Mohamed El Garah

Djallel Eddine Touaibia

Sofiane Achache

Alexandre Michau

Elizaveta Sviridova

Pavel S Postnikov

Mohamed M Chehimi

Frederic Schuster

Frédéric Sanchette

Mohamed El Garah
email: mohamed.el_garah@utt.fr

Frederic Sanchette

Effect of nitrogen content on structural and mechanical properties of AlTiZrTaHf(-N) high entropy films deposited by reactive magnetron sputtering

Keywords: Thin films, high entropy nitrides, magnetron sputtering, surface analysis

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

Introduction

High entropy alloys (HEAs) break the design of the conventional alloys which are based on one principal element. HEAs are defined as quasi-or equimolar alloys consisting, at least, of five elementary components with an atomic percentage varying from 5 to 35 at. % [START_REF] Yeh | Nanostructured highentropy alloys with multiple principal elements: novel alloy design concepts and outcomes[END_REF][START_REF] Cantor | Microstructural development in equiatomic multicomponent alloys[END_REF] .

They become a great research hotspot in the field of metallic materials. The reported HEAs often reveal a single-phased solid solution, free of intermetallic phases. In the literature, these particular alloys are coined Multi-element Alloys MEA, Complex Component Alloys CCA, or Multielement High Entropy MHE. The ability of HEAs in forming simple solid solution phases (BCC, FCC, HCP or a mixture) make them stable and therefore reduces considerably the free-energy [START_REF] Li | A review on high entropy alloys coatings: fabrication processes and property assessment[END_REF][START_REF] Yan | Physics, A brief review of high-entropy films[END_REF]. Various investigations have been achieved to explore their physicochemical properties revealing potential features in use of some applications [START_REF] Zhang | Microstructures and properties of high-entropy alloys[END_REF][START_REF] Gludovatz | A fracture-resistant high-entropy alloy for cryogenic applications[END_REF]. They exhibit excellent mechanical and physical properties such as wear resistance [START_REF] Cheng | Effect of Nb addition on the structure and mechanical behaviors of CoCrCuFeNi high-entropy alloy coatings[END_REF], corrosion resistance [START_REF] Hsueh | Effect of nitrogen content and substrate bias on mechanical and corrosion properties of high-entropy films (AlCrSiTiZr) 100-xNx[END_REF] and thermal stability [START_REF] Sheng | Nano-crystallization of high-entropy amorphous NbTiAlSiWxNy films prepared by magnetron sputtering[END_REF]. This is why HEA are potential candidates in different critical sectors: aerospace, turbine or nuclear industries [START_REF] Li | A review on high entropy alloys coatings: fabrication processes and property assessment[END_REF][START_REF] Yan | Physics, A brief review of high-entropy films[END_REF].

Since they have been considered as promising material compared to traditional alloys, they have been investigated for high temperature applications. Like bulk HEAs, thin films of five elements or more are developed and exhibit high performance physico-chemical properties.

The choice of the constituting elements is of considerable interest to control their properties.

Oxidation resistance can be improved by incorporation of different elements such as Al, Cr, Ta and Si. Adding Al to other systems, e.g. FeCrCoNi [START_REF] Middleburgh | Segregation and migration of species in the CrCoFeNi high entropy alloy[END_REF], leads to transition from FCC to BCC with modification of mechanical properties. HEA containing aluminum are reported to have an enhanced oxidation resistance. Butler et al. point out that the increase of Al content enhances the oxidation resistance of AlFeCoCrNi alloy [START_REF] Butler | Oxidation behavior of arc melted AlCoCrFeNi multicomponent high-entropy alloys[END_REF]. Shen et al. have deposited (Al0.34Cr0.22Nb0.11Si0.11Ti0.22)50N50, by using magnetron sputtering technique, pointing out also the role of Al element on the oxidation resistance of the film. Superior oxidation resistance is attributed to the formation of a dense Al2O3 to layer on the film surface [START_REF] Shen | Superior oxidation resistance of (Al0. 34Cr0. 22Nb0. 11Si0. 11Ti0. 22) 50N50 high-entropy nitride[END_REF].

High entropy metal-sublattice nitrides [START_REF] Schneider | How high is the entropy in high entropy ceramics[END_REF][START_REF] Kirnbauer | Mechanical properties and thermal stability of reactively sputtered multi-principal-metal Hf-Ta-Ti-V-Zr nitrides[END_REF] have been extensively studied during the last years because these alloys combine good mechanical behaviors with high thermal stabilities. These films can be deposited in reactive mode in order to control the nitrogen content.

Nitrogen content has obviously significant influence on both structure and mechanical properties. Recent investigations showed the nitrogen effect on the self-organization of metallic atoms leading to phase transition [START_REF] Cui | Effects of nitrogen content on microstructures and mechanical properties of (AlCrTiZrHf) N high-entropy alloy nitride films[END_REF][START_REF] Khan | High entropy nitride (HEN) thin films of AlCoCrCu0. 5FeNi deposited by reactive magnetron sputtering[END_REF][START_REF] Sha | FeMnNiCoCr-based high entropy alloy coatings: Effect of nitrogen additions on microstructural development, mechanical properties and tribological performance[END_REF][START_REF] Garah | Structural and mechanical properties of AlTiTaZr (N) medium entropy films (MEF) obtained by DC magnetron sputtering in dynamic mode[END_REF]. Nitrides films are chemically inert and harder than the corresponding metallic alloys. Some films, prepared with strong nitride forming elements (for example Al, Ti, Hf, etc.), have hardness ranging from 10 to 40 GPa [START_REF] Chang | Nitride films deposited from an equimolar Al-Cr-Mo-Si-Ti alloy target by reactive direct current magnetron sputtering[END_REF][START_REF] Chang | Effect of microstructure on the nanomechanical properties of TiVCrZrAl nitride films deposited by magnetron sputtering[END_REF][START_REF] Cheng | Structural and mechanical properties of multi-element (AlCrMoTaTiZr)Nx coatings by reactive magnetron sputtering[END_REF], much higher than those of films containing non-nitride forming element (Co, Cu, Fe, Ni, etc.). Moreover, the nitrides films can have not only enhanced mechanical properties but also good corrosion resistance [START_REF] Chen | Effects of nitriding on the microstructure and properties of VAlTiCrMo high-entropy alloy coatings by sputtering technique[END_REF].

The most used techniques to deposit the nitride films are laser cladding, magnetron sputtering, spraying and electrodeposition [START_REF] Li | Microstructures and properties of high-entropy alloy films and coatings: a review[END_REF]. Magnetron sputtering allows to deposit nitride films with tuned physical and chemical properties. Bombardment energy of impinging particles can be adjusted via, for instance, the bias voltage or the working pressure, which have a great effect on morphology as well as on structure and microstructure. Besides, the technique, versatile, offers the possibility to change the reactive gas (N2, O2, CH4, C2H2).

Previous research on high-entropy nitride coatings mainly focused on their properties such as microstructure, mechanical, tribological performance and corrosion resistance are reported.

The present study is focusing on the investigation of a new AlTiTaZrHf material in different Ar+N2 atmospheres. Microstructure, chemical and mechanical properties of the high entropy metal-sublattice are discussed. The material is based on AlTiTaZrHf system, consisting of forming-nitride metal elements. Results on nitride thin films based on the Al-Ti-Ta-Zr medium entropy films have been published [START_REF] Garah | Structural and mechanical properties of AlTiTaZr (N) medium entropy films (MEF) obtained by DC magnetron sputtering in dynamic mode[END_REF]. The addition of Hf metal is intended to promote also the formation of HfN with FCC structure [START_REF] Bab | Nanocrystalline HfN produced by mechanical milling: Kinetic aspects[END_REF]. Besides, HfN has been reported to influence significantly the hardness and the oxidation resistance of films [START_REF] Fernandez | Effect of hafnium contaminant present in zirconium targets on sputter deposited ZrN thin films[END_REF]. Even various high entropy nitrides have already been reported in the literature, it is difficult to provide a general design rule. Investigating new high entropy nitrides become essential in the current phase of high entropy development. This work discusses how the nitriding effect influence the mechanical and tribological performance of high entropy alloy composed of strong-nitride forming elements.

Experimental details

Deposition

The films are deposited on glass substrates and silicon wafers by using DC magnetron sputtering of 99.99% pure high entropy equiatomic AlTiTaZrHf alloy target. The reactor is a Deph4 (DEPHIS, Etupes, France) system equipped with 4 circular cathodes 200 mm in diameter. Substrates are cleaned by subsequent ultra-sonication in acetone and ethanol (20 minutes each) followed by drying process with warm air. Afterward, the substrates are ion etched in argon (0.5 Pa) using RF power of 300 watts for a duration of 10 minutes.

The process starts with a target cleaning in argon (0.5 Pa) for a duration of 10 minutes before deposition. The discharge current on AlTiTaZrHf alloy target is kept at 2A. The targetsubstrate distance is kept at a distance of 10 cm. The flow rates ratio RN2=N2/Ar+N2 is varied from 0 to 50%. Total pressure is maintained at 0.5 Pa. No bias or substrate heating is applied during deposition.

X-Ray Diffraction

All films were analyzed by X-ray diffraction using D8-Advance Bruker diffractometer with Cu radiation (λ=1.54 Å), θ-2θ (20-90°), step size=0.02° and step times of 1s. X-ray source is operated at 40 kV. Ni filter is used to absorb Kβ signal from monochromatic beam.

X-ray Photoelectron Spectroscopy

X-ray photoelectron spectroscopy (XPS) measurements were performed using a NEXSA apparatus (Thermo, East Grinsted, UK) fitted with a monochromatic Al Kα source (hν = 1486.6 eV). The instrument is based at TPU, Tomsk, Russian Federation. All measurements were conducted at a pressure of 10 -9 mbar. The survey and narrow scans were recorded in the following conditions: pass energy/step size = 200/1 eV, and 40/0.1 eV, respectively. The materials are conductive and no spectral calibration was needed. In these conditions, the adventitious hydrocarbon contamination exhibited a C1s peak centered at 284.8 eV, whereas for individual nitrides the adventitious hydrocarbon contamination C1s peak slightly shift from one nitride to another [START_REF] Greczynski | C 1s peak of adventitious carbon aligns to the vacuum level: dire consequences for material's bonding assignment by photoelectron spectroscopy[END_REF]. We have processed the data as acquired without any charge calibration. As a matter of fact, SEM images were recorded without any metallization to compensate for static charging. Note that the samples were analyzed by XPS in Tomsk Polytechnic University implying that they have been exposed to ambient air therefore inducing some extent of unavoidable oxidation, discussed in Section 3.4. For the peak-fitting procedure, the constraints were as follows: Shirley baseline, ± 0.1 eV for binding energy position, ± 0.1 eV for FWHM, and the Laurentian to Gaussian (L/G) peak shape ratio was adjusted to get the best fit. For doublets (Hf4f, Ta4f, Zr3d and Ti2p), we have considered the binding energy splitting as well as the theoretical peak area ratios expected for f, d, and p orbitals (4/3 for 4f7/2/4f5/2; 3/2 for 3d5/2/3d3/2; 2/1 for 2p3/2/2p1/2) as indicated in Section 3.4.

Scanning Electron Microscopy

The cross-sectional morphology of films was caried out using Hitachi SU 8030 SEM (Scanning Electron Microscope) FEG (Field Emission Gun). Their chemical composition is assessed by using SEM Hirow SH-4000M equipped with an energy dispersive x-ray spectrometer (EDS). The measurements were carried out using voltage varying from 10 to 25KV with a working distance of 8 mm.

Nano indentation

Hardness and Young's modulus are measured by using the Nano Indenter XP, MTS Systems equipped with a three-sided pyramidal diamond tip (Berkovitch indenter). The maximum penetration depth is limited to less than 10% of the film thickness, to avoid the influence of the substrate stiffness. Hundred indents were performed for each sample and an average was calculated.

Tribological analysis

The tribological properties of the deposited high entropy metal-sublattice nitrides were investigated at room temperature by using a tribometer (CSM Instruments, High-Temperature Tribometer) under a ball-on-disk configuration. The applied normal load and sliding speed were 2 N and 0.1 m/s, respectively with a frequency of 4 Hz. The sliding distance was 96 m and the counter face was 6 mm Al2O3 balls.

Results and discussion

Deposition rate

Figure 1 shows the deposition rate as a function of RN2. At RN = 0% it is shown that the film sputtering deposition rate is 12.43 ± 0.2 nm/min. When nitrogen content in the gas mixture is increased until RN = 30%, the deposition rate of AlTiZrTaHf(-N) film drastically decreases and reaches 3.65 ± 0. 2 nm/min when RN2 is 30%. The main reason of this decrease is obviously target's poisoning when the nitrogen flow increases [START_REF] Chang | Nitride films deposited from an equimolar Al-Cr-Mo-Si-Ti alloy target by reactive direct current magnetron sputtering[END_REF][START_REF] Lai | Preparation and characterization of AlCrTaTiZr multi-element nitride coatings[END_REF]. Above 30% of the flow rates ratio RN, The deposition rate is quite constant, which means that the target's surface is entirely poisoned by a nitride layer, which has a lower sputtering yield; a factor 3 compared to the metallic sputtering yield. The target's poisoning effect seems to be the predominant.

Elemental composition

Chemical composition of AlTiTaZrHf(-N) films is evaluated using scanning electron microscope equipped with Energy Dispersive Spectroscopy (EDS). The results are shown Figure 2 where an equiatomic target was used for the disposition. Without nitrogen, the film composition shows an atomic percentage varying between 19 and 22 at.% that is estimated according to EDS analysis. The metal content in the film decreases as the nitrogen flow (RN2) increases up to RN2 = 15%. Target poisoning occurs and the sputtering yield of nitrides are very smaller than that of metals resulting in decreasing of metal amount in the films [START_REF] Chen | Microstructure and mechanical properties of (AlCrTiZrV) Nx high-entropy alloy nitride films by reactive magnetron sputtering[END_REF].

Anyway the incorporation of nitrogen in the films comes from a complex phenomenon: the target poisoning, the combination of nitrogen and metallic species in the homogeneous phase and the adsorption of nitrogen on the surface of the growing film. It is believed that the target poisoning is the preponderant phenomenon. For that reason, the deposition rate directly depends on the nitriding conditions of the target. The figure 2 tends to show that the film is a nitride for a value of RN2 of about 10%.

X-ray diffraction analysis

Diffraction patterns of AlTiTaZrHf(-N) HEF, as a function of the nitrogen content in the gas mixture, are shown figure 3. The films obtained at RN2=0% and exhibits an amorphous structure as illustrated by the broader peaks. When the nitrogen flow is increased to 5%, AlTiTaZrHf nitride is still amorphous or nanocrystal. It is clearly seen that this halo shifts towards the lower angles at RN2=5%, which means that nitrogen addition in the film leads to increase the mean distance between atoms first neighbor. High mixing entropy associated with high cooling rates of the condensation are the main reasons of the growth of amorphous phase. The high entropy can increase the mutual solubility between the element constituting the film while the large atomic size can lead to the topological instability promoting the formation amorphous structure [START_REF] Tsai | Effect of nitrogen flow ratios on the structure and mechanical properties of (TiVCrZrY)N coatings prepared by reactive magnetron sputtering[END_REF]. However, as expected (section 3.2), the increase of nitrogen flow rate leads to the growth of a single-phased crystallized material and FCC solid solution is clearly detected above RN2=10%. The diffractograms of this structure show five peaks (111), (200), (220), (311) and (222) as presented in the figure 3. It is no surprise knowing that Al, Ti, Ta and Hf elements can form FCC-structured nitrides with nitrogen. The same trend has been reported in the literature for various nitrides as (TiAlCrSiV)N, (AlCrTaTiZr)N or (AlCrMoSiTi)N [START_REF] Cheng | Structural and mechanical properties of multi-element (AlCrMoTaTiZr)Nx coatings by reactive magnetron sputtering[END_REF][START_REF] Lai | Preparation and characterization of AlCrTaTiZr multi-element nitride coatings[END_REF][START_REF] Ren | Structure and properties of (AlCrMoNiTi) Nx and (AlCrMoZrTi) Nx films by reactive RF sputtering[END_REF][START_REF] Lin | Multi-component nitride coatings derived from Ti-Al-Cr-Si-V target in RF magnetron sputter[END_REF]. Increasing the nitrogen content promotes nitrides growth that can improve the crystallinity of the films. Many studies investigated the effect of nitrogen content in HEFs and revealed that the growth of FCC solid solution phase could be due to the formation of nitrides [START_REF] Cheng | Structural and mechanical properties of multi-element (AlCrMoTaTiZr)Nx coatings by reactive magnetron sputtering[END_REF][START_REF] Chen | Effects of nitriding on the microstructure and properties of VAlTiCrMo high-entropy alloy coatings by sputtering technique[END_REF][START_REF] Ren | Structure and mechanical properties of multi-element (AlCrMnMoNiZr)Nx coatings by reactive magnetron sputtering[END_REF]. In our case, the (AlTiTaZrHf(-N) films exhibit NaCl type structure. This phenomenon can be attributed to high entropy effect that enhance the mutual solubility of different elements. As shown in diffractograms, the nitride films have an out-of plane preferential orientation. The reflection (200) appears for RN2 values higher than 15%. The preferential orientation can be explained by the extended structure zone model (ESZM) [START_REF] Mahieu | Biaxial alignment in sputter deposited thin films[END_REF]. When the energy flow toward the substrate is low, the adatoms hit the surface with nearly negligible mobility and amorphous structure can be formed. However, as the energy flow increases, the mobility of adatoms into the surface increases and crystalline structure will be formed. Mathieu et al. reported that preferred {111} orientation occurs at low N/Ti flow and {100} orientation occurs at high N/Ti flow [START_REF] Mahieu | Biaxial alignment in sputter deposited thin films[END_REF]. According to that, the control of N/Metal atomic flow could promote the preferred {111} or {100} orientation. Diffraction peaks of the NaCl type structure shift towards lower angles (Figure 4a), which means that the lattice distortion increases (with higher lattice parameters of AlTiTaZrHf(-N)) as the nitrogen flow increases. This phenomenon is more pronounced for higher values of RN2. Figure 4b shows a linear increase of the lattice parameter with RN2, which varies between 4.43 and 4.48Å. The peak shift linked to the distortion effect could be due to the residual compressive stress formed after adding the nitrogen [START_REF] Li | Hard and tough (NbTaMoW)Nx high entropy nitride films with sub-stoichiometric nitrogen[END_REF].

X-ray photoelectron spectroscopy (XPS) analyses

X-ray photoelectron spectroscopy was used to probe the surface chemistry of AlTiTaZrHf(-N) films. Figure 5 displays survey spectra and high-resolution core level peaks of the individual elements of the films, obtained at RN2= 0% (red) and RN2= 15% (blue). The survey spectra show peaks assigned to Al2p (~74 eV), Ti2p, H4f-Ta4f, O1s and N1s. Carbon s is assigned to adventitious hydrocarbon contamination whilst O1s is due to metal oxide and contamination of the films surface. Nitrogen is absent in the surface of the film obtained at RN2= 0% while at RN2= 15% a strong peak of N1s is observed (Figures 5a-b). To get a deep knowledge on the surface chemistry of these films, high resolution spectra were recorded for each individual element (Figures 5b-f). XPS measurements were performed one month after deposition.

Oxides formed on the top of the film surfaces upon exposure of the sample to air. On a thermodynamic point of view, oxides can be formed on the extreme surface of the films since they have low enthalpies (ΔHf°) compared to nitrides (Table 1). For this reason, in this section, we focus only on the formation of nitrides.

Without nitrogen, Al2p spectrum (Figure 5c, red spectrum) shows the presence of metallic Al2p. At RN2= 15%, a transformation of metallic Al to AlN can be clearly seen. The peak of metallic Al2p disappears and AlN is observed at 73.2. Compared to red peak, the blue one is broad, therefore showing the presence of AlN (Al2O3 is present because of contamination).

Figure 5d shows that Ti is present at metallic state as judged from the Ti2p3/2 peak position at 453.0 eV [START_REF] Simon | ESCA study of Nb and Ti oxides. Applications to the determination of the nature of the superficial films formed during the oxidation of Nb-Ti and Nb-Ti alloys[END_REF], when RN2= 0%. At RN2= 15%, TiN is detected at 455.1 eV [START_REF] Bagdasaryan | A new type of (TiZrNbTaHf) N/MoN nanocomposite coating: Microstructure and properties depending on energy of incident ions[END_REF]. The Ti2p3/2 peak of metallic Ti disappears after increasing the nitrogen. Zr high resolution spectra (Figure 5e) show the presence of metallic Zr3d5/2 at 177.9 eV [START_REF] Takano | Nitrogenation of various transition metals by N+ 2-ion implantation[END_REF], when RN2= 0%. Increasing the nitrogen flow rate leads to the formation of ZrN at 180.5 eV. The value for ZrN is close to 180.9 eV, reported by Badrinarayanan et al. [START_REF] Badrinarayanan | XPS studies of nitrogen ion implanted zirconium and titanium[END_REF]. Quasi no Zr3d from metallic Zr was detected at RN2= 15% (Figure 5e). Ta exhibits a Ta4f peak doublet assigned to the metallic state; i.e. Ta4f7/2 centred at 21.0 eV [START_REF] Badrinarayanan | X-ray photoelectron spectroscopy studies of the reaction of N+ 2-ion beams with niobium and tantalum metals[END_REF], when RN2= 0% (Figure 5f). At RN2= 15%, metallic Ta disappears and TaN is formed and show Ta4f7/2 peaks centered at 24.3 eV. The same tendency is observed for Hf (Figure 5f). Hf4f7/2 peak due to metallic Hf is observed at 13.2 eV for RN2= 0%. At RN2= 15%, a transformation of metallic Hf to HfN is recorded. This compound is corresponding to Hf4f7/2 peaks centred at 15.6 eV.

For the sake of clarity, we have displayed only the untreated and one HEA doped with nitrogen. For all samples, we have peak fitted all important peaks taking into account the constraints of peak separation in doublets (Hf4f, Ta4f, Zr3d and Ti2p) and the theoretical ratios (4/3 for 4f7/2/4f5/2; 3/2 for 3d5/2/3d3/2; 2/1 for 2p3/2/2p1/2). This permitted to peak fit all spectra and determine surface chemical composition of all samples. By introducing the nitrogen during the deposition process, atomic percent of the elements decreases due to the formation of nitrides. Nitrided elements are then sputtered and deposited into the surface leading to the formation of textured films depending on the introduced nitrogen flow.

Morphology

Fracture cross-sectional SEM images of all film, in function of the nitrogen flow rate, are presented in the figure 6. The films with metallic structure, obtained at RN2= 0% and 5%, have a compact morphology, related the low deposition pressure. Indeed, the mean free path in the plasma is high at low deposition pressure. Thus, the adatoms have enough energy to fill the voids between grains, which increases the film density.

However, from RN2= 10% to 50% a columnar structure emerges. The film at 10% presents columns less dense compared to that of the film at 40%. The density increases as the nitrogen flow increases. Mathieu et al. [START_REF] Mahieu | Biaxial alignment in sputter deposited thin films[END_REF] calculated the energy flux incident at the growing film in the case of TiN and revealed that the mobility of the growing plane increases as the nitrogen flow rate increases. This could be the main reason for the increasing of the density of the film when RN2 varies from 10 to 50%.

Mechanical and tribological properties

Hardness and Young's modulus

The mean grain size of the nitride phase is estimated using the Scherrer equation from the most intense (111) reflection (Figure 7a). The mean grains size is ranged from 300 to 480 nm and 3 zones (R1, R2 and R3) can be distinguished. The mean grain size decreases when RN2 is between 10% and 15% (R1 zone) and above RN2=30% (R3 zone). It increases when RN2 is in the range 15% to 30%.

Evolution of mechanical properties (Hardness and Young Modulus) of AlTiTaZrHf(-N) is presented in figure 7b. Hardness and Young's modulus are in the ranges 7.85 GPa to 27.67 RN2=20% to 30%, the friction coefficient increases. However, when the nitrogen continuous to increase > 30% the friction coefficient decreases. This trend my due to increasing of nitrogen content in the film increases leading to the refining of the structure.

As mentioned above, XRD analysis (cf. figure 4) revealed a widening of peaks when RN2>30%.

The results are in good agreement with the calculated H/E and H 3 /E 2 reports revealing the high mechanical and tribological properties of the film obtained at RN2=10%.

Conclusion

AlTiTaZrHf(-N) HEFs are deposited by magnetron sputtering of a target with equiatomic composition in different Ar+N2 atmospheres. The increase of flow rates ratio (RN2=N2/Ar+N2) in the chambre leads to a drop in the deposition rate up to RN2 of 30% and a decrease in metallic elements content up to RN2 of 15% and these values stabilize for the higher nitrogen contents in the gas mixture. Films are amorphous single-phased without nitrogen and a FCC single phased material is detected for higher RN2 flow rates ratios. All the as-deposited films have a non-columnar compact morphology.

Maximum values of hardness and Young's modulus are measured at 27.67 GPa and 205.56

GPa respectively and they are obtained for film deposited at RN2=10%. The increase of hardness and Young's modulus is mainly explained by nitride growth and evolution of the mean grain size. The film obtained at RN2=10% exhibits the highest H/E and H 3 /E 2 ratios and good tribological performances are then expected. This film has the lowest friction coefficient against aluminum oxide ball (Φ 6 mm) due to the compact film. XPS analysis revealed the formation of nitrides of all elements constituting the films. Oxides are formed on the surface of films due to their low enthalpy energy compared to that of nitrides.

Figure 1 .

 1 Figure 1. Sputtering deposition rate of AlTiZrTaHf(-N) high entropy films as a function of RN2.

Figure 2 .

 2 Figure 2. Chemical composition of AlTiTaZrHf(-N) films as a function of RN2.

Figure 3 .

 3 Figure 3. Diffractograms of AlTiTaZrHf(-N) in function of RN2.

Figure 4 .

 4 Figure 4. a) Zoom of diffractograms showing the shift of (111) and (200) peaks. b) Lattice parameter evolution as a function of RN2.

Figure 5 .

 5 Figure 5. XPS spectra of AlTiTaZrHf(-N) films at RN2 = 0% (red, bottom spectra) and RN2 =15% (blue, upper spectra): survey (a), N1s (b), Al2p (c), Ti2p (d), Zr3d (e), and Hf4f-Ta4f (f).

Figure 6 .

 6 Figure 6. Cross-sectional SEM image of AlTiTaZrHf(-N) as a function of RN2.

Figure 9 .

 9 Figure 9. Fiction coefficient of AlTiTaZrHf(-N) films as a function of RN2.

Table 1 .

 1 Enthalpy of formation of oxides and nitrides of different elements constituting the films[START_REF] Elder | Thermodynamics of ternary nitride formation by ammonolysis: application to lithium molybdenum nitride (LiMoN2), sodium tungsten nitride (Na3WN3), and sodium tungsten oxide nitride (Na3WO3N)[END_REF][START_REF] Ushakov | Carbides and nitrides of zirconium and hafnium[END_REF][START_REF] Kornilov | The enthalpy of formation of hafnium dioxide[END_REF].

Acknowledgements

This works was financially supported by Groupement d'Intérêt Public Haute-Marne and Commissariat à l'Energie Atomique et aux Energies Alternatives (CEA). Annual funding (no funding number).

GPa and 109.33 GPa to 205.5 GPa, respectively. The microstructure refinement and the lattice distortion seem to drive the mechanical properties when the nitrogen content in the gas mixture increases.

Toughness and crack resistance

In general way, the tribological properties of films can be predicted via their mechanical properties under nanoindentation. Toughness and crack resistance can be predicted from two ratios: elasticity index (H/E) and plasticity index (H 3 /E 2 ) respectively [START_REF] Leyland | On the significance of the H/E ratio in wear control: a nanocomposite coating approach to optimised tribological behaviour[END_REF]. Higher values of both H/E and H 3 /E 2 ratios may indicate good tribological performance of the films. The maximum values of H/E and H 3 /E 2 (0.135 and 0.5 respectively) are obtained for AlTiTaZrHf(-N) film deposited at RN2=10% (Figure 8). As RN2 increases from 0% to 10%, both H/E and H 3 /E 2 ratios increase from 0.071 to 0.135 and from 0.041 to 0.050, respectively. However, when RN2 increase H/E remains constant until RN2=30% followed by a decreasing to 0.111 while H 3 /E 2 starts to decrease and reaches 0.253 (Figure 8).

Ren et al. reported a variation of tribological properties of (AlCrMnNiZrB0.1)Nx films as a function of nitrogen flow rate [START_REF] Ren | Structure and properties of (AlCrMnMoNiZrB0.1)Nx coatings prepared by reactive DC sputtering[END_REF]. The results reveal a variation of H/E and H 3 /E 2 ratios depending on the variation of N/Ar flow rate. High value of H/E and H 3 /E 2 are obtained of (AlCrMnNiZrB0.1)Nx deposited with N2/Ar=0.5. Furthermore, the same film i.e. deposited at N2/Ar=0.5, exhibits a low friction coefficient. This result confirms that the high values of H/E and H 3 /E 2 ratios are associated with good tribological properties of nitrides films. In our case, high H/E and H 3 /E 2 are obtained for film RN2=10% which could present excellent property for the resistance to plastic deformation.

Friction coefficient


Mirjana Mladenović
email: mirjana.mladenovic@univ-eiffel.fr

Marius Lewerenz
email: marius.lewerenz@univ-eiffel.fr

Accurate equilibrium structures of linear triatomic molecules from a combined theoretical-experimental method: The protonated nitrogen molecule, HN + 2

Keywords: HN + 2, Equilibrium structure, r 0 structure, Rovibrational calculations, α constants, Theoretical spectroscopy

This article examines several strategies which can be used to derive molecular equilibrium structure from experimental data and extends our previously suggested method to HN + 2 . Full-dimensional rovibrational calculations are carried out for eight isotopologues of HN + 2 using two ab initio potential energy representations. Rovibrational energies computed for rotational angular momenta J as high as 15 in both parities are used to derive theoretical vibration-rotation corrections to ground-state rotation constants B 0 . These results are combined with the available experimentally derived B 0 values to determine the r 0 , r α , and r e structures of the ion. Higher-order corrections beyond the vibration-rotation α constants are found to be essential for the structure determination. Our analysis is supported by graphical representations, illustrating the anticorrelation of the structural parameters. The r

Introduction

Determination of the equilibrium structure of molecules is considered as one of the most basic goals of spectroscopy. Geometric parameters can be unambiguously deduced only from equilibrium rotational constants B e , which are unfortunately not directly accessible experimentally. Theoretical models have to be used to account for the effect of the vibration-rotation coupling and to provide corrections to the effective rotational constants B v for a vibrational state v, with a special interest in the zero-point correction, ∆B 0 = B e -B 0 .

Except for diatomics and nonlinear triatomics the conversion of equilibrium rotational constants into an equilibrium structure requires informations on more than one isotopologue. The standard procedure assumes a common mass-independent structure, which is equivalent to the Born-Oppenheimer picture.

The concept of the equilibrium structure is theoretically well founded within the Born-Oppenheimer approximation, that is, a molecule is described by a mass-independent potential energy surface (PES), so that the minimum of the PES uniquely defines the equilibrium structure of the molecule. Numerically exact quantummechanical calculations are then needed to establish a direct link between a given PES and spectroscopic observables. Such an approach employs the exact rovibrational kinetic energy operator (no dynamical approximation) and the full PES (no re-expansion) in combination with numerically convenient methods, yielding a complete answer about molecular rovibrational dynamics associated with the applied PES. A common representation connecting experimental approaches and theory can be obtained by fitting the computed rovibrational energies to appropriate spectroscopic Hamiltonians.

We have recently developed a two-step procedure for deriving molecular equilibrium structures from numerically exact rovibrational energies [START_REF] Mladenović | Theoretical spectroscopic parameters for isotopic variants of HCO + and HOC +[END_REF][START_REF] Mladenović | Comparison of spectroscopic strategies to determine molecular geometries and the impact of nuclear versus atomic masses: the example of HCO + and HOC +[END_REF]. This approach was applied to determine spectroscopic parameters for isotopic variants (isotopologues) of the formyl cation, HCO + , and the isoformyl cation, HOC + , in excellent agreement with available experimental results [START_REF] Mladenović | Theoretical spectroscopic parameters for isotopic variants of HCO + and HOC +[END_REF].

The aim of the present work is to provide a similar spectroscopic characterization of the protonated nitrogen molecule, HN + 2 , also known as diazenylium. This is a linear molecular ion, characterized by a degenerate bending ν 2 vibration, a symmetric stretching ν 1 vibration, and an antisymmetric stretching ν 3 vibration.

In the present paper, we use two three-dimensional potential energy surfaces, developed for HN + 2 in its ground electronic state, X 1 Σ + , by ab initio computation. The first is the CCSD(T)/cc-pVQZ PES of Schmatz [START_REF] Schmatz | Intramolecular dynamics and density of states for HN + 2[END_REF][START_REF] Schmatz | Globale Potentialhyperflächen und genaue Schwingungszustandsdichten am Beispiel der Kationen HN + 2 und HCO +[END_REF], hereafter called SM PES. This is a global potential energy representation with two equivalent colinear minima, separated by an isomerization barrier 17 137 cm -1 above the energy minima at the linear geometries. Low-lying states of HN + 2 are localized in one of the two wells. The double-well and nuclear spin symmetries are lifted for mixed nitrogen isotope forms. These issues are relevant when studying, for instance, interstellar isotope fractionation [START_REF] Mladenović | Ion-molecule reactions involving HCO + and N 2 H + : Isotopologue equilibria from new theoretical calculations and consequences for interstellar isotope fractionation[END_REF].

As the second PES, we use the ACTQ5+rel+ACPF quartic internal coordinate force field (QFF) of Huang, Valeev, and Lee [START_REF] Huang | Comparison of one-particle basis set extrapolation to explicitly correlated methods for the calculation of accurate quartic force fields, vibrational frequencies, and spectroscopic constants: Application to H 2 O, N 2 H + , NO + 2 , and C 2 H 2[END_REF], hereafter called HVL QFF. This PES includes one-particle basis set extrapolation, core-correlation, and scalar relativity, as well as higher-order correlation terms obtained by the averaged coupled-pair functional (ACPF) method. Huang et al. also reported corresponding vibrational frequencies and rotational parameters, computed by second-order vibrational perturbation theory. For our rovibrational variational calculations, HVL QFF was first transformed to Morse-cosine coordinates, as recommended previously [START_REF] Huang | Comparison of one-particle basis set extrapolation to explicitly correlated methods for the calculation of accurate quartic force fields, vibrational frequencies, and spectroscopic constants: Application to H 2 O, N 2 H + , NO + 2 , and C 2 H 2[END_REF][START_REF] Dateo | An accurate quartic force field and vibrational frequencies for HNO and DNO[END_REF].

Quantum mechanical calculations of rovibrational energies for the protonated nitrogen molecules are performed combining a discrete variable representation for the angular motion with distributed Gaussian basis functions for the radial coordinates [START_REF] Mladenović | Highly excited vibration-rotation states of floppy triatomic molecules by a localized representation method: The HCN/HNC molecule[END_REF][START_REF] Mladenović | Theoretical study of the rovibrational energy spectrum and the numbers and densities of bound vibrational states for the system HCO + /HOC +[END_REF]. Atomic masses are applied, as commonly done in similar studies. The computed rovibrational energies are used to extract the parameters of appropriate spectroscopic Hamiltonians in a least-squares analysis and to determine theoretical vibration-rotation corrections to the zero-point rotational constant B 0 (Section 2). The results for HN + 2 and DN + 2 are first employed to derive the stucture of the ion (Section 3) and to analyse the correlation between structural parameters (Section 4).

The scope of the study is then broadened to include eight experimentally observed isotopologues (Section 5). To analyze the notion of the r 0 struc-ture, obtained directly from the B 0 values, a comparative study of the isoelectronic series of linear triatomic molecules HCO + , HCN, HN + 2 , HNC, and HOC + is carried out, revealing that the r 0 structure can be related to a structure projected onto the principal a axis (Section 6).

Our theoretical procedure produces a highly consistent set of data, obtained within the Born-Oppenheimer approximation. Limitations arising from this approximation are also reviewed (Section 7).

Corrections to zero-point rotational constants

Following common practice of experiment, the rovibrational energies computed for a vibrational Σ state v are fitted to a standard spectroscopic power series expansion,

E v (J) =T v + B v J(J + 1) -D v J 2 (J + 1) 2 + • • • , (1)
where T v is the vibrational term energy, B v the effective rotational constant, D v the quartic centrifugal distortion constant, and so on, in a vibrational state v. For a vibrational Π state, we use

E v (J) = T v + B v J(J + 1) -2 -D v J(J + 1) -2 2 + • • • (2) ± 1 2 q v J(J + 1) + q J v J 2 (J + 1) 2 + • • • ,
where the rotational dependence of the -type doubling contribution is expressed in terms of J(J + 1), using the constants q v , q J v , and so on. We define the ground state vibrational correction ∆B 0 to the equilibrium rotational constant B e as

∆B 0 = B th e -B th 0 , (3)
where B th e and B th 0 are theoretical values of the rotational constant at equilibrium and in the ground vibrational state, respectively. In keeping with the notation previously used [START_REF] Mladenović | Theoretical spectroscopic parameters for isotopic variants of HCO + and HOC +[END_REF][START_REF] Mladenović | Comparison of spectroscopic strategies to determine molecular geometries and the impact of nuclear versus atomic masses: the example of HCO + and HOC +[END_REF], the spectroscopic correction S 0 to B e is computed as

S 0 = 1 2 (α 1 + α 3 ) + α 2 , where α i = B 0 -B i , (4)
with α i denoting the vibration-rotation interaction constant for the ith vibrational mode. Replacing B 0 and B i with their theoretical values, we obtain the theoretical estimate S th 0 . The spectroscopic correction S 0 is accessible experimentally when the rotational constants for the ground state and the three fundamental vibrations are all known.

In the present work, rovibrational calculations for HN + 2 and DN + 2 are carried out for total rotational angular momenta as high as J = 15 for both SM PES and HVL QFF. The rovibrational energies are fitted using the thirdorder, fourth-order, and fifth-order spectroscopic Hamiltonians of Eqs. [START_REF] Mladenović | Theoretical spectroscopic parameters for isotopic variants of HCO + and HOC +[END_REF] and (2), producing three sets of parameters called Fit 3, Fit 4, and Fit 5, respectively. The fitted rotational energies go up to approximately 370 cm -1 (11 THz). The fit standard deviations are better than 50 kHz for Fit 3, 40 Hz for Fit 4, and 5 Hz for Fit 5. A linear unweighted least-squares method [START_REF] Press | Numerical Recipes[END_REF] is applied to obtain Fit 3-Fit 5, implicitly assuming that the computed rovibrational energies form consistent sets of data of uniform quality. This assumption is reasonable for the modelling of the rotational transitions within a single vibrational state.

The potential energy minima of SM PES and HVL QFF are collected in Table 1, along with the associated equilibrium rotational constants B e .

The equilibrium bond distances r e for SM PES are approximately 0.003 Å Comparison of the values for ∆B 0 and S th 0 from Fit 4 and Fit 5 shows that the physical uncertainties related to the choice of the fitting model (number of centrifugal distortion constants) are at most 5 •10 -6 MHz. This indicates an extremely high internal consistency in the set of computed rovibrational energy levels but is unrelated to differences with respect to experiment.

In Table 1, the values of ∆B 0 obtained for the two potential energy surfaces agree within 0.4 MHz and the values of S th 0 within 1.8 MHz. For a given PES, the difference between ∆B 0 and S th 0 is smaller than 10 Mz, ∆B 0 being smaller (larger) than S th 0 for HN + 2 (DN + 2 ). In our previous studies, we found that ∆B 0 and S th 0 differ by up to 50 MHz for the isotopic variants of HCO + and even 100 MHz for the isotopic variants of HOC + [START_REF] Mladenović | Theoretical spectroscopic parameters for isotopic variants of HCO + and HOC +[END_REF][START_REF] Mladenović | Comparison of spectroscopic strategies to determine molecular geometries and the impact of nuclear versus atomic masses: the example of HCO + and HOC +[END_REF].

From the experimental rotational constants B exp v reported by Yu et al. [START_REF] Yu | Rotational spectroscopy of vibrationally excited N 2 H + and N 2 D + up to 2.7 THz[END_REF] for the ground vibrational state and three fundamental vibrations, the experimental vibration-rotation interaction constants [ α exp 1 , α exp 2 , α exp 3 ] are determined to be [ 377. 9, 104.5, 330.2 ] ) is somewhat fortuitous in this case.

3. The equilibrium structure from the measurements on two isotopologues

The moment of inertia of a linear triatomic molecule ABC, with masses m A , m B , and m C lying along the molecular axis, is

I = m A m BC M d 2 1 + m AB m C M d 2 2 + 2 m A m C M d 1 d 2 , (5)
where c m = m C /m BC , the moment of inertia I reads

I = µ r r 2 + µ R R 2 , (7)
with

µ r = m B m C /m BC and µ R = m A m BC /M . Geometric
I (i) e = 2 2 B (i) e (8)
for i = 1 and 2. The equilibrium distances follow as

r 2 e =F 1 m µ (2) R I (1) e -µ (1) R I (2) e , R 2 e =F 1 m µ (1) r I (2) e -µ (2) r I (1) e , (9)
where

F m = µ (1) r µ (2) R -µ (2) r µ (1) R , (10)
implying

d 2e = r e and d 1e = R e -1 2 r e . (11)
Uncertainties of the equilibrium distances can be estimated from the uncertainties of the equilibrium rotational constants by applying rules for error propagation. The corresponding results are summarized in Table 2, which Table 2: Parameter uncertainties from the error propagation, with δ q denoting the uncertainty of q and σ (i) the uncertainty of the equilibrium rotational constant

B (i)
e of the ith isotopologue. The mass factor F m is defined by Eq. ( 10).

Parameter Uncertainty

I (i) e δ (i) I = I (i) e [B (i) e ] 1 σ (i) r e δ r = 1 2 r 1 e F 1 m µ (2)
R δ

(1)

I + µ (1) R δ (2) I R e δ R = 1 2 R 1 e F 1 m µ (2)
r δ

(1)

I + µ (1) r δ (2) I d 1e δ d1 = δ R + 1 2 δ r
shows that the uncertainties of the structural parameters are linearly proportional to the uncertainties σ (i) of B

e . Rotational constants from microwave measurements are obtained regularly with much better accuracy than those from infrared measurements, so they are the preferred choice for the structure determination.

In the derivation of geometric parameters from experimentally determined B exp 0 values, three types of corrections to ground-state rotation constants are considered, leading to three different types of structure definitions. The use of B exp 0 , uncorrected for vibrational effects, provides the so-called r 0 structure. Applying the spectroscopic correction S 0 to B exp 0 , we generate a structure called the r α structure. Finally, B exp 0 + ∆B 0 yields the r e structure.

The geometric r 0 , r α , and r e parameters derived using only data for HN + 2 and DN + 2 are summarized in Table 3. The millimeter wave spectroscopy values of Yu et al. [START_REF] Yu | Rotational spectroscopy of vibrationally excited N 2 H + and N 2 D + up to 2.7 THz[END_REF] 1. The structural parameters were then obtained using the analytical solutions of Eqs. ( 9)- [START_REF] Yu | Rotational spectroscopy of vibrationally excited N 2 H + and N 2 D + up to 2.7 THz[END_REF].

The error limits in Table 3 are determined with the help of Table 2. The accuracy of the experimental B exp 0 values is in order of 10 -4 MHz. As the accuracy of S 0 and ∆B 0 from Fit 5 in Table 1 is 10 -7 MHz or better, the uncertainties of the structural parameters in Table 3 are primarily defined by

σ i of B exp 0 .
This is important so as not to spoil the information contained in B exp 0 . We also note that Fit 4 provides values identical to those obtained using Fit 5. The results derived for Fit 3, which have approximately twice the uncertainty, are found to be statistically identical to the values shown in Table 3. The accuracy of the bond lengths is estimated to be much better than 10 -6 Å in Table 3. This information should be viewed only as a measure of quality of the data sets and the procedure applied in the derivation according to the chosen mathematical model.

In Table 3, the values of r α (HN) are longer than r e (HN) by approximately 0.0015 Å, whereas r α (NN) are shorter than r e (NN) by approximately 0.0003 Å. The r 0 structure, computed from the uncorrected experimental B exp 0 values, differ from r α and r e by up to 0.004 Å. The results derived for SM PES and HVL QFF agree with each other within 9.5•10 -5 Å for the r α structure and within 4•10 -5 Å for the r e structure, differing thus beyond the stated error estimates. Bearing in mind that the two ab initio potential energy surfaces are of different nature, the agreement of the resulting structural parameters can be viewed as very satisfying.

The last two rows in Table 3 show experimental equilibrium structures.

Owrutsky et al. [START_REF] Owrutsky | Determination of the equilibrium structure of protonated nitrogen by high resolution infrared laser spectroscopy[END_REF] combined their infrared measurements with microwave a Values in parentheses show error limits obtained using Table 2.

and infrared data coming from different experimental groups, and fitted all spectral data in a global least-squares analysis to a general spectroscopic expansion of the form

G v, ,J = 3 i v i + 1 2 d i ω i + 3 i,k≥i v i + 1 2 d i v k + 1 2 d k x ik + B v ± 1 2 q v J(J + 1) -2 -D v J(J + 1) -2 2 , ( 12
)
where the effective rotational constant B v for the vibrational state v is given by

B v =B e - 3 i β i v i + 1 2 d i + 3 i,k γ ik v i + 1 2 d i v k + 1 2 d k . (13)
In the model of Eq. ( 12), the equilibrium rotational constant B e appears as a fitting parameter. The constant β i in Eq. ( 13) is sometimes called α i and is close to, but actually not identical with the α i constant defined in Eq. ( 4)

as α i = B 0 -B i .
For the constant β 1 , for instance, we have

β 1 = B 0 -B 1 + 2γ 11 + γ 12 + 1 2 γ 13 = α 1 + 2γ 11 + γ 12 + 1 2 γ 13 . (14)
Another set of experimental data in Table 3 is due to Amano et al. [START_REF] Amano | Submillimeter-wave spectroscopy of HN + 2 and DN + 2 in the excited vibrational states[END_REF],

who reported submillimeter-wave spectroscopy measurements on HN + 2 and DN + 2 in the ground state and for all fundamentals. To estimate higher-order corrections to the ground-state rotation constants, Amano et al. [START_REF] Amano | Submillimeter-wave spectroscopy of HN + 2 and DN + 2 in the excited vibrational states[END_REF] applied higher-order vibration-rotation coupling constants γ ij available from other experimental studies on HN + 2 and a similar system (HCN). The experimental results in Table 3 differ from each other by up to 0.001 Å for the distances and by 8 MHz for B e , i.e. considerably beyond the stated error limits. To investigate this situation, we compare in Table 4 the structural r α parameters obtained using three sets of experimental data, which include in addition to B 0 also the values of B 1 , B 2 , and B 3 [START_REF] Yu | Rotational spectroscopy of vibrationally excited N 2 H + and N 2 D + up to 2.7 THz[END_REF][START_REF] Owrutsky | Determination of the equilibrium structure of protonated nitrogen by high resolution infrared laser spectroscopy[END_REF][START_REF] Amano | Submillimeter-wave spectroscopy of HN + 2 and DN + 2 in the excited vibrational states[END_REF]. As seen there, the three experimental r α structures are statistically identical, implying that the difference between the experimentally derived equilibrium bond distances r e in Table 3 is solely due to the use of higher-order vibrationrotation contributions.

Table 4 also provides the parameters of the pure theoretical r α structure, determined from the theoretical B th 0 , B th 1 , B th 2 , and B th 3 values calculated for SM PES and HVL QFF. Comparing the r α distances of Table 4 with the corresponding equilibrium r e bond lengths shown in Table 1, we see that r α (HN)-r e (HN)=0.0015 Å and r α (NN)-r e (NN)=-0.0003 Å for both of the PESs. The r α results do not reproduce the r e values, implying that S 0 is not sufficient as a zero-point correction. This finding is important for both experimental and theoretical spectroscopic studies. At the same time, the rotational constant B α = B 0 + S 0 in Table 4 differs from the true B e of Table 1 by only a few MHz (up to 6 MHz).

The higher-order vibration-rotation contribution C 0 to the ground-state rotational constant is defined here by The explicit values of C 0 are shown in Table 5. There, C 0 are up to about 12 MHz (0.000 4 cm -1 ), so they are much smaller than S 0 of Table 1. Nevertheless, these higher-order correction terms C 0 have important consequences on the structural parameters, as easily visible in Table 3.

C 0 = B α -B e , so that C 0 = S 0 -∆B 0 . ( 15
)
Whereas the theoretical C 0 corrections for HN + 2 and DN + 2 have different signs in Table 5, the experimental C 0 values for the two isotopic variants have the same sign. Since no sufficient information was available for DN + 2 , Amano et al. [START_REF] Amano | Submillimeter-wave spectroscopy of HN + 2 and DN + 2 in the excited vibrational states[END_REF] estimated C 0 (DN + 2 ) from the value C 0 (HN + 2 ) by assuming its proportionality to the square of the rotational constant, so that C 0 (HN + 2 ) and C 0 (DN + 2 ) are both positive in Table 5. In the fitting procedure to Eq. ( 12), Owrutsky et al. [START_REF] Owrutsky | Determination of the equilibrium structure of protonated nitrogen by high resolution infrared laser spectroscopy[END_REF] effectively included only one of the six higher-order coupling constants γ ij , namely γ 12 , which is reported to be γ 12 (HN + 2 )= 9.31 [START_REF] Kraitchman | Determination of molecular structure from microwave spectroscopic data[END_REF] MHz and γ 12 (DN + 2 )= 24.60(78) MHz. In view of Eq. ( 13), we then have C 0 = -γ 12 /2, so that C 0 (HN + 2 ) = -4.6 MHz and C 0 (DN + 2 ) = -12.3 MHz, as seen in Table 5.

Correlation between the structural parameters

Replacing I, d 1 , and d 2 in Eq. ( 5) with their values at equilibrium, we easily find that

d 1e = - m C d 2e m BC + M m A m BC I e - m B m C m BC d 2 2e ( 16
)
or equivalently [12] and Amano et al. [START_REF] Amano | Submillimeter-wave spectroscopy of HN + 2 and DN + 2 in the excited vibrational states[END_REF] for HN + 2 and DN + 2 . The values of B e (HN + 2 ) are listed in Table 3. The corresponding B e (DN + 2 ) values are reported to be 38 722.9 [START_REF] Špirko | Energies and electric dipole moments of the bound vibrational states of HN + 2 and DN + 2[END_REF] and 38 708.38(58) MHz in [START_REF] Owrutsky | Determination of the equilibrium structure of protonated nitrogen by high resolution infrared laser spectroscopy[END_REF] and [START_REF] Amano | Submillimeter-wave spectroscopy of HN + 2 and DN + 2 in the excited vibrational states[END_REF], respectively. The case involving B exp e of Owrutsky et al. [START_REF] Owrutsky | Determination of the equilibrium structure of protonated nitrogen by high resolution infrared laser spectroscopy[END_REF] is denoted with X Ow and of Amano et al. [START_REF] Amano | Submillimeter-wave spectroscopy of HN + 2 and DN + 2 in the excited vibrational states[END_REF] with X Am , using X=H and X=D for HN + 2 and DN + 2 , respectively. The point of intersection of two lines is a structure (d c 1e , d c 2e ) common to both isotopic forms, meaning that (d c 1e , d c 2e ) simultaneously reproduces the rotational constants B e (HN + 2 ) and B e (DN + 2 ). The triangular points ( ) marked with Ow and Am in Fig. 1 are the experimental equilibrium distances reported by Owrutsky et al. [START_REF] Owrutsky | Determination of the equilibrium structure of protonated nitrogen by high resolution infrared laser spectroscopy[END_REF] and Amano et al. [START_REF] Amano | Submillimeter-wave spectroscopy of HN + 2 and DN + 2 in the excited vibrational states[END_REF], respectively, given explicitly in Table 3. The equilibrium bond distances derived from the results of Owrutsky et al. [START_REF] Owrutsky | Determination of the equilibrium structure of protonated nitrogen by high resolution infrared laser spectroscopy[END_REF] using the analytical solutions of Eqs. ( 9)-( 11 The experimental r e (NN) distances from [START_REF] Owrutsky | Determination of the equilibrium structure of protonated nitrogen by high resolution infrared laser spectroscopy[END_REF] and [START_REF] Amano | Submillimeter-wave spectroscopy of HN + 2 and DN + 2 in the excited vibrational states[END_REF] agree within the larger error bar of the earlier study, while the r e (HN) distances are statistically incompatible. In order to quantitatively account for the intrinsic strong correlation between these structural parameters we used the stated error bars of the experimental B e values and a Gaussian error distribution model and Eqs. ( 16) and [START_REF] Kraitchman | Determination of molecular structure from microwave spectroscopic data[END_REF] to construct the resulting probability densities P (r e (HN), r e (NN)) for the two experiments. The results are shown in Fig. 2.

d 2e = - m A d 1e m AB + M m AB m C I e - m A m B m AB d 2 1e . (17)
For an easier comparison both probability densities have been scaled such that their maximum value is equal to one. These most likely r e (HN), r e (NN) combinations correspond to the line intersections in Fig. 1 The r 0 , r α , and r e values computed from the data for eight isotopologues are summarized in Table 6. They are obtained in weighted least-squares fits.

The laboratory results of Yu et at. [START_REF] Yu | Rotational spectroscopy of vibrationally excited N 2 H + and N 2 D + up to 2.7 THz[END_REF] and Dore et al. [START_REF] Dore | Doubly 15 N-substituted diazenylium: THz laboratory spectra and fractionation models[END_REF] are employed, with experimental uncertainties varying from 0.000 13 MHz (D 15 N +

2 ) [START_REF] Dore | Doubly 15 N-substituted diazenylium: THz laboratory spectra and fractionation models[END_REF] to 0.000 93 MHz (DN +

2 ) [START_REF] Yu | Rotational spectroscopy of vibrationally excited N 2 H + and N 2 D + up to 2.7 THz[END_REF]. The uncertainties for the theoretical vibrationrotation corrections are taken from Table 1. To test the sensitivity of our results to these values, we also carried out unweighted nonlinear least-squares fits, denoted by Fit U in Table 6. The equilibrium distances found in the original and Fit U agree within 2•10 -6 Å. In Set C, the value for B exp 0 (HN + 2 ) given as 46 586.875 29(49) MHz in [START_REF] Yu | Rotational spectroscopy of vibrationally excited N 2 H + and N 2 D + up to 2.7 THz[END_REF] is replaced with the value of Cazzoli et al. [START_REF] Cazzoli | Precize THz measurements of HCO + , N 2 H + , and CF + for astrophysical observations[END_REF] reported to be 46 586.875 49 [START_REF] Costain | Determination of molecular structures from ground state rotational constants[END_REF] MHz. The resultant r e values for Set C agree within 8•10 -6 Å with r e from the original fits and are compatible with them within one standard deviation. In Fit F i , we employed r e associated with one PES in combination with the corrections ∆B 0 from the other PES. In Fit F 2 for SM PES, for instance, r e (NN) is kept constant at the value 1.093 017 Å associated with HVL QFF, yielding the bond distance r e (HN)=1.033 219(5) Å, which agrees with r e (HN)=1.033 209 [START_REF] Mladenović | Theoretical study of the rovibrational energy spectrum and the numbers and densities of bound vibrational states for the system HCO + /HOC +[END_REF] Å

for HVL QFF within one standard deviation. This increase of r e (HN) resulting from the shortening of r e (NN) is consistent with the fact that r e (NN)

and r e (HN) are anticorrelated.

The equilibrium distances in Table 6 show spreads in the r e (HN) and r e (NN) values of 5•10 -5 Å and 2•10 -5 Å, respectively. The r e (HN) values for b Root-mean-square error (rms) of the residuals is in MHz. one of the deuterium-containing forms, cross each other in the neighbourhood of r g and define a compatible molecular geometry. The spreads of these crossing points around r g become smaller when viewing Fig. 3 from the top to the bottom. Theoretical corrections computed for SM PES were applied in Fig. 3. Graphical representations of the results obtained using HVL QFF are identical to Fig. 3.

The Born-Oppenheimer approximation defines a mass independent common structure for all isotopologues. Consequently the correlation lines for all isotopologues should intersect in a single point. A residual spread could be indicative of non Born-Oppenheimer effects. It is clear that a search for such effects requires highly consistent equilibrium structure reconstructions and our proposed mixed experimental-theoretical method visibly shows the best internal consistency of the three strategies.

A pair determination of the molecular structure is frequently used in the literature. In the case when experimental data are known for n isotopic species, they can be paired in n(n -1)/2 different ways. From eight isotopic variants of HN + 2 , we thus have 28 possible pair solutions. These are shown for the case when B est = B exp 0 + ∆B 0 in Fig. 3 (bottom). Only 18 solutions are, however, visible over the range shown there. The mean values r ap of all 28 pair solutions are 1.033 27 Å for r ap (HN) and 1.093 01 Å for r ap (NN), yielding B ap (HN + 2 )=46 827.5 MHz. Hence, r ap (HN) is 10 -4 Å longer and r ap (NN) is 2•10 -5 Å shorter than their r e counterparts in Table 6. The r ap structure is sensitive to the choice of pairs or to the choice of parent species when Kraitchman-Costain substitution formulas [START_REF] Kraitchman | Determination of molecular structure from microwave spectroscopic data[END_REF][START_REF] Costain | Determination of molecular structures from ground state rotational constants[END_REF] are applied. In view of Fig. 3, we see that nonlinear least-squares methods appear more appro-priate for the determination of molecular structures than the pair solution approach. This conclusion is in line with our previous finding for HCO + and HOC + [START_REF] Mladenović | Comparison of spectroscopic strategies to determine molecular geometries and the impact of nuclear versus atomic masses: the example of HCO + and HOC +[END_REF]. 15 N-substituted forms of HN + 2 were first observed by Szanto et al. [START_REF] Szanto | A microwave substitution structure for protonated nitrogen N 2 H +[END_REF]. From the rotational J = 0 -1 transition available for six isotopic forms at the time, these authors deduced a r 0 structure given by r s (HN) = 1.032 00( 10 in good agreement with the r 0 parameters of Table 6. The r 0 values derived here from the experimental data of Szanto et al. [START_REF] Szanto | A microwave substitution structure for protonated nitrogen N 2 H +[END_REF] and of Warner [START_REF] Warner | The microwave spectroscopy of ions and other transient species in dc glow and extended negative glow discharges[END_REF] are recommended for future reference.

The r 0 structure

To a first approximation, a molecule can be viewed as a set of oscillators, each of which executes vibrations about a well-defined equilibrium position r e . The actual bond lengths in a vibrating molecule are then expected to be larger than the corresponding r e values, that is, r 0 > r e for the ground vibrational state. This relation does not necessarily hold for linear molecules. We see, for instance, that r 0 (HN) < r e (HN) for HN + 2 in Tables 3 and6. In the present rovibrational calculations, we have employed orthogonal (Jacobi) internal coordinates (r, R, θ) in the body-fixed formulation, where r is the NN bond length, R the distance from the proton to the center of mass of the N-N subunit, and θ the angle between the Jacobi vectors r and R. The body-fixed z axis is aligned with the bond-distance vector r and the bodyfixed z ∧ x plane coincides with the molecular plane [START_REF] Mladenović | Rovibrational hamiltonians for general polyatomic molecules in spherical polar parametrization. I. Orthogonal representations[END_REF]. Our coordinates are schematically depicted in Fig. 4 for the case of a general triatomic molecule ABC.

The rotational contribution to the kinetic energy is strictly diagonal for the principal axis system. The direction perpendicular to the molecular plane is always a principal axis for triatomic molecules, that is, c = y. As seen in Fig. 4, the principal a and b axes are rotated in the molecular plane by an angle ψ relative to the body-fixed z and x axes, where

tan(2ψ) = - µ R R 2 sin(2θ) µ R R 2 cos(2θ) + µ r r 2 . ( 18
)
The angle ψ is the third Euler angle defining the instantaneous principal axis reference frame.

The vibrationally averaged structure of a linear molecule is nonlinear.

This is seen in Table 7, where we compare the vibrationally averaged geometries in the ground vibrational state for the isoelectronic series of linear triatomic molecules HCO + , HCN, HN + 2 , HNC, and HOC + . Both hydrogen containing and deuterium containing isotopic forms are considered. In these calculations, SM PES is used for HN + 2 , a global three-dimensional CCSD(T)/cc-pVQZ PES for HCO + /HOC + [START_REF] Mladenović | Theoretical study of the rovibrational energy spectrum and the numbers and densities of bound vibrational states for the system HCO + /HOC +[END_REF], and a CCSD(T)/aug-cc-pCVQZ PES for HCN/HNC [22]. The experimental r 0 parameters derived for HCN and HNC by Pearson et al. [START_REF] Pearson | The molecular structures of HNC and HCN derived from the eight stable isotopic species[END_REF] are used in Table 7.

In addition to the expectation value θ of the Jacobi angle, Table 7 also provides the expectation value of the Legendre polynomial P 2 = (3 cos 2 θ -1)/2 and the expectation value of the Euler angle ψ. The amplitude of the bending motion varies between 7 • for DCO + and 16 • for HOC + . This evolution is best demonstrated by Fig. 5, showing the minimum energy paths along the Jacobi angle. The stiffest profile there belongs to HCO + and the softest to HOC + . The effective bending wavenumbers ∆E 2 in Table 7 are computed as E(0, 1 1 , 0; 1, 1) -E(0, 0, 0; 1, 1), with E(v 1 , v 2 , v 3 ; J, p) denoting the energy of the vibrational state (v 1 , v 2 , v 3 ) for rotational angular momentum J and parity p. The highest ∆E 2 value is associated with HCO + and the lowest with DOC + .

In spectroscopic studies, the r 0 structure is derived directly from the experimentally determined zero-point rotational constants B 0 . For linear triatomic molecules this approximation effectively means that

I 0 = µ r r 2 0 + µ R R 2 0 , (19)
where I 0 = 2 /2B 0 . The values of r 0 and R 0 , computed using Eq. ( 19), differ from the expectation values r and R since the vibrationally averaged structure is nonlinear, as exemplified in Table 7. As a way of linearizing a real vibrating molecule, we introduce here a structure projected onto the principal a axis (the axis with least moment of inertia), for which we use the parameters r AB and r BC , with r XY denoting the projection of the bonddistance vector r XY onto the a axis. The r AB and r BC values in Table 7 are calculated from the expectation values r , R , θ , and ψ obtained in

J = 0 calculations.
For the isoelectronic series of linear triatomic molecules, Table 7 reveals that r > r e actually holds for the vibrationally averaged bond lengths for both r AB and r BC . We also see that r 0 (AB) < r e (AB) and r 0 (BC) > r e (BC),

where AB is the bond distance involving a light atom (H or D), placed farthest away from the a axis in Fig. 4. The values of r AB are longer than r 0 (AB), with differences ranging from 0.017 Å for DCO + to 0.046 Å for HOC + . At the same time, r AB and r 0 (AB) are closer to each other, deviating by at most 0.006 Å. We also observe that r HB and r DB differ by 0.005-0.006 Å, whereas the difference between r HB and r DB is only 0.001-0.002 Å. The bond-distance vector r BC between heavy atoms is tilted by the Euler angle ψ relative to the a axis in Fig. 4. Since ψ = 2 6 • , r BC and r BC are similar, differing by at most 0.006 Å for HOC + .

In view of Table 7, we may infer that r AB and r BC are good approximations for the structural r 0 parameters. This statement should, however, be viewed only as a qualitative rationalization since vibrationally averaged quantities are J-dependent [START_REF] Mladenović | Comparison of spectroscopic strategies to determine molecular geometries and the impact of nuclear versus atomic masses: the example of HCO + and HOC +[END_REF]. This in itself is to be perceived as an inherent conceptual problem for the definition of the r 0 structure.

The difficulties in determining molecular structures from zeropoint rotational constants B 0 are clearly visible in Fig. 3 (top): the intersection points of the correlation lines there show a large spread and a geometric structure common to different (actually all) iso-topologues is impossible to find. In the mass-dependence (r m ) method proposed by Watson [START_REF] Watson | The estimation of equilibrium molecular structures from zero-point rotational constants[END_REF], the equilibrium moment of inertia is approximated with 2I s -I 0 , where I s is the moment of inertia computed for the substituted r s geometry. This approach was, however, found unsatisfactory for hydrogen containing species [START_REF] Watson | The estimation of equilibrium molecular structures from zero-point rotational constants[END_REF]. , a global PES based on the CCSD(T)/cc-pVQZ approch [START_REF] Schmatz | Intramolecular dynamics and density of states for HN + 2[END_REF] and a quartic force field based on the ACTQ5+rel+ACPF approach [START_REF] Huang | Comparison of one-particle basis set extrapolation to explicitly correlated methods for the calculation of accurate quartic force fields, vibrational frequencies, and spectroscopic constants: Application to H 2 O, N 2 H + , NO + 2 , and C 2 H 2[END_REF]. Both PESs lead to very similar ∆B 0 estimates which therefore appears to be a rather robust and useful quantity. It is reasonable to expect that similar conclusions will be drawn also for other potential energy representations [START_REF] Špirko | Energies and electric dipole moments of the bound vibrational states of HN + 2 and DN + 2[END_REF] available for HN + 2 in the literature. The rovibrational energies for eight isotopologues of HN + 2 and two ab initio PESs are obtained in numerically exact full-dimensional rovibrational calculations for rotational angular momentum J as high as 15 in both parities.

The computed energies are easily fitted to highly accurate low-order effective spectroscopic Hamiltonians. This can be viewed as a hint towards regular rovibrational behaviour, in agreement with conclusions from our wavefunction analysis.

The spectroscopic correction S 0 of approximately 200 MHz for eight isotopic variants of HN + 2 is the dominant contribution to the full correction ∆B 0 . The higher-order corrections C 0 = S 0 -∆B 0 amount to several MHz and are thus much smaller than S 0 . Although small, the contribution C 0 is found to be essential for the determination of the molecular structure. This is exemplified by Fig. 3, which graphically represents the solution for the structural parameters common to eight isotopologues of HN + 2 obtained using three different models for the estimates B est of the equilibrium rotational constants B e . Visibly the highest degree of consistency for the molecular geometry is obtained by combining our variationally computed ∆B 0 values with accurate experimental data. It is also evident that the accuracy of the geometry determination is not limited by the experimental accuracy but by shortcomings in the theoretical model for the B 0 → B e correction. The corrections derived on the basis of low order perturbational formulas clearly do not provide sufficient accuracy.

A similar representation for HCO + is presented in Fig. 6. For both molecular ions, that is, HN + 2 in Fig. 3 and HCO + in Fig. 6, a situation of a welldefined intersection point is approached for the case when B est = B exp 0 + ∆B 0 .

The model based on the spectroscopic correction S 0 leads to less consistent structural parameters, as verified by Table 6 for HN + 2 . The residual contributions C 0 of the zero-point motion in B est = B exp 0 + S 0 are more pronounced for the deuterium containing forms of both HN + 2 in Fig. 3 (middle) and HCO + in Fig. 6 (top). We also found that a combined analysis of all data using a nonlinear least squares procedure provides a more balanced solution and thus more accurate structural parameters than pairwise approaches (like Kraichman's equations). This conclusion fully corroborates our previous recommendation [START_REF] Mladenović | Comparison of spectroscopic strategies to determine molecular geometries and the impact of nuclear versus atomic masses: the example of HCO + and HOC +[END_REF].

In experimental studies, the spectroscopic correction S 0 is obtained from the effective rotational constants for the ground vibrational state and singly excited vibrational ν 1 , ν 2 , and ν 3 states, as seen by Eq. ( 4). Experimental data for the fundamental vibrations are not always available also for (many) different isotopic forms. To access information relevant for the higher-order vibration-rotation contribution term C 0 by experimental means, measure- ments on vibrational overtones and combination bands are additionally required. But, this is not an easy task due to the increasing number of perturbations of the rovibrational structures which have to be identified and accounted for.

In electronic-structure program packages, the spectroscopic correction S 0 obtained by second-order vibrational perturbation theory [START_REF] Puzzarini | Quantum-chemical calculation of spectroscopic parameters for rotational spectroscopy[END_REF] has become a standard tool for the computation of the zero-point rotational constant B 0 from the calculated equilibrium rotational constant B e . As shown in the present study for HN + 2 and previously for HCO + and HOC + [START_REF] Mladenović | Theoretical spectroscopic parameters for isotopic variants of HCO + and HOC +[END_REF][START_REF] Mladenović | Comparison of spectroscopic strategies to determine molecular geometries and the impact of nuclear versus atomic masses: the example of HCO + and HOC +[END_REF], this correction appears to have its own fundamental deficiency due to the neglect of higher-order effects. Electronic-structure (single-point) computations are also employed to obtain other rotational parameters but only at equilibrium, such as D e and H e . Their comparison with the actual experimental situation must be handled with great care, otherwise the theory may be led to improper conclusions, as demonstrated before for the case of C 3 H + [START_REF] Mladenović | The B11244 story: Rovibrational calculations for 44 C 3 H + and C 3 Hrevisited[END_REF].

In view of the preceding analysis, full-dimensional rovibrational calculations reemerge naturally as helpful. To gather the maximum amount of information possible in such a calculation, it is essential to have a good functional form for the potential, satisfying at least the fundamental symmetries [START_REF] Mladenović | The B11244 story: Rovibrational calculations for 44 C 3 H + and C 3 Hrevisited[END_REF].

Cyanocarbene, HCCN, for instance, is a radical with a bent equilibrium structure [ ∠(HCC) e =147 • , ∠(CCN) e =175 • ], whose rovibrational energy pattern is well modeled by the linear-molecule convention [START_REF] Mladenović | Six-dimensional potential energy surface and rovibrational energies of the HCCN radical in the ground electronic state[END_REF]. A quartic internal coordinate force field for HCCN, covering only a narrow (near-equilibrium) part of the configuration space, may then be a serious drawback for a proper description of the dynamical behaviour of the angular degrees of freedom [START_REF] Mladenović | The B11244 story: Rovibrational calculations for 44 C 3 H + and C 3 Hrevisited[END_REF].

The proposed strategy for the computation of ∆B 0 stands out as the most consistent method to derive a r e structure. It can be also applied to larger molecules, since only vibrational ground state information for a sufficient set of J values is required which is much easier to compute than a full spectrum. Such a calculation requires only a moderate region of the PES which has to be combined with an exact rovibrational kinetic energy operator to build a full Hamiltonian which can then be treated by any numerically convenient and accurate ground state method. The observed robustness of the computed ∆B 0 values for two different PESs suggests that the level of electronic structure treatment does not have to be extremely high. This expectation deserves computational verification. However, the total number of isotopologues required in the reconstruction procedure for larger molecules may be fairly high.

The equilibrium geometry and spectral behaviour are not always correlated in a simple fashion, as seen in the example of HCCN. This is also manifested when determining the molecular structure from the spectral data.

A molecule is defined as linear when the potential energy minimum is at a linear geometry. This definition is applied in traditional spectroscopy under the assumption that the out-of-line motion is executed so that positive and negative excursions from the line leave the molecule on average in a linear configuration. This interpretation in essence neglects the rotational aspect in the doubly degenerate bending mode. The vibrationally averaged structure of linear molecules is bent even in the ground vibrational state [START_REF] Winterhoff | Empirical potential energy surface and bending angle probability densities for the electronic ground state of HCO +[END_REF]. In connection with this, the present work also provides a qualitative rationalization of the so-called r 0 structure, explained in terms of a structure projected onto the principal a axis. This is shown by the example of the isoelectronic series of linear triatomic molecules HCO + , HCN, HN + 2 , HNC, and HOC + . Our approach for determining the equilibrium structure of the protonated nitrogen molecule is self-consistent, and the issues under our control are detailed down to the last instance. We solve it as a well-posed mathematical problem with the help of least-squares techniques, which produce equilibrium bond lengths accurate to 10 5 -10 6 Å as the least-squares solution. The physical significance of such structural parameters should not be overinterpreted, however. The most commonly used assumption in studies like the present one is to use the Born-Oppenheimer approach in combination with atomic masses (the electron masses attached to the nuclei under the point mass assumption). It is easy to verify that the influence of one single electron mass on the rotational constant is approximately 0.5 MHz. Rotational constants known to five significant digits produce structural parameters known also to five significant digits, that is, to 10 5 Å, in accordance with Table 2 which shows that the uncertainty of r e is linearly proportional to the uncertainty of B e . In our previous study on HCO + and HOC + , we found that the replacement of the atomic masses with the nuclear masses affects the structural parameters in order of 10 4 Å, which is an effect much larger than the accepted statistical uncertainties of 10 5 Å found there for r e [START_REF] Mladenović | Comparison of spectroscopic strategies to determine molecular geometries and the impact of nuclear versus atomic masses: the example of HCO + and HOC +[END_REF]. This is consistent with the electron to proton mass ratio of 5•10 4 . We also note that the equilibrium structure obtained for HCO + and HOC + using atomic masses is found to be statistically somewhat better than the nuclear-mass results [START_REF] Mladenović | Comparison of spectroscopic strategies to determine molecular geometries and the impact of nuclear versus atomic masses: the example of HCO + and HOC +[END_REF]. Highly consistent methods for the derivation of r e structures are a mandatory step on the way to identify effects beyond the Born-Oppenheimer picture and our method appears to be valuable for this enterprise.

To bring experiment/experimental observations and theory even closer, challenges ahead are associated with the manifestation of non-Born-Oppenheimer effects in real molecules, which always resolve their own eigenvalue problem exactly. In rotational spectra, this will bring in a small unknown mass-dependent contribution ∆B e to the experimentally observed B 0 , whose dominant part is the Born-Oppenheimer B e value. This implies further that the true equilibrium parameters will be isotopologue dependent and distributed about the equilibrium Born-Oppenheiimer structure, assuming that non-Born-Oppenheimer effects are such that the notion of the potential energy surface will be sustained and the true equilibrium structure will be defined by its minimum. For diatomic molecules, there are models to incorporate the mass-dependent nonadiabatic effects into ab initio potential energy curves by morphing them separately for each of the isotopologues, like, for instance, Ref. [START_REF] Augustovičová | Radial molecular property functions of CH in its ground electronic state[END_REF]. Further conceptual developments are, however, required to address this issue for the case of triatomic molecules from the theoretical and experimental viewpoints in order to properly include translational and rotational invariances of entire molecular Hamiltonians. α constants are insufficient to obtain accurate r e structures from B 0 .

Graphical Abstract

Rovibrational variational calculations for HN + 2 for two potential energy surfaces.

Theoretical zero-point corrections are insensitive to ab initio level.

using d 1

 1 = r(AB) and d 2 = r(BC). For the mass factors, we have m XY = m X + m Y and M = m A + m BC . In terms of Jacobi coordinates r and R, r = d 2 and R = d 1 + c m d 2 ,

  Accordingly, a manifold of pairs (d 1e , d 2e ) will reproduce the same I e value, that is, the same equilibrium rotational constant B e . The structural parameters d 1e and d 2e are anticorrelated by Eqs. (16) and (17), as a decrease of d 1e in (d 1e , d 2e ) can be compensated by an increase of d 2e for a given B e .

Figure 1

 1 Figure 1 is a graphical representation of the pairs (d 1e , d 2e ) which reproduce the experimental equilibrium rotational constants of Owrutsky et al.

  ) are r o e (HN)=1.033 48(43) Å and r o e (NN)=1.092 793(93) Å. This pair is marked with Ow corr (•) in Fig. 1. The two solutions, Ow and Ow corr , appear as different in Fig. 1. Taking into account their uncertainties, they are, strictly speaking, statistically compatible. Given Fig. 1, we recommend to use our results r o e (HN) and r o e (NN) for future reference to the experimental equilibrium structure of Owrutsky et al. [12].

Figure 1 :Figure 2 :

 12 Figure 1: Pairs (d 1e , d 2e ) reproducing experimental rotational constants of Owrutsky et al. [12] and Amano et al. [13] for HN + 2 and DN + 2 , where d 1e = r e (HN) and d 2e = r e (NN). For more details, see Section 4 of the main text.

Figure 3 :

 3 Figure 3: Pairs of the bond distances r(HN) and r(NN) reproducing B est modeled as B exp 0 (top), B exp 0 + S th 0 (middle), and B exp 0 + ∆B th 0 (bottom) for eight isotopologues of the protonated molecular nitrogen. Pair solutions for B exp 0 + ∆B th 0 (bottom) are shown by triangular points ( ).

  ) Å and r s (NN) = 1.094 70(40) Å. Using their rotational constants, we, however, find in an unweighted least-squares procedure that r 0 (HN) = 1.030 90(40) Å and r 0 (NN) = 1.096 70(8) Å. These improved r 0 values are in excellent agreement with the r 0 parameters shown in Table 6. Another r 0 estimate is due to Warner [20], who applied Kraitchman's relations for different parent species formed from a set of eight isotopologues. His pair solutions for eight parent species exhibit a large spread of 3•10 4 Å, with a mean at r ap (HN) = 1.030 41 Å and r ap (NN) = 1.094 90 Å. Our least-squares solution derived from the data of Warner is r 0 (HN) = 1.030 90(20) Å and r 0 (NN) = 1.096 70(6) Å,

Figure 4 :

 4 Figure 4: Jacobi coordinates (r, R, θ), the body-fixed z axis, the principal a and b axes, and the third Euler angle ψ for a triatomic molecule ABC.

2 , ν 1

 21 , and ν 3 , as well as r e and r 0 values for the isoelectronic series of linear triatomic molecules HCO + , HCN, HN + 2 , HNC, and HOC + . Distances are in Å, angles in degree, and energies in cm -1 . For detailed parameter definitions, see Section 6 of the main text.

bFigure 5 :

 5 Figure 5: Minimum energy paths along the Jacobi angle θ measured relative to the energy of the respective optimum equilibrium configuration for the isoelectronic series of linear triatomic molecules HCO + , HCN, HN + 2 , HNC, and HOC + .

7 .

 7 Final remarks Microwave rotational spectroscopy measures the energies of rotational transitions, frequently with stunning precision. The experimental observations are fitted to effective spectroscopic Hamiltonians following the linearmolecule or nonlinear-molecule formalism and the derived spectroscopic parameters are commonly applied to derive the molecular equilibrium structure. Yet this task is far from straightforward. In real vibrating-rotating molecules, corresponding to the actual experimental situation, the ground vibrational state is described by a wavefunction, extended over other-than-equilibrium arrangements. The effects of zero-point vibrational motion produce an effective rotational constant B 0 , which is different from the equilibrium B e value, which is essential for the determination of the molecular structure. The rotational constant B 0 is not an observable, but a parameter arising within the model used to describe the rotation. There is no obvious way to derive ∆B 0 = B e -B 0 experimentally. In the present work, the methods commonly used to correct the experimental rotational constant B 0 for the vibration-rotation interaction effects are examined by theoretical means for the case of the protonated nitrogen molecule, HN + 2 . The concepts of B 0 and B e are both well founded in the Born-Oppenheimer picture, providing a clear definition for the full vibration-rotation correction to B 0 , given simply by the difference ∆B 0 = B e -B 0 . Different potential energy representations (may) support different minima, however. This issue is addressed here by employing two PESs for HN + 2

Figure 6 :

 6 Figure 6: Pairs of the bond distances r(HC) and r(CO) that reproduce B est e modeled as

2 Mirjana 2 Mirjana

 22 Accurate equilibrium structures of linear triatomic molecules from a combined theoretical-experimental method: The protonated nitrogen molecule, HN + of linear triatomic molecules from a combined theoretical-experimental method: The protonated nitrogen molecule, HN + Mladenović, Marius Lewerenz Strategies to derive molecular equilibrium structures from experimental data.

Table 1 :

 1 Potential energy minima r th than those for HVL QFF. Accordingly, B e for SM PES is approximately 200 MHz smaller than B e for HVL QFF. This trend is expected, given MHz for Fit 3 to 10 -8 MHz for Fit 5.

			e (in Å), associated ro-
	tational constants B th e (in MHz), and vibrational-rotation
	corrections ∆B 0 and S th 0 (both in MHz) obtained for the
	two potential energy representations. a
	Parameter	SM PES	HVL QFF
	r th e (HN)	1.034 02	1.033 28
	r th e (NN)	1.095 62	1.092 78
	B th e (HN + 2 )	46 627.52	46 844.35
	B th e (DN + 2 )	38 560.87	38 727.41
	Fit 3
	∆B 0 (HN + 2 ) 240.150 5(2)	240.550 5(2)
	∆B 0 (DN + 2 ) 160.658 8(2)	160.648 7(2)
	S th 0 (HN + 2 )	245.748 1(11)	244.370 4(11)
	S th 0 (DN + 2 )	153.498 9(11)	151.743 7(12)
	Fit 4
	∆B 0 (HN + 2 ) 240.148 807 97(29)	240.548 864 05(31)
	∆B 0 (DN + 2 ) 160.657 278 90(33)	160.647 195 45(36)
	S th 0 (HN + 2 )	245.746 513 70(159) 244.368 722 13(174)
	S th 0 (DN + 2 )	153.496 193 94(240) 151.740 733 12(325)
	Fit 5
	∆B 0 (HN + 2 ) 240.148 806 33(5)	240.548 862 29(4)
	∆B 0 (DN + 2 ) 160.657 277 07(1)	160.647 193 41(2)
	S th 0 (HN + 2 )	245.746 511 51(20)	244.368 722 57(17)
	S th 0 (DN + 2 )	153.496 193 55(12)	151.740 738 02(18)

a Values in parentheses show one standard error to the last significant digit from the least-squares procedure. longer

Table 3 :

 3 Parameters (in Å) of the r 0 , r α , and r e structures determined using the experimental ground state rotational constants reported for HN + 2 and DN + 2 by Yu et al. [11]. The theoretical corrections ∆B 0 and S th 0 obtained in Fit 5 of Table 1 are applied. The experimental equilibrium structures (Exp) reported by Owrutsky et al. (OGM) [12] and Amano et al. (AHT) [13] are additionally shown. Rotational constants are in MHz. a

		r(HN)	r(NN)	B(HN + 2 )
	r 0	1.030 878 97(17) 1.096 748 74(4) 46 586.8753(5)
	SM PES
	r α	1.034 635 22(17) 1.092 689 15(4) 46 832.6218(5)
	r e	1.033 169 52(17) 1.093 033 35(4) 46 827.0241(5)
	HVL QFF
	r α	1.034 730 33(17) 1.092 690 63(4) 46 831.2440(5)
	r e	1.033 209 28(17) 1.093 020 62(4) 46 827.4242(5)
	Exp
	OGM	1.033 59(43)	1.092 766(94)	46 840.40(110)
	AHT	1.034 60(14)	1.092 698(26)	46 832.45(71)

Table 4 :

 4 Structural r α parameters (in Å) and associated rotational constants B α (in MHz).

		r α (HN)	r α (NN)	B α (HN + 2 )
	Experiment
	OGM	1.034 6(15) 1.092 6(3)	46 835.8(35)
	AHT	1.034 632(3) 1.092 637 2(6) 46 836.462(7)
	YPD	1.034 635(1) 1.092 636 7(2) 46 836.463(4)
	Theory
	SM PES	1.035 50	1.095 27	46 633.12
	HVL QFF	1.034 80	1.092 45	46 848.17

a

The experimental values are obtained using the spectroscopic data of Owrutsky et al. (OGM)

[START_REF] Owrutsky | Determination of the equilibrium structure of protonated nitrogen by high resolution infrared laser spectroscopy[END_REF]

, Amano et al.

(AHT)

[START_REF] Amano | Submillimeter-wave spectroscopy of HN + 2 and DN + 2 in the excited vibrational states[END_REF]

, and Yu et al. (YPD)

[START_REF] Yu | Rotational spectroscopy of vibrationally excited N 2 H + and N 2 D + up to 2.7 THz[END_REF]

. The theoretical values are derived for both SM PES and HVL QFF.

a Values in parentheses show error limits obtained using Table

2

.

Table 5 :

 5 Higher-order correction term C 0 (in MHz) effectively used by Owrutsky et al.

	(OGM) [12] and Amano et al. (AHT) [13], along with the theoretical counterparts calcu-
	lated for both SM PES and HVL QFF.
		Experiment	Theory
		OGM AHT	SM PES HVL QFF
	C 0 (HN + 2 )	-4.6	4.0	5.6	3.8
	C 0 (DN + 2 ) -12.3	2.7	-7.2	-8.9

Table 6 :

 6 Parameters of the r 0 , r α , and r e structures (in Å) and the corresponding rotational constants B (in MHz), derived using the data for eight isotopologues of the protonated nitrogen molecule. a

	Structure	r(HN)	r(NN)	B(HN + 2 )	rms b
	r 0	1.030 86(21)	1.096 67(5)	46592.5(65)	2.93
	SM PES
	r α	1.035 01(16)	1.092 63(4)	46 831.9(52)	2.06
	r e	1.033 173(9)	1.093 029(2) 46 827.28(28) 0.13
	r e (Fit U)	1.033 172(10) 1.093 031(2) 46 827.20(32) 0.11
	r e (Set C)	1.033 166(11) 1.093 032(3) 46 827.21(34) 0.11
	r e (Fit F 1 ) 1.033 209	1.093 020(1) 46 827.44(8)	0.22
	r e (Fit F 2 ) 1.033 219(5)	1.093 017	46 827.56(8)	0.29
	HVL QFF
	r α	1.035 19(19)	1.092 62(5)	46 830.6(62)	2.43
	r e	1.033 209(9)	1.093 017(2) 46 827.70(28) 0.14
	r e (Fit U)	1.033 210(11) 1.093 018(3) 46 827.61(34) 0.13
	r e (Set C)	1.033 201(12) 1.093 019(3) 46 827.62(36) 0.12
	r e (Fit F 1 ) 1.033 173	1.093 026(1) 46 827.42(8)	0.20
	r e (Fit F 2 ) 1.033 163(5)	1.093 029	46 827.43(7)	0.23

a Values shown in parentheses are one standard deviation to the last significant digit from the least-squares procedure. Distances shown without error limits are kept constant in the fit.

Table 7 :

 7 Vibrationally averaged structural parameters θ , P

	AB , r AB , r BC , and r BC in the ground vibrational

2 , ψ , r

the two PESs agree within 4•10 -5 Å and the r e (NN) values within 1•10 -5 Å.

The estimates for r e (HN) and r e (NN) reproduce the fitted B est = B exp 0 +∆B 0 values with root-mean-square (rms) deviations in order of 0.1 MHz.

The r e results in Table 6, obtained in our combined theoretical-experimental approach, represent best estimates of the structural parameters. The r th e values for the high-quality ab initio HVL QFF surface deviate from these estimates by 7 • 10 -5 Å for the HN bond length and by 24 • 10 -5 Å for the NN bond length. Our best estimate structures for HCO + and HOC + [1, 2] show comparable agreement with the ab initio structure determined by Koput [START_REF] Koput | Ab initio structure and vibration-rotation dynamics of the formyl and isoformyl cations, HCO + /HOC +[END_REF] at the CCSD(T)/aug-cc-pV7Z level of theory with several high order corrections.

Table 6 is next compared with Table 3, which was derived by applying the rotational data for only two isotopologues. The equilibrium bond distances r e (HN) and r e (NN) from the two tables agree better than 4•10 -6 Å. The corresponding agreement for r 0 (HN), r 0 (NN), and r α (NN) is within 8•10 -5 Å and for r α (HN) within 5•10 -4 Å.

The three models for the rotational constant B est are graphically compared in Fig. 3. Ranges of 0.004 Å along the abscissa and 0.001 Å along the ordinate are used in the two-dimensional graphs at the top and in the middle.

The graph on the bottom covers a range of only 0.0005 Å along the abscissa and 0.0001 Å along the ordinate. The intersection of the horizontal and vertical dashed lines indicates the structural r g parameters, that is, r 0 (top), r α (middle), and r e (bottom) from the all isotope least-squares fits. Their explicit values are listed in Table 6. Analogous to Fig. 1, a (red) line, representing one of the hydrogen-containing forms, and a (blue) line, representing