Parallels in Structural Chemistry between the Molecular and Metallic

Article Cite This: Acc. Chem. Res. XXXX, XXX, XXX−XXX

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Parallels in Structural Chemistry between the Molecular and Metallic Realms Revealed by Complex Intermetallic Phases Published as part of the Accounts of Chemical Research special issue “Advancing Chemistry through Intermetallic Compounds”. Daniel C. Fredrickson* Department of Chemistry, University of WisconsinMadison, 1101 University Avenue, Madison, Wisconsin 53706, United States CONSPECTUS: 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 Fe8Al17.6Si7.4 compound obtained in these syntheses was traced to a fragmentation of the fluorite-type structure (as adopted by NiSi2), driven by the maintenance of 18electron 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 Tsaitype 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.

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which a face-centered ∼30 Å cubic unit cell is populated with more than a thousand atoms. Intense synthetic and structural investigations have since revealed intermetallic phases to be a realm of astounding diversity,3 including numerous examples of structurally complex phases with giant unit cells or long-rangeordered patterns that are incompatible with a 3D unit cell: quasicrystals and incommensurately modulated phases.4,5 Even as this structural diversity was first emerging, links to molecular systems were apparent. Early on, Hume-Rothery recognized three factors determining the stability range of an intermetallic phase: electron concentration (number of valence

INTRODUCTION

Much like the white powders of organic chemistry, the outer appearance of intermetallic phases, often gray solids with a metallic sheen, do little justice to the unique structural features that define the compounds at the atomic scale or their unique properties. This theme is vividly illustrated by the early work of Linus Pauling, who collected X-ray diffraction patterns of crystals of NaCd2 in the 1920s.1 Crystals of this phase several millimeters in size are readily obtained from Na−Cd melts, and aside from their remarkable facets, the crystals appear to be made of a typical metal (see the cover of this issue). However, their X-ray diffraction patterns are of such complexity that it took nearly 40 years before Pauling’s associate Sten Samson was able to propose the first structural model for NaCd2,2 in © XXXX American Chemical Society

Received: December 19, 2017

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DOI: 10.1021/acs.accounts.7b00625 Acc. Chem. Res. XXXX, XXX, XXX−XXX

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Accounts of Chemical Research

Figure 1. In pursuit of intermetallic host−guest chemistry. (a) The Fe25Al78Si20 structure. (b) The Fe8Al17.6Si7.4 structure viewed in terms of fragments of the fluorite-type phase NiSi2. (c) The preservation of 18-electron configurations on going from NiSi2 to Fe8Al17.6Si7.4. Adapted with permission from ref 18. Copyright 2012 American Chemical Society.

fragmentation.19−21 A side-by-side comparison between the electronic structures of these complex structures and their simpler parent structures has the potential to guide us to productive ways of analyzing the electronic structure of these compounds. In this Account, I will tell two stories about how an iterative approach along these lines combining theory and experiment has provided schemes for the ways two of HumeRothery’s factors shape intermetallic structures: the electron concentration and atomic size factors.

electrons/atom), atomic size ratio, and electrochemical potentials,6 with strong parallels to molecular chemistry’s electron counting rules, steric effects, and electronegativity differences, respectively. The Zintl concept also framed the bonding of a limiting case of intermetallic compounds, those with sufficiently large electronegativity differences, in terms of ionic electronic configurations and covalent bonds to create filled octets.7 Semiempirical tight-binding approaches (such as the extended Hückel method8) and density functional theory9 provided the opportunity to base such bonding schemes on electronic structure calculations, with a range of theoretical tools now being available: crystal orbital overlap and Hamilton populations,10,11 the electron localization and indicators,12,13 the quantum theory of atoms in molecules,14 Wannier functions,15,16 and the distinction between directional and nondirectional contributions to the energy.17 The level of detail provided by modern theoretical calculations, however, has made it difficult to use them to build further principles for the prediction of new structural phenomena. In our research group, we have found that complex intermetallic compounds offer windows into which of these numerous details matter most for controlling intermetallic structures. In particular, many complicated structures can be described as variations on simpler structure types, which are derived through such modifications as interface insertion, and

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THE ELECTRON CONCENTRATION FACTOR The first story begins with inspiration provided by the Fe25Al78Si20 structure,22 a compound we found in the structural literature when searching for examples of complexity. While visualizing this structure, we saw that it could be understood in a way suggestive of new compounds: here, an Al−Si primitive cubic network, which serves as a host to Fe−Al units derived from the face-centered-cubic structure (Figure 1a), which we imagined could be distributed in different ways. After noting that the Fe−Al−Si phase diagram contains several phases within unknown structures, we set out to see if we could synthesize other variations on this host−guest theme. Through these efforts, we obtained a new compound, Fe8Al17Si8.18 In its structure, cubes of Si/Al atoms similar to those in Fe25Al78Si20 were indeed found, but instead of forming B


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out this analysis is then to use the mathematical framework of quantum mechanics but reverse the basis set approximation: we used the simple functions as the eigenfunctions of a model Hamiltonian operator, and the occupied wave functions of the full system as a basis set. We implemented this approach in the reversed approximation Molecular Orbital (raMO) method.24 In Figures 2 and 3, we show two simple examples of the raMO process: the simple 18-electron carbonyl compound

a full grid, they appeared as lower-dimensional patterns: columns of edge-sharing cubes, whose faces are capped with Fe atoms (Figure 1b). These columns run parallel to each other in a rod-like packing, with additional Si/Al atoms filling in the gaps. This arrangement gained greater significance when we discerned its structural relationship to a much simpler phase, NiSi2. In NiSi2, the same columns of transition metal (T) atomcapped main group (E) atom cubes can be found, but here they are fully integrated into a fluorite-type lattice: the Si atoms lie at the corners of a primitive cubic network with the Ni atoms occupying every other cubic hole. Following the compositional and structural relationship between NiSi2 and Fe8Al17Si8, we can see the outlines of a story begin to take shape: if we begin with NiSi2 and replace Ni with Fe and some of the Si with Al, the structure responds through fragmentation of the fluorite type into columns. We then considered why these substitutions should drive such a transition. We noticed in DFT-calculations that the valence electron density distribution around the Si atoms indicated sp3 hybridization, such that each Ni atom was surrounded by a cubic field of σ-ligands. Using DFT-calibrated Hückel calculations, we obtained a local MO diagram for the Ni atoms and saw that a gap opened at 18-electrons per Ni atom, building an isolobal analogy with molecular transition metal complexes, in which an electron pair is associated with each of the 9 s, p, and d valence atomic orbitals of the Ni centers. This gap coincided with both the Fermi energy (EF) and a shallow pseudogap in the density of states (DOS) for the full compound, as well as the 18 valence electrons per formula unit given by the stoichiometry of NiSi2. These conclusions mirrored closely those obtained previously for the related halfHeusler phases.23 It then became clear how replacing Ni with Fe and Si with Al would cause problems for the stability of the fluorite-type structure. Each of these substitutions lowers the electron count, pulling it away from 18. The clearest way for the compound to maintain the electron count during these substitutions is to incorporate more Si/Al atoms relative to NiSi2 ’s 1:2 stoichiometry. The observed fragmentation of the fluorite type in Fe8Al17Si8 would thus appear to provide a path to the insertion of these additional Si/Al atoms. Indeed, the stoichiometry of Fe8Al17Si8 corresponds to an electron count of 18.3/Fe atom, and calculations of its DOS and local MO diagrams for its Fe atoms confirms the role of 18 electron configurations in its stability. The structure of Fe8Al17Si8 thus illustrates the lengths to which a compound might go to preserve 18-electron configurations on its T atoms. These results suggested to us a broader principle. The notion that a transition metal atom would have a filled shell at 18 valence electrons is based on its number of valence atomic orbitals, not its particular structural context. In thinking about how we could more directly test our isolobal analogies between intermetallic compounds and molecular complexes, we realized that the analysis we envisioned was in fact the opposite of a typical quantum mechanical calculation. Usually, we desire the true wave function of a system, and as the exact solution is too difficult, we calculate an approximant using a basis set of simple functions. In our case, we have wave functions we deem reliable (either from DFT, or a DFT-calibrated Hückel model) and are asking how well they can be recombined to make simple, chemically meaningful functions, such as functions centered by a T atom’s valence s, p, and d orbitals. A simple way of carrying

Figure 2. Reversed approximation Molecular Orbital (raMO) approach, illustrated with the reconstruction of the Cr 18-electron configuration in Cr(CO)6 from the molecule’s occupied MOs.

Cr(CO)6 and the intermetallic CrGa4 with 18 electrons per formula unit. In both cases, a simple hypothesis for the bonding is that the Cr atoms have 18-electron configurations. To test this idea, we build a model Hamiltonian operator in which Cr s, p, and d valence orbitals of one such atom are the eigenstates (Figure 2). The occupied molecular or crystal orbitals are then used as a basis set for the construction of a Hamiltonian matrix, the diagonalization of which returns the best reproductions of the original target functions possible from the basis set (plus additional states that are orthogonal to the target functions). In both cases, these functions are fairly well-localized to the first coordination environment of the Cr atom, but in-phase combinations can sometimes be seen on the neighboring C


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and then examining the eigenfunctions generated that are orthogonal to the Ir-centered functions, a set we call remainder raMOs. The resulting functions showed Sn−Sn bonding interactions along specific contacts. The role of this Sn−Sn bonding is then testable by using the remainder raMOs as a basis set for a second round of raMO analysis with Sn−Sn σ and σ* MOs as the model eigenstates, which reveals the presence of localized Sn−Sn bonds at these contacts. Populating these Sn−Sn bonds requires 1.33 electrons/Ir atom, accounting for the remaining electrons in the structure. The nearly electron-precise nature of the phase is verified by the EF lying near a pseudogap. A key feature in this electron counting scheme is the assignment of shared electron pairs to the Ir−Ir dimers. We should note, however, that these are not Ir−Ir bonds in the classic sense of the term. The experimental Ir−Ir distance here is 2.94 Å,25 longer than the sum of the metallic radii (2.71 Å). A closer look at the raMO-derived function for the shared electron pair gives a clearer picture of the interaction (Figure 5): the Ir lobes directed along the Ir−Ir contact exhibit substantial stabilization by contributions from bridging Sn atoms. The result is a multicenter function that shows significant Ir−Sn bonding but shares the shape and symmetry properties of a classical Ir−Ir σ bond, permitting us to formally count the electron pair as shared directly between the Ir atoms. For this reason, we refer to this interaction as an Ir−Ir isolobal bond. With the recognition that such isolobal bonds allow electron pairs to be shared between T atoms to complete 18 electron configurations, we were able to build analogous schemes for a much wider range of T-E intermetallic structures. The overall results are summarized in the 18 − n rule: the T atoms in a T-E intermetallic compound will require 18 − n electrons to achieve a closed shell, where n is the number of isolobal bonds to other T atoms in which it participates. With this approach, we have been able to derive bonding schemes for more than 34 structure types, including the Nowotny chimney ladders, an intriguing family of compounds in which T helices serve as the host for a separate set of E helices.26,27 Almost simultaneously, Kitahara and co-workers began drawing similar schemes based on the Wannier analysis of quasicrystal approximants, hinting at the broad applicability of molecular-like bonding schemes even in intermetallic compounds traditionally understood from the Mott−Jones approach.28,29 It has been exciting to test the limits of the 18 − n bonding scheme. For example, we have seen that the need for Co−Co isolobal bonds provides the driving force for a low-temperature superstructure of the CeNiSi2 type in GdCoSi2,30 which can be reversed through a high-temperature phase transition. In another investigation, we demonstrated that the BCC structures of group 6 transitions metals are understandable through 18 − n resonance structures.31

Figure 3. Application of the raMO approach to the reconstruction of 18-electron configurations in CrGa4. (a) The crystal structure. (b) The target orbitals and their raMO reconstructions. (c) Alignment of the raMOs with the DOS distribution for the full CrGa4 structure. Adapted with permission from ref 24. Copyright 2014 American Chemical Society.

atoms, reflecting that in the occupied MOs these Cr atomic orbitals often occur as parts of bonding orbitals. For Cr(CO)6, the raMO analysis shows clearly the separation between Crand ligand-based functions, as well as the key back-donation from the Cr center to the CO π* orbital. For CrGa4 (Figure 3), the raMO reconstructions of the Cr atomic orbitals highlight the filled 18 electron configurations on the Cr centers, accounting for the deep pseudogap at the EF. The raMO method can also be used to breakdown a more complicated electronic structures in a hypothesis-driven fashion, as illustrated in Figure 4 for Ir3Sn7. Here, the Ir atoms occur as dimers, so we begin by using the MOs of the Ir2 dumbbells as targets for the raMO analysis. When doing so, we obtain excellent reproductions of the first 17 MOs, while the recreation of the highest energy σ* level fails. This corresponds to there being a net bonding interaction between the Ir atoms, with an electron pair shared between them. The sharing of this electron pair allows for 18-electron configurations to be achieved by the Ir atoms with only 17 electrons of their own. So far we have accounted for 17 of the 18.33 electrons/Ir atom in Ir3Sn7. The placement of the remaining electrons in the structure can be determined by first carrying out a raMO analysis to reproduce using the full set of MOs for all of the Ir2 dimers in the structure (except for the unoccupied σ* MOs)

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THE ATOMIC SIZE FACTOR In the previous story, we saw how noticing a connection between a simple structure (NiSi2) and a more complex one (Fe8Al17.4Si7.6) opened a path to a broadly applicable bonding scheme for T-E intermetallic compounds. Now, let us consider a case where a complex structure holds the key to analyzing another chemical factor: atomic size. While gleaning examples of structural diversity from the structural literature, we encountered the structure of Ca2Ag7,32 in whose structure one can perceive slabs of the common CaCu5 type (Figure 6a). D


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Figure 4. Dissection of the electronic structure of Ir3Sn7 using the raMO method. Adapted with permission from ref 24. Copyright 2014 American Chemical Society.

Figure 5. Ir−Ir isolobal bond and Sn−Sn σ bond in Ir3Sn7 derived from raMO analysis. Adapted with permission from ref 24. Copyright 2014 American Chemical Society.

In thinking about the origins of this structure, we saw that no CaCu5-type phase actually exists in the Ca−Ag system, but that one does occur in the Sr−Ag system. In other words, if we begin with the experimentally synthesizable CaCu5-type SrAg5 phase and replace the Sr with Ca, the insertion of interfaces becomes necessary to maintain the stability of CaCu5-type features. This observation immediately suggests a role for atomic size in the structural transition. Sr and Ca are both alkaline earth metals, sharing the same electropositive character and tendency to become 2+ cations. Their major difference is in their atomic sizes, with the metallic radius of Sr being moderately larger than that of Ca (2.15 vs 1.97 Å). It would seem then the replacement of a smaller atom on the Sr sites of SrAg5 leads to a stress that is severe enough that a structural response becomes necessary. After seeing this effect, we realized that we faced a difficult challenge: how can one test such a hypothesis with theoretical calculations? Atomic size is an inherently empirical concept, and as such it is difficult to extract the energetic costs of one atom encroaching upon the space of another atom. A possible solution to this problem began to take shape when we started to consider the idea of local forces acting between atoms: intermetallic compounds generally exhibit dense atomic

Figure 6. Chemical Pressure (CP) driven transition between the CaCu5- and Yb2Ag7-type structures. (a) Structural relationship between the CaCu5 type and Ca2Ag7 (Yb2Ag7 type). (b) CP relief upon going from CaCu5-type CaAg5 (hypothetical) to the observed Ca2Ag7 structure. Adapted with permission from ref 34. Copyright 2014 American Chemical Society.

packing, where coordination numbers of higher than 10 are common. Along with these high coordination numbers come E


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through layer deletions, allowing the Ca atom to strengthen its interactions with Ag atoms on the other side. At the same time, a shifting of the CaCu5-type slabs resulting from the layer deletion relative to each other places a pair of Ag atoms directly on the opposite side of the Ca atom, supporting its motion. The net result is that 6 overly long Ca−Ag contacts are replaced with 2 short ones, and the coordination number by Ag goes from 18 to 14. A look at the CP scheme of the observed Ca2Ag7 structure confirms that this environment provides a much better fit. This contraction of the Ca coordination environment in response to negative pressures reminded us of similar trends in pressure-induced phase transitions. In this case, however, the pressures involved were all induced by local packing constraints, rather than any external force. As such, we can refer to this effect as chemical pressure (CP)-induced transition. As our methods for calculating and analyzing CP maps have improved, we have been able to interpret an increasing number of complex structures as the result of CP release. Intriguingly, we have seen that very different structural mechanisms can be followed for relieving similar CP schemes. For example, the CaCu5 type can be stabilized through the interfaces to form not only the Yb2Ag7 type (as observed for Ca2Ag7) but also the more complicated Gd14Ag51 type. Analogous CP features also underlie the substitution of cations with transition metal dumbbells to create the Th2Ni17 and Th2Zn17 structure types (Figure 7a).35 An even more striking avenue to CP relief is the contraction of hexagonal polyhedra to pentagonal ones, inducing curvature in the structure and the formation of local

correlations between interatomic distances and challenges in independently optimizing interatomic interactions. When the formation of a bond between two atoms is impeded by unbonded contacts elsewhere, local stresses can arise. If we could analyze those local stresses using quantum mechanical calculations, we would have a way of evaluating the impact atomic sizes have on such phenomena as the interfaces in Ca2Ag7. Drawing on the notions of a quantum mechanical stress density, we then formulated chemical pressure (CP) analysis.33 Here, the output of DFT calculations, such as the kinetic energy densities, electron densities, and components of the Kohn−Sham potential, is used to construct grids of energy values spread throughout a structure, the sums over which give the correct total energies. The creation of such grids for expanded and contracted versions of a structure allows for the calculation of a pressure map through the relation P = −∂E/∂V. The result are maps of a pressure distributed throughout a crystal structure (the average over which leads to the macroscopic pressure, usually 0 GPa for a geometrically optimized structure), which can then be analyzed in terms of interatomic interactions by projecting contributions assigned to pairs of atoms onto spherical harmonics. The result of applying this approach to Ca2Ag7 is shown in Figure 6, where the local pressures experienced by each atom are represented by radial plots: the radial extent of a lobe protruding from an atom is proportional to the sum of the pressure contributions experienced by the atom along that direction. The color of the lobe gives the sign of the pressure. Black indicates negative pressure, meaning that a contraction of the structure is favorable locally; that is, these interatomic contacts would prefer to be shorter. Lobes denoting positive pressures, indicating that expansion of the structure is favored, i.e., the atoms are too close together. We begin in Figure 6b with a hypothetical structure of a CaCu5-type CaAg5 phase: the Ca atom occurs here in a large 18-coordinate hexagonal polyhedron defined by Ag atoms. Within this environment, the Ca atom exhibits large negative pressure lobes pointing up and down, which are met with corresponding lobes on the hexagons of Ag atoms at the top and bottom of the polyhedron. These Ca−Ag contacts are overly long and call for the contraction of the structure. Such contraction is prevented, however, by positive pressures elsewhere: note the prevalence of white lobes between the Ag atoms surrounding the Ca atom. These contacts are already too short and would be further strained by the contraction of the structure. The overall result is that the Ca atom appears to be too small for the coordination environment provided by a CaCu5-type CaAg5 phase, which would be consistent with the observation that replacing the Sr in SrAg5 with Ca should lead to structural instability. The CP approach also provides a rationalization for the features of the observed structure of Ca2Ag7. The Ca atoms in the hypothetical CaAg5 structure are teetering between two sets of contacts that would like to contract: those to the Ag atoms in the hexagons above and below. Moving the Ca atom either up or down would solve half of the problems it experiences, while making the other half worse; as it moves toward one set of atoms, it is naturally weakening its interactions on the other side. Adopting a superstructure provides the opportunity to remove this competition. In the observed Ca2Ag7 structure, the Ag atoms on one side of each Ca atom are removed

Figure 7. Mechanisms of CP relief. (a) Y/Co2 substitution in Y2Co17. (b) Local icosahedral order in the Tsai-type quasicrystal approximant CaCd6. Adapted with permission from refs 35 and 36. Copyright 2017 and 2013, respectively, American Chemical Society. F


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Accounts of Chemical Research icosahedral order in the quasicrystal approximant CaCd6 (Figure 7b).36 As we gazed at CP schemes such as that shown for the hypothetical CaCu5-type CaAg5 in Figure 6, we became aware that these images are suggestive of more than just the tendency of a compound to undergo superstructure formation. For example, the negative CP features pointing up and down from the Ca atom suggest that Ca atom has much more room to move along this axis than in the perpendicular directions, implying that its vibrations should be quite anisotropic. We then began to trace more directly the relationships between the CP scheme for a solid state structure and the features of that material’s phonon band structure.37 We first noted that the anharmonicity of interatomic interactions makes it so that motions along positive pressures should encounter significantly more resistance that those along negative CPs, which then allowed us to compare the expectations of our CPs schemes with phonon band structures. For the case of one model system, CaPd2, these comparisons highlighted to us the need for a consideration of the ionicity of the atoms in considering the degree to which each atom influences the space around it (Figure 8). Once we considered the Ca as somewhat cationic, with the Pd atoms being anionic, we obtained a CP scheme that aligned closely with some of the more peculiar features of the phonon band structure. In particular, both the lowest and highest frequency vibrational modes correspond to motions of the Pd atoms. This was readily interpretable in terms of the quadrupolar distribution of CP around these atoms (Figure 8), in which positive pressures are oriented perpendicularly to negative CPs, to create a shape similar to a dz2 orbital. Motions along the negative CPs not only shorten some overly long contacts but also lengthen contacts experiencing positive CP. These motions are thus expected to be extremely soft. Motions along the positive CPs, however, shorten contacts that are already too short, while lengthening contacts that are too long. These are should lie at high frequencies. This correlation between CP quadrupoles and anisotropic vibrations could then be used to explain how the lowest optical modes of the superconductor Nb3Ge and the CaCu5-type structure reported for CaPd5 emerge from their atomic arrangements. In this way, the relationship between CP schemes and phonon band structures provides a way to both validate our CP schemes and derive structure−properties relationships for atomic vibrations. The CP distributions we obtained for intermetallic phases also hinted to us the possibilities of new high-pressure chemistry. The appearance of large negative CP features pitted against positive CPs elsewhere strongly indicates that a structure is filling space with poor efficiency and that it should become less favorable relative to more densely packed variants at increased pressures.38 We thus carried out calculations of the pressures at which structures reported to adopt the CaCu5 type would become unfavorable relative to a structure with denser packing (the Yb2Ag7 type, as adopted by Ca2Ag7) with the exclusion of some transition metal atoms. Surprisingly, we found that for the CaPd5, the transition was predicted to be favorable even at atmospheric pressure, suggesting that it should be realizable at very modest pressures. We thus synthesized CaPd5 as the starting material for a high pressure experiment. Our synthesis of CaPd5 yielded beautiful crystals with a hexagonal plate-like morphology; however X-ray diffraction experiments quickly demonstrated that they did not belong to

Figure 8. Relationship between CP schemes and soft phonon modes. (a) The phonon density of states of CaPd2 interpreted in terms of its CP scheme. (b) CP quadrupoles as signature of soft atomic motions. (c) Interpretation of the phonon modes of CaPd2 in terms of its CP scheme. Adapted with permission from ref 37. Copyright 2016 American Chemical Society.

the simple CaCu5-type. The c-axis appeared as tripled, and rows of satellites emanated from the strongest reflections along three different directions in the a*−b*-plane. The selection of smaller crystals revealed that the hexagonal symmetry of the parent structure was lowered to orthorhombic by a modulation (Figure 9a) and that the pseudo-symmetry of the basic cell leads to facile twinning via 60° rotations around c. After carrying out the (3 + 1)D structure solution and refinement of phase, a structural picture emerged that was in close agreement with the expectation of CP analysis.38 The G


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Figure 9. CP-driven incommensurability in CaPd5+x (x = 0.22). (a) Diffraction pattern of a CaPd5+x crystal illustrating satellite reflections. (b) The average structure of CaPd5+x. (c) Comparison of CP schemes for CaCu5-type CaPd5 and commensurate model of observed structure. (d) Correlation of residual CPs in commensurate model and atomic positions in the modulated structure. Adapted with permission from ref 38. Copyright 2016 American Chemical Society.

one seemingly obscure structure, Ca2Ag7, prompted the development of the CP approach to analyzing atomic size effects, allowing us to visualize how atomic packing issues can give rise to complex structures, incommensurability, and soft phonon modes. We are continuing to test and extend the raMO and CP approaches to elucidating an ever broader range of structural phenomena in the metallic state. For example, investigating how the 18 − n rule and other bonding schemes, such as the Zintl and nearly free electron models, intersect (or compete?) at the borders of their applicability ranges could lead to a more comprehensive view of the electron concentration factor. Similarly, work is underway to improve the robustness of the CP analysis so that it can be carried out across solid state structures simply and generally. However, it is already clear that a single electron counting rule or CP scheme can give rise to a variety of different structures. An emerging question is which of these numerous paths to closed electronic shells or more efficient atomic packing will be chosen by any particular compound. One hint here is provided by Hume-Rothery’s notion that in many phases the observed structure is not controlled by just one of the factors he identified (atomic size, electron concentration, electrochemical potentials) but instead emerges through the combination of these factors.6 We are currently examining how the CP and raMO analyses can be combined to explore the interactions between these two factors, while remaining on the lookout for model systems where the third and less explored factor, electrochemical potential, plays a leading role.

CaCu5-type parent structure had become divided into slabs similar to those found in the Ca2Ag7 type, with layers of Pd atoms running between the slabs, placing pairs or triangles of Pd atoms in place of the hexagons of overly distant ones above or below the Ca atoms. In a CP calculation on a commensurate model of this arrangement, the CP relief on the Ca is clearly evident (Figure 9c), as is the presence of CP quadrupoles on the Pd atoms in the interface layer. These quadrupoles point toward the favorability of the contraction of the Pd layer relative to the rest of the structure, in line with observed incommensurability (Figure 9d). In addition, the positive CP components of the quadrupole highlight the need for the Pd atoms to undulate during this contraction to avoid overly close contacts to atoms in the CaCu5-type slabs above and below. In this way, the detailed features of this incommensurate structure become clearly interpretable, illustrating how CP quadrupoles can be indicators of not only soft phonon modes but the potential for structural modulations.

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FUTURE DIRECTIONS In this Account, we have explored how focusing on complex intermetallic compounds can reveal driving forces at work throughout intermetallic chemistry and prompt the development of new theoretical tools for analyzing these effects. In one case, we saw how the synthesis of a complicated Fe−Al−Si phase led us to recognize that the stability of 18-electron configurations can drive structural rearrangements, and the generalization of this scheme brought us the 18 − n bonding picture and raMO analysis. In the second story line, we explored how tracing the role of atomic size in the stability of H


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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. ORCID

Daniel C. Fredrickson: 0000-0002-3717-7008 Notes

The author declares no competing financial interest. Biography Daniel C. Fredrickson’s fascination with orbitals and crystals began with his undergraduate studies at the University of Washington (B.S., 2000), where he explored the face-selective incorporation of nucleotides into KH2PO4 crystals under the guidance of Prof. Bart Kahr. This interest only grew during his Ph.D. research with Profs. Stephen Lee and Roald Hoffmann at Cornell University into the origins of the 14-electron rule of the Nowotny Chimney Ladder phases (Ph.D., 2005) and his postdoctoral research with Prof. Sven Lidin at Stockholm University (2005−2008), in which he gained his first experiences with solid state synthesis and crystallography. Since 2009, he has been a faculty member at the University of Wisconsin Madison, where his group combines theory and experiment to understand and (eventually) predict structural phenomena in intermetallic compounds.

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ACKNOWLEDGMENTS This Account draws together work done by many students and collaborators that I have had the pleasure to work with in my group at UW-Madison, including (in alphabetical order) Veronica M. Berns, Joshua Engelkemier, Rie T. Fredrickson, Yiming Guo, Amelia E. Hadler, Katerina Hilleke, Brandon J. Kilduff, Timothy E. Stacey, Anastasiya I. Vinokur, and Vincent J. Yannello. I also thank many other students whose projects are less-aligned with the themes presented here. I gratefully acknowledge the financial support of the National Science Foundation through Grant DMR-1508496.

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REFERENCES

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Accounts of Chemical Research (33) Berns, V. M.; Engelkemier, J.; Guo, Y.; Kilduff, B. J.; Fredrickson, D. C. Progress in Visualizing Atomic Size Effects with DFT-Chemical Pressure Analysis: From Isolated Atoms to Trends in AB5 Intermetallics. J. Chem. Theory Comput. 2014, 10, 3380−3392. (34) Berns, V. M.; Fredrickson, D. C. Structural Plasticity: How Intermetallics Deform Themselves in Response to Chemical Pressure, and the Complex Structures That Result. Inorg. Chem. 2014, 53, 10762−10771. (35) Hilleke, K. P.; Fredrickson, R. T.; Vinokur, A. I.; Fredrickson, D. C. Substitution Patterns Understood through Chemical Pressure Analysis: Atom/Dumbbell and Ru/Co Ordering in Derivatives of YCo5. Cryst. Growth Des. 2017, 17, 1610−1619. (36) Berns, V. M.; Fredrickson, D. C. Problem Solving with Pentagons: Tsai-Type Quasicrystal as a Structural Response to Chemical Pressure. Inorg. Chem. 2013, 52, 12875−12877. (37) Engelkemier, J.; Fredrickson, D. C. Chemical Pressure Schemes for the Prediction of Soft Phonon Modes: A Chemist’s Guide to the Vibrations of Solid State Materials. Chem. Mater. 2016, 28, 3171− 3183. (38) Kilduff, B. J.; Fredrickson, D. C. Chemical Pressure-Driven Incommensurability in CaPd5: Clues to High-Pressure Chemistry Offered by Complex Intermetallics. Inorg. Chem. 2016, 55, 6781− 6793.

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Parallels in Structural Chemistry between the Molecular and Metallic

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