properties of these derivatives.

58,59

Taking these aspects into account, indirect simulations of these packing were performed by constraining the copper atomic positions in the metallacycles to those of the X-ray structures. These partially constrained geometry optimizations lead to A opt-const (S 0 ) and B opt-const (S 0 ). The main structural differences between the full and constrained geometries are observed within the metallacycle with an interaction only occurring in the fully optimized structures between one of the two copper atoms and the carbon atom of the neighboring cyano groups (Fig. 2d andS31). This leads to the shortening of the Cu⋯Cu distances of 0.24 Å and 0.19 Å respectively observed in A opt-full (S 0 ) and B opt-full (S 0 ) compared to the X-ray structural data. Yet, these differences between the constrained and unconstrained geometries affect only moderately the electronic structures and the Franck Condon (FC) excited states in both systems (Table S7 and Fig. S32-S33). The lowest excited states T 1 -FC and S 1 -FC are MLCT character for both systems as it was already pointed out in the previous communication for B. [START_REF] Sayed Moussa | A solid state highly emissive Cu(I) metallacycle: promotion of cuprophilic interactions at the excited states[END_REF] Interestingly, no major difference in the optical properties is calculated between A (mixed arsenic and phosphorus) and B (only phosphorus) systems. Spin-orbit coupling between S 1 -FC and T 1 -FC is moderate in all cases (6 to 10 cm -1 ). Therefore, the substitutions of phosphorus by arsenic atoms do not lead to the strengthening of the calculated SOC values as it was initially anticipated. [START_REF] Artem'ev | Pyridylarsine-based Cu(I) complexes showing TADF mixed with fast phosphorescence: a speeding-up emission rate using arsine ligands[END_REF][START_REF] Kobayashi | Stimuli-Responsive Emission of Dinuclear Rhombic Copper(I) Iodide Complexes Having Triphenylarsine and N-Heteroaromatic Co-Ligands[END_REF][START_REF] Kobayashi | Homo-and hetero-metallophilicity-driven synthesis of highly emissive and stimuli-responsive Au(I)-Cu(I) double salts[END_REF][START_REF] Galimova | Luminescent Cu 4 I 4cubane clusters based on N-methyl-5,10-dihydrophenarsazines[END_REF][START_REF] Galimova | Cu 4 I 4 -cubane clusters based on 10-(aryl)phenoxarsines and their luminescence[END_REF] The intersystem crossing is thus susceptible to be similar in both systems. TD-DFT geometry relaxations of these excited states were performed. The most important structure features rely in all systems on the significant shortening of one or two Cu⋯Cu distances (Table S6) and the increased bending of Cu-cyano angles, leading to less symmetrical rectangular shape of the metalacyclic backbones. Interestingly enough, the Cupnictogen bonds are hardly affected upon relaxation after excitation. The main energies and characteristics of these fully optimized excited states are reported in Table S8. The SOC at the geometry of T 1 between T 1 and S 0 is slightly smaller than between S 1 and T 1 , which leads to a long lifetime for T 1 if only spontaneous emission is considered (from 1 to 4 ms, Table S8).

Considering that our calculations are done at 0 K and the level of theory used, the agreement between calculated phosphorescent and low temperature experimental emission wavelengths is satisfactory when considering the ground phases A g and B g luminescence properties (vide supra) as experimental solid-state phase references. This set of calculated data reveals, in these Cu(I) polymetallic assemblies bearing pnictogen ligands, that the substitution of a phosphorus atom by an arsenic atom does not imply systematically an alteration of the SOC values. Indeed, if the electronic density lying on the pnictogen atoms is not implied in the electronic excitation processes, the impact of the substitution is inoperant. Importantly, in such situation, the intermolecular constraints applied on the gross metalacyclic backbone can induce subtle but noticeable structural alteration impacting optical changes, that may differentiate the photophysics of the assemblies regarding the pnictogen atom present, in particular in the case of congener species bearing different crystal packing organizations. Therefore, the emission properties of such assemblies are highly related with their molecular environment alteration, and can probe the application in the solid-state of diverse external stress offering potential access to attractive stimulisensitive luminescent molecular materials.

Stimuli-sensitive behavior

In a very general point of view, Cu(I) coordination complexes are endowed with labile and non-directional coordination spheres. This feature renders the gross conformation of Cu(I) species very sensitive to external stimuli and, concomitantly, photophysical properties can be impacted, affording potential new stimuli responsive multifunctional luminescent materials. Among the different classes of photochromic materials that exhibit promising potentials in versatile light emission related applications, a growing number of stimuli-responsive luminescent sensors is reported comprising rich thermo-, mechano-, vapo-or solvato-chromic luminescence properties. More specifically, mechanochromic luminescent derivatives constitute a class of chromogenic materials that can reversibly generate variation of their emission spectra in response to mechanical external stress, providing a direct visual detection of the environmental alteration. These optical variations primarily originate from force-induced transformations of molecular structure, disruption of structural arrangements and/or packing modes (molecular conformations and/or intermolecular interactions) in the solid-state. In the field of such mechanochromic molecular materials, most of the systems that have been reported are based on organic dyes 60-

68

, gold [START_REF] Ito | Mechanical stimulation and solid seeding trigger single-crystal-to-single-crystal molecular domino transformations[END_REF][START_REF] Schneider | Strong Intra-and Intermolecular Aurophilic Interactions in a New Series of Brilliantly Luminescent Dinuclear Cationic and Neutral Au(I) Benzimidazolethiolate Complexes[END_REF][START_REF] Laguna | Combining Aurophilic Interactions and Halogen Bonding To Control the Luminescence from Bimetallic Gold-Silver Clusters[END_REF][START_REF] Osawa | Vapochromic and Mechanochromic Tetrahedral Gold(I) Complexes Based on the 1,2-Bis(diphenylphosphino)benzene Ligand[END_REF][START_REF] Koshevoy | Octanuclear gold(i) alkynyl-diphosphine clusters showing thermochromic luminescence[END_REF][START_REF] Deák | Anion-, Solvent-, Temperature-, and Mechano-Responsive Photoluminescence in Gold(I) Diphosphine-Based Dimers[END_REF][START_REF] Chen | A novel carbazole-based gold(i) complex with interesting solid-state, multistimuli-responsive characteristics[END_REF][START_REF] Seki | A gold isocyanide complex with a pendant carboxy group: orthogonal molecular arrangements and hypsochromically shifted luminescent mechanochromism[END_REF][START_REF] Leung | Metal-Metal and π-π Interactions Directed End-to-End Assembly of Gold Nanorods[END_REF][START_REF] Wu | Photochromic Benzo[b]phosphole Alkynylgold(I) Complexes with Mechanochromic Property to Serve as Multistimuli-Responsive Materials[END_REF][START_REF] Seki | Aurophilicity-Mediated Construction of Emissive Porous Molecular Crystals as Versatile Hosts for Liquid and Solid Guests[END_REF] or platinum [START_REF] Nishiuchi | A Polymorphic Platinum(II) Complex: Yellow, Red, and Green Polymorphs and X-ray Crystallography of [Pt(fdpb)Cl] [Hfdpb = 1,3-Bis(5-trifluoromethyl-2-pyridyl)benzene[END_REF][START_REF] Choi | Two-step Changes in Luminescence Color of Pt(II) Complex Bearing an Amide Moiety by Mechano-and Vapochromism[END_REF][START_REF] Zhang | Vapochromic and Mechanochromic Phosphorescence Materials Based on a Platinum(II) Complex with 4-Trifluoromethylphenylacetylide[END_REF][START_REF] Han | Reversible Mechanochromic Luminescence at Room Temperature in Cationic Platinum(II) Terpyridyl Complexes[END_REF][START_REF] Chowdhury | Mechanofluorochromic Pt II Luminogen and Its Cysteine Recognition[END_REF][START_REF] Cuerva | Multi-Stimuli-Responsive Properties of Aggregation-Enhanced Emission-Active Unsymmetrical Pt II Metallomesogens through Self-Assembly[END_REF][START_REF] Zhang | Twostep phosphorescent mechanochromism due to intramolecular deformation[END_REF] metal-complexes bearing rich variety of mechanochromic behaviors. Regarding the fascinating structural flexibility and luminescence properties that can be associated to Cu(I)-based molecular assemblies, these derivatives carry appealing potential for designing such stimuli-sensitive material, 1 and indeed attractive mechanochromic luminescent mononuclear or mainly polynuclear Cu(I) complexes have been reported in the recent years.

1, 28,38,43,44,58 The geometrical parameter differences between the crystal structures of the experimental derivative A and the optimized phase for the isolated species A opt-full (in vacuum) (Fig. 2d) suggest that the crystal packing and the intermolecular interactions dictate the geometrical configuration adopted by this tetrametallic assembly in the solid-state crystalline structure, and concomitantly, the electronic structure, the relaxed geometries in the excited state and the photophysical properties. The influence of these intermolecular parameters in the photophysics of related Cu(I)-based stimuli-sensitive luminescent polymetallic assemblies were highlighted in related 'Cu 8 Pd 1 ' assemblies 58 as a consequence of conformational changes involving non-covalent interaction alterations upon either grinding or melting of the bulk crystalline samples leading to emission properties modifications. Triggered by the TD-DFT calculations' conclusions that have highlighted such structural differences between the experimental and calculated ground state discrete structures (Fig. 2d), it can be anticipated that the photophysical processes lying in the A and B scaffolds should be impacted upon external constraints that could alter the crystal packing, affording new luminescent phases. This has motivated us to study the mechanochromic and thermolysis responses of solid-state polycrystalline samples of derivative A. Importantly, derivatives A and B crystallize in different space groups (respectively monoclinic and triclinic) in spite of their molecular structure similarities which will allow exploring the impact of the extended crystal structure organization differences on the stimuli-sensitive luminescence responses of such molecular solids. Mechanical stress. Mechanical grinding of the polycrystalline powder of the derivatives A and B affords the colorless powder ground phases A g and B g (Fig. S12) that do not exhibit under visible light any eye-perceived change in the solid-state morphologies regarding the pristine phases. However, if this grinding is applied at RT under a UV-vis light irradiation of 365 nm, an instantaneous change of the eye-perceived cyan (A) or blue (B) light emitted is observed, affording eyeperceived greenish light emission at the places where the mechanical stress was applied. The infrared spectra of the phases A g and B g show a broadening and a shift to higher energy of the (C≡N) band reaching 2126 cmˉ1 and 2121 cmˉ1, respectively (Fig. 1 and S4-S5). This reveals that structural alterations are generated in the metalacyclic backbones upon application of a mechanical stress, and might be attributed to metallacycle reorganizations leading to alternative conformers such as those observed in the theoretical study (interactions between the copper and the carbon atoms of the cyano bridging units in the geometry optimized ground state structures (Fig. 2d). The X-ray powder diffraction patterns conducted on the A g and B g phases, to get insights about the structural modifications potentially induced by the application of the mechanical stress on A and B derivatives, show two broad peaks around 7° and 20° revealing a major amorphization regarding the original crystalline A and B phases (Fig. 5 andS8). However,

P{

1 H} and 1 H NMR spectra of the A g and B g phases dissolved in CD 2 Cl 2 exhibit the same chemical shifts as derivatives A and B, respectively (Fig. S1 andS2). Moreover, the original crystalline phases of A and B are obtained after recrystallisation of A g and B g from pentane vapors diffusions in CH 2 Cl 2 solutions (as confirmed by the crystals lattice parameters determination). Finally, the infrared spectra of these two recrystallized phases are identical with the initial spectra of phases A and B (Fig. S4 andS5). This set of measurements highlights that no chemical transformations are induced as a mechanical grinding is applied. Solid-state absorption spectra at room temperature of derivatives A g and B g show a broad band in the UV region centered at 297 nm and 309 nm, respectively (Fig. S13 and S14) and their excitation spectrum shows two maxima at 318/372 nm (A g ) and at 329/374 nm (B g ) (Fig. S15). The room temperature solid-state emission spectrum of the phase A g presents a large featureless band centered at 486 nm (Fig. 4 and S17, FWHM = 4833 cmˉ1) upon excitation at 372 nm, which is significantly red-shifted ( em(Ag-A) = 0.168 eV (1345 cmˉ1)) compared to the emission spectrum of derivative A. Similarly, the room temperature solid-state emission spectrum of the phase B g is characterized by a large featureless band with a maximum at 490 nm ( ex = 374 nm, FWHM = 4257 cm -1

), which has been red-shifted of  em(Bg- B) = 0.123 eV (991 cmˉ1) compared to the emission of B (Fig. 4 and S20). For both phases A g and B g , the emission lifetimes at 300 K were treated by a tri-exponential decay affording  1 (24%) = 67 µs,  2 (59%) = 24 µs,  3 (17%) = 6 µs and  1 (20%) = 66 µs,  2 (65%) = 26 µs,  3 (15%) = 7 µs, respectively while RT EQY reach 47 % (A g ) and 74 % (B g ). Upon cooling from 300 K to 80 K, emission profiles of derivatives A g and B g exhibit a gradual red-shift reaching a green-yellow intense luminescence centered at 520 nm and 515 nm at 80 K respectively (compared to 483 nm for A and 475 nm for B), with emission lifetimes of 348 µs ( 1 (30%)), 167 µs ( 2 (63%) and 44 µs ( 3 (7%)) for A g and 439 µs ( 1 (33%)), 238 µs ( 2 (60%)), 80 µs ( 3 (7%)) for B g . The temperature dependent shift of the solid-state emission spectra of these two phases is reversible and the greenish luminescence observed at 300 K is recovered once the samples are warmed up. This manual grinding applied on the crystalline solid-state batches of derivatives A and B induces therefore a major amorphization of the pristine phases and likely a local release of the non-covalent interaction networks formed between the metallacycles, generating multiple species bearing distinct emission properties. [START_REF] Utrera-Melero | Mechanically responsive luminescent films based on copper iodide clusters[END_REF][START_REF] Huitorel | Luminescence mechanochromism of copper iodide clusters: a rational investigation[END_REF] However, the observed photophysical properties of the A g and B g phases (including the red-shift upon cooling of the single large band in the emission spectra and the general profile of the temperature dependance of the observed decay times of the excited states) support a preservation of the TADF relaxation processes evidenced in the pristine crystalline phases A and B. Thermolysis. Thermal gravimetric analysis and differential thermal analysis (TGA-DTA) under nitrogen flux from RT to 950 °C reveal that compounds A and B decompose at 305 °C and 350 °C, respectively (Fig. S25 andS28). Beforehand, from 30 °C to 110 °C (A) and from 80 °C to 140 °C (B), a loss of mass of 3 and 4 %, respectively, takes place, together with endothermic peaks at 61 °C and 105°C (A) and at 126°C (B) that are tentatively assigned to the desorption of included guest solvent molecules and/or adsorbed molecular water (these derivatives are air-stable and were manipulated under ambient atmosphere). In addition, at 237 °C for A and 233 °C for B, an exothermic peak is observed, associated with no mass variation (Fig. S26 and 29), indicating that these thermal phenomena do not correspond to a decomposition process. Then, when heating the samples up to 270 °C and directly cooling down to RT, no thermal event is observed probing that an irreversible process occurred (Fig. S26 and29). The solidstate infrared spectra of the new phases A' and B' present a single (CN) vibration observed at 2129 cm -1 and 2126 cm -1

, respectively (Fig. 1) shifted to higher frequencies compared to the values for the initial phases A and B (2118 cm -1 ). Significant differences are also highlighted in the powder X-ray diffraction patterns of the phases A' and B' that show markedly different profiles compared to the native phases A and B, respectively (Fig. 5 andS8). These observations suggest that the exothermic peaks observed at 237 °C (A') and 233 °C (B') witness a crystalline phase transition associated with a conformational change of the molecular architecture of the tetrametallic metalacyclic backbone of the phases A and B, preserving the chemical structure of these derivatives, but impacting their spectroscopic signature, crystalline phases and also likely photophysical properties. This assumption is confirmed by the quantitative obtention of single crystals of native phases A and B, (evidenced by the determination of their unit cell parameters and infrared spectrum, Fig. S4-S5) from pentane vapor diffusions within CH 2 Cl 2 solutions in which, respectively, the phases A' and B' were dissolved.

The two new phases, denoted A' and B', are collected as polycrystalline white solids with an eye-perceived solid-state aspect very similar to the initial samples A and B. Under excitation at 365 nm and 325 nm, the transited solid phases A' and B' exhibit a clear-blue eye-perceived emission, characterized by a large band centered at 462 nm (FWHM = 2345 cm -1

) and 453 nm (FWHM = 3608 cm -1 ), respectively, very reminiscent of the solid-state luminescence of A and B at room temperature (Fig. 4 andS18-S21). Yet, their excitation spectrum at RT is markedly different from the one of pristine phases A and B, showing two broad bands centered at 327 nm and 365 nm for A' and at 325 nm and 372 nm for B' (Fig. S15). In addition, the temperature dependent photophysical properties of A' and B' evidence slight but significant changes compared to A and B, such as a larger bathochromic shift in the emission spectra (Fig. 4) upon cooling from RT to 80 K, with  em(300-80K) = 0.227 eV (1832 cm -1

) for A' and  em(300-80K) = 0.204 eV (1645 cm -1

) for B', while  em(300-80K) = 0.176 eV (1420 cm -1 ) for A and  em(300-80K) = 0.151 eV (1219 cm -1 ) for B. However, examination of the thermal variation of the decay time of the excited state reveals similar biexponential lifetime profiles for A' compared to A, for which  1 (63%) and  2 (37%) are equal to 45 µs and 12 µs at RT and reaching 328 µs (33%) and 262 µs (67%) at 80 K (Table 2). The thermal variation of the decay time of the excited state recorded for B' is mono-exponential with values of 52 µs at 300 K and 199 µs at 80 K (Table 2). Finally, the values of the solid-state RT EQY for both phases A' and B' are ca. 80 % (Table 2). The temperature dependent shift of the solid-state emission spectra of these two phases is reversible and the eyeperceived blue-colored luminescence observed at 300 K is recovered once the samples are warmed up. Very likely, these data suggest that the radiative relaxation process assigned to TADF admixed with phosphorescence lying in the initial phases A and B are preserved but modulated in the thermally processed phases A' and B'.

Remarkably, these thermally transited phases A' and B' can also be formed by heating the amorphous ground phases A g and B g up to 230°C (see TGA-DTA curves, Fig. S27-S30). White solids whose PXRD diffractograms are similar to those of the phases A' and B' are obtained (Fig. S8-S9), demonstrating that upon solid-state thermolysis treatment of the amorphous A g and B g phases, the same crystalline states than obtained from the pristine A and B phases can be regenerated. This is confirmed by the IR spectrum of these phases that show the typical thinner and redshifted (C≡N) bands at 2129 cm -1 (A') and 2126 cm -1 (B') compared to A g and B g , respectively (Fig. S6-S7).

Conclusions

With the preparation of the new metalacyclic assembly A based on the mixed arsenic-phosphorus L As,P ligands, we demonstrated that the synthetic procedure applied to obtain the derivative B from the phosphorous dppm ligand can be extended to other ditopic pnictogen-based assembling ligands. While moderate intramolecular metric data alterations are determined in the discrete solid-state structure of the tetrametallic backbone of A compared to those of B, different extended crystalline packing are observed. Importantly, the luminescence properties of these two derivatives recorded on homogeneous polycrystalline powders are comparable, which is quite unexpected considering the generally admitted assumption that changing phosphorus atoms coordinated on the Cu(I) ions by arsenic atoms in similar molecular scaffolds should induce more efficient radiative relaxation processes as the SOC values would increase. TD-DFT calculations reveal that both A and B assemblies present very comparable electronic structures in which no electronic density lying on the pnictogen atoms is implied in the electronic excitation processes, explaining the neglectable impact of the substitution of the phosphorus centers by arsenic atoms on the recorded photophysical properties of A compared to B. Therefore, differences in the solid-state crystal structures may have in such systems an influence in the luminescence properties that can be competitive to those of the single atom SOC values. This questions in general the interpretation of the photophysical properties of such families of Cu(I) luminescent assemblies based on different organo-pnictogen ligands. A thorough analysis of both intramolecular metric data and intermetallic interaction networks must be conducted since antagonistic or complement effects may take place, in order to allow supplying an in-depth rationalization of the luminescence properties generated. The stimuli-sensitivity of such photophysical properties toward intermolecular interaction alterations resulting of the application of external mechanical or thermal stress is also highlighted. This reveals that upon the preparation of a specific Cu(I) polymetallic assembly, various luminescent phases can be readily generated, whose photophysical parameters can greatly vary from a phase to another one (Fig. S24). Moreover, interconversion of all these species occurs reversibly, giving an access to a family of luminescent sensors from one pristine derivative. This set of results suggest that beside the molecular design of specific Cu(I)-based molecular assemblies bearing targeted luminescence properties, the consideration of the intermolecular interactions and extended packing can also lead to the preparation of manifold new luminescent species, including performant solid-state luminophores and stimuli-sensitive sensors.

  Scheme 2 a) Molecular structure of the ligands LAs,P and dppm, b) Syntheses and schematic description of the tetrametallic metallacycles A and B.

Fig. 1

 1 Fig. 1 Infra-red spectra centered on (C≡N) vibrational bands of a) derivatives A, A', Ag and b) derivatives B, B', Bg.

Fig. 2

 2 Fig. 2 a) View of the molecular X-ray structure of the derivative A; b) and c) Comparative views and selective metric parameters of the derivatives A and B respectively; d) view of the Aopt-full(S0) calculated structure. (Counteranions, H atoms (b), c) and d)) and solvent molecules have been omitted for clarity; location of the C and N atoms of the cyano ligands and of the P and As atoms of the LAs,P ligands have been chosen arbitrarily). The asymmetric unit observed in the single crystals of the derivative A contains one [Cu 2 (L As,P ) 2 CN] + fragment, one PF 6 ˉ counter-anion and three disordered CH 2 Cl 2 solvent molecules. The derivative A is a tetrametallic metallacycle generated by the connection of two [Cu 2 (µ 2 -(L As,P ) 2 ] 2+ fragments by two cyanide ligands acting as ditopic linkers. The two Cu(I) ions of the [Cu 2 (µ 2 -(L As,P ) 2 ]

Fig. 3

 3 Fig. 3 Views of the extended stacking network observed in the singlecrystals of A and B (the metalacyclic backbones are highlighted by colored ellipses, counter-anions, H atoms and solvent molecules have been omitted for clarity; carbon atoms (excepted the carbon atoms of the cyano ligands and pnictogen atoms are shown in line style).The cyano linker are located within the plane of the metallacycle and have a slightly tilted coordination angle on the metal centers (Cu(1)-C≡N = 170.6(6)°; Cu(2)-N≡C = 173.5(3)°). Comparatively to the metallacycle B which exhibits short intermetallic distance (d Cu-Cu = 2.866(9) Å), the intermetallic distance Cu(I)⋯Cu(I) for compound A is slightly larger (d Cu-Cu = 2.957(1) Å), which is close to the upper limit of 2.9 Å generally considered for cuprophilic interactions. The PF 6 ˉ anions are stabilized by hydrogen bonds with the phenyl rings of the L As,P ligand (d(F⋯H) = 2.515 -2.642 Å).

Fig. 5

 5 Fig. 5 PXRD patterns of derivatives A, Ag, and A'.

  heating Ph 2 AsCl at its reflux point with

	Ph 2 PCH 2 -TMS (TMS: trimethylsilyl) under inert atmosphere
	affording without additive purification L As,P as a white powder
	collected in 92 % yield. As for compound B based on the dppm
	ligand, a synthetic approach adapted of coordination driven
	supramolecular chemistry was conducted. The pre-assembled
	"U-shape" molecular clip [Cu 2 (µ 2 -(L As,P ) 2 (CH 3 CN) 4 ](PF 6 ) 2 was
	formed in situ in a CH 2 Cl 2 solution reacting two equivalents of
	[Cu(CH 3 CN) 4 ]PF 6 with two equivalents of the L As,P ligand. One
	equivalent of KCN was then added affording a clear colorless
	solution from which air stable single crystals of A were isolated
	by slow diffusion of pentane vapors through the crude mother
	solution, leading to a homogeneous batch of colorless
	polycrystalline powder in 89 % yield.	56 The RT	31 P{ 1

Table 1 .

 1 Crystal data and structure refinement of compound A

	CCDC number	2284699
	Empirical formula	C54H50As2Cl6Cu2F6NP3
	Formula weight	1409.48
	Temperature/K	150(2)

Goodness-of-fit on F 2 1.099 Final R indexes [I>=2σ (I)] R1 = 0.0839, wR2 = 0.2307 Final R indexes [all data] R1 = 0.0925, wR2 = 0.2372 Largest diff. peak/hole / e Å -3 3.37/-2.89

Table 2 .

 2 Photophysical data for species A ( ex = 356 nm), A k r =  em / obs ; b k nr = (1- em )/ obs ; Values for k r and k nr have been calculated considering the main component of RT τ obs .

			d(Cu-Cu) (Å)	 em 300K (nm)	 em 80K (nm)	τ obs 300K (µs)	τ obs 80K (µs)	 em (%)	k r (sˉ1)	a	k nr (sˉ1)	b
	A	2.957(1)	456	483	38 16	224 111	84	2.2*10 4	4.2*10 3
	A g	-	486	520	67 24	348 167	47	7.0*10 3	7.9*10 3
						6	44
	A'	-	462	500	45 12	328 262	82	-		-
	B	32	2.866(9)	449	475	40 10	200 40	72	1.8*10 4	7.0*10 3
	B g	-	490	515	66 26	439 238	74	1.1*10 4	3.9*10 3
						7	80
	B'	-	453	494	52	199	80	-		-

g ( ex = 372 nm), A' ( ex = 365 nm), B

[START_REF] Kritchenkov | Efficient one-pot green synthesis of tetrakis(acetonitrile)copper(i) complex in aqueous media[END_REF]

( ex = 330 nm), B g ( ex = 374 nm) and B' ( ex = 325 nm) in the solid-state. a

C. Bizzarri, E. Spuling, D. M. Knoll, D. Volz and S. Bräse, Sustainable metal complexes for organic light-emitting diodes (OLEDs), Coord. Chem. Rev., 2018, 373,

49-82. 4 I. O. Koshevoy, M. Krause and A. Klein, Non-covalent intramolecular interactions through ligand-design promoting

Acknowledgements

This work was supported by the CNRS, the ANR (ANR PRC SMAC and ANR PRCI SUPRALUM), the French 'Ministère de l'Enseignement Supérieur, de la Recherche et de l'Innovation' and the French 'Ministère des Affaires Etrangères' as well as by the Deutsche Forschungsgemeinschaft within the project Sche 384/42-1. C.L. thanks the Alexander von Humboldt Foundation for a fellowship for experienced researcher. R.U., K.C., and C.P. acknowledge support by the "Grand équipement national de calcul intensif (GENCI)" through HPC resources of CINES, TGCC and IDRIS (grants AD010800649R1).

Author Contributions

The manuscript was written through contributions of all authors. All authors have given approval to the final version of the manuscript.

Conflicts of interest

There are no conflicts to declare.

Notes and references


Mauro Boero
email: mauro.boero@ipcms.unistra.fr

Cheick Oumar Diarra
email: cheick-oumar.diarra@etu.unistra.fr

Mohammed Guerboub
email: mohammed.guerboub@ipcms.unistra.fr

Kana Ishisone
email: kana.ishisone@ipcms.unistra.fr

Achille Lambrecht
email: achille.lambrecht@etu.unistra.fr

Évelyne Martin

Carlo Massobrio
email: carlo.massobrio@ipcms.unistra.fr

Irene Amiehe Essomba

Evelyne Martin
email: evelyne.martin@unistra.fr

Icare Morrot-Woisard
email: icare.morrotwoisard@etu.unistra.fr

Guido Ori
email: guido.ori@ipcms.unistra.fr

Christine Tugène
email: christine.tugene@ipcms.unistra.fr

Steve Dave

Wansi Wendji

On the actual difference between the Nosé and the Nosé-Hoover thermostats : a critical review of canonical temperature control by molecular dynamics

Keywords: Statistical Mechanics, Molecular Dynamics, Computational Materials Science, Temperature control, Car-Parrinello method

On the actual difference between the Nosé and the Nosé-Hoover thermostats : a critical review of canonical temperature control by molecular dynamics

Carlo Massobrio, Irene Amiehe Essomba,

This paper has to be intended both as a review and as an historical tribute to the ideas developed almost forty years ago by S. Nosé establishing the theoretical foundations of the implementation and use of thermostats in molecular dynamics. The original motivation of this work is enriched by an observation related to the connection between the Nosé seminal expression of temperature control and the extension proposed in 1985 by W. G. Hoover, known as the Nosé-Hoover thermostat. By carefully rederiving the equations of motion in both formalisms, it appears that all features of Nosé-Hoover framework (replacement of the Nosé variables by a single friction coefficient in the equations of motion) are already built in the Nosé approach. Therefore, one is able to work directly within the Nosé formalism with the addition of a single variable only, by greatly extending its general impact and simplicity and somewhat making redundant the Nosé-Hoover extension. Having been implicitly (and somewhat inadvertently) put to good use by a multitude of users over the past 40 years (in the context of classical and first-principles molecular dynamics), this finding does not need any specific application to be assessed.

Purpose and context

This paper reviews concepts and ideas inherent in the use of thermostats in molecular dynamics (MD). It has been stimulated by the author's continued use of the Nosé and Nosé-Hoover formalism (for some of them over the past thirty years), both in the classical and first-principles MD frameworks. Also, interactions with young colleagues and more established scientists have prompted us to realize that, unfortunately, running into users partially (or even totally) unaware of the basics of control temperature in molecular dynamics is not infrequent. Therefore, one of the purposes is to be pedagogical and informative while explaining where the "best" thermostat within MD comes from and why. This work has come to maturity during the final stages of writing of a recent book (monography) on molecular dynamics in disordered systems [START_REF] Massobrio | The Structure of Amorphous Materials using Molecular Dynamics[END_REF]. 1 It occurred that treating theory appropriately in the monography required not only a thorough account of past contributions in the area of temperature control within MD but also a careful rehearsal of the underlying concepts. This exercise has proved highly profitable, since allowing to take a "fresh" look at the Nosé and Nosé-Hoover formalism. It was a big surprise for us to realize the existence of unmistakable evidence proving that the Nosé formalism, in its very first formulation, includes already all features of the supposedly "more manageable" Nosé-Hoover scheme. This is by itself a valuable complement that gives to the present article the multifold flavor of historical review, critical analysis and insightful revisiting of theory. We would like to stress that, in this context, there is no need to show any practical applications, since we are neither proposing a new scheme nor extending the validity of an existing one. The rational developed here is purely analytical and does not affect the validity of any of the results published over the past 40 years by using the strategies referred to above. All this will appear clearly in what follows.

Introduction

The implementation of the notion of thermostat within molecular dynamics has marked an important step toward the proper use of statistical mechanics concepts to produce atomic-scale temporal trajectories. It has to be recognized that molecular dynamics have long faced the challenge of obtaining a de-sired target temperature (as a macroscopic quantity) by acting on the equations of motion (the microscopic counterpart) for a collection of N p atoms (or, equivalently, 3N p degrees of freedom). Early attempts showed the need of a rigorous theoretical framework to achieve a consistent realization of the canonical NVT ensemble (with N playing the role of the above N p , V being the volume and T the temperature) in the framework of a computer simulation based on time evolution [START_REF] Berendsen | [END_REF]3]. Also, the expected formalism has to comply with the attainment of a thermodynamic limit. This means that all concepts relevant to thermodynamics have to find their applications within a description purely based on the atomicscale behavior of a collection of degrees of freedom giving access to properties averaged on time.

The Nosé thermostat

This issue has been solved in the eighties by Shuichi Nosé [4,5]. Nosé showed that a molecular dynamics trajectory can be created to produce a canonical ensemble within a formalism that preserves the Hamiltonian character of the equations of motion. This goal can be achieved by coupling a system of N ind independent degrees of freedom to a new dynamical variable s in a way compatible with the Hamiltonian character of the extended global dynamical quantity, provided the time t inherent in Equations 1 through 6 can be considered as a virtual physical parameter. The virtual time t and the real time t r are linked by the equality t = st r . Accordingly, the Nosé Hamiltonian reads:

H N osé = Np i=1 p 2 i 2ms 2 + E pot [r i ] + p 2 s 2η + (3N p + 1)K b T T G ln s (1)
with the equations of motion derived straightforwardly as follows for the coordinates r i , the momentum p i , the variable s and its momentum p s

dr i dt = ṙi = ∂H ∂p i = p i ms 2 dp i dt = ṗi = ∂H ∂r i = - ∂E pot [r i ] ∂r i
(2)

ds dt = ṡ = ∂H ∂p s = p s η dp s dt = ṗs = - ∂H ∂s = Np i=1 p i 2 ms 3 - (3N p + 1)K b T T G s (3)
Note that η is the mass associated with s and p s and T T G is the target temperature to be obtained (in average on time) via application of the thermostat. For sake of completeness, it is worth pointing out that the above formalism can be also expressed in Lagrangian terms by maintaining the same framework well adapted to be checked numerically, i.e. the existence of a constant of motion from which the time behavior of ṙi , ṙi , ṡ, s can be directly derived:

L N osé = Np i=1 m ṙi 2 s 2 2 -E pot [r i ] + ṡ2 η 2 -(3N p + 1)K b T T G ln s (4)
d dt (ms 2 ṙi ) = - ∂E pot [r i ] ∂r i or ri = - 1 ms 2 ∂E pot [r i ] ∂r i -2 ṡ ṙi s -1 (5)
d dt (η ṡ) = Np i=1 m ṙi 2 s - (3N p + 1)K b T T G s (6)
The presence of the potential energy, namely ln s, pertaining to the variable s is critical to ensure that the time average taken over the extended Nosé phase space (the one defined with respect to the virtual time t and the associated variables r i , p i ) is equivalent to a time average in the canonical ensemble for the real variable in real time t r , namely p r i and r r i . This assertion has been explicitly demonstrated by Nosé in Ref. [4] and Ref. [5] to which we refer to interested reader, together with other relevant papers by the same author (see, for instance, Ref. [START_REF] Nosé | Constant-temperature molecular dynamics[END_REF] and Ref. [START_REF] Nosé | An improved symplectic integrator for nos-poincar thermostat[END_REF]). The quoted proof shows also explicitly that the above equivalence holds provided the factor (3N p +1) multiplying K b T T G ln s in Equation 1 is employed in conjunction with the virtual variables at the place of 3N p , to be selected when real variables p r i and r r i evolving in real time t r are adopted.

4 From the Nosé to the Nosé-Hoover thermostat 4.1 Origin and motivations: from virtual time to real time When it came out in 1984, the breakthrough by Nosé was unanimously considered as the most important achievement of the late 20th century for both statistical mechanics and atomic-scale modelling. As such, it was readily implemented in molecular dynamics codes available or on the verge of being developed as softwares open to the whole community. However, some experts acknowledged to feel uncomfortable with the appearance of additional variables and the notion of virtual time by pointing out that their physical meaning could be hard to grasp since all statistical averages are expected to refer to a real time. This objection was partially counteracted by Nosé himself in Ref. [4] by showing how Equation 2and Equation 3 can be expressed by working in a real time framework, the time evolution of the variable s and its momentum p s to be followed along with atomic coordinates r r i and momenta p r i in real time. The first step consists in defining the relationships involving the dynamical variables in the descriptions of the Nosé formalism based on real and virtual time, respectively. The basic definitions to begin with are the following:

r r i = r i p r i = p i /s s r = s t r = t dt/s p r s = p s /s (7)
where we have introduced the superscript r for all variables evolving in real time. Given the above relationship, and by employing the previous Equations 5 and 6 for evolution in virtual time, we obtain the time derivative of the above quantities and the related expression ready to use to implement numerically the Nosé formalism in the real, physical time.

dr r i dt r = s dr r i dt = s dr i dt = p i ms = p r i m (8) dp r i dt r = s dp r i dt = s d(p i /s) dt = dp i dt - 1 s ds dt p i (9) dp r i dt r = - ∂E pot [r r i ] ∂r r i - 1 s p s η p i = - ∂E pot [r r i ] ∂r r i - s r p r s p r i η (10)
ds r dt r = s ds r dt = s ds dt = s r p s η = s r2 p r s η (11) dp r s dt r = s dp r s dt = s d(p s /s) dt = dp s dt - 1 s ds dt p s (12) dp r s dt r = Np i=1 (p i r ) 2 ms r - 3N p K b T T G s r - s r (p r s ) 2 η ( 13
)
Therefore, by using these new set of equations of motion one can totally disregard the notion of virtual time and work in a framework strictly equivalent to the one employed to follow the evolution in the microcanonical (NVE) ensemble. A straightforward rewriting allows obtaining the equivalent of Equation 1in terms of the real variables:

H r N osé = Np i=1 (p i r ) 2 2m + E pot [r r i ] + (s r ) 2 (p r s ) 2 2η + 3N p K b T T G ln s r (14)
and the corresponding equivalent of Equation 4 in the form

L r N osé = Np i=1 m( ṙi r ) 2 2 -E pot [r r i ] + ( ṡr ) 2 η 2(s r ) 2 -3N p K b T T G ln s r (15)
together with the Lagrangian equations of motion expressed in real time:

d s r dt r ( m(s r ) 2 ṙi r s r ) = - ∂E pot [r r i ] ∂r r i d dt r (m(s r ) ṙi r ) = -s r ∂E pot [r r i ] ∂r r i (16) d dt r ( η s r ds r dt r ) = Np i=1 m( dr r i dt r ) 2 -3N p K b T T G (17)
It is important to realize that the Nosé equations as depending on the real variables change the purely Hamiltonian character of the formalism, since Equations 8 through 13 cannot be derived directly from Equation 14. In other terms, Equation 14 is not a Hamiltonian (i.e. Equation 15is not a Lagrangian), both expressing now "mere" constants of motion. Also, what we have obtained above as the Hamiltonian and Lagrangian equations of motion in real time have been derived directly from the equations of motion in virtual time, since one could obtain them from Equation 14and Equation 15. Once again, the reason being that the Nosé formalism is a Hamiltonian one only in virtual time.

Can we attach to the Nosé variables a physical meaning ?

We have seen that the Nosé framework, although established in virtual time and holding a fully hamiltonian character in this framework only, can be reformulated within the customary temporal scale (the real one) inherent in any molecular dynamics application. Is this going to settle the issue of its straightforward and manageable use to implement a temperature control ? What makes it still cumbersome to the eyes of pratictioners is the presence of the s variable to which one cannot attach a transparent physical meaning. However, it appears that careful anlysis of the equations provides a clear hint to address favorably this issue and recover a more palatable analytical expression. The route followed in this paper is alternative to the one proposed by Hoover [8]

dp r i dt r = - ∂E pot [r r i ] ∂r r i - s r p r s p r i η = - ∂E pot [r r i ] ∂r r i -ξp r i (18)
by showing that an actual friction term can be identified as acting on the time evolution of p r i . At first sight, this postulate does not correspond to any simplification in the equations of motions, since one is bound to follow the time evolution of both s r and p r s . However, it turns out that focusing on the product of the two variables s r and p r s (defined above as proportional to ξη) has a quite insightful meaning when attention is drawn on its time derivative ξ as it results from the following equations:

d(s r p r s ) dt r = s r dp r s dt r + p r s ds r dt r = s r ( Np i=1 (p i r ) 2 ms r - 3N p K b T T G s r - s r (p r s ) 2 η ) + p r s s r2 p r s η (19)
d(s r p r s ) dt r = s r dp r s dt r + p r s ds r dt r = s r ( Np i=1 (p i r ) 2 ms r - 3N p K b T T G s r ) = ( Np i=1 (p i r ) 2 m -3N p K b T T G ) ( 20
)
dξ dt r = ( Np i=1 (p i r ) 2 m -3N p K b T T G )/η (21)
Equation 21 has a form clearly reminiscent of the time evolution of a friction variable, as it appears when grouping and considering as a whole the three equations expressing the Nosé evolution in real time of the variables r r i , p r i and ξ:

dr r i dt r = s dr r i dt = s dr i dt = p i ms = p r i m ( 22
)
dp r i dt r = - ∂E pot [r r i ] ∂r r i - s r p r s p r i η = - ∂E pot [r r i ] ∂r r i -ξp r i (23) dξ dt r = ( Np i=1 (p i r ) 2 m -3N p K b T T G )/η (24)
What is the form taken by the constant of motion as a result of the introduction of the ξ variable ? It appears that the first expression one can come up with (obtained from Equation 14with the appropriate changes) can contain ξ explicitly and yet, the s r variable is still present, undermining the postulate that only ξ has to be followed in time:

H r N osé = Np i=1 (p i r ) 2 2m + E pot [r r i ] + η ξ 2 2 + 3N p K b T T G ln s r (25)
However, it is possible to eliminate the s r variable by exploiting Equation 11

ξ = 1 s r ds r dt r = s r p r s η 1 s r ds r dt r dt r = ln s r (26)
This gives an expression for the Nosé constant of motion in real time in which the ξ variable appears both in the kinetic and in the potential part of the time evolution of the thermostat, the only inconvenient feature being the presence of a time integral that, in principle, one should perform over the running time trajectory to make sure that the constant of motion is well conserved.

H r N osé = Np i=1 (p i r ) 2 2m + E pot [r r i ] + η ξ 2 2 + 3N p K b T T G ξdt r (27)
The rationale developed in this section shows unambiguously that the Nosé formalism as developed in Refs. [4,5] does not imply (as claimed in the literature, see Ref. [8] and reiterated over the years) any mandatory use of the s variable in the equation of motions. Also the Nosé formalism is not characterized by an unnecessary "scaling of the time", since a set of equations in real variables (time and coordinates) can be straightforwardly obtained, by reducing the application of a thermostat to a skillful use of the ξ variable.

To summarize what shown in the above sections, the following points have to be underlined

• To preserve the Hamiltonian (or, equivalently, Lagrangian) character of the equations of motion, the Nosé formalism has to be expressed, at the outset, in virtual time by connecting virtual and real time in the form dt r = dt v /s = dt/s through the s variable, where the superscript r stands for real, and we do not use throughout the paper the superscript v standing for virtual (note also that for "real" and "virtual" we use both italic and standard characters depending on the context).

• The Nosé formalism can be expressed in real time by applying the suitable relationship connecting these two frameworks (Equations 7 through 13). However this can be done at the price of the Hamiltonian character of Eq. 1, that becomes a constant of motion only, quite useful to be followed in time to monitor the correctness of the time integration of the equations of motion.

• By focusing on the variable ξ = s r p r s /η and on its time derivative, one can reformulate the entire Nosé formalism by using three equations of motion for the variables r r i , p r i and ξ.

The Hoover derivation of the Nosé thermostat

Shortly after the publication of the two main papers by Nosé (Refs. [4,5]) featuring the breakthrough methodology of rigorous thermostats for molecular dynamics, W. G. Hoover proposed alternative expressions for the equations of motion meeting exactly the same purposes (see Ref. [8]). He pointed out that friction coefficients could be employed at the place of the original s and p s , by claiming that the set of equations developed were "free of time scaling". The formalism by Hoover, known as Nosé-Hoover thermostat, became immediately very popular in the field of molecular dynamics, to the point of (almost) supplanting the original Nosé ideas as if the actual discovery of right thermostats for molecular dynamics were the offspring of Ref. [8]. This is somewhat unfair since, Hoover has only rewritten a set of equation already existing in Ref. [4] by interpreting in his own way the concepts of real time and variables. This will become quite evident in the following.

In practice, Hoover introduces first a change in the time variable by imposing dt N osé = s dt Hoover (dt Hoover = dt hereafter for simplicity, for instance as in dr i /dt). This amounts to rewriting Equation 2 and Equation 3 as multiplied by s:

dr i dt = ṙi = ∂H ∂p i = p i ms dp i dt = ṗi = ∂H ∂r i = -s ∂E pot [r i ] ∂r i
(28)

ds dt = ṡ = ∂H ∂p s = sp s η dp s dt = ṗs = - ∂H ∂s = Np i=1 p i 2 ms 2 -(3N p + 1)K b T T G (29) ri = ṗi ms - p i ms ṡ s = - 1 m ∂E pot [r i ] ∂r i - p s η ṙi (30)
p i 2 ms 2 -(3N p + 1)K b T T G ) = 1 η ( Np i=1 mṙ i 2 -(3N p + 1)K b T T G ) (32)
At this point, the formalism expressed in Ref. [8] can be made equivalent to the original one by Nosé by redefining ṙi = p i /m and by replacing 3N p +1 by 3N p . The final set of equations according to Hoover's strategy reads:

ṙi = p i m mr i = - ∂E pot [r i ] ∂r i -mζ ṙi ṗi = - ∂E pot [r i ] ∂r i -ζp i ζ = 1 η ( Np i ṗi 2 m -3N p K b T T G ) (33)
It appears that the three relationships contained in Equation 33 have the same physical meaning than what we have expressed through Equation 22, Equation 23 and Equation 24, with ξ in the Nosé formalism acting as ζ in the Hoover's one. However, we reiterate that the route taken by Hoover to obtain Equation 33 does not follow any specific protocol based on the introduction of a real time together with real variables, the only advantage of it being the simpler link ζ = p s /η between friction and Nosé variables.

5 Extending the Nosé ideas: multiple thermostats and first-principles molecular dynamics

Constant of motion and the friction variable made simpler

As a prerequisite to any other thoughts on the Nosé scheme, particularly in the context of first-principles molecular dynamics (that is to say, a scheme where the potential energy depends on the electronic structure, unlike in classical molecular dynamics) it is useful to focus on the expression for the conserved quantity given in Equation 27 that we rewrite in the form

H Conserved = Np i=1 (p i ) 2 2m + E pot [r i ] + η ξ 2 2 + 3N p K b T T G ξdt (34)
by dropping the upperscript r standing for real since from now on we shall refer to real time. Careful analysis of Equation 34 reveals that the cumbersome presence of a time integral within a constant of motion can be eliminated by resorting to a new variable α such as α = ξ. Accordingly, Equation 33 and Equation 34 take the form:

ṙi = p i m mr i = - ∂E pot [r i ] ∂r i -m α ṙi ṗi = - ∂E pot [r i ] ∂r i
-

αp i α = 1 η ( Np i=1 ṗi 2 m -3N p K b T T G ) (35) H Conserved = Np i=1 p i 2 2m + E pot [r i ] + η α2 2 + 3N p K b T T G α (36)
The above equations, linking a quantity constant in time (H Conserved ) to non-Hamiltonian equations of motion can be easily generalized to the case of multiple thermostats. One can consider the case of two subsystems made of N p1 and N p2 atoms of coordinates and momenta r l , p l (for the subclass made of N p1 atoms) and r k , p k ((for the subclass made of N p2 atoms) kept at two different temperatures T T G1 , T T G2 .

Here we made the choice of two subsystems only for simplicity, the number of subsystems not being limited. Calculations of thermal conductivity with the AEMD are revealing examples of application of multiple thermostats to sub-sets of atoms belonging to the same system but kept at different (initial) temperatures [9,10,11,12]. The corresponding equations of motion read (note that the friction variables α and β have masses η and γ, respectively):

ṙl = p l m mr l = - ∂E pot [r l ] ∂r l -m α ṙl ṗl = - ∂E pot [r l ] ∂r l -αp l α = 1 η ( N p1 l=1 ṗl 2 m -3N p1 K b T T G1 ) (37) ṙk = p k m mr k = - ∂E pot [r k ] ∂r k -m β ṙk ṗk = - ∂E pot [r k ] ∂r k -βp k β = 1 γ ( N p2 k=1 ṗk 2 m -3N p2 K b T T G2 )
(38) with the constant of motion H C12 accounting for the presence of two different friction variables α and β and their associated kinetic and potential energy:

H C12 = N p1 l=1 p l 2 2m + N p2 k=1 p k 2 2m + E pot [r l , r k ] + η α2 2 + 3N p1 K b T T G1 α + γ β2 2 + 3N p2 K b T T G2 β (39)
5.2 First-principles (Car-Parrinello) molecular dynamics with thermostats: part 1

The ideas developed so far have found a natural application within first-principles molecular dynamics (in the version pioneered by Roberto Car and Michele Parrinello [13], FPMD-CP herafter). First, it appears that the formalism by Nosé is perfectly suited to be employed in the FPMD-CP framework to control the temperature of the ionic degrees of freedom (corresponding to the atomic degrees of freedom in the classical MD description). Also, the combination of Nosé and FPMD-CP ideas is essential to achieve an overall manageable use of the whole FPMD-CP approach, by ensuring adiabatic separation between ionic and electronic dynamical sub-systems. This stems from the possibility to control the temperature of different families of degrees of freedom, as we have explicitly shown via Equation 37 and Equation 38.

In what follows we shall slightly change our previous notation to allow for the introduction of electronic degrees of freedom coupling to the ionic ones. Let's consider a Lagrangian collection of ions of coordinates R I with masses M I and orbitals ψ i (r) accounting for the electronic structure. µ is a mass inherent in the so-called fictitious movements of the orbitals, its value, in Hartree atomic units, being much smaller than the one of M I . In this way, one has ions and orbitals dynamically decoupled since moving with drastically different frequencies. This postulate and its correct implementation are at the very heart of the FPMD-CP methodology, for which these conditions are best fulfilled by insulators and, at not too high temperatures, semiconductors, due to the the presence of a gap separating valence and conductions states. By assuming that the total energy is obtained within the density functional Kohn-Sham framework (and by imposing as it should be the constraint that the orbitals are orthonormal via E orcnstr (ψ i (r))), one has:

L CP = E f ic (ψ i (r)) + E kin ({R I }) -E tot [{ψ i }; {R I }] + E orcnstr (ψ i (r)) (40)
L CP = µ i | ψi (r)| 2 dr + 1 2 N I=1 M I Ṙ2 I -E tot [{ψ i }; {R I }] - ij λ ij ψ * i (r)ψ j (r)dr -δ ij (41)
For future purposes (as it will be the case when making explicit the action of thermostats), it is convenient to express E tot [{ψ i }; {R I }] in the form E tot [{ψ i }; {R I }, α l ], by implying that l variables can be included in the dynamical evolution of the system. Therefore, in analogy with what developed in Sec. 5.1 we can first rewrite Equation 41 in the generic form

L CP Cons = µ i | ψi (r)| 2 dr+ 1 2 N I=1 M I Ṙ2 I -E tot [{ψ i }; {R I }, α l ]- ij λ ij ψ * i (r)ψ j (r)dr -δ ij + 1 2 l η l α2 l ( 42
) that contains explicitly the potential and the kinetic parts due to a thermostat as follows:

L CP Cons = µ i | ψi (r)| 2 dr+ 1 2 N I=1 M I Ṙ2 I -E tot [{ψ i }; {R I }]- ij λ ij ψ * i (r)ψ j (r)dr -δ ij + 1 2 η I α2 I +F (α I )
(43) This is exactly the FPMD-CP equivalent of the constant of motion made explicit in the previous sections for the case of a thermostat applied to the ionic (atomic) degrees of freedom (see Equation 36). Note that we have termed the new constant of motion L CP Cons to underline the conceptual difference between L CP Cons and L CP , the former having lost, due to the presence of the thermostat variables, the Lagrangian character of the latter, i.e. no Lagrangian equation of motion can be derived directly from it. In analogy with the above rationale on the Nosé formalism in real time (see Sec. 4.2) one can associate to the variable α I the kinetic term 1 2 η I α2 I as well as the potential energy term in the form

F (α I ) = 2E I-T G kin α I (44)
where 2E I-T G kin is the targetted average ionic kinetic energy, typically expressed via the target temperature so as 2E I-T G kint = N ind k B T T G , N ind being the number of degrees of freedom (the strict equivalent of 3N p employed to introduce the Nosé formalism within classical molecular dynamics). This holds true provided one keeps in mind that only the average value of the kinetic energy can be taken to correspond to a temperature, even though often both kinetic energy and temperature are employed as instantaneous and average quantities. Having obtained the expression for a conserved quantity within FPMD-CP after application of a thermostat on the ionic family of degrees of freedom, it is now straightforward to write down the expression for the FPMD-CP equations of motions along the same lines followed in the classical case via Equation 35:

µ ψi (r, t) = - δE tot [{ψ i }, {R I }] δψ i * (r, t) + j λ ij ψ j (r, t) (45)
M I RI = -∇ I E tot [{ψ i }, {R I }] -M I ṘI αI (46) η I αI = 2( 1 2 N I=1 M I Ṙ2 I - 1 2 N ind k B T T G ) (47)
5.3 First-principles (Car-Parrinello) molecular dynamics with thermostats: part 2

In addition to the use of Nosé formalism to control the temperature of a system (as expressed through the average kinetic energy of the atoms, or, better, of the ions if one considers also electronic degrees of freedom), one can take advantage of the whole scheme to make sure the FPMD-CP time evolution is as much as possible adiabatic. This means that the notion of temperature control can be employed to hamper the increase of E f ic (ψ i (r)) that favors unwanted transferof energy from the ionic to the fictitious electronic degrees of freedom. Based on what we have seen in Sec. 5.1, Equation 37, Equation 38 and Equation 39 this can be achieved by considering a variable α l acting as a thermostat for E f ic (ψ i (r)) (as first proposed in Ref. [14]). While this approach is not bound to work on extended intervals for dramatic cases of gapless systems (as metals), it remains quite valid to ensure adiabaticity for a wide class of systems featuring electronic density of states at least depleted (exhibiting pseudogaps) around the Fermi energy level. The corresponding FPMD-CP conserved quantity reads

L CP Cons = E f ic (ψ i (r)) + E kin ({R I }) -E tot [{ψ i }; {R I }, α I , α e ] + E orcnstr (ψ i (r)) + E ext (α I ) + E ext (α e ) (48) L CP Cons = E f ic (ψ i (r))+E kin ({R I })-E tot [{ψ i }; {R I }]+E orcnstr (ψ i (r))+ 1 2 η I α2 I +2E I-T G kin α I + 1 2 η e α2
e +2E e-T G kin α e (49) in which E e-T G kin plays the role of a value to be reached (in average) by µ i | ψi (r)|2 dr. The corresponding equations of motion are as follows

µ ψi (r, t) = - δE tot [{ψ i }, {R I }] δψ i * (r, t) + j λ ij ψ j (r, t) -µ ψi αe (50) M I RI = -∇ I E tot [{ψ i }, {R I }] -M I ṘI αI (51) η I αI = 2( 1 2 N I=1 M I Ṙ2 I - 1 2 N ind k B T T G ) (52) η e αe = 2(µ i | ψi (r)| 2 dr -E e-T G kin ) (53)
where we have simply adapted and rewritten for the combined ionic/electronic degrees of freedom of the FPMD-CP method the set of equations shown in Sec. 5.1 when introducing the notion of multiple temperature control 2 .

Conclusions

The main motivation of this paper was to review and highlight some very fundamental features of early temperature control approaches within molecular dynamics. Despite the fact that we are referring here to a formalism proposed back in 1985, we were driven by the conviction that some refocusing on the foundations and the underlying theory of thermostats application in atomic-scale dynamical simulation was very much needed. While reviewing the Nosé ideas for the benefit of those that did not have the opportunity to go through the original derivation, we realized that, quite surprisingly, the so-called extension and improvement due to Hoover (the Nosé-Hoover thermostat) can be easily traced back and identified in the original Nosé expressions. This is way the present paper, in addition to its predominant review character, contains also some new observation that can be taken as an valuable contribution to the temperature control ideas. We showed that the Nosé formalism as it stands does not require, to increase its manageability and acquire a sound physical interpretation, any of the rearrangements put forth by Hoover and leading to the so-called Nosé-Hoover thermostat. Therefore, our analysis provides a form for the equations of motion easily understandable, containing the required friction force and yet based uniquely on a skillful reworking on what Nosé has presented in his seminal papers. As a complement to our rationale, we include the explicit expressions for the use of multiple thermostats by considering the case of separate families of degrees of freedom as it occurs in the Car-Parrinello version of first-principles molecular dynamics. We can also observe that the temperature of any additional dynamical variable, not necessarily atoms or fictitious electronic degrees of freedom, can be controlled with the scheme proposed by Nosé and revisited here. This is the case, for instance, of reaction coordinates or collective variables driving an activated process from reactants to products and included in the so-called metadynamics scheme [START_REF] Iannuzzi | Efficient exploration of reactive potential energy surfaces using car-parrinello molecular dynamics[END_REF][START_REF] Boero | Atomistic insight into the initial stage of graphene formation on sic(0001) surfaces[END_REF].

It would be unfair to conclude this paper without doing justice to those that, over the years, contributed to a better understanding of the theoretical foundations of the Nosé thermostat and the underlying nonhamiltonian dynamics. The references that follow (and those quoted therein) are not only highly representative of these theoretical efforts but provide important clues to capture the differences between microcanonical and canonical dynamics. As such, they have to be considered of the outmost importance when reviewing concepts and ideas related to temperature control within molecular dynamics [START_REF] Liu | Generalized Gaussian moment thermostatting: A new continuous dynamical approach to the canonical ensemble[END_REF]19,[START_REF] Zhang | A unified thermostat scheme for efficient configurational sampling for classical/quantum canonical ensembles via molecular dynamics[END_REF][START_REF] Tuckerman | Non-Hamiltonian molecular dynamics: Generalizing Hamiltonian phase space principles to non-Hamiltonian systems[END_REF][START_REF] Tuckerman | On the classical statistical mechanics of nonhamiltonian systems[END_REF].

Such an accomplishment appeared at the end of

shortly after having received, by this Editor, the invitation to publish this paper in the context of an invited talk at the EMRS 2022 Fall meeting.

for a more complete and detailed view on the methodology and applications of FPMD-CP see Refs.[START_REF] Marx | Ab Initio Molecular Dynamics: Basic Theory and Advanced Methods[END_REF][START_REF] Massobrio | The Structure of Amorphous Materials using Molecular Dynamics[END_REF]

Acknowledgements

We are grateful to Prof. Giovanni Ciccotti (Istituto per le Applicazioni del Calcolo, Consiglio Nazionale delle Ricerche, Roma (Italia)) for instructive comments.