Progress in Visualizing Atomic Size Effects with DFT-Chemical Pressure Analysis: From Isolated Atoms to Trends in AB5 Intermetallics

Interface Nucleus Complementarity: An Iterative Process for the Discovery of Modular Intermetallics Guided by Chemical Pressure

We present an iterative process for the discovery of modular intermetallic structures based on a recently developed interface nucleus approach. The process begins with the proposal of a suitable geometric motif that may serve as an interface nucleus. We then screen crystallographic databases for structures containing this motif as potential intergrowth partners. The extent to which pairs of these structures are likely to combine into more complicated assembles is then assessed with a new chemical pressure-based metric, interface nucleus complementarity (INC). Promising combinations of structures are translated into systems for synthesis, with new compounds providing either support for the importance of the original interface nucleus or new geometrical motifs for the next round of analysis. We demonstrate this process using a fragment derived from the σ-phase structure as a starting point, leading to the synthesis of PrMg4.13Zn10.20 and a new motif to seed the next cycle.

Navigating the 18-n+m Isomers of PdSn2: Chemical Pressure Relief through Isolobal Bonds and Main Group Clustering

The 18-n electron-counting rule provides structural guidelines for electronically feasible transition metal (T)-main group (E) phases, contributing toward the goal of material design. However, the availability of numerous potential structure types at any electron count creates a challenge for the prediction of the preferred structures of specific compounds, as is illustrated by the concept of 18-n+m isomerism. In this Article, we explore the driving forces stabilizing one 18-n+m isomer over another with an analysis of the structure of PdSn2, a layered intergrowth of the fluorite and CuAl2 structure types. The DFT-reversed approximation Molecular Orbital (DFT-raMO) method reveals that PdSn2 and its hypothetical parent structures all adhere to bonding schemes approximating the electronic configurations expected from the 18-n rule, with various degrees of isolobal Pd-Pd bonding and Sn clustering. However, partial electron transfer between the Pd 5p orbitals to the Sn 5s orbitals contributes to the absence of convincing electronic pseudogaps near their Fermi energies. As such, there is no clear electronically driven preference among the structure types. This situation allows for atomic packing effects to prevail: DFT-Chemical Pressure (DFT-CP) analysis illustrates that in the fluorite-type parent structure, positive Pd-Sn CPs lead to overcompression of the Pd atoms and a stretching of the relatively open Sn sublattice. In contrast, in the CuAl2-type parent structure, Sn atoms cluster into tetrahedra, opening space for an expanded Pd environment and the formation of Pd-Pd interactions. However, the tetrahedral packing of the Sn atoms here leads to frustration between negative and positive Sn-Sn CPs. Through the development of the angular CP correlation function (CPcor+) as a tool to quantify frustration among interatomic interactions, we demonstrate how the observed PdSn2 structure balances these effects by tuning the degree of Sn-Sn clustering and expansion of the Pd environment. These observations point to generalizations for most 18-n+m isomers, where increased main group ligand clustering (+m) and isolobal bonds (+n) can accommodate compositions with different T and E atomic sizes.

Interface Nuclei in the Y-Ag-Zn System: Three Chemical Pressure-Templated Phases with Lamellar Mg2Zn11- and CaPd5+x-Type Domains

The interface nucleus approach was recently presented as a framework for understanding and predicting the emergence of modular intermetallic phases, i.e., complex structures derived from the assembly of units from simpler parent structures. Here, we present the synthesis and crystal structures of three new modular intermetallics in the Y-Ag-Zn system that support this strategy: YAg2.79Zn2.80 (I), YAg2.44Zn3.17 (II), and YAg2.71Zn2.71 (III). Each of these structures is derived from an intergrowth of slabs of the Mg2Zn11 and CaPd5+x types, with the chief differences being in the thickness and degree of disorder within the CaPd5+x-type domains. The merging of the parent structure domains is facilitated by their sharing a common geometrical unit, a double hexagonal antiprism. The use of this motif as an interface nucleus mirrors its role in another family of structures: an intergrowth series combining the CaCu5 and Laves phase structure types, as in the PuNi3-type phase YNi3. However, there is a key difference between the two series. While in the CaCu5/Laves intergrowths, the interface between the parent structures arises perpendicular to the interface nucleus's unique (hexagonal) axis, in the Mg2Zn11/CaPd5+x-type intergrowths revealed here, the interfaces run parallel to this axis. Using CP analysis of the Mg2Zn11/CaPd5+x-type parent structures, we trace this behavior to the different directions of high-CP compatibility of the interface nuclei in the Mg2Zn11/CaPd5+x and CaCu5/Laves structure type pairs. In this way, the Y(Ag/Zn)5+x phases highlight the role that interface nuclei play in directing the domain morphologies of modular intermetallic phases.

Toward Predicting the Assembly of Modular Intermetallics from Chemical Pressure Analysis: The Interface Nucleus Approach

Complex intermetallic phases are often constructed from domains derived from simpler structures arranged into hierarchical assemblies. These modular arrangements offer intriguing prospects, such as the integration of the properties of distinct compounds into a single material or for the emergence of new properties from the interactions among different domains. In this article, we develop a strategy for the design of such complex structures, which we term the interface nucleus approach. Within this framework, the assembly of complex structures is facilitated by interface nuclei: geometrical motifs shared by two parent structures that serve as a region of overlap to nucleate or seed the formation of a combined structure. Our central hypothesis is that the formation of an interface between structures at these motifs creates opportunities for the relief of atomic packing stresses, as revealed by Density Functional Theory-Chemical Pressure (DFT-CP) analysis: when corresponding interatomic contacts in two structures exhibit complementarity─negative CP with positive CP or intense CP with mild CP─the intergrowth allows for a more balanced packing arrangement. To illustrate the application of the interface nucleus concept, we analyze three modular intermetallic structures, the σ-phase (FeCr), PuNi3, and Ca6Cu6Al5 types. In each case, the assembly of the structure can be connected to complementary CP features in an interface nucleus shared by its parent structures, while the distribution of the interface nuclei in the parents serves to template the geometry of the overall framework. In this way, the interface nucleus approach points toward avenues for the design of modular intermetallics from the CP schemes of potential partner structures.

Twists and Puckers: Tuning Crystal Chemistry in the La(AuxGe1-x)2 Compositional Series

The physical properties of solid-state materials are closely tied to their crystal structure, yet our understanding of how competing structural arrangements energetically compare is limited. In this work, we explore how small differences in composition affect the structure in the La(AuxGe1-x)2 series of compounds, which comprises four unique structure types between LaGe2 and LaAu2. This family includes the previously unknown AlB2-type compound with stoichiometry La(Au0.375Ge0.625)2 as well as La(Au0.25Ge0.75)2, an intergrowth of the AlB2 and ThSi2 structure types. We then study the chemical forces driving the structure changes and use phonon band structure calculations and DFT-Chemical Pressure to evaluate atomic-size effects. These calculations show that the parent AlB2 structure type is disfavored in Au-rich compounds due to soft atomic motions along the c axis. The instability of AlB2-type LaAuGe is confirmed by the presence of imaginary modes in the phonon band structure that correspond to a "puckering" of the hexagonal AlB2-type lattice, resulting in the experimentally observed LiGaGe structure type. The impact of size effects is less clear for Au-poor compositions; instead, twisting the AlB2 structure type to form the ThSi2 type opens a pseudogap at the Fermi level in the electronic density of states. This investigation demonstrates how crystal structure in solid-state materials can be compositionally tuned based on balancing size and electronics when multiple structure types are in close thermodynamic competition.

Emergent Transitions: Discord between Electronic and Chemical Pressure Effects in the REAl3 (RE = Sc, Y, Lanthanides) Series

Atomic packing and electronic structure are key factors underlying the crystal structures adopted by solid-state compounds. In cases where these factors conflict, structural complexity often arises. Such is born in the series of REAl₃ (RE = Sc, Y, lanthanides), which adopt structures with varied stacking patterns of face-centered cubic close packed (FCC, AuCu₃ type) and hexagonal close packed (HCP, Ni₃Sn type) layers. The percentage of the hexagonal stacking in the structures is correlated with the size of the rare earth atom, but the mechanism by which changes in atomic size drive these large-scale shifts is unclear. In this Article, we reveal this mechanism through DFT-Chemical Pressure (CP) and reversed approximation Molecular Orbital (raMO) analyses. CP analysis illustrates that the Ni₃Sn structure type is preferable from the viewpoint of atomic packing as it offers relief to packing issues in the AuCu₃ type by consolidating Al octahedra into columns, which shortens Al-Al contacts while simultaneously expanding the RE atom's coordination environment. On the other hand, the AuCu₃ type offers more electronic stability with an 18-n closed-shell configuration that is not available in the Ni₃Sn type (due to electron transfer from the RE d_z² atomic orbitals into Al-based states). Based on these results, we then turn to a schematic analysis of how the energetic contributions from atomic packing and the electronic structure vary as a function of the ratio of FCC and HCP stacking configurations within the structure and the RE atomic radius. The minima on the atomic packing and electronic surfaces are non-overlapping, creating frustration. However, when their contributions are added, new minima can emerge from their combination for specific RE radii representing intergrowth structures in the REAl₃ series. Based on this picture, we propose the concept of emergent transitions, within the framework of the Frustrated and Allowed Structural Transitions principle, for tracing the connection between competing energetic factors and complexity in intermetallic structures.

Conventional High-Temperature Superconductivity in Metallic, Covalently Bonded, Binary-Guest C-B Clathrates

Inspired by the synthesis of XB₃C₃ (X = Sr, La) compounds in the bipartite sodalite clathrate structure, density functional theory (DFT) calculations are performed on members of this family containing up to two different metal atoms. A DFT-chemical pressure analysis on systems with X = Mg, Ca, Sr, Ba reveals that the size of the metal cation, which can be tuned to stabilize the B-C framework, is key for their ambient-pressure dynamic stability. High-throughput density functional theory calculations on 105 Pm3̅ symmetry XYB₆C₆ binary-guest compounds (where X, Y are electropositive metal atoms) find 22 that are dynamically stable at 1 atm, expanding the number of potentially synthesizable phases by 19 (18 metals and 1 insulator). The density of states at the Fermi level and superconducting critical temperature, T_c, can be tuned by changing the average oxidation state of the metal atoms, with T_c being highest for an average valence of +1.5. KPbB₆C₆, with an ambient-pressure Eliashberg T_c of 88 K, is predicted to possess the highest T_c among the studied Pm3̅n XB₃C₃ or Pm3̅ XYB₆C₆ phases, and calculations suggest it may be synthesized using high-pressure high-temperature techniques and then quenched to ambient conditions.

Buffering Octahedra in Mo4Zr9P: Intergrowth as a Solution to the Frustrated Packing of Tricapped Trigonal Prisms and Icosahedra

Atomic packings based on icosahedra and tricapped trigonal prisms are prone to frustration─indeed, these polyhedra represent common configurations in metallic glasses. In this Article, we illustrate how these packing issues can serve as a driving force for the formation of modular intermetallic structures. Using Density Functional Theory-Chemical Pressure (DFT-CP) analysis, we relate the Hf₉Mo₄B-type structure of Mo₄Zr₉P to interatomic pressures experienced by the atoms in two parent structures: Zr₃P, whose structure is built from tricapped trigonal prisms, and ZrMo₂, a Laves phase containing icosahedra. CP analysis of Zr₃P reveals that it has particularly frustrated packing because of the entangling of its tricapped trigonal prisms. In the ternary phase, the frustration is significantly relieved as the units become isolated from each other. Further analysis points to the stabilizing effect of a face-sharing network of octahedra in Mo₄Zr₉P that largely separates the structure into Zr-Mo and Zr-P domains and serves as a buffering region for the relaxation of interatomic distances. These conclusions are generalized to the broader members of this structure type with the examination of the CP schemes for the isostructural Mo₄Zr₉B, Al₅Co₂, and Mg₅Pd₂ phases. Finally, we screen the structural literature using the ToposPro software to identify three additional structure types that have similar intergrowth patterns: the Dy₄CoCd, La₂₃Ni₇Mg₄, and Gd₁₄Co₃In₃ types. An analysis of the interatomic distances within the octahedral networks of these structures suggests that these networks commonly facilitate the reconciliation of packing incompatibilities in intermetallic chemistry.

Metavalent bonding in chalcogenides: DFT-chemical pressure approach

Understanding the chemical bond nature has attracted considerable attention as it is crucial to analyze and comprehend the different physical and chemical properties of materials. This work is considered a complementary part of our previous work in studying the nature of different types of bonding interactions in a wide variety of molecules and materials using the DFT Chemical Pressure (CP) approach. Recently, a new type of chemical bond, the metavalent bond (MVB), has been defined. We show how the CP formalism can be used to analyze and study the establishment of MVB in two chalcogenides, GeSe and PbSe, in a similar fashion as the electron localization function (ELF) profiles. This is accomplished by analyzing the CP maps of these two chalcogenides at different pressures (up to 40 GPa for GeSe and 10 GPa for PbSe). The CP maps show distinctive features related to the MVB, providing insights into the existence of such chemical interaction in the crystal structure of the two compounds. Similar to ELF profiles, CP maps can visualize and track the strength of the MVB in GeSe and PbSe under pressure.

Connecting Gas-Phase Computational Chemistry to Condensed Phase Kinetic Modeling: The State-of-the-Art

In recent decades, quantum chemical calculations (QCC) have increased in accuracy, not only providing the ranking of chemical reactivities and energy barriers (e.g., for optimal selectivities) but also delivering more reliable equilibrium and (intrinsic/chemical) rate coefficients. This increased reliability of kinetic parameters is relevant to support the predictive character of kinetic modeling studies that are addressing actual concentration changes during chemical processes, taking into account competitive reactions and mixing heterogeneities. In the present contribution, guidelines are formulated on how to bridge the fields of computational chemistry and chemical kinetics. It is explained how condensed phase systems can be described based on conventional gas phase computational chemistry calculations. Case studies are included on polymerization kinetics, considering free and controlled radical polymerization, ionic polymerization, and polymer degradation. It is also illustrated how QCC can be directly linked to material properties.

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.

Frustrated and Allowed Structural Transitions at the Limits of the BaAl4 Type: The (3 + 2)D Modulated Structure of Dy(Cu0.18Ga0.82)3.71

While elemental substitution is the most common way of tuning properties in solid state compounds, this approach can break down in fantastic ways when the stability range of a structure type is exceeded. In this article, we apply the Frustrated and Allowed Structural Transitions (FAST) principle to understand how structural complexity, in this case incommensurate modulations, can emerge at the composition limits of one common intermetallic framework, the BaAl₄ type. While the Dy-Ga binary intermetallic system contains no phases related to the BaAl₄ archetype, adding Cu to form a ternary system creates a composition region that is rich in such phases, including some whose structures remain unknown. We begin with an analysis of electronic and atomic packing issues faced by the hypothetical BaAl₄-type phase DyGa₄ and a La₃Al₁₁-type variant (in which a fraction of Ga₂ pairs are substituted by single Ga atoms). Through an inspection of its electronic density of states (DOS) distribution and DFT-Chemical Pressure (CP) scheme, we see that the stability of BaAl₄-type DyGa₄ is limited by an excess of electrons and overly large coordination environments around the Dy atoms, with the latter factor being particularly limiting. The inclusion of Cu into the system is anticipated to soothe both issues through the lowering of the valence electron count and the release of positive CPs between atoms surrounding the Dy atoms. With this picture in mind, we then move to an experimental investigation of the Dy-Cu-Ga system, elucidating the structure of Dy(Cu_0.18Ga_0.82)_3.71(1). In this compound, the BaAl₄ type is subject to a 2D incommensurate modulation (q₁ = 0.31a* + 0.2b*, q₂ = 0.31a* - 0.2b*), which can be modeled in the (3+2)D superspace group Pmmm(αβ0)000(α-β0)000. The resulting structure solution contains blocks of the La₃Al₁₁ type, with the corners of these domains serving to shrink the Dy coordination environments. These results highlight how the addition of a well-chosen third element to a binary system with a missing-but plausible-compound (BaAl₄-type DyGa₄) can bring it to the cusp of stability with intriguing structural consequences.

Varying Electronic Configurations in Compressed Atoms: From the Role of the Spatial Extension of Atomic Orbitals to the Change of Electronic Configuration as an Isobaric Transformation

A quantum chemical model for the study of the electronic structure of compressed atoms lends itself to a perturbation-theoretic analysis. It is shown, both analytically and numerically, that the increase of the electronic energy with increasing compression depends on the electronic configuration, as a result of the variable spatial extent of the atomic orbitals involved. The different destabilization of the electronic states may lead to an isobaric change of the ground-state electronic configuration, and the same first-order model paves the way to a simple thermodynamical interpretation of this process.

Mn39Si9Nx: Epitaxial Stabilization as a Pathway to the Formation of Intermetallic Nitrides

The realization of the full potential of nitrogen-containing solid-state materials is limited by the inert and gaseous nature of N₂. In this Communication, we describe the simple synthesis yet complex structure of the new phase Mn₃₉Si₉N_x (x = 0.84). The formation of this intermetallic subnitride appears to be facilitated by the high solubility of nitrogen in manganese metal, while its structural features are guided by the complementary internal packing strains of Mn-Si and Mn-N domains, an effect known as epitaxial stabilization. These domains intergrow into a composite structure based on the interpenetration of tetrahedrally close-packed (TCP) and Mackay cluster-like modules. We anticipate that other systems combining nitrogen with the TCP packing of metals will be similarly driven toward intergrowth, opening a path to a broader family of intermetallic nitrides.

Intermetallic Reactivity: Ca3Cu7.8Al26.2 and the Role of Electronegativity in the Stabilization of Modular Structures

The structural chemistry of intermetallic phases is generally viewed in terms of what crystal structure will be most stable for a given combination of metallic atoms. Yet, individual atoms do not always make the best reference points. As the number of elements involved in compounds increases, their structures can often appear to be assembled from structural motifs derived from simpler compounds nearby in the phase diagram rather than fundamentally new arrangements of atoms. In this Article, we explore the notion that complex multinary phases can be viewed productively in terms of motif-preserving reactions between binary compounds, as opposed to direct reactions of the component elements. We present the targeted synthesis and structure solution of Ca₃Cu_7.8Al_26.2, an intermetallic phase whose placement in the phase diagram is suggestive of a reaction between CaAl₄ and CuAl₂. Single-crystal X-ray diffraction analysis reveals that this compound crystallizes in the Y₃TaNi_6+xAl₂₆ (or stuffed BaHg₁₁) structure type and is constructed from three modules: Ca@(Cu/Al)₁₆ polyhedra derived from the BaAl₄ type, Cu@Al₈ cubes, and Al₁₃ cuboctahedra. To help understand this arrangement, we identify forces driving the reactivity of one of the supposed starting materials, CaAl₄, through visualization of its atomic charge distribution and chemical pressure (CP) scheme, which suggest that the Al sites closest to the Ca atoms should show a high affinity for substitution by Cu atoms. Such a process on its own, however, would lead to overly long Ca-Cu distances and electron deficiency. When Cu is made available to CaAl₄ in the Ca-Cu-Al ternary system, its incorporation in the Ca coordination environments instead nucleates domains of a fluorite-like CuAl₂ phase, which act as nodes in the primitive cubic framework of CaAl₄- and fluorite-like units. The cubic holes created by this framework are occupied by Al₁₃ face-centered-cubic fragments that donate electrons while also resolving negative CPs in the Ca environments. This structural chemistry illustrates how new elements added to a binary compound at sites with conflicting electronic and atomic size preferences can serve as anchor points for the growth of domains of different bonding types, a notion that can be applied as a more general design strategy for new intermetallic intergrowth structures.

Entropy-Driven Incommensurability: Chemical Pressure-Guided Polymorphism in PdBi and the Origins of Lock-In Phenomena in Modulated Systems

Incommensurate order, in which two or more mismatched periodic patterns combine to make a long-range ordered yet aperiodic structure, is emerging as a general phenomenon impacting the crystal structures of compounds ranging from alloys and nominally simple salts to organic molecules and proteins. The origins of incommensurability in these systems are often unclear, but it is commonly associated with relatively weak interactions that become apparent only at low temperatures. In this article, we elucidate an incommensurate modulation in the intermetallic compound PdBi that arises from a different mechanism: the controlled increase of entropy at higher temperatures. Following the synthesis of PdBi, we structurally characterize two low-temperature polymorphs of the TlI-type structure with single crystal synchrotron X-ray diffraction. At room temperature, we find a simple commensurate superstructure of the TlI-type structure (comm-PdBi), in which the Pd sublattice distorts to form a 2D pattern of short and long Pd-Pd contacts. Upon heating, the structure converts to an incommensurate variant (incomm-PdBi) corresponding to the insertion of thin slabs of the original TlI type into the superstructure. Theoretical bonding analysis suggests that comm-PdBi is driven by the formation of isolobal Pd-Pd bonds along shortened contacts in the distorted Pd network, which is qualitatively in accord with the 18-n rule but partially frustrated by the population of competing Bi-Bi bonding states. The emergence of incomm-PdBi upon heating is rationalized with the DFT-Cemical Pressure (CP) method: the insertion of TlI-type slabs result in regions of higher vibrational freedom that are entropically favored at higher temperatures. High-temperature incommensurability may be encountered in other materials when bond formation is weakened by competing electronic states, and there is a path for accommodating defects in the CP scheme.

Frustrated and Allowed Structural Transitions: The Theory-Guided Discovery of the Modulated Structure of IrSi

To the experienced molecular chemist, predicting the geometries and reactivities of a system is an exercise in balancing simple concepts such as sterics and electronics. In this Article, we illustrate how recent theoretical developments can give this combination of concepts a similar predictive power in intermetallic chemistry through the anticipation and discovery of structural complexity in the nominally MnP-type compound IrSi. Analysis of the bonding scheme and DFT-Chemical Pressure (CP) distribution of the reported MnP-type structure exposes issues pointing toward new structural behavior. The placement of the Fermi energy below an electronic pseudogap indicates that this structure is electron-poor, an observation that can be traced via the 18-n rule to the structure's Ir-Ir connectivity. In parallel with this, the structure's CP scheme highlights facile paths of atomic motion that could enable a structural response to this electronic deficiency. Combined, these analyses suggest that IrSi may adopt a more complex structure than previously recognized. Through synthesis and detailed structural investigation of this phase, we confirm this prediction: single-crystal X-ray diffraction reveals an incommensurately modulated structure with the (3+1)D superspace group P2₁/n(0βγ)00 and q ≈ -0.22b* + 0.29c*. The structural modulations increase the average number of Ir-Ir bonds to nearly match the 18-n expectations of the phase through Ir-Ir trimerization along negative CPs with the incommensurability arising from the difficulty of contracting and stretching the Ir-Ir contacts in a regular pattern without expanding the structure along directions of negative Si-Si CP. Integrating these results with prior analyses of related systems points to a simple guideline for materials design, the Frustrated and Allowed Structural Transitions (FAST) principle: the ease with which competing structural phenomena can be experimentally realized is governed by the degree to which they are supported by the coordination of the atomic packing and electronic factors.

Chemical pressure-chemical knowledge: squeezing bonds and lone pairs within the valence shell electron pair repulsion model

The valence shell electron pair repulsion (VSEPR) model is a demanding testbed for modern chemical bonding formalisms. The challenge consists in providing reliable quantum mechanical interpretations of how chemical concepts such as bonds, lone pairs, electronegativity, or hypervalence influence (or modulate) molecular geometries. Several schemes have been developed thus far to visualize and characterize these effects; however, to the best of our knowledge, no scheme has yet incorporated the analysis of the premises derived from the ligand close-packing (LCP) extension of the VSEPR model. Within the LCP framework, the activity of the lone pairs of the central atom and ligand-ligand repulsions constitute the two key features necessary to explain certain controversial molecular geometries that do not conform to the VSEPR rules. Considering the dynamical picture obtained when electron local forces at different nuclear configurations are evaluated from first-principles calculations, we investigate the chemical pressure distributions in a variety of molecular systems, namely, electron-deficient molecules (BeH₂, BH₃, BF₃), several AX₃ series (A: N, P, As; X: H, F, Cl), SO₂, ethylene, SF₄, ClF₃, XeF₂, and nonequilibrium configurations of water and ammonia. Our chemical pressure maps clearly reveal space regions that are totally consistent with the molecular and electronic geometries predicted by VSEPR and provide a quantitative correlation between the lone pair activity of the central atom and electronegativity of ligands, which are in agreement with the LCP model. Moreover, the analysis of the kinetic and potential energy contributions to the chemical pressure allows us to provide simple explanations on the connection between ligand electronegativity and electrophilic/nucleophilic character of the molecules, with interesting implications in their potential reactivity. NH₃, NF₃, SO₂, BF₃, and the inversion barrier of AX₃ molecules are selected to illustrate our findings.

Templating Structural Progessions in Intermetallics: How Chemical Pressure Directs Helix Formation in the Nowotny Chimney Ladders

In the structural diversity of intermetallic phases, hierarchies can be perceived relating complex structures to relatively simple parent structures. One example is the Nowotny Chimney Ladder (NCL) series, a family of transition metal-main group (T-E) compounds in which the T sublattices trace out helical channels populated by E-atom helices. A sequence of structures emerges from this arrangement because the spacing along the channels of the E atoms smoothly varies relative to that of the T framework, dictated largely by optimization of the valence-electron concentration. In this Communication, we show how this behavior is anticipated and explained by the Density Functional Theory-Chemical Pressure (DFT-CP) schemes of the NCLs. A CP analysis of the RuGa₂ parent structure reveals CP quadrupoles on the Ga atoms (telltale signs of soft atomic motion) that arise from overly short Ru-Ga contacts along one axis and underutilized spaces in the perpendicular directions. In their placement and orientation, the CP quadrupoles highlight a helical path of facile movement for the Ga atoms that avoids further compression of the already strained Ru-Ga contacts. The E atoms of a series of NCLs (in their DFT-optimized geometries) are all found to lie along this helix, with the CP quadrupole character being a persistent feature. In this way, the T sublattice common to the NCLs encodes helical paths by which the E-atom spacing can be varied, creating a mechanism to accommodate electronically driven compositional changes. These results illustrate how CP schemes can be combined with electron-counting rules to create well-defined structural sequences, potentially guiding the discovery of new intermetallic phases.

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.

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.

Discerning Chemical Pressure amidst Weak Potentials: Vibrational Modes and Dumbbell/Atom Substitution in Intermetallic Aluminides

The space requirements of atoms are generally regarded as key constraints in the structures, reactivity, and physical properties of chemical systems. However, the empirical nature of such considerations renders the elucidation of these size effects with first-principles calculations challenging. DFT-chemical pressure (DFT-CP) analysis, in which the output of DFT calculations is used to construct maps of the local pressures acting between atoms due to lattice constraints, is one productive approach to extracting the role of atomic size in the crystal structures of materials. While in principle this method should be applicable to any system for which DFT is deemed an appropriate treatment, so far it has worked most successfully when semicore electrons are included in the valence set of each atom to supply an explicit repulsive response to compression. In this Article, we address this limiting factor, using as model systems intermetallics based on aluminum, a key component in many structurally interesting phases that is not amenable to modeling with a semicore pseudopotential. Beginning with the Laves phase CaAl₂, we illustrate the difficulties of creating a CP scheme that reflects the compound's phonon band structure with the original method due to minimal core responses on the Al atoms. These deficiencies are resolved through a spatial mapping of three energetic terms that were previously treated as homogeneous background effects: the Ewald, E_α, and nonlocal pseudopotential components. When charge transfer is factored into the integration scheme, CP schemes consistent with the phonon band structure are obtainable for CaAl₂, regardless of whether Ca is modeled with a semicore or valence-only pseudopotential. Finally, we demonstrate the utility of the revised method through its application to the La₃Al₁₁ structure, which is shown to soothe CPs that would be encountered in a hypothetical BaAl₄-type parent phase through the substitution of selected Al₂ pairs with single Al atoms. La₃Al₁₁ then emerges as an example of a more general phenomenon, CP-driven substitutions of simple motifs.

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.

Chemical Pressure Maps of Molecules and Materials: Merging the Visual and Physical in Bonding Analysis

The characterization of bonding interactions in molecules and materials is one of the major applications of quantum mechanical calculations. Numerous schemes have been devised to identify and visualize chemical bonds, including the electron localization function, quantum theory of atoms in molecules, and natural bond orbital analysis, whereas the energetics of bond formation are generally analyzed in qualitative terms through various forms of energy partitioning schemes. In this Article, we illustrate how the chemical pressure (CP) approach recently developed for analyzing atomic size effects in solid state compounds provides a basis for merging these two approaches, in which bonds are revealed through the forces of attraction and repulsion acting between the atoms. Using a series of model systems that include simple molecules (H₂, CO₂, and S₈), extended structures (graphene and diamond), and systems exhibiting intermolecular interactions (ice and graphite), as well as simple representatives of metallic and ionic bonding (Na and NaH, respectively), we show how CP maps can differentiate a range of bonding phenomena. The approach also allows for the partitioning of the potential and kinetic contributions to the interatomic interactions, yielding schemes that capture the physical model for the chemical bond offered by Ruedenberg and co-workers.

Inducing Complexity in Intermetallics through Electron-Hole Matching: The Structure of Fe14 Pd17 Al69

We illustrate how the crystal structure of Fe₁₄ Pd₁₇ Al₆₉ provides an example of an electron-hole matching approach to inducing frustration in intermetallic systems. Its structure contains a framework based on IrAl_2.75 , 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 IrAl_2.75 phase, the electronic structure of Fe₁₄ Pd₁₇ Al₆₉ 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.

On the Functionality of Complex Intermetallics: Frustration, Chemical Pressure Relief, and Potential Rattling Atoms in Y11Ni60C6

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 YNi₃C_x 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.

Epitaxial Stabilization between Intermetallic and Carbide Domains in the Structures of Mn16SiC4 and Mn17Si2C4

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.

Exo-bonded six-membered heterocycle in the crystal structures of RE7Co2Ge4(RE = La–Nd)

The stability of the heterocyclic {Co₄Ge₆} clusters in RE₇Co₂Ge₄(RE = La–Nd) is determined by strong interactions with the surrounding RE atoms in the structures.