Optimality guarantees for crystal structure prediction.

Crystalline materials enable essential technologies, and their properties are determined by their structures. Crystal structure prediction can thus play a central part in the design of new functional materials1,2. Researchers have developed efficient heuristics to identify structural minima on the potential energy surface3-5. Although these methods can often access all configurations in principle, there is no guarantee that the lowest energy structure has been found. Here we show that the structure of a crystalline material can be predicted with energy guarantees by an algorithm that finds all the unknown atomic positions within a unit cell by combining combinatorial and continuous optimization. We encode the combinatorial task of finding the lowest energy periodic allocation of all atoms on a lattice as a mathematical optimization problem of integer programming6,7, enabling guaranteed identification of the global optimum using well-developed algorithms. A single subsequent local minimization of the resulting atom allocations then reaches the correct structures of key inorganic materials directly, proving their energetic optimality under clear assumptions. This formulation of crystal structure prediction establishes a connection to the theory of algorithms and provides the absolute energetic status of observed or predicted materials. It provides the ground truth for heuristic or data-driven structure prediction methods and is uniquely suitable for quantum annealers8-10, opening a path to overcome the combinatorial explosion of atomic configurations.

Chemical control of structure and guest uptake by a conformationally mobile porous material.

Metal-organic frameworks (MOFs) are crystalline synthetic porous materials formed by binding organic linkers to metal nodes: they can be either rigid1,2 or flexible3. Zeolites and rigid MOFs have widespread applications in sorption, separation and catalysis that arise from their ability to control the arrangement and chemistry of guest molecules in their pores via the shape and functionality of their internal surface, defined by their chemistry and structure4,5. Their structures correspond to an energy landscape with a single, albeit highly functional, energy minimum. By contrast, proteins function by navigating between multiple metastable structures using bond rotations of the polypeptide6,7, where each structure lies in one of the minima of a conformational energy landscape and can be selected according to the chemistry of the molecules that interact with the protein. These structural changes are realized through the mechanisms of conformational selection (where a higher-energy minimum characteristic of the protein is stabilized by small-molecule binding) and induced fit (where a small molecule imposes a structure on the protein that is not a minimum in the absence of that molecule)8. Here we show that rotation about covalent bonds in a peptide linker can change a flexible MOF to afford nine distinct crystal structures, revealing a conformational energy landscape that is characterized by multiple structural minima. The uptake of small-molecule guests by the MOF can be chemically triggered by inducing peptide conformational change. This change transforms the material from a minimum on the landscape that is inactive for guest sorption to an active one. Chemical control of the conformation of a flexible organic linker offers a route to modifying the pore geometry and internal surface chemistry and thus the function of open-framework materials.

Cation Distribution and Anion Transport in the La3Ga5-xGe1+xO14+0.5x Langasite Structure.

Exploration of compositional disorder using conventional diffraction-based techniques is challenging for systems containing isoelectronic ions possessing similar coherent neutron scattering lengths. Here, we show that a multinuclear solid-state Nuclear Magnetic Resonance (NMR) approach provides compelling insight into the Ga3+/Ge4+ cation distribution and oxygen anion transport in a family of solid electrolytes with langasite structure and La3Ga5-xGe1+xO14+0.5x composition. Ultrahigh field 71Ga Magic Angle Spinning (MAS) NMR experiments acquired at 35.2 T offer striking resolution enhancement, thereby enabling clear detection of Ga sites in different coordination environments. Three-connected GaO4, four-connected GaO4 and GaO6 polyhedra are probed for the parent La3Ga5GeO14 structure, while one additional spectral feature corresponding to the key (Ga,Ge)2O8 structural unit which forms to accommodate the interstitial oxide ions is detected for the Ge4+-doped La3Ga3.5Ge2.5O14.75 phase. The complex spectral line shapes observed in the MAS NMR spectra are reproduced very accurately by the NMR parameters computed for a symmetry-adapted configurational ensemble that comprehensively models site disorder. This approach further reveals a Ga3+/Ge4+ distribution across all Ga/Ge sites that is controlled by a kinetically governed cation diffusion process. Variable temperature 17O MAS NMR experiments up to 700 °C importantly indicate that the presence of interstitial oxide ions triggers chemical exchange between all oxygen sites, thereby enabling atomic-scale understanding of the anion diffusion mechanism underpinning the transport properties of these materials.

Local Structure in Disordered Melilite Revealed by Ultrahigh Field 71Ga and 139La Solid-State Nuclear Magnetic Resonance Spectroscopy.

Multinuclear Nuclear Magnetic Resonance (NMR) spectroscopy of quadrupolar nuclei at ultrahigh magnetic field provides compelling insight into the short-range structure in a family of fast oxide ion electrolytes with La1+xSr1-xGa3O7+0.5x melilite structure. The striking resolution enhancement in the solid-state 71Ga NMR spectra measured with the world's unique series connected hybrid magnet operating at 35.2 T distinctly resolves Ga sites in four- and five-fold coordination environments. Detection of five-coordinate Ga centers in the site-disordered La1.54Sr0.46Ga3O7.27 melilite is critical given that the GaO5 unit accommodates interstitial oxide ions and provides excellent transport properties. This work highlights the importance of ultrahigh magnetic fields for the detection of otherwise broad spectral features in systems containing quadrupolar nuclei and the potential of ensemble-based computational approaches for the interpretation of NMR data acquired for site-disordered materials.

Statistically Derived Proxy Potentials Accelerate Geometry Optimization of Crystal Structures.

The crystal structures of known materials contain the information about the interatomic interactions that produced these stable compounds. Similar to the use of reported protein structures to extract effective interactions between amino acids, that has been a useful tool in protein structure prediction, we demonstrate how to use this statistical paradigm to learn the effective inter-atomic interactions in crystalline inorganic solids. By analyzing the reported crystallographic data for inorganic materials, we have constructed statistically derived proxy potentials (SPPs) that can be used to assess how realistic or unusual a computer-generated structure is compared to the reported experimental structures. The SPPs can be directly used for structure optimization to improve this similarity metric, that we refer to as the SPP score. We apply such optimization step to markedly improve the quality of the input crystal structures for DFT calculations and demonstrate that the SPPs accelerate geometry optimization for three systems relevant to battery materials. As this approach is chemistry-agnostic and can be used at scale, we produced a database of all possible pair potentials in a tabulated form ready to use.

Inferring energy-composition relationships with Bayesian optimization enhances exploration of inorganic materials.

Computational exploration of the compositional spaces of materials can provide guidance for synthetic research and thus accelerate the discovery of novel materials. Most approaches employ high-throughput sampling and focus on reducing the time for energy evaluation for individual compositions, often at the cost of accuracy. Here, we present an alternative approach focusing on effective sampling of the compositional space. The learning algorithm PhaseBO optimizes the stoichiometry of the potential target material while improving the probability of and accelerating its discovery without compromising the accuracy of energy evaluation.

Control of Polarity in Kagome-NiAs Bismuthides.

A 2×2×1 superstructure of the P63/mmc NiAs structure is reported in which kagome nets are stabilized in the octahedral transition metal layers of the compounds Ni0.7Pd0.2Bi, Ni0.6Pt0.4Bi, and Mn0.99Pd0.01Bi. The ordered vacancies that yield the true hexagonal kagome motif lead to filling of trigonal bipyramidal interstitial sites with the transition metal in this family of "kagome-NiAs" type materials. Further ordering of vacancies within these interstitial layers can be compositionally driven to simultaneously yield kagome-connected layers and a net polarization along the c axes in Ni0.9Bi and Ni0.79Pd0.08Bi, which adopt Fmm2 symmetry. The polar and non-polar materials exhibit different electronic transport behaviour, reflecting the tuneability of both structure and properties within the NiAs-type bismuthide materials family.

Enhancement of Low Temperature Superionic Conductivity by Suppression of Li Site Ordering in Li7Si2-xGexS7I.

Ge4+ substitution into the recently discovered superionic conductor Li7Si2S7I is demonstrated by synthesis of Li7Si2-xGexS7I, where x ≤ 1.2. The anion packing and tetrahedral silicon location of Li7Si2S7I are retained upon substitution. Single crystal X-ray diffraction shows that substitution of larger Ge4+ for Si4+ expands the unit cell volume and further increases Li+ site disorder, such that Li7Si0.88Ge1.12S7I has one Li+ site more (sixteen in total) than Li7Si2S7I. The ionic conductivity of Li7Si0.8Ge1.2S7I (x = 1.2) at 303 K is 1.02(3) × 10-2 S cm-1 with low activation energies for Li+ transport demonstrated over a wide temperature range by AC impedance and 7Li NMR spectroscopy. All sixteen Li+ sites remain occupied to temperatures as low as 30 K in Li7Si0.88Ge1.12S7I as a result of the structural expansion. This differs from Li7Si2S7I, where the partial Li+ site ordering observed below room temperature reduces the ionic conductivity. The suppression of Li+ site depopulation by Ge4+ substitution retains the high mobility to temperatures as low as 200 K, yielding low temperature performance comparable with state-of-the-art Li ion conducting materials.

Accessing Mg-Ion Storage in V2PS10 via Combined Cationic-Anionic Redox with Selective Bond Cleavage.

Magnesium batteries attract interest as alternative energy-storage devices because of elemental abundance and potential for high energy density. Development is limited by the absence of suitable cathodes, associated with poor diffusion kinetics resulting from strong interactions between Mg2+ and the host structure. V2PS10 is reported as a positive electrode material for rechargeable magnesium batteries. Cyclable capacity of 100 mAh g-1 is achieved with fast Mg2+ diffusion of 7.2 × ${\times }$ 10-11-4 × ${\times }$ 10-14 cm2 s-1. The fast insertion mechanism results from combined cationic redox on the V site and anionic redox on the (S2)2- site; enabled by reversible cleavage of S-S bonds, identified by X-ray photoelectron and X-ray absorption spectroscopy. Detailed structural characterisation with maximum entropy method analysis, supported by density functional theory and projected density of states analysis, reveals that the sulphur species involved in anion redox are not connected to the transition metal centres, spatially separating the two redox processes. This facilitates fast and reversible Mg insertion in which the nature of the redox process depends on the cation insertion site, creating a synergy between the occupancy of specific Mg sites and the location of the electrons transferred.

Disorder and Oxide Ion Diffusion Mechanism in La1.54Sr0.46Ga3O7.27 Melilite from Nuclear Magnetic Resonance.

Layered tetrahedral network melilite is a promising structural family of fast ion conductors that exhibits the flexibility required to accommodate interstitial oxide anions, leading to excellent ionic transport properties at moderate temperatures. Here, we present a combined experimental and computational magic angle spinning (MAS) nuclear magnetic resonance (NMR) approach which aims at elucidating the local configurational disorder and oxide ion diffusion mechanism in a key member of this structural family possessing the La1.54Sr0.46Ga3O7.27 composition. 17O and 71Ga MAS NMR spectra display complex spectral line shapes that could be accurately predicted using a computational ensemble-based approach to model site disorder across multiple cationic and anionic sites, thereby enabling the assignment of bridging/nonbridging oxygens and the identification of distinct gallium coordination environments. The 17O and 71Ga MAS NMR spectra of La1.54Sr0.46Ga3O7.27 display additional features not observed for the parent LaSrGa3O7 phase which are attributed to interstitial oxide ions incorporated upon cation doping and stabilized by the formation of five-coordinate Ga centers conferring framework flexibility. 17O high-temperature (HT) MAS NMR experiments capture exchange within the bridging oxygens at 130 °C and reveal coalescence of all oxygen signals in La1.54Sr0.46Ga3O7.27 at approximately 300 °C, indicative of the participation of both interstitial and framework oxide ions in the transport process. These results further supported by the coalescence of the 71Ga resonances in the 71Ga HT MAS NMR spectra of La1.54Sr0.46Ga3O7.27 unequivocally provide evidence of the conduction mechanism in this melilite phase and highlight the potential of MAS NMR spectroscopy to enhance the understanding of ionic motion in solid electrolytes.

Control of Ionic Conductivity by Lithium Distribution in Cubic Oxide Argyrodites Li6+xP1-xSixO5Cl.

Argyrodite is a key structure type for ion-transporting materials. Oxide argyrodites are largely unexplored despite sulfide argyrodites being a leading family of solid-state lithium-ion conductors, in which the control of lithium distribution over a wide range of available sites strongly influences the conductivity. We present a new cubic Li-rich (>6 Li+ per formula unit) oxide argyrodite Li7SiO5Cl that crystallizes with an ordered cubic (P213) structure at room temperature, undergoing a transition at 473 K to a Li+ site disordered F4̅3m structure, consistent with the symmetry adopted by superionic sulfide argyrodites. Four different Li+ sites are occupied in Li7SiO5Cl (T5, T5a, T3, and T4), the combination of which is previously unreported for Li-containing argyrodites. The disordered F4̅3m structure is stabilized to room temperature via substitution of Si4+ with P5+ in Li6+xP1-xSixO5Cl (0.3 < x < 0.85) solid solution. The resulting delocalization of Li+ sites leads to a maximum ionic conductivity of 1.82(1) × 10-6 S cm-1 at x = 0.75, which is 3 orders of magnitude higher than the conductivities reported previously for oxide argyrodites. The variation of ionic conductivity with composition in Li6+xP1-xSixO5Cl is directly connected to structural changes occurring within the Li+ sublattice. These materials present superior atmospheric stability over analogous sulfide argyrodites and are stable against Li metal. The ability to control the ionic conductivity through structure and composition emphasizes the advances that can be made with further research in the open field of oxide argyrodites.

Predicting spinel solid solutions using a random atom substitution method.

Exploration of chemical composition and structural configuration space is the central problem in crystal structure prediction. Even in limiting structure space to a single structure type, many different compositions and configurations are possible. In this work, we attempt to address this problem using an extension to the existing ChemDASH code in which variable compositions can be explored. We show that ChemDASH is an efficient method for exploring a fixed-composition space of spinel structures and build upon this to include variable compositions in the Mn-Fe-Zn-O spinel phase field. This work presents the first basin-hopping crystal structure prediction method that can explore variable compositions.

Stable amorphous georgeite as a precursor to a high-activity catalyst.

Copper and zinc form an important group of hydroxycarbonate minerals that include zincian malachite, aurichalcite, rosasite and the exceptionally rare and unstable--and hence little known and largely ignored--georgeite. The first three of these minerals are widely used as catalyst precursors for the industrially important methanol-synthesis and low-temperature water-gas shift (LTS) reactions, with the choice of precursor phase strongly influencing the activity of the final catalyst. The preferred phase is usually zincian malachite. This is prepared by a co-precipitation method that involves the transient formation of georgeite; with few exceptions it uses sodium carbonate as the carbonate source, but this also introduces sodium ions--a potential catalyst poison. Here we show that supercritical antisolvent (SAS) precipitation using carbon dioxide (refs 13, 14), a process that exploits the high diffusion rates and solvation power of supercritical carbon dioxide to rapidly expand and supersaturate solutions, can be used to prepare copper/zinc hydroxycarbonate precursors with low sodium content. These include stable georgeite, which we find to be a precursor to highly active methanol-synthesis and superior LTS catalysts. Our findings highlight the value of advanced synthesis methods in accessing unusual mineral phases, and show that there is room for exploring improvements to established industrial catalysts.

Machine-Learning Prediction of Metal-Organic Framework Guest Accessibility from Linker and Metal Chemistry.

The choice of metal and linker together define the structure and therefore the guest accessibility of a metal-organic framework (MOF), but the large number of possible metal-linker combinations makes the selection of components for synthesis challenging. We predict the guest accessibility of a MOF with 80.5 % certainty based solely on the identity of these two components as chosen by the experimentalist, by decomposing reported experimental three-dimensional MOF structures in the Cambridge Structural Database into metal and linker and then learning the connection between the components' chemistry and the MOF porosity. Pore dimensions of the guest-accessible space are classified into four ranges with three sequential models. Both the dataset and the predictive models are available to download and offer simple guidance in prioritization of the choice of the components for exploratory MOF synthesis for separation and catalysis based on guest accessibility considerations.

One Site, Two Cations, Three Environments: s2 and s0 Electronic Configurations Generate Pb-Free Relaxor Behavior in a Perovskite Oxide.

The piezoelectric devices widespread in society use noncentrosymmetric Pb-based oxides because of their outstanding functional properties. The highest figures of merit reported are for perovskites based on the parent Pb(Mg1/3Nb2/3)O3 (PMN), which is a relaxor: a centrosymmetric material with local symmetry breaking that enables functional properties, which resemble those of a noncentrosymmetric material. We present the Pb-free relaxor (K1/2Bi1/2)(Mg1/3Nb2/3)O3 (KBMN), where the thermal and (di)electric behavior emerges from the discrete structural roles of the s0 K+ and s2 Bi3+ cations occupying the same A site in the perovskite structure, as revealed by diffraction methods. This opens a distinctive route to Pb-free piezoelectrics based on relaxor parents, which we demonstrate in a solid solution of KBMN with the Pb-free ferroelectric (K1/2Bi1/2)TiO3, where the structure and function evolve together, revealing a morphotropic phase boundary, as seen in PMN-derived systems. The detailed multiple-length-scale understanding of the functional behavior of KBMN suggests that precise chemical manipulation of the more diverse local displacements in the Pb-free relaxor will enhance performance.

Extended Condensed Ultraphosphate Frameworks with Monovalent Ions Combine Lithium Mobility with High Computed Electrochemical Stability.

Extended anionic frameworks based on condensation of polyhedral main group non-metal anions offer a wide range of structure types. Despite the widespread chemistry and earth abundance of phosphates and silicates, there are no reports of extended ultraphosphate anions with lithium. We describe the lithium ultraphosphates Li3P5O14 and Li4P6O17 based on extended layers and chains of phosphate, respectively. Li3P5O14 presents a complex structure containing infinite ultraphosphate layers with 12-membered rings that are stacked alternately with lithium polyhedral layers. Two distinct vacant tetrahedral sites were identified at the end of two distinct finite Li6O1626- chains. Li4P6O17 features a new type of loop-branched chain defined by six PO43- tetrahedra. The ionic conductivities and electrochemical properties of Li3P5O14 were examined by impedance spectroscopy combined with DC polarization, NMR spectroscopy, and galvanostatic plating/stripping measurements. The structure of Li3P5O14 enables three-dimensional lithium migration that affords the highest ionic conductivity (8.5(5) × 10-7 S cm-1 at room temperature for bulk), comparable to that of commercialized LiPON glass thin film electrolytes, and lowest activation energy (0.43(7) eV) among all reported ternary Li-P-O phases. Both new lithium ultraphosphates are predicted to have high thermodynamic stability against oxidation, especially Li3P5O14, which is predicted to be stable to 4.8 V, significantly higher than that of LiPON and other solid electrolytes. The condensed phosphate units defining these ultraphosphate structures offer a new route to optimize the interplay of conductivity and electrochemical stability required, for example, in cathode coatings for lithium ion batteries.

Highly Absorbing Lead-Free Semiconductor Cu2AgBiI6 for Photovoltaic Applications from the Quaternary CuI-AgI-BiI3 Phase Space.

Since the emergence of lead halide perovskites for photovoltaic research, there has been mounting effort in the search for alternative compounds with improved or complementary physical, chemical, or optoelectronic properties. Here, we report the discovery of Cu2AgBiI6: a stable, inorganic, lead-free wide-band-gap semiconductor, well suited for use in lead-free tandem photovoltaics. We measure a very high absorption coefficient of 1.0 × 105 cm-1 near the absorption onset, several times that of CH3NH3PbI3. Solution-processed Cu2AgBiI6 thin films show a direct band gap of 2.06(1) eV, an exciton binding energy of 25 meV, a substantial charge-carrier mobility (1.7 cm2 V-1 s-1), a long photoluminescence lifetime (33 ns), and a relatively small Stokes shift between absorption and emission. Crucially, we solve the structure of the first quaternary compound in the phase space among CuI, AgI and BiI3. The structure includes both tetrahedral and octahedral species which are open to compositional tuning and chemical substitution to further enhance properties. Since the proposed double-perovskite Cs2AgBiI6 thin films have not been synthesized to date, Cu2AgBiI6 is a valuable example of a stable Ag+/Bi3+ octahedral motif in a close-packed iodide sublattice that is accessed via the enhanced chemical diversity of the quaternary phase space.

Ordered Oxygen Vacancies in the Lithium-Rich Oxide Li4CuSbO5.5, a Triclinic Structure Type Derived from the Cubic Rocksalt Structure.

Li-rich rocksalt oxides are promising candidates as high-energy density cathode materials for next-generation Li-ion batteries because they present extremely diverse structures and compositions. Most reported materials in this family contain as many cations as anions, a characteristic of the ideal cubic closed-packed rocksalt composition. In this work, a new rocksalt-derived structure type is stabilized by selecting divalent Cu and pentavalent Sb cations to favor the formation of oxygen vacancies during synthesis. The structure and composition of the oxygen-deficient Li4CuSbO5.5□0.5 phase is characterized by combining X-ray and neutron diffraction, ICP-OES, XAS, and magnetometry measurements. The ordering of cations and oxygen vacancies is discussed in comparison with the related Li2CuO2□1 and Li5SbO5□1 phases. The electrochemical properties of this material are presented, with only 0.55 Li+ extracted upon oxidation, corresponding to a limited utilization of cationic and/or anionic redox, whereas more than 2 Li+ ions can be reversibly inserted upon reduction to 1 V vs Li+/Li, a large capacity attributed to a conversion reaction and the reduction of Cu2+ to Cu0. Control of the formation of oxygen vacancies in Li-rich rocksalt oxides by selecting appropriate cations and synthesis conditions affords a new route for tuning the electrochemical properties of cathode materials for Li-ion batteries. Furthermore, the development of material models of the required level of detail to predict phase diagrams and electrochemical properties, including oxygen release in Li-rich rocksalt oxides, still relies on the accurate prediction of crystal structures. Experimental identification of new accessible structure types stabilized by oxygen vacancies represents a valuable step forward in the development of predictive models.

Polymorph of LiAlP2O7: Combined Computational, Synthetic, Crystallographic, and Ionic Conductivity Study.

We report a new polymorph of lithium aluminum pyrophosphate, LiAlP2O7, discovered through a computationally guided synthetic exploration of the Li-Mg-Al-P-O phase field. The new polymorph formed at 973 K, and the crystal structure, solved by single-crystal X-ray diffraction, adopts the orthorhombic space group Cmcm with a = 5.1140(9) Å, b = 8.2042(13) Å, c = 11.565(3) Å, and V = 485.22(17) Å3. It has a three-dimensional framework structure that is different from that found in other LiMIIIP2O7 materials. It transforms to the known monoclinic form (space group P21) above ∼1023 K. Density functional theory (DFT) calculations show that the new polymorph is the most stable low-temperature structure for this composition among the seven known structure types in the AIMIIIP2O7 (A = alkali metal) families. Although the bulk Li-ion conductivity is low, as determined from alternating-current impedance spectroscopy and variable-temperature static 7Li NMR spectra, a detailed analysis of the topologies of all seven structure types through bond-valence-sum mapping suggests a potential avenue for enhancing the conductivity. The new polymorph exhibits long (>4 Å) Li-Li distances, no Li vacancies, and an absence of Li pathways in the c direction, features that could contribute to the observed low Li-ion conductivity. In contrast, we found favorable Li-site topologies that could support long-range Li migration for two structure types with modest DFT total energies relative to the new polymorph. These promising structure types could possibly be accessed from innovative doping of the new polymorph.

Chemical Control of the Dimensionality of the Octahedral Network of Solar Absorbers from the CuI-AgI-BiI3 Phase Space by Synthesis of 3D CuAgBiI5.

A newly reported compound, CuAgBiI5, is synthesized as powder, crystals, and thin films. The structure consists of a 3D octahedral Ag+/Bi3+ network as in spinel, but occupancy of the tetrahedral interstitials by Cu+ differs from those in spinel. The 3D octahedral network of CuAgBiI5 allows us to identify a relationship between octahedral site occupancy (composition) and octahedral motif (structure) across the whole CuI-AgI-BiI3 phase field, giving the ability to chemically control structural dimensionality. To investigate composition-structure-property relationships, we compare the basic optoelectronic properties of CuAgBiI5 with those of Cu2AgBiI6 (which has a 2D octahedral network) and reveal a surprisingly low sensitivity to the dimensionality of the octahedral network. The absorption onset of CuAgBiI5 (2.02 eV) barely changes compared with that of Cu2AgBiI6 (2.06 eV) indicating no obvious signs of an increase in charge confinement. Such behavior contrasts with that for lead halide perovskites which show clear confinement effects upon lowering dimensionality of the octahedral network from 3D to 2D. Changes in photoluminescence spectra and lifetimes between the two compounds mostly derive from the difference in extrinsic defect densities rather than intrinsic effects. While both materials show good stability, bulk CuAgBiI5 powder samples are found to be more sensitive to degradation under solar irradiation compared to Cu2AgBiI6.