Which inorganic structures are the most complex?

The wondrous world of ABX3 molecular perovskites

The substitution of atoms with molecular building blocks to form hybrid organic-inorganic networks has been an important research theme for several decades. ABX3 molecular perovskites (MolPs) are a subclass of hybrid networks, adopting the perovskite structure with cationic and anionic molecules on the A-site and X-site. MolPs such as ((CH3)2NH2)Zn(HCOO)3 or ((n-C3H7)4N)Mn(C2N3)3 show a range of fascinating structure-chemical properties, including temperature-driven phase transitions that include a change of polarity as interesting for ferroelectrics, pressure-driven order-disorder phase transitions as interesting for barocaloric solid-state refrigeration, and most recently, melting-behaviour before decomposition with subsequent glass formation after cooling. In this feature article, we take a more personal perspective, overviewing the field's current state and outlining future directions. We start by comparing the MolPs' structural chemistry with their inorganic parents, a comparison that helps us identify opportunities for material design. After discussing the MolPs' potential as barocalorics, ferroelectrics, and in the area of glasses, we outline some challenges that lie ahead. Beyond their relevance as a hybrid analogue of inorganic perovskites, we find that MolPs' chemical parameter space provides exciting opportunities for systematically developing design guidelines for functional materials.

Inorganic Metal Thiocyanates

Metal thiocyanates were some of the first pseudohalide compounds to be discovered and adopt a diverse range of structures. This review describes the structures, properties, and syntheses of the known binary and ternary metal thiocyanates. It provides a categorization of their diverse structures and connects them to the structures of atomic inorganic materials. In addition to this description of characterized binary and ternary thiocyanates, this review summarizes the state of knowledge for all other binary metal thiocyanates. It concludes by highlighting opportunities for future materials development.

Comparing transmission- and epi-BCARS: a round robin on solid-state materials

Broadband coherent anti-Stokes Raman scattering (BCARS) is a powerful spectroscopy method combining high signal intensity with spectral sensitivity, enabling rapid imaging of heterogeneous samples in biomedical research and, more recently, in crystalline materials. However, BCARS encounters spectral distortion due to a setup-dependent non-resonant background (NRB). This study assesses BCARS reproducibility through a round robin experiment using two distinct BCARS setups and crystalline materials with varying structural complexity, including diamond, 6H-SiC, KDP, and KTP. The analysis compares setup-specific NRB correction procedures, detected and NRB-removed spectra, and mode assignment. We determine the influence of BCARS setup parameters like pump wavelength, pulse width, and detection geometry and provide a practical guide for optimizing BCARS setups for solid-state applications.

Organically Templated Uranyl Sulfates and Selenates: Structural Complexity and Crystal Chemical Restrictions for Isotypic Compounds Formation

This paper reviews the state of the art in the structural chemistry of organically templated uranyl sulfates and selenates, which are considered as the most representative groups of U-bearing synthetic compounds. In total, there are 194 compounds known for both groups, the crystal structures of which include 84 various organic molecules. Structural studies and topological analysis clearly indicate complex crystal chemical limitations in terms of the isomorphic substitution implementation, since the existence of isotypic phases has to date been confirmed only for 24 compounds out of 194, which is slightly above 12%. The structural architecture of the entire compound depends on the combination of the organic and oxyanion parts, changes in which are sometimes realized even while maintaining the topology of the U-bearing complex. An increase in the size of the hydrocarbon part and number of charge functional groups of the organic cation leads to the formation of rare and more complex topologies. In addition, the crystal structures of two novel uranyl sulfates and one uranyl selenate, templated by isopropylammonium cations, are reported.

Entropy by Neighbor Distance as a New Measure for Characterizing Spatiotemporal Orders in Microscopic Collective Systems

Collective systems self-organize to form globally ordered spatiotemporal patterns. Finding appropriate measures to characterize the order in these patterns will contribute to our understanding of the principles of self-organization in all collective systems. Here we examine a new measure based on the entropy of the neighbor distance distributions in the characterization of collective patterns. We study three types of systems: a simulated self-propelled boid system, two active colloidal systems, and one centimeter-scale robotic swarm system. In all these systems, the new measure proves sensitive in revealing active phase transitions and in distinguishing steady states. We envision that the entropy by neighbor distance could be useful for characterizing biological swarms such as bird flocks and for designing robotic swarms.

Polytypism of Ln(SeO3)(HSeO3)·2H2O compounds: synthesis and crystal structure of the first monoclinic modification of Nd(SeO3)(HSeO3)·2H2O, DFT calculations and order/disorder description

Compounds with the general formula Ln³⁺(SeO₃)(HSeO₃)·2H₂O, where Ln = Sm³⁺, Tb³⁺, Nd³⁺ and Lu³⁺, are characterized by orthorhombic symmetry with space group P2₁2₁2₁ and unit-cell parameters in the ranges a ∼ 6.473-6.999, b ∼ 6.845-7.101, c ∼ 16.242-16.426 Å. Light-purple irregularly shaped crystals of a new monoclinic polytype of neodymium selenite Nd(SeO₃)(HSeO₃)·2H₂O have been obtained during a mild-condition hydrothermal synthesis. The monoclinic unit-cell parameters are: a = 7.0815 (2), b = 6.6996 (2), c = 16.7734 (5) Å, β = 101.256 (1)°, V = 780.48 (6) ų; space group P2₁/c. The crystal structures of Nd(SeO₃)(HSeO₃)·2H₂O polymorphs show order-disorder (OD) character and can be described using the same OD groupoid family, more precisely a family of OD structures built up from two kinds of non-polar layers (category IV). The first monoclinic maximum degree order (MDO) structure (MDO1-polytype) with space group P2₁/c can be obtained when the inversion centre is active in the L_2n-type layers, while the second MDO structure (MDO2-polytype) is orthorhombic with space group P2₁2₁2₁ and can be obtained when the [2₁--] operation is active in the L_2n-type layers. The structural complexity parameters and DFT calculations of both polytypes show that the polytype structures are extremely close energy-wise and almost equally viable from the point of total energy of the structure.

Modular Principle for Complex Disordered Tetrahedral Frameworks in Quenched High-Pressure Phases of Phosphorus Oxide Nitrides

The crystal structures of the new phosphorus oxide nitrides P₄₀ O₃₁ N₄₆ and P₇₄ O₅₉ N₈₄ , which were synthesized from amorphous phosphorus oxide nitride imide, exhibit complex frameworks built up from P(O,N)₄ tetrahedra. The latter form various chain-like building units with various degrees of branching. These modular units can be combined and arranged in different ways, which leads to closely related structures and several disordered configurations in each compound. As the material was obtained by high-pressure high-temperature synthesis, the disorder is most likely a consequence of quenching a high-pressure phase with P(O,N)₅ trigonal bipyramids. Under ambient conditions, P atoms are expected to relax by moving to the centers of the face-sharing tetrahedra that constitute the bipyramid. Diffraction patterns acquired with microfocused synchrotron radiation reveal that domains of both compounds are intergrown with H₃ P₈ O₈ N₉ , whose tetrahedral framework represents a cutout of the structures of both P₄₀ O₃₁ N₄₆ and P₇₄ O₅₉ N₈₄ . Powder diffraction patterns do not indicate any further phases.

Finding a Deep-UV Borate BaZnB4 O8 with Edge-sharing [BO4 ] Tetrahedra and Strong Optical Anisotropy

The polarization modulation of deep-UV light is an important process that incorporates functionality to selectively respond to light-mater interaction. Typically, optical anisotropy is foremost to the use efficiency of deep-UV birefringent crystals. Herein, a new congruently melting polyborate with extremely large birefringence (Δn₍₀₀₁₎ =0.14@589.3 nm) and band gap (6.89 eV) is discovered as a high performance birefringent crystal, which breaks the current deadlock of deep-UV polyborates that usually show small birefringence. The rigid tetrahedra, including [ZnO₄ ] and edge-sharing [BO₄ ] tetrahedra, make all the planar [BO₃ ] triangles in the lattice adopt preferential arrangement and thereby lead to an extraordinary large birefringence that is larger than all the deep-UV borates with experimentally measured values. Structural analyses with the additional theoretical calculations were used to study the origin of strong optical anisotropy in BaZnB₄ O₈ .

Na2Cu+[Cu2+ 3O](AsO4)2Cl and Cu3[Cu3O]2(PO4)4Cl2: two new structure types based upon chains of oxocentered tetrahedra

Single crystals of Na2Cu+[Cu2+ 3O](AsO4)2Cl (1) and Cu3[Cu3O]2(PO4)4Cl2 (2) were prepared by chemical vapor transport reactions. Both crystal structures are based upon the same [O2Cu6]8+ chains formed by corner-sharing (OCu4)6+ tetrahedra and interconnected by (TO4)3− (T = P, As) tetrahedra into porous {[OCu3](TO4)2Cl}3− frameworks. The channels within the frameworks are occupied by Na+, Cu+ and Cl− ions in the crystal structure of 1, whereas the channels in the structure of 2 contain edge-sharing CuO4Cl tetragonal pyramids. Both compounds are structurally related to the previously described synthetic Na2Cu+[Cu2+ 3O](PO4)2Cl and NaCu2+[Cu2+ 3O](PO4)2Cl. The compound 2 is structurally and chemically related to yaroshevskite, Cu3[Cu3O]2(VO4)4Cl2, a mineral discovered in volcanic fumaroles, but the two structure types are drastically different. The crystal chemical analysis of the title and related compounds allows to recognize a family of at least four compounds based upon {[OCu3](TO4)2Cl}3− frameworks with channels occupied by different chemical constituents.

One of Nature's Puzzles Is Assembled: Analog of the Earth's Most Complex Mineral, Ewingite, Synthesized in a Laboratory

Through the combination of low-temperature hydrothermal synthesis and room-temperature evaporation, a synthetic phase similar in composition and crystal structure to the Earth's most complex mineral, ewingite, was obtained. The crystal structures of both natural and synthetic compounds are based on supertetrahedral uranyl-carbonate nanoclusters that are arranged according to the cubic body-centered lattice principle. The structure and composition of the uranyl carbonate nanocluster were refined using the data on synthetic material. Although the stability of natural ewingite is higher (according to visual observation and experimental studies), the synthetic phase can be regarded as a primary and/or metastable reaction product which further re-crystallizes into a more stable form under environmental conditions.

Pascoite Minerals and Potential Application of NMR Spectroscopy

The 20 minerals encompassing the pascoite family of decavanadate isopolyanion-containing [V10O28]6− minerals include a few minerals, such as rakovanite, that have been described as containing a protonated decavanadate anion. Rakovanite was originally assigned the formula Na3[H3V10O28]•15H2O and now is redefined with an ideal formula (NH4)3Na3[V10O28]•12H2O. Nuclear magnetic resonance (NMR) and particularly 51V NMR spectroscopy is an informative method used to describe the protonation state and speciation in both solid and solution states of materials in the chemical and life sciences. However, 51V NMR spectroscopy has not yet been used experimentally to distinguish the protonation state of the decavanadate ion of leaching solutions and thus contributing to the discussion regarding the controversial protonation states of decavanadate ions in gunterite, rakovanite, and nashite. In contrast, the morphology and crystal structure for apatites, vanadinite, pyromorphite, and mimetite was related to 207Pb NMR chemical shifts, assisting in describing the local environments of these minerals. NMR spectroscopy could be a useful method if used in the future for decavanadate-containing minerals. Currently, partial reduction of two Pascoite minerals (caseyite and nashite) is proposed and accordingly could now effectively be investigated using a different magnetic resonance technique, EPR spectroscopy.

Composition-driven archetype dynamics in polyoxovanadates

Molecular metal oxides often adopt common structural frameworks (i.e. archetypes), many of them boasting impressive structural robustness and stability. However, the ability to adapt and to undergo transformations between different structural archetypes is a desirable material design feature offering applicability in different environments. Using systems thinking approach that integrates synthetic, analytical and computational techniques, we explore the transformations governing the chemistry of polyoxovanadates (POVs) constructed of arsenate and vanadate building units. The water-soluble salt of the low nuclearity polyanion [V6As8O26]4- can be effectively used for the synthesis of the larger spherical (i.e. kegginoidal) mixed-valent [V12As8O40]4- precipitate, while the novel [V10As12O40]8- POVs having tubular cyclic structures are another, well soluble product. Surprisingly, in contrast to the common observation that high-nuclearity polyoxometalate (POM) clusters are fragmented to form smaller moieties in solution, the low nuclearity [V6As8O26]4- anion is in situ transformed into the higher nuclearity cluster anions. The obtained products support a conceptually new model that is outlined in this article and that describes a continuous evolution between spherical and cyclic POV assemblies. This new model represents a milestone on the way to rational and designable POV self-assemblies.

Multiple dimensionalities in A2M3(SO4)4 (A = Rb, Cs; M = Co, Ni) analogues

New representatives of the A₂M₃(SO₄)₄ (A = Rb and Cs, M = Co, Ni) family were found, inspired by the discovery and characterization of itelmenite, a mineral of composition Na₂CuMg₂(SO₄)₄. Four new compounds were obtained by high-temperature solid-state reactions in air. All new compounds were structurally characterized by single-crystal and powder X-ray diffraction. Rb₂Ni₃(SO₄)₄ and Rb₂Co₃(SO₄)₄ crystallize in the monoclinic space group P2₁/c, Cs₂Ni₃(SO₄)₄ in P2₁/n whereas Rb₂Co₃(SO₄)₄ crystallizes in the orthorhombic space group P2₁2₁2₁. In order to determine the temperature of crystallization of the new phases DTA and TG were performed for the mixtures of the precursors. Several synthesis strategies were tested and discussed. The investigation of the reactivity upon heating highlights the stability of the precursors before they collapse, explaining the difficulties to get pure powder samples.

Polymorphism, polytypism and modular aspect of compounds with the general formula A 2 M 3(TO4)4 (A = Na, Rb, Cs, Ca; M = Mg, Mn, Fe3+, Cu2+; T = S6+, P5+): order–disorder, topological description and DFT calculations

The crystal structure of Na2Mn3(SO4)4 [unit-cell parameters a = 14.8307 (18), b = 9.9107 (18), c = 8.6845 (12) Å, space group Cmc21] displays order–disorder (OD) character and can be described using the OD groupoid family, more precisely a family of OD structures built up by two types of non-polar layers, with layer symmetry P(m)c21 (L 2n+1 type) and P(b)cm (L 2n type) (category IV). A new hypothetical MDO2 polytype has been proposed and the geometry optimization demonstrates its reasonability as another possible stable polytype. Compounds Na2Mn3–x Mg x (SO4)4 with the unit-cell parameters a ∼ 29.2–29.7 Å, b ∼ 9.5–9.9 Å, c ∼ 8.7 Å and space group Pbca can be described in terms of modularity as a sequence of A, S 1 and S 2 modules:…|AS 1 AS 2 AS 1 AS 2|… or (AS 1 AS 2), together with MDO1 (AS 1 AS 1) and MDO2 (AS 2 AS 2). The crystal structures of itelmenite, NaCaFe3+ 3(PO4)4, and Ca2MgFe3+ 2(PO4)4 are crystal-chemical isotypic to Na2Mn3–x Mg x (SO4)4 and should be considered as (A*S 1 A*S 2) derivatives of the (AS 1 AS 2)-type structure.

Complexity of Molecular Nets: Topological Approach and Descriptive Statistics

The molecular net complexity (HmolNet) is an extension of the combinatorial complexity (Hmol) of a crystal structure introduced by Krivovichev. It was calculated for a set of 4152 molecular crystal structures with the composition of CxHyOz characterized by the structural class P21/c, Z = 4 (1). The molecular nets were derived from the molecular Voronoi–Dirichlet Polyhedra (VDPmol). The values of the molecular coordination number (CNmol) and critical coordination number (CNcrit) are discussed in relation with the complexity of the crystal structures. A statistical distribution of the set of molecular crystals based on the values of CNmol, CNcrit, and the complexity parameters is obtained. More than a half of the considered structures has CNmol = 14 and CNmol′ = 9 with the Wyckoff set of edges e5dcba. The average multiplicity of intermolecular contacts statistically significantly decreases from 1.58 to 1.51 upon excluding all contacts except those bearing the molecular net. The normalized value of HmolNet is of the logistic distribution type and is distributed near 0.85HmolNet with a small standard deviation. The contribution of Hmol into HmolNet ranges from 35 to 95% (mean 79%, SD 6%), and the subset of bearing intermolecular contacts accounts for 41 to 100% (mean 62%, SD 11%) of the complexity of the full set of intermolecular contacts.

Order and information in the patterns of spinning magnetic micro-disks at the air-water interface

The application of the Shannon entropy to study the relationship between information and structures has yielded insights into molecular and material systems. However, the difficulty in directly observing and manipulating atoms and molecules hampers the ability of these systems to serve as model systems for further exploring the links between information and structures. Here, we use, as a model experimental system, hundreds of spinning magnetic micro-disks self-organizing at the air-water interface to generate various spatiotemporal patterns with varying degrees of order. Using the neighbor distance as the information-bearing variable, we demonstrate the links among information, structure, and interactions. We establish a direct link between information and structure without using explicit knowledge of interactions. Last, we show that the Shannon entropy by neighbor distances is a powerful observable in characterizing structural changes. Our findings are relevant for analyzing natural self-organizing systems and for designing collective robots.

Crystallographic complexity partition analysis

We present an illustrative analysis of the complexity of a crystal structure based on the application of Shannon’s entropy formula in the form of Krivovichev’s complexity measures and extended according to the contributions of distinct discrete probability distributions derived from the atomic numbers and the Wyckoff multiplicities and arities of the atoms and sites constituting the crystal structure, respectively. The results of a full crystallographic complexity partition analysis for the intermetallic phase Mo3Al2C, a compound of intermediate structural complexity, are presented, with all calculations performed in detail. In addition, a partial analysis is discussed for the crystal structures of α- and β-quartz.

The Structural Complexity of Perovskites

A recent research direction related to ABX3 perovskites is the use of molecules on the A and/or X-site, a development that has proved fruitful for photovoltaics, (improper) ferroelectrics and barocalorics....

Molybdenum Disorder in Hydrated Sedovite, Ideally U(MoO4)2·nH2O, a Microporous Nanocrystalline Mineral Characterized by Three-Dimensional Electron Diffraction, Density Functional Theory Computations, and Complexity Analysis

Sedovite, U⁴⁺(Mo⁶⁺O₄)₂·nH₂O, is reported as being one of the earliest supergene minerals formed of the secondary zone. The difficulty of isolating enough pure material limits studies to techniques that can access the nanoscale combined with theoretical analyses. The crystal structure of sedovite has been solved and refined using the dynamical approach from three-dimensional electron diffraction data collected on natural nanocrystals found among iriginite. At 100 K, sedovite is monoclinic a ≈ 6.96 Å, b ≈ 9.07 Å, c ≈ 12.27 Å, and V ≈ 775 ų with space group C2/c. The microporous structure presents a characteristic framework built from uranium polyhedra and disordered Mo pyramids creating pore hosting water molecules. To confirm the formula U⁴⁺(Mo⁶⁺O₄)₂·nH₂O, the possible presence of a hydroxyl group that would promote Mo⁵⁺ was tested with density functional theory (DFT) computations at the ambient temperature. DFT predicts that sedovite is a ferromagnetic insulator with a fundamental bandgap of E_g ∼ 1.7 eV with its chemical and physical properties dominated by U⁴⁺ rather than Mo⁶⁺. The structural complexity, I_G,tot, of sedovite was evaluated in order to get indirect information about the missing formation conditions.

Metal-metal bonds in polyoxometalate chemistry

Half a century ago, F. Albert Cotton emphasized the relevance of metal-metal bonding in the constitution of cluster materials. Based on his description, nanoscale polyoxometalates (POMs) normally would not be regarded as cluster materials. One reason is that metal-metal bonding is typically associated with inorganic systems featuring metal centres in low oxidation states, a feature that is not common for POMs. However, over the past decades, there have been increasing reports on POMs integrating different types of metal-metal bonding. This article conceptualises and reviews the area of metal-metal bonded POMs, and their preparation and physicochemical properties. Attention is given to the changes in the electronic structure of POMs, the emergence of covalent dynamics and its impact on the development of applications in catalysis, nanoswitches, donor-acceptor systems, electron storage materials and nanoelectronics (i.e., "POMtronics").

Complexity Parameters for Molecular Solids

Structural complexity measures based on Shannon information entropy are widely used for inorganic crystal structures. However, the application of these parameters for molecular crystals requires essential modification since atoms in inorganic compounds usually possess more degrees of freedom. In this work, a novel scheme for the calculation of complexity parameters (HmolNet, HmolNet,tot) for molecular crystals is proposed as a sum of the complexity of each molecule, the complexity of intermolecular contacts, and the combined complexity of both. This scheme is tested for several molecular crystal structures.

Crystal chemistry of lamprophyllite-group minerals from the Murun alkaline complex (Russia) and pegmatites of Rocky Boy and Gordon Butte (USA): single crystal X-ray diffraction and Raman spectroscopy study

Specific features of the crystal chemistry of lamprophyllite-group minerals (LGMs) are discussed using the available literature data and the results of the single-crystal X-ray diffraction and a Raman spectroscopic studies of several samples taken from the Murun alkaline complex (Russia), and Rocky Boy and Gordon Butte pegmatites (USA) presented here. The studied samples are unique in their chemical features and the distribution of cations over structural sites. In particular, the sample from the Gordon Butte pegmatite is a member of the barytolamprophyllite-emmerichite solid solution series, whereas the samples from the Murun alkaline complex and from the Rocky Boy pegmatite are intermediate members of the solid solution series formed by lamprophyllite and a hypothetical Sr analogue of emmerichite. The predominance of O^2- over OH^- and F^- at the X site is a specific feature of sample Cha-192 from the Murun alkaline complex. New data on the Raman spectra of LGMs obtained in this work show that the wavenumbers of the O-H stretching vibrations depend on the occupancies of the M2 and M3 sites coordinating with (OH)^- groups. Cations other than Na⁺ and Ti⁴⁺ (mainly, Mg and Fe³⁺) can play a significant role in the coordination of the X site occupied by (OH)^-. Data on polarized Raman spectra of an oriented sample indicate that the OH groups having different local coordinations have similar orientations with respect to the crystal. The calculated measures of similarity (Δ) for lamprophyllite and ericssonite are identical (0.157 and 0.077 for the 2M- and 2O-polytypes, respectively), which indicates that these minerals are crystal-chemically isotypic and probably should be considered within the same mineral group by analogy to the other mineralogical groups which combine isotypic minerals.

Expanding the Averievite Family, (MX)Cu5O2(T5+O4)2 (T5+ = P, V; M = K, Rb, Cs, Cu; X = Cl, Br): Synthesis and Single-Crystal X-ray Diffraction Study

Averievite-type compounds with the general formula (MX)[Cu₅O₂(TO₄)], where M = alkali metal, X = halogen and T = P, V, have been synthesized by crystallization from gases and structurally characterized for six different compositions: 1 (M = Cs; X = Cl; T = P), 2 (M = Cs; X = Cl; T = V), 3 (M = Rb; X = Cl; T = P), 4 (M = K; X = Br; T = P), 5 (M = K; X = Cl; T = P) and 6 (M = Cu; X = Cl; T = V). The crystal structures of the compounds are based upon the same structural unit, the layer consisting of a kagome lattice of Cu²⁺ ions and are composed from corner-sharing (OCu₄) anion-centered tetrahedra. Each tetrahedron shares common corners with three neighboring tetrahedra, forming hexagonal rings, linked into the two-dimensional [O₂Cu₅]⁶⁺ sheets parallel to (001). The layers are interlinked by (T⁵⁺O₄) tetrahedra (T⁵⁺ = V, P) attached to the bases of the oxocentered tetrahedra in a “face-to-face” manner. The resulting electroneutral 3D framework {[O₂Cu₅](T⁵⁺O₄)₂}⁰ possesses channels occupied by monovalent metal cations M⁺ and halide ions X⁻. The halide ions are located at the centers of the hexagonal rings of the kagome nets, whereas the metal cations are in the interlayer space. There are at least four different structure types of the averievite-type compounds: the P-3m1 archetype, the 2 × 2 × 1 superstructure with the P-3 space group, the monoclinically distorted 1 × 1 × 2 superstructure with the C2/c symmetry and the low-temperature P2₁/c superstructure with a doubled unit cell relative to the high-temperature archetype. The formation of a particular structure type is controlled by the interplay of the chemical composition and temperature. Changing the chemical composition may lead to modification of the structure type, which opens up the possibility to tune the geometrical parameters of the kagome net of Cu²⁺ ions.

crystIT: complexity and configurational entropy of crystal structures via information theory

The information content of a crystal structure as conceived by information theory has recently proved an intriguing approach to calculate the complexity of a crystal structure within a consistent concept. Given the relatively young nature of the field, theory development is still at the core of ongoing research efforts. This work provides an update to the current theory, enabling the complexity analysis of crystal structures with partial occupancies as frequently found in disordered systems. To encourage wider application and further theory development, the updated formulas are incorporated into crystIT (crystal structure and information theory), an open-source Python-based program that allows for calculating various complexity measures of crystal structures based on a standardized *.cif file.

Structural flexibility in crystalline coordination polymers: a journey along the underlying free energy landscape

In this perspective, we discuss structural flexibility in crystalline coordination polymers. We identify that the underlying free energy landscape unites scientific disciplines, and discuss key areas to advanced the field.

Embedding parallelohedra into primitive cubic networks and structural automata description

The main goal of the paper is to contribute to the agenda of developing an algorithmic model for crystallization and measuring the complexity of crystals by constructing embeddings of 3D parallelohedra into a primitive cubic network (pcu net). It is proved that any parallelohedron P as well as tiling by P, except the rhombic dodecahedron, can be embedded into the 3D pcu net. It is proved that for the rhombic dodecahedron embedding into the 3D pcu net does not exist; however, embedding into the 4D pcu net exists. The question of how many ways the embedding of a parallelohedron can be constructed is answered. For each parallelohedron, the deterministic finite automaton is developed which models the growth of the crystalline structure with the same combinatorial type as the given parallelohedron.

Producing highly complicated materials. Nature does it better

Through the years, mineralogical studies have produced a tremendous amount of data on the atomic arrangement and mineral properties. Quite often, structural analysis has led to elucidate the role played by minor components, giving interesting insights into the physico-chemical conditions of mineral crystallization and allowing the description of unpredictable structures that represented a body of knowledge critical for assessing their technological potentialities. Using such a rich database, containing many basic acquisitions, further steps became appropriate and possible, into the directions of more advanced knowledge frontiers. Some of these frontiers assume the name of modularity, complexity, aperiodicity, and matter organization at not conventional levels, and will be discussed in this review.

Polyoxometalate clusters in minerals: review and complexity analysis

Most research on polyoxometalates (POMs) has been devoted to synthetic compounds. However, recent mineralogical discoveries of POMs in mineral structures demonstrate their importance in geochemical systems. In total, 15 different types of POM nanoscale-size clusters in minerals are described herein, which occur in 42 different mineral species. The topological diversity of POM clusters in minerals is rather restricted compared to the multitude of moieties reported for synthetic compounds, but the lists of synthetic and natural POMs do not overlap completely. The metal-oxo clusters in the crystal structures of the vanarsite-group minerals ([As³⁺V⁴⁺₂V⁵⁺₁₀As⁵⁺₆O₅₁]^7-), bouazzerite and whitecapsite ([M³⁺₃Fe₇(AsO₄)₉O_8-;n(OH)_n]), putnisite ([Cr³⁺₈(OH)₁₆(CO₃)₈]^8-), and ewingite ([(UO₂)₂₄(CO₃)₃₀O₄(OH)₁₂(H₂O)₈]^32-) contain metal-oxo clusters that have no close chemical or topological analogues in synthetic chemistry. The interesting feature of the POM cluster topologies in minerals is the presence of unusual coordination of metal atoms enforced by the topological restraints imposed upon the cluster geometry (the cubic coordination of Fe³⁺ and Ti⁴⁺ ions in arsmirandite and lehmannite, respectively, and the trigonal prismatic coordination of Fe³⁺ in bouazzerite and whitecapsite). Complexity analysis indicates that ewingite and morrisonite are the first and the second most structurally complex minerals known so far. The formation of nanoscale clusters can be viewed as one of the leading mechanisms of generating structural complexity in both minerals and synthetic inorganic crystalline compounds. The discovery of POM minerals is one of the specific landmarks of descriptive mineralogy and mineralogical crystallography of our time.

On an extension of Krivovichev's complexity measures

An extension is proposed of the Shannon entropy-based structural complexity measure introduced by Krivovichev, taking into account the geometric coordinational degrees of freedom a crystal structure has. This allows a discrimination to be made between crystal structures which share the same number of atoms in their reduced cells, yet differ in the number of their free parameters with respect to their fractional atomic coordinates. The strong additivity property of the Shannon entropy is used to shed light on the complexity measure of Krivovichev and how it gains complexity contributions due to single Wyckoff positions. Using the same property allows for combining the proposed coordinational complexity measure with Krivovichev's combinatorial one to give a unique quantitative descriptor of a crystal structure's configurational complexity. An additional contribution of chemical degrees of freedom is discussed, yielding an even more refined scheme of complexity measures which can be obtained from a crystal structure's description: the six C's of complexity.

Crystal Chemistry and Structural Complexity of Uranium(IV) Sulfates: Synthesis of U3H2(SO4)7(H2O)5·3H2O and U3(UO2)0.2(SO4)6(OH)0.4·2.3H2O with Framework Structures by the Photochemical Reduction of Uranyl

Two uranium(IV) sulfate framework compounds were crystallized at room temperature from aqueous solutions containing uranyl ions by photochemical reduction in the presence of 2-propanol. U₃H₂(SO₄)₇(H₂O)₅·3H₂O (1) crystallizes in space group P6₅ with a = 9.3052(17) Å, c = 53.515(10) Å, V = 4012.9(13) ų, and Z = 6, and U₃(UO₂)_0.2(SO₄)₆(OH)_0.4·2.3H₂O (2) is tetragonal, with space group P4₂/nmc, a = 25.624(3) Å, c = 8.9435(10) Å, V = 5872.2(11) ų, and Z = 8. The structures of 1 and 2 are the most complex among uranium(IV) sulfates.

The Fedorov-Groth law revisited: complexity analysis using mineralogical data

The Fedorov-Groth law points out that, on average, chemical simplicity corresponds to higher symmetry, and chemically complex compounds usually have lower symmetry than chemically simple compounds. Using mineralogical data, it is demonstrated that the Fedorov-Groth law is valid and statistically meaningful, when chemical complexity is expressed as the amount of Shannon chemical information per atom and the degree of symmetry as the order of the point group of a mineral.

Observation of a Large Magnetic Anisotropy and a Field-Induced Magnetic State in SrCo(VO4)(OH): A Structure with a Quasi One-Dimensional Magnetic Chain

A new member of the descloizite family, a cobalt vanadate, SrCo(VO₄)(OH), has been synthesized as large single crystals using high-temperature and high-pressure hydrothermal methods. SrCo(VO₄)(OH) crystallizes in the orthorhombic crystal system in space group P2₁2₁2₁ with the following unit cell parameters: a = 6.0157(2) Å, b = 7.645(2) Å, c = 9.291(3) Å, V = 427.29(2) ų, and Z = 4. It contains one-dimensional Co-O-Co chains of edge-sharing CoO₆ octahedra along the a-axis connected to each other via VO₄ tetrahedra along the b-axis forming a three-dimensional structure. The magnetic susceptibility of SrCo(VO₄)(OH) indicates an antiferromagnetic transition at 10 K as well as unusually large spin orbit coupling. Single-crystal magnetic measurements in all three main crystallographic directions displayed a significant anisotropy in both temperature- and field-dependent data. Single-crystal neutron diffraction at 4 K was used to characterize the magnetically ordered state. The Co²⁺ magnetic spins are arranged in a staggered configuration along the chain direction, with a canting angle that follows the tipping of the CoO₆ octahedra. The net magnetization along the chain direction, resulting in ferromagnetic coupling of the a-axis spin components in each chain, is compensated by an antiferromagnetic interaction between nearest neighbor chains. A metamagnetic transition appears in the isothermal magnetization data at 2 K along the chain direction, which seems to correspond to a co-alignment of the spin directions of the nearest neighbor chain. We propose a phenomenological spin Hamiltonian that describes the canted spin configuration of the ground state and the metamagnetic transition in SrCo(VO₄)(OH).

Targeted crystal growth of uranium gallophosphates via the systematic exploration of the UF4–GaPO4–ACl (A = Cs, Rb) phase space

The flux synthesis of a uranium gallophosphate and a uranium gallate, Cs₄[UO₂Ga₂(PO₄)₄] and Cs₂UO₂Ga₂O₅, and 4 uranium phosphates, [Rb₂Rb_3.93Cl_0.93][(UO₂)₅(PO₄)₅], Rb₁₁[(UO₂)₈(PO₄)₉], Rb_7.6[(UO₂)₈O_8.6F_0.4(PO₄)₂], and Rb₆[(UO₂)₅O₂(PO₄)₄], is reported.

Extraordinary structural complexity of ilmajokite: a multilevel hierarchical framework structure of natural origin

The crystal structure of ilmajokite, a rare Na-K-Ba-Ce-titanosilicate from the Khibiny mountains, Kola peninsula, Russia, has been solved using single-crystal X-ray diffraction data. The crystal structure is based on a 3D titanosilicate framework consisting of trigonal prismatic titanosilicate (TPTS) clusters centered by Ce3+ in [9]-coordination. Four adjacent TPTS clusters are linked into four-membered rings within the (010) plane and connected via ribbons parallel to 101. The ribbons are organized into layers parallel to (010) and modulated along the a axis with a modulation wavelength of csinβ = 32.91 Å and an amplitude of ∼b/2 = 13.89 Å. The layers are linked by additional silicate tetrahedra. Na+, K+, Ba2+ and H2O groups occur in the framework cavities and have different occupancies and coordination environments. The crystal structure of ilmajokite can be separated into eight hierarchical levels: atoms, coordination polyhedra, TPTS clusters, rings, ribbons, layers, the framework and the whole structure. The information-based analysis allows estimation of the complexity of the structure as 8.468 bits per atom and 11990.129 bits per cell. According to this analysis, ilmajokite is the third-most complex mineral known to date after ewingite and morrisonite, and is the most complex mineral framework structure, comparable in complexity to paulingite-(Ca) (11 590.532 bits per cell).

Flux Crystal Growth of Lanthanide Tungsten Oxychlorides, La8.64W6O30.45Cl, Ce8.64W5.74O30Cl, and Ln8.33W6O30Cl (Ln = Pr, Nd): Structural Stability in the Presence of Extreme Cation and Anion Disorder

A series of lanthanide tungsten oxychlorides with compositions of La_8.64W₆O_30.45Cl, Ce_8.64W_5.74O₃₀Cl, Pr_8.33W₆O₃₀Cl, and Nd_8.33W₆O₃₀Cl was synthesized as single crystals via a high-temperature flux growth method. The reduction of Ce(IV) to Ce(III) was performed in the synthesis of Ce_8.64W_5.74O₃₀Cl using Zn metal as the reducing agent. All four compounds crystallize in the tetragonal space group P4₂/nmc with a highly disordered and rather unusual arrangement of Ln(III) and W(VI) cations. The three-dimensional crystal structure consists of a complex network of Ln cations occupying multiple coordination environments, including LnO₈ cubes. The level of complexity the disorder adds to the overall structure was considered using calculations for the total information content of the crystal. The temperature dependence of the magnetic susceptibility of Pr_8.33W₆O₃₀Cl and Nd_8.33W₆O₃₀Cl was measured, and both compounds exhibit paramagnetic behavior across the entire 2-300 K temperature range measured.

Aggregation Patterns in Low- and High-Charge Anions Define Opposite Solubility Trends

Molecular dynamics simulations in aqueous solution reveal the existence of two distinct patterns of aggregation in low and high charge density Lindqvist-type polyoxometalates (POMs). Our results indicate the presence of contact and solvent-shared ion pairs and specific and preferential interactions of alkalis with POMs. Highly charged POMs are capable of breaking apart the Li⁺ and Cs⁺ solvation shell, thus enhancing the formation of long-lived alkali-POM contact ion pairs, where alkalis act as an electrostatic "glue" forming large oligomers. Stronger ion pair interactions for Li⁺ than for Cs⁺ promote lower solubility for Li⁺ than for Cs⁺, evoking anomalous solubility trends. Lower charge density POMs are not capable of disrupting the Li⁺ solvation shell and only solvent-shared ion pairs are formed, whereas for Cs⁺, contact ion pairs exist. The large number of oxygen atoms in the POM surface enhances the hydrogen bonds between POM and water, thus promoting aggregation. In this case, aggregation follows normal solubility trends. Thus, aggregation depends on the strength of ion pair interactions, the capacity of POM to disrupt alkali's solvation shell, and the contact surface area between the solvent and POM.

Crystal Chemistry and Structural Complexity of Natural and Synthetic Uranyl Selenites

Comparison of the natural and synthetic phases allows an overview to be made and even an understanding of the crystal growth processes and mechanisms of the particular crystal structure formation. Thus, in this work, we review the crystal chemistry of the family of uranyl selenite compounds, paying special attention to the pathways of synthesis and topological analysis of the known crystal structures. Comparison of the isotypic natural and synthetic uranyl-bearing compounds suggests that uranyl selenite mineral formation requires heating, which most likely can be attributed to the radioactive decay. Structural complexity studies revealed that the majority of synthetic compounds have the topological symmetry of uranyl selenite building blocks equal to the structural symmetry, which means that the highest symmetry of uranyl complexes is preserved regardless of the interstitial filling of the structures. Whereas the real symmetry of U-Se complexes in the structures of minerals is lower than their topological symmetry, which means that interstitial cations and H2O molecules significantly affect the structural architecture of natural compounds. At the same time, structural complexity parameters for the whole structure are usually higher for the minerals than those for the synthetic compounds of a similar or close organization, which probably indicates the preferred existence of such natural-born architectures. In addition, the reexamination of the crystal structures of two uranyl selenite minerals guilleminite and demesmaekerite is reported. As a result of the single crystal X-ray diffraction analysis of demesmaekerite, Pb2Cu5[(UO2)2(SeO3)6(OH)6](H2O)2, the H atoms positions belonging to the interstitial H2O molecules were assigned. The refinement of the guilleminite crystal structure allowed the determination of an additional site arranged within the void of the interlayer space and occupied by an H2O molecule, which suggests the formula of guilleminite to be written as Ba[(UO2)3(SeO3)2O2](H2O)4 instead of Ba[(UO2)3(SeO3)2O2](H2O)3.