Bonding in elemental metals and simple alloys has long been thought of as involving intense delocalization, with little connection to the localized bonds of covalent systems. In this Article, we show that the bonding in body-centered cubic (bcc) structures of the group 6 transition metals can in fact be represented, via the concepts of the 18-n rule and isolobal bonding, in terms of two balanced resonance structures. We begin with a reversed approximation Molecular Orbital (raMO) analysis of elemental Mo in its bcc structure. The raMO analysis indicates that, despite the low electron count (six valence electrons per Mo atom), nine electron pairs can be associated with any given Mo atom, corresponding to a filled 18-electron configuration. Six of these electron pairs take part in isolobal bonds along the second-nearest neighbor contacts, with the remaining three (based on the t2g d orbitals) interacting almost exclusively with first-nearest neighbors. In this way, each primitive cubic network defined by the second-nearest neighbor contacts comprises an 18-n electron system with n = 6, which essentially describes the full electronic structure of the phase. Of course, either of the two interpenetrating primitive cubic frameworks of the bcc structure can act as a basis for this discussion, leading us to write two resonance structures with equal weights for bcc-Mo. The electronic structures of CsCl-type variants with the same electron count can then be interpreted in terms of changing the relative weights of these two resonance structures, as is qualitatively confirmed with raMO analysis. This combination of raMO analysis with the resonance concept offers an avenue to extend the 18-n rule into other transition metal-rich structures.

AbstractWe illustrate how the crystal structure of Fe14Pd17Al69 provides an example of an electron–hole matching approach to inducing frustration in intermetallic systems. Its structure contains a framework based on IrAl2.75, a binary compound that closely adheres to the 18−n rule. Upon substituting the Ir with a mixture of Fe and Pd, a competition arises between maintaining the overall ideal electron concentration and accommodating the different structural preferences of the two elements. A 2×2×2 supercell results, with Pd- and Fe-rich regions emerging. Just as in the original IrAl2.75 phase, the electronic structure of Fe14Pd17Al69 exhibits a pseudogap at the Fermi energy arising from an 18−n bonding scheme. The electron–hole matching approach's ability to combine structural complexity with electronic pseudogaps offers an avenue to new phonon glass–electron crystal materials.

AbstractWe illustrate how the crystal structure of Fe14Pd17Al69 provides an example of an electron–hole matching approach to inducing frustration in intermetallic systems. Its structure contains a framework based on IrAl2.75, a binary compound that closely adheres to the 18−n rule. Upon substituting the Ir with a mixture of Fe and Pd, a competition arises between maintaining the overall ideal electron concentration and accommodating the different structural preferences of the two elements. A 2×2×2 supercell results, with Pd- and Fe-rich regions emerging. Just as in the original IrAl2.75 phase, the electronic structure of Fe14Pd17Al69 exhibits a pseudogap at the Fermi energy arising from an 18−n bonding scheme. The electron–hole matching approach's ability to combine structural complexity with electronic pseudogaps offers an avenue to new phonon glass–electron crystal materials.

The vibrational modes of inorganic materials play a central role in determining their properties, as is illustrated by the importance of phonon–electron coupling in superconductivity, phonon scattering in thermoelectric materials, and soft phonon modes in structural phase transitions. However, the prediction and control of these vibrations requires an understanding of how crystal structure and the stiffness of interatomic interactions are related. For compounds whose relationships between bonding and structure remain unclear, the elucidation of such structure–property relationships is immensely challenging. In this Article, we demonstrate how the Chemical Pressure (CP) approach can be used to draw visual and intuitive schemes relating the structure and vibrational properties of a solid state compound using the output of DFT calculations. We begin by illustrating how phonon band structures can validate the DFT-CP approach. For some intermetallic crystal structures, such as the Laves phases, the details of the packing geometries make the resulting CP scheme very sensitive to assumptions about how space should be partitioned among the interatomic contacts. Using the Laves phase CaPd2 (MgCu2 type) as a model system, we demonstrate how the phonon band structure provides a reference against which the space-partitioning method can be refined. A key parameter we identify is the ionicity of the crystal structure: the assumption of some electron transfer from the Ca to the Pd leads to a close agreement between the CP distribution and the major features of its phonon band structure. In particular, atomic motions along directions of positive CP (indicative of overly short interatomic distances) contribute to high frequency modes, while those along negative CPs (corresponding to overly long distances) make up the lowest frequency modes. Finally, we apply this approach to Nb3Ge (Cr3Si type) and CaPd5 (CaCu5 type), for which low-frequency phonon modes correlate with superconductivity and a rich variety of superstructures, respectively. Through these examples, CP analysis will emerge as a means of predicting the presence of soft phonon modes in a crystal structure and a guide to how elemental substitutions will affect the frequencies of these modes.

While composition and pressure are generally considered orthogonal parameters in the synthesis and optimization of solid state materials, their distinctness is blurred by the concept of chemical pressure (CP): microscopic pressure arising from lattice constraints rather than an externally applied force. In this article, we describe the first cycle of an iterative theoretical/experimental investigation into this connection. We begin by theoretically probing the ability of physical pressure to promote structural transitions in CaCu5-type phases that are driven by CP in other systems. Our results point to the instability of the reported CaCu5-type CaPd5 phase to such a transition even at ambient pressure, suggesting that new structural chemistry should arise at only modest pressures. We thus attempted to synthesize CaPd5 as a starting material for high-pressure experiments. However, rather than obtaining the expected CaCu5-type phase, we encountered crystals of an incommensurately modulated variant CaPd5+q/2, whose composition is related to its satellite spacing, q = qbbasic* with q ≈ 0.44. Its structure was solved and refined in the (3 + 1)D superspace group Cmcm(0β0)s00, revealing CaCu5-type slabs separated by distorted Pd hexagonal nets with an incommensurate periodicity. DFT-CP analysis on a commensurate model for CaPd5+q/2 indicates that the new Pd nets serve to relieve intense negative CPs that the Ca atoms would experience in a CaCu5-type CaPd5 phase but suffer from a desire to contract relative to the rest of the structure. In this way, both the Pd layer substitution and incommensurability in CaPd5+q/2 are anticipated by the CP schemes of simpler model systems, with CP quadrupoles tracing the paths of the favorable atomic motions. This picture offers predictions for how elemental substitution and physical pressure should affect these structural motifs, which could be applicable to the magnetic phase Zr2Co11 whose previously proposed structures show close parallels to CaPd5+q/2.

Intermetallic carbides provide excellent model systems for exploring how frustration can shape the structures and properties of inorganic materials. Combinations of several metals with carbon can be designed in which the formation of tetrahedrally close-packed (TCP) intermetallics conflicts with the C atoms’ requirement of trigonal prismatic or octahedral coordination environments, as offered by the simple close-packings (SCP) of equally sized spheres. In this Article, we explore the driving forces that lead to the coexistence of these incompatible arrangements in the Yb11Ni60C6-type compound Y11Ni60C6 (cI154), as well as potential consequences of this intergrowth for the phase’s physical properties. Our focus begins on the structure’s SCP regions, which appear as C-stuffed versions of a AuCu3-type YNi3 phase that is not observed on its own in the Y–Ni system. DFT-Chemical Pressure (DFT-CP) calculations on this hypothetical YNi3 phase reveal large negative pressures within the Ni sublattice, as it is stretched to accommodate the size requirements of the Y atoms. In the Y11Ni60C6 structure, two structural mechanisms for addressing these CP issues appear: the incorporation of interstitial C atoms, and the presence of interfaces with CaCu5-type domains. The relative roles of these two mechanisms are investigated with the CP analysis on a hypothetical YNi3Cx series of C-stuffed AuCu3-type phases, the Y–Ni sublattice of Y11Ni60C6, and finally the full Y11Ni60C6 structure. Through these calculations, the C atoms appear to play the roles of relieving positive Y CPs and supporting relaxation at the AuCu3-type/CaCu5-type interfaces, where the cancellation occurs between opposite CPs experienced by the Y atoms in the two parent structures (following the epitaxial stabilization mechanism). The CP analysis of Y11Ni60C6 also highlights a sublattice of Y and Ni atoms with large negative CPs (and thus the potential for soft vibrational modes), illustrating how frustrated structures could lead to the full realization of the phonon glass–electron crystal concept.

Diffusionless (or displacive) phase transitions allow inorganic materials to show exquisite responsiveness to external stimuli, as is illustrated vividly by the superelasticity, shape memory, and magnetocaloric effects exhibited by martensitic materials. In this Article, we present a new diffusionless transition in the compound GdCoSi2, whose origin in frustrated bonding points toward generalizable design principles for these transformations. We first describe the synthesis of GdCoSi2 and the determination of its structure using single crystal X-ray diffraction. While previous studies based on powder X-ray diffraction assigned this compound to the simple CeNi1–xSi2 structure type (space group Cmcm), our structure solution reveals a superstructure variant (space group Pbcm) in which the Co sublattice is distorted to create zigzag chains of Co atoms. DFT-calibrated Hückel calculations, coupled with a reversed approximation Molecular Orbital (raMO) analysis, trace this superstructure to the use of Co–Co isolobal bonds to complete filled 18 electron configurations on the Co atoms, in accordance with the 18–n rule. The formation of these Co–Co bonds is partially impeded, however, by a small degree of electron transfer from Si-based electronic states to those with Co–Co σ* character. The incomplete success of Co–Co bond creation suggests that these interactions are relatively weak, opening the possibility of them being overcome by thermal energy at elevated temperatures. In fact, high-temperature powder and single crystal X-ray diffraction data, as well as differential scanning calorimetry, indicate that a reversible Pbcm to Cmcm transition occurs at about 380 K. This transition is diffusionless, and the available data point toward it being first-order. We expect that similar cases of frustrated interactions could be staged in other rare earth–transition metal–main group phases, providing a potentially rich source of compounds exhibiting diffusionless transformations and the unique properties these transitions mediate.

We explore the factors stabilizing one member of the diverse structures encountered in Ln–T–E systems (Ln = lanthanide or similar early d-block element, T = transition metal, E = p-block element): the HoCoGa5 type, an arrangement of atoms associated with unconventional superconductivity. We first probe the boundaries of its stability range through the growth and characterization of ScTGa5 crystals (T = Fe, Co, Ni). After confirming that these compounds adopt the HoCoGa5 type, we analyze their electronic structure using density functional theory (DFT) and DFT-calibrated Hückel calculations. The observed valence electron count range of the HoCoGa5 type is explained in terms of the 18-n rule, with n = 6 for the Ln atoms and n = 2 for the T sites. The role of atomic sizes is investigated with DFT-chemical pressure (DFT-CP) analysis of ScNiGa5, which reveals negative pressures within the Ga sublattice as it stretches to accommodate the Sc and T atoms. This CP scheme is consistent with HoCoGa5-type gallides only being observed for relatively small Ln and T atoms. These conclusions account for the relative positions of the HoCoGa5, BaMg4Si3, and Ce2NiGa10 types in a structure map, demonstrating how combining the 18-n and CP schemes can guide our understanding of Ln–T–E systems.

Despite being one of the most common minerals in the earth's crust the crystal structure of intermediate e-plagioclase remains only partially understood, due in a large part to its complex diffraction patterns including satellite reflections. In this article we present a detailed analysis of the structure of e-plagioclase (An44) using single-crystal X-ray diffraction measured at ambient and low temperature (T = 100 K), in which the full modulated structure is successfully refined. As in earlier studies, the diffraction pattern exhibits strong main a-reflections and weak e-satellite reflections. The average structure could be solved in terms of an albite-like basic cell with the triclinic centrosymmetric and non-centrosymmetric space groups and P1 (treated in its and C1 setting, respectively, to follow conventions in the literature), while the incommensurately modulated structure was modeled in (3 + 1)D superspace, employing both the centro- and non-centrosymmetric superspace groups (αβγ)0 and X1(αβγ)0, where X refers to a special (3 + 1)D lattice centering with centering vectors (0 0 ½ ½), (½ ½ 0 ½), and (½ ½ ½ 0). Individual positional and occupational modulations for Ca/Na were refined with deeper insights being revealed in the non-centrosymmetric structure model. Through the structural details emerging from this model, the origin of the modulation can be traced to the communication between Ca/Na site positions through their bridging aluminosilicate (Si/Al)O4 tetrahedra.

The concept of frustration between competing geometrical or bonding motifs is frequently evoked in explaining complex phenomena in the structures and properties of materials. This idea is of particular importance for metallic systems, where frustration forms the basis for the design of metallic glasses, a source of diverse magnetic phenomena, and a rationale for the existence of intermetallics with giant unit cells containing thousands of atoms. Unlike soft materials, however, where conflicts can be synthetically encoded in the molecular structure, staging frustration in the metallic state is challenging due to the ease of macroscopic segregation of incompatible components. In this Article, we illustrate one approach for inducing the intergrowth of incompatible bonding motifs with the synthesis and characterization of two new intermetallic carbides: Mn16SiC4 (mC42) and Mn17Si2C4 (mP46). Similar to the phases Mn5SiC and Mn8Si2C in the Mn–Si–C system, these compounds appear as intergrowths of Mn3C and tetrahedrally close-packed (TCP) regions reminiscent of Mn-rich Mn–Si phases. The nearly complete spatial segregation of Mn–Si (intermetallic) and Mn–C (carbide) interactions in these structures can be understood from the differing geometrical requirements of C and Si. Rather than macroscopically separating into distinct phases, though, the two bonding types are tightly interwoven, with most Mn atoms being on the interfaces. DFT chemical pressure analysis reveals a driving force stabilizing these interfaces: the major local pressures acting between the Mn atoms in the Mn–Si and Mn–C systems are of opposite signs. Joining the intermetallic and carbide domains together then provides substantial relief to these local pressures, an effect we term epitaxial stabilization.

Simple sphere packings of metallic atoms are generally assumed to exhibit highly delocalized bonding, often visualized in terms of a lattice of metal cations immersed in an electron gas. In this Article, we present a compound that demonstrates how covalently shared electron pairs can, in fact, play a key role in the stability of such structures: Mo2CuxGa6–x (x ≈ 0.9). Mo2CuxGa6–x adopts a variant of the common TiAl3 structure type, which itself is a binary coloring of the fcc lattice. Electronic structure calculations trace the formation of this compound to a magic electron count of 14 electrons/T atom (T = transition metal) for the TiAl3 type, for which the Fermi energy coincides with an electronic pseudogap. This count is one electron/T atom lower than the electron concentration for a hypothetical MoGa3 phase, making this structure less competitive relative to more complex alternatives. The favorable 14 electron count can be reached, however, through the partial substitution of Ga with Cu. Using DFT-calibrated Hückel calculations and the reversed approximation Molecular Orbital (raMO) method, we show that the favorability of the 14 electron count has a simple structural origin in terms of the 18 – n rule of T–E intermetallics (E = main group element): the T atoms of the TiAl3 type are arranged into square nets whose edges are bridged by E atoms. The presence of shared electron pairs along these T–T contacts allows for 18 electron configurations to be achieved on the T atoms despite possessing only 18 – 4 = 14 electrons/T atom. This bonding scheme provides a rationale for the observed stability range of TiAl3 type TE3 phases of ca. 13–14 electrons/T atom, and demonstrates how the concept of the covalent bond can extend even to the most metallic of structure types.

Intermetallic phases exhibit a vast structural diversity in which electron count is known to be one controlling factor. However, chemical bonding theory has yet to establish how electron counts and structure are interrelated for the majority of these compounds. Recently, a simple bonding picture for transition metal (T)–main group (E) intermetallics has begun to take shape based on isolobal analogies to molecular T complexes. This bonding picture is summarized in the 18-n rule: each T atom in a T–E intemetallic phase will need 18-n electrons to achieve a closed-shell 18-electron configuration, where n is the number of electron pairs it shares with other T atoms in multicenter interactions isolobal to T–T bonds. In this Article, we illustrate the generality of this rule with a survey over a wide range of T–E phases. First, we illustrate how three structural progressions with changing electron counts can be accounted for, both geometrically and electronically, with the 18-n rule: (1) the transition between the fluorite and complex β-FeSi2 types for TSi2 phases; (2) the sequence from the marcasite type to the arsenopyrite type and back to the marcasite type for TSb2 compounds; and (3) the evolution from the AuCu3 type to the ZrAl3 and TiAl3 types for TAl3 phases. We then turn to a broader survey of the applicability of the 18-n rule through a study of the following 34 binary structure types: PtHg4, CaF2 (fluorite), Fe3C, CoGa3, Co2Al5, Ru2B3, β-FeSi2, NiAs, Ni2Al3, Rh4Si5, CrSi2, Ir3Ga5, Mo3Al8, MnP, TiSi2, Ru2Sn3, TiAl3, MoSi2, CoSn, ZrAl3, CsCl, FeSi, AuCu3, ZrSi2, Mn2Hg5, FeS2 (oP6, marcasite), CoAs3 (skutterudite), PdSn2, CoSb2, Ir3Ge7, CuAl2, Re3Ge7, CrP2, and Mg2Ni. Through these analyses, the 18-n rule is established as a framework for interpreting the stability of 341 intermetallic phases and anticipating their properties.

In the formation of binary compounds, heteroatomic interactions are generally expected to play the leading role in providing stability. In this Article, we present a series of gallides, T4Ga5 (T = Ta, Nb, and Ta/Mo), which appear to defy this expectation. Their complex crystal structures represent a new binary structure type (to the best of our knowledge),, which can be visualized in terms of a host lattice of T@T8 body centered cubic (bcc) clusters linked through face-capping Ga2 dumbbells to form a primitive cubic framework. The cubic spaces that result are alternately filled by distorted T pentagonal dodecahedra (sharing atoms with the host lattice) and dimers of bcc fragments, leading to a √2 × √2 × 2 supercell of the host framework structure. Ga tetrahedra and icosahedral units fill the remaining void spaces. Underlying these structural features is a strong tendency for homoatomic clustering of Ta and Ga, which is evident in all of the coordination polyhedra. Electronic structure calculations using density functional theory (DFT) and DFT-calibrated Hückel models reveal possible origins for this elemental segregation and the factors stabilizing the structure as a whole. A deep pseudogap is present at the Fermi energy of Ta4Ga5 (as well as at that of Nb4Ga5), corresponding to the near-optimization of Ta–Ta and Ta–Ga interactions. This pseudogap emerges as a result of the ability of extensive Ta–Ta bonding to provide local 18-electron configurations to the Ta atoms, despite the electron concentration being only 8.75 electrons per Ta atom. Support for these Ta–Ta interactions is provided by Ga bridging atoms, whose valence orbitals’ low number of angular nodes confers preferential stabilization to Ta–Ta bonding functions over antibonding ones. The observed spatial separation of the structure into Ta and Ga domains occurs as a consequence of the Ga atoms being pushed toward the periphery of the Ta clusters to play this supporting role.

Carbometalates are a diverse family of solid state structures formed from transition metal (TM)–carbon polyanionic frameworks whose charges are balanced by rare earth (RE) cations. Remarkable structural features, such as transition metal clusters, are often encountered in these phases, and a pressing challenge is to explain how such features emerge from the competing interaction types (RE–TM, TM–TM, TM–C, etc.) in these systems. In this Article, we describe a joint experimental and theoretical investigation of two compounds, Gd13Fe10C13 and its oxycarbide Gd13Fe10C13-xOx (x ≈ 1), which add a new dimension to the structural chemistry of carbometalates: π-conjugation through both TM–C and TM–TM multiple bonds. The crystal structures of both compounds are built from layers of Fe-centered Gd prisms stacked along c and surrounded by an Fe–C network, and differ chiefly in the stacking sequence of these layers. The phases’ identical local structures have two types of Fe environment: trigonal planar FeC3 sites and H-shaped Fe2C4 sites, with unusually short Fe–Fe and Fe–C bonds. 57Fe Mössbauer spectroscopy and DFT-calibrated Hückel calculations on Gd13Fe10C13 build a picture of covalent Fe–C σ bonds and conjugated π systems for which Lewis structures can be drawn. Using the reversed approximation Molecular Orbital approach, we can draw isolobal analogies between the Fe centers of this compound and molecular TM complexes: 18-electron configurations could be achieved through σ and π bonds with 18 electrons/Fe for the FeC3 site and 18-n (n = 2 for an Fe═Fe double bond) electrons/Fe for the Fe2C4 site. In this way, the vision of a unified bonding scheme of carbometalates and organometallics proffered by earlier studies is realized in a visual manner, directly from the 1-electron wave functions of the Hückel model. The bonding analysis predicts that Gd13Fe10C13 is one electron/formula unit short of an ideal electron count, explaining the tendency of the system toward a small degree of oxygen substitution. Analogies between the π bonding in Gd13Fe10C13 and that of the allyl anion help rationalize the presence of trigonal planar Fe and linear C units in the structure. The isolobal analogy between Gd13Fe10C13 and an 18-electron coordination complex is expected to apply to carbometalates as a whole, and will be extended to other examples in our future work.

Single crystals of LnMnxGa3 (Ln = Ho–Tm; x < 0.15) were grown from a Ga self-flux. These compounds crystallize in a variant of the AuCu3 structure type where Mn partially occupies the Ga6 octahedral holes. Introduction of the Mn guest atoms allows for modulation of the structures and magnetic properties of their hosts: While TmGa3 orders antiferromagnetically at ∼4.2 K, TmMnxGa3 (x = 0.05, 0.10) remains paramagnetic down to 1.8 K. Ho and Er analogs order antiferromagnetically, with effective moments and Néel temperatures, respectively, decreasing and increasing as a function of Mn concentration. DFT–chemical pressure analysis elucidates the trends in the stability of LnGa3 AuCu3-type phases and their stuffed derivatives. Guest atom insertion supports expansion of the filled octahedra, allowing the relief of negative chemical pressures in the surrounding Ga–Ga contacts.Keywords: AuCu3 structure type; Flux growth; gallium; lanthanides; Y4PdGa12 structure type;

The notion of atomic size poses an important challenge to chemical theory: empirical evidence has long established that atoms have spatial requirements, which are summarized in tables of covalent, ionic, metallic, and van der Waals radii. Considerations based on these radii play a central role in the design and interpretation of experiments, but few methods are available to directly support arguments based on atomic size using electronic structure methods. Recently, we described an approach to elucidating atomic size effects using theoretical calculations: the DFT-Chemical Pressure analysis, which visualizes the local pressures arising in crystal structures from the interactions of atomic size and electronic effects. Using this approach, a variety of structural phenomena in intermetallic phases have already been understood in terms that provide guidance to new synthetic experiments. However, the applicability of the DFT-CP method to the broad range of the structures encountered in the solid state is limited by two issues: (1) the difficulty of interpreting the intense pressure features that appear in atomic core regions and (2) the need to divide space among pairs of interacting atoms in a meaningful way. In this article, we describe general solutions to these issues. In addressing the first issue, we explore the CP analysis of a test case in which no core pressures would be expected to arise: isolated atoms in large boxes. Our calculations reveal that intense core pressures do indeed arise in these virtually pressure-less model systems and allow us to trace the issue to the shifts in the voxel positions relative to atomic centers upon expanding and contracting the unit cell. A compensatory grid unwarping procedure is introduced to remedy this artifact. The second issue revolves around the difficulty of interpreting the pressure map in terms of interatomic interactions in a way that respects the size differences of the atoms and avoids artificial geometrical constraints. In approaching this challenge, we have developed a scheme for allocating the grid pressures to contacts inspired by the Hirshfeld charge analysis. Here, each voxel is allocated to the contact between the two atoms whose free atom electron densities show the largest values at that position. In this way, the differing sizes of atoms are naturally included in the division of space without resorting to empirical radii. The use of the improved DFT-CP method is illustrated through analyses of the applicability of radius ratio arguments to Laves phase structures and the structural preferences of AB5 intermetallics between the CaCu5 and AuBe5 structure types.

Interfaces between periodic domains play a crucial role in the properties of metallic materials, as is vividly illustrated by the way in which the familiar malleability of many metals arises from the formation and migration of dislocations. In complex intermetallics, such interfaces can occur as an integral part of the ground-state crystal structure, rather than as defects, resulting in such marvels as the NaCd2 structure (whose giant cubic unit cell contains more than 1000 atoms). However, the sources of the periodic interfaces in intermetallics remain mysterious, unlike the dislocations in simple metals, which can be associated with the exertion of physical stresses. In this Article, we propose and explore the concept of structural plasticity, the hypothesis that interfaces in complex intermetallic structures similarly result from stresses, but ones that are inherent in a defect-free parent structure, rather than being externally applied. Using DFT-chemical pressure analysis, we show how the complex structures of Ca2Ag7 (Yb2Ag7 type), Ca14Cd51 (Gd14Ag51 type), and the 1/1 Tsai-type quasicrystal approximant CaCd6 (YCd6 type) can all be traced to large negative pressures around the Ca atoms of a common progenitor structure, the CaCu5 type with its simple hexagonal 6-atom unit cell. Two structural paths are found by which the compounds provide relief to the Ca atoms’ negative pressures: a Ca-rich pathway, where lower coordination numbers are achieved through defects eliminating transition metal (TM) atoms from the structure; and a TM-rich path, along which the addition of spacer Cd atoms provides the Ca coordination environments greater independence from each other as they contract. The common origins of these structures in the presence of stresses within a single parent structure highlights the diverse paths by which intermetallics can cope with competing interactions, and the role that structural plasticity may play in navigating this diversity.

Despite significant progress in the structural characterization of the quasicrystalline state, the chemical origins of long- and short-range icosahedral order remain mysterious and a subject of debate. In this Article, we present the crystal structure of a new complex intermetallic phase, Ca10Cd27Cu2 (mC234.24), whose geometrical features offer clues to the driving forces underlying the icosahedral clusters that occur in Bergman-type quasicrystals. Ca10Cd27Cu2 adopts a C-centered monoclinic superstructure of the 1/1 Bergman approximant structure, in which [110] layers of Bergman clusters in the 1/1 structure are separated through the insertion of additional atoms (accompanied by substantial positional disorder). An examination of the coordination environments of Ca and Cu (in the ordered regions) reveals that the structure can be viewed as a combination of coordination polyhedra present in the nearest binary phases in the Ca–Cd–Cu compositional space. A notable feature is the separation of Ca–Cd and Cu–Cd interactions, with Bergman clusters emerging as Ca–Cd Friauf polyhedra (derived from the MgZn2-type CaCd2 phase) encapsulate a Cu–Cd icosahedron similar to those appearing in Cu2Cd5. DFT chemical pressure calculations on nearby binary phases point to the importance of this segregation of Ca–Cd and Cu–Cd interactions. The mismatch in atomic size between Cu and Cd leads to an inability to satisfy Ca–Cu and Ca–Cd interactions simultaneously in the Friauf polyhedra of the nearby Laves phase CaCd2. The relegation of the Cu atoms to icosahedra prevents this frustration while nucleating the formation of Bergman clusters.

The space requirements of atoms are empirically known to play key roles in determining structure and reactivity across compounds ranging from simple molecules to extended solid state phases. Despite the importance of this concept, the effects of atomic size on stability remain difficult to extract from quantum mechanical calculations. Recently, we outlined a quantitative yet visual and intuitive approach to the theoretical analysis of atomic size in periodic structures: the DFT-Chemical Pressure (DFT-CP) analysis. In this Article, we describe the methodological details of this DFT-CP procedure, with a particular emphasis on refinements of the method to make it useful for a wider variety of systems. A central improvement is a new integration scheme with broader applicability than our earlier Voronoi cell method: contact volume space-partitioning. In this approach, we make explicit our assumption that the pressure at each voxel is most strongly influenced by its two closest atoms. The unit cell is divided into regions corresponding to individual interatomic contacts, with each region containing all points that share the same two closest atoms. The voxel pressures within each contact region are then averaged, resulting in effective interatomic pressures. The method is illustrated through the verification of the role of Ca–Ca repulsion (deduced earlier from empirical considerations by Corbett and co-workers) in the long-period superstructure of the W5Si3 type exhibited by Ca36Sn23.

Incommensurate modulations are increasingly being recognized as a common phenomenon in solid-state compounds ranging from inorganic materials to molecular crystals. The origins of such modulations are often mysterious, but appear to be as diverse as the compounds in which they arise. In this Article, we describe the crystal structure and bonding of Co3Al4Si2, the δ phase of the Co–Si–Al system, whose modulated structure can be traced to a central concept of inorganic chemistry: the 18 electron rule. The structure is monoclinic, conforming to the 3 + 1D superspace group C/2m(0β0)s0. The basis of the crystal structure is a rod packing of columns of the fluorite (CaF2) type, a theme that is shared by the recently determined structure of Fe8Al17.4Si7.6. The columns are arranged into sheets, within which the fluorite structure’s primitive cubic network of Si/Al atoms continues uninterrupted from column to column. Between the sheets, layers of interstitial Si/Al atoms occur, some of which are arranged with a periodicity incommensurate with that of the fluorite-type columns. Strong modulations in the interstitial layers result. Electronic structure calculations, using a DFT-calibrated Hückel model on a commensurate approximate structure, reveal that the complex pattern of atoms within these interstitial layers serves to distribute Si/Al atoms around the Co atoms in order to reach 18 electron counts (filled octadecets). Central to this bonding scheme is the covalent sharing of electron pairs between Co atoms. The shared electron pairs occupy orbitals that are isolobal to classical Co–Co σ and π bonds, but whose stability is tied to multicenter character involving bridging Si/Al atoms. Through these features, Co3Al4Si2 expands the structural and electronic manifestations of the 18 electron rule in solid-state inorganic compounds.

Intermetallic phases remain a large class of compounds whose vast structural diversity is unaccounted for by chemical theory. A recent resurgence of interest in intermetallics, due to their potential in such applications as catalysis and thermoelectricity, has intensified the need for models connecting their compositions to their structures and stability. In this Article, we illustrate how the μ3-acidity model, an extension of the acid/base concept based on the Method of Moments, offers intuitive explanations for puzzling structural progressions occurring in intermetallics formed between transition metals. Simple CsCl-type structures are frequently observed for phases with near 1:1 ratios of transition metals. However, in two compounds, TiCu and Ti21Mn25, structures are adopted which deviate from this norm. μ3-Acidity analysis shows that the formation of CsCl-type phases in these exceptional systems would yield an imbalance in the acid/base strength pairing, resulting in overneutralization of the weaker partner and thus instability. Intriguing geometrical features emerge in response, which serve to improve the neutralization of the constituent elements. In both TiCu and Ti21Mn25, part of the structure shields weaker acids or bases from their stronger partners by enhancing homoatomic bonding in the sublattice of the weaker acid or base. In TiCu, this protection is accomplished by developing doubled layers of Ti atoms to reduce their heteroatomic contacts. In Ti21Mn25 the structural response is more extreme: Ti-poor TiMn2 domains are formed to guard Mn from the Ti atoms, while the remaining Ti segregates to regions between the TiMn2 domains. The geometrical details of this arrangement fine-tune the acid/base interactions for an even greater level of stability. The most striking of these occurs in the Ti-rich region, where a paucity of Mn neighbors leads to difficulty in achieving strong neutralization. The Ti atoms arrange themselves in helical tubes, maximizing the surface area for Ti–Mn interactions. Through these examples, we show how the μ3-acidity model provides simple explanations for some of the beautiful structural motifs observed in intermetallic crystals. The foundation of the model in the Method of Moments makes it applicable to a variety of other contexts, including glasses, defects, and nanostructured surfaces.

Even after significant advances in the structural characterization of quasicrystals—phases whose diffraction patterns combine the sharp peaks normally associated with lattice periodicity and rotational symmetries antithetical to such periodicity—this new form of long-range order remains enigmatic. Here, we present DFT–chemical pressure calculations on the Tsai-type quasicrystal approximant CaCd6, which reveal how its icosahedral clusters can be traced to simple CaCu5-type (hP6) intermetallics. The results indicate that the Tsai-type clusters emerge from an atomic-size-driven transformation from planar arrangements to spherical clusters, recalling the relationship between graphene and C60.

Atomic size effects have long played a role in our empirical understanding of inorganic crystal structures. At the level of electronic structure calculations, however, the contribution of atomic size remains difficult to analyze, both alone and relative to other influences. In this paper, we extend the concepts outlined in a recent communication to develop a theoretical method for revealing the impact of the space requirements of atoms: the density functional theory-chemical pressure (DFT-CP) analysis. The influence of atomic size is most pronounced when the optimization of bonding contacts is impeded by steric repulsion at other contacts, resulting in nonideal interatomic distances. Such contacts are associated with chemical pressures (CPs) acting upon the atoms involved. The DFT-CP analysis allows for the calculation and interpretation of the CP distributions within crystal structures using DFT results. The method is demonstrated using the stability of the Ca2Ag7 structure over the simpler CaCu5-type alternative adopted by its Sr-analogue, SrAg5. A hypothetical CaCu5-type CaAg5 phase is found to exhibit large negative pressures on each Ca atom, which are concentrated in two symmetry-related interstitial spaces on opposite sides of the Ca nucleus. In moving to the Ca2Ag7 structure, relief comes to each Ca atom as a defect plane is introduced into one of these two negative-pressure regions, breaking the symmetry equivalence of the two sides and yielding a more compact Ca coordination environment. These results illustrate how the DFT-CP analysis can visually and intuitively portray how atomic size interacts with electronics in determining structure, and bridge theoretical and experimental approaches toward understanding the structural chemistry of inorganic materials.

We report the synthesis and crystal structure of the carbide Gd13Fe10C13. This compound adopts a new structure type that is remarkable for its “H”-shaped C2FeFeC2 units, which have some of the shortest Fe–Fe contacts known. A bonding analysis using DFT-calibrated Hückel calculations hints that Fe–Fe multiple bonding underlies these short distances. Gd13Fe10C13 undergoes ferromagnetic ordering at ∼55 K.

This Article presents the synthesis, structure determination, and bonding analysis of Fe8Al17.4Si7.6. Fe8Al17.4Si7.6 crystallizes in a new monoclinic structure type based on columns of the fluorite (CaF2) structure type. As such, the compound can be seen as part of a structural series in which the fluorite structure—adopted by several transition metal disilicides (TMSi2)—is fragmented by the incorporation of Al. Electronic structure analysis using density functional theory (DFT) and DFT-calibrated Hückel calculations indicates that the fluorite-type TMSi2 phases (TM = Co, Ni) exhibit density of states (DOS) pseudogaps near their Fermi energies. An analogous pseudogap occurs for Fe8Al17.4Si7.6, revealing that its complex structure serves to preserve this stabilizing feature of the electronic structure. Pursuing the origins of these pseudogaps leads to a simple picture: the DOS minimum in the TMSi2 structures arises via a bonding scheme analogous to those of 18 electron transition metal complexes. Replacement of Si with Al leads to the necessity of increasing the (Si/Al):TM ratio to maintain this valence electron concentration. The excess Si/Al atoms are accommodated through the fragmentation of the fluorite type. The resulting picture highlights how the elucidating power of bonding concepts from transition metal complexes can extend into the intermetallic realm.

A central challenge in the design of new metallic materials is the elucidation of the chemical factors underlying the structures of intermetallic compounds. Analogies to molecular bonding phenomena, such as the Zintl concept, have proven very productive in approaching this goal. In this Article, we extend a foundational concept of molecular chemistry to intermetallics: the Lewis theory of acids and bases. The connection is developed through the method of moments, as applied to DFT-calibrated Hückel calculations. We begin by illustrating that the third and fourth moments (μ3 and μ4) of the electronic density of states (DOS) distribution tune the properties of a pseudogap. μ3 controls the balance of states above and below the DOS minimum, with μ4 then determining the minimum’s depth. In this way, μ3 predicts an ideal occupancy for the DOS distribution. The μ3-ideal electron count is used to forge a link between the reactivity of transition metals toward intermetallic phase formation, and that of Lewis acids and bases toward adduct formation. This is accomplished through a moments-based definition of acidity which classifies systems that are electron-poor relative to the μ3-ideal as μ3-acidic, and those that are electron-rich as μ3-basic. The reaction of μ3 acids and bases, whether in the formation of a Lewis acid/base adduct or an intermetallic phase, tends to neutralize the μ3 acidity or basicity of the reactants. This μ3-neutralization is traced to the influence of electronegativity differences at heteroatomic contacts on the projected DOS curves of the atoms involved. The role of μ3-acid/base interactions in intermetallic phases is demonstrated through the examination of 23 binary phases forming between 3d metals, the stability range of the CsCl type, and structural trends within the Ti–Ni system.

Qualitative molecular orbital theory is central to our understanding of the bonding and reactivity of molecules and materials across chemistry. Advances in computational technology and methodology, however, have made ab initio or density functional theory calculations a simpler alternative, offering reliable results on increasingly large systems in a reasonable time-scale without the need for concerns about the approximations and parameterization of semi-empirical one-electron based methods. In this perspective, we illustrate how the availability of higher-level computational results can augment, rather than supplant, the insights provided by approaches such as the simple and extended Hückel methods. We begin by describing a way to parameterize Hückel-type Hamiltonians against DFT results for intermetallic systems. The potential for chemical understanding embodied by such orbital-based models is then demonstrated with two schemes of bonding analysis that originated in them (but can be extended to DFT results): the μ3-acid/base model and the μ2-Hückel chemical pressure analysis, which translate the molecular concepts of acidity and electronic/steric competition, respectively, into the context of intermetallic chemistry.

The assignment of distinct roles to electronics and sterics has a long history in our rationalization of chemical phenomena. Exploratory synthesis in the field of intermetallic compounds challenges this dichotomy with a growing list of phases whose structural chemistry points to an interplay between atomic size effects and orbital interactions. In this paper, we begin with a simple model for how this interdependence may arise in the dense atomic packing of intermetallics: correlations between interatomic distances lead to the inability of a phase to optimize bonds without simultaneously shortening electronically under-supported contacts, a conflict we term electronic packing frustration (EPF). An anticipated consequence of this frustration is the emergence of chemical pressures (CPs) acting on the affected atoms. We develop a theoretical method based on DFT-calibrated μ2-Hückel calculations for probing these CP effects. Applying this method to the Ca2Ag7 structure, a variant of the CaCu5 type with defect planes, reveals its formation is EPF-driven. The defect planes resolve severe CPs surrounding the Ca atoms in a hypothetical CaCu5-type CaAg5 phase. CP analysis also points to a rationale for these results in terms of a CP analogue of the pressure-distance paradox and predicts that the impetus for defect plane insertion is tunable via variations in the electron count.

Qualitative molecular orbital theory is central to our understanding of the bonding and reactivity of molecules and materials across chemistry. Advances in computational technology and methodology, however, have made ab initio or density functional theory calculations a simpler alternative, offering reliable results on increasingly large systems in a reasonable time-scale without the need for concerns about the approximations and parameterization of semi-empirical one-electron based methods. In this perspective, we illustrate how the availability of higher-level computational results can augment, rather than supplant, the insights provided by approaches such as the simple and extended Hückel methods. We begin by describing a way to parameterize Hückel-type Hamiltonians against DFT results for intermetallic systems. The potential for chemical understanding embodied by such orbital-based models is then demonstrated with two schemes of bonding analysis that originated in them (but can be extended to DFT results): the μ3-acid/base model and the μ2-Hückel chemical pressure analysis, which translate the molecular concepts of acidity and electronic/steric competition, respectively, into the context of intermetallic chemistry.