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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.

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: Mo2Cu(x)Ga(6-x) (x ≈ 0.9). Mo2Cu(x)Ga(6-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.

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, T(4)Ga(5) (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@T(8) body centered cubic (bcc) clusters linked through face-capping Ga(2) 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 Ta(4)Ga(5) (as well as at that of Nb(4)Ga(5)), 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.

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.

Intermetallic phases offer a wealth of unique and unexplained structural features, which pose exciting challenges for the development of new bonding concepts. In this article, we present a straightforward approach to rapidly building bonding descriptions of such compounds: the reversed approximation Molecular Orbital (raMO) method. In this approach, we reverse the usual technique of using linear combinations of simple functions to approximate true wave functions and employ the fully occupied crystal orbitals of a compound as a basis set for the determination of the eigenfunctions of a simple, chemically transparent model Hamiltonian. The solutions fall into two sets: (1) a series of functions representing the best-possible approximations to the model system's eigenstates constructible from the occupied crystal orbitals and (2) a second series of functions that are orthogonal to the bonding picture represented by the model Hamiltonian. The electronic structure of a compound is thus quickly resolved into a series of orthogonal bonding subsystems. We first demonstrate the raMO analysis on a familiar molecule, 1,3-butadiene, and then move to illustrating its use in discovering new bonding phenomena through applications to three intermetallic phases: the PtHg4-type CrGa4 and the Ir3Ge7-type compounds Os3Sn7 and Ir3Sn7. For CrGa4, a density of states (DOS) minimum coinciding with its Fermi energy is traced to 18-electron configurations on the Cr atoms. For Os3Sn7 and Ir3Sn7, 18-electron configurations also underlie DOS pseudogaps. This time, however, the 18-electron counts involve multicenter interactions isolobal with classical Ir-Ir or Os-Os covalent bonds, as well as Sn-Sn single bonds serving as electron reservoirs. Our results are based on DFT-calibrated Hückel calculations, but in principle the raMO analysis can be implemented in any method employing one-electron wave functions.