Unified cluster-based description of valence bands in AlIr, RuAl2, RuGa3 and Al–TM quasicrystalline approximants

Tutorial on Chemical Pressure Analysis: How Atomic Packing Drives Laves/Zintl Intergrowth in K3Au5Tl

The tight atomic packing generally exhibited by alloys and intermetallics can create the impression of their being composed of hard spheres arranged to maximize their density. As such, the atomic size factor has historically been central to explanations of the structural chemistry of these systems. However, the role atomic size plays structurally has traditionally been inferred from empirical considerations. The recently developed DFT-Chemical Pressure (CP) analysis has opened a path to investigating these effects with theory. In this article, we provide a step-by-step tutorial on the DFT-CP method for non-specialists, along with advances in the approach that broaden its applicability. A new version of the CP software package is introduced, which features an interactive system that guides the user in preparing the necessary electronic structure data and generating the CP scheme, with the results being readily visualized with a web browser (and easily incorporated into websites). For demonstration purposes, we investigate the origins of the crystal structure of K3Au5Tl, which represents an intergrowth of Laves and Zintl phase domains. Here, CP analysis reveals that the intergrowth is supported by complementary CP features of NaTl-type KTl and MgCu2-type KAu2 phases. In this way, K3Au5Tl exemplifies how CP effects can drive the merging for geometrical motifs derived from different families of intermetallics through a mechanism referred to as epitaxial stabilization.

At the Limits of Isolobal Bonding: π-Based Covalent Magnetism in Mn2Hg5

Magnetic ordering in inorganic materials is generally considered to be a mechanism for structures to stabilize open shells of electrons. The intermetallic phase Mn₂Hg₅ represents a remarkable exception: its crystal structure is in accordance with the 18-n bonding scheme and non-spin-polarized density functional theory (DFT) calculations show a corresponding pseudogap near its Fermi energy. Nevertheless, it exhibits strong antiferromagnetic ordering virtually all the way up to its decomposition temperature. In this Article, we examine how these two features of Mn₂Hg₅ coexist through the development of a DFT implementation of the reversed approximation Molecular Orbital (raMO) analysis. In the non-spin-polarized electronic structure, the DFT-raMO approach confirms that Mn₂Hg₅ adheres to the 18-n rule: its chains of Mn atoms are linked through isolobal triple bonds, with three electron pairs being shared at each Mn-Mn contact in one σ-type and two π-type functions. Because each Mn atom has 6 isolobal Mn-Mn bonds, it achieves a filled 18-electron count at the compound's electron concentration of 18 - 6 = 12 electrons/Mn. A pseudogap thus occurs at the Fermi energy. Upon the introduction of antiferromagnetic order, the original pseudogap widens and deepens, suggesting enhancement of a stabilizing effect already present in the nonmagnetic state. A raMO analysis reveals that antiferromagnetism enlarges the gap by allowing diradical character to enter into the Mn-Mn isolobal π bonds, reminiscent of the dissociation of a classic covalent bond. Antiferromagnetism is accompanied by residual bonding in the π system, making Mn₂Hg₅ a vivid realization of the concept of covalent magnetism.

Electron-Precise Semiconducting ReGa2Ge: Extending the IrIn3 Structure Type to Group 7 of the Periodic Table

Intermetallic compounds with semiconducting properties are rare, but they give rise to advanced materials for energy conversion and saving applications. Here, we present ReGa₂Ge, a new electron-precise narrow-gap intermetallic semiconductor. The compound crystallizes in the IrIn₃ structure type (space group P4₂/mnm, a = 6.5734(3) Å, c = 6.7450(8) Å, and Z = 4), where Re atoms occupy the Ir site, while Ga and Ge jointly populate the In sites. ^69,71Ga nuclear quadrupole resonance spectroscopy indicates nonstatistical partially ordered distribution of Ga and Ge over two available crystallographic sites; however, the Ga:Ge ratio is exactly 2:1 without noticeable homogeneity range. The stoichiometry of ReGa₂Ge ensures its precise valence electron count, which is 17 e^- per formula unit. Accordingly, a narrow energy gap opens up at the Fermi energy in the electronic structure. Electrical resistivity, Seebeck coefficient, and thermal conductivity are in agreement with the semiconducting behavior deduced from the electronic structure calculations and point to prospective thermoelectric properties at high temperatures. Bonding analysis reveals dominant covalency in Re-E (E = Ga, Ge) and Re-Re interactions.

Cation Clustering in Intermetallics: The Modular Bonding Schemes of CaCu and Ca2Cu

Electropositive metals such as the alkaline earths or lanthanides are generally assumed to act largely as spectator cations in solid state compounds. In polar intermetallic phases, atoms of such elements are indeed often placed at the peripheries of anions or polyanionic fragments. However, they also show a pronounced tendency to cluster with each other in these peripheral regions in a manner suggestive of multicenter bonding. In this Article, we theoretically investigate the bonding schemes that underlie these cationic cluster arrangements, focusing on CaCu (whose two polymorphs are based on the intergrowth of the FeB- and CrB-types) and Ca₂Cu (a Ca-intercalated derivative of CaCu). The structures of these phases are based on Cu zigzag chains embedded in matrices of Ca atoms arranged into increasingly well-developed fragments of closest-packed arrangements. Using reversed approximation Molecular Orbital (raMO) analysis, the Cu chains of both structures are revealed to be connected via nearly fully occupied Cu-Cu isolobal σ-bonds, such that the Cu atoms control 11.67 of the 13 and 15 electrons/formula unit of CaCu and Ca₂Cu, respectively. Most of the remaining electrons are drawn to multicenter bonding functions in the Ca sublattices despite the availability of additional Cu 4p orbitals, indicating that the electronegativity difference between Ca and Cu is insufficient to achieve formal Cu oxidation states far beyond -1. The metallic nature of the Ca-based bonding subsystem is reflected in the raMO analysis by a plurality of resonance structures that can be generated from the occupied crystal orbitals. Across these bonding schemes, a separation of the electronic structure into largely self-contained Ca-Ca and Ca-Cu states is a consistent theme. This modularity in the bonding can be correlated to the ease with which this and related systems rearrange FeB- and CrB-type features, which may provide clues to identifying other intermetallic families with similar degrees of structural versatility.

Principles of weakly ordered domains in intermetallics: the cooperative effects of atomic packing and electronics in Fe2Al5

Many complex intermetallic structures feature a curious juxtaposition of domains with strict 3D periodicity and regions of much weaker order or incommensurability. This article explores the basic principles leading to such arrangements through an investigation of the weakly ordered channels of Fe2Al5. It starts by experimentally confirming the earlier crystallographic model of the high-temperature form, in which nearly continuous columns of electron density corresponding to disordered Al atoms emerge. Then electronic structure calculations on ordered models are used to determine the factors leading to the formation of these columns. These calculations reveal electronic pseudogaps near 16 electrons/Fe atom, an electron concentration close to the Al-rich side of the phase's homogeneity range. Through a reversed approximation Molecular Orbital (raMO) analysis, these pseudogaps are correlated with the filling of 18-electron configurations on the Fe atoms with the support of isolobal σ Fe-Fe bonds. The resulting preference for 16 electrons/Fe requires a fractional number of Al atoms in the Fe2Al5 unit cell. Density functional theory-chemical pressure (DFT-CP) analysis is then applied to investigate how this nonstoichiometry is accommodated. The CP schemes reveal strong quadrupolar distributions on the Al atoms of the channels, suggestive of soft atomic motions along the undulating electron density observed in the Fourier map that allow the Al positions to shift easily in response to compositional changes. Such a combination of preferred electron counts tied to stoichiometry and continuous paths of CP quadrupoles could provide predictive indicators for the emergence of channels of disordered or incommensurately spaced atoms in intermetallic structures.

Pseudogap Control of Physical and Chemical Properties in CeFeSi-Type Intermetallics

We describe the synthesis of the new ternary compound CaRuSi whose chemical and physical properties help draw a clear picture of how electronic structure controls the behavior of an isostructural series of intermetallics. DFT calculations reveal that an electronic pseudogap arises near the Fermi level ( E_F), corresponding to 14 valence electrons per RuSi unit. The closed-shell-like character is further investigated by comparisons with the electronic structures of CaCoSi (15 electrons), where the E_F lies above the corresponding pseudogap, and its hydride CaCoSiH, where formation of H anions restores the 14-electron count on the metal sublattice, returning the E_F to the pseudogap. The chemical origin of the 14-electron pseudogap is interpreted with a reversed approximation Molecular Orbital analysis. Here, the pseudogap is shown to coincide with the filling of Ru 16 electron configurations isolobal to the d⁸ square planar complexes of coordination chemistry (but where 4 electron pairs are shared covalently between Ru atoms such that only 12 electrons are required), and the occupation of Si lone pairs (2 electrons). Experimentally, the pseudogap is confirmed with heat capacity measurements, which indicate that the 14-electron systems CaRuSi and CaCoSiH each exhibit  a smaller electronic density of states at the E_F than the 15-electron system CaCoSi. Importantly, the 14-electron pseudogap also significantly affects the chemical properties of the compounds, as evidenced by the difference in the stabilities of CaCoSiH and CaRuSiH observed in hydrogen desorption measurements. These results may support the design of functional materials for superconductivity, hydrogen storage, and catalysis involving hydrogenation.

Paths to Stabilizing Electronically Aberrant Compounds: A Defect-Stabilized Polymorph and Constrained Atomic Motion in PtGa2

The structures and properties of intermetallic phases are intimately connected to electron count; unfavorable electron counts can result in structural rearrangements or new electrical or magnetic behavior when no such transformation is available. The compound PtGa₂ appears to teeter on the border between these two scenarios with its two polymorphs: a cubic fluorite type form (c-PtGa₂) and a complex tetragonal superstructure (t-PtGa₂) whose Pt-Pt connectivity aligns with the 18- n electron counting rule. Here, we investigate the factors underlying this polymorphism. Electronic structure calculations show that the transition to t-PtGa₂ opens a pseudogap at the Fermi energy that can be traced to Pt-Pt isolobal bond formation, in line with the 18- n bonding scheme. Conversely, DFT-chemical pressure (CP) analysis reveals a network of positive local pressures along Pt-Ga contacts, requiring that the c-PtGa₂ to t-PtGa₂ transition follows tightly concerted atomic motions. Experimentally, a series of samples with varying Pt:Ga ratios were synthesized to examine the stability ranges of the polymorphs. Ga-poor samples yield exclusively the cubic polymorph over the full range of temperatures studied, which can be correlated to the enhanced incorporation of interstitial Pt atoms (at points of negative pressure in the CP scheme). At more Ga-rich compositions, however, t-PtGa₂ emerges as a low-temperature form. In these samples, the t-PtGa₂ to c-PtGa₂ transition is found to be reversible, but with a large hysteresis that in single crystals can exceed 100 °C. Together, the theoretical and experimental results indicate that the c-PtGa₂ phase is buttressed at its unfavorable electron count by the interstitial atoms and networks of positive CPs that restrict atomic motion, suggesting more general strategies for achieving exotic electronic structures in intermetallic materials.

Parallels in Structural Chemistry between the Molecular and Metallic Realms Revealed by Complex Intermetallic Phases

The structural diversity of intermetallic phases poses a great challenge to chemical theory and materials design. In this Account, two examples are used to illustrate how a focus on the most complex of these structures (and their relationships to simpler ones) can reveal how chemical principles underlie structure for broad families of compounds. First, we show how experimental investigations into the Fe-Al-Si system, inspired by host-guest like features in the structure of Fe₂₅Al₇₈Si₂₀, led to a theoretical approach to deriving isolobal analogies between molecular and intermetallic compounds and a more general electron counting rule. Specifically, the Fe₈Al_17.6Si_7.4 compound obtained in these syntheses was traced to a fragmentation of the fluorite-type structure (as adopted by NiSi₂), driven by the maintenance of 18-electron configurations on the transition metal centers. The desire to quickly generalize these conclusions to a broader range of phases motivated the formulation of the reversed approximation Molecular Orbital (raMO) approach. The application of raMO to a diverse series of compounds allowed us recognize the prevalence of electron pair sharing in multicenter functions isolobal to classical covalent bonds and to propose the 18 - n electron rule for transition metal-main group (T-E) intermetallic compounds. These approaches provided a framework for understanding the 14-electron rule of the Nowotny Chimney Ladder phases, a temperature-driven phase transition in GdCoSi₂, and the bcc-structure of group VI transition metals. In the second story, we recount the development of the chemical pressure approach to analyzing atomic size and packing effects in intermetallic structures. We begin with how the stability of the Yb₂Ag₇-type structure of Ca₂Ag₇ over the more common CaCu₅ type highlights the pressing need for approaches to assessing the role of atomic size in crystal structures, and inspired the development of the DFT-Chemical Pressure (CP) method. Examples of structural phenomena elucidated by this approach are then given, including the Y/Co₂ dumbbell substitution in the Th₂Zn₁₇-type phase Y₂Co₁₇, and local icosahedral order in the Tsai-type quasicrystal approximant CaCd₆. We next discuss how deriving relationships between the CP features of a structure and its phonon modes provided a way of both validating the method and visualizing how local arrangements can give rise to soft vibrational modes. The themes of structural mechanisms for CP relief and soft atomic motions merge in the discovery and elucidation of the incommensurately modulated phase CaPd₅. In the conclusion of this Account, we propose combining raMO and CP methods for focused predictions of structural phenomena.

Local cluster networks and the number of valence states in aluminium–transition metal face-centred icosahedral quasicrystals

Coordination environments of a special class of clusters which are related to the valence states in aluminium–transition metal face-centred icosahedral quasicrystals were analysed on the basis of the six-dimensional model of Katz and Gratias for the atomic sites and that of Yamamoto for the centres of the clusters. The number of the clusters per atom in the model (≈0.263) is comparable with that of Al–Cu–Ir cubic approximant crystal (≈0.258), and it indicates that the model reasonably represents the distribution of the clusters. We identified all the possible local cluster networks of short distances, which are related to the inter-cluster bonds, and evaluated the number of the valence states by assuming two possible configurations of inter-cluster bonds. The number of the valence states per atom was derived as a function of the fraction of transition-metal atoms, and the value is in good agreement with that expected from the compositions of real quasicrystals. Notably, the number of the part of the valence states which do not depend on the fraction of transition-metal atoms is numerically similar to the electron to atom ratio known as e/a, which is phenomenologically related to the stability of quasicrystals.