Chloe A Fuller

Quentin Berrod

Bernhard Frick

Mark R Johnson

Maxim Avdeev

John S O Evans

Ivana Radosavljević Evans

Oxide Ion and Proton Conductivity in Highly Oxygen-Deficient Cubic Perovskite SrSc 0.3 Zn 0.2 Ga 0.5 O 2.4

Keywords: Oxide ion conductor, neutron spectroscopy, local structure

A series of Zn-substituted compounds, Sr2Sc1-xZnxGaO5-0.5x, based on the brownmillerite-type oxide ion conductor Sr2ScGaO5 have been synthesized and a single-phase region identified at 0.4 ≤ x < 0.6. The structure and dynamics of Sr2Sc0.6Zn0.4GaO4.8 were investigated by X-ray and neutron diffraction, neutron total scattering and pair distribution function (PDF) analysis, impedance spectroscopy and neutron spectroscopy. The material was found to be a highly-disordered cubic perovskite with a remarkable level of oxygen deficiency across a large temperature range. These structural properties lead to an increase of oxide ion conductivity of about two orders of magnitude relative to the parent Sr2ScGaO5. The presence of proton conductivity and some water uptake was suggested by the impedance data and corroborated by thermogravimetric analysis (TGA), solid state nuclear magnetic resonance (NMR), variable temperature X-ray diffraction and neutron spectroscopy. Both proton and oxide ion conductivity produced a measurable quasi-elastic neutron scattering (QENS) signal and the onset of each dynamic process could be observed by monitoring the temperature dependence of the elastic and inelastic scattering intensities measured in fixed window scans. Neutron total scattering and PDF studies revealed a local structure that is markedly different from the perovskite average structure, and we propose it contains a rare one-coordinate or terminal oxygen site.

INTRODUCTION

The perovskite structure, often adopted by mixed metal oxides of the general formula ABO3, is one of the most ubiquitous in solid state chemistry, owing to the vast number of combinations of the A and B cations it can support. As a consequence of this chemical versatility, perovskite-type oxides display a number of technologically important physical properties: ferroelectricity, piezoelectricity, colossal magnetoresistance, catalytic activity, fast ion conductivity, mixed ionic electronic conductivity and others. Oxide ion conductivity in perovskites is readily induced by aliovalent (acceptor) doping, generating oxygen-deficient materials. At relatively low doping levels, the cubic perovskite structure can be retained, the vacancies are distributed randomly and three-dimensional oxide ion diffusion proceeds via a vacancy-hopping mechanism. [START_REF] Adler | Chemical Structure and Oxygen Dynamics in Ba2In2O5[END_REF][START_REF] Ishihara | Doped LaGaO3 Perovskite Type Oxide as a New Oxide Ionic Conductor[END_REF][START_REF] Li | A Family of Oxide Ion Conductors Based on the Ferroelectric Perovskite Na0.5Bi0.5TiO3[END_REF] At higher defect concentrations, ordered structures such as the brownmillerite structure can be formed. With general formula A2B2O5, this can be viewed as oxygen-deficient perovskite with one-sixth of the oxygen sites vacant. The oxide ion vacancies are ordered along one crystallographic direction, resulting in alternating corner-sharing octahedral and tetrahedral layers formed by BO6/2 and BO4/2 groups, respectively. This structural arrangement means that the oxide ion conductivities of brownmillerites are typically lower than their perovskite counterparts, as well as being predominantly one-dimensional. For example, orthorhombic brownmillerite-type Sr2ScGaO5 is a modest oxide ion conductor (~ 10 -6 S cm -1 at 600 °C), whereas the conductivity of this material in the high-temperature cubic form is two orders of magnitude higher. [START_REF] Corallini | Cubic Sr2ScGaO5 Perovskite: Structural Stability, Oxygen Defect Structure, and Ion Conductivity Explored on Single Crystals[END_REF][START_REF] Fuller | Brownmillerite-Type Sr2ScGaO5 Oxide Ion Conductor: Local Structure, Phase Transition, and Dynamics[END_REF] We have recently demonstrated that introducing a small amount of additional (extrinsic) vacancies into the octahedral layer of Sr2ScGaO5 by Zn-doping increases the oxide ion conductivity relative to the parent material by up to two orders of magnitude, while retaining the orthorhombic brownmillerite structure. We attributed this increase in conductivity and the lowering of the activation energy to the extrinsic vacancies disordered in the octahedral layers, providing additional oxide ion migration pathways in the structure. [START_REF] Fuller | Brownmillerite-Type Sr2ScGaO5 Oxide Ion Conductor: Local Structure, Phase Transition, and Dynamics[END_REF] An alternative way to introduce disorder into perovskite derivatives is to change the ratio of numbers of octahedral and tetrahedral building blocks in the structure. For example, Chernov et al. [START_REF] Chernov | Sr2GaScO5, Sr10Ga6Sc4O25, and SrGa0.75Sc0.25O2.5: A Play in the Octahedra to Tetrahedra Ratio in Oxygen-Deficient Perovskites[END_REF] have described how increasing the GaO4 tetrahedra to ScO6 octahedra ratio in the Sr/Sc/Ga/O system leads to an evolving family of structures. Sr10Sc4Ga6O25 (Sr2Sc0.8Ga1.2O5), for example, is perovskite-related, but the connectivity balance means that coordination polyhedra can no longer share all their corners and the structure contains a mix of ScO6/2, GaO4/2 and GaO3/2O units. Tetrahedra that are not fully corner-linked have been shown to facilitate oxide ion mobility in different structural families. [START_REF] Kendrick | Developing Apatites for Solid Oxide Fuel Cells: Insight into Structural, Transport and Doping Properties[END_REF][START_REF] Evans | Phase Transitions, and Oxygen Migration in Zirconium Tungstate[END_REF][START_REF] Kuang | Interstitial Oxide Ion Conductivity in the Layered Tetrahedral Network Melilite Structure[END_REF][START_REF] Tallentire | Systematic and Controllable Negative, Zero, and Positive Thermal Expansion in Cubic Zr1-xSnxMo2O8[END_REF] In this work, we have used this design approach to prepare new oxide ion conductors of general formula Sr2Sc1-xZnxGaO5-0.5x (x = 0.1-0.6). We show that by increasing the number of tetrahedral sites in a brownmillerite and increasing the oxygen vacancies beyond the 1/6 level, we can reform highly-disordered and highly oxygen-deficient cubic perovskite materials. We report the proton and oxide ion conductivity in this system, and the use of variable-temperature powder X-ray and neutron diffraction, neutron total scattering and pair distribution function (PDF) analysis, and inelastic/quasielastic neutron scattering to elucidate in detail the structure-property relationships in SrSc0.3Zn0.2GaO2.4.

EXPERIMENTAL

Synthesis. Polycrystalline samples of Sr2Sc1-xZnxGaO5-0.5x (x = 0.1-0.6) were synthesized from stoichiometric amounts of SrCO3 (Fisher Chemical, >99.9%), Sc2O3 (Sigma-Aldrich, >99.9%), Ga2O3 (Sigma-Aldrich, >99.99%), and ZnO (Alfa Aesar, >99.99%). The starting materials were ground together and heated at 1200 °C for 12 h. Samples were reground, pressed into 10 mm pellets, and heated at the same temperature for a further 24 h. This was repeated until the powder X-ray diffraction (PXRD) patterns no longer changed with additional heating. A Bruker D8 Advance diffractometer equipped with a Lynx-eye detector and Cu Kα radiation, was used to monitor the progress of the solidstate reactions. The same instrument was used over an extended 2θ range with a Si internal standard (a = 5.431195(9) Å at room temperature) for accurate cell parameter determination. Analysis of the diffraction data was performed using the Rietveld method 11 implemented in the Topas Academic software. [START_REF] Coelho | The TOPAS Symbolic Computation System[END_REF][START_REF] Leineweber | Practical Powder Diffraction Pattern Analysis Using TOPAS[END_REF] Within this work, specific samples will be referred to by an abbreviated name of the form SSGO (for Sr2ScGaO5) or SSZ40GO (for Sr2Sc0.6Zn0.4GaO4.8) where the number in the abbreviation indicates the percentage of Sc that has been replaced by Zn.

Variable-Temperature X-ray Diffraction. VT-XRD patterns of single phase compositions were recorded on a Bruker D8 Advance (Lynx-eye detector, Cu Kα radiation) with an Anton Paar HTK1200 furnace attachment over a temperature range of 25-1000 °C. Furnace temperatures were calibrated based on an external Al2O3 standard. [START_REF] Stinton | Parametric Rietveld Refinement[END_REF] Data were collected every 20 °C on heating and cooling, and resulting patterns were analyzed by Rietveld fitting in Topas Academic. [START_REF] Coelho | The TOPAS Symbolic Computation System[END_REF][START_REF] Leineweber | Practical Powder Diffraction Pattern Analysis Using TOPAS[END_REF] Neutron Powder Diffraction (NPD). Diffraction data were collected on a powdered, 6 g sample of SSZ40GO in a 9 mm vanadium sample can on the high-resolution powder diffractometer ECHIDNA [START_REF] Avdeev | ECHIDNA: A Decade of High-Resolution Neutron Powder Diffraction at OPAL[END_REF] at the OPAL reactor at ANSTO, Sydney. Diffraction patterns were measured using neutrons with a wavelength of 1.6215 Å at room temperature, 200, 300, 400, 500, 700, 800, and 900 °C as well as at -253 °C in a vacuum furnace and cryofurnace, respectively. Chemical Analysis. The elemental composition of SSZ40GO was confirmed by inductively coupled plasma mass spectrometry (ICPMS) analysis using ∼50 mg of powdered sample dissolved in nitric acid with a Thermo Scientific Neptune Plus Higher Resolution Multicollector.

Impedance Spectroscopy. SSZ40GO powder was uniaxially pressed into a 10 mm pellet and sintered for 24 h at 1200 °C yielding a pellet that was 84% dense. The pellet was coated in platinum ink and mounted on a Probostat A-6 cell. The electrodes were set by heating at 1000 °C for 1 h. A Solartron 1260 frequency-response analyser was used to measure and record impedance spectra every 20 degrees on heating and cooling in air, dry N2, and wet N2 (bubbled through water) using frequency ranges of 0.1-10 7 Hz and voltages of 100-2500 mV.

Thermogravimetric Analysis. A PerkinElmer TGA 8000 instrument was used to measure mass changes on heating and cooling cycles. Multiple samples of SSZ40GO were heated from 30 to 1000 °C and back to ambient temperature at 10 °C min -1 under an air atmosphere.

Solid state NMR. Static 45 Sc and 71 Ga NMR spectra were recorded at 97.21 and 122.04 MHz respectively, for both SSZ40GO and SSGO on a Bruker Avance III HD spectrometer using 1 M Ga(NO3)3 and ScCl3 solutions as references. WURST-qCPMG pulse sequences [START_REF] O'dell | QCPMG Using Adiabatic Pulses for Faster Acquisition of Ultra-Wideline NMR Spectra[END_REF] were used for both experiments yielding spikelet spectra. The 1 H NMR spectrum of SSZ40GO was recorded on a powder sample that had been exposed to air for 2 weeks following synthesis. A Bruker Avance III HD spectrometer with tetramethylsilane as a reference was used for the measurement.

Neutron Total Scattering. Total scattering data on SSZ40GO were collected on the POLARIS instrument at the ISIS facility at Rutherford Appleton Laboratory. A ∼6 g powder sample was packed into an 8 mm diameter vanadium can. Rietveld-quality data sets were collected at 45 temperatures on heating for 4 mins each. High-quality data sets for pair distribution function (PDF) analysis were acquired at room temperature, 600 and 1000 °C in 8 sets of 175 μAh collections i.e. 8 h at each temperature. The scattering due to the empty furnace and the empty can were measured at room temperature for 1-2 h each for background subtraction. Neutron total scattering data were normalized onto an absolute scale using GudrunN software; [START_REF] Soper | Programs for Correcting Raw Neutron and X-Ray Diffraction Data to Differential Scattering Cross Section[END_REF] the D(r) functions used for fitting were extracted using a Qmax of 40 Å -1 using standard formalisms [START_REF] Keen | A Comparison of Various Commonly Used Correlation Functions for Describing Total Scattering[END_REF] and Fourier ripples < r = 1.6 Å were removed. Bragg data were extracted using the established analysis routines in Mantid. [START_REF] Arnold | Mantid-Data Analysis and Visualization Package for Neutron Scattering and μ SR Experiments[END_REF] Rietveld and large-box local structure analyses were performed using Topas Academic. [START_REF] Coelho | The TOPAS Symbolic Computation System[END_REF][START_REF] Leineweber | Practical Powder Diffraction Pattern Analysis Using TOPAS[END_REF] The three highest resolution data banks (banks 3, 4, and 5) of POLARIS were used for variable-temperature neutron refinements covering a total d-spacing range of 0.15-5.75 Å. Bank 3 was selected for local structure analysis as it offers a good compromise between resolution and Q-range. For PDF fitting, large-box models were constructed by converting appropriate small box models onto a perovskite basis and expanding them to a close-to isotropic supercell containing between 2000 and 8000 atoms. Atomic coordinates were then refined against the data subject to polyhedral constraints to keep local geometry sensible. Bragg and PDF datasets were fitted simultaneously using a fitting range of 2000-20000 µs for Bragg data and 0-19.5 Å for PDF D(r) data. In initial refinement cycles, coordinate changes were constrained to prevent large atomic shifts; these constraints were relaxed during fitting. An equal weighting was applied to each point in the PDF such that the PDF and Bragg data gave a similar contribution to the overall χ 2 . PDF patterns were calculated directly from the atomic coordinates of the large box model and the Bragg scattering was calculated by folding the largebox coordinates back into a single crystallographic unit cell. Refinements converged within a couple of hours on a standard desktop PC. Multiple starting configurations were tested for each model to ensure the final fit could be replicated.

Inelastic and Quasielastic Neutron Scattering. Inelastic neutron scattering data on SSZ40GO were measured on both the backscattering spectrometer IN16B, and the timeof-flight spectrometer IN6-SHARP (run by the Laboratoire Léon Brillouin), at the Institut Laue Langevin.

The experiment on IN16B used an incident neutron wavelength of 6.271 Å and an energy resolution of 0.75 μeV, to give access to nanosecond dynamics. A 10 g powdered sample of SSZ40GO was placed in a cylindrical 8 mm diameter Nb-foil sample holder and measurements of elastic intensity (E = 0 μeV) and inelastic intensity at 2 μeV, were performed on heating and cooling from 20 to 1000 °C using a heating rate of 0.04 °C s -1 . Data points were collected in pairs with an elastic intensity (EFWS-elastic fixed window scan) measured for 30 s, immediately followed by an inelastic intensity (IFWS-inelastic fixed window scan) measurement lasting 2 minutes. [START_REF] Frick | New Possibilities with Inelastic Fixed Window Scans and Linear Motor Doppler Drives on High Resolution Neutron Backscattering Spectrometers[END_REF] Quasielastic neutron scattering (QENS) measurements were performed at room temperature, 300 and 400 °C for 6 h at each temperature (as 12 0.5h repetitions) and 900 and 1000 °C for 1.5 h each. An energy transfer window of ±20 μeV was used for all QENS data collections. The background scattering from the empty can was measured for 5 h at both room temperature and 400 °C, and a vanadium resolution function was acquired for 4 h at 40 °C. The data were reduced and analysed using established routines implemented in the Mantid code. [START_REF] Arnold | Mantid-Data Analysis and Visualization Package for Neutron Scattering and μ SR Experiments[END_REF] The IN6-SHARP experiment was performed with an incident neutron wavelength of 5.12 Å and an elastic energy resolution of 70 μeV to give access to picosecond dynamics. Data were collected at 20, 200, 400, 600, 800 and 1000 °C, with a collection time of 6 h at each temperature on the same sample. Background scattering from the empty sample holder was measured for 6 h at 20 °C and a vanadium standard was measured at 20 °C for 6 h for normalization and to describe the instrument resolution function. Data were processed using Mantid and LAMP software packages [START_REF] Richard | the Large Array Manipulation Program[END_REF] , and analyzed using QENSH software. 22

RESULTS AND DISCUSSION

PXRD patterns for each of the samples are shown in Figure S1. They suggest that single-phase compounds with nominal compositions Sr2Sc1-xZnxGaO5-0.5x (x = 0.4 and 0.5) were synthesized successfully. Previous work showed a x = 0.05 sample retained the brownmillerite-type structure adopted by Sr2ScGaO5 5 with the same room-temperature space group I2mb. Evidence for successful substitution was provided by small changes in the crystallographic a-and baxes and in the change in conduction behaviour. [START_REF] Fuller | Brownmillerite-Type Sr2ScGaO5 Oxide Ion Conductor: Local Structure, Phase Transition, and Dynamics[END_REF] For compositions with 0.4 ≤ x < 0.6, the single-phase product is a cubic perovskite. Increasing the Zn content and B-site disorder therefore appears to lead to the room-temperature stabilization of the high-temperature disordered phase that has been seen in multiple brownmillerites, including Sr2Fe2O5 and Ba2In2O5. [START_REF] Corallini | Cubic Sr2ScGaO5 Perovskite: Structural Stability, Oxygen Defect Structure, and Ion Conductivity Explored on Single Crystals[END_REF][START_REF] Speakman | In-Situ Diffraction Study of Ba2In2O5[END_REF][START_REF] Shin | Order-Disorder Transition of Sr2Fe2O5 from Brownmillerite to Perovskite Structure at an Elevated Temperature[END_REF] Compositions with x = 0.1, 0.2 and 0.3 were two-phase, containing the brownmillerite and an increasing proportion of the perovskite. Rietveld refinement shows that the quantities of perovskite and brownmillerite in the 0.05 < x < 0.4 two-phase region are consistent with phase separation into Sr2Sc0.95Zn0.05GaO4.975 (SSZ05GO) and Sr2Sc0.6Zn0.4GaO4.8 (SSZ40GO). Finally, the x = 0.6 sample contained mainly the perovskite phase but also a very small amount of an unidentified impurity (top plot in Figure S1).

Since SSZ40GO is a cubic perovskite, we can write its formula as SrSc0.3Zn0.2Ga0.5O2. 4. Literature examples of perovskites with oxygen contents lower than ABO2.5 are uncommon and typically found for compounds containing variable oxidation state transition metals [START_REF] Dong | Surprisingly High Activity for Oxygen Reduction Reaction of Selected Oxides Lacking Long Oxygen-Ion Diffusion Paths at Intermediate Temperatures: A Case Study of Cobalt-Free BaFeO3-Δ[END_REF][START_REF] Kuklja | Combined Theoretical and Experimental Analysis of Processes Determining Cathode Performance in Solid Oxide Fuel Cells[END_REF][START_REF] Istomin | A Novel Complex Cobalt Gallium Oxide Ca2Co0.8Ga1.2O4.8: Synthesis and High-Temperature Electron Transport Properties[END_REF][START_REF] Waidha | BaCoO2+δ : A New Highly Oxygen Deficient Perovskite-Related Phase with Unusual Co Coordination Obtained by High Temperature Reaction with Short Reaction Times[END_REF][START_REF] Mentré | 2.22: The Most Oxygen-Deficient Certified Cubic Perovskite[END_REF] (e.g. Ba0.5Sr0.5Co0.8Fe0.2O3-δ where δ can be up to 1 depending on the pO2) or in compounds like SrCuO2 and SrFeO2 which have the square planar-based infinite layer structure. [START_REF] Takano | ACuO2(A: Alkaline Earth) Crystallizing in a Layered Structure[END_REF][START_REF] Tsujimoto | Infinite-Layer Iron Oxide with a Square-Planar Coordination[END_REF][START_REF] Crespin | Reduced Forms of LaNiO3 Perovskite. Part 1.-Evidence for New Phases: La2Ni2O5 and LaNiO2[END_REF] These are frequently made by low-temperature topotactic reduction of oxygen-rich phases. On the other hand, Takeda et al. [START_REF] Takeda | Oxide Ion Conductivity in Perovskite Type Sr2ScAlO5 and Related Compounds[END_REF] reported oxygen deficiency corresponding to the oxygen content of 2.45 (normalized to perovskite) in the Sr/Ba/Sc/Al/O system. The cubic perovskite materials reported here display even higher oxygen deficiency, corresponding to an oxygen content as low as ABO2.375 (SSZ50GO), while containing only fixed-oxidation-state cations.

Average Structure. Powder X-ray and neutron diffraction patterns of SSZ40GO (Figure 1) can be fully fitted using an a = 3.9871 Å perovskite model in the cubic space group Pm-3m. The structural model included Sr on the A-site, Sc, Ga and Zn on the B-site and O sites in their nominal proportions, confirmed by ICP elemental analysis of the cations. The neutron diffraction pattern shows some structured diffuse scattering that is particularly noticeable at Q = 4.5 Å -1 . This suggests that there are local deviations from the average structure. The fact that this is seen in the neutron pattern but not the X-ray implies that the deviation occurs mainly on the oxygen sublattice. The local structure of this material is discussed later. Conductivity. The conductivity of SSZ40GO was measured using impedance spectroscopy between 200 and 1000 °C on both heating and cooling. The complex impedance plots (Figure S4) across the whole temperature range comprised two responses: a large semicircle at high frequency and an electrode response at low frequency. At high temperature the electrode response turned over into a semicircle indicating diffusion through a finite layer, which suggests conductivity is ionic in nature. The capacitance of the high frequency semicircle was found to be ~10 -12 F cm -1 using the relationship ωRC=1 at the arc maximum, and is consistent with bulk transport [START_REF] Irvine | Electroceramics: Characterization by Impedance Spectroscopy[END_REF] . Bulk resistance values were therefore extracted from the high-frequency intercept of the semicircular signal and the derived conductivities are plotted in Figure 2a along with those of the parent material SSGO and related brownmillerite SSZ05GO. [START_REF] Fuller | Brownmillerite-Type Sr2ScGaO5 Oxide Ion Conductor: Local Structure, Phase Transition, and Dynamics[END_REF] There is an up to 2 order-of-magnitude increase in total conductivity in SSZ40GO relative to the parent, and the largest increase occurs at lower temperatures, below 500 o C.

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In air, the log(σ) vs 1/T plot for SSZ40GO (red curve in Figure 2a) is non-linear, and can be divided into three broad sections: low-temperature (<300 °C), intermediate-temperature (300-650 °C) and high-temperature (>650 °C). Apparent activation energies for each section were extracted from a plot of ln(σT) vs. 1/T and are 0.94(2), 0.57(1) and 1.51(3) eV, respectively. The changes in slope indicate variation in the conduction behavior as temperature is increased. This could arise from structural transitions or from a change in the dominant conduction mechanism. Since no crystallographic phase transitions were observed by variable-temperature diffraction, the changes in slope in the conductivity plot are more likely due to changes in the dominant charge carriers.

We also measured the conductivity of SSZ40GO under dry and wet flowing N2. The results are summarized in Figure 2b. Above ~700 °C, the conductivity is the same in all the atmospheres, suggesting that SSZ40GO is a purely oxide ion conductor in the high-temperature regime. In the intermediate-and low-temperature regions the conductivity in dry atmospheres is significantly lower than that in moist atmospheres. This, and the curvature of the moist-atmosphere plots, are suggestive of the presence of proton conductivity (with a magnitude comparable to the Gd-doped BaZrO3, SrZrO3 and SrCeO3 proton conductors [START_REF] Zając | Applicability of Gd-Doped BaZrO3, SrZrO3, BaCeO3 and SrCeO3 Proton Conducting Perovskites as Electrolytes for Solid Oxide Fuel Cells[END_REF] ). Similar behavior has been reported for the related Ga-rich perovskite Sr2Sc0.5Ga1.5O5•0.13H2O. [START_REF] Chernov | Sr2GaScO5, Sr10Ga6Sc4O25, and SrGa0.75Sc0.25O2.5: A Play in the Octahedra to Tetrahedra Ratio in Oxygen-Deficient Perovskites[END_REF] The conductivity behavior under air or wet N2 can be understood in terms of a low-temperature proton-conducting region, a decreasing proton contribution at intermediate temperature as water is lost under heating, and a purely oxide ion conducting region at high temperature once all water has been lost from the sample. The difference in low-temperature conductivity between air and water-saturated N2 is small, indicating similar water uptake under these conditions. Also, the absence of hysteresis in the heating and cooling cycles in moist N2 suggests that SSZ40GO picks up water very quickly on cooling and starting from high temperatures (~700 °C).

To directly compare only the oxide ion conductivity of SSGO and the zinc-substituted phases, their conductivities under dry nitrogen are plotted in Figure 2c. SSZ40GO is a better oxide ion conductor than SSGO across the entire temperature range, but has a larger activation energy possibly due to cation disorder and/or defect trapping. [START_REF] Perriot | Intrinsic Carriers: Contrasting Effects of Cation Chemistry and Disorder on Ionic Conductivity in Pyrochlores[END_REF] The lowand intermediate-temperature oxide ion conductivity of SSZ40GO is lower than that of the 5%-substituted material (SSZ05GO), despite SSZ40GO having a higher-symmetry, more disordered structure and more oxygen vacancies. This again suggests the presence of local ordering that traps oxide ions. Comparisons between the conductivity curves obtained in wet and dry atmospheres show an approximately ½ an order of magnitude increase due to proton conductivity in SSGO and SSZ05GO. SSZ40GO shows a much larger increase (~3 orders of magnitude, Figure 2c) suggesting that either there are more protons present or they are much more mobile.

In comparison with other families where the conductivity mechanism is well established, the oxide ion conductivity of SSZ40GO at 800 °C (1.3 × 10 -4 S cm -1 ) is around two orders of magnitude lower than the benchmark vacancy-type conductor yttria-stabilized-zirconia [START_REF] León-Reina | High Oxide Ion Conductivity in Al-Doped Germanium Oxyapatite[END_REF] and also Bi46V8O89. [START_REF] Kuang | Polymorphism and Oxide Ion Migration Pathways in Fluorite-Type Bismuth Vanadate, Bi46V8O89[END_REF] The latter exhibits a complex oxide ion migration mechanism reliant on vacancy-hopping through the fluorite-related Bi-O sublattice, and fast oxide ion exchanges with the V-O sublattice. [START_REF] Kuang | Polymorphism and Oxide Ion Migration Pathways in Fluorite-Type Bismuth Vanadate, Bi46V8O89[END_REF][START_REF] Dunstan | Variable-Temperature Multinuclear Solid-State NMR Study of Oxide Ion Dynamics in Fluorite-Type Bismuth Vanadate and Phosphate Solid Electrolytes[END_REF] It is also about two orders of magnitude lower than the oxygen-excess apatite-type Bi2La8(GeO4)6O3, in which the ionic migration proceeds mainly via interstitial oxygen exchanges between GeO4 and GeO5 groups, with a smaller contribution from diffusion along the hexagonal channels in the structure [START_REF] Tate | New Apatite-Type Oxide Ion Conductor, Bi2La8(GeO4)6O3: Structure, Properties, and Direct Imaging of Low-Level Interstitial Oxygen Atoms Using Aberration-Corrected Scanning Transmission Electron Micros[END_REF][START_REF] Peet | Location and Orientation of Lone Pairs in Apatite-Type Materials: A Computational Study[END_REF][START_REF] Peet | Dynamics in Bi(III)-Containing Apatite-Type Oxide Ion Conductors: A Combined Computational and Experimental Study[END_REF] , and one order of magnitude lower than oxygen-excess scheelite-type LaNb0.92W0.08O4.04 . [START_REF] Toyoura | Oxide-Ion Conduction: Via Interstitials in Scheelite-Type LaNbO4: A First-Principles Study[END_REF][START_REF] Li | Crystal Structure and Potential Interstitial Oxide Ion Conductivity of LnNbO4and LnNb0.92W0.08O4.04[END_REF] It compares favorably with perovskite-related Nd0. 9Sr0.1BaInO3.95, which exhibits strongly anisotropic vacancy-hopping-based oxide ion diffusion. [START_REF] Li | Identification of Oxygen Diffusion Mechanisms in Nd1-xAExBaInO4-x/2 (AE = Ca, Sr, Ba) Compounds through Molecular Dynamics[END_REF] While SSZ40GO is therefore a moderately good oxide ion conductor in absolute terms, the most significant direct comparison is the two order of magnitude oxide ion conductivity increase relative to its parent compound SSGO, likely related to the changes of local structure described later in the paper.

Water incorporation into SrSc0.3Zn0.2Ga0.5O2.4. To support the conclusion of a proton conductivity contribution in SSZ40GO, its water uptake was probed with a variety of methods. [START_REF] Adler | Chemical Structure and Oxygen Dynamics in Ba2In2O5[END_REF] H NMR showed a weak resonance at 3.1 ppm indicating the presence of a small number of protons. TGA measurements showed two regions of mass loss: between 100 and 400 °C and at 700-800 °C. Multiple TGA measurements (Figure S5) on different SSZ40GO samples gave a formula of SrSc0.3Zn0.2Ga0.5O2.4•xH2O, with x between 0.025 and 0.1, depending on the exact thermal history of the sample.

Unit cell parameters extracted from VT-XRD also show a deviation from their extrapolated high-temperature behavior between 100 and 400 °C (Figure 3). This is more apparent in the derived thermal expansion coefficient, which also reveals a smaller change in slope at ~800 °C. These discontinuities coincide with the mass losses in the TGA and are therefore likely to be due to the same process. They are also consistent with the moist-atmosphere impedance data, where we proposed the proton contribution starts decreasing above ~300 °C and disappears completely from 800 °C.

On first inspection there is an apparent discrepancy between the TGA data, which show no, or very limited, water uptake on cooling on the time scale of the experiment (~100 minutes, Figure 3c), and the conductivity data which suggest rapid water uptake even at high temperature (Figure 2b). Interestingly, the VT-XRD data show both behaviors, depending on how the experiment is conducted. When data are recorded on loosely sprinkled powder sample on heating and cooling, the cell parameter changes are irreversible, suggesting no moisture uptake on cooling, even at the very slow cooling rate of 0.1 K min -1 . In contrast, when data are recorded on a pressed pellet (similar to those used for impedance measurements) we see clear reversible cell parameter behavior, suggesting moisture uptake below 300 o C. Unit cell parameter changes illustrating the reversible and irreversible behavior are given in Figure S6. Our diffraction and impedance experiments therefore consistently show that pelletized samples undergo reversible water uptake, whereas loose powders show little or very slow water uptake. This can be rationalized in terms of the enhanced capillary effects caused by the concave pores on the surface of the pellet versus the convex surface of the loose powder grains. [START_REF] Jain | Effect of Pore Size Distribution on Humidity Sensing Properties of MgO Doped ZrO2-TiO2 Ceramic[END_REF][START_REF] Sidhu | Diffusion and Capillary Flow in Track-Etched Membranes[END_REF][START_REF] Yamazoe | Humidity Sensors: Principles and Applications[END_REF][START_REF] Kim | Water Uptake of Nanocrystalline Ceria: Weight and Conductance Effects[END_REF][START_REF] Carrott | Ex-Hydroxide Magnesium Oxide as a Model Adsorbent for Investigation of Micropore Filling Mechanisms[END_REF] 4a, and as color maps of S(Q,ω) in Figure 4b. The appearance of the QENS signal at high temperatures is evident from the broadening of the elastic peak between 800 and 1000 °C (red and orange curves in Figure 4a, respectively). Figure 4b shows that this broadening is largest at high Q values. This indicates the onset of a high-temperature dynamic process in SSZ40GO on the picosecond timescale. In order to demonstrate that this process corresponds to long-range jump diffusion and calculate jump distance and lifetime, we considered the Q-dependence of the QENS signal (Figure S7). However, the data quality at low Q values precludes us from drawing a definitive conclusion. While picosecond dynamics are observed in SSZ40GO, we did not record clear QENS broadening signals at a sufficient number of temperatures to allow us to extract quantitative parameters such as the activation energy for this process.

The phonon density of states (PDOS) plots extracted from the spectra measured at 20, 200, 400, 600, 800 and 1000 °C are shown in Figure S8. The most prominent change is the disappearance of a broad feature centered around 80 meV between 200 and 400 °C. This broad band in the INS spectra corresponds to librational motions of water molecules. [START_REF] Loong | Hydrogen Vibrational Density of States of Adsorbed Water on Rare-Earth Modified Zirconia[END_REF][START_REF] Loong | Crystal Phases, Defects, and Dynamics of Adsorbed Hydroxyl Groups and Water in Pure and Lanthanide-Modified Zirconia: A Neutron-Scattering Study[END_REF] Water loss in SSZ40GO in this temperature range is consistent with the conductivity, unit cell parameter and TGA observations discussed above.

Elastic and inelastic fixed window scans carried out on the backscattering spectrometer IN16B, which is sensitive to motions on the nanosecond timescale, are shown in Figure 5a and5b, respectively. These measurements record the intensity of the elastic peak (at 0 µeV) and the inelastic intensity at fixed energy offset as a function of temperature. An inelastic energy offset of 2 µeV was chosen based on our previous measurements of mixed metal oxide systems. [START_REF] Peet | Insight into Design of Improved Oxide Ion Conductors: Dynamics and Conduction Mechanisms in the Bi0.913V0.087O1.587 Solid Electrolyte[END_REF] On heating, we see a decrease of ~30% in the elastic intensity up to 500 °C. This is a steeper decrease in intensity than would be expected due to Debye-Waller effects alone. This can be seen from the shallower gradient at higher temperatures on heating and on cooling (dashed line in the EFWS in Figure 5a). This loss of elastic intensity coincides with a peak in the inelastic fixed window scans. Together, these data suggest a dynamic process occurring in SSZ40GO at these lower temperatures, which is irreversible on cooling in vacuum. Based on our other experimental evidence, this dynamic process likely arises from the mobility of protons and the subsequent water loss. A second decrease of the elastic intensity and concurrent increase in the inelastic intensity at 2 µeV is observed from 600 °C onwards (Figure 5b). This can be attributed to oxide ion dynamics which dominate the conductivity at high temperatures.

Full inelastic spectra measured on IN1B spectrometer after the initial sample heating (i.e., after water had been removed from the sample) are shown in Figure S9. As expected, no QENS signals were seen in the lower temperature region. Only a weak indication of broadening was observed at the highest temperature in our experiment (1000 °C, red vs blue data points in the bottom panel of Figure S9), despite the increase in inelastic intensity indicating that the dynamics have entered the energy window of the instrument. In other words, the second dynamic process, corresponding to oxide ion conduction, is either too fast or too weak at 1000 °C to be unambiguously and reliably observed from IN16B data. It is worth noting that in this study we observed oxide ion dynamics directly by neutron scattering for SSZ40GO, while they were not observed for SSGO on either IN6-SHARP or IN16B spectrometer. [START_REF] Fuller | Brownmillerite-Type Sr2ScGaO5 Oxide Ion Conductor: Local Structure, Phase Transition, and Dynamics[END_REF] This is consistent with impedance measurements, which show that oxide ion conductivity of SSZ40GO at high temperatures (2.9 × 10 -2 S cm -1 at 900 °C) is about two orders of magnitude higher than that of undoped SSGO (3 × 10 -4 S cm -1 at 900 °C). These findings are also consistent with our neutron scattering study of La2Mo2O9, which exhibits oxide ion conductivity of 2 × 10 -2 S cm -1 at 600 °C (i.e. the same order of magnitude as SSZ40GO at 900 °C), where QENS signals were observed on IN16B from 570 o C onwards. [START_REF] Peet | Direct Observation of Oxide Ion Dynamics in La2Mo2O9 on the Nanosecond Timescale[END_REF]

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Local Structure of SrSc0.3Zn0.2Ga0.5O2.4. Powder X-ray and neutron diffraction show that SSZ40GO is a disordered perovskite on average, but from a chemical perspective we expect there to be significant local ordering due to the different coordination preferences of the cations, the large number of oxygen vacancies relative to the ABO3 perovskite and the diffuse scattering seen in the NPD pattern. In addition, [START_REF] Li | Identification of Oxygen Diffusion Mechanisms in Nd1-xAExBaInO4-x/2 (AE = Ca, Sr, Ba) Compounds through Molecular Dynamics[END_REF] Sc and 71 Ga solid state NMR spectra (Figure S10) suggest that both species are present in a range of local environments.

The local structure was studied in more detail using neutron total scattering and PDF analysis. As a starting point, an average perovskite model was fitted to the Bragg and PDF data simultaneously, yielding the D(r) fit shown in Figure 6a. The D(r) data at large r are described fairly well by the perovskite model, but there are significant deviations below ~8 Å. Since these features could not be described using a simple perovskite unit cell, a series of large-box models were tested. In each case, an approximately isotropic simulation box containing 2000-8000 atoms was created. Topas Academic was used to fit the atomic coordinates of the models to the PDF and Bragg data simultaneously, subject to polyhedral restraints to maintain sensible local chemical environments. By including the Bragg data, the local structural model remains consistent with the average long-range structure. Models were judged based on the visual fits to both datasets and the corresponding R-factors, alongside coordination number, bond length and bond angle distributions that assess chemical plausibility.

The closely related brownmillerite structural model was tested first. All Sc atoms were placed on the octahedral sites, occupying 60% of them. The remaining 40% of octahedral sites and the tetrahedral sites were populated by Ga and Zn in their nominal ratios. This distribution reflects the coordination environment preferences of the atoms based on the statistical analysis by Waroquiers et al. [START_REF] Waroquiers | Statistical Analysis of Coordination Environments in Oxides[END_REF] and what would be expected based on related oxides. The calculated D(r) shows significant differences to the experimental data, but the positions of the first two peaks are well-described (Figure 6b). Here, the shorter bond lengths at ~1.85 Å come from tetrahedra and the longer bonds at ~2.05 Å from octahedra. This suggests that SSZ40GO contains these (or similar) polyhedral units. The octahedral:tetrahedral group ratio (Oh:T) in brownmillerite is 1:1 and this (when combined with the different scattering lengths) overestimates the peak due to the longer, octahedral bonds compared to the experimental data. This calculated overestimate relative to the experimental data implies that SSZ40GO must contain more tetrahedral than octahedral groups, and therefore more of the Zn and Ga occupy tetrahedral sites. The next model tested therefore aimed to maximize the number of tetrahedra while retaining the polyhedral corner-connectivity of the brownmillerite and perovskite structures. An in-house Monte Carlo code was used to construct the polyhedral framework. Starting from a perovskite structure, oxygen atoms were randomly removed to produce an oxygen content corresponding to ABO2.4; oxygen atoms and vacancies were then swapped to create a network of chemically-sensible corner-linked polyhedra. The number of 1,2,3 and 5-coordinate polyhedra were minimized by the algorithm, and the resulting configuration comprised ACS Paragon Plus Environment Chemistry of Materials 60% 4-coordinate polyhedra, 39% octahedra and 1% 5-coordinate polyhedra, with a ratio of octahedra to 4-coordinate polyhedra of 1:1.5. One consequence of this minimization algorithm is the creation of square planar sites alongside tetrahedra (approximately 20% of the 4-coordinate sites; the statistically expected number). This is somewhat undesirable as none of the B-site cations are expected to occupy square planar environments, but is difficult to avoid. (Configurations in which we explicitly minimized the number of square planar sites had a higher number of 1, 2 and 3-coordinate polyhedra, which is chemically less likely.) After the construction of the polyhedral network, cations were initially arranged randomly across all sites to give the correct overall composition.

This model gave a good fit to the data both locally and on longer length scales; however, it is not chemically sensible as the cations don't have their preferred coordination environment. [START_REF] Waroquiers | Statistical Analysis of Coordination Environments in Oxides[END_REF] For example, analysis of 103 Sc-containing compounds by Waroquiers et al. [START_REF] Waroquiers | Statistical Analysis of Coordination Environments in Oxides[END_REF] found no occurrences of 4coordinate Sc so we can be confident that this random model is not a likely representation of the real structure. When a more sensible cation distribution was imposed, i.e. Sc atoms were exclusively placed on octahedral sites and Ga and Zn were arranged over the remaining octahedral and 4coordinate sites, the local fit became significantly worse with the model again overestimating the peak due to 2.05 Å bond lengths (see Figure 6c).

These observations suggest that SSZ40GO contains more tetrahedral groups than the Oh:T ratio 1:1.5 indicates. The only way to attain this with the oxygen content ABO2.4 or ABO2.5 is to break the corner-connectivity of the polyhedra and include some with one-coordinate or terminal oxygen sites (e.g. MO3/2O1 tetrahedral groups in Niggli formalism). A small number of perovskite-related compounds with such units have been reported. The compound Sr10Sc4Ga6O25 described by Chernov et al. [START_REF] Chernov | Sr2GaScO5, Sr10Ga6Sc4O25, and SrGa0.75Sc0.25O2.5: A Play in the Octahedra to Tetrahedra Ratio in Oxygen-Deficient Perovskites[END_REF] has an Oh:T ratio of 1:1.5 in which one third of the tetrahedra have terminal O sites (MO3/2O1). Several compounds with general formula A3Oh1T2O7.5 (where A = Ba, Sr, Na, Ca; Oh = Y, Mn, Er and T = Ga, Co, V, Al, Si) [START_REF] Abakumov | The Phase Transitions and Crystal Structures of Ba3RM2O7.5 Complex Oxides (R = Rare-Earth Elements, M = Al, Ga)[END_REF][START_REF] Wang | Crystal Structure of Sr6Y2Al4O15: XRD Refinements and First-Principle Calculations[END_REF][START_REF] Luo | Complex Cation Order in Anion-Deficient BanYFen-1O2.5n Perovskite Phases[END_REF][START_REF] Luo | Synthesis and Selective Topochemical Fluorination of the Cation and Anion-Vacancy Ordered Phases Ba2YCoO5 and Ba3YCo2O7.5[END_REF][START_REF] Arroyabe | K2Ca6Si4O15-Structural and Spectroscopical Studies on a Mixed Tetrahedral-Octahedral Framework[END_REF][START_REF] Ben Yahia | Structure and Magnetic Properties of the New Oxygen-Deficient Perovskite Na3MnV2O7.5[END_REF] have a Oh:T ratio of 1:2, with half the tetrahedra being MO3/2O1. Structures of Sr10Sc4Ga6O25 and Ba3ErGa2O7.5 are included as examples in Figure S11. The compound Sr2Sc0.5Ga1.5O5, also reported by Chernov, [START_REF] Chernov | Sr2GaScO5, Sr10Ga6Sc4O25, and SrGa0.75Sc0.25O2.5: A Play in the Octahedra to Tetrahedra Ratio in Oxygen-Deficient Perovskites[END_REF] is proposed to have an even higher tetrahedron content (Oh:T of 1:3), although its structure was simply reported as a disordered perovskite.

Large box models corresponding to A2Oh0.8T1.2O5 (OT1.5) and A2Oh0.66T1.33O5 (OT2) were constructed based on these literature structures [START_REF] Chernov | Sr2GaScO5, Sr10Ga6Sc4O25, and SrGa0.75Sc0.25O2.5: A Play in the Octahedra to Tetrahedra Ratio in Oxygen-Deficient Perovskites[END_REF][START_REF] Abakumov | The Phase Transitions and Crystal Structures of Ba3RM2O7.5 Complex Oxides (R = Rare-Earth Elements, M = Al, Ga)[END_REF] and fitted to the PDF and Bragg data of SSZ40GO. Both models provide a good description of the local structure below 8 Å. OT1.5 still slightly overestimates the proportion of the 2.05 Å bond lengths (Figure 6d), whereas OT2 gives a much better agreement (Figure 6e). The close agreement between this final model and the observed data suggests that terminal oxygen sites are present in SSZ40GO. Distributions of bond length and angles of the final model are available in Figure S12.

It is, of course, possible that there are some 5-coordinate species present in SSZ40GO. This has been suggested, for example, in structural models of disordered BaY0.25Fe0.75O2.5. [START_REF] King | Identifying the Local Structural Units in La0.5Ba0.5MnO2.5 and BaY0.25Fe0.75O2.5 through the Neutron Pair Distribution Function[END_REF] Although Zn and Ga don't have the strong electronic driving force for 5-coordination exhibited by some transition metals, 5-coordination is found in some of their oxides. It is, however, significantly less common than 4-or 6-coordination; Waroquiers give values of 48%:21%:20% and 61%:6%:33% 4:5:6 for Zn and Ga, respectively. [START_REF] Waroquiers | Statistical Analysis of Coordination Environments in Oxides[END_REF] This tendency is also reflected in molecular compounds. The 2020 Cambridge Structural Database [START_REF] Groom | The Cambridge Structural Database[END_REF] , shows a ~40%:20%:40% ratio of 4:5:6 coordination for Zn coordinated solely by O, and 30%:10%:60% for Ga. We also tested two structural models containing more 5-coordinate metal sites (a random perovskite model with a 39%:42%:19% 4:5:6 coordination number ratio, and the favored Ba3YFe2O7.5 model of King et al. [START_REF] King | Identifying the Local Structural Units in La0.5Ba0.5MnO2.5 and BaY0.25Fe0.75O2.5 through the Neutron Pair Distribution Function[END_REF] with 25%:25%:50% Td:SqPy:Oh, both with the 6-coordinate sites being exclusively occupied by Sc and Zn/Ga on other sites); both models gave significantly worse fits to the data. Given the complexity of SSZ40GO and the fact that the contributions from all the M-O distances contribute to a relatively narrow region of the PDF, it is not possible to definitively rule out the possibility of a small number of five coordinate Zn/Ga, but we favor our all octahedral-tetrahedral model.

There is a very small overestimation of the peak due to 2.05 Å bonds in our final OT2 model, but this is likely to be due to the fact that the model has an oxygen stoichiometry of O2.5 (normalized to perovskite) whereas SSZ40GO has O2.4. This could mean that the real material has an even higher proportion of tetrahedra. Indeed, if all Ga and Zn atoms in SSZ40GO occupied a tetrahedral site the Oh:T ratio would be 1:2.33, but we are not aware of a simple perovskite derivative structure corresponding to this composition. Polyhedral connectivity leading to an Oh:T ratio of 1:3, as would be expected in the single-phase limit SSZ50GO, is geometrically possible; however, we are again unaware of structurally characterized compounds with this ratio. The local structure of the closely related compound, Sr2Sc0.5Ga1.5O5, has not been fully investigated but evidence of local ordering was observed as diffuse scattering in electron diffraction patterns [START_REF] Chernov | Sr2GaScO5, Sr10Ga6Sc4O25, and SrGa0.75Sc0.25O2.5: A Play in the Octahedra to Tetrahedra Ratio in Oxygen-Deficient Perovskites[END_REF] indicating that small domains of this connectivity may be plausible. This material was also shown to decompose in moist atmospheres suggesting this connectivity pattern is very not stable.

The final OT2 model was also fitted to datasets recorded at 600 and 1000 °C. No major structural changes are expected from room temperature since the diffraction patterns show no phase transitions. The structural diffuse scattering in the NPD pattern is also unchanged up to at least 900 o C, so it is unlikely that the local structure undergoes significant changes. We find that, with the exception of thermal broadening, the experimental PDFs at higher temperatures are comparable with that at room temperature. Accordingly, the OT2 model gives a good fit to both the 600 and 1000 °C datasets (Figure S13).

CONCLUSIONS

We have investigated Zn substitution for Sc in Sr2ScGaO5, resulting in the formation of a new Pm-3m perovskite phase for certain Zn concentrations. The composition ACS Paragon Plus Environment Chemistry of Materials SrSc0.3Zn0.2Ga0.5O2.4 (SSZ40GO) was studied in detail. This phase is a highly disordered perovskite on average, has a remarkably low oxygen content, and has a two order-of-magnitude increase in conductivity over its brownmillerite parent at high temperature. It can incorporate a limited amount of water reversibly into the structure resulting in a measurable change in low temperature cell parameters, as well as a significant proton contribution to the conductivity. QENS signals were observed for both proton and oxygen dynamics, but they were not large enough to extract quantitative parameters such as activation energy, jump length and frequency.

The local structure of SSZ40GO is markedly different from its average structure. Large-box fitting to neutron total scattering data suggests that local polyhedral environments include 50% of the tetrahedra with a terminal oxygen site. The terminal oxygen site is an unusual structural feature, but consistent with observations in related crystallographically characterized materials. There is significant precedence from our work and that of others that this type of lowlinked tetrahedron can lead to oxygen dynamics at low temperatures, [START_REF] Kendrick | Developing Apatites for Solid Oxide Fuel Cells: Insight into Structural, Transport and Doping Properties[END_REF][START_REF] Evans | Phase Transitions, and Oxygen Migration in Zirconium Tungstate[END_REF][START_REF] Kuang | Interstitial Oxide Ion Conductivity in the Layered Tetrahedral Network Melilite Structure[END_REF][START_REF] Tallentire | Systematic and Controllable Negative, Zero, and Positive Thermal Expansion in Cubic Zr1-xSnxMo2O8[END_REF] and this could be a contributing factor to why SSZ40GO has a higher oxide ion conductivity than its parent material, SSGO, by two orders of magnitude.

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Figure 1 .

 1 Figure 1. Rietveld fits of a simple Pm-3m perovskite model against the room temperature X-ray (top) and neutron (bottom) data for SSZ40GO. Blue, red and grey curves show the observed, calculated and difference curves respectively. Blue tick marks indicate the expected peak positions. The peak highlighted with an asterisk is as a result of W Lα incident X-ray contamination. X-ray data are shown on a square root scale to emphasize weaker features. Variable-temperature PXRD was performed across a temperature range of 16-1273 K and some of the data are displayed in Figure S2. No major change (peak splitting / appearance / disappearance) is observed, indicating that there are no structural phase transitions in this temperature range. The same conclusion can be drawn from the NPD

Figure 2 .

 2 Figure 2. Conductivity as a function of temperature for SSZ40GO and related compounds. a) comparison of conductivities in air for the different compounds. b) conductivity under different atmospheres for SSZ40GO c) comparison between conductivities in dry nitrogen for the different compounds. Data shown for SSGO and SSZ05G are from both heating and cooling cycles.

Figure 3 .

 3 Figure 3. SSZ40GO cell parameter evolution with temperature from variable temperature X-ray diffraction (top panel) and derived thermal expansion (da/dT) (middle). Thermogravimetric analysis over the same temperature range (bottom) on both heating and cooling cycles. Cell parameters on cooling are shown in the SI. Probing dynamics in SrSc0.3Zn0.2Ga0.5O2.4 by inelastic neutron scattering. Inelastic neutron scattering data were collected on SSZ40GO on the time-of-flight spectrometer IN6-SHARP, which probes dynamics on picosecond timescales. QENS spectra were recorded at 20, 200, 400, 600, 800 and 1000 °C. Plots of the scattering function S(Q=1.92,ω) against energy transfer are shown in Figure4a, and as color maps of S(Q,ω) in Figure4b. The appearance of the QENS signal at high temperatures is evident from the broadening of the elastic peak between 800 and 1000 °C (red and orange curves in Figure4a, respectively). Figure4bshows that this broadening is largest at high Q values.

Figure 4 .

 4 Figure 4. a) Plot of the scattering function S(ω) at Q = 1.92 Å showing the QENS broadening of the elastic peak. b) Colour maps of S(Q,ω) at different temperatures to show the evolution of the QENS broadening at high temperatures and that this is enhanced at high Q.

Figure 5 .

 5 Figure 5. a) Elastic intensity as a function of temperature summed over all Q-values (0.2-1.75 Å -1 ). Multiple heating and cooling cycles are shown. The dashed line is a guide to the eye showing the expected decrease in intensity due to Debye-Waller effects. Temperatures where QENS spectra were subsequently measured are indicated by the red markers on the temperature axis. b) Inelastic intensity at 2 µeV energy transfer as a function of temperature summed over all Q-values (0.2-1.75 Å -1 ). Key features are labelled with the dynamic processes to which they have been attributed.

Figure 6 .

 6 Figure 6. Fit of various local structure models (red curve) to the measured PDF (black markers) data of SSZ40GO. Difference curves are shown in grey. The Rwp values given are calculated for 1.5 Å ≤ r ≤ 8 Å to reflect the fit to the local structure.

ASSOCIATED CONTENT

Supporting Information. This material is available free of charge via the Internet at http://pubs.acs.org Diffraction patterns of all synthesized compounds; variable temperature XRD on SSZ40GO; additional variable temperature NPD patterns of SSZ40GO; a representative complex impedance plot; TGA and cell parameter cycling plots; Q-depend-


Éric Brémond

Ilaria Ciofini
email: ilaria.ciofini@chimie-paristech.fr

Juan Carlos

Carlo Adamo
email: carlo.adamo@chimie-paristech.fr

Double-Hybrid Functionals and Tailored Basis Set: Fullerene (C 60 ) Dimer and Isomers as Test Cases

A computational protocol making use of Double Hybrid functionals in conjunction with a recently developed basis set tailored to reproduce non covalent interactions (hereafter named DH-SVPD) is here applied and tested for the evaluation of properties C 60 fullerenes namely intermolecular interactions in the weakly bound C 60 dimer and relative stabilities of C 60 isomers (as described by the C60ISO and iso-C60 datasets).

The obtained results suggest that the DH-SVPD performance is very close to that obtained with empirical corrections and larger quadruple- basis for the C 60 dimer. In contrast, both approaches (tailored basis set and larger basis with empirical potential) do not reach the envisaged accuracy for the relative stabilities of C 60 isomers.

Nevertheless, this test well underlined how the DH-SVPD basis set is able to recover the performance obtained by coupling the DH functionals with empirical dispersion corrections and larger basis set, significantly reducing the computational effort for double hybrids and thus enabling to expand their application domain to larger molecular systems.

1.Introduction

Double hybrid (DH) functionals [START_REF] Sancho-García | Double-hybrid density functionals: merging wavefunction and density approaches to get the best of both worlds[END_REF] , that are exchange-correlation functionals casting a fraction of correlation computed using a second order perturbative (PT2) approach [START_REF] Görling | Exact Kohn-Sham scheme based on perturbation theory[END_REF] , represent a further step forward with respect to the global hybrids (GH), introduced by Becke more than 20 years ago [START_REF] Becke | A new mixing of Hartree-Fock and local density-functional theory[END_REF] . Indeed, modern DH schemes, firstly proposed by Truhlar [START_REF] Zhao | Doubly hybrid meta DFT: new multi-coefficient correlation and density functional methods for thermochemistry and thermochemical kinetics[END_REF] and then popularized by Grimme [START_REF] Grimme | Semiempirical hybrid density functional with perturbative second-order correlation[END_REF] , provide enhanced performance for a number of properties, ranging from molecular structures to thermochemistry, reaction barriers, and even excited-state energies [START_REF] Sancho-García | Double-hybrid density functionals: merging wavefunction and density approaches to get the best of both worlds[END_REF] .

Contrarily to what can be expected from an error compensation between the (overbinding) PT2 and the (underbinding) DFT contributions, weak dispersion interactions are yet not very accurately reproduced by DHs [START_REF] Goerigk | A thorough benchmark of density functional methods for general main group thermochemistry, kinetics, and noncovalent interactions[END_REF][START_REF] Calbo | Accurate treatment of large supramolecular complexes by double-hybrid density functionals coupled with nonlocal van der Waals corrections[END_REF][START_REF] Sancho-García | Partnering dispersion corrections with modern parameter-free double-hybrid density functionals[END_REF][START_REF] Pérez-Jiménez | Accurate description of interaction energies and three-body effects in weakly bound molecular complexes by PBE-QIDH models[END_REF] An effective way to overcome this problem is to add an empirical correction based on pairwise classical potential [START_REF] Wu | Empirical correction to density functional theory for van der Waals interactions[END_REF][START_REF] Grimme | Accurate description of van der Waals complexes by density functional theory including empirical corrections[END_REF] . This simple and chemically sound approach meets a large success since in such a way interaction energies of non-covalently bonded or of weakly interacting systems can be reproduced with minimal computational extra cost and high accuracy [START_REF] Risthaus | Benchmarking of London dispersion-accounting density functional theory methods on very large molecular complexes[END_REF] . Such potentials depend, of course, both on the atom types involved in the interactions and on the exchange-correlation functional considered [START_REF] Grimme | A consistent and accurate ab initio parametrization of density functional dispersion correction (DFT-D) for the 94 elements H-Pu[END_REF][START_REF] Grimme | Effect of the damping function in dispersion corrected density functional theory[END_REF] . Empirical corrections accounting for three-body interactions have been also developed, enabling to further increase the overall accuracy [START_REF] Von Lilienfeld | Two-and three-body interatomic dispersion energy contributions to binding in molecules and solids[END_REF] .

When such empirical potentials are coupled to DHs, small errors are found on dispersionruled systems [START_REF] Sancho-García | Partnering dispersion corrections with modern parameter-free double-hybrid density functionals[END_REF] , even if their contribution is lower than what found when these corrections are used in conjunction with more traditional functionals, such as GHs. Indeed, DHs provide a better estimation of the dispersion effects at intermediate intermolecular distances, which shows the delicate interplay between electronic effects at all ranges. [START_REF] Brauer | The S66x8 benchmark for noncovalent interactions revisited: explicitly correlated ab initio methods and density functional theory[END_REF] In this paper we will focus on the behavior of two different double hybrids: the B2-PLYP [START_REF] Grimme | Semiempirical hybrid density functional with perturbative second-order correlation[END_REF] and the PBE-QIDH [START_REF] Brémond | Double-hybrid functionals from adiabatic-connection: the PBE-QIDH model[END_REF] functional. The first was developed in analogy to the parent B3LYP approach [START_REF] Becke | Density-functional thermochemistry. III. The role of exact exchange[END_REF] , that is fitting the (few) coefficients contained its functional form on some reference dataset [START_REF] Grimme | Semiempirical hybrid density functional with perturbative second-order correlation[END_REF] . The second, instead, has been developed choosing a number of theoretical constraints among those that the universal functional should fulfill [START_REF] Brémond | Double-hybrid functionals from adiabatic-connection: the PBE-QIDH model[END_REF][START_REF] Cohen | Assessment and formal properties of exchangecorrelation functionals constructed from the adiabatic connection[END_REF][START_REF] Toulouse | Rationale for a new class of doublehybrid approximations in density-functional theory[END_REF][START_REF] Brémond | Nonempirical double-hybrid functionals: An effective tool for chemists[END_REF] .

For the sake of clearness, hereafter we will denoted as non-empirical all the exchangecorrelation functionals whose internal coefficients are not determined by an error minimization procedure with respect to external reference datasets, but rather fixed using only theoretical arguments. The PBE-QIDH functional is thus ascribed to this category. By opposition, semi-empirical are those functionals whose coefficients are determined by a parametrization procedure, albeit their functional form could respect some theoretical constraints [START_REF] Tsuneda | A reexamination of exchange energy functionals[END_REF] . The B2-PLYP is the most widely used DH for this category.

Potentials developed for weak interactions are, instead, empirical by their own definition, since both their form and coefficients are determined on the basis of the best agreement with respect to external reference values.

It then clearly appears how the present situation is vexing, since non-empirical DH functionals must be partnered with empirical potential in order to provide accurate results for dispersion interactions.

With this inconsistency in mind, we have recently developed a basis set, called DH-SVPD, which significantly improves the performances of DHs in reproducing interaction energies of noncovalent systems, without requiring further corrections and thus allowing to restore a full non-empirical DFT approach [START_REF] Brémond | Small basis set allowing the recovery of dispersion interactions with double-hybrid functionals[END_REF] . This philosophy is even more stringent, since the DH-SVPD basis set is developed using a "self-consistent" approach where only energy contributions of the monomers and dimer computed at the same DH level of theory are considered, without need of any external reference values [START_REF] Varandas | Zeroth-order exchange energy as a criterion for optimized atomic basis sets in interatomic force calculations[END_REF] .

The first tests carried out on standard benchmarks, such as S22 [START_REF] Gráfová | Comparative study of selected wave Function and density functional methods for noncovalent interaction energy calculations using the extended S22 data set[END_REF] , S66 [START_REF] Řezáč | A well-balanced database of benchmark interaction energies relevant to biomolecular structures[END_REF] and L7 [START_REF] Sedlak | Accuracy of quantum chemical methods for large noncovalent complexes[END_REF] dataset, suggested that the PBE-QIDH/DH-SVPD model is well suited for noncovalent interactions, providing results close (or even better) than those obtained when the same DH is coupled to larger basis set and empirical dispersion corrections [START_REF] Brémond | Small basis set allowing the recovery of dispersion interactions with double-hybrid functionals[END_REF] . They also showed the transferability of this basis set to other DHs.

Here we want firstly to verify the robustness of this approach extending our investigation to larger systems, namely the weakly bound dimer of C 60 . This system represents indeed a challenging playground for any method rooted in Density Functional Theory (DFT) as pointed out by a recent work [START_REF] Sharapa | Accurate intermolecular potential for the C60 dimer: the performance of different levels of quantum theory[END_REF] , due to the dispersion forces which dominate the intermolecular interaction. Next, two sets concerning the relative stabilities of C 60 isomers were considered [START_REF] Sure | Comprehensive theoretical study of all 1812 C60 isomers[END_REF][START_REF] Karton | Performance of DFT for C60 isomerization energies: a noticeable exception to acob's ladder[END_REF] . Albeit composed by the same molecule, the relative stabilities of the C 60 isomers seem not affected by dispersion, but their accurate evaluation at the DFT level is still troublesome, being related to an accurate description of the aromatic resonance structures. It is therefore interesting to verify the performance of our model on these two challenging sets in order to fully validate this approach beyond purely noncovalent interactions.

Computational details

Calculations were carried out with the non-empirical PBE0-DH [START_REF] Brémond | Seeking for parameter-free double hybrid functionals: the PBE0-DH model[END_REF] and PBE-QIDH [START_REF] Brémond | Double-hybrid functionals from adiabatic-connection: the PBE-QIDH model[END_REF] functionals and the semi-empirical B2-PLYP [START_REF] Grimme | Semiempirical hybrid density functional with perturbative second-order correlation[END_REF] and DSD-PBEP86 [START_REF] Kozuch | PBEP86: in search of the best double-hybrid DFT with spin-component scaled MP2 and dispersion corrections[END_REF] approaches. In some cases, these functionals have been coupled to the empirical D3(BJ) [START_REF] Becke | Exchange-hole dipole moment and the dispersion interaction revisited[END_REF] dispersion correction whose paramters are reported in Refs 34 and 35 for non-empirical DHs. Preliminary tests showed that, in contrast to our previous experience with smaller noncovalent systems (see comments in reference 21), the effect of three body-interactions, modelled by the Axilrod-Teller-Muto (ATM) potential [START_REF] Von Lilienfeld | Two-and three-body interatomic dispersion energy contributions to binding in molecules and solids[END_REF] , were negligible for the C 60 dimer. Indeed, this contribution was computed to be 0.18 kcal/mol, that is about 1% of the PBE-QIDH interaction energy. Therefore, this contribution will be not considered in the following.

Note that the DH-SVPD basis set has been developed to be used in connection with any DH functional, by optimizing the diffuse functions of the small Def2-SVPD basis set [START_REF] Weigend | Balanced basis sets of split valence, triple zeta valence and quadruple zeta valence quality for H to Rn: design and assessment of accuracy[END_REF] . Since this basis set has been developed considering an error compensation between Basis Set Superposition Error (BSSE) and Basis Set Incompleteness Error (BSIE), the former will be not explicitly taken into account, if not explicitly mentioned. This approach is consistent with some recent works which do not recommended BSSE corrections for small or medium basis set [START_REF] Sheng | Counterpoise correction is not useful for short and van der Waals distances but may be useful at long range?[END_REF][START_REF] Alvarez-Idaboy | Counterpoise corrected interaction energies are not systematically better than uncorrected ones: comparison with CCSD(T) CBS extrapolated values[END_REF] due to an unbalanced behavior between BSSE and BSIE. Moreover, the so-called halfcounterpoise approach, computed as the average of BSSE corrected and uncorrected energy, was recommended for MP2 calculations with small (lower than quadruple- quality) instead of a full BSSE corrections [START_REF] Burns | Comparing counterpoise-corrected, uncorrected, and averaged binding energies for benchmarking noncovalent interactions[END_REF] .

For purposes of comparison with literature data, calculations were also carried out using the large Def2-TZVPP or nearly complete Def2-QZVP basis sets [START_REF] Weigend | Balanced basis sets of split valence, triple zeta valence and quadruple zeta valence quality for H to Rn: design and assessment of accuracy[END_REF] .

The structures of the C 60 dimer were taken from reference 28, while the structures of the C60ISO and iso-C60 datasets are those reported in reference 29 and computed at PBE-D3-Def2-TZVPP level of theory. For the sake of clarity, they are sketched in Figure 1.

All calculations were performed with the Gaussian code 40 .

Results and discussion

C 60 dimer

The first system considered in this study is the weakly bound C 60 dimer for which accurate calculations have been recently reported [START_REF] Sharapa | Accurate intermolecular potential for the C60 dimer: the performance of different levels of quantum theory[END_REF] . As mentioned by Clark and collaborators [START_REF] Sharapa | Accurate intermolecular potential for the C60 dimer: the performance of different levels of quantum theory[END_REF] , the interaction between the two molecules in this dimer is largely dominated by dispersion interactions, but, in contrast with other systems, such as those collected in the L7 [START_REF] Sedlak | Accuracy of quantum chemical methods for large noncovalent complexes[END_REF] dataset, the many-body interactions are negligible. This dimer is, therefore, a perfect playground to verify the quality of the proposed PBE-QIDH/DH-SVP model where two-body interactions should be improved with respect to other density functionals by the presence of a PT2 contribution.

In Table 1 are reported the binding energies for the C 60 dimer, computed at different DFT levels using the same reference structure [START_REF] Sharapa | Accurate intermolecular potential for the C60 dimer: the performance of different levels of quantum theory[END_REF] , and completed with reference values obtained with wave-function correlated methods (all these latter values are taken from reference 28). These reference values are relatively homogeneous, ranging between -11.97 (CCSD(T)) and -12.94 (CEPA-1) kcal/mol [START_REF] Sharapa | Accurate intermolecular potential for the C60 dimer: the performance of different levels of quantum theory[END_REF] . As expected, the interaction energy computed at the MP2 level is largely overestimated (-23.46 kcal/mol), whereas the SOS-MP2 result is in better agreement with the reference value (-14.17 kcal/mol). Concerning DFT approaches, it clearly appears from the data collected in Table 1, that all the methods missing an empirical potential significantly underestimate the interaction energy. The obtained values range between 6.14 kcal/mol at the B3LYP level, i.e. corresponding to an unbound dimer, to -3.95 kcal/mol for the B2-PLYP functional, where the negative value well underlines the role of the dynamical correlation contribution.

The empirical corrections have, as expected, a beneficial effect on the binding energies leading in all reported cases to a bound dimer, albeit the estimated energies are roughly one half of the reference values (all between -6 and -7 kcal/mol). The only exceptions are represented by M06-2X-D3(BJ) (-9.7 kcal/mol) and B2-PLYP-D3(BJ) (-11.07 kcal/mol), even if the discrepancy with respect to the reference values is still large. In all these cases (with and without dispersion corrections), these values have been obtained using the Def2-TZVP basis set.

In this context, the PBE-QIDH functional coupled with the Def2-TZVP and the D3(BJ) dispersion provides a better agreement with respect to the reference values, its estimation being -12.29 kcal/mol. A very close result, -12.27 kcal/mol, is then found using the nonempirical PBE-QIDH/DH-SVPD protocol. Noteworthy, this small basis leads also to significant improvements for two other DHs, namely B2-PLYP (-8.00 kcal/mol) and DSD-PBEP86 (-17.40 kcal/mol), thus further proving its transferability.

It should be also remarked that the Def2-QZVP basis has 3720 functions (6120 primitives) for the C 60 dimer, while in the case of the smaller DH-SVPD basis only 1200 functions (1920 primitives) are present. Consequently, single point PBE-QIDH calculations with the latter basis set are roughly 4 times faster than those performed with the larger basis (for the C 60 dimer at the equilibrium distance) if all other technical factors remain the same.

Beside the interaction energy at a fixed geometry, it is also interesting to analyze the full potential energy profile along the distance between the centers of mass of the two moieties constituting the C 60 dimer. This test is even more stringent since the dispersion corrections could fail to reproduce the energy profile far from energy minima [START_REF] Gould | Are dispersion corrections accurate outside equilibrium? A case study of benzene[END_REF] , sometime giving spurious discontinuities [START_REF] Savarese | Exploring the limits of recent exchangecorrelation functionals in modeling lithium/benzene interaction[END_REF] . In our case, however, the validity of this test should be modulated, since the available reference values are calculated at the CEPA-1/cc-pVDZ level, while larger basis set are normally used to reach energies at convergence with this level of theory (triple-or quadruple-, see for instance reference 43 and 44). Indeed, the dissociation energy at the CEPA-1 level is -12.94 kcal/mol with the cc-pVTZ basis set, and -6.88 kcal/mol with the smaller cc-pVDZ basis set [START_REF] Sharapa | Accurate intermolecular potential for the C60 dimer: the performance of different levels of quantum theory[END_REF] .

The profile computed at the PBE-QIDH/DH-SVPD level is reported in Figure 2 together with other results taken from reference 28. This profile provides the deepest well with respect to the other ones reported, the reference CEPA-1 being the shallowest one. However, the PBE-QIDH/DH-SVPD potential does not show any unphysical discontinuities, as it is also the case for the others approaches tested, and, more interesting, its minimum is shifted toward a shorter distance (9.75 Å ) in slight better agreement with the CEPA-1 results than B3LYP-D3(BJ) or VV10-SCNL (9.95 and 9.85 Å , respectively). Equilibrium distances and binding energies are also reported in Table 2 for sake of clearness.

In order to give an estimate of the correlation between BSIE and BSSE, this latter was evaluated using different basis sets. Indeed, the BSSE for the DH-SVPD (5.9 kcal/mol) is, as expected, intermediate between that computed for the smaller Def2-SVPD (9.8 kcal/mol) and that for the larger Def2-TZVP basis set (1.7 kcal/mol). In other words, the optimized exponents in the DH-SVPD are the results of the balance between the underestimation of the interaction energy due to BSIE set and its overestimation coming from BSSE.

C60ISO and iso-C60 benchmarks

The second test concerns the relative stability of selected isomers of the C 60 . Here two different sets have been considered. The first is the so-called C60ISO, composed by the five energetically lowest and by the five energetically highest isomers of C 60 selected among the 1812 ones identified by Grimme and co-workers [START_REF] Sure | Comprehensive theoretical study of all 1812 C60 isomers[END_REF] . The second set, named iso-C60, contains 8 isomers chosen so that their relative energies cover a large interval (from 73.5 to 256.7 kcal/mol) [START_REF] Karton | Performance of DFT for C60 isomerization energies: a noticeable exception to acob's ladder[END_REF] . In the first case, reference energies are computed at CCSD(T)/CBS level, while in the second set reference values are given using a composite method, namely the G4(MP2) approach. In view of their accuracy and the high energy differences, the reference values obtained by these two methods can be considered of a comparable precision.

Results obtained for the C60ISO dataset are reported in Table 3. As it can be easily seen from the data obtained with the Def2-QZVP basis, results obtained adding the empirical dispersion correction (here D3(BJ)) are very similar to those not including it. In particular, a slight deterioration of the MADs is observed for most of the functionals, including B3LYP (+0.22 kcal/mol), PBE0 (+0.19 kcal/mol) and PBE0-DH (+ 0.21 kcal/mol). An opposite behavior, but always with marginal variations, is found for B2-PLYP (-0.26 kcal/mol) while PBE-QIDH is practically constant (-0.01 kcal/mol). This trend, already commented in literature [START_REF] Sedlak | Accuracy of quantum chemical methods for large noncovalent complexes[END_REF] , indicates that dispersion interactions play a negligible role in such systems.

It is not therefore surprising that the results obtained with the DH-SVPD basis are also very close. Indeed, these results globally show the same trend observed when the empirical correction is added to the functionals using the larger Def2-QZVP basis. Some exceptions can however be noticed. For instance, the mPW2-PLYP functional is not affected by the D3(BJ) correction when the Def2-QZVP basis set is used (variations of -0.01 kcal/mol upon adding it), while the small basis set significantly increases the MAD with respect to the value obtained with the larger basis set (5.10 vs. 4.66 kcal/mol, i.e. + 0.44 kcal/mol).

More in general it should be remarked that, despite the apparent high deviations obtained for some functionals, all the functionals provide quite accurate results. Indeed, considering the percent error with respect to the sum of all the total energies, the deviations range between 0.2 % (B3LYP/Def2-QZVP) and 0.8 % (B2-PLYP/DH-SVPD). Therefore, the overall error is clearly less dramatic than that appears at a first sight. The same behavior is observed for the other set, iso-C60. Here the MADs computed using the Def2-TZVPP basis set are significantly larger, as it could expected in view of how the isomers have been selected. Indeed, they range between 3.3 (mPW2-PLYP) and 13.9 (PBE0) kcal/mol. A small decrease of the error is observed when the DH-SVPD basis set is considered, with the PBE-QIDH and mPW2-PLYP functionals providing the lowest MADs, of 3.8 kcal/mol and 1.5 kcal/mol, values which correspond to a deviation of less than 0.4% with respect to the sum of all the energies. Such a small deviation is close to those found for the C60ISO set and can be, in our opinion, considered as acceptable for any chemical applications involving this kind of challenging systems.

Despite the fact that the relative stabilities of the C60 isomers for both sets seem not to be ruled by dispersion interactions, these results show the interest in the developed computational protocol. Indeed, in both datasets, the results obtained with the DH-SVPD basis set are very close to (or better) than those obtained with the larger Def2-QZVP or Def2-TZVPP basis coupled to empirical corrections showing that this basis set do not deteriorate the description of covalent interactions. The DH-SVPD basis is therefore a valid and fast alterative to more time-consuming protocols.

It is also interesting to remark that there is not a systematic improvement in going from GGA to hybrids to double hybrids for the relative stabilities of the C60 isomers. Indeed, for the C60ISO set the PBE/Def2-QZVP deviation is 11.06 kcal/mol 30 , a very low value is found for PBE0 (2.16 kcal/mol) and finally the PBE-QIDH functional gives a slightly high error (3.48 kcal/mol). A similar trend can be found for the BLYP family. However, the behavior is different for the two sets. Indeed, for the iso-C60 set the MAD increases in going from GGA to GH and then decreases back when DHs are considered (3.0, 13.9 and 5.5 kcal/mol for the PBE, PBE0 and PBE-QIDH functionals, respectively, with the Def2-TZVPP basis set).

Again, a similar behavior is found with the corresponding functional casting the BLYP model (BLYP, B3LYP and B2-PLYP). At the best of our knowledge [START_REF] Brémond | Systematic improvement of density functionals through parameter-free hybridization schemes[END_REF] , these are one of the very few sets for which there is not a systematic improvement in going from GGA to GH to DH.

It has been argued for the iso-C60 set that the oscillating behavior for MAD is related to the quantity of HF (or exact) exchange introduced in the functional, more than 3%-5% deteriorating the results [START_REF] Karton | Performance of DFT for C60 isomerization energies: a noticeable exception to acob's ladder[END_REF] . This argument is, however, not fully applicable to the C60ISO set, where the introduction of a not negligible quantity of HF exchange (≥ 20%) significantly improves the results. It is however true that a larger quantity of HF exchange significantly deteriorates the performance. Indeed, the MAD computed for the PBE-QIDH functional without taking into account the PT2 contribution (the so called PBE-QIH model) is 42.5 kcal/mol for the iso-C60 set and using the DH-SVPD basis set. This well illustrate the increasing in going from PBE0 (11.9 kcal/mol) to PBE-QIH (42.5 kcal/mol), which parallels their HF contribution, 25 and ~70 %, respectively. This trend is also found for the C60ISO, where the values obtained with the DH-SVPD basis set are 2.2 and 18.3 kcal/mol for PBE0 and PBE-QIH, respectively. However, in DHs the observed error results from a strong compensation between the large HF contribution and the PT2 counterpart. For instance, the MAD for the iso-C60 set is 42.5 kcal/mol for PBE-QIH and only 3.8 kcal/mol for PBE-QIDH (DH-SVPD basis set). A similar behavior can be found for the other DHs when the PT2 contribution is not considered (see Table 3 and4). As a consequence, the different order found for the computed MADs, GGA > GH < DH for the C60ISO or GGA > GH > DH for iso-C60 depends on how large is the error at GGA level for the given set, an error that could not be efficiently compensated by the PT2 contribution in the final DH. .

Conclusions

Fullerene represents a challenging and interesting molecule to test new DFT-based approaches: in the tackled dimer, the intermolecular interaction is ruled by weak noncovalent forces, which, instead, do not affect the relative stabilities of its isomers, for which an accurate description of the electronic structure is fundamental. On such difficult playground we have tested how double-hybrid functionals coupled to a small basis set tailored for weak interactions (the so-called DH-SVPD basis) may work. Beyond of a detailed comparison with reference values obtained with accurate wave-function methods, we have used as relevant feature the deviations obtained with respect to functionals incorporating empirical dispersion potentials, since these latter represent the most-common approach in the field. Indeed, the DH-SVPD basis set has been specifically developed to avoid their use with double hybrids, without loss in accuracy.

The obtained results confirm that the DH-SVPD values are always very close to those obtained with empirical corrections. Indeed, the PBE-QIDH/DH-SVPD model provides an accuracy for the interaction of the C 60 dimer comparable to those obtained with the best functionals using empirical dispersion and larger quadruple- basis. In contrast, both approaches (tailored basis set and larger basis with empirical potential) do not reach the same level of accuracy in the prediction of relative stabilities of C 60 isomers. Nevertheless, this test well underlined how the DH-SVPD basis set can efficiently replace the use of dispersion potentials coupled with larger basis set, at a fraction of the computational cost, thus significantly reducing the computational effort for double hybrids and enabling the description of larger molecular systems. References energies (DLPNO-CCSD(T)/CBS for the former and G4(MP2) for the latter) and point group symmetries are also reported (in kcal/mol).

Figure 1 .

 1 Figure 1. Structures of the fullerene isomers in the C60ISO (top) and iso-C60 (bottom) set.

Figure 2 .

 2 Figure 2. Potential energy profile of the fullerene dimer computed as function of the distance between the C 60 centers of mass. CEPA-1, B3LYP-D3(BJ), VV10-SCNL and BP86-D3(BJ) data are taken from reference 28.

Table 1 .

 1 Binding energies (E, kcal/mol) for the C 60 dimer.

	Method	Basis set	E
	DLPNO-CCSD a	CBS	-12.15
	DLPNO-CCSD(T) a	CBS	-11.97
	DLPNO-CEPA-1 a	CBS	-12.94
	MP2 a	Def2-TZVP	-23.46
	SOS-MP2 a	Def2-TZVP	-14.17
	B2-PLYP a	Def2-TZVP	-3.95
	B2-PLYP-D3(BJ) a	Def2-TZVP	-11.07
	DSD-PBEP86-NL a	Def2-TZVP	-0.19
	B3LYP a	Def2-TZVP	6.14
	B3LYP-NL a	Def2-TZVP	-7.81
	B3LYP-D3(BJ) a	Def2-TZVP	-7.09
	B97X-D3 a	Def2-TZVP	-6.03
	M06-2X a	Def2-TZVP	-2.64
	M06-2X-D3(BJ) a	Def2-TZVP	-9.69
	PBE-QIDH-D3(BJ)	Def2-TZVP	-12.29
	B2-PLYP	DH-SVPD	-8.00
	DSD-PBEP86	DH-SVPD	-17.40
	PBE-QIDH	DH-SVPD	-12.27
	a) From reference 28

Table 2 .

 2 Equilibrium distances (Re, Å ) and binding energies (E, kcal/mol) of the C 60 dimer, extracted from the energy profiles reported in Figure1.

	Method	Basis set	R e	E
	DLPNO-CEPA-1 a	cc-pVDZ	9.75	-6.88
	B3LYP-D3(BJ) a	cc-pVDZ	9.85	-7.73
	VV-SCNL a	cc-pVDZ	9.85	-9.27
	BP86-D3(BJ) a	Def2-TZVP	9.75	-10.07
	PBE-QIDH	DH-SVPD	9.75	-12.27
	a) From reference 28

Table 3 .

 3 Mean Absolute Deviations (kcal/mol) for the C60ISO set by Grimme. Reference values are computed at the DLPNO-CCSD(T)/CBS level[START_REF] Sure | Comprehensive theoretical study of all 1812 C60 isomers[END_REF] . In parenthesis are reported the values obtained without inclusion of the PT2 contribution to the exchange-correlation energy.

		Def2-QZVP	Def2-QZVP+D3(BJ)	DH-SVPD
	B3LYP	2.01	2.22	2.27
	PBE0	2.16	2.35	2.21
	PBE0-DH	2.91	3.12	2.84
	B2-PLYP	7.02	6.76	7.49 (12.72)
	mPW2-PLYP	4.66	4.65	5.10 (13.49)
	DSD-PBEP86	7.86	7.56	7.47
	PBE-QIDH	3.48	3.47	4.26 (18.32)

Table 4 .

 4 Mean Absolute Deviations (kcal/mol) for the iso-C60 set by Karton et co-workers[START_REF] Karton | Performance of DFT for C60 isomerization energies: a noticeable exception to acob's ladder[END_REF] .Reference values are computed at the G4(MP2) level. In parenthesis are reported the values obtained without inclusion of the PT2 contribution to the exchange-correlation energy.

		Def2-TZVPP	Def2-TZVPP+D3(BJ)	DH-SVPD
	B3LYP	11.06	11.16	9.35
	PBE0	13.29	13.30	11.93
	PBE0-DH	13.70	13.85	12.18
	B2-PLYP	3.59	3.81	4.83 (32.68)
	mPW2-PLYP	3.39	2.38	1.50 (34.11)
	DSD-PBEP86	4.83	4.82	6.21
	PBE-QIDH	5.54	5.58	3.82 (42.50)

Acknowledgement

We thank the authors of references 28, 29 and 30 for proving the row data of their results in the Supplementary Information of the articles. This work has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (grant agreement No 648558). E.B. gratefully acknowledges the GENCI-CINES for HPC resources (Projects AP010810360 and A0040810359). ANR (Agence Nationale de la Recherche) and CGI (Commissariat à l'Investissement d'Avenir) are gratefully acknowledged for their financial support of this work through Labex SEAM (Science and Engineering for Advanced Materials and devices) ANR 11 LABX 086, ANR 11 IDEX 05 02.

Supporting information

The Supporting Information is available free of charge on the ACS Publications website: raw data of Table 3 and