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 MnPtype structure exposes issues pointing toward new structural behavior. The placement of the Fermi energy below an electronic pseudogap indicates that this structure is electronpoor, 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 P2 1 /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.

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.

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.

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 BaAl 4 type. While the Dy-Ga binary intermetallic system contains no phases related to the BaAl 4 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 BaAl 4 -type phase DyGa 4 and a La 3 Al 11 -type variant (in which a fraction of Ga 2 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 BaAl 4 -type DyGa 4 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 0.18 Ga 0.82 ) 3.71(1) . In this compound, the BaAl 4 type is subject to a 2D incommensurate modulation (q 1 = 0.31a* + 0.2b*, q 2 = 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 La 3 Al 11 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 (BaAl 4 -type DyGa 4 ) can bring it to the cusp of stability with intriguing structural consequences.

In lead­(II) halide compounds including virtually all lead halide perovskites, the Pb2+ 6s lone pair results in distorted octahedra, in accordance with the pseudo-Jahn–Teller effect, rather than generating hemihedral coordination polyhedra. Here, in contrast, we report the characterization of an organic–inorganic hybrid material consisting of one-dimensional edge-sharing chains of Pb–Br square pyramids, separated by [Mn­(DMF)6]2+ (DMF = dimethylformamide) octahedra. Molecular orbital analysis and density-functional theory calculations indicate that square pyramidal coordination about Pb2+ results from the occupancy of the empty ligand site by a Pb2+ lone pair that has both s and p orbital character rather than the exclusively 6s lone pair. These results demonstrate that a Pb2+ lone pair can be exploited to behave like a ligand in lead halide compounds, greatly expanding the realm of possible lead halide materials to include extended solids with nonoctahedral coordination environments.