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Article Cite This: Inorg. Chem. XXXX, XXX, XXX−XXX

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Cation Clustering in Intermetallics: The Modular Bonding Schemes of CaCu and Ca2Cu Sang-Won Park,†,‡ Hideo Hosono,*,† and Daniel C. Fredrickson*,†,‡ †

Materials Research Center for Element Strategy, Tokyo Institute of Technology, 4259 Nagatsuta, Midori-ku, Yokohama 226-8503, Japan ‡ Department of Chemistry, University of Wisconsin-Madison, 1101 University Avenue, Madison, Wisconsin 53706, United States

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S Supporting Information *

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

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segregation of bonding types can be detected,16−19 but the origins of these features are often unknown. One intriguing structural theme hinting at such segregation of bonding modes is found in the compounds that are rich in electropositive metals such as the lanthanides or alkaline earth metals. These presumably cationic atoms frequently condense into fragments of simple sphere packings, filling in space left over by the anionic components of the structure.20−28 Striking examples include a series of Mg-rich Mg−Y−Cu phases,21 e.g. Mg8Y5Cu5, where matrixes of Mg atoms forming an fcc pattern enclose columns of the Fe2P type; the structure of Sc44Ir7, where Keggin units of Sc atoms emerge in the spaces between Sc−Ir Mackay clusters;20,26 and Ca10Pt7Si3,24,25 where Ca6 octahedra are immersed in a network of Pt6Si3Pt trefoil units. The recurrence of such features opens the questions of what role they play in the bonding and stability of these compounds.

INTRODUCTION The structures and stability of intermetallic phases are governed by an immense diversity of bonding interactions that has yet to be fully comprehended by chemical theory.1 Indeed, in drawing from elements across the periodic table, intermetallic systems sample a wide range of bond polarities and degrees of electron localization. The structures of the simplest Zintl phases2,3 are well-rationalized with the view of electrons populating covalent bonds between the anions to complete filled octets or cluster shells, while the Hume− Rothery phases are considered in terms of a weakly perturbed free-electron gas,4,5 and a broad class of transition-metal/pblock intermetallics appear to be governed by a variation on the 18-electron rule.6−11 However, there is also an emerging realm of intermetallics where several bonding or packing types coexist within the same crystal structure, sometimes leading to immense complexity as seeming mutually exclusive interactions (say, polar and nonpolar) are accommodated within a single unit cell.12−15 In other structures, geometrical hints of such a © XXXX American Chemical Society

Received: May 21, 2019

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DOI: 10.1021/acs.inorgchem.9b01486 Inorg. Chem. XXXX, XXX, XXX−XXX

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Inorganic Chemistry

raMO analysis.8 The raMO analyses and visualization of their results were performed with the in-house MATLAB programs raMOmovie and Figuretool2.

In this Article, we begin to explore these questions with the analysis of alkaline earth-transition metal intermetallics with a particular interest in Ca2Cu, in which zigzag chains of Cu atoms occur embedded a host lattice of Ca (Figure 1).29 We

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RESULTS AND DISCUSSION In this Article, we will explore the bonding of the Ca2Cu phase and its parent compound CaCu to probe its separation of Carich regions from the Ca−Cu interactions (Figure 1) and the electronic consequences of these geometrical arrangements. Let’s begin with a closer inspection of the Ca2Cu structure. When viewed down the b-axis, the structure exhibits a striped pattern in which double-layers of Ca atoms (pink) alternate with layers containing Cu zigzag chains. The Ca-containing layers can be viewed as a two-atom thick slab of the hcp structure (Figure 1c) sliced along the (110) plane. The Cucontaining regions also follow a recognizable pattern, as can be seen by examining the structure of another Ca−Cu phase, CaCu. CaCu appears at first to be quite complex, but the structures of both its α- and β-polymorphs are derived from intergrowths of the common FeB and CrB types (Figure 2, where we focus

Figure 1. Ca-clustering in the crystal structure of Ca2Cu. (a) The full crystal structure. (b) Columnar fragments of fcc Ca. (c) Sheets of hcp Ca.

show that the electronic structures of Ca2Cu and its parent compound CaCu can be simply separated into Ca- and Cucentered components using the reversed approximation Molecular Orbital (raMO) approach,8,9 which complements such widely used methods as crystal orbital Hamilton population (COHP)30,31 and the electron localization function/localizability indicator (ELF and ELI)32−34 analyses by generating local MO diagrams from a system’s crystal orbitals. The Cu atoms exhibit a 3d10 closed shell and nearly filled Cu σ-band (approximately Cu1−) which are supported through Lewis acid/base interactions with their surrounding Ca atoms. This leaves additional valence electrons, though, which the Zintl or 18-electron schemes would direct to the remaining unused orbitals on the Cu atoms. Instead, these electrons will be found in a largely separate bonding system based on multicenter Ca−Ca interactions (which can be represented with a series of resonance structures). In this way, the Ca-rich domains may be seen as a reservoir for electrons that are insufficiently drawn to the Ca−Cu regions.

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EXPERIMENTAL SECTION

Figure 2. Common structural motifs shared by the (a) FeB, (b) CrB, and (c) β-CaCu structure types.

To provide reference electronic structures for the parametrization of Hückel models, density functional theory (DFT) calculations were carried out on CaCu (in the FeB, CrB, and β-CaCu structure types) and Ca2Cu using the Vienna Ab initio Simulation Package (VASP)35 in the accurate-precision mode with the Perdew−Burke−Ernzerhof formulation of the generalized gradient approximation (GGA)36,37 and the projector augmented wave (PAW) method.38,39 First, geometry optimizations were performed in which the relaxation of the atomic positions within a fixed unit cell were followed by a release of all structural parameters. Single-point energy calculations were then done on the optimized geometries to obtain electronic band energies and projected density of states (DOS) curves. The DFT band energies and DOS distributions next served as reference data for the calibration of Hückel models of the structures with the eHtuner program.40 Using the best-fit parameters, simple Hückel calculations were carried out with YAeHMOP41 on supercells of FeB-type CaCu (3 × 3 × 3), CrB-type CaCu (4 × 2 × 4), and Ca2Cu (3 × 3 × 2) to map multiple k-points to the Γ-point, the Hamiltonian matrix for which was extracted as the main input for the

on the β-form).42 In all of these structures, the relatively electronegative and presumably anionic atoms (Cu or B) form zigzag chains that are well-isolated from each other. The more electropositive and likely cationic atoms (Ca, Cr, or Fe) then collect around the anions to create columns of trigonal prisms whose shared rectangular faces bisect the anion−anion contacts. The third rectangular faces of the trigonal prisms are capped by a cation such that, when looking down on the columns, the zigzag chains appear to occupy channels shaped like flattened hexagons (see the dashed oval in Figure 2a). The structures differ in how these columns are arranged. In the CrB type (Figure 2b), the triangular faces of the trigonal prisms of neighboring columns fuse together to create sheets in which the planes of the zigzag chains are all parallel. These layers are then joined through shared Ca−Ca edges to build B


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Inorganic Chemistry

suggests the compounds are electron-rich in keeping with their Ca-rich composition. The close correspondences in the DOS features for these three structural models for CaCu hint that they exhibit similar bonding and stabilities. An examination of their total energies indicates that they are all within 6 meV/atom of each other. As such, we will focus our attention on one of the simpler members of this series in our analysis of CaCu, the FeB-type version. A quick comparison with results obtained for the CrBtype version will then show that the electronic features are transferable between structures, allowing us to understand the complex structures observed for CaCu through interpolation. At first glance, the DOS features for Ca2Cu (Figure 3d) are similar to those of the CaCu variants. Two main differences may be noted: the EF appears higher relative to the top of the Cu 3d states, and no analogue is evident to the DOS minimum near −7 eV in the CaCu DOS distributions. Both changes seem consistent with the addition of slabs of electropositive atoms in a geometry associated with elemental metals. As we look in more detail into the electronic features of this compound, we see that the bonding in the Cu-containing slabs of Ca2Cu can be directly transferred from the CaCu phase with the Ca-rich layer then having a distinct bonding system. Electron Counting in FeB-type CaCu. Now that we have oriented ourselves to the overall features of the electronic structures of Ca2Cu and CaCu, let’s explore how they arise from the phases’ geometrical motifs. A method for associating preferred electron counts with specific structural units is given by the raMO analysis.8 In this approach, the occupied crystal orbitals of a compound are used as a basis set for constructing a local MO diagram for a part of the structure. This allows the development of bonding schemes that connect structure to ideal electron counts, providing distinct information from the strengths and optimization of pairwise interatomic interactions (as given by COHP analysis)30,31 or the spatial separation of electrons of the same spin as a proxy for electron localization (ELF and ELI).32−34 The raMO process is initiated with a hypothesis about the types of orbitals that are occupied in the electronic structure, such as a set of atomic orbitals or bonding functions. Model Hamiltonian operators are then created in which the proposed local orbitals of interest are the eigenstates. Next, the occupied crystal orbitals are used as an approximate basis set for the construction of a model Hamiltonian matrix, whose diagonalization leads to the best approximations to the target functions possible as well as remainder functions that correspond to electrons in states orthogonal to the target functions. Based on these results, one may revise the hypothesis−such as moving from the use of isolated atomic orbitals as targets to bonding combinations of them−and repeat the process. When satisfactory reproductions are obtained, the remainders can be used as basis functions for additional rounds of analysis with another set of targets, creating an iterative loop in which the occupied crystal orbitals are gradually recombined into chemically meaningful localized functions (closely resembling Wannier functions)44,45 that recreate the same overall multielectron wave function. Consider CaCu in the FeB type. From the locations of the sharp peaks of Cu d states far below the EF, the Cu 3d10 subshells appear to be completely filled. This forms a starting point for raMO analysis, where repeated raMO cycles can reproduce each of the Cu 3d orbitals atom by atom in the structure, leaving behind a set of wave functions which have

the full structure. In the FeB type (Figure 2a), the triangular faces of the trigonal prisms instead join with the faces of the cation pyramids of their neighboring columns, leading to two sets of zigzag chains lying in different orientations. CaCu represents an intergrowth of these two arrangements with slabs of the FeB-type being separated by CrB-type interfaces, recalling the HfFe6Ge6/ScFe6Ga6 intergrowth nature of many complex stuffed CoSn-type phases.43 When comparing these arrangements to that of the Ca2Cu structure in Figure 1, the same columns of capped trigonal prisms become apparent. The Cu-containing layer of Ca2Cu can in fact be derived from any of the three structures in Figure 2, including that of β-CaCu. We begin with the CaCu phase and then break it up into slabs through the intercalation of additional Ca atoms. These newly inserted Ca then form domains of hcp Ca in the spaces between the CaCu slabs. While the clustering of Ca atoms into close-packed arrangements is particularly vivid in Ca2Cu, a similar theme can already be noted in CaCu. In the spaces between Cucentered trigonal prisms, double layers of Ca can be perceived in the white spaces between the blue polyhedra in Figure 2. When extracted, these appear as two square nets staggered relative to each other to create a section of the fcc structure (as shown for an analogous unit in Ca2Cu in Figure 1b). In this way, CaCu joins Ca2Cu as a compound in which atoms of the electropositive elements segregate out into clusters rather than fully commit themselves to binary interactions. Preliminary Electronic Structure Analysis. We now turn to developing bonding schemes for Ca2Cu and CaCu with the goal of understanding the origins and electronic features of their Ca-rich domains. First, in Figures 3a−c, we compare the

Figure 3. GGA-DFT electronic DOS distributions for CaCu in the (a) FeB-, (b) CrB-, and (c) CaCu-types as well as for (d) Ca2Cu. The Cu 3d contributions to the DOS distributions are shaded in black.

electronic density of states (DOS) of CaCu in the FeB, CrB, and β-CaCu types. Here, the remarkably similar local geometrical structures are reflected in nearly identical DOS curves. In all three, a block of Cu d states centered between ca. −9.5 and −8.0 eV jumps out from a disperse distribution of sprich states whose tail stretches down to around −12 eV. The Fermi energy (EF) for each structure lies more than 2 eV above both the top of Cu d states, indicating that the Cu 3d orbitals comprise a filled shell. The EF is also placed above the most pronounced DOS minimum in the sp-distribution, centered near −7 eV. As DOS minima are often associated with special electronic stability, the location of the EF’s above these features C


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indicating it is orthogonal to the basis set of occupied functions. As such, the Cu−Cu σ-band can be interpreted as populated at a level of about 5/6 banding filling, accounting for 10/6 = 1.67 electrons/formula unit. Together with the Cu 3d10 configurations, we have now accounted for about 11.67 of the compound’s 13 electrons/formula unit. The remaining Cu 4p raMOs provide clues to where the additional electrons reside. The association of electron density with both the Cu 4 pz and 4 py orbitals suggests that neither one individually is a clear focal point for the remaining electrons. In fact, these raMOs are strongly polarized toward the Ca atoms in the Cu coordination environment, suggesting little direct involvement from the Cu atoms themselves. Instead, Ca-based orbitals may provide a better starting point. To isolate such Ca-centered functions, we build raMO functions for the full set of Cu−Cu isolobal bonds to yield a set of remainders that can be analyzed directly. Based on the assumption that the remaining electrons are associated with electron-deficient, multicenter bonding in the Ca-sublattice, we can analyze them through the use of target functions associated with interstitial spaces between the Ca atoms. As the Ca atoms are aggregated into fragments of the fcc structure, two obvious choices for such spaces are the remnants of the fcc octahedral or tetrahedral holes. For the fragmentation of the fcc structure, the octahedral holes are truncated down to the Ca5 square pyramids in the coordination environment of each Cu atom. A localized state centered on any given square pyramid can then be templated by using a Cu sp2 hybrid orbital pointing into the cage as a raMO target. As shown in the lower left of Figure 6, the raMO function resulting for one such target is well-localized within the corresponding Ca5 square pyramid (though it is more diffuse along the Cu chain). The substantial Ca 4s character involved here means, however, that there is substantial orbital overlap between the Ca5 raMOs that would be obtained for neighboring cages that share vertices. We must proceed with care then to explore how many of these cages can be filled independently. This process is illustrated in Figure 6, where we successively reproduce a series of Ca5 cages in the structure. Each cage is fused to several other Ca5 square pyramids: two through shared basal edges, and two through shared edges connecting to the apices of the pyramids. A sequence of raMOs can be generated for neighboring cages that share basal edges, as shown in the first three raMOs from the left in Figure 6. Indeed, if beginning with one cage in each Ca fcc fragment, we follow a path through shared basal edges, we can generate raMOs containing electron pairs for half of the Ca5 pyramids in the structure. As there is one Ca5 pyramid per formula unit, this accounts for (2 electrons/occupied pyramid) × (1 occupied pyramid/2 pyramids) × (1 pyramid/formula unit) = 1 electron per formula unit. If we then continue and try to develop raMOs for the remaining pyramids, however, the resulting raMOs become substantially more diffuse, indicating that they are far from fully occupied (Figure 6, right-most three raMOs). At this point, we have developed a nearly complete accounting of the electrons in FeB-type CaCu, as illustrated with DOS projections in Figure 7. Our raMO analysis began with assigning 10 of the 13 electrons per formula unit to Cu 3d10 configurations, which captures the dominant group of peaks in the DOS centered near −8.75 eV. From the remainder states, we were then able to reconstruct the majority

been orthogonalized to the occupied Cu 3d orbitals. In the bottom panel of Figure 4, we present the raMO reconstruc-

Figure 4. raMO reconstructions of 4s, 4p, and 3d orbitals for a Cu atom in the FeB type CaCu. Bottom panel: raMOs for the 3d orbitals of one Cu atom created following the reconstruction of the 3d subshells for every other Cu atom in the structure. Top panel: raMOs for the 4s and 4p (labeled in terms of a local coordinate system) for a Cu atom after the reconstruction of the full Cu 3d band.

tions for the 3d orbitals of a Cu atoms made after recreating every other Cu 3d orbital in the structure (thus representing the completion of the Cu 3d bands). In the plots, each Cu 3d orbital is tightly localized to the central Cu atom, indicating that 10 electrons fill the 3d orbitals without noticeable interactions with the surrounding atoms. This is in accord with the narrowness of the Cu 3d peak in the DOS curve. In this first step, we have accounted for 10 of the 13 electrons per formula unit of CaCu, leaving 3 electrons left to locate in the electronic structure. As the assignment of 18 electron configurations to transition metals has been fruitful for many intermetallic phases,9 we look next at how well the remaining valence orbitals of the Cu, the 4s and 4p, can be reconstructed from the remainder states. The results for the one Cu atom are shown in the upper panel of Figure 4. In each case, some contribution from the target function on the central atom is apparent, but in general, the raMOs are much more spread out into the neighboring Ca and Cu atoms through bonding interactions. Importantly, the contributions from the Cu neighbors indicate electron sharing between the Cu atoms; these raMO functions for neighboring Cu atoms cannot be independently occupied. A closer inspection of the Cu 4s and 4p raMOs is revealing. The Cu 4s- and Cu 4px-centered raMOs show strong σbonding interactions along the Cu−Cu chain, with bridging contributions from the Ca atoms. Together, these functions resemble the symmetric and antisymmetric combinations of two Cu−Cu isolobal bonds.46 The Cu−Cu chain then appears to be connected through a series of isolobal bonds, which would account for two electrons/Cu atom (and thus 2 electrons/formula unit). From here, we generate target functions corresponding to the full bands of Cu−Cu σ-bonding in the chains, as shown in Figure 5a with the k-points sampled by our supercell indicated (the ±kx points are combined, where necessary, to create real functions). The raMO reconstructions of these functions are given in Figure 5b, where aside from the very top of the band, each target appears well-reproduced and augmented by Cabridging contributions. The target for the top of the point of the σ-band, curiously, yields no raMO reconstruction, D


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Figure 5. raMO reconstructions of the σ-bands of a Cu zigzag chain in FeB-type model of CaCu. (a) The target functions. (b) The corresponding raMO functions, illustrating that each target is well-reproduced (with the support of bridging interactions from Ca neighbors) except for the top of the band.

of the Cu−Cu σ-bonding bands, accounting for 1.67 electrons/ formula unit. Of the remaining electrons, 1.00 electron/ formula unit was assigned to the placement of an electron pair in every other Ca5 cage. This leaves 0.33 electrons per formula unit left over for the Cu 4pz-based π-band and the remaining Ca5 cages. Repeating this analysis for CaCu in the CrB type, the other end-member of the FeB/CrB intergrowth series of structures, yields essentially identical conclusions (see the Supporting Information), suggesting that the more complex observed structure of CaCu will obey the same scheme. This rather precise assignment of electrons may at first appear to contradict the highly delocalized nature of the electronic structures indicated by the DOS curves and the absence of pseudogaps at their EF’s. How is the metallic nature of CaCu reflected in these results? Delocalization enters into

this picture through the multiple ways that the raMO reconstructions can be made. When we built raMOs for each Ca fcc fragment, we made a choice of which set of Ca5 cages we would use for the construction. An equivalent construction could have been made using the other set of Ca5 cages. Other possibilities exist as well. For example, we can use instead the tetrahedral holes of the Ca fcc fragments, which can be divided into two sets that share only single vertices, reducing the overlap between their potential cage functions (Figure 8). raMOs corresponding to electron pairs in each tetrahedron can be produced for either of these sets. This feature is illustrated in the top row of Figure 8, where we construct raMO functions for 54 of the 108 tetrahedra in the CaCu supercell. Performing these reconstructions then leaves little occupied orbital character for the creation of raMOs for the remaining set of tetrahedra (bottom row). This multiplicity E


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Figure 6. Successive raMO reconstructions for Cu sp2 hybrid orbitals pointing into the Ca5 square-pyramids of a Ca fcc fragment in a FeB-type model of CaCu. Localized functions can be obtained for one set of edge-sharing square-pyramids (pink), but the remainder states are insufficient for the creation of analogous functions centered on the second set of pyramids (gray). In some panels, the repetition of the raMOs across the upper and lower edges of the fragment is visible; this coincides with the period of the supercell used in the analysis.

Curiously, though, the electron configurations of the Ca fcc units are also largely unaffected. Instead, the electron removal is accommodated by a depopulation of the Cu σ-bonding bands from 5/6 filled to approximately 1/2 filled with the bottom half of the σ-band being predominately based on the Cu 4s orbitals. The 12 electron pseudogap then corresponds to filled Cu 3d10 configurations, a half-filled Cu 4s band, and on average one electron per Ca tetrahedron or Ca5 pyramid. The observation that going from 13 to 12 electrons leads to the depopulation of Cu−Cu interactions rather than the Ca-based states highlights the strength of the multicenter Ca−Ca interactions. Electron Counting in Ca2Cu. In our analysis of CaCu, we saw that each of the key geometrical motifs, the Cu zigzag chains and Ca fcc fragments, can be associated with a specific electron count. The Cu zigzag chain holds nearly 12 electrons per Cu atom in Cu 3d10 configurations and Cu−Cu σ-bonds (shared to some degree with the bridging Ca atoms). The Ca fcc fragments, meanwhile, contain one electron per Ca atom, which can be viewed in terms of resonance structures in which pairs of electrons occupy every other Ca5 square pyramid or Ca4 tetrahedron (with small Cu contributions, as seen in Figures 6 and 8). Let’s now examine how this picture transfers to the more Ca-rich compound Ca2Cu, in which layers of the CaCu structure are separated by the insertion of hcp Ca. First, we carry out the same sequence of raMO reconstructions as before for the CaCu layers in Ca2Cu, deriving raMOs for the Cu 3d orbitals, Cu−Cu σ-band, and Ca5 cage orbitals. The resulting functions are closely analogous to those we saw earlier, leading to the projected DOS contributions in Figure 9. Summing these together (right panel of the Figure 9) leads to a shaded region that accounts for most of the DOS curve, reflecting that we have assigned the majority of the electrons in the structure, 12.67 of the 15 per formula unit, to the CaCu-derived layers of Ca2Cu. The remaining electrons are associated with the hashed portion of

Figure 7. Projected DOS distributions for raMO functions derived for FeB-type CaCu. (a) The Cu 3d raMOs. (b) The Cu σ-band. (c) Ca5 cages/Ca4 tetrahedra.

of raMO schemes can be understood in terms of resonance structures where separate electron pairs can be placed in either half of the Ca5 cages or Ca4 tetrahedra with each configuration corresponding to an average count of 1 electron/Ca atom. It is also interesting to note that while no pseudogap occurs at the EF for CaCu (Figure 3), a DOS minimum is apparent about 1 eV below that level. This minimum coincides roughly with 12 electrons/formula unit and would thus be expected to be relevant to the bonding in such related compounds as SrPd. The origins of this feature can be investigated by watching how the raMO results change when we lower the electron count of our Hückel model such that the EF lies within this pseudogap, at −7.13 eV. As may be expected, the drop in the electron count has little effect on the Cu 3d10 reconstructions. F


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Figure 8. raMO reconstructions of electron pairs in Ca4 tetrahedra in FeB-type CaCu. The top panel shows the successive reconstructions for a series of tetrahedral cluster orbitals linked through corner sharing, accounting for half of the tetrahedra. In the lower panel, the process is continued through the remaining tetrahedra, with the diffuse nature of the resulting functions indicating that full electron pairs cannot be assigned to these Ca4 units.

tried using various target functions focused on these sheets, in which linear combinations of Ca sp-hydrid orbitals are used to create s-orbital like functions centered on Ca4 tetrahedra, Ca6 octahedra, or Ca trigonal bipyramids. In doing this, the most localized functions appear for the raMO reconstructions of Ca trigonal bipyramid. In fact, as is shown in Figure 10, an electron pair function can be derived for each of the Ca5 bipyramids of the Ca hcp layers in a sequential fashion. As there is one Ca5 bipyramid per formula unit in the structure, the placement of an electron pair in each of them accounts for 2 electrons/formula unit. When combined with the 12.67 electrons/formula unit already assigned to the CaCu fragments of the structure, almost all electrons in the structure are now accounted for. The Ca5 bipyramid raMOs are then the largest contributors to the hashed region of the DOS distribution in right panel of Figure 9. The remaining 0.33 electrons/formula unit are likely distributed over the Cu−Cu π-system and Ca FCC units, following our considerations of CaCu. Our investigations into the CaCu and Ca2Cu structures were originally motivated by an interest in the origins and bonding of their Ca clusters. We are now in a position to draw some conclusions regarding the nature of these Ca arrays. In these structures, two types of Ca-based motifs appear: Ca fcc fragments (in both CaCu and Ca2Cu) and layers of the hcp structure (in Ca2Cu). In the fcc fragments, multiple raMOderived resonance structures can be drawn, but on average,

Figure 9. Projected DOS curves for the raMO reconstructions of the Cu 3d orbitals, Cu σ-bands, and Ca cages in the CaCu fragments of Ca2Cu, by analogy to Figure 7. The remaining states, corresponding to approximately 2 electrons/formula unit, are concentrated just below the Fermi energy, EF (dashed line).

the DOS distribution, which is concentrated higher in energy, just below the EF. How are these remaining electrons distributed through the structure? At least some of these electrons should be associated with the Ca hcp sheets. raMO reconstructions can then be G


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Figure 10. Reconstructions of s-like cage orbitals centered on the Ca trigonal bipyramids in the hcp Ca sheets of Ca2Cu. The targets and raMOs of the first and last cages analyzed in the sequence of raMO steps through the supercell are shown.

atoms. This is in contrast to the transition-metal isolobal bonds that form in other intermetallics with the support of bridging contributions from p-block elements such as Sn, Ga, or Al.8 The inability of such an s-block metal to stabilize a transitionmetal p-based function is reminiscent of the situation we found earlier for CaRuSi and the 16 electron configurations on its Ru atoms.48 In CaRuSi, the relatively high energy of the Ru 5pz orbital led to its being depopulated in favor of the filling of Si lone pairs. Here, the Cu 4p orbitals are similarly being outcompeted, this time by cage orbitals based on Ca−Ca multicenter bonding. It will be interesting to explore how these themes extend to systems with other cluster arrangements of cations such as the flattened stella quadrangula in the Cr5B3-type A5T3 (A = Ca, Sr, T = Au, Ag, Hg, Cd, Zn) phases studied previously by Lee and coworkers.22

they correspond to an electron count of 1 per Ca4 tetrahedron. In the Ca hcp layers, the raMO analysis is more clear-cut: electrons pairs closely associated with each trigonal bipyramid can be created. When we recall that the trigonal bipyramids of the hcp structure are derived from the fusion of two tetrahedral holes, though, a correspondence emerges: the placement of an electron pair in each trigonal bipyramid amounts to one electron per tetrahedron as in the fcc fragments. The difference is then in the freedom with which this 1 electron/tetrahedron assignments can be made. For the hcp sheets of Ca2Cu, one single resonance structure is much more representative of the electronic structure than the alternatives that can be drawn (see the Supporting Information). The higher localization of the electron pairs in the hcp layers of Ca2Cu is supported by the ELF function (see the Supporting Information, Figure S5). ELF maxima with a peak magnitude of 0.67 are present in the trigonal bipyramids of these layers, comparable to those found by Silvi and Gatti for the tetrahedral holes of fcc Ca.47 In the interstitial regions of the fcc units of Ca2Cu and FeB-type CaCu, however, the ELF function is below the 0.5 value for a free electron gas.

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ASSOCIATED CONTENT

* Supporting Information S

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.inorgchem.9b01486. DFT-optimized geometries, additional computational details, DFT-calibrated Hückel parameters, raMO results for CrB-type CaCu and SrPd, raMO reconstructions of electron pairs in Ca4 tetrahedra and Ca6 octahedra in Ca2Cu, and an ELF isosurface plot of Ca2Cu (PDF)

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CONCLUSIONS Over the course of this Article, we have examined the chemical bonding interactions underlying the tendency of electropositive metal atoms to cluster in the structures of intermetallic phases. The atomic packing in both Ca2Cu and its parent CaCu compound is suggestive of a segregation of Ca−Cu and Ca domains being built from Ca-capped Cu−Cu chains and blocks of close-packed Ca atoms. raMO analysis supports this viewpoint in its division of the electronic structure into (1) essentially single-bonded Cu−Cu chains supported by bridging Ca atoms and (2) electrons delocalized over the Ca sublattice that can be organized into variety of resonance structures with one electron occupying each Ca4 tetrahedron, on average. In their raMO reconstructions, analogous geometrical features between Ca2Cu and CaCu are mirrored in very similar bonding schemes, highlighting a modular electronic structure that makes sense of the small energy costs to rearranging CaCu into FeB- or CrB-type variants. Implicit in these results is a chemical story related to electronegativity: the Cu sublattices in these structures do not have a sufficient number of low-lying orbitals to completely oxidize the Ca atoms to Ca2+. In particular, the raMO analysis shows only minor involvement of bonding interactions between Cu 4p orbitals, despite the availability of Ca bridging

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AUTHOR INFORMATION

Corresponding Authors

*E-mail: [email protected]. *E-mail: [email protected]. ORCID

Sang-Won Park: 0000-0002-2843-9803 Daniel C. Fredrickson: 0000-0002-3717-7008 Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS We thank Katerina Hilleke, Gordon Peterson, and Erdong Lu for insightful discussions as well as Hillary Mitchell Warden, Kendall Kamp, and Mary Hawgood for helpful comments on the presentation of this research. This work was supported by the World Research Hub Initiative (WRHI) program. We gratefully acknowledge the financial support of the Department H


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Inorganic Chemistry

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of Energy, Office of Basic Energy Sciences through Grant DESC0018938 and MEXT Element Strategy Project Initiative to form a research core.

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