Super Atomic Clusters: Design Rules and Potential for Building Blocks

Review Cite This: Chem. Rev. XXXX, XXX, XXX−XXX

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Super Atomic Clusters: Design Rules and Potential for Building Blocks of Materials Puru Jena*,† and Qiang Sun‡,§,† †

Physics Department, Virginia Commonwealth University, Richmond, Virginia 23284-2000, United States Department of Materials Science and Engineering, and §Center for Applied Physics and Technology, Peking University, Beijing 100871, China

‡

ABSTRACT: Atomic clusters, consisting of a few to a few thousand atoms, have emerged over the past 40 years as the ultimate nanoparticles, whose structure and properties can be controlled one atom at a time. One of the early motivations in studying clusters was to understand how the properties of matter evolve as a function of size, shape, and composition. Over the past few decades, more than 200 000 papers have been published in this field. These studies have not only led to a considerable understanding of this evolution from clusters to crystals, but also have revealed many unusual size-specific properties that make cluster science an interdisciplinary field on its own, bridging physics, chemistry, materials science, biology, and medicine. More importantly, the possibility of creating a new class of materials, composed of clusters instead of atoms as building blocks, has fueled the hope that one can synthesize materials from the bottom-up with unique and tailored properties. This Review focuses on the properties that set clusters apart from their corresponding bulk. Furthermore, this Review describes how different electron-counting rules can lead to the design of stable clusters, mimicking the chemistry of atoms. We highlight the potential of these “superatoms” as building blocks of clusterassembled materials. Specifically, we emphasize cluster-inspired materials for energy applications. The concluding section includes a summary of the salient features of clusters, potential challenges that remain, and an outlook for the future of cluster science.

CONTENTS 1. Introduction 2. Unique Properties of Clusters 2.1. Stability and Magic Numbers 2.1.1. Noble Gas Clusters 2.1.2. Simple Metal Clusters 2.1.3. Transition Metal Clusters 2.1.4. Covalently Bonded Clusters 2.1.5. Ionically Bonded Clusters 2.1.6. Metal−Carbide Clusters 2.2. Geometries 2.2.1. Noble Gas Clusters 2.2.2. Simple and Noble Metal Clusters 2.2.3. Transition Metal Clusters 2.2.4. Covalently Bonded Clusters 2.2.5. Clusters with Cage Structures 2.3. Electronic Structure 2.3.1. Simple and Transition Metal Clusters 2.3.2. Covalently Bonded Clusters 2.4. Magnetic Properties 2.4.1. Simple Metal Clusters 2.4.2. Clusters of Ferromagnetic Elements 2.4.3. Ferromagnetism in Clusters of Otherwise Nonmagnetic Transition Metal Elements

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2.4.4. Magnetic Coupling in Clusters of Antiferromagnetic Elements 2.4.5. Magnetic Transition Induced by Heteroatoms 2.5. Multiply Charged Clusters: Stability and Fragmentation 2.5.1. Interaction between Two Positively Charged Transition Metal Atoms 2.5.2. Interaction between Two Negatively Charged Clusters 2.5.3. Symmetric versus Asymmetric Fission of Doubly Charged Metal Clusters 3. Super Atomic Clusters and the Role of ElectronCounting Rules 3.1. Jellium Shell Closure Rule 3.1.1. Na Clusters 3.1.2. Mg Clusters 3.1.3. Al13− Cluster 3.1.4. Au20 Cluster 3.1.5. Ligated Clusters 3.1.6. Metal Doped Si Clusters 3.1.7. Real-Space Representation of Shell Structure in Jellium-like Clusters

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Received: August 31, 2017

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DOI: 10.1021/acs.chemrev.7b00524 Chem. Rev. XXXX, XXX, XXX−XXX

Chemical Reviews 3.2. 3.3. 3.4. 3.5.

Octet Shell Closure Rule 18-Electron Shell Closure Rule Wade−Mingos Shell Closure Rule Zintl-like Ions beyond the Wade−Mingos Rule 3.6. Huckel’s Rule of Aromaticity 3.7. Multiple Electron-Counting Rules 3.8. Magnetic Superatoms 3.8.1. VLi8 3.8.2. VNa8 and VCs8 3.8.3. FeMg8, FeCa8, and TcMg8 3.8.4. Ligated Magnetic Superatoms 4. Cluster Assembled Materials (CAMs) 4.1. Fullerenes-Based CAMs 4.1.1. Self-Assembly of FCC-C60 and Related Structures 4.1.2. Assembling C60 through Other ElectronRich Clusters 4.1.3. C60-Based Dendritic Assemblies 4.1.4. Protein-Directed C60 Assembly 4.2. Silicon-Based CAMs 4.2.1. Assembly of Metal Encapsulated Si12 Clusters 4.2.2. Assembly of Metal Encapsulated Si16 Clusters 4.3. ZnO-Based CAMs 4.4. Al-Based CAMs 4.5. Zintl Cluster-Based CAMs 4.5.1. As73−- and As113−-Based Assemblies 4.5.2. Icosahedral Matryoshka Zintl ClusterBased Assemblies 4.5.3. [Ti@Au12]2−-Based Assemblies 4.6. Ligated Cluster-Based Assemblies 4.7. Chevrel-like Cluster-Based Assemblies 4.8. Assembling through Support or Confinement 5. Clusters for Energy Applications 5.1. Clusters for Hydrogen Storage 5.1.1. Hydrogen Interaction Mechanism 5.1.2. Irreversibility of Hydrogen Storage in Light-Metal Borohydrides 5.1.3. Role of Catalysts in the Dehydrogenation of NaAlH4 5.1.4. Increasing Safety of Complex MetalHydrides 5.2. Clusters for Energy Harvesting 5.3. Clusters for Energy Efficiency 5.4. Clusters for Energy Storage 5.4.1. Halogen-Free Electrolytes in Li- and Other Metal-Ion Batteries 5.4.2. Superhalogen-Based Li-Ion Superionic Conductors 5.4.3. Cluster-Inspired Materials for High Capacity Anodes for Li-Ion Batteries 5.5. Conversion and Cracking of CO2 5.5.1. Cu Atom 5.5.2. Cu4 Cluster 5.5.3. Cu8 Cluster 5.5.4. Cu55 Cluster 5.5.5. Mo6S8 Cluster 5.5.6. Superalkalis for CO2 Cracking 5.6. Clusters for Multiferroics 6. Summary and Outlook

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1. INTRODUCTION Unusual properties of matter emerge when its size is reduced to a few nanometers. Because of the large surface to volume ratio and quantum confinement, the structure and properties of these nanoparticles differ substantially from those of their bulk counterparts. The ability to tailor the properties of materials, by changing the size, shape, and composition, offers an unprecedented opportunity for science and technology to uncover new phenomena and synthesize novel materials. Consequently, nanoscience and nanotechnology has become a major field of research during the past few decades. Much of the progress in these fields is due to the advancement of new experimental and theoretical techniques. These advances enable researchers to synthesize and probe the properties of matter at the atomic scale. We note that many biological functions occur at the nanoscale, and so nanotechnology is not new to nature. For example, for billions of years, magnetospirilium magneticum bacteria have used a built-in nanomagnetic needle1 to navigate. Similarly, man has made use of nanotechnology since the medieval ages; for example, the brilliant colors of the stained glass, used in cathedrals in Europe, originate from the scattering of light from embedded metal nanoparticles.2 Despite the vast range of experimental techniques, starting from bottom-up to top-down approaches that are now available, the synthesis of nanoparticles with atomic precision has not been possible. Furthermore, what constitutes a nanoparticle remains imprecise, as its range is generally regarded3 to be between 1 and 100 nm. Thus, nanoparticles do not yet provide a platform to probe how properties of matter evolve one atom at a time. Atomic clusters bridge this gap. Atomic clusters are aggregates, consisting of a few to a few thousand atoms,4 which form when a hot plume of atoms cools through collisions with rare gas atoms in near vacuum conditions. Clusters can be composed of homo- or heteroatom species, and, once produced, they can be mass isolated and studied individually. Thus, atomic clusters are the ultimate nanoparticles where every atom and every electron count. Although clusters have been of interest since the 1930s,5 the majority of the advancements within this field have occurred over the past 40 years. In the field’s adolescent years,6 experimental studies were confined to low melting point metals, rare gas species, and molecular clusters. On the theoretical side, first-principles studies were mostly confined to very small clusters,7 consisting of only a few atoms; larger clusters were studied using phenomenological models.8 The advent of new synthesis techniques in the 1980s, such as laser vaporization9,10 and pulsed arc,11 enabled researchers to produce clusters of transition and refractory metals, as well as those of semiconductors. Use of two or more targets further made the study of heteroatomic clusters, with varying size and composition, possible.12 Inevitably, the field developed rapidly. High-speed computers, along with efficient computer codes based on density functional theory, permitted studies of clusters, consisting of hundreds of atoms.13 Atomic clusters

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clusters both in the title and in the topic, shows that during the past 40 years nearly 200 000 papers have appeared in this field, with about 10 000 papers appearing per year, in the past decade. Through extensive research, many unusual properties of clusters have been discovered, and the potential of clusterassembled materials has been vastly explored. Additionally, researchers have not only focused on clusters in the gas phase, but also on clusters supported on substrates, embedded in matrixes, and decorated with ligands. Clusters bridge the disciplines of physics, chemistry, biology, materials science, and medicine. Many books,17−28 conference proceedings,14,29−36 and review articles37−146 have been published, dealing with various aspects of cluster science. It is impossible to discuss all of the interesting properties of clusters, such as melting,147−149 that have been discovered. We apologize to the reader if topics of his/her interest are not covered and hope that the references provided here will help to overcome some of the shortcomings. The objective of this Review is not to build upon these reviews; rather, the objective is to focus on some of the unique properties of clusters that have emerged, and use the knowledge gained from these studies to show how cluster science can play a role in the emerging science of nanomaterials. We discuss how stable clusters can be created to mimic the chemistry of atoms and how these “superatoms” can be assembled to form a new class of cluster-assembled materials. Primarily, the focus is on the potential of clusters as the building blocks of energy materials. In section 2, we highlight some of the unique properties of clusters. First, we consider the relative stability of clusters as studied through the distribution of mass-ion intensity and their reactivity pattern. The pronounced peaks in the mass spectra are often interpreted as clusters that are more stable than their neighbors. These are usually referred to as “magic numbers”. The relationship between the magic numbers and the underlying electronic structure of Na clusters helped bring the effect of electronic shell closure150,151 into focus, much as the magic numbers in nuclei gave rise to the nuclear shell model.152 Geometries of atomic clusters, and the nature of their evolution with size, depend on their underlying electronic structure and rarely resemble the atomic arrangements in corresponding crystals. Because of the unique structure− property relationships, clusters provide a fascinating system for study. To highlight some of these unique cluster properties, we primarily focus on structural, electronic, magnetic, and optical properties. For example, it is well-known that dimensionality plays an important role in the underlying magnetic behavior of materials. Surface atoms of magnetic materials exhibit larger magnetic moments than those in the interior.153 Because clusters are low dimensional systems with a large surface to volume ratio, one can expect the magnetic moments in clusters to exceed their respective values in crystals. Clusters composed of nonmagnetic elements could also exhibit magnetic ordering.154 Both theoretical and experimental results, related to the magnetism of simple, transition, and rare-earth metals, are discussed. Similarly, the optical properties in bulk matter are governed by their energy band gap. In clusters, these gaps are usually referred to as the HOMO−LUMO (highest occupied and lowest unoccupied molecular orbitals) gaps and can be tailored,155,156 as they depend upon size, composition, and structure. Therefore, clusters provide a fertile ground to design materials with novel optical properties.155 More so, this section highlights the interaction between charged clusters. According to Coulomb’s law, like charges repel and unlike

emerged as a new phase of matter, intermediate between atoms and bulk matter. Initially, the motivation to study clusters was to understand how the structure and collective properties of matter, such as the electrical conductivity, color, and magnetism, evolve as atoms come together. Although significant progress has been made to answer some of these questions, scientists did not anticipate, in the early stages, that the properties of clusters could vary nonmonotonically and nonpredictively. Furthermore, it has been shown that not all properties evolve in the same manner. This finding gave cluster science its own identity, and so atomic clusters emerged as a new phase of matter with unprecedented opportunities. The ability to probe structure− property relationships with atomic precision enables the design and synthesis of novel materials with tailored properties, in a manner that nature had never seen. Size regimes are crucial when dealing with atomic clusters. A schematic view of how properties of matter evolve from a single atom to the bulk solid is shown in Figure 1.14 As the size of a

Figure 1. Size dependence of a cluster property χ(n) on the number n of the cluster constituents. The data are plotted versus n−β where 0 ≤ β ≤ 1. “Small” clusters reveal specific size effects, while “large” clusters are expected to exhibit for many properties a “smooth” dependence of χ(n), which converges for n →∞ to the bulk value χ(∞). Adapted with permission from ref 14. Copyright 1992 Springer.

solid is continually reduced, the properties initially vary monotonically. This is the so-called scalable regime. Below a critical size (∼100 nm), properties change from being monotonic to nonmonotonic; however, these properties are not sensitive to the addition or removal of a single atom. Nanoparticles belong to this nonscalable regime. As the particle size becomes still smaller, the properties change abruptly and nonpredictively, a stage in which even the addition of a single atom or electron may cause a drastic change. Atomic clusters, whose size falls within this range, are considered the ultimate nanoparticles. The underlying reason for this nonmonotonic behavior is quantum confinement, a regime where the electron wavelength becomes comparable to the particle size. The fact that properties of matter at this length scale are fundamentally different from their bulk behavior can be effectively used to produce materials with tailored properties. For example, novel materials may be created by assembling clusters, much as conventional crystals are created by assembling atoms.15,16 Such cluster-assembled materials, with their unique properties, can expand the scope of materials science. Over the years, considerable research has been carried out on homo- and heteroatom clusters of noble gases, simple, transition, and rare earth metals, semiconducting elements, as well as molecular species. A search in the web of science, with C


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charges attract. We give examples157 where two clusters, carrying like charges, indeed attract and form a bound state. Section 3 is devoted to a special class of clusters called “superatoms”.15,16 They can be composed of either homo- or heteroatoms; both share the common feature that they mimic the chemistry of atoms in the periodic table. This allows one to conceive of a three-dimensional periodic table, where these superatoms constitute the third dimension. Such a periodic table may usher a new era in materials science, as superatoms composed of earth-abundant elements can replace “rare” or “expensive” elements. Furthermore, different electron-counting rules that can aid in the design of these superatoms are discussed. It is hoped that this will pave the way for their experimental discovery. In section 4, we discuss a new class of materials called clusterassembled materials.16 Here, clusters, instead of atoms, serve as the building blocks. As mentioned before, clusters are produced and analyzed in the gas phase under near vacuum conditions; however, they are metastable and reactive. Once removed from this near vacuum environment, clusters react with the ambient gases and/or coalesce, and so they lose their identity. To protect the clusters from reactions and/or coalescence, they need to be coated with ligands or isolated on a substrate. Conversely, this solution of adding the ligands and substrate also affects the properties of otherwise bare clusters. Therefore, one must understand how clusters interact with their ligands and support, as well as the effects on their structure and properties. The hope is cluster-assembled materials will have unusual properties because of intracluster and intercluster interactions. Can one tailor the properties of cluster-assembled materials? Because the focus on clusters for decades has been to understand their fundamental science, one may wonder if clusters have any practical applications within the modern industry. In section 5, we focus on clusters as building blocks of energy materials. We discuss three different areas where clusters are known to have an impact. These include hydrogen storage materials,158 electrolytes and anode materials in metal ion batteries,159 and solar cells.155,156 We discuss the material challenges in these technologies and show how knowledge gained from studies of clusters in the gas phase may provide a paradigm shift in materials design and synthesis. Finally, in section 6, we outline some of the challenging problems that remain and an outlook for future research.

geometry of clusters drives their electronic, magnetic, and optical properties. We outline this structure−property relationship by focusing on a few selected clusters. Furthermore, we illustrate some unusual properties that are associated with the interaction between charged clusters; some clusters carrying like charges may attract, in defiance of the Coulomb’s law. 2.1. Stability and Magic Numbers

There are two factors that govern cluster stability: atomic packing160 and electron shell closure.150,161 Although these two factors are intertwined, they can be distinguished from one another in specific cases. In the following, we discuss the relative stability of clusters composed of noble gas atoms where atomic packing is the dominant factor, simple metals where electronic structure is the driving factor, transition metals and semiconductors where it is difficult to separate between these two factors, and ionic systems where the bulk identity emerges in very small clusters. We pay particular emphasis to magic numbers that correspond to clusters, which are significantly more stable than their neighbors. 2.1.1. Noble Gas Clusters. Clusters of noble gas atoms offer an extreme example where the magic numbers are goverened by atomic shell closure. This is because noble gas atoms, due to their closed electronic shells (ns2np6), are chemically inert. Therefore, the bonding between these atoms is weak and is governed by the van der Waal’s interaction. Clusters of noble gas atoms form close-packed structures, and the most stable clusters (i.e., magic numbers) are formed when their atomic shells are closed. One such packing is given by MacKay icosahedra. The number N of atoms, which are needed to complete successive icosahedric shells, is given by N = (10/ 3)K3 − 5K2 + (11/3)K − 1, where K is the number of shells. These correspond to 13, 55, 147, ... atoms that complete 2, 3, 4, ... shells, respectively. This is indeed what was observed by Recknagel and co-workers.162 In Figure 2, we plot the mass spectra of Xe clusters observed by these authors. Conspicuous peaks at 13, 55, 147 can be seen from this figure. 2.1.2. Simple Metal Clusters. One of the seminal experiments that brought the concept of magic numbers into

2. UNIQUE PROPERTIES OF CLUSTERS Clusters are very different from crystals in terms of their stability, structure, and properties. A considerable amount of work has been carried out in both of these areas, although it should be noted that our intention here is not to review all that has been done. Rather, we focus on specific examples where clusters exhibit unique properties. Two of the early observations of this uniqueness deal with stability and structure. Mass spectra revealed that some peaks are more pronounced than others. Clusters exhibiting conspicuous peaks in the mass spectra are referred to as magic numbers, which are different for different elements, and their origin is intimately tied to their underlying chemistry. Similarly, the equilibrium geometries of most clusters bear no resemblance to their bulk structures, and their evolution with size differs from one element to another. The size range that clusters mimic their bulk lattice structure and how this relates to the underlying electronic structure have been the subjects of investigation for many years. The unusual

Figure 2. Mass spectrum of xenon clusters. Observed magic numbers are marked in bold; brackets are used for numbers with less pronounced effects. Numbers below the curve indicate predictions or distinguish sphere packings. Adapted with permission from ref 162. Copyright 1981 American Physical Society. D


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focus is the experiment on Na clusters by Knight and coworkers.150 In Figure 3a, we show the mass spectra of Na

Figure 4. Self-consistent effective potential of jellium sphere corresponding to Na40 with the electron occupation of the energy levels. Adapted with permission from ref 140. Copyright 1993 American Physical Society.

geometry. Thus, the degenerate electronic orbitals are further split. This is referred to as the Jahn−Teller distortion. Thus, the relative stability of Na clusters can be explained purely on the basis of their electronic structure. Martin and co-workers165 studied the mass spectra of very large Na clusters containing up to 22 000 atoms, to determine if the electronic shell closure continues to account for the magic numbers, within this size range. The mass spectra of these clusters are shown in Figure 5. The authors noted that the Figure 3. Sodium cluster abundance spectrum: (a) Experimental data of Knight et al.150 (b) Second derivative of the total energy, with dashed line using Woods−Saxon potential, and solid line using the ellipsoidal shell (Clemenger−Nilsson model). Adapted with permission from ref 140. Copyright 1993 American Physical Society.

clusters obtained by these authors. One immediately notices conspicuous peaks at cluster sizes corresponding to 8, 20, 40, and 58 Na atoms, which imply that these clusters are far more abundant, and hence are likely to be more stable than their neighbors. Similar features were seen earlier in nuclear physics where nuclei with the 8, 20, 40, ... nucleons were found to be more stable than others, hence the term magic numbers.163 Their origin was explained to be due to nuclear shell closure. In analogy with nuclear physics, Knight et al.150 also termed the most stable Na clusters as magic clusters and performed the same analysis to see if the electronic shell closure could give rise to these magic numbers. They constructed a simple model where a Na cluster is modeled by a sphere of radius R, with the positive charge on each of the Na atoms distributed homogeneously over this sphere. Such a model is called the spherical “jellium” model. Next, the authors calculated the corresponding energy levels of electrons, which are ordered, as in nuclear shell structure, in increasing energies as 1S21P61D102S21F142P6. Note that the number of electrons needed to close successive electronic shells are 2, 8, 18, 20, 34, 40. In Figure 4, we show schematically the energy level ordering in a Na potential well140 that contains 40 electrons. The magic numbers in Figure 3a correspond to electrons needed to close successive electronic shells. Further examination of the counting rates in Figure 3a shows that evennumbered clusters are more abundant than odd-numbered clusters, which could not be explained within the spherical jellium model. Later, Clemenger164 showed that this phenomenon could be explained by the Nilsson model, originally developed in nuclear physics. Clusters with an insufficient number of electrons, to fill the electronic shells, undergo distortion from a spherical geometry into an ellipsoidal

Figure 5. Average mass spectra of Nan clusters photoionized with 415 and 423 nm light. Well-defined minima occur at values of n corresponding to the total number of atoms in close-packed cuboctahedra and nearly close-packed icosahedra (listed at top). Adapted with permission from ref 165. Copyright 1990 Elsevier.

electronic shell closure holds for up to 1500 Na atoms. Beyond this size, atomic shell closure takes effect, allowing the authors to observe magic numbers associated with icosahedric and cuboctahedric shell closure. Another subtle aspect of physics is inherent in the spherical jellium model. Note that magic numbers seen in Figure 3 correspond to neutral clusters where an N-atom Na cluster has N-number of electrons. However, the mass spectra only correspond to positively charged clusters. What if clusters are born in the positively charged state? Will the mass spectra of such clusters be different from that seen for clusters born neutral? Rao et al.166 were the first to theoretically address this E


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Mg10, and Al13− clusters, which have been identified as magic numbers as they contain 8, 20, and 40 electrons, respectively. 2.1.3. Transition Metal Clusters. The jellium model has been successfully applied to study the relative stability of clusters composed of simple metals, such as the alkali metals, alkaline earth metals, and noble metals. However, this model does not apply to clusters composed of transition metal and rare earth metal atoms, or those composed of semiconductors and insulators. Unlike simple metals, where electrons are nearly free, the electrons in transition and rare earth metals are quasilocalized. On other hand, the electrons in semiconductors form covalent bonds. This is reflected in the mass spectra of these elements. Sakurai et al.168 studied the mass spectra of a number of transition metals such as Fe, Ti, Zr, Nb, and Ta using timeof-flight (TOF) mass spectrometry. It was found that the TOF intensities of 7, 13, and 15 atom clusters are larger than those of their neighbors, in all cases. For illustration, in Figure 8, we

and show how clusters that are born positively charged will have magic numbers shifted by one; that is, clusters of 3, 9, and 21 (see Figure 6) atoms will show more pronounced peaks than clusters of 2, 8, and 20. This is, indeed, what was obeserved experimentally for Rbn+ clusters167 (see Figure 7).

Figure 6. Second dervative of the total energy (EN+1 + EN−1 − 2EN) in hartree units as a function of N for LiN, LiN+, LiN2+, and LiN3+. Adapted with permission from ref 166. Copyright 1987 American Physical Society.

Figure 8. Time-of-flight (TOF) spectra of Fe clusters. (a) “Auto ionized” means the positive ion clusters directly ejected from the source, and (b) “post ionized” means the neutral clusters ionized by an ArF laser. Adapted with permission from ref 168. Copyright 1999 American Institute of Physics.

show the mass spectra of Fe clusters, obatined by these authors. The “magic numbers” here are different from those predicted by the jellium model or by the hard sphere packing model for noble gas atoms. The results indicate that more realistic calculations, based on first-principles, are needed to understand the underlying electronic structure and stability of transition metal clusters. We will address these aspects in the following sections. 2.1.4. Covalently Bonded Clusters. To highlight the relative stability of clusters formed of covalently bonded systems, we focus on C and Si, two of the most important elements in the periodic table. Carbon forms the basis of all life on the Earth, and silicon is the primary basis for electronics. Furthermore, the most celebrated mass spectra of clusters are probably those of carbon. The discovery of the magic peak of C60, in the carbon mass spectra by Smalley and co-workers,169 not only gave new meaning to the carbon chemistry but also to nanoscience. In Figure 9, we have reproduced the original mass spectra of carbon clusters taken under different experimental

Figure 7. Mass spectrum of Rbn+ (n = 1−9). Adapted with permission from ref 167. Copyright 1987 American Physical Society.

The electron shell-closing rule has been able to explain the magic numbers seen in the mass spectra of other simple metal clusters such as Cu, Mg, Al, etc. A few examples of this are Mg4, F


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Figure 10. A spectrum of small to medium Si cluster ions. An example of the preselection of cluster size is shown on the upper right where a single ion mass, Si12+, is isolated and fragmented to produce the fragmentation pattern. Adapted with permission from ref 176. Copyright 1995 American Physical Society. Figure 9. Time-of-flight mass spectra of carbon clusters prepared by laser vaporization of graphene and cooled in a supersonic beam. The three mass spectra shown differ in the extent of helium collisions occurring in the supersonic nozzle. Adapted with permission from ref 169. Copyright 1985 Nature Publishing Group.

A different class of covalently bonded systems, where mass spectrometry has been able to shed light on cluster stability, is the transition metal oxides. This has been illustrated in the works of Castleman, Duncan, and their groups. As an example, we show the mass distribution of V, Nb, and Ta oxide cluster ions178 in Figure 11. Although these metals belong to the same group of transition metals, the first peak of V-oxide cluster ions occurs at VO+, whereas for Nb and Ta they occur at MO2+ (M = Nb, Ta). These continue to differ at higher stoichiometry. To establish the relative stability of mass-selected ion clusters further, we note the progress made by photodissociation experiments. For example, in the V-oxide group, VO2+ and V3O7+ are favored, while MO2, MO3, and M2O5 are the favored neutral products. Photodissociation experiments of V, Nb, and Ta, by Duncan and co-workers,179 showed dissociation occurring either by eliminating O or by fission. In the Vsystem, the cations tend to lose O2, while Nb- and Ta-species lose O. For each metal increment, oxygen elimination proceeds until a terminal stoichiometry is reached. Clusters having terminal stoichiometry may no longer eliminate O, but these clusters undergo fission. Similar results have been obtained for Fe-, Cr-, and Zr-based systems.180,181 2.1.5. Ionically Bonded Clusters. As previously discussed, in general, small clusters do not mimic their respective bulk behavior. Conversely, clusters bonded by an ionic interaction such as in alkali halides and metal nitrides do bear resemblance to their bulk. Figure 12a shows the mass spectra of (TiN)n+ clusters.182 We notice the pronounced peaks, corresponding to cubic structures, which resemble pieces of the FCC lattice of a TiN crystal. In Figure 12b, the proposed geometries of (TiN)n+ clusters, based on the magic numbers observed in the mass spectra of Figure 12a, are given. Sun et al.183 showed that even a dimer of tungsten oxide (WO3)2 can possess bulk-like features, and the geometry of a small cluster, containing only 4 W and 12 O atoms, bears the hallmarks of crystalline tungsten oxide, WO3. This observation was validated by measuring the mass distributions under quasisteady-state conditions (see Figure 13). It is important to note

conditions. The singularly most prominent C60 peak, obtained by optimizing thermalization and cluster−cluster reaction conditions, pointed to the unusual stability of this cluster. Smalley and co-workers suggested a soccer ball structure for this cluster and termed it as fullerene, after Buckminster Fuller. With 12 pentagons and 20 hexagons, the fullerene structure has 60 vertices, one for each carbon atom. The unusual stability arises as the valence of all C atoms, with two single bonds and one double bond, is satisfied. The bulk production of C60 by Kratschmer et al.170 led to the experimental validation of the C60 geometry, as predicted by Smalley and co-workers. The unusual geometry of C60, having a cavity large enough to store a metal atom, gave rise to an explosive growth in the new carbon science and technology, an era that could not have been possible without the gift of cluster science. The ability to embed atoms in C60, endohedrally or exohedrally, has opened new possibilities to explore novel science as well as new technologies. Silicon, although it belongs to the same group as carbon, has an entirely different chemistry. Unlike C that forms sp1, sp2, and sp3 bonding, Si prefers sp3 bonding. This is manifested in both the bulk and the cluster properties of Si. For example, attempts171 to find Si60 in the same fullerene structure as C60 have failed. Similarly, Si clusters do not possess the chain or ring structures seen in carbon.172−175 In contrast to C, Si6 and Si10 are magic clusters that were discovered through photodissociation experiments. This was demonstrated176 by isolating a Si12+ cluster and examining its fragments (see Figure 10). It is important to note that as Si12+ fragments, the two most prominent products are Si 10 + and Si 6 + . It has been demonstrated both theoretically and experimentally that fragmentation of a cluster always leads to magic numbers.166,177 G


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Figure 11. Mass distributions of (a) vanadium, (b) niobium, and (c) tantalum oxide cluster cations. Adapted with permission from ref 178. Copyright 2002 American Chemical Society. Figure 12. Growth patterns of (TiN)n+. (a) TOF mass spectrum of (TiN)n+ clusters. Abundance patterns indicate the clusters have cubic structures resembling pieces of the FCC lattice of solid TiN. (b) Proposed structures of (TiN)n+ clusters based on magic numbers observed in the mass spectrum. Adapted with permission from ref 182. Copyright 1993 American Institute of Physics.

that all major peaks, in the figure, correspond to clusters whose building blocks are WO3 or multiples thereof, having the same stoichiometry as in the bulk. A recent experiment by Bowen and co-workers184 has confirmed the existence of these “baby” crystals in PbS system where clusters containing 32 PbS monomers resemble the bulk crystalline form. 2.1.6. Metal−Carbide Clusters. Similar to C60, another class of clusters that exhibited unusual stability is a compound cluster containing 8 Ti and 12 C atoms. This cluster, discovered by Castleman et al.185 during a study of the reactions of hydrocarbons with transition metals, is known as mettallocarbohedrene or “met-cars” for short. In Figure 14, we present the mass spectra of TimCn+ clusters generated from reactions of Ti with CH4 and C2H2. Note the “super magic” peak with Ti8C12+ stoichiometry. Subsequently, similar clusters have been seen186 with Zr, Hf, V, Nb, Mo, and Cr as metal components. However, Ti−carbide clusters containing higher Ti concentration showed a different trend. Pilgrim and Duncan187,188 as well as Castleman’s group found Ti14C13+ and V14C13+ to be abundant with stabilities comparable to those of M8C12+ species. Thus, the chemsitry of met-cars is not the same as what has been observed for carbon fullerenes.

Figure 13. Experimental mass spectra. Adapted with permission from ref 183. Copyright 2004 American Institute of Physics.

2.2. Geometries

Unique geometries set most clusters apart from any other matter; that is, they are very different from their respective bulk crystal structures, they evolve nonpredictively as a function of size, and they differ from one element to another. For example, the interplay of surface tension and compressibility in metal clusters results in the preference of icosahedral or decahedral atomic structures. We note that no current experimental

techniques exist that can independently determine the geometry of gas-phase clusters, including the bond lengths and bond angles. Experimental techniques such as ion mobility,189 electron diffraction,190 Raman spectroscopy,191,192 and infrared spectroscopy193−209 provide information on cluster geometries; however, they do require theoretical input. H


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appearance of bulk-like crystal NbC structure for small clusters down to 12 atoms.209 The geometries of clusters may also be validated, indirectly, by comparing calculated electronic properties with experiment. One such technique that has been very useful in this regard is photoelectron spectroscopy.210 Here, a mass selected negative ion is interrogated with a fixed frequency laser, which ejects the extra electron. The binding energy of the ejected electron is determined by measuring its kinetic energy and subtracting it from the photon energy. Both electron affinity and vertical detachment energy are directly measured in experiments. The former is the energy difference between the ground-state geometry of the negative cluster ion and the corresponding neutral, whereas the latter is the energy difference between the ground state of the anion and its neutral, at the anion geometry. A good agreement between these quantities validates the calculated geometry. X-ray diffraction can also be used to determine cluster structure, once the clusters are assembled in crystalline form. Recent synthesis of ligated clusters using conventional chemical methods has made this possible. Some recent examples include chiral bimetallic nanoclusters, Ag28Cu12(SR)24]4−,211 large boxshaped Ag67 nanocluster protected by a mixed shell of thiolate (2,4-dimethylbenzenethiolate, SPhMe2) and phosphine (triphenylphosphine, PPh3) ligands, Ag67(SPhMe2)32(PPh3)8]3+,212 and Pd or Pt atom incorporated into thiolated Ag-rich 25-metal-atom nanoclusters, [MAg24(SR)18]2− (M = Pd, Pt).213 Several first-principles techniques, based on quantum chemical and density functional formalism, provide information on cluster geometry and properties. Although the former can yield quantitatively accurate results, due to computational cost they are not suitable for large clusters. Density functional theory, on the other hand, can treat large clusters; however, the accuracy of the calculation is limited by the choice of energy functional. Thus, synergy between theory and experiment is necessary to validate the computed geometry and properties of clusters. Determining cluster geometries from theory is a nontrivial task. Unlike crystals, clusters lack translational symmetry constraints. Equally important, the number of isomers increases exponentially with cluster size.214 For example, there are 487 635 isomers of C60H4, of which 4165 isomers are symmetrically distinct. However, when one more H is added to form C60H5, the number of isomers increases to 5 461 512. Among the additional isomers, only 45 010 are symmetrically distinct.215 This huge increase in the number of isomers, just by adding a single atom, creates an enormous challenge when determining the ground-state atomic structure. A reliable and efficient approach is to use global search methods for finding a configuration, which can minimize the total energy. For a cluster of N atoms, a (3N − 3)-dimensional vector is needed to describe its atomic positions. The potential energy surface of these N atoms, as a function of their position, will have innumerable local minima, among which the global minimum corresponds to the ground-state geometry of the cluster. Search for this minimum in a large multidimensional space is a difficult mathematical problem, especially if the landscape is complex. Irrespective of the starting point, mathematically, a true global minimization procedure should be able to locate the global minimum. To guarantee the global optimum of a function, one must visit the entire search space. In practice, however, this minimization turns out to be extremely expensive or even

Figure 14. (a) Mass distribution of TimCn+ clusters generated from reactions of titanium with CH4. (b) Mass distribution of TimCn+ clusters generated from reactions with C2H2. Note the “super magic” peak corresponding to Ti8C12+. Adapted with permission from ref 185. Copyright 1992 American Association for the Advancement of Science.

Infrared (IR) spectroscopy involves the interaction of infrared radiation with matter and has been recently used to shed light on many cluster geometries. The infrared spectrum of a sample is recorded by passing a beam of infrared light through the sample. When the frequency of the IR is the same as the vibrational frequency of a bond or a collection of bonds, absorption occurs. By comparing the experimental IR spectrum with the calculated one, the geometric structure can be assigned. Note that the geometric structures of clusters can vary for different charge states. Photoelectron spectroscopy (PES) is used to provide information on the geometry of the anions. Any change in geometry, following electron ejection, can be inferred by studying the sharpness of the photoemission peaks. IR spectroscopy, on the other hand, can directly provide information on the geometry of neutral clusters as well as charged clusters and has been widely used in cluster science.193 Using IR spectroscopy combined with DFT calculations, several authors have recently determined the preferred geometry of monatomic and heteroatomic clusters. For example, Fielicke and collaborators studied Au clusters,194 Si6X clusters (X = Be, B, C, N, O),195 SinC clusters (n = 3− 8),196 Si6B cluster,197 B13+ cluster,198 CoxO+y (x = 3−6, y = 3− 8) clusters,199 SinBm (n = 3−8, m = 1−2) clusters,200 and Nb clusters.201 Meijer and co-workers have also studied structures of neutral Nbn (n = 5−20) clusters,202 in addition to charged Tb+n (n = 5−9) clusters203 and hydrated bisulfate anion clusters HSO4−(H2O)1−16.204 Bakkeret al. studied C60,205 neutral and cationic niobium clusters,206 and ferrimagnetic cage-like Fe4O6 cluster.207 Geometries of partially oxidized rhodium cluster cations, Rh6Om+ (m = 4, 5, 6), were illustrated by Koyama et al.208 In the study of neutral niobium carbide clusters NbnCm (n = 3−6, m = 3−7), Chernyy et al. confirmed experimentally the I


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Figure 15. Icosahedric structures of noble gas atoms containing 13, 55, and 147 atoms.

the geometries of small Nan clusters (n < 8, n = 13), from firstprinciples, without any symmetry constraint. The results, shown in Figure 16, created a great deal of excitement in the cluster community as it was shown that Na

impossible in many cases. Global optimization, therefore, is all about finding clever ways to “judge” finite patches of search space without visiting all points in them.216 To perform this optimization, one needs to assume that the potential energy surface is smooth and the global minimum is not a sharp valley; that is, its vicinity should also have relatively lower energies. Several search techniques such as genetic algorithm,217 basin hopping,218−221 and particle swarm optimization (PSO)222,223 techniques have been introduced to expedite the determination of ground-state geometry. In the methods listed above, the total energy, or the free energy, is the target function for the global search. This is primarily done by assuming that an experimentally synthesizable cluster is in the ground-state structure, if it is kinetically accessible. However, when the energy barrier is too high, the cluster may stay at local minima, with even more attractive properties. This is the case for diamond versus graphite, with the former being energetically metastable. Therefore, in the design of cluster-based functional materials, the focus is oriented toward the properties, instead of the global minima. In this context, property-directed search (or inverse search) becomes exciting.224 Numerous studies have been carried out on the geometries of clusters composed of noble gas atoms, metal (simple, noble, transition, and rare earth) atoms, semimetals, semiconductors, and insulators. It would be impossible to discuss the structures of all of the studied clusters in this Review. To demonstrate the uniqueness of cluster geometries and their variation from one class of elements to another, we only discuss a few selected cases. 2.2.1. Noble Gas Clusters. We begin with the geometries of clusters of noble gas atoms. As pointed out earlier, these geometries can be determined by using the close-packing model, because here electronic structure does not play a significant role. These clusters assume the icosahedral pattern shown in Figure 15. Here, clusters with 13, 55, 147, ... atoms correspond to the atomic shell closure of the second, third, fourth, ... shells, respectively. For a cluster composed of 19 atoms, the structure is that of a double icosahedron. We note that the mass spectra are taken on positively charged clusters, and this change can have a strong influence on the cluster geometry. For example, it may lead to the formation of a core molecule (dimer, trimer, or tetramer) within the cluster, which can then lead to magic sizes that are different from those of the neutral clusters. 2.2.2. Simple and Noble Metal Clusters. Clusters of simple metals, such as Na, fall within another extreme, where geometries show very different growth patterns than were discussed for noble gas systems. Here, the underlying electronic structure plays the dominant role. Martins et al.7 determined

Figure 16. Equilibrium geometries of Nan clusters. The internuclear distances are given in atomic units. Adapted with permission from ref 7. Copyright 1984 American Physical Society.

clusters containing 4 and 5 atoms were planar. Even when the clusters assume a three-dimensional shape, the results did not resemble the bulk structure. Today, it is possible to compute the geometries of clusters containing several hundred atoms. Studies of the geometries of Na clusters225 containing up to 60 atoms showed the emergence of 5-fold symmetry (see Figure 17) and an icosahedral growth pattern, akin to what is seen in quasicrystals. Although the geometry of the largest cluster studied here does not resemble the bcc crystal structure of Na, the nearest neighbor distance approaches the bulk value to within 10%, when the cluster contains as few as 10 atoms. For divalent simple metals, such as Be and Mg, threedimensional geometries emerge in clusters starting with 4 J


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Figure 17. Geometries of Na clusters (courtesy of Dr. M. S. Lee). Adapted with permission from ref 225. Copyright 2007 American Physical Society.

Kappes and co-workers.229 Bonačić-Koutecký et al.230 also used DFT and determined electronic structure and geometries of bimetallic clusters composed of Au and Ag and compared their results to those of pure Ag and Au clusters. Bimetallic tetramer as well as hexamer are planar and have common structural properties with corresponding one-component systems. Ag4Au4 and Ag8 clusters, on the other hand, have three-dimensional forms. In contrast, Au8 cluster is planar. In the range of 14−18 atoms, these clusters assume a cage structure. A compact pyramid structure, mimicking the (111) Au surface, emerges for Au20. In Figure 19, we show the geometries of Au cluster anions.231 2.2.3. Transition Metal Clusters. The electronic structures of transition metals are primarily governed by their d electrons, which are more localized than the s and p electrons of simple metals. Because of the unfilled d-shells, transition metal atoms also carry magnetic moments. The presence of geometric and magnetic isomers introduces further complications in identifying the global minimum geometries of transition metal clusters. Consequently, the geometries as well as their electronic and magnetic properties significantly differ from those of simple metals. For example, as discussed before, the relative stabilities of transition metal clusters do not

atoms, and the structures assume compact shapes. We show the geometries of Be clusters in Figure 18.226 The coinage group of metals, such as Cu, Ag, and Au, has the same nearly free electron character as the alkali metals; however, the growth pattern of their cluster geometries, especially Au, is very different. Note that alkali metal clusters assume three-dimensional form with as few as six atoms. Using density functional theory, Hakkinen et al.227 studied anionic coinage metal clusters, focusing on an extensive set of isomers of Cu7 −, Ag7−, and Au7 −. While the ground states of Cu7 − and Ag7− are three-dimensional (3D), that of Au7− is planar, separated from the optimal 3D isomer by 0.5 eV. The propensity of AuN− clusters to favor planar structures (with N as large as 13) was correlated with strong hybridization of the atomic 5d and 6s orbitals due to relativistic effects. Using a combination of ion mobility measurements and density functional theory (DFT) and comparison with vertical detachment energies, obtained from photoelectron spectroscopy, Furche et al.228 determined the structures of small gold cluster anions containing up to 13 atoms. According to the mobility data, geometries of small cluster anions are planar and three-dimensional structures set in at N = 12, which was further confirmed by trapped ion electron diffraction experiment by K


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Figure 18. Geometries corresponding to the ground state of Be clusters obtained in the genetic algorithm simulations. The number of reflection planes for each cluster is given in parentheses. Clusters are plotted such that the symmetry as well as atomic shell structures are clearly visible. Adapted with permission from ref 226. Copyright 2005 American Institute of Physics.

follow any particular pattern and there are no clear-cut magic numbers, as seen in the case of simple metals. However, the geometries of transition metal clusters show some similarity to other metal clusters, such as the emergence of icosahedral structures in clusters containing 13 or more atoms. In Figure 20, we illustrate the geometries of small Ni clusters.232 2.2.4. Covalently Bonded Clusters. Here, we discuss three different classes of clusters, composed of boron, carbon, and silicon. Clusters of these elements show very different structural patterns with size. Although boron belongs to the same group as aluminum, its properties are very different. Elemental boron is a metalloid with properties intermediate between a metal and a nonmetal and shows metallic and superconducting properties only when exposed to high pressures. Carbon in bulk form shows metallic behavior in graphite but semiconducting behavior in diamond. Silicon is always a semiconductor. In clusters, however, C and Si form covalent bonds. On the other hand, B, being electron-deficient, exhibits different bonding pattern. Numerous studies of the geometries and electronic properties of clusters of these elements have been carried out. Here, we only highlight the evolution of their geometries.

Elemental boron does not exist in nature. Thus, all known forms of boron are manmade and exhibit a variety of atomic arrangements, composed of B12 icosahedra. Following the seminal work of Boustani233 in 1994, on the geometries of small B clusters, many studies234−271 of their structural evolution have been carried out, both theoretically and experimentally. The mass spectra of B cluster ions reveal high stability of 5, 7, 8, 10, and 11 atom clusters; however, the one case, containing 13 atoms, has anomalously high intensity. The geometries evolve from planar and quasi-planar to tubular and cage structures. In Figure 21a, we show the geometries243 of neutral Bn clusters, containing up to 13 atoms. The geometries are either planar or quasi-planar structures, composed of triangular units. Although the geometry of B13 is quasi-planar, the dissociation energy of B13+, with three-dimensional geometry, agrees better with experiment than does the ground-state quasi-planar structure. This suggests the coexistence of both planar and three-dimensional isomers. Moreover, the ground-state geometries of Bn cluster anions (n = 3− 25, 27, 36),272 identified through the synergy between photoelectron spectroscopy (PES) and theory, are plotted in L


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Figure 19. Geometries of Au cluster anions (courtesy of Prof. X. C. Zeng). Adapted with permission from ref 231. Copyright 2010 American Institute of Physics.

dangling bonds, that are composed of a bulk-like 5-atom core, surrounded by a fullerene-like surface. Insight gained from a valence-bond interpretation of the bonding in small Si clusters led Patterson and Messmer340 to propose the structures of the above clusters, as composed of a 17-atom bulk-like core, whose surfaces resemble reconstructed Si surfaces. Using tight binding and density functional theory, Ho and co-workers341 recently studied the geometries and relative stability of Si nanoclusters containing up to 220 atoms. The geometries of the clusters studied included bulk-like, icosahedral, bucky-diamond, and onion-like structures. The researchers found the preferred geometry of Si220 to be bulklike, with the transition to bulk-like structure, emerging around 173 Si atoms. In Figure 25, the geometries and the corresponding cohesive energies are compared. 2.2.5. Clusters with Cage Structures. One of the first experiments that brought into focus clusters with hollow cage structures is that of C60 fullerene. As previously mentioned, the unusual stability of C60 cluster was first suggested to originate from its unique geometry, that of a soccer ball structure with 12 pentagons and 20 hexagons (see Figure 26). The synthesis of C60 by Kratschmer et al.170 permitted direct experimental validation of this geometry. The smallest carbon fullerene consists of 20 carbon atoms, all arranged in pentagons (see Figure 23). However, the fullerene structure is not the ground state of C20. Furthermore, two more isomers exist; one is shaped like a bowl, while the other is shaped like a ring. Both isomers are close in energy, with the bowl shape being the lowest energy structure, followed by the cage and ring structures. These structures have been verified by photoelectron spectroscopy experiments,317 and the relative stability of the structure has been studied theoretically.345 The advantage of having a cluster in the form of a hollow cage is that it can accommodate an atom inside it, giving potential for applications in electronics and medicine. The large surface area of the cage can also be used to functionalize its exterior for applications in catalysis. An example of such a structure is the C60 fullerene, with a cage diameter of 6.5 Å. Moreover, there

Figure 21b. We will discuss the fullerene structures of select boron clusters in a later part of this section. The discovery of C60 fullerene has led to considerable interest in the study of carbon clusters,172,174,273−320 which were found to exhibit a rich variety of geometries, starting with linear chains and continuing on to rings172,174,273 and fullerene cages. In Figure 22, we show some of these geometries. For clusters containing up to 10 atoms, the odd-numbered ones form linear chains, while the even-numbered ones form rings. Cage structures begin to emerge in carbon clusters containing 20 atoms, and in the most celebrated of all carbon clusters, of course, the C60 fullerene. While we will discuss C60 in more detail in a later part of this section, we should point out that C20 is the smallest fullerene. In Figure 23, we show select, low-lying isomers of C20. Although the cage structure of C20 is not the ground state, Prinzbach et al.318 identified the C20 isomer in the respective photoelectron spectroscopy experiment. Silicon clusters are also among the most studied systems.171,173,321−344 The geometries of these clusters have been studied experimentally using ion mobility and Raman spectroscopy and theoretically using tight binding models, quantum chemical and density functional theories, as well as quantum Monte Carlo methods. Unlike boron and carbon, silicon clusters exhibit three-dimensional form with as few as five atoms and do not form hollow cage structures. Small Si clusters, up to 27 atoms, were found to assume a prolate shape, whereas larger clusters assume more of a spherical oblate shape. First-principles calculations, based on density functional theory, showed321 that a transition from elongated to compact structures, as size increases beyond a critical value, becomes possible when the interior atoms become stable. Elongated shapes of clusters, with low energy, are obtained by stacking stable subunits; however, optimization of the surface to volume ratio and surface structure leads to a compact structure. In Figure 24, we show the geometries of Si clusters, containing up to 70 Si atoms. Structures of particular interest are Si clusters containing 21, 25, 33, 39, and 45 atoms, which Smalley and coworkers337−339 found to be unreactive toward ammonia. Pan and Ramakrishna342 proposed unique structures, without M


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Figure 20. Equilibrium geometries of Nin (2 < n < 23) clusters. Adapted with permission from ref 232. Copyright 1997 American Chemical Society.

with two adjacent hexagonal holes, the lowest energy structure of neutral B40 is a fullerene-like cage with heptagonal face (see Figure 27). In contrast to C60, the authors noted that the surface of the all-boron fullerene, bonded uniformly via delocalized σ and π bonds, is not perfectly smooth and exhibits unusual hexagonal faces. Using ab initio calculations, Yakobson and co-workers241 predicted B80 to have a hollow cage structure similar to that of C60; however, an extra B atom occupies the center of each of the 20 hexagons (Figure 28). The structure was found to be lower in energy than the ring structure, which is the building block of B nanotubes. With a HOMO−LUMO gap of ∼1 eV and Ih symmetry, the prediction of a B80 cage prompted studies, where researchers explored its potential application for CO2

have been numerous studies of endohedral- and exohedraldecorated C60 fullerenes over the years. The technological potential of cage clusters has led to a constant experimental and theoretical search for similar cage structures in other elements. This search has led to the discovery of cage clusters composed of B, Au, Sn, and metallocarbohedrenes (met-cars). Using a combination of photoelectron spectroscopy and density functional theory, Zhai et al.239 identified a fullerene-like structure of B40. The authors observed the added electron in B40− to have an extremely low binding energy. This is characteristic of a very stable cluster with a large HOMO−LUMO gap. Theoretical calculations revealed a cage structure with a large energy gap. While the ground state of B40− was found to have a quasi-planar structure, N


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Figure 21. (a) Geometries of small neutral B clusters.239 Copyright 2011 American Physical Society. (b) Global minima geometries of Bn anions (n = 3−25, 27, 36) confirmed by experiment. The close-lying isomer is also shown for n = 25 and 27. Adapted with permission from ref 272. Copyright 2016 Taylor & Francis.

Figure 22. (a) Geometries of carbon clusters up to 10 atoms. Adapted with permission from ref 172. Copyright 1987 American Institute of Physics. (b) Geometries of larger carbon clusters. Adapted with permission from ref 174. Copyright 1993 Elsevier.

capture and separation. However, a subsequent unbiased search250 for the ground-state geometry of B80, using simulated annealing, showed the preferred structure in fact not to be a hollow cage, but rather a B12-centered core−shell structure. Jemmis and co-workers242 demonstrated how stuffing improves the stability of larger B clusters containing 98−102 atoms (see Figure 29). A similar conclusion has also been reached by Li et al.346 Unlike carbon and boron clusters, metal clusters tend to form compact structures. However, the observation of the Au16− cluster as a hollow cage, by Wang and co-workers,347 provided an exception to the compact structures. As pointed out earlier, Au20− is a compact pyramidal cluster with tetrahedral symmetry. However, when four of the vertex

atoms are removed, and the structure is optimized by allowing the face-centered atoms to relax outward, Au16− forms a hollow cage structure. Wang and co-workers showed that the photoelectron spectrum of Au16− is consistent with this hollow structure. Using the electron diffraction technique, Xing et al.348 later confirmed the validity of the structure as well as demonstrated that Aun−, with n = 14−17, also possesses hollow structures. Thus, Au clusters evolve, with size, from planar to hollow to compact structures. The neutral Au16 cluster, however, is a compact cluster with Td symmetry. This suggests that the electron affinity (EA) and the vertical detachment energy (VDE) should be different. This is, notably, due to the former being a measure of the difference between the ground-state geometries of the neutral and the anion, O


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whereas the latter is the energy difference between the neutral and the anion, both having the anion geometry. The experimental PES result of Bulusu et al.,347 however, did not agree with the results above. Both the EA and the VDE were the same, within the experimental error. The discrepancy was clarified in a later molecular dynamics study,349 demonstrating that Au16−, once formed in the hollow-cage structure, is prevented from assuming the Td symmetry of the neutral structure, due to an energy barrier. The authors also identified several isomers of Au16− (see Figure 30) that lie within a narrow energy range. However, due to the accuracy of calculations, it is difficult to predict the exact ground-state geometry of a cluster. Experiments, capable of determining directly and unambiguously the geometry of a gas-phase cluster, remain a desired goal. Another metal cluster, found to have a cage structure, is Sn122−, known as stannaspherene (see Figure 31).350 While studying the semiconductor to metal transition, Wang and coworkers350 observed a very simple PES of Sn12−, which was very different from the Ge12− spectrum. The authors performed a structure optimization, starting with an icosahedral (Ih) cluster. This led to a slightly distorted cage, with C5v symmetry. Adding an extra electron resulted in a closed-shell Sn122− cluster, with Ih symmetry. The authors synthesized this cluster in the form of KSn12−, which can be viewed as K+[Sn122−]. Furthermore, they found its PES spectrum to be similar to that of Sn12−. With a diameter of 6.1 Å, Sn122− offers enough space to embed a metal atom. Subsequent studies have shown that magnetic atoms can

Figure 23. (a) The fullerene cage and bowl isomers. Prinzbach et al. (ref 313) have created these structures for the first time in minute quantities. Note that the fullerene is expected to be distorted from ideal icosahedral (12-fold) symmetry. (b) The ring and chain isomers, all of which have been observed previously. Several different forms of the “tadpole” (a chain attached to a ring) and the “bow-tie” isomers exist. Adapted with permission from ref 317. Copyright 2000 Nature Publishing Group.

Figure 24. Geometries of Si clusters. Adapted with permission from ref 24. Copyright 2010 Elsevier, Amsterdam. Original figures from refs 324−328. P


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Figure 28. Optimized structure of B80. Adapted with permission from ref 241. Copyright 2007 American Physical Society.

Figure 25. (a) Bulk-like and (b) Ih isomers for the Sin NCs (n = 216, 220, and 224), and the values in brackets are the DFT-PBE cohesive energies in eV/atom; (c) a comparison of the DFT-PBE energies of the Ih (red ●) and bulk-like (black ◆) structures. Adapted with permission from ref 341. Copyright 2016 American Chemical Society.

Figure 29. Optimized boron clusters based on icosahedron units: (a) B12, (b) B12H122−. (c) B84 is extracted from boron. In B98 (d), B100 (e), and B102 (f), the added boron atoms are on top of the B84 at the respective positions shown in green (dark areas). B99 and B101 are generated by removal of three and one central hexagonal boron atoms from B102. Adapted with permission from ref 242. Copyright 2008 American Physical Society.

Figure 26. Geometry of C60.

Hollow cage clusters, composed of more than one element, have been known for a long time. A classic example of such a cluster is B12H122−, which is stable as a dianion (see Figure 32). We will discuss more about this cluster and how it can be functionalized to meet the challenges of energy storage in section 5 of this Review. Ti8C12, known as metallo-carbohedrene or “met-car”,185 is another example of a cluster composed of two elements that was initially thought185,352 to be a cage structure with cuboctahedral symmetry. However, later calculations,353,354 based on an exhaustive search of phase space, led to a global equilibrium structure with tetrahedral symmetry354 (see Figure 33). This structure was found to be consistent with the infrared

Figure 27. Top and side views of the global minimum and low-lying isomers of B40− and B40 at the PBE0/6-311+G* level. Adapted with permission from ref 239. Copyright 2014 Nature Publishing Group.

be embedded within the hollow metal cage. Embedding a magnetic atom will produce a “molecular” magnet.351 Q


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Figure 32. Geometry of closo-borane B12H122−.

Figure 33. Optimized tetrahedral structure of the Ti8C12 met-car. Titanium atoms are shown as dark spheres, and carbon atoms are shown as light spheres. Adapted with permission from ref 354. Copyright 2001 The Royal Society of Chemistry.

Figure 30. Optimized geometries of anionic Au16− cluster for the first nine low-lying isomers. The relative energies are measured in eV with respect to the 3D hollow cage structure A2. Adapted with permission from ref 349. Copyright 2010 American Institute of Physics.

spectroscopy experiment Ti8C12+ by Meijer and co-workers355 as well as with PES experiments.356 2.3. Electronic Structure

As previously mentioned, there are three general classes of materials in terms of their electronic structure. The materials composed of noble gas atoms form weakly bound systems, where electronic structures display very little overlap between atomic orbitals. In metals, the valence electrons of the atoms become free from their ions, and the energy gap vanishes at the Fermi level. In semiconductors and insulators, the valence and the conduction bands are separated by a significant energy gap at the Fermi level. In clusters, however, there are no energy bands, and the atomic energy levels form molecular energy levels as atoms come together. A schematic diagram of the electronic structure357 evolution is shown in Figure 34. As clusters form, the atomic energy levels hybridize and broaden with size. All clusters, irrespective of their elemental origin, show an energy gap between HOMO and LUMO. The HOMO−LUMO gaps of these clusters change with size and are expected to show bulk behavior at some critical size. The

Figure 34. Schematic of the evolving electronic structure of magnesium. Adapted with permission from ref 357. Copyright 2002 American Physical Society.

Figure 31. Optimized structures: (a) Sn12−, (b) Sn122−, and (c) KSn12−. The bond distances and cage diameters are in angstroms. Adapted with permission from ref 350. Copyright 2006 American Chemical Society. R


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size at which a metal cluster becomes metallic or a nonmetal cluster shows the properties of its bulk semiconductor/insulator remains a difficult question to answer. Despite numerous attempts to answer these fundamental questions, no unique answer has emerged. In the following, we discuss the current understanding. 2.3.1. Simple and Transition Metal Clusters. The jellium model explaining the origin of magic numbers in simple metal clusters implies that Na and Mg clusters, containing as few as two atoms, should be metallic. If vanishing of the energy gap is an indication of metallic behavior, then the HOMO−LUMO gap of these clusters at the smallest size should be zero. Bowen and co-workers357 measured the HOMO−LUMO gaps of Mg clusters containing up to 35 atoms, using photoelectron spectroscopy experiment. These results are shown in Figure 35. We note that the HOMO−

Figure 36. Ionization potentials of LinO clusters as a function of cluster size n. Adapted with permission from ref 358. Copyright 1999 American Institute of Physics.

clusters cannot show magic numbers until n = 44. However, charged Al clusters can exhibit magic bavior at much smaller sizes. For example, Al13− is a 40-electron system that satisfies the jellium electronic shell closure. Because of the closed-shell structure, Al13− should be chemically inert. This is indeed what was observed by Castleman and co-workers.359 The authors found that, when Aln− clusters are exposed to oxygen, clusters other than Al13− and Al23− were etched (Figure 37). Note that these two clusters contain 40 and 70 electrons, respectively. However, the conspicuous peak observed for Al7− in the mass spectra posed an interesting challenge as this cluster would have 22 electrons that do not satisfy electronic shell closure. Earlier, Jarrold et al.360 had observed a conspicuous peak for Al7+ cluster. Having 20 electrons, Al7+ should be a magic cluster. Thus, the abundance of Al7+ is consistent with the mass spectra. The unusual stability of both Al7− and Al7+ clusters can be explained within the context of the jellium model only if Al can be assumed to exhibit multiple valences of 1 and 3. Here, we note that the energy gap between the 3s2 and 3p1 valence orbitals of Al atom is 4.99 eV. Unless both orbitals hybridize, which is what happens in bulk Al, Al in small clusters will behave as a monovalent species. Thus, it appears that Al exhibits both mono- and trivalent character in Al7. This is indeed the case. Rao and Jena361 studied the evolution of this sp hybridization as a function of size and determined that the transition from monovalent to trivalent character would happen around n = 7 (see Figure 38). This result is consistent with photoelectron spectroscopy experiments of Wang and coworkers.362 Taylor et al.363 had studied the evolution of the electronic structure of coinage metal (Cu, Ag, Ag) clusters (see Figure 39) by measuring their ultraviolet photoelectron spectra. The authors observed large even−odd alternation of electron affinity in small clusters, which largely disappeared as the cluster size exceeded 40 atoms. The authors noted that starting around 20 atoms the clusters could be regarded as small metallic particles that are perturbed by their small size. The evolution of the electronic structure of transition metal clusters is more difficult to interpret, because these systems do not confirm to the jellium model of simple metal clusters that show distinct shell closure effects. An example of this is shown in Figure 40 where Wang and co-workers364 measured the PES of Vn clusters (n = 3−65). Small clusters containing up to 12

Figure 35. Plot of Mgn− gap values versus their sizes, n. Adapted with permission from ref 357. Copyright 2002 American Physical Society.

LUMO gaps show nonmonotonic behavior, with peaks at 4, 10, 20... Mgn clusters corresponding to electron shell closures at 8, 20, 40... electrons. The HOMO−LUMO gaps begin to vanish for Mg clusters, containing 16 atoms, but reopen again at Mg20. The tendency to reopen the HOMO−LUMO gaps at magic cluster sizes continues. Thus, one cannot say conclusively that Mg clusters, containing 16 or more atoms, are metallic. The electronic shell effects may also be seen through the vertical ionization potential, which measures the energy it takes to remove an electron from a neutral cluster. This can be calculated by taking the energy difference between the neutral and cationic cluster, both clusters having the geometry of the neutral. This is shown for LinO (2 < n < 70) clusters in Figure 36.358 The evolution of the ionization potenial, as a function of cluster size, shows disctinct steps at n = 10, 22, and 42 with the step heights becoming significantly smaller as sizes grow. Note that these steps are also consistent with the jellium model, when one considers that LinO clusters have (n − 2) electrons, as oxygen localizes 2 valence electrons from the Li atoms. The ionization potential shows no size dependence beyond 40 atoms. This result can be used to interpret that the electronic structure mimics the bulk behavior at this size. Aluminum, a free-electron metal, offers an intersting example. With three valence electrons per atom, neutral Al S


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Figure 39. Growth of the 6s and 5d band onsets of gold clusters to bulk values (marked with solid rectangles) as a function of cluster size. The dashed lines are the predictions of the electron-drop model. Adapted with permission from ref 363. Copyright 1992 American Institute of Physics.

Figure 37. Series of mass spectra showing progression of the etching reaction of aluminum anions (Al5−Al24−) with oxygen. Note that these are all on the same intensity scale clearly showing production of Al13− and Al23−: (a) 0.0 sccm oxygen, (b) 7.5 sccm oxygen, and (c) 100.0 sccm oxygen. Adapted with permission from ref 359. Copyright 1989 American Institute of Physics.

Figure 38. Concentration of s- and p-type electrons in the highest occupied molecular orbital (HOMO) of aluminum clusters. Adapted with permission from ref 361. Copyright 1999 American Institute of Physics.

Figure 40. Photoelectron spectra of V7−, V27−, V43−, and V65− at 6.42 eV photon energy, as compared to the bulk photoelectron spectrum of V(100) surface at 21.21 eV photon energy. It shows the appearance of bulk features at V17 and how the cluster spectral features evolve toward bulk. Adapted with permission from ref 364. Copyright 1996 American Physical Society.

atoms were found to be molecular-like, with strong size dependence of the PES spectra. For clusters containing 17 atoms, bulk-like features emerge, and the PES spectrum of V65 agrees rather well with the bulk spectrum. The electron affinity

trend in Figure 41 is consistent with the spectral shape. For clusters containing more than 17 V atoms, the electron affinity becomes linear as a function of n−1/3 and approaches the bulk work function. T


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between Ge and Sn, while in clusters, it occurs between Sn and Pb. 2.3.2. Covalently Bonded Clusters. We discuss the electronic structure of silicon and carbon clusters as examples of systems characterized by covalent bonding. Although silicon and carbon belong to the same group of elements in the periodic table, they have very different electronic structures in crystals and clusters. Unlike metal clusters, where HOMO− LUMO gaps show nonmonotonic behavior in small clusters and decrease as clusters grow, the behavior in Si is different. Ganteför and co-workers330 have measured the PES of Sin− clusters containing up to 20 atoms and showed that the band gaps do not decrease with cluster size, as seen in metal clusters (see Figure 43). By comparing the measured vertical detach-

Figure 41. Electron affinity of the vanadium clusters as a function of cluster size n. Inset: Electron affinity as a function of n−1/3, showing that EAs above V17 become approximately linear and extrapolate to the bulk work function at infinite size (dot on the EA axis). Adapted with permission from ref 364. Copyright 1996 American Physical Society.

The transition from covalent to metallic behavior in clusters versus crystals can be studied in a different way. Consider, for example, group 14 elements. Bulk C, Si, and Ge are covalent, whereas Sn and Pb are metallic. The question is as follows: Do clusters, composed of these elements, behave the same way? Shvartsburg and Jarrold365 examined this problem by carrying out ion mobility experiments in Pb clusters containing up to 32 atoms. The authors compared their results with C, Si, Ge, and Sn (see Figure 42). Pb clusters were found to assume near spherical morphologies for all of the clusters studied. This behavior is a characteristic of metallic elements. However, Si, Ge, and Sn clusters, within the same size range, were found to adopt prolate geometries and are produced by stacking tricapped trigonal prism (TTP) units. Thus, while in the bulk phase, transition from covalent to metallic behavior occurs

Figure 43. (A) Vertical detachment energies of Sin− anions (n ≤ 20). The line indicates experiment, while symbols mark values calculated at the LDA level with various geometries. (B) Band gaps calculated (by PWB) for the lowest energy Sin neutrals. Adapted with permission from ref 330. Copyright 2000 American Physical Society.

ment energies (VDE) with those calculated from optimized anion geometries, the authors provided spectroscopic evidence for the tricapped tetragonal prism structure of Si clusters. They also showed that the structures of anionic clusters are different from those of their neutral or cationic counterparts. The electronic structure of C clusters is exemplified by the uniqueness of C60 fullerene. Soon after the discovery of C60, Haddon et al.366 applied the Hückel molecular orbital theory for nonplanar conjugated organic molecules to study the electronic structure of icosahedral C60. The authors suggested that this may represent the first example of a spherical aromatic molecule, although the debate still continues. To account for the unusual stability of icosahedral C60, Kroto367 proposed that all stable fullerenes should have pentagons isolated by hexagons. The isolated pentagon rule (IPR) is currently widely accepted as the guiding principle for stabilizing a fullerene.368 The destabilization of non-IPR fullerenes was explained by Kroto to be due to an increase in local steric strain caused by fused pentagons. Extensive conjugation in C60 implies that

Figure 42. Relative mobilities of group-14 cluster cations in the n ≤ 32 size range measured at room temperature. Filled symbols are for Sin (◆), Gen (■), and Snn (▲). “○” are for lead clusters. Adapted with permission from ref 365. Copyright 2000 Elsevier. U


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electrons should be delocalized. Poater et al.369 showed that the electron delocalization per C atom is smaller for C60 than it is in benzene or naphthalene. However, as the electron charge is shared with a larger number of C atoms, the global electron delocalization per C atom is almost the same as that of the aromatic systems, such as benzene.

singlet rhombus being the ground state. Similar results were also seen for the Na4 cluster. It was later found371−373 that when Li4 is allowed to form a three-dimensional structure, the preferred spin is also a triplet, resulting in a total magnetic moment of 2 μB, as expected from the jellium model. This suggests that there must be a spin cross over as the planar structure of Li4 transforms into a threedimensional structure. Indeed, this is what was seen (Figures 44 and 45) from first-principles calculations.373

2.4. Magnetic Properties

Two of the most fundamental quantities that control the magnetic property of a material are magnetic moment/atom and the nature of their coupling. Although one-half of the atoms in the periodic table possess a magnetic moment, there are only five elements, Fe, Co, Ni, Gd, and Dy, which show an intrinsic ferromagnetic order in the bulk. Other elements are either nonmagnetic, diamagnetic, antiferromagnetic, or ferrimagnetic. Simple metals containing sp valence electrons are paramagnetic, while those containing localized d or f electrons show magnetic behavior. However, magnetism in atomic clusters shows very different behavior. Clusters of simple metals, as well as those of nonmagnetic and antiferromagnetic transition metals, can become ferromagnetic. Topology and composition of clusters may also induce a magnetic transition. Additionally, the magnetic moment per atom, in most clusters, is larger than in the respective bulk, although there are examples where the reverse is possible. A considerable amount of work has been carried out, exploring the uniqueness of cluster magnetism. Instead of reviewing the entire literature on the magnetism of clusters, in the following section, we focus on specific examples that have presented unexpected results and posed some interesting questions and challenges. 2.4.1. Simple Metal Clusters. Properties of simple metals, such as the alkali metals, aluminum, and gold, are governed by their s and p valence electrons. These electrons tend to be delocalized, and the atoms in the bulk phase do not carry a magnetic moment; these metals are paramagnetic. However, in clusters, the situation can be very different, as the geometry plays an important role in the underlying electronic structure. An example of such system is a Li4 cluster. In section 2.2.2 of the jellium model, two of the electrons occupy the 1S2 state, while the other two occupy the 1P2 state. In keeping with Hund’s rule, the electrons in the 1P2 shell would prefer to have their spins parallel, and hence the ground state of Li4 should be a spin triplet, with the cluster carrying a total magnetic moment of 2 μB. However, as mentioned earlier, Li4 has a planar geometry and is nonmagnetic; that is, it has a magnetic moment of 0 μB. The reason for this is that energy may be lowered due to Jahn−Teller distortion. This distortion would split the degenerate energy levels, allowing the two-1P2 electrons to have antiparallel spins, resulting in the ground state becoming nonmagnetic. Therefore, whether a cluster will be magnetic or nonmagnetic depends upon the competition between Hund’s rule coupling and the Jahn−Teller distortion. The relationship between cluster geometry and preferred spin configuration was brought into focus by Beckman et al.370 Using a number of ab initio methods such as self-consistent field, coupled electron pair approximation, multireference double excitation configuration interaction, and limited configuration interaction, the authors studied the influence of continuous deformation of a Li4 cluster, from the square to the rhombus geometry, on the underlying spin multiplicity. The preferred spin multiplicity of the rhombus structure is found to be a singlet, while that of the square structure is a triplet, with

Figure 44. Possible structures of Li4 clusters: (a) The ground state is a planar singlet. (b) The gradual conversion of the planar geometry to the three-dimensional phase through the displacement of atom D as shown. Adapted with permission from ref 373. Copyright 1985 Elsevier.

No experiments are available, in alkali metal clusters, to verify the result given above. However, the conclusion that clusters of sp elements could be magnetic has been shown to hold for Al clusters. Cox et al.374 showed, by measuring the deflection in a Stern−Gerlach field, that even-numbered Al clusters, contain-

Figure 45. Total energy of the optimized Li4 cluster as a function of the dihedral angle. The solid and the dashed curves are for the singlet and triplet states, respectively. Adapted with permission from ref 373. Copyright 1985 Elsevier. V


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moments of Fe clusters, containing 2−20 atoms. They observed a nearly monotonic growth of the magnetic moment as a function of size; furthermore, for n = 13, a sharp drop in the magnetic moment, by 9 μB, yielded an average moment of 2.4 ± 0.4 μB/atom. This value is close to the bulk value of 2.2 μB/ atom (see Figure 47). Because the orbital contribution to the

ing less than 10 atoms, have a spin multiplicity of 3, and thus a magnetic moment of 2 μB. However, one would normally expect this to be 0 μB. Theoretical studies of the optimized geometries of these clusters, in both spin singlet and triplet states, confirmed361 the above experimental observation. Furthermore, it has been shown that larger Aln clusters, containing even number of electrons, are nonmagnetic. 2.4.2. Clusters of Ferromagnetic Elements. A systematic study375,376 of the parameters that determine the magnetic moment has shown that it increases as the interatomic distance increases and the coordination number decreases. In metal clusters, the nearest neighbor distance and the coordination number are smaller than they are in the bulk; furthermore, these values increase with cluster size. Although the two factors have an opposite effect on the magnetic moment, the role of coordination number dominates over that of the nearest neighbor distances. It has been known for a long time153 that the magnetic moments of surface atoms are larger than those of bulk atoms. Because, in a small cluster, there are more surface atoms than interior atoms, the magnetic moment per atom is expected to be larger than that in the crystal. Naturally, the initial observation by de Heer et al.,377 that the average magnetic moment of Fe clusters containing 15−650 atoms is smaller than the bulk value, posed an interesting dilemma. This conclusion conflicted with an earlier experiment on Fe clusters378 as well as with earlier theoretical calculations.379,380 Merikowski et al.382 addressed this unexpected result by applying the Ising model and Monte Carlo technique. They attributed the apparent contradiction to the statistical behavior at finite temperatures. Furthermore, they noted that the temperature and size dependencies of the magnetization of these clusters are large enough to mask any possible change in the value of the magnetic moment. Later, Khanna and Linderoth383 proposed the superparamagnetic model for cluster magnetism that explained the anomalous result observed experimentally. Later experiments381,384,385 have used the superparamagnetic model to analyze their data. It has been confirmed that the magnetic moments of Fe, Co, and Ni clusters are larger than those of their bulk values and decrease with cluster size (see Figure 46). Recent studies on the magnetism of charged Fe and Gd clusters continue to reveal interesting and unexpected results. Using X-ray magnetic circular dichroism experiment, Niemeyer et al.386 studied the orbital and spin components of magnetic

Figure 47. Spin (●), orbital (○), and total (□) magnetic moments per 3d hole (left axis) and per atom (right axis) of Fen+ clusters. In Fe13+, antiferromagnetic coupling of the central atom to the surrounding shell and reduced spin magnetic moments lead to a significantly lower total magnetic moment. Adapted with permission from ref 386. Copyright 2012 American Physical Society.

total magnetic moment was found to be less than 0.4 μB and nearly constant across the series, the sharp drop is attributed solely to the spin contribution. This was an unexpected result because, if one assumes a simple tight binding model, removal of an electron can only change the magnetic moment of the cluster by ±1.0 μB, depending upon whether electron is removed from the spin down or spin up channel. This was, indeed, the case for Fe2+ to Fe6+. The authors suggested that the anomalous result might be due to the antiferromagnetic coupling between the magnetic moments of the central atom and those on the icosahedron surface. It should be noted that, for the neutral Fe13 cluster, such a state was earlier predicted by Dunlap.387 State of the art calculations later carried out, however, told a different story.388 While studying several possible isomers of neutral and positively charged Fe13 cluster with gradient corrected density functional theory, Wu et al.388 noticed that the origin of this anomalous behavior is due to the quenching of the moment of the central atom, as well as the reduced magnetic moments of the surface atoms (see Figure 48). In the case of Gd13 cluster, the situation is different. Contrary to the results of transition metal clusters, where the magnetic moments per atom are larger than those in the bulk, the magnetic moment per atom in Gd13 was found to be much smaller than the respective bulk value. Additionally, there was a long-standing controversy between the experimental and the theoretical values. One experiment389 reported a value of 3 μB/ atom, whereas another390 reported a value of 5.4 μB/atom. Interestingly, both experiments used the Stern−Gerlach technique. Note that these values are 8.82 μB/atom in the dimer and 7.63 μB/atom in the bulk. Similarly, previous theoretical calculations provided conflicting mechanisms for the reduced magnetic moment. One group,391 using DFT/GGA level of theory, found the central atom to be antiferromagnetically coupled to the outer atoms in the Gd13 icosahedron;

Figure 46. Iron cluster magnetic moments per atom at T = 120 K. Horizontal bars indicate cluster ranges. Adapted with permission from ref 381. Copyright 1997 American Physical Society. W


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the Gd13 cluster. They stressed the necessity to include Hubbard U in the calculation, because the correct magnetic state cannot be obtained without doing so. The authors successfully showed that (see Figure 49) the correct ground state of the Gd13 cluster is icosahedral. Moreover, the reduced magnetic moment is primarily due to noncollinear spins.

Figure 49. Noncollinear spin configurations of the Gd13 cluster for two configurations: (a) AFM state with U = 0 eV. (b) FM state with U = 5.5 eV. Adapted with permission from ref 393. Copyright 2014 American Physical Society.

2.4.3. Ferromagnetism in Clusters of Otherwise Nonmagnetic Transition Metal Elements. In Figure 50,

Figure 50. Spin magnetic moments of free atoms and the bulk magnetic moments for 3d transition metal elements. Adapted with permission from ref 154. Copyright 1991 American Physical Society. Figure 48. Geometries of the lowest total energy state and two antiferromagnetic states of Fe13 and Fe13+. Bond lengths are in angstroms, local magnetic moments are in μB, and G.S. denotes the ground state. The blue (light) color is used for the atoms whose local spin magnetic moments are antiferromagnetically coupled to the local spin magnetic moments of atoms marked with the red (dark) color. Adapted with permission from ref 388. Copyright 2012 American Physical Society.

we compare the magnetic moments of isolated 3d transition metals atoms with those in the crystal phase.154 Note that the magnetic moments of isolated atoms increase until a peak is reached at Cr, and then the moments decrease. However, as the atoms form a crystal, only Fe, Co, and Ni are ferromagnetic, Mn is paramagnetic (although the CRC database lists its Neel temperature to be 100 K), Cr is antiferromagnetic, and Sc, Ti, and V are nonmagnetic. Liu et al.154 studied how clusters of nonmagnetic elements can become magnetic, due to low coordination and larger interatomic spacing. The study was confined to V clusters, having both bcc and linear chain structures. The authors found the small clusters to be magnetic. However, for clusters with bcc structure, the magnetic moments vanish abruptly as the cluster size increases, whereas they retain their moment in a linear-chain configuration. Later, Reddy et al.394 examined the magnetic properties of 4d (Ru, Rh, Pd) 13-atom clusters, having icosahedral geometry. Note that these atoms carry no magnetic moment in the crystalline phase. The clusters were found to have nonzero magnetic moments, with Rh13 carrying a magnetic moment of 21 μB. Magnetism was shown to arise when the dimensionality

conversely, another group,392 using a phenomenological model, attributed the reduced magnetic moment to spin canting. It should be emphasized that the study of Gd clusters offers considerable challenges, not only due to the interplay between geometries, charges, and spin degrees of freedom, but also in regards to strong correlation, spin−orbit coupling, and noncollinear spins. Tao et al.393 carried out a comprehensive theoretical study. This was done with a self-consistently determined on-site Coulomb repulsion parameter, Hubbard U to account for strong correlation, spin−orbit coupling, and noncollinear spins, all within the DFT/GGA framework. More so, the authors also examined various competing isomers, for X


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dimensional structures. The most interesting prediction was that the V atoms couple ferromagnetically, with a magnetic moment of 1 μB localized at each V site. This prediction was later verified by Miyajima et al.399 The authors suggested that soft-landing of Vn(C6H6)n+1 clusters onto nanoscale designed surfaces may open new applications in magnetic recording and spintronic devices. Mn@Sn12 also provides an interesting example where a transition metal atom embedded inside a metal cluster can give rise to a cluster with a large magnetic moment. Previously, we mentioned that Sn122−, the so-called stannaspherene, is a very stable cluster. Because Mn is divalent, neutral Mn@Sn12 can be viewed as Mn2+@Sn122−. This cluster should have a magnetic moment of 5 μB, arising from the five unfilled Mn 3d orbitals. Indeed, this was found theoretically. Kandalam et al.351 carried out first-principles calculations of the preferred geometry of Mn@Sn12 and determined its ground-state structure to be that of Mn encapsulated in an icosahedral Sn12 cage (see Figure 52); the cluster has a total magnetic moment of 5 μB. As the two species approach each other, their respective geometries are retained and the endohedral Mn atoms couple antiferromagentically. The ferromagnetic state, with a total moment of 10 μB, lies 0.11 eV higher in energy. 2.4.4. Magnetic Coupling in Clusters of Antiferromagnetic Elements. Now, we pose the question: Can clusters, composed of Mn and Cr, couple ferromagentically, and, if so, at what size would they assume their bulk antiferromagnetic behavior? Here, we will only focus on Mn, which, with half-filled 3d and filled 4s (3d54s2), predominantly exists in a +2 oxidation state. This leaves five unfilled 3d orbitals, and hence the magnetic moment of Mn is 5 μB. Thus, if all of the Mn atoms would couple ferromagnetically, a Mn cluster containing N atoms could have a total magnetic moment of 5N μB. Early experiments by Weltner and coworkers400 on Mn2, isolated in rare gas matrix, showed that the atoms are coupled antiferromagnetically, whereas a cluster, believed to be Mn5, proved to have a magnetic moment of 25 μB. An earlier experiment by Ludwig et al.401 on Mn4 cluster, isolated in a Si matrix, suggested a moment of 20 μB. Furthermore, the removal of a single electron from Mn2 was shown402 to have a pronounced effect on the underlying magnetic order. The coupling in Mn2+ was shown to be ferromagnetic, with a total magnetic moment of 11 μB.403−406 The complexity of treating transition metal clusters computationally delayed theoretical development in the systems, mentioned above. For example, the first theoretical work on a Mn5 cluster was carried out by Shillady et al.407 at the unrestricted Hartree−Fock level of theory, while confining the cluster to a planar pentagon with fixed interatomic distance.

is reduced and electronic degeneracy is increased. Both aspects are due to the symmetry of the cluster. Subsequently, Cox et al.395 experimentally verified this prediction, by studying Rh clusters containing 12−32 atoms. They found the clusters to be superparamagnetic below 93 K, with magnetic moments varying between 0.3 and 1.1 μB/atom. Nayak et al.396 later showed that magnetism, of otherwise nonmagnetic clusters, is sensitive to cluster geometry, the interaction with substrates, and the interaction with gas molecules. For example, Rh atoms, in an Rh4 cluster, carry magnetic moments when their structure is planar; however, in a tetrahedral configuration, the magnetic moments vanish. The authors further demonstrated that the spin multiplicity of both the planar and the tetrahedral isomers of Rh4 change as they interact with hydrogen, thus providing a link between magnetism and reactivity. Pd, which is nonmagnetic in the bulk, was also found to be magnetic in small clusters.397 This was demonstrated for both neutral and anionic Pd clusters, containing up to 13 atoms, by using theory and photoelectron spectroscopy experiments. First-principles studies398 of Vn(C6H6)m (n = 1−3, m = 1−4), where the V atoms are sandwiched between the benzene rings (see Figure 51), revealed that these complexes prefer one-

Figure 51. Ground-state geometry of the V3(Bz)4 complex. Adapted with permission from ref 398. Copyright 2004 American Institute of Physics.

Figure 52. Ground state and higher energy isomers of Mn@Sn12 cluster. The energies are relative to the ground state Ih cage. The symmetry and spin multiplicities are also shown. Adapted with permission from ref 351. Copyright 2008 American Institute of Physics. Y


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obtained earlier by Kabir et al.412 for n > 7 (see Figure 54). Therefore, the level of theory used should be calibrated against other experiments. To do this, it is particularly helpful to compare results with photoelectron spectroscopy.

Subsequent theoretical studies, using local density approximation to density functional theory, were carried out by Fujima and Yamaguchi.408 In these calculations, the authors optimized the interatomic distance, with fixed geometries. Nayak and Jena403 were the first ones to carry out full gemoetry optimization of Mn clusters containing up to 5 atoms at the DFT/GGA level of theory. It was shown that all of these clusters, including Mn2, are ferromagnetic, with each Mn atom carrying a magnetic moment of 5 μB. How large does a Mn cluster have to be before ferrogmagentic order gives way to ferri- and antiferromagnetic order? This problem was addressed409 in a synergistic study involving Stern−Gerlach magnetic deflection experiments and first-principles calculations on fully optmized geometries. The authors demonstrated that ferrimagnetism occurs when Mn clusters contain as few as seven atoms. The calculated magnetic moment of 0.71 μB /atom agreed very well with the experimental value of 0.72 ± 0.42 μB/atom. The substantial reduction, in the magnetic moment, resulted not from the reduction of individual moments at the Mn sites, but from ferrimagnetic coupling (see Figure 53).

Figure 54. Spin multiplicities of the lowest total energy states of the neutral Mnn clusters: from (a) ref 412 (Copyright 2006 American Physical Society) and (b) ref 411. Adapted with permission from ref 411. Copyright 2015 American Institute of Physics.

2.4.5. Magnetic Transition Induced by Heteroatoms. As pointed out before, Mn2+ is ferromagnetic with a total magnetic moment of 11 μB, while Mn2 is antiferromagnetic with a total magnetic moment of 0 μB. Wu et al.413 showed that Mn2 can be made ferromagnetic by reacting it with a Cl atom. Note that because Cl is electronegative, it draws an electron from Mn2; therefore, the dimer is left in a Mn2+ state. Hence, reactivity can be used as a tool to induce magnetic transition. Rao and Jena414 had earlier used this concept to show that clustering of Mn around N can give rise to ferromagnetic coupling among the Mn atoms. Note that while Mn atoms interact weakly with each other, they interact strongly with N, due to the overlap of nitrogen’s p orbitals with the d orbitals of Mn. These results illustrated why Mn atoms cluster around N. Additionally, a single N atom, with an odd number of electrons, carries a magnetic moment. It was found that the coupling between N and Mn spins becomes antiferromagnetic. Thus, Mn atoms, clustered around a single N atom, are ferromagnetically coupled. Using DFT-based calculations, Rao and Jena demonstrated that the magnetic moment of MnN is 4 μB, while that of Mn5N is 24 μB. Their results are presented in Figure 55. Although the results are obtained in free clusters, it has been shown that they also have implications in bulk systems. For example, the discovery of ferromagnetism415,416 in Mn doped InAs and GaAs, as well as a subsequent theoretical prediction417 of room-temperature ferromagnetic ordering in Mn doped GaN, created considerable excitement, due to the potential applications in spintronics devices. A considerable amount of experimental effort was made to verify these theoretical predictions. This was done by varying the concentration of Mn impurities, as well as varying the synthesis conditions.418,419 Unfortunately, these experiments led to conflicting results. The measured Curie temperatures ranged from 10 to 940 K. Furthermore, some experiments420 revealed antiferromagnetic and spin glass behavior in these materials. It was suggested379 that the conflicting experimental results may have been due to the experimental conditions. It was also argued that the observation of ferromagnetism in Mn doped GaN was likely due to Mn clustering around N.

Figure 53. Geometries corresponding to the (a) ground state and (b and c) low-lying isomers of Mn7. The bond lengths are in angstroms. The arrows ↑ and ↓ indicate the direction of spin polarization at each site. Adapted with permission from ref 409. Copyright 2003 Elsevier.

Recent calculations by Gutsev et al.410 place the ferro- to ferri-magnetic transition at Mn5, Mn6−, and Mn3+. When going beyond Mn6, the spin multiplicities, between neutral and charged species, differ by ±1. In a recent study, Gutsev et al.411 combined theory with photoelectron spectroscopy to show that Mnn− ions (n = 2−16) are ferrimagnetic, beginning with n = 6. Again, it is emphasized that the large number of nearly degenerate isomers, accompanied by different spin couplings, provides an enormous challenge to theory. Furthermore, the problems become more difficult when the energy differences lie within the range of computational accuracy. We note that the spin multiplicities obtained by Gutsev et al.371 differ from those Z


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In the case of multiply negatively charged clusters, the electron−electron repulsion will also lead to either autoejection of the added electrons or fragmentation of the cluster. What is the critical size of a cluster that can hold two extra electrons? The answer again will depend upon the cluster composition and the underlying electronic structure. Compton and coworkers425,426 addressed this problem using mass spectrometry and showed that multiply charged negative ions of small molecules and clusters can exist as isolated entities in metastable states. However, in the following we discuss a related problem, that is, the interaction between two equally charged atoms/clusters to highlight the interesting science that has emerged. 2.5.1. Interaction between Two Positively Charged Transition Metal Atoms. To study the stability of doubly positively charged transition metal clusters, we note that a dimer, composed of two atoms, is the smallest size a cluster can have. If a dimer were doubly positively charged, due to Coulomb repulsion, one would expect it to spontaneously fragment into two singly charged atoms. However, it came as a surprise when a Mo22+ dimer was observed experimentally,427 to be stable, with a binding energy of at least 1.2 eV. Liu et al.428 carried out a systematic study of the stability of doubly positively charged 3d transition metal dimers (Sc to Ni) as well as those of W22+ and Mo22+ dimers. Using both tight binding and density functional theory, the authors calculated the total energy as a function of the interatomic distance. The occupancy of the d orbitals was found to be important when determining the variation in the total energy. While most of the doubly charged dimers were found to be metastable, those with nearly filled d-shells underwent spontaneous fission. Certain dimers, such as Cr22+, W22+, and Mo22+, have a barrier for spontaneous fission that may be high enough for the doubly charged dimer to remain in a metastable state for an extended period. This may give a false sense of stability. These results are shown in Figure 56. It is important to note that the energy barrier increases from Cr22+ to Mo22+. As interatomic distances increase beyond 10 au, classical Coulomb repulsion dominates; however, for distances shorter than 10 au, due to charge polarization, quantum mechanics dominates.

Figure 55. Geometries of MnxN (left column) clusters in their ground states. The bond lengths are given in angstroms. The spin density surfaces corresponding to 0.005 au for these clusters are plotted in the right column. The green surfaces represent negative spin densities around the N site, while the blue represents positive spin density around Mn sites. Adapted with permission from ref 414. Copyright 2002 American Physical Society.

2.5. Multiply Charged Clusters: Stability and Fragmentation

As clusters are charged, by either adding or removing an electron, their stability, geometry, and corresponding energy gain, or loss, will change as the size of the cluster changes, with the latter evolving toward the bulk work function. However, a more interesting aspect is the effect on cluster’s stability as the cluster is multiply charged, For example, the electrostatic repulsion between like charges would increase as the size of a cluster is decreased. When this repulsion becomes greater than the binding energy of the cluster, the cluster will fragment. The above phenomenon was first studied by Recknagel and co-workers421 in doubly positively charged Pb, NaI, and Xe clusters and is referred to as Coulomb explosion. The critical size for Coulomb explosion is dependent on the chemistry of the cluster and the nature of its binding. For example, doubly charged clusters of noble gas atoms, bound by a weak van der Waals force, will not be stable until it reaches a very large size. The situation can be different for clusters bound by strong covalent or ionic bonds, as well as for clusters bound by metallic bonds. The above authors observed the critical size for Coulomb explosion of Xen, Pbn, and (NaI)n clusters to be 52, 30, and 20, respectively. Note that the critical size decreases as the binding strength increases. An equally important question one may ask is: “What is the nature of the fragments as a cluster explodes?” Are they of equal size and does each fragment carry equal charge? A considerable amount of work has been carried out on the stability and fragmentation of charged clusters, and we refer the reader to some review articles.422−424

Figure 56. Interaction energies of Cr22+, Mo22+, and W22+ as a function of interparticle separation. Adapted with permission from ref 428. Copyright 1987 American Physical Society. AA


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interaction due to the dipole−dipole term and a repulsive interaction between the like charges due to the Coulomb’s law. Note that because there is aspherical geometry, B12X9− moieties possess large dipole moments, ranging between 130 and 153 D. 2.5.3. Symmetric versus Asymmetric Fission of Doubly Charged Metal Clusters. A doubly charged metal cluster can fragment in three different ways; it can eject a charged atom, break asymmetrically, or break symmetrically. When the cluster breaks symmetrically, both fragments carry equal charge. Fission channels, where one fragment is neutral and the other carries both of the charges, are rare. Jena et al.177 studied this problem by calculating only the energy that is needed to fragment a charged cluster along different channels. Focusing on small alkali clusters, the authors noticed that the preferred channel always consisted of a magic cluster as a product of fission. This observation is similar to what was already known in the fission of nuclei. Later, the authors166 verified this conclusion using self-consistent molecular orbital theory-based calculations. Three years later, Brechignac and coworkers432 experimentally observed the asymmetric fission of doubly charged Nan2+ clusters where one of the fragments, as predicted by the above theory, indeed consisted of a magic cluster. In later publications, involving both theory and experiment, Landman and co-workers433,434 analyzed the effect of an energy barrier on the fission of charged Nan metal clusters. The authors found a barrierless fission for clusters consisting of n < 6 atoms, whereas fission in larger clusters involved barriers. Similar results were obtained for the fission of Kn2+ clusters.434

The stability of doubly charged clusters also depends upon the nature of the metal atoms. For example, in a recent experiment, Renzler et al.429 showed that the smallest doubly charged Na, K, and Cs clusters, that could be observed, contain 9, 11, and 9 atoms, respectively. For a triply charged Cs cluster, the smallest observable size is 19. The numbers of atoms, presented above, are a factor of 2−3 times smaller than the number of atoms that had been observed earlier. The authors emphasized that experimental conditions, such as the temperature of the cluster and the mechanism by which ions form, are important to observe long-lived multiply charged clusters with reduced size. It was noted that the presence of helium droplets, allowing the neutral cluster to cool to 0.37 K, may also help the clusters to cool after ionization. 2.5.2. Interaction between Two Negatively Charged Clusters. Here, we address the possibility of two negatively charged clusters attracting at short distances; more so we ponder the possibility of the clusters remaining in a metastable state long enough to be detected in an experiment. We note that two electrons mediated by phonons can form a bound state known as Cooper pairs.430 Motivated by the experimental observation that B12I9− anions, formed during the deiodination of B12I122−, spontaneously combined to form a stable bound B24I182− moiety,431 Zhao et al.157 recently studied the interaction between two B12X9− moieties (X = F, Cl, Br, I, H, Au, and CN) using density functional theory and ab initio molecular dynamics. The calculated interaction energies as a function of distance, separating the centers of the two B12X9− icosahedra, are plotted in Figure 57. It is important to note that

3. SUPER ATOMIC CLUSTERS AND THE ROLE OF ELECTRON-COUNTING RULES As demonstrated in the previous section, a cluster’s properties often, uniquely, depend on the size, shape, and composition of the cluster. However, once removed from the vacuum environment, clusters react with their environment and/or coalesce with one another, and so the unique properties are destroyed. To use clusters as building blocks of matter, the clusters must be stable and have the ability to retain its identity. This section deals with such clusters. Imagine that one can design and synthesize a cluster with atomically precise size and composition such that it mimics the chemistry of an atom on the periodic table. Such a cluster may be regarded as a superatom, which can then be used to build a new threedimensional periodic table (see Figure 58), with superatoms constituting the third dimension. Because there are limitless ways to design these superatoms, the third dimension of this new periodic table, in principle, can be infinite. This is significant, because select elements in the periodic table, which are critical to industry, are either scarce or expensive (Figure 59). It will be highly desirable to identify stable superatoms composed of Earth-abundant elements that can replace scarce or expensive elements. If superatoms can be assembled to build bulk matter, where the individual identities are retained, a new class of materials may emerge. Such materials, referred to as cluster-assembled materials, would have unique properties, because superatoms, instead of atoms, form the building blocks. In this section, we concentrate on the design and synthesis of superatoms. In section 4, we will outline the progress that has been made in assembling these super atomic clusters, to form bulk materials. The concept of superatoms was first introduced by Khanna and Jena15,16 nearly a quarter century ago. However, the term

Figure 57. Energy as a function of distance between the boron cage centers of two B12X9− (X = F, Cl, Br, I; H, Au, CN) icosahedra. Adapted with permission from ref 157. Copyright 2016 American Chemical Society.

the energy barrier, separating the two B12X9− moieties, decreases as the size of the anion increases from F to I. The energy barrier, separating two B12I9− moieties, is so small that it may be surpassed at a temperature of 400 K, thus enabling the two anions to combine spontaneously. The attraction between these two negatively charged species is a direct result of the competition between the two opposing terms: an attractive AB


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maximal valence of the metal atom by 1 (e.g., LiF2), would behave as a halogen atom. A year later, the same authors showed440 that a cluster, with one extra electron than needed for its electronic shell closure (e.g., Li3O), would behave as an alkali atom. They termed the former species as superhalogens and the later species as superalkalis. The electron affinities of superhalogens are larger than those of the halogens, while the ionization potentials of superalkalis are smaller than those of the alkali atoms. One can thus regard superhalogens and superalkalis as superatoms, because they mimic the chemistry of halogen and alkali atoms, respectively. To illustrate the analogy between an atom and a superatom, we begin again with the jellium model, where a sphere carrying a uniform distribution of positive charge approximates a cluster. The electrons surround this “nucleus” of finite size, in spherical shells (see Figure 60), just as they surround a point nucleus in

Figure 58. Three-dimensional periodic table with superatoms mimicking the chemistry of halogens and alkalis. Examples are those of superalkalis designed using the octet rule and superhalogens designed using different electron-counting rules: Al13− and Al20 (jellium rule), MnCl3 and PtF6 (octet rule), WSi12 (18 electron rule), and Al4H6 (Wade−Mingos rule).

Figure 60. Schematic diagram of an atom and atomic orbitals (left panel) where the positively charged nucleus is localized at a point, and jellium model of a cluster where the positive charge is smeared over a sphere of finite radius with corresponding electronic orbitals (right panel). Adapted with permission from ref 435. Copyright 2013 American Chemical Society.

an atom. The difference between the two descriptions lies in the ordering of the shells. Therefore, the chemistry of the superatoms originates in the same manner as it does in atoms, in the order that successive shells are filled. In simple metal clusters, the super atomic orbitals, in the jellium model, are qualitatively similar to those obtained using molecular orbital theory. In clusters of transition metals and semiconductor elements, however, the geometry and electronic structure play a significant role, and the molecular orbitals are not the same as those obtained from the jellium model. In the past decade, a considerable amount of work441−453 has been done to illustrate the fundamental science behind the concept of superatoms. Calculations show that the valence electrons of the atoms, constituting the superatom cluster, occupy a new set of orbitals, defined by all of the atoms in the cluster as a single unit, rather than as individual set of atoms. As filling of electronic shells in clusters is central to the understanding of superatoms, we discuss how different electron-counting rules can be used to design these new species.

Figure 59. Relative abundance of elements in Earth’s crust. Source U.S. Geological Service.

superatom, or its variant, has been used by other authors before,435 although in different contexts. In 1986, Watanabe and Inoshita436 used the term superatom to describe a semiconductor heterostructure consisting of a “spherical core, modulation-doped with donors, and a surrounding impurityfree matrix with larger electron affinity”. Using a semiclassical calculation, the authors demonstrated that the “formation of a superatom with well-defined atomic orbitals is possible under reasonable conditions”. A year later, based on the jellium model calculations, Saito and Ohnishi437 showed that Na8, with closed electronic shell (1S21P6), is chemically inert, whereas it was shown that Na19, with open electronic shells (1S21P61D102S1), is reactive, mimicking the chemistry of alkali atoms. The authors termed Na8 and Na19 as “giant atoms”. However, later calculations,438 based on the actual geometry of the Na8 cluster, showed that while it retains its structure up to 600 K on an insulating NaCl(001) surface, it spontaneously collapses on a Na(110) surface, forming an epitaxial adlayer. In vacuum, two Na8 clusters interact, forming a deformed Na16 cluster. In 1981, Gutsev and Boldyrev439 showed that a metal atom, decorated with halogen atoms whose number exceeds the

3.1. Jellium Shell Closure Rule

The jellium model treats a metal cluster as a sum of two interacting subsystems: the valence electron subsystem and the subsystem of a positively charged ionic core. The valence AC


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electron subsystem moves in the field, created by itself, as well as the field created by the subsystem of a positively charged ionic core. Within the ordinary jellium model, the detailed ionic structure of the cluster is smeared out and substituted by a uniform and spherically symmetric distribution of the positive charge. The electrons respond to this uniform positive background and fill the successive angular momentum states in accordance with the rules of quantum mechanics. In reality, this is not the case, even for simple metals. In addition to screening effects, the electrostatic potential arising from the discrete ions in a cluster is not spherically symmetric. Hence, the energy shells and degeneracies given by a spherical jellium model will be affected by the crystal field, the so-called Jahn− Teller effect. This could cause a reordering of the energy levels and consequently change the shell structure, leading to a modified jellium picture such as ellipsoidal jellium. The interplay between structure and electron filling can bring out new physics, such as in magnetism.454 As clusters begin to grow, the energy gaps between the angular momentum states near the Fermi level become comparable to the energy level shift due to the Jahn−Teller distortion. In addition, in a large cluster, the geometry could play a more important role in the magic numbers as the sp hybridizations are cluster size-dependent. The jellium model was originally developed for clusters of alkali metals, wherein each atom contributes one electron and the shell closures occur in clusters containing the numbers of atoms in this series. Later, it was extended to alkaline-earth metal clusters, Al clusters, coinage metal clusters, and ligated clusters. We note that in crystalline form the Fermi surfaces of simple metals such as Mg and Al are not spherical. In the following section, we present a select few examples of nearly free-electron systems. 3.1.1. Na Clusters. A seminal experiment, which brought the concept of magic numbers into focus, is the experiment, discussed earlier, on Na clusters by Knight and co-workers.150 In the mass spectra of Na clusters, in Figure 3a, one immediately notices the conspicuous peaks at cluster sizes corresponding to 8, 20, 40, 58, ... Na atoms. These peaks were termed as “magic” clusters. The origin of these magic numbers was explained in terms of the electronic shell closure in the spherical jellium model. On the basis of the assumption of the jellium model that the electrons in Na clusters are free-electron-like, one would ask the following questions: How metallic are the Na clusters, and what is the smallest size of a Na cluster that displays metallicity? One approach to finding answers to these questions is to study the dipole moment of Na clusters. This is a sound starting point because the free electrons in a metal, likely, rearrange themselves to eliminate any inhomogeneity and to quench any electric dipoles. This means that a metal cluster cannot support a nonzero dipole in its interior, and the interior atoms, in metallic clusters, should be strongly screened from an applied external field. Screening occurs, due to the charge induced at the cluster’s surface. de Heer and his collaborators carried out cryogenic cluster beam experiments455 as well as electric deflection measurement to study this problem. By analyzing the response of small sodium clusters to static electric fields, the authors found the electric dipole moments to nearly vanish (see Figure 61), even for clusters as small as the sodium trimer. This provided the experimental evidence that the electric fields surrounding alkali clusters are very small, as expected for a classical metallic object. To better understand

Figure 61. Dipole moments for Na clusters at 20 K. Adapted with permission from ref 455. Copyright 2011 American Physical Society.

the experimental results, Ma and her co-workers performed comprehensive calculations,456 focusing on the site-specific static polarizability response of Nan clusters for n up to 80. The cluster structures stem from extensive searches of the total energy landscape. The analysis involved partitioning the total cluster polarizability exactly into site (or atomic) contributions, and decomposing the polarizability into local (or dipole) and charge transfer contributions. The computed total polarizabilities, found by the authors, were in excellent agreement with recent experimental measurements, albeit a small overall shift. The site analysis provided clear evidence that interior atoms in sodium clusters are strongly screened from an applied external field, induced by the charge on the cluster surface. Therefore, Na clusters are well described by the jellium model, and the metallic behavior exists down to the smallest sizes. In the following, we focus on the Na20 cluster, corresponding to the highest peak in the mass spectra in Figure 3, and we analyze its structure−property relationship. Recently, using particle swarm optimization (CALYPSO) method, Sun at al.457 identified the ground-state geometry of Na20 to have C3 symmetry, as shown in Figure 62. The molecular orbitals are

Figure 62. Ground-state geometry of Na20. Adapted with permission from ref 457. Copyright 2017 American Chemical Society.

plotted in Figure 63. In this plot, one can see that the nondegenerate HOMO is primarily a 2s-type atomic orbital, which corresponds to the spherical shell structure. However, the LUMO resembles an f-type atomic orbital. The molecular orbitals of HOMO−q (q = 1−5) are d-type atomic orbitals: the HOMO−3 orbital exhibits the distinctive dz2 direction, while the other directions of d-atomic orbitals (such as dxy, dyz, dx2−y2, and dxz) are featured in HOMO−1, HOMO−2, HOMO−4, and HOMO−5, respectively. The HOMO−6 to HOMO−8 orbitals have p-type atomic orbital character (HOMO−8, for instance, can be assigned the pz orbital). Finally, the HOMO−9 AD


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magic numbers are followed by strong dips in the spectrum. Especially, clusters having 22, 37, 57, 58, 60, 71, 93, 108, and 148 atoms appear with extremely low intensity. In the lower size range, the abundance distributions can be explained by electronic shell structure. The associated electron delocalization, that is, metallic bonding, is found to occur around n = 20 atoms. Mgn clusters have resolved crossings of electronic levels at the highest-occupied molecular orbital, which result in additional magic numbers, as compared to the alkali metals, for example, Mg40 with 80 electrons. This specific electronic shell structure is also present in the intensity pattern of doubly charged Mgn. For larger clusters with n > 92, coexistence of electronic shell effects and geometrical packing is observed. This is shown in Figure 64.

Figure 63. Molecular orbitals and energy levels of neutral Na20 cluster. The HOMO−LUMO energy gap is indicated (in green). Adapted with permission from ref 457. Copyright 2017 American Chemical Society.

orbital is a classical s-type atomic orbital within the spherical shell structure, and the orbital is formed by σ Na−Na bonds. Therefore, the neutral Na20 cluster’s electronic structure is best described as 1S21P61D102S2, which is consistent with the HOMO, LUMO, and lower occupied molecular orbitals. The level degeneracies are broken by the nonspherical cluster geometry. However, some degeneracies remain; the molecular orbitals of HOMO−1, HOMO−2, as well as HOMO−4, HOMO−5, HOMO−6, and HOMO−7 are degenerate. The large gap between the superatom’s 2S and 1F orbitals leads to a sizable HOMO−LUMO gap of 1.43 eV. In fact, in the studied size range of 10−25 atoms, Na20 was found to have the largest HOMO−LUMO gap. This is consistent with the mass spectra where Na20 has the highest peak. Thus, Na20 cluster can be viewed as a typical model system described by the spherical jellium model. 3.1.2. Mg Clusters. Next, we discuss Mg, which is adjacent to Na, in the periodic table. Although bulk Mg is metallic, small Mg clusters are not necessarily metallic. Because of the 3s2 valence electron configuration of the Mg atom, a van der Waalstype bonding exists, not only for the dimer but also for larger clusters. A point of interest is the size at which an Mg cluster becomes metallic, with jellium shell structures. Because the electrical conductivity cannot be directly investigated for free clusters, it is difficult to define an appropriate experimental observable, and so it is necessary to use a special technique. Thomas et al.357 measured the photoelectron spectra of massselected magnesium cluster anions, Mgn (n = 3−35), and found that the s−p band gaps vanished at n = 18, signaling the onset of metallic behavior. A theoretical study, by Acioli and Jellinek,458 quantitatively reproduced most of the pertinent spectroscopic results derived from photoelectron experiments. They found that the degree of p character of the charge distribution in the neutrals is generally different from that in the anions, and the degree of p character in the neutrals approached that in their anionic counterparts, as the size of the clusters increased. Later, Diederich et al.459 prepared Mg clusters in helium nanodroplets and measured their abundance with high precision. Intense peaks were observed at n = 20, 30, 35, 40, 47, 56, 59, 69, 74, 99, 106, 147, and 178. Some of these

Figure 64. Development of the electronic levels for Mg cluster (the lower panel). Empty levels are represented by white bars, whereas the color gradient indicates a rising or shrinking occupation. Completely filled shells are indicated by solid bars. In the case of a level crossing, partly filled shells transfer all of their electrons to a high-S state whose binding energy increases. This is shown in the diagram as an overlapping of levels. If an energy level is fully occupied, the corresponding number of electrons is given, and the position is marked by a line extending into the mass spectra. Comparing both diagrams in the picture, one finds that almost all intensity maxima in the range up to n = 135 can be explained by electronic shell effects. Adapted with permission from ref 459. Copyright 2005 American Physical Society.

3.1.3. Al13− Cluster. Another classic example of a metal cluster, which can be described by the jellium model, Al13−, has unique stability and was originally observed by Castleman and co-workers.359 Al13, with 39 electrons, needs one extra electron to fill its 2P5 shell, just as a halogen atom needs one extra electron to close its np5 atomic shell. Thus, Al13 should behave as a halogen atom and thus could be regarded as a superatom. Indeed, the measured electron affinity460 of Al13, at 3.62 eV, confirmed that Al13 behaves as a Cl atom, whose electron affinity is also 3.6 eV. It should be mentioned that Al13− is sometimes referred to as a superhalogen in the literature. Because its electron affinity does not exceed that of Cl, it is more appropriate to term Al13 as a superatom, mimicking halogens. To better understand the electronic shell structure of an Al13− cluster, Figure 65 shows the energy level and orbitals, calculated using the density functional theory at the B3LYP level with 6-31G(D) (valence) double-ζ quality basis functions, AE


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lead to the existence of low-lying electronic states with strong collective character. The charge analysis of the ground and excited states suggests that most of the computed transitions are due to delocalized electrons on the cluster surface, with a minute participation from the electrons on the central aluminum. Thus, the low energy, collective excitations in Al13− may be characterized as surface plasmon-like transitions. Furthermore, the excitations can be rationalized by the jellium model. The main features of the simulated spectrum are found to be associated with transitions to triply degenerate excited states, having T1u symmetry. For example, the peak at 5.2 eV, with an oscillator strength of f = 0.84, is mainly obtained as electronic excitations from the highest occupied t1u and gu orbital sets (corresponding to the F orbitals in the jellium model), to hg and ag empty orbitals (corresponding to D and S orbitals in the jellium model). The next intense peak at 6.5 eV ( f = 0.93) corresponds to orbital transitions from the highest occupied gu orbitals to higher hg virtual orbitals, while the peak at 8.5 eV (f = 1.21) may be described as a variety of linear combinations of hg → gu and t1u → ag orbital transitions. This state corresponds to the collective excitation of ∼7 electron− hole pairs, that is, with the effective participation of 14 electrons, corresponding to one electron per Al atom plus the additional electron of the anionic Al13− cluster. 3.1.4. Au20 Cluster. Another cluster, having closed electronic shells and identified as a superatom, is Au20. Figure 66 shows the photoelectron spectra from which one can see that the HOMO−LUMO gap of Au20465 is 1.77 eV, about 0.2 eV greater than that of C60 (1.57 eV). Moreover, the electron affinity (EA) of Au20 is 2.745 ± 0.015 eV, which is higher than

Figure 65. Energy levels and orbitals of Al13− cluster. Adapted with permission from ref 461. Copyright 2008 The Royal Society of Chemistry.

extended by adding one set of polarization (d) functions.461 One can see that for the 40 valence electrons in the icosahedral Al13− cluster, the inner 20 valence electrons sequentially occupy 1S (A1g), 1P (T1u), 1D (Hg), and 2S (A1g). Furthermore, it can be seen that the remaining 20 electrons in Al13− are in the ungerade 1F and 2P MOs. These are very closely spaced and intermixed, relating to their similar ungerade parities. A natural consequence of Al13 mimicking the chemistry of a halogen is that it could react with an alkali atom, such as K, to form a salt-like molecule, K+Al13−. This is similar to Cl, reacting with K, to form K+Cl−. One can view this from another perspective; K acts as a ligand that stabilizes the Al13 metallic core, by donating an electron. In the later part of this Review, we will discuss how the stability of ligated Au clusters is currently being explained by the role of ligands. Khanna and Jena462 predicted that KAl13 can be a stable cluster, bound by ionic interaction. This was later experimentally verified by Bowen and co-workers463 using photoelectron spectroscopy. These authors observed KAl13 to have a large HOMO−LUMO gap, characteristic of a closed shell system. This led the authors to remark that this ionic molecule is a stepping-stone toward cluster-assembled materials. Another example, confirming the superatom property of Al13, is derived from the work of Bergeron et al.441 on Al13I2−. This cluster was shown to behave as an I3− ion with Al13 mimicking I. Similarly, Al14I3− can be viewed441 as Al142+(I−)3, where Al14 has two electrons more than are needed for the jellium shell closure; it behaves the same way as an alkaline-earth element. On the basis of the finding that Al nanocompounds exhibit plasmon resonances from the visible to the deep UV regions, these resonances in the Al13− cluster were recently explored by using time-dependent density functional theory (TDDFT).464 Because of the icosahedral symmetry of Al13 − and the large density of states, electronic excitations, from the highest occupied energy levels to the lowest unoccupied energy levels,

Figure 66. Photoelectron spectra of Au20−. (A) At 355 nm (3.496 eV). (B) At 266 nm (4.661 eV). (C) At 193 nm (6.424 eV). The 355 and 266 nm photons were from a Nd−yttrium−aluminum−garnet laser, and the 193 nm photons were from an ArF excimer laser. Adapted with permission from ref 465. Copyright 2003 American Association for the Advancement of Science. AF


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that of C60 (2.689 eV). This suggests that Au20 is even more electronegative than C60. By comparing the energies of different isomers, the ground-state geometry of Au20 was identified as a tetrahedral structure with Td symmetry, as shown in Figure 67. In the figure, the four faces of the Td structure possess an atomic configuration, similar to that of the Au(111) surface.

Figure 68. Structure and super atomic-molecule models of Au20 (TAu4). (b) Schematic representation for the superatom−atom D3S−s bonding of Au20 (TAu4). Adapted with permission from ref 467. Copyright 2014 The Royal Society of Chemistry.

Figure 67. Selected optimized Au20 structures. (A) Tetrahedral structure (Td). (B) Amorphous structure (C1). (C) Capped decahedron (C2v). (D) Planar structure (C2h). (E) Octahedral structure (Oh). (F) Dodecahedral structure (Ih). Adapted with permission from ref 465. Copyright 2003 American Association for the Advancement of Science.

comparing the canonical Kohn−Sham MO diagrams of TAu4 and OsH4, as given in Figure 69, where the electronic

The nonspherical geometry of Au20 poses a dilemma: How can one reconcile the cluster’s stability from the electronic structure point of view? Recall that the shell structure of a spherical jellium cluster is |1S2|1P6|1D102S2|1F142P6|..., which leads to magic numbers 2, 8, 20, 40. However, for an ellipsoidal shell model, the ordering of shells is different. The resulting difference in shell ordering yields different magic numbers.164 Cheng et al. addressed this by introducing a super valence bond (SVB) model.466 According to this model, a nonspherical metallic cluster can be viewed as a superatomic molecule, bonded by several spherical superatoms. The valence electrons are mainly delocalized over the region of each superatom, as opposed to the whole cluster volume. This follows the rule set forth by the jellium model. The superatoms can be open shell. Molecule-like shell-closure is achieved by sharing the electron pairs between the superatoms via super bonding. For Au20, the truncated tetrahedron Au16 core, of the Au20 pyramid, is sufficiently spherical, and may be viewed as an open-shell 16esuperatom (abbreviated as T), with electronic shells of T (1S21P61D8). Again, this follows the rules set forth by the jellium model. Molecule-like electronic shell-closure is achieved by four superatom−atom super bondings (T−Au). Furthermore, Au20 can be viewed as a super atomic molecule TAu4, as shown in Figure 68.467The 1S21P6 shells of T are super lonepairs (LPs) of the inner core. The five 1D orbitals are split into two sets in a tetrahedral field: a set of lower double-degenerate orbitals (1Dx2−y2,z2) and a set of higher triple-degenerate orbitals (1Dxy,yz,zx). The lower two 1Dx2−y2,z2 orbitals are fully filled as super LPs. The higher three 1Dxy,yz,zx orbitals form D3S hybridization with the 2S orbital. The four D3S super orbitals are bonded with the four-6s1 orbitals of the vertex Au atoms. This splits into four occupied lower bonding orbitals and four higher antibonding orbitals. Such a bonding pattern is analogous to the simple molecule OsH 4 , where Os [5s25p6(5d6s)8] is in the d3s hybridization state, bonded with four H atoms. Further confirmation may be achieved by

Figure 69. Comparison of the Kohn−Sham MO diagrams of (a) Au20 (TAu4) and (b) OsH4. Adapted with permission from ref 467. Copyright 2014 The Royal Society of Chemistry.

configurations and orbital shapes of TAu4 are in good agreement with those of OsH4. (1a1)2 and (1t2)6 MOs correspond to the 1S and 1P (5s and 5p) orbitals of T (Os). The (1e)4 MOs correspond to 1Dx2−y2,z2 (5dx2−y2,z2) orbitals of T (Os). The (2a1)2(2t2)6 MOs correspond to the four D3S−6s (d 3 s−1s) T−Au (Os−H) bonding orbitals. The four antibonding D3S−6s (d3s−1s) MOs are (3t2)0(3a1)0 for T− Au and (4t2)0(3a1)0 for Os−H. Recently, Muñoz-Castro and King revisited this system.468,469 They suggested that Au20 can, conveniently, be viewed as the combination of concentric structures, denoted by [(Au4@ Au12)Au4], with considerable sharing of the electron density between the different concentric layers. The results are plotted in Figures 70 and 71. As the number of concentric structures AG


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Figure 70. Electronic structure of Td-[Au12], [Au16]4−, and [Au4]4−, denoting the contribution of the constituent polyhedrons to the formation of [Au4@Au16]4−. Adapted with permission from ref 468. Copyright 2017 American Chemical Society.

Figure 71. Electronic structure of the capping Au atoms, [Au20] and [Au16]4−, denoting the contribution of the constituent fragments to the formation of [Au4]. White boxes denote the 5d block. Adapted with permission from ref 468. Copyright 2017 American Chemical Society.

increases from [Au4] → [Au4 @Au12] → Au20, the super atomic shells are, consequently, expanded as 1S1P → 1S1P1D2S2P1F → 1S1P2S1D2P3S1F3P. Au20 is formed by capping the Au16 core, with four independent Au atoms. The resulting neutral Au20 cluster retains an electronic configuration, similar to that of [Au16]4−, but with a slightly different shell order, given by the a12t26a12e4t26 electronic configuration under the Td point group. This corresponds to a 1S21P62S21D10 superatomic electronic configuration. Furthermore, natural population analyses indicate an overall charge distribution of Au4+0.187Au12+0.272Au4−0.460, suggesting that the Au16 core is able to share its electron density with the four capping gold atoms. Because of tetrahedral geometry, Au20 exhibits coupling between super atomic shells of different angular momenta, driven by symmetry restrictions of the Td point group. Here, the very symmetric s-type super atomic shell (l = 0) spans as the a1 irreducible representation (irrep), which can potentially couple with the a1 part of the F-type shell (Fxyz). In the t2 irrep, the coupling between the p-type, part of the d-type, the dγ orbitals (dxy, dxz, and dyz), and some f-type functions is allowed. Because of the electronic shell closure, with 20 valence electrons, Au20 is highly stable on some substrates. For example, when soft-landed on an amorphous carbon substrate, the cluster retained its gas-phase geometry (see Figure 72). However, it is unlikely that the Au20 clusters could be used to form a cluster-assembled material. Because of its metallic character, Au20 clusters are expected to coalesce when brought

Figure 72. Direct atomic imaging and dynamical fluctuations of the tetrahedral Au20 cluster soft-landed on amorphous carbon substrate. Adapted with permission from ref 470. Copyright 2013 The Royal Society of Chemistry.

into the vicinity of each other, destroying their individual identity. 3.1.5. Ligated Clusters. Atomic clusters can be prevented from coalescing, if they are protected by ligands. Here too, AH


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Table 1. Examples of Ligated Clusters with Effective Number (ne) of Valence Electrons for Jellium Shell Closure ne 2 8 18 34 40 58 138

examples

ref

Ag14(SR)12(PR)8, Ag16(SR)14(dppe)4, {Au34[Fe(CO)3]6[Fe(CO)4]8}8− Au11X3(PR3)7, [Au13Cl2(PR3)10]3+, [Au25(PET)18]−, Au28(SR)20, Al4(C5Me5)4, Al4(SiC(CH3)3)4, [Au13Cu2(PR)6(SPy)6]+, [Au13Cu4(PR2Py)4(SR)8]+, [Au13Cu8(SPy)12]+, [Au11(dpdp)6]3+, [Ag21(S2P(O/Pr)2)12]+ [Ag44(SR)30]4−, [Ag44(SR)30]4−, [Au12Ag32(SR)30]4−, [Au12+nCu32(SR)30+n]4− (n = 0,2,4,6), Au24Ag20(SR)4(C2Ph)20Cl2 [Au39(PR3)14Cl6]−, Au68(SR)34, [Au67(SR)35]2−, Au39Cl6(PH3)14 {Ge9[Si(SiMe3)3]3}−, SiAl14(C5Me5)6 Au102(p-MBA)44, Au102(SMe)44, [GaGa11{GaN(SiMe3)2}11] Al50(C5Me5)12

474−476 477−484 485−488 489−491 492,493 494,495 453

occupying the central site, reminiscent of a stuffed fullerene structure. The stability of Ti@Si16 was attributed to the electronic shell closure, in the jellium model, by assuming that each Si atom donates one electron and the Ti atom donates four electrons. With 20 electrons, the 1S21P61D102S2 orbitals are full. Later, the chemical stability and large HOMO−LUMO gap of this cluster were confirmed by the experimental work of Nakajima and co-workers.497 The authors measured the HOMO−LUMO gap to be 1.9 eV and found the cluster to be unreactive toward F2. More so, the authors showed that Ti@ Si16 could be formed by fine-tuning the experimental conditions to resemble the conditions required for the formation of C60. Clusters such as Sc@Si16− and V@Si16+ are isoelectronic with Ti@Si16. Furthermore, the clusters were found to exhibit pronounced peaks in the mass spectra (see Figure 73). In this

electron-counting rules play a role in determining the preferred cluster size. This field has attracted considerable attention, as the number of ligands and the size of the metallic core can be simultaneously varied to design a superatom. Consider, for example, a simple metal cluster with a core corresponding to a magic number of atoms, NC, and having NS number of surface atoms. One can decorate it with ligands that can interact with the valence electrons from these NS surface atoms, forming a covalent or ionic bond. Such a ligated cluster will then have unusual stability, because it has the right number of electrons in its core to close electronic shells. Hakkinen and co-workers453,471 have used this approach to explain the unusual stability of some ligated Al and Au clusters. For example, the unusual stability of Al50Cp*12 can be explained by realizing that 12 Cp* ligands withdraw 12 electrons from the Al50 core, thus leaving 50 × 3 − 12 = 138 electrons. This is just enough electrons to close the 1I superatom jellium shell. Similarly, the authors471 showed that ligand-protected Au clusters, satisfying electron shell closure at 8, 34, and 58, can, indeed, be synthesized. The superatom concept, of ligated Au clusters, has been verified experimentally. For example, using differential scanning calorimetry, Tofanelli and Ackerson447 showed that oxidation of Au25(SR)18−, which has 8 (=25 + 1 − 18) electrons and hence a closed jellium shell, led to open-shell radical 1S21P5 and diradical 1S21P4, with decreased thermal stability. Experiments472 on the structure of thiol monolayer-protected Au nanoparticle provided support to the concept of superatoms. Following a similar rule, the design of As7 and As11, with cryptated alkali atoms, has also led to the synthesis of bulk cluster-assembled materials.442,444 Examples of ligated clusters consistent with the jellium model are given in Table 1. We should alert the readers that extending the superatom concept to ligated cluster is not immune from controversy. It depends upon how strongly the ligands interact with the metal cluster and how they affect the geometry of the metal core. For example, Han and Jung445 pointed out that the geometrical symmetry of a cluster is a crucial factor in counting electrons. Shafai et al.450 showed that an Au13 cluster, when ligated to phosphines, has an icosahedral structure, while pure Au13 has a planar geometry. Zhang et al.,473 who studied the interaction of Aln clusters with S in AlnS− (n = 3−15) and AlnS2− (n = 7−15), showed that Al13 cannot be considered as a superatom when it interacts with S. 3.1.6. Metal Doped Si Clusters. The jellium shell closure rule, for designing superatoms, was not intended for application to clusters of transition metals, or Si and Ge, because these do not constitute a simple metal. Therefore, it came as a surprise when Kumar and Kawazoe335,496 found Ti@Si16 to be a very stable cluster, with a HOMO−LUMO gap of 2.36 eV. The cluster’s geometry was found to be a cage, with the Ti atom

Figure 73. Mass spectra showing size-selective formation of (A) TiSi16 neutrals, (B) ScSi16 anions, and (c) VSi16 cations. Adapted with permission from ref 497. Copyright 2005 American Chemical Society.

aspect, Sc@Si16 may be regarded as mimicking the chemistry of a halogen atom and V@Si16 mimicking an alkali atom. One can then imagine [Sc@Si16−][V@Si16+] forming a salt-like molecule. Recently, a study conducted by Jackson and Jellinek498 revealed a mechanism that may help us better understand why the jellium model can be applied to Si-based clusters. Using DFT, the authors calculated the total effective polarizabilities of silicon clusters, over a broad range of sizes up to n = 147 atoms. They found Si clusters possessing a higher degree of metallicity than bulk Si, as well as the existence of strong electrostatic screening in the cluster interior. This was confirmed by the calculated atomic polarizabilities, leading to a metallic-like response for Si clusters. 3.1.7. Real-Space Representation of Shell Structure in Jellium-like Clusters. It is believed that the electronic shell AI


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structure, in simple metal clusters, is a consequence of the freeelectron-like nature of the bonding in these systems. Furthermore, a jellium model provides a simple but helpful tool for understanding the stability of clusters of predominantly sp bonded metals. These metals include alkali (Li, Na, K), divalent (group II B), trivalent (group III), noble metals (Cu, Ag, Au), as well as their alloys. Therefore, understanding the behavior of electrons in jellium-like clusters is important to answer the following basic questions: How do “free” electrons really behave in jellium-type clusters, and is it possible that we may find some direct evidence for the free-electron behavior in jellium clusters? Recall that the most striking feature of jellium clusters is the shell structure of electrons. Can this be represented in real space? Sun and co-workers499 answered these questions by using the electron localization function (ELF),499which takes values in the range 0 < ELF < 1. Within this range, 1 corresponds to perfect localization, 0.5 corresponds to perfect delocalization, and 0 corresponds to very low density regions. For a single electron or paired electrons of antiparallel spins, the excess kinetic energy diminishes and the value of the ELF approaches 1. For a free-electron gas (jellium), ELF equals 0.5 at any density. Therefore, ELF provides a quantitative criterion to study the jellium-like behavior in metal clusters as well as the variation of the bonding nature, in different regions on an absolute scale. This is very important in clusters, because there may be significant variations in bonding with size as well as variations due to different bond lengths and/or atom distributions. A graphical representation of ELF provides a vivid description of electron localization in space, instead of the atom-centered description, such as seen in a Mulliken population analysis. To visualize the free-electron-like character in jellium clusters, ELF analyses were performed for Al6, Al12Si, Al12C, and Al54Si clusters having 18, 40, and 166 valence electrons, respectively, with closed-shell features. The results are in agreement with those expected from the jellium model. Figure 74 shows the ELF for Al12Si and Al12C clusters. For comparison, ELF for bulk aluminum, in the [100] crystalline plane, is also given. The jellium behavior of bulk aluminum, in the regions between the ions, can be seen clearly in clusters. The maximum value of ELF is 0.63; therefore, the deviation from the jellium behavior is small. This result may be used as a reference for understanding the behavior in clusters. Figure 74b shows the orientation of the plane (labeled as plane h) containing the central and four other vertex ions. The plane perpendicular to h is labeled as plane v, which contains only the central ion. The contour plots of ELF through planes h and v are also shown in Figure 74 as well as the isosurfaces, labeled as iso, for ELF = 0.9. In both clusters, one can see that in regions (labeled as J in the contour plots) between the vertex and the central ions, ELF has values close to 0.5, and it displays the jellium-like behavior with nearly delocalized charge distribution. Because bonding in both clusters is metallic, we can regard this as another direct evidence (besides the site polarizability as we discussed for Na clusters above) for the observed free-electronlike behavior. Outside of the vertex ions, electrons are highly localized, with a maximum ELF value of 0.94 and 0.92 for Al12Si and Al12C, respectively. This suggests that the surface valence electrons in Al12Si are more localized than those in the Al12C cluster. Similarly, ELF resulted in high values for Al surfaces,500 slightly off from the outermost surface plane, with a maximum value of 0.86 and 0.73 for the [110] and the close-packed [111]

Figure 74. Contour plots and iso-surfaces (denoted by iso) of ELF. h represents the horizontal plane [see panel (b)], and v is the the vertical plane perpendicular to h and having only the central atom. In isosurfaces, red color is for ELF, and the black balls are for ions. Adapted with permission from ref 499. Copyright 2001 American Physical Society.

surfaces. Therefore, electrons are much more localized on the cluster surface than on the crystal surface. The electronic shell structure of atoms has been clearly demonstrated in real space by using ELF.501,502 For example, ELF shows six peaks separated by five minima, corresponding to six electronic shells in Rn, as one may expect from the highest occupied principle quantum number state, 6p6. In Al6, Al12Si, Al12C, and Al54Si, states with 1, 2, 2, and 3 principle quantum numbers, respectively, are occupied. Accordingly, we expect 1, 2, 2, and 3 shells in ELF, respectively. This is indeed the case, as shown in Figure 75; the dotted line shows the center of the cluster. The Jahn−Teller distortions in Al6 and Al54Si make ELF asymmetric. However, an interesting finding is, even in Al6, the value of ELF at the center is close to 0.5; therefore, the behavior is very close to the behavior it would have in the jellium model. It is interesting to note that the free-electron-like behavior in the jellium clusters, provided by ELF, is consistent with the behavior obtained from the dipole moment. Note that it is only in the interior region of the cluster that ELF shows jellium behavior. In the same region, the dipole moment becomes zero as the free-electrons rearrange themselves to eliminate any inhomogeneity. Additionally, we have found the radius (R) of the jellium sphere to increase linearly with the subshell numer ns. This is shown in Figure 76 where Al6, Al12Si, and Al54Si have 3, 6, and 15 subshells according to their electronic configurations of 1S21P61D10, 1S21P61D101F142S22P6, and 1S21P61D101F141G181H221I262S22P62D102F142G183S23P63D10, respectively. As Jena435 previously discussed, there are other electroncounting rules, in addition to the jellium shell closure rule, which can be used to design superatoms. These rules are the octet rule, the 18-electron rule, the Wade−Mingos rule, and the AJ


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noble gas atoms arises because of the closing of their outer s2 and p6 shells. This entails large HOMO−LUMO gaps for the atoms. Because halogen and alkali atoms have, respectively, one less and one more electron than is needed to complete this octet rule, they are chemically reactive. In chemical reactions, halogens and alkalis combine to give each of the atoms eight electrons, so that a noble gas configuration may be achieved. The design of superhalogens and superalkalis, by Gutsev and Boldyrev,439,440 made use of this octet electron-counting rule. Consider, for example, LiF2. This cluster lacks one electron for shell closing; hence, LiF2 comparatively has the same chemistry as F. The exception is that the extra electron is distributed over a larger phase space, distributed over two F atoms. This distribution reduces electron−electron repulsion and lowers the total energy, thus stabilizing the negative ion further. For example, the electron affinity (EA) of LiF2 is 5.45 eV,506 which is larger than that of the F atom, 3.4 eV. In Figure 77, we show

Figure 75. Real-space representation of shell structures by plotting ELF along the diagonal direction of the cubic supercell, and the radial distance (R) is measured from a cubic cell corner. The dotted lines specify the center of the cluster. Adapted with permission from ref 499. Copyright 2001 American Physical Society. Figure 77. Electron affinity of coinage metal atoms decorated with F. Adapted with permission from ref 507. Copyright 2010 American Chemical Society.

the electron affinities of coinage metal (M = Cu, Ag, Au) atoms decorated with multiple F atoms.507 Note, as soon as the clusters contain two or more F atoms, the EAs of MFn (n ≥ 2) clusters exceed the EA of F, and the clusters behave like superhalogens. It is possible to attach up to seven F atoms, with electron affinities continuing to rise until reaching an asymptotic value; for AuF6, the EA is as high as 8.6 eV. It is important to note that Au exhibits multiple valences, 1, 3, and 5. Indeed, CsAuF6, where Au is penta-valent, does exist. The use of the octet rule, when designing superhalogens, can also be extended to nonmetallic species. Consider, for example, BH4. B, being trivalent, needs an extra electron to satisfy the octet rule. With a vertical detachment energy of 4.42 eV,508 BH4 can be regarded as a superhalogen. Similarly, CN behaves as a halogen atom and, in the literature, is referred to as a pseudohalogen. However, according to the definition above, with an electron affinity of 3.86 eV,509 CN could be considered as a superhalogen. In a similar vein, consider Li3O. This cluster has one electron more than needed to satisfy the octet rule; hence, it should behave as an alkali atom. In analogy with the superhalogens, the energy needed to remove this electron, that is, the ionization potential (IP), should be less than that of an alkali atom.

Figure 76. Radius (R) of jellium sphere as a function of the subshell number ns.

aromatic rule, which have existed in chemistry for a long time and account for the stability and/or reactivity of molecular species. In the following, we discuss how superatoms may be designed, by applying these rules. 3.2. Octet Shell Closure Rule

Developed in the early 1900s, the octet rule503−505 is a simple electron-counting rule that describes the chemistry of elements with a low atomic number ( 1. Figure 79. Al(BH4)3 (left panel) and KAl(BH4)4 (right panel). Adapted with permission from ref 435. Copyright 2013 American Chemical Society (courtesy of D. Knight and R. Zidan, private communication).

Figure 78. Electron affinity (EA) of Au(BO2)n as a function of n (black line). Adapted with permission from ref 435. Copyright 2013 American Chemical Society.

Here, note that BO2, with an electron affinity of 4.32 eV, is a superhalogen due to the octet rule and BO2−, being isoelectronic with CO2, is a very stable anion. One can synthesize novel salts by using hyperhalogens as building blocks. This was demonstrated in the recent synthesis532 of KAl(BH4)4. We note that Al(BH4)3, which contains 16.8 wt % hydrogen, could, in principle, be a good hydrogen storage material; however, this is a volatile and pyrophoric liquid, making it unsafe for commercial use. Knight et al.532 showed that Al(BH4)3 can be converted into a safer material by adding an additional BH4 unit to the cluster and reacting the resulting Al(BH4)4 hyperhalogen with K. In Figure 79, we show that KAl(BH4)4 is a stable solid at room temperature. The crystal structure of KAl(BH4)4 confirms that the geometry of BH4, in the crystal, is almost identical to the geometry of BH4− in the gas phase. The structure of a superatom in the gas phase can remain close to that in a bulk matter. This was also demonstrated earlier in the experiment by Kasuya et al.533 on ultrastable nanocluster (CdSe)34 (see Figure 80). Peppernick et al.534 used a different electron-counting rule to design superatoms. Using photoelectron spectroscopy of negatively charged ions, of group 10 elements, the authors showed that Ni−, Pd−, and Pt− are very similar to their isoelectronic molecular counterpart MX− (M = Ti, Zr, W; X =

Figure 80. Structure of (CdSe)34. Adapted with permission from ref 533. Copyright 2004 Nature Publishing Group.

O, C). To illustrate this electron-counting rule, consider Pd versus ZrO. Note that the electronic configurations of Pd and Zr atom, respectively, are [Kr]4d10 and [Kr]5s24d2. Thus, Zr lacks the six electrons needed to be isoelectronic with Pd. The needed six electrons can be provided by O, whose electronic configuration is [He] 2s22p4. If one can demonstrate that ZrO has the same chemistry as Pd, then ZrO, instead of a more expensive Pd, can be used as a catalyst. Later, Castleman and co-workers535 demonstrated the similarity of the interaction between organic molecules and Pd+ to that with ZrO+. However, a subsequent calculation,536 involving larger clusters of Pd and ZrO, showed that the conclusions from dimer studies cannot always be translated to realistic catalysts that contain many atoms. As described in the jellium model, the octet rule also has limitations; molecules can form even though they do not satisfy the octet rule. These include, for example, NO (which has an odd number of valence electrons), BH3 and BF3 (which are electron-deficient), and PCl5, SF4, and SF6 (which are electron rich). AL


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Figure 81. PES spectrum of Al4H6 anion (left panel) and mass spectra of Al4Hm−. The inset shows the geometry of Al4H6−. Adapted with permission from ref 551. Copyright 2007 American Association for the Advancement of Science.

3.3. 18-Electron Shell Closure Rule

from the metal atom and the rest of the electrons come from the ligands. Therefore, only a limited number of combinations, of metal atoms and ligands, can satisfy the 18e count. This rule works best for hydrides and carbonyls, because these are sterically small, high-field ligands, while it works least for highvalent metals with weak-field ligands. However, there are many cases where a stable complex does not obey the 18-electron rule; examples are WMe6, Pt(PCy3)2, [Mn(H2O)6]2+, and CoCp2, having 12, 14, 17, and 19 electrons, respectively.

Transition metal complexes, containing 18-electrons, are considered to have achieved the electronic shell closures,537 analogous to the noble gas atoms. In the former, 18 electrons can fill the s2p6d10 electronic configuration, just as eight electrons fill the s2p6 electronic configuration of noble gas atoms discussed in the above. Therefore, one can use the 18electron rule to design superatoms involving transition metal atoms. Well-known examples of the transition metal-based compounds, stabilized by the 18-electron rule, are Cr(C6H6)2, Fe(C5H5)2, [Co(NH3)5Cl]2+, Mo(CO)6, [Fe(CN)6]4−, etc. Pyykko and Runenberg538 used the 18-electron rule to show that Au12W, with a HOMO−LUMO gap of 3.0 eV, can be a very stable cluster. This prediction was later verified by Wang and co-workers539 using photoelectron spectroscopy experiments. Although the measured HOMO−LUMO gap (1.68 eV) was found to be much smaller than the predicted value, it is very close to that of another very stable cluster, C60. This design rule was further extended538 to charged clusters, such as TaAu12−, which also contains 18-electrons. Thus, such clusters can be very stable. Note that here the extra electron can delocalize over 12 Au atoms, similar to the superhalogens discussed in the previous section. Because Au, with an electron affinity of 2.31 eV, is the most electronegative metal atom in the periodic table, TaAu12 should be a superhalogen. The measured electron affinity540 of 3.76 eV indeed confirms that TaAu12 is an all-metal superhalogen. Hiura et al.333 used the 18-electron rule to explain the unusual stability of WSi12, which they observed in a systematic study of hydrogenated Si clusters, interacting with a single metal atom M (M = Hf, Ta, W, Re, Ir, etc.). We should mention that Beck541 earlier had studied the mass spectra of MSi15 (M = Cr, Mo, W) clusters, but not the geometries of these species. Hiura et al. determined the geometry of WSi12 to be similar to that of Cr(C6H6)2, a W atom being sandwiched between two Si6 hexagons. The authors assumed each Si atom to contribute one electron, thus making the total electron count of WSi12 to be 18. The authors further suggested that such a cluster might be a suitable building block for cluster-assembled materials. The 18-electronic shell closure was later shown334 to be responsible for the total quenching of the magnetic moment of CrSi12, which is isoelectronic with WSi12. In the next section, we discuss, in more detail, a different origin for the stability of WSi12. It is worth noting that in ligated clusters, for the 18 electrons to fill the nine orbitals, some electrons would have to come

3.4. Wade−Mingos Shell Closure Rule

Wade542,543 and Mingos544,545 developed a set of new electroncounting rules to account for the structure and bonding of polyhedral borane clusters (BnHm). Because B is electron deficient, it does not have a sufficient number of electrons for conventional covalent bonding between adjacent pairs of atoms, and so the concept of multicenter bonds was introduced. In this concept, the valence electrons are separated into external and skeletal electrons. The former form covalent bonds with external ligands (in this case H), while the latter are responsible for cage bonding. According to this polyhedral skeletal electron pair theory (PSEPT), popularly known as the Wade−Mingos rule, 2n + 2, 2n + 4, and 2n + 6 electrons are needed to form closo, nido, and arachno boranes, where n is the number of vertices in the borane polyhedron. A well-known example of the closo-borane cluster is B12H122−, which has a perfect icosahedral form (Figure 32) and is among the most stable dianions, in the gas phase. Each BH pair contains four valence electrons, two of which are involved in the covalent bond, while the other two contribute to cage bonding. Thus, the 12 BH pairs contribute 12 × 2 = 24 electrons to cage bonding. According to the Wade−Mingos rule, 26 (=2 × 12 + 2) electrons are needed to stabilize the cage; hence, B12H122− is a very stable cluster. The origin of the 2n + 2 electron-rule for closo-boranes is that they contain n bonding σ (in-plane) and one bonding π-like (out-of-plane) orbital, which are formed from the s-p-hybridized boron atomic orbitals. A consequence of the Wade−Mingos rule is that if one of the B atoms in BnHn2− is substituted with a C atom, a very stable monocarborane, CBn−1Hn−, could form. Indeed, this occurs, as monocarboranes with n = 7−12 have been isolated and structurally characterized.546−549 As an example of a superatom, we focus on CB11H12. Because this cluster requires one electron to satisfy the Wade−Mingos rule, CB11H12 should mimic the chemistry of a halogen. The added electron will be distributed over a large phase space, causing one to wonder if the electron AM


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Note that Mn is divalent and is expected to remain as Mn2+ in Mn@Sn12. Because of the 3d54s2 configuration of the Mn atom, the magnetic moment of Mn@Sn12 is expected to be 5 μB. Indeed, this is what Kandalam et al.351 found. To examine whether Mn@Sn12 could be used as the building block of a magnetic cluster assembled material, the authors351 studied the stability of the cage structure as two of these moieties are brought together, as well as the preferred magnetic configuration. Their results are shown in Figure 82. While

affinity of CB11H12 could exceed that of Cl, making it a superhalogen. This was found to be the case,550 and the calculated electron affinity of CB11H12 is 5.39 eV. Pathak et al.550 also found that MB12H12 (M = Li, Na, K, Rb, Cs) is representative of superhalogens. The authors continued on to illustrate how CB11H12 moieties can be used to produce hyperhalogens, and their computed electron affinities of M[CB11H12]2 indeed confirmed this expectation. We will discuss, in section 5, the roles these superhalogens can play in the design of halogen-free electrolytes for metal−ion batteries. Although Al and B belong to the same group of elements, the chemistry of Al is very different from the chemistry of B; Al does not form the same wide range of complexes with H as does B. To see if this rule applies to small Al clusters, Bowen and co-workers551 studied the interaction of H with small Al clusters produced in a PACIS source. The authors observed a vast array of, previously unknown, hydrogenated Al clusters. Among these, they noted the particular stability of Al4H6 in the mass spectra (Figure 81). The large HOMO−LUMO gap, 1.9 eV, obtained from photoelectron spectroscopy experiment, further confirmed its stability. First-principles calculation explained the stability of Al4H6, originating from the Wade− Mingos rule. Note that the geometry of Al4H6 is that of an Al tetrahedron, where four H atoms are radially bonded while the other two H atoms are bridge bonded, on opposite sides of the tetrahedron. There are 10 skeletal electrons in Al4H6, with eight electrons coming from the four AlH pairs and two electrons coming from the two bridge bonded H atoms. Thus, Al4H6 may be viewed as Al4H42−, mimicking the closo-boranes. Later, Henke et al.552 were able to synthesize a bulk material with Al4R6 (R = Br dimethylphosphine) as the building block. Sn122− and Pb122−, which are all-metal clusters, are also found to obey the Wade−Mingos rule. This can be understood by noting that Sn and Pb are isoelectronic with a BH pair. Accordingly, Sn122− and Pb122− are isoelectronic with B12H122−. In fact, experiment has confirmed the similarity of bonding pattern in Sn122− with that of B12H122− cage, where the 12 5s2 localized electron pairs in Sn12 replace the 12 B−H bonds.553 This discovery was the surprising result of a systematic study of Sn clusters, by Cui et al.,350 where the authors attempted to understand the nonmetal to metal transition of Sn clusters, as a function of size. In the process, they observed a remarkably simple photoelectron spectrum of Sn12−. The observed spectrum was different from its isoelectronic cousin Ge12−. The structure of Sn12− is a slightly distorted icosahedral cage, with C5v symmetry. On the other hand, Sn122−, synthesized in the form of KSn12− (K+Sn122−), is a very stable icosahedral cage with Ih symmetry. The PES spectrum was very similar to that of Sn12−. The authors termed Sn122− as stannaspherene, showing that its bonding pattern is similar to that of B12H122−. Note, with 26 valence electrons, Sn122− would satisfy the Wade− Mingos rule; however, unlike B12H122−, the diameter of the Sn122− cage was found to be 6.1 Å, large enough to accommodate a metal atom inside it. Four years prior, Kumar and Kawazoe554 showed M@Sn12 (M = Zn, Cd) to be a perfect icosahedron, with a large HOMO−LUMO gap, signifying its high stability. Later, Cui et al.555 synthesized M@Sn12 (M = Ti, V, Cr, Fe, Co, Ni, Cu, Y, Nb, Gd, Hf, Ta, Pt, Au), where the metal atom was found to occupy the endohedral position. The possibility that Mn@Sn12 could be magnetic, and serve as a building block of a magnetic cluster assembled material, was studied by Kandalam et al.351

Figure 82. Optimized structures of Mn@Sn12 dimer. (a) The lowest energy structure; (b and c) the higher energy isomers. The relative energies ΔE of the isomers are also shown. Adapted with permission from ref 351. Copyright 2008 American Institute of Physics.

the cage structures undergo a slight distortion, the integrity of the cage is maintained and the two icosahedra form a stable dimer. The coupling between the two cages is, however, antiferromagnetic, with each Mn atom carrying a magnetic moment of 4 μB. The ferromagnetic state lies 0.11 eV above the antiferromagnetic ground state. Cui et al.556 also reported that Pb122−, which is isoelectronic with Sn122−, has a hollow cage structure. They termed this cluster plumbaspherene. With a diameter of 6.29 Å, plumbaspherene is large enough to accommodate an endohedral metal atom. This is consistent with the earlier gas-phase synthesis557 of AlPb12+, which can be viewed as Al3+Pb122−. Furthermore, we note that a series of endohedral cage compounds, M@Pb122− (M = Ni, Pd, Pt), stabilized by K+ counterions, have been synthesized in solution and in crystalline form. Both of their icosahedral symmetries have been confirmed558 by X-ray diffraction and NMR experiments. Inspired by these compounds, Wang and co-workers559 were later able to synthesize a new Pd2@Sn184− cluster (Figure 83), which was crystallized as a [(2,2,2-crypt)K]4[Pd2@Sn18]·3ED salt and was characterized by X-ray diffraction. AN


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groups IV and V metals. Later, Castleman and co-workers566 observed [Na4Bi3]+ as a magic number in the mass spectra of NanBim+ clusters. They attributed its unusual stability to the presence of a Bi33− ion, which was identified as a Zintl ion. Subsequent photoelectron spectroscopy experiments and calculations have identified350,556,567,568 the presence of Zintl ions, Sn44−, Ga42−, Sn122−, and Pb122−, in cluster anions of [Na3Sn4]−, [NaGa4]−, [KSn12]−, and [KPb12]−, respectively. As mentioned in the previous section, the stability of deltahedral Zintl anions is usually rationalized by the Wade− Mingos rule, but this is not always the case. The exceptions to this rule include Ni2Sn7Bi53−,569 [Ge9(η4-Ni(CO))]3−,570 and [Ge9(η4-Pd(PPh3))]3−.571 Here, we present Zintl-like ions that have been designed, using jellium and 18-electron rules, and subsequently have been experimentally confirmed. Consider, for example, In117−, Si44−, and Pd44−, which have already been studied.572−574 These clusters contain 40, 20, and 20 electrons, respectively, and hence satisfy the electron shell closure rule in the jellium model. Because the ions are multiply charged, one can regard them as Zintl-like. Note, in the jellium model, all of the valence electrons are involved in skeletal bond formation. Using a synergistic combination of photeletron spectroscopy and density functional theory, Bowen and co-workers575showed that aluminum moieties, with a selected set of sodium− aluminum clusters, are Zintl-like anions. The authors arrived at this conclusion by calculating the geometries of these clusters in both neutral and anionic states, validating these geometries by comparing the calculated vertical detachment energy and electron affinity with experiment, and then analyzing charge distribution in the clusters. As an example, consider Na2Al6. The calculated geometry of this cluster is given in Figure 84. Note that in the ground-state structures of both the neutral and the anion, the clusters are composed of an Al6 prism, with two Na atoms capping the faces of two, adjacent, four-member rings, which are composed of Al atoms. Because Na is more

Figure 83. Comparison of the structural evolution from Pd@Sn122− to Pd2@Sn184− to that from C60 to C70. Adapted with permission from ref 559. Copyright 2007 American Chemical Society.

3.5. Zintl-like Ions beyond the Wade−Mingos Rule

Zintl ions are multiply negatively charged clusters, belonging to the post-transition metal elements in groups 13, 14, and 15. Examples of Zintl ions, that have been studied since the 1930s,560,561 include Sn52−, Pb52−, Pb94−, Sb73−, and Bi42−.560,561 When countered with positively charged alkali or alkaline-earth atoms, Zintl ions form Zintl phase compounds. The Zintl ions can be extracted from the lattice and, when dissolved in appropriate solvents, can be used as reactants in solution chemistry. The potential use of Zintl compounds, as thermoelectric materials,562,563 has fueled renewed research interest in the search for new Zintl ions. The role of cluster science in this search was highlighted by the early works of Recknagel564 on Pb, Sb, and Bi clusters and by Duncan565 on binary clusters of

Figure 84. Three lowest energy isomers of the Na2Al6− cluster anion and the Na2Al6 neutral cluster along with their relative energies, ΔE (eV). Isomers 1−3 correspond to the Na2Al6− anion, while 4−6 correspond to the lowest energy isomers of neutral Na2Al6. The gray spheres represent aluminum atoms, and the purple spheres represent sodium atoms. All of the bond lengths are given in angstroms. The calculated Natural Population Analysis (NPA) charges are given in italics below each isomer. Adapted with permission from ref 575. Copyright 2014 American Institute of Physics. AO


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Figure 85. Optimized geometries of four possible [Ti@Au12]2− isomers with relative energies. Adapted with permission from ref 578. Copyright 2014 The Royal Society of Chemistry.

Figure 86. Left panel: Photoelectron spectra at 355 nm (3.496 eV) for (A) LiAl4−, (B) NaAl4−, and (C) CuAl4− and at 266 nm (4.661 eV) for (D) LiAl4−, (E) NaAl4−, and (F) CuAl4−. Right panel: Optimized structures of LiAl4−, NaAl4−, Al42− [at the CCSD(T)/6-311+G* level of theory], and CuAl4−. Adapted with permission from ref 592. Copyright 2001 American Association for the Advancement of Science.

Zhou et al.578 found that a large electron transfer still persists from Mg to [FeAu6], raising the possibility that the 18-electron rule can be used to design transition metal-based Zintl ions.The authors calculated the geometries and examined the stability of two clusters, [Ti@Au12]2− and [Ni@Au6]2−, both having 18valence electrons. In Figure 85, we present the ground-state geometries as well as those of higher lying isomers of [Ti@ Au12]2−. The ground-state structure has the icosahedric symmetry. To further confirm the stability, as a dianion, Zhou et al. studied the structure and electron distribution in Na2Ti@Au12. They found that the Na atoms remain in a +1 charge state. In section 4, we discuss the potential of the transition metal-based Zintl compounds in forming magnetic materials.

electro-positive than Al, it is expected that Na atoms will donate their charge to the Al6 moiety, making the cluster look like Na22+Al62−. The natural population analysis shows that a net charge of 1.54e, instead of 2.0e, is transferred to the Al6 moiety. Similar results are obtained from Na4Al5− and Na3Al12−. These clusters can be viewed as Na44+[Al55−] and Na33+[Al124−]. The aluminum moieties [Al55−] and [Al124−] appear Zintl-like, having 20 and 40 electrons, respectively. Moreover, they correspond to jellium shell closure. As described in the previous section, much of the earlier works on Zintl ions concentrated on groups 13, 14, and 15 post transition metal atoms. In the past decade, a few endohedral clusters with transition metal atoms embedded inside posttransition metal cages, such as Nb@As83−, Pd2@Sn184−, Ni@ Pb122−, Pt@Pb122−, and Ir@Sn123−, have been experimentally synthesized.558,559,576,577 While some of these may be identified as Zintl anions, obeying the Wade−Mingos rule, others may not. Here, we discuss the recent work of Zhou et al.,578 who used the 18-electron rule to design Zintl-like ions composed of transition metal atoms. The authors noted that Mg2FeH6 has already been synthesized.579,580 By using density functional theory, they showed that Mg2FeH6 can be viewed as a Zintl phase compound with the [FeH6]4− behaving as a Zintl anion, obeying the 18-electron rule. By replacing H with isovalent Au,

3.6. Huckel’s Rule of Aromaticity

Aromaticity is a chemical property associated with planar conjugated molecules, such as benzene, C6H6. Hückel581,582 developed a molecular orbital theory to describe the stability of benzene and showed that any planar conjugated monocyclic polyene that has (4n + 2) (n = 0, 1, 2, ...) π or nonbonding electrons will be aromatic and stable. For benzene, n = 1. One can also view the Huckel model as being equivalent to a ringlike jellium where the six electrons from the carbon atoms in benzene are delocalized over the ring. However, the debate for AP


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an understanding of the physical origin of aromaticity still continues, and various criteria of aromaticity have been put forth to gain a deeper insight into aromaticity.583−592 In a recent study of bimetallic clusters with chemical composition MAl4− (M = Li, Na, Cu), Li et al.592 reported evidence of aromaticity in all-metal systems. Using photoelectron spectroscopy and ab initio calculations, the authors showed that all MAl4− species possess pyramidal structures, with the M+ cation interacting with a square Al42− moiety (Figure 86). The authors computed the geometry of an isolated Al42−. Although such a cluster is unstable against the autodetachment of an electron, it was found to have the same square geometry as in MAl4− (M = Li, Na, Cu). The HOMO of MAl4−, as well as that of Al42−, was found to be doubly occupied and composed of a delocalized π orbital, with other molecular orbitals being either σ-type or lone pairs. The presence of two valence electrons is characteristic of aromaticity, with n = 0. Because of the high stability, aromatic molecules have low electron affinities (EA). In fact, the EA of benzene is negative, −1.15 eV,583 and the EAs of organic molecules, composed of carbon rings, seldom exceed 1.17 eV. In contrast, as discussed before, electron affinities of inorganic systems can exceed those of halogens, and superhalogen moieties, obeying the octet, 18electron, and Wade−Mingos rules, have been known to exist. Jena and co-workers593,594 wondered if it was possible to use the aromaticity rule to design organic molecules, with electron affinities that can approach or even exceed those of halogens. The authors noted that cyclopentadienyl (C5H5) has five π electrons, meaning it needs one more electron to be aromatic. Note that the electron affinity of C5H5, +1.80 eV, is significantly larger than that of C6H6. Thus, using C5H5 as the backbone, the authors calculated593,594 the electron affinity by replacing the H ligands of cyclopentadienyl with X = F, CF3, NCO, CN, and BO2, as well as by multiple benzo-annulations of cyclopentadienyl, in conjunction with substituting CH groups with isoelectronic N atoms. The electron affinities of these ligandmodified C5X5 moieties were found to be consistently higher than the electron affinities of C5H5. In the case of C5(CN)5, the electron affinity, at the CCSD level of theory, was found to approach a value of 5.30 eV. The electron affinities may increase when successively adding a benzene ring to C5H5, while replacing the CH group with N. The results are given in Figure 87. The figure shows that polyaromatic hydrocarbons, with CH groups replaced by N and two benzene rings attached to a C5H5 moiety, can reach the status of a superhalogen. Similar results are obtained by substituting a C atom in C6H6, with a trivalent atom (B, Al, Ga), as well as replacing the H ligand with F and CN, separately or simultaneously. These results are presented in Table 2.

Figure 87. EA (◆) and VDE (■) values of larger nitrogen replaced PAH molecules. Adapted with permission from ref 594. Copyright 2014 John Wiley & Sons, Inc.

Table 2. Calculated EA, VDE, and NICS (0/1) for Different 2π and 6π Electronic Systems at the B3LYP/6-31+G(d,p) Level of Theorya systems

EA [eV]

VDE [eV]

NICS(0) [ppm]

NICS(1) [ppm]

BC2H3 B2CH3 B2CF3 B2C(CN)3 C6H6 BC5H6 BC5F6 BC5(CN)6 AlC5H6 AlC5F6 AlC5(CN)6 GaC5H6 GaC5F6 GaC5(CN)6

−1.09 1.70 2.98 4.66 −1.30 2.27 3.19 5.87 1.79 2.70 5.21 1.85 2.88 5.31

−1.08 2.39 3.92 5.32 −1.10 2.35 3.49 5.93 1.84 2.92 5.27 1.91 3.11 5.36

−18.94 −16.70 −38.66 −23.28 −8.12 −5.85 −17.47 −8.71 −4.00 −15.00 −6.58 −4.20 −15.43 −6.83

−13.99 −14.27 −10.89 −13.34 −10.19 −7.85 −10.87 −8.40 −5.23 −9.57 −5.82 −5.81 −9.95 −6.30

a

Adapted with permission from ref 593. Copyright 2014 John Wiley & Sons, Inc.

cluster’s stability is attributed to Hückel’s rule of aromaticity. However, unlike C6H6, whose electron affinity is negative (−1.15 eV), the electron affinity of C6(CN)6 is 3.53 eV.551,554 Because BC5(CN)6 lacks one electron, it can gain high stability, once an extra electron is attached. Indeed, the electron affinity of BC5(CN)6 is 5.87 eV,593 making it a hyperhalogen. Note that the total number of electrons in Mn[BC5(CN)6]2 is 17. Of these 17 electrons, Mn, with an electronic configuration of 3d54s2, contruibutes 7 electrons, whereas each BC5(CN)6 only contributes 5 electrons. Thus, Mn[BC5(CN)6]2 needs an extra electron to satisfy the 18-electron rule. It is interesting that the addition of a single electron, simultaneously, satisfies the octet rule for CN, the aromaticity rule for BC5(CN)6, and the 18electron rule of Mn[BC5(CN)6]2. This results in an electron affinity of 6.40 eV, making Mn[BC5(CN)6]2 a superhyperhalogen. In Figure 88, we present the geometry of a neutral and an anionic Mn[BC5(CN)6]2 cluster. The extraordinary stability of Mn[BC5(CN)6]2− is further confirmed by noting that it takes 6.2 eV of energy to dissociate the cluster into Mn+ and two [BC5(CN)6]− ions. This is indicative of a strong ionic bond. The above discussion provides a recipe for creating moieties

3.7. Multiple Electron-Counting Rules

In the above sections, we outlined how different electroncounting rules can be individually used to design a cluster that mimicks the properties of an atom. Here, we discuss how Giri et al.595 applied multiple electron-counting rules, simultaneouly, to design extremely stable clusters. We focus on two examples, Mn[BC5(CN)6]2− and Cr[BC5(CN)6]22−. The design of these two clusters, simultaneously, makes use of the octet rule, the aromatic rule, and the 18-electron rule. As pointed out before, CN requires one extra electron to satisfy the octet rule. With an electron affinity of 3.86 eV,509 CN is regarded as a pseudo- or superhalogen. C6(CN)6 is isoelectronic with C6H6, and the AQ


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electron is bound by 5.3 eV. To the best of our knowledge, this is the most stable dianion, in the gas phase, to date. The reason for its enermous stability can again be traced to the fact that two electron-counting rules (octet and Wade−Mingos rule) are simultaneously satisfied. The reason why B12(CN)122− is more stable than Cr[BC5(CN)6]22− may be related to the cluster size as well as its symmetry. In Figure 90, we show that

Figure 88. Equilibrium geometries of (a) neutral and (b) anionic Mn[BC5(CN)6]2 cluster. Adapted with permission from ref 595. Copyright 2015 The Royal Society of Chemistry.

with ever increasing electron affinity. Another interesting aspect of the electron shell closure is that the magnetic moment of a Mn atom is completely quenched. The above approach can be extended to design very stable dianions. As pointed out before, a small isolated cluster, carrying multiple charges, may fragment or eject extra electrons. This is due to Coulomb repulsion. B12H122− is a rare example of a cluster that is stable in the gas phase. It takes 0.9 eV to detach the second electron. Is it possible, by taking advantage of multiple electron-counting rules, to stabilize a dianion that is even more stable than B12H122− in the gas phase? To examine this possibility, Giri et al.595 studied the structure as well as the stability of neutral and charged Cr[BC5(CN)6]2. It is necessary to note that because Cr has an electronic configuration of 3d54s1, Cr[BC5(CN)6]2 has 16 electrons. Having 16 electrons means the cluster will require two more electrons to, simultaneously, satisfy the octet, Hückel, and 18-electron rules. This cluster was found to be extremely stable as a dianion; it requires 2.58 eV to eject the second electron from Cr[BC5(CN)6]22−. Cr[BC5(CN)6]2 can thus be classified as a superhyperhalogen. The stability of Cr[BC5(CN)6]22− was further confirmed by molecular dynamics simulation at 300 K. Snapshots of the geometries, at different time intervals, are plotted in Figure 89. Note that there is very little change in the geometry, and Cr[BC5(CN)6]22− is thermally stable. In a recent study, Zhao et al.596 demonstrated the extraordinary stability of B12(CN)122−. Here, the second

Figure 90. Geometry of B12(CN)122− with Ih symmetry.

B12(CN)122− has a perfect icosahedral symmetry. The role of these stable negatively charged clusters, in forming the building blocks of energy materials, will be discussed in section 5. 3.8. Magnetic Superatoms

The superatoms with closed electronic shells have zero magnetic moment; hence, these superatoms are nonmagnetic. Clusters may also be designed to be magnetic by suitably choosing their size, shape, and composition. As illustrated earlier, clusters of nonmagnetic elements can become magnetic. The word “magnetic superatom” was first coined by Kumar and Kawazoe.597 The authors studied divalent metal (M)-doped XnM clusters (X = Si, Ge, and Sn; M = Be, Mg, Zn, Cd, and Mn; n = 8−14, and 14), using the pseudopotential plane wave method. They found Mn@X12 to be magnetic, with a magnetic moment of 5 μB, localized at the Mn site. Reveles et al.598 proposed a framework for designing magnetic superatoms by invoking systems that have both localized and delocalized electronic states. Localized electrons stabilize magnetic moments, while filled delocalized electrons contribute to stability. This idea has been used to design and synthesize a variety of magnetic superatoms. 3.8.1. VLi8. Lithium is the first alkali metal that may be considered an ideal prototype for simple metals; more so, Li presents a good starting point for a theoretical understanding of metal clusters. According to the jellium model, the eight valence electrons, from Li8, satisfy the electron shell closure requirement, and so it would be possible to construct magnetic superatoms by introducing transition metal (TM) atoms into a Li8 cluster. This expectation was confirmed in detailed DFT studies.599 Among the 3d transition metal atoms, from Sc to Zn, it was discovered that V, interacting with Li8, and having a square-pyramidal structure, exhibits the highest moment of 5 μB, whereas the moment on Ni in NiLi8 is quenched, as shown in Figure 91a. It is very intriguing that V, in the VLi8 cluster, has a magnetic moment of 5 μB, when considering the fact that a single V atom, having three unpaired electrons in the atomic configuration of 3d34s2, has a magnetic moment of 3 μB. Moreover, the VLi8 cluster (see Figure 91b) also has a large HOMO−LUMO gap of 0.779 eV and a high fragmentation energy of 1.80 eV. To better undertand its stability and properties, it is helpful to analyze the electronic structure. A VLi8 cluster contains 13 valence electrons, which does not

Figure 89. Molecular dynamics simulation of Cr[BC5(CN)6]22− cluster. Adapted with permission from ref 595. Copyright 2015 The Royal Society of Chemistry. AR


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Figure 91. (a) Magnetic moment of MLi8 (M = Sc, Ti, V, Cr, Mn, Fe, Co, Ni, Cu, Zn); (b) fluctuation of the potential energy of VLi8 over time; and (c) orbitals of VLi8. Adapted with permission from ref 599. Copyright 2013 American Chemical Society.

satisfy any electron-counting rules, discussed above. However, the electronic configuration of VLi8 can be considered within the jellium model. It consists of the five vanadium electrons, in excess of the closed-shell eight-electron configuration. The latter are paired, occupying the 1S and 1P shells. The extra five electrons, from the V atom, are then singly placed into each of the five orbitals of the 1D shell. This individual placement causes each of the electrons to be unpaired. Therefore, the VLi8 cluster has a half-filled D subshell, with a magnetic moment of 5 μB, and the remaining electrons form a filled shell of 1S21P6 electrons. With filled 1S21P6 and a half-filled 1D5 orbital, VLi8 is stable. Figure 91c shows the first two electrons occupying the lowest state, which is spread out over the whole VLi8 cluster, having a 1S superorbital character. The next majority states are two degenerate 1P states, having a 1Px and 1Py character, whereas the 1Pz state is pushed 0.43 eV higher in energy, and so the VLi8 cluster can be regarded as a compression of the spherical jellium in the z direction. The next higher energy orbitals, above the 1P orbitals, are 1D orbitals. The oblate shape of the cluster breaks the degeneracy of the 1D orbitals into three groups: two (Dxy and Dx2−y2), two (Dxz and Dyz), and one (Dz2) orbitals. The three groups are due to the crystal field splitting. In the minority states, after the Pz state, there are five unfilled 1D orbitals, with degeneracies similar to the majority states. The filling of majority and minority orbitals leads to a 1S21P61Dα5 configuration, with a 5 μB magnetic moment in the VLi8 cluster. This is equivalent to the Mn2+ ion, which, because of its half-filled 3d shell, has a magnetic moment of 5 μB. 3.8.2. VNa8 and VCs8. Because the jellium model was first proposed for Na clusters, magnetic superatoms based on Na8 are especially attractive. Theoretical studies598 in 2009 predicted that VNa8 is a magnetic superatom. This was experimentally confirmed, in 2013, by using a pulsed arc cluster ionization source and anion photoelectron spectroscopy.600 The underlying mechanism here is very similar to that of VLi8; the VNa8 cluster has 13 valence electrons, and the ground-state geometry is a square antiprism of Na atoms with a V atom encapsulated in the center. The electronic ground state of the cluster has a 1S21P61D5 distribution of electrons (see Figure 92). Because the 1D-state is derived from the atomic d-state of V, it undergoes large exchange splitting, leading to a cluster with a spin magnetic moment of 5.0 μB. To illustrate the general trend of magnetism in transition metal atoms doped in Na8 cluster, Guo et al.601 carried out a

Figure 92. Orbitals of VNa8. Adapted with permission from ref 600. Copyright 2013 American Chemical Society.

comprehensive study by including 3d, 4d, and 5d elements. The results are listed in Table 3. It was found that (1) nearly all of the ground states of TM@Na8 keep the same structure, where the TM atom is at the center of a square antiprism Na8 cluster. (2) The magnetic moments of TM@Na8 clusters vary from 0 to 6 μB, with CrNa8 having the highest magnetic moment, whereas NiNa8, PdNa8, and PtNa8 proved to be nonmagnetic. (3) The electron filling of most TMNa8 clusters, because of the exchange splitting in the superatom shell, does not preserve Hund’s rule. (4) Among the studied magnetic superatoms, VNa8 has the largest HOMO−LUMO gap, showing high chemical stability, as observed in experiment. To examine the potential of magnetic superatoms in spinpolarized transport, first-principles calculations were carried out for a transition metal atom, doped Na8 cluster (TM@Na8, TM = Sc, Ti, V, Cr, Mn, Fe, Co, Ni, Cu, Zn).602 The electrode materials were found to have a notable influence on the AS


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Table 3. Spin Magnetic Moments (MM), Valence Electron Configurations, Mulliken Spin Density (MSD) of TM Atom, Binding Energies (BE), Embedding Energies (EE), LUMO−HOMO Energy Gaps (Gap) of TMNa8 Clusters, and Atomic Hirshfeld Charges (HC) of TM Atoma

a

cluster

MM (μB)

configuration

MSD

EE (eV)

BE (eV)

gap (eV)

HC

ScNa8 TiNa8 VNa8 CrNa8 MnNa8 FeNa8 CoNa8 NiNa8 YNa8 ZrNa8 NbNa8 MoNa8 TcNa8 RuNa8 RhNa8 PdNa8 LaNa8 HfNa8 TaNa8 WNa8 ReNa8 OsNa8 IrNa8 PtNa8

3 4 5 6 5 4 1 0 3 4 5 4 3 2 1 0 3 4 5 4 3 2 1 0

1S21P61D3 1S21P61D4 1S21P61D5 1S21P61D52S1 1S21P61D62S1 1S21P61D72S1 1S21P61D9 1S21P61D10 1S21P61D3 1S21P61D4 1S21P61D5 1S21P61D6 1S21P61D7 1S21P51D9 1S21P51D10 1S21P61D10 1S21P61D3 1S21P61D4 1S21P61D5 1S21P61D6 1S21P61D7 1S21P61D8 1S21P61D9 1S21P61D10

1.753 3.135 4.649 5.332 4.894 3.567 1.7941 0.0 2.339 2.795 4.730 4.702 3.443 1.522 −0.097 0.0 2.339 2.795 4.730 4.702 3.443 1.522 −0.097 0.0

1.542 2.407 3.231 1.755 1.381 2.101 2.930 3.750 2.108 2.406 3.084 1.975 2.937 4.318 5.434 4.104 2.086 2.722 3.431 3.411 2.545 4.206 5.820 6.363

7.100 7.965 8.789 7.313 6.939 7.659 8.488 9.308 7.666 7.964 8.642 7.533 8.495 9.876 10.992 9.662 7.644 8.280 8.989 8.969 8.103 9.764 11.378 11.921

0.250 0.330 0.623 0.380 0.258 0.305 0.352 0.614 0.246 0.343 0.751 0.270 0.493 0.358 0.380 0.959 0.329 0.411 0.539 0.160 0.354 0.040 0.195 0.655

−0.283 −0.374 −0.427 −0.186 −0.296 −0.337 −0.446 −0.281 −0.112 −0.325 −0.331 −0.286 −0.523 −0.461 −0.533 −0.206 −0.207 −0.224 −0.390 −0.372 −0.440 −0.664 −0.647 −0.455

Adapted with permission from ref 601. Copyright 2014 Elsevier.

high transmission spin polarization when they are connected to Li electrodes. Further studies are needed to explore the best electrode materials for magnetic superatoms, with large moments, to exhibit high transmission spin polarization, while maintaining good stability. Similar studies have also been carried out for the VCs8 cluster.598 It was found that the geometry and magnetism of this cluster are similar to those of VLi8 and VNa8. The special point for a VCs8 cluster is that as two VCs8 clusters were allowed to interact with one another, they do not spontaneously collapse to form a V2Cs16 cluster. Rather they remain as a meta “dimer”. The mechanism preventing this collapse is the energy barrier of at least 0.3 eV. Interestingly, the total magnetic moment of [V2Cs16]2 was found to be 12 μB, with V atoms coupling ferromagnetically. This encouraged the study of quantum spin transport through magnetic superatom dimer (VCs8−VCs8).603,604 The results suggested that the equilibrium conductance of the system shows an oscillatory behavior in both spin channels, linked to the coupling strength as well as the energy level shift. A high spin polarized conductance in FM state is also found. This increases when the distance becomes large, exceeding 80% when the distance reaches 2.258 Å. This result serves as an indication that the magnetic superatom VCs8 dimer could be used as a highly efficient spin polarizer. 3.8.3. FeMg8, FeCa8, and TcMg8. Medel et al.605 calculated the electronic structure of FeMgn (n ≤ 12) and TMMg8 (TM = 3d transition metal atoms) clusters, using firstprinciples theory. They found that the exchange-split atomic d states of TM hybridize with D orbitals of the superatom, leading to 1D and 2D superatom orbitals. These orbitals have large exchange splitting and fill to maximize the total spin, in

structural stability of superatoms. Pauling’s electronegativity can be used as a guide to choose the electrode. Materials, such as gold and graphene, which have large electronegativity, are not suitable candidates for electrodes; however, lithium may be a good choice, due to the geometrical stability of the junction. Moreover, the results show that, among all of the studied systems (see Figure 93), TiNa8 and NiNa8 have the highest transmission spin polarization. This can be attributed to the back scattering of passing electrons, with the same spin, by localized electrons. Notably, it is surprising that the high moment clusters, composed of VNa8 and CrNa8, do not show

Figure 93. Transmission spin polarization (TSP) of different Li− TM@Na8−Li junctions and magnetic moment of corresponding isolated superatoms. Adapted with permission from ref 602. Copyright 2014 Springer. AT


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Figure 94. Magnetic moment (a) and HOMO−LUMO gap (b) of TMMg8 clusters for 3d elements. (c) Superatom orbitals of FeMg8. Adapted with permission from ref 605. Copyright 2011 National Academy of Sciences.

Figure 95. Relativistic density functional calculations of spin moment (a), orbital moment (b) of TMMg8 clusters. (c) Superatom orbitals of FeMg8 cluster. Adapted with permission from ref 606. Copyright 2016 Taylor & Francis.

HOMO−LUMO gaps, with the latter two being nonmagnetic (see Figure 94b). FeMg8 is a magnetic superatom, with a magnetic moment of 4.0 μB and a HOMO−LUMO gap of 0.64 eV. The superatom orbitals are plotted in Figure 94c. One can see that the lowest state is spread out over several atoms and has 1S superorbital character. The first two electrons occupy majority and minority S states. The next two states have 1P character. These are followed by four D orbitals. Mulliken

accordance with Hund’s rule in atoms. As the TM atoms change from Sc to Ni, sequential filling, as shown in Figure 94, occurs in the superatom orbitals of 2S 2D. This results in nonmagnetic TiMg8 and NiMg8 clusters, with electronic configurations of 2S2 and 2S22Dα32Dβ3, respectively. On the other hand, MnMg8 shows the largest magnetic moment of 5 μB, with a configuration of 2Sα12Dα4. However, among all of the studied clusters, FeMg8, TiMg8, and NiMg8 display large AU


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Figure 96. Total electron density (a), net spin electron density (isovalue 0.01 coulomb m−3) (b), and DOS obtained by broadening the molecular levels by Gaussians of width 0.017 eV (c). Blue, gold, yellow, and white circles represent, respectively, manganese, gold, sulfur, and hydrogen atoms. The delocalized 1S and 1P electronic shells and localized 3d atomic orbitals are marked in orange, green, and blue areas, respectively. Adapted with permission from ref 612. Copyright 2009 American Physical Society.

supershells. More so, the splitting yields an exchange energy of 1.01 eV for the 2D state, and an induced, 2D orbital, crystal field splitting energy of 0.16 eV. Because the exchange energy of the 2D orbital is much larger than the crystal field splitting energy in TcMg8, where the outer five valence electrons occupy the 2Dxy, 2Dxz, 2Dyz, 2Dx2−y2, and 2S majority states, consequently this leads to a total magnetic moment of 5 μB, with a configuration of 2Sα12Dα4. In addition, when going from Mg and Ca to Sr, in the same group of elements in the periodic table, CrSr9 and MnSr10 have also been proposed as magnetic superatoms, with magnetic moments of 4 μB and 5 μB, respectively.609 When research on the divalent alkali-earth element is extended to Zn, it was found that CrZn17 has 40 valence electrons. Furthermore, CrZn17 was reported to be a magnetic superatom with high stability, low ionization potential, and a high total spin magnetic moment, exactly the same as an isolated Cr atom,610,611 6 μB.610,611 3.8.4. Ligated Magnetic Superatoms. As a representative of ligated clusters, we considerAu25(SR)−18. This cluster has a centered icosahedral shell of Au13, protected by six RS(AuSR)2 motifs (RS- being an alkylthiolate group). Au25(SR)−18 is a superatom with eight valence electrons. If the central Au, in the Au13 core, is replaced with a Mn atom, the resulting neutral MnAu24(SR)18 cluster would be a stable magnetic superatom. Because of the 3d54s2 configuration of the Mn atom, the two 4s electrons would maintain shell closure, while the half-filled 3d5 electrons would contribute to the moment. This expectation was confirmed, as shown in Figure 96, by detailed DFT calculations.611,612 When Mn is replaced with Cr and Fe, the resulting anion and cation clusters remain magnetic superatoms, with closed electron shells. The corresponding data are given in Table 4.612 The above idea, for the design of magnetic superatoms, can be applied to many ligated superatoms, as listed in Table 1.

analysis indicates that these D orbitals derive 20% contribution from the Fe site, and the remaining contribution comes from the Mg atoms. A Pz orbital and a D orbital, of Dz2 symmetry, follow the set of D orbitals. These states are followed by the 2S state. The 1S2, 1P6, 1D10, and 2S2 states of both spins are filled with 20 valence electrons that come from the Fe and Mg sites. Of the total 24 valence electrons, one is then left with four electrons. In a confined, homogeneous, nearly free-electron gas (jellium model for a homogeneous cluster), this sequence is followed by a 1F supershell. Here, the presence of the Fe site leads to a 2D set of orbitals. Recently, 3d transition metal atom-doped Mg8 clusters were revisited, using relativistic density functional theory.606 It was found that the spin−orbit coupling could change the order of orbital filling, resulting in different magnetic states. For example, CrMg8 has the configuration of 2S22Dα2 instead of 2Sα12Dα3, while NiMg8 becomes magnetic with a configuration of 2S22Dα42Dβ2, as shown in Figure 95a. V and Co, doped in Mg8 clusters, show substantial orbital moments, in comparison to the almost quenched orbital moments for the other 3dtransition metal in a Mg8 cluster (see Figure 95b). For FeMg8, although the magnetic state remains 2S22Dα4, the order of orbital filling is changed (see Figure 95c). Similar to Mg8, Chauhan et al.607 systematically studied 3d elements doped in a Ca8 cluster. They found that TiCa8, with 20 valence electrons, has the 1S21P61D102S2 closed electronic shell configuration. When Fe is introduced to Ca8, the enhanced exchange splitting in supershells leads to a stable FeCa8 magnetic superatom, where the filling of majority super orbitals results in a magnetic species, with a spin magnetic moment of 4.0 μB. This result mimics the properties of the Fe atom. Calculations of the (FeCa8)2 dimer, starting from two motifs, indicate that the individual clusters retain their shape and have a ferromagnetic (FM) configuration with a spin magnetic moment of 8.0 μB as the lowest state. An antiferromagnetic (AF) state is only 0.021 eV higher in energy. In the TMMg8 cluster, when a 4d transition metal is introduced, TcMg8 has been identified as a magnetic superatom,608 where the d states of Tc hybridize with the superatomic D orbital states. This causes splitting between the majority and minority sets of the nearly free-electron gas

4. CLUSTER ASSEMBLED MATERIALS (CAMs) One of the main ideas in atomic theory, proposed by Dalton in 1803, was that atoms are the building blocks of matter. However, the properties of molecular crystals, where molecules form the building blocks, are very different from those of AV


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a

Adapted with permission from ref 615. Copyright 2015 Springer.

no hydrophobic effect

varies with metal, ligand, and coordination number Co > Ni > Cu > Rh. All M atoms prefer to adsorb on the surface of the cluster; K prefers

Figure 175. Predicted reaction paths for CO2 hydrogenation on Mo6S8 cluster, where the reaction energies (top) and barriers (bottom in parentheses) are expressed in eV (Mo, big cyan; S, small yellow; C, small gray; O, small red; H, small white). Adapted with permission from ref 913. Copyright 2010 American Chemical Society.

the S−S 2-fold bridge site, while other metal dopants favor the S−Mo−Mo−S 4-fold hollow site. When doped with K, Rh, Ni, and Co, the cluster exhibits higher activity in CH3OH production than Cu in an increasing sequence: Ti−Mo6S8 < Cu−Mo6S8 < Cu(111) < Cu29 < Co−Mo6S8 < Ni−Mo6S8 < Mo6S8 < Rh−Mo6S8 < K−Mo6S8.914 Among the systems studied, K is unique for accelerating the CH3OH synthesis from CO2 and H2 on Mo6S8. In part, this is due to the largest charge transfer in the cluster complex. The reaction pathways are given in Figure 176, where K + helps to stabilize *H x CO y intermediates adsorbed at the Mo sites via the electrostatic interaction between K and the O of *HxCOy. In particular, it leads to the high selectivity of *HOCO, promoting the reaction via *HOCO and hindering that via *HCOO. 5.5.6. Superalkalis for CO2 Cracking. Recall that the first step involved in converting CO2 into fuel is to activate CO2; that is, it must transform from the linear to a bent structure. As seen in Figure 177, CO2− has a bent structure, while neutral CO2 is a linear chain. However, with an electron affinity of −0.6 eV, CO2− is metastable. Thus, stabilization of CO2− requires a compensating cation, which may be an atom or a molecule. Recently, Bowen and co-workers860 studied M−CO2− anions (M = Cu, Ag, Au) using photoelectron spectroscopy. By comparing the measured vertical detachment energies with those calculated from density functional theory, they arrived at the geometry of these negative ions. While Ag−CO2− and Cu− CO2− exist only in physisorbed and chemisorbed states, respectively, they found Au−CO2− to exist in both physisorbed and chemisorbed states. As the geometry of a CO2 molecule can change by charge transfer, a better catalyst is one that can transfer an electron easily. In other words, it should have a low ionization potential. In this context, superalkalis may activate the CO2 bond more easily than can an alkali atom. Recently, Zhao et al.915 studied this possibility by considering a large number of superalkali clusters created using different electron-counting rules and reacting these with CO2. Using the octet rule, the 18-electron rule, the Wade−Mingos rule, and the Hückel’s rule, respectively, they designed Al3, Mn(B3N3H6)2, CN


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Figure 176. Geometries of the reaction intermediates and transition states (TS) involved in methanol synthesis on a K−Mo6S8 cluster. Big purple, K; big cyan, Mo; small yellow, S; small red, O; small white, H; small gray, C. Adapted with permission from ref 914. Copyright 2015 American Chemical Society.

Ferroelectricity is relatively common in bulk compound crystals, where it usually results from a spontaneous displacement of an ionic sublattice from its symmetric position, so that the unit cell acquires a dipole. de Heer and collaborators917 measured the ferroelectric properties of free niobium (Nb) clusters, in molecular beams. They observed a transition from a metallic to a ferroelectric state when going from room to low temperature. Furthermore, this transition temperature (T) was found to depend on the cluster size (n). As n increases, the characteristic temperature TG(n) decreases from 110 to 10 K. In the metallic state, the polarizability measurements of the same clusters indicate efficient metallic screening, so that the valence electrons are now free to respond to external fields. In the ferroelectric state, on the other hand, the center of charge for the valence electrons of the clusters is displaced from that of the ions by about 1 Å. This produces the observed dipoles; furthermore, the shift leads to an unusual electron charge density distribution, as compared to the metallic state. The absence of free carriers and the inability of the electron charge distribution to neutralize the dipole indicate that the conduction electrons are rigid (because they do not redistribute to alleviate the stress caused by the large electric fields) and collective (because all of the conduction electrons must participate).917 In addition, the ferroelectric state showed pronounced even−odd alternations for 38 < n < 102, while for small size, the Nbn clusters with n = 9, 11, 14, 18, 20, 21, 24, and 28 display outstanding ferroelectricity, as shown in Figure 179. To further understand the experimental results, Fa et al.919 carried out DFT calculations using the geometries obtained from an unbiased global search with guided simulated annealing. It was found that the electric dipole moment (EDM) existed in most of the Nbn clusters and varied

Figure 177. Optimized geometries with bond angle and CO bond distance, and NBO charge distribution in (a) neutral CO2 and (b) CO2− anion. Adapted with permission from ref 915. Copyright 2017 The Royal Society of Chemistry.

B9C3H12, and C5NH6 clusters. Their superalkali properties were confirmed from the calculated ionization potentials of 4.75, 4.35, 3.64, and 3.95 eV, respectively. To study their role in activating a CO2 molecule, they calculated the geometries of these clusters interacting with CO2. These are shown in Figure 178. The O−C−O bond angles in Al3, Mn(B3N3H6)2, B9C3H12, and C5NH6 clusters are 126°, 131°, 129°, and 133°, respectively. These angles are smaller than they would otherwise be in the case of CuCO2, which is 139°. Thus, superalkalis may serve as better catalysts for CO2 activation. 5.6. Clusters for Multiferroics

Multiferroics refers to a class of materials that exhibit, simultaneously, more than one primary ferroic order parameter, in a single phase.916 Because of their applications in sensors, transducers, memories, spintronics, energy conversion, and energy harvesting, multiferroics have become an important area of research in condensed matter physics and materials science. Here, we discuss the contribution of cluster science to this field. CO


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Figure 178. (a−d) Optimized geometries (left panel) and NBO charge distribution (right panel) of four different kinds of neutral superalkalis interacting with CO2 molecule. Adapted with permission from ref 915. Copyright 2017 The Royal Society of Chemistry.

moment of 1 μB due to the odd number of electrons. This is shown in Figure 180. The observed ferroelectricity and

Figure 180. Size evolution of the calculated EDM and magnetic moment of TaN (N = 2−23), represented by the “●” and “○”, respectively. The inset shows the dependence of the inverse coordination number (ICN) function on cluster size. Adapted with permission from ref 918. Copyright 2006 American Institute of Physics. Figure 179. Characteristic dipole moments P0/N and energies TG(N) of ferroelectric clusters. Adapted with permission from ref 917. Copyright 2003 American Association for the Advancement of Science.

ferromagnetism in Tan can be understood from its reduced atomic coordination, leading to the stronger electron localization and inequivalent interatomic distances, which easily induce an asymmetric charge distribution.919 To confirm the theoretical predictions of the coexistence of ferroelectricity and ferromagnetism in metal clusters, de Heer’s group performed experimental studies920 using a cryogenic molecular beam apparatus, where magnetic and electric deflection measurements can be performed under identical conditions. The clusters are produced in a cryogenic pulsed laser vaporization cluster source and are detected in a highresolution, position-sensitive time-of-flight mass spectrometer. Here, the mass and deflection of the clusters in the beam are individually measured and recorded. Temperature-independent magnetic moments (up to 1 μB per atom), superparamagnetic

considerably with their size. This is especially true for the EDM values of Nbn (n > 38), which exhibited an extraordinary even− odd oscillation. This is consistent with the experimental observation, showing a close relationship with the geometries of clusters. Later, this research group explored, theoretically, the coexistence of ferroelectricity and ferromagnetism in tantalum clusters (Tan, n = 2−23).918 They found that, except for some symmetric Tan at n = 2, 4−6, 10, 15, and 22, most of the Tan clusters have a moderate EDM value, and those with 9, 11, 12, and 20 atoms attain a large EDM. What is more interesting is that the odd-numbered Tan clusters acquired a magnetic CP


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It is difficult to predict the future of any field of science, let alone for those that are emerging. We envision that current efforts for a fundamental understanding of the structure− property relations of homo- and heteroatomic clusters will continue and novel results will emerge. Equally important, we anticipate that the understanding gained in cluster science will have a major impact on the development of nanomaterials. Keeping in mind the possibility that our prediction for the future of this field may be wrong, we outline some of the current challenges and opportunities, moving forward. As pointed out in the beginning, one of the striking features of clusters is their unique geometry. A cluster does not look like a baby crystal (with possible exceptions in some ionically or covalently bonded systems, such as TiN and PbS), and the properties depend strongly on its atomic structure. However, no current experimental technique exists that can independently determine the cluster geometry, in the gas phase. Theoretical input is always necessary. This remains a challenge for future experiments. Similarly, despite considerable progress in theoretical methods and computational techniques, determining geometries of large clusters from theory is a difficult task, due to numerous local minima in the potential energy surface. Additionally, achieving chemical accuracy in large clusters, weakly interacting clusters, and clusters composed of strongly correlated elements also remains a challenge. Further developments of theoretical methods and computational codes are needed. Understanding of the effect of ligands and cluster−support interaction is key to developing cluster-assembled materials. In this context, a property-directed search for selecting cluster size and composition holds promise for the future. Consider, for example, Mn clusters. Small clusters, containing 3−5 atoms, are ferromagnetic, and they carry large magnetic moments. However, the total magnetic moment of Mn7 cluster is small, as the coupling becomes ferrimagnetic. Could suitable ligands or support make these clusters ferromagnetic with each Mn cluster carrying a giant magnetic moment? A similar strategy is also required for developing ideal catalysts. Here, one may vary size, composition, and support, individually or collectively, to identify efficient catalysts. Can one identify clusters to promote a given chemical reaction selectively, and can such clusters survive conditions far from the vacuum conditions where they exist? For the synthesis of cluster-assembled materials, for practical applications, starting with clusters in the gas phase is not the ideal route. Methods need to be developed for bulk production of clusters, using conventional techniques, once their size and composition are identified from gas-phase studies. In other words, one needs to marry the bottom-up design and top-down synthesis. An example of this approach, we discussed, is the recent design and synthesis of halogen-free electrolytes for Liion batteries. Here, lessons learned from gas-phase clusters helped to identify the anion component of the halogen-free electrolytes, which were then synthesized using conventional experimental techniques. The superatom concept needs to be generalized so that clusters of Earth-abundant materials can replace rare or expensive elements. Some of the other areas in materials science, where clusters can have an impact, include clusters with unique functionality for applications in spintronics, sensors, hard magnets, and cluster-inspired materials for energy harvesting, storage, and conversion. Other areas where clusters have the potential to make a significant impact are biology and

blocking temperatures up to 20 K, and ferroelectric dipole moments of about 1D (with transition temperatures up to 30 K) are observed. For the measured size range, ferromagnetism and ferroelectricity do coexist in rhodium clusters, whereas their transition temperatures TLM(N) and TFE(N) tend to be anticorrelated, as seen in Figure 181. This inverse relationship

Figure 181. Size dependence of transition temperatures from locked magnetic moments to unlocked moments TLM(N) (blue) and the ferroelectric transition temperature TFE(N) (red). Adapted with permission from ref 920. Copyright 2014 American Physical Society.

implies that the magnitude of the vibronic coupling causing ferroelectricity is inversely related to the magnitude of the spin−orbit coupling that ultimately causes the magnetic anisotropy (and the coupling of the spin to cluster). As bulk Rh is neither magnetic nor ferroelectric, one can expect that as the cluster size becomes large enough, ferromagnetism and ferroelectricity would vanish. Experiment found such a transition to occur at N = 40.920

6. SUMMARY AND OUTLOOK Atomic clusters, once considered an annoyance because they contaminated mass peaks and made it difficult to resolve and identify species, have evolved into a robust field, intermediate between atoms and solids. The ability to produce and analyze clusters of specific size and composition has shed light on phenomena such as nucleation, catalysis, and evolution of structure and properties of matter, one atom at a time. Equally important, studies of clusters have unveiled properties and phenomena not seen in bulk systems, as well as highlighted the importance of surface to volume ratio, quantum confinement, geometry, and size on electronic structure and properties. These studies have helped to gain a fundamental understanding of the unique phenomena in matter at the atomic scale. In this Review, we discussed several novel properties of clusters such as their geometries, relative stability, electronic structure, and properties, as well as ways in which stable clusters may be designed and assembled to form a new class of cluster-assembled materials. We focused on superatomic clusters that mimic the chemistry of atoms and discussed how various electron-counting rules facilitate their design. We also highlighted the significant impact of cluster science in the design and synthesis of materials for energy harvesting, energy storage, and energy conversion. CQ


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(3) http://www.nano.gov/about-nnihttp://www.nano.gov/aboutnni. (4) Campbell, E. E. B. Proceedings of Nobel Symposium; World Scientific Publishing Co.: Sweden, 2011. (5) Pfund, A. H. Infrared filters of controllable transmission. Phys. Rev. 1930, 36, 71−76. (6) Becker, E. W.; Bier, K.; Henkes, W. Strahlen aus kondensierten atomen und Molekeln im hochvakuum. Eur. Phys. J. A 1956, 146, 333−338. (7) Martins, J. L.; Buttet, J.; Car, R. Equilibrium geometries and electronic structures of small sodium clusters. Phys. Rev. Lett. 1984, 53, 655−658. (8) Kubo, R. Electronic properties of metallic fine particles. I. J. Phys. Soc. Jpn. 1962, 17, 975−986. (9) Dietz, T. G.; Duncan, M. A.; Powers, D. E.; Smalley, R. E. Laser production of supersonic metal cluster beams. J. Chem. Phys. 1981, 74, 6511−6512. (10) Duncan, M. A. Invited review article: Laser vaporization cluster sources. Rev. Sci. Instrum. 2012, 83, 041101. (11) Ganteför, G.; Siekmann, H. R.; Lutz, H. O.; Meiwes-Broer, K. H. Pure metal and metal-doped rare-gas clusters grown in a pulsed ARC cluster ion source. Chem. Phys. Lett. 1990, 165, 293−296. (12) Sattler, K. Cluster Assembled Materials; CRC Press: New York, 1996. (13) Ciobanu, C. V.; Wang, C.-Z.; Ho, K.-M. Atomic Structure Prediction of Nanostructures, Clusters and Surfaces; John Wiley & Sons: New York, 2013. (14) Jena, P.; Khanna, S.; Rao, B. Physics and Chemistry of Finite Systems: From Clusters to Crystals; Springer Science & Business Media: New York, 1992. (15) Khanna, S. N.; Jena, P. Assembling crystals from clusters. Phys. Rev. Lett. 1992, 69, 1664−1667. (16) Khanna, S. N.; Jena, P. Atomic Clusters: Building blocks for a class of solids. Phys. Rev. B: Condens. Matter Mater. Phys. 1995, 51, 13705−13716. (17) Sugano, S. Microcluster Physics; Springer: New York, 1991. (18) Duncan, M. A. Advances in Metal and Semiconductor Clusters: Cluster Materials; Elsevier: New York, 1998. (19) Jellinek, J. Theory of Atomic and Molecular Clusters: With a Glimpse at Experiments; Springer Science & Business Media: New York, 2012. (20) Meiwes-Broer, K.-H. Metal Clusters at Surfaces: Structure, Quantum Properties, Physical Chemistry; Springer Science & Business Media: New York, 2012. (21) Milani, P.; Iannotta, S. Cluster Beam Synthesis of Nanostructured Materials; Springer Science & Business Media: New York, 2012. (22) Kawazoe, Y.; Kondow, T.; Ohno, K. Clusters and Nanomaterials: Theory and Experiment; Springer Science & Business Media: New York, 2013. (23) Castleman, A. W.; Khanna, S. N. Quantum Phenomena in Clusters and Nanostructures; Springer Berlin Heidelberg: Berlin, Heidelberg, 2003. (24) Jena, P.; Castleman, A., Jr. Introduction to Atomic Clusters Nanoclusters−A Bridge across Disciplines; Elsevier: New York, 2010. (25) Chattaraj, P. K. Aromaticity and Metal Clusters; CRC Press: New York, 2010. (26) Alonso, J. A. Structure and Properties of Atomic Nanoclusters, 2nd ed.; World Scientific: River Edge, NJ, 2012. (27) Heiles, S.; Schäfer, R. Dielectric Properties of Isolated Clusters: Beam Deflection Studies; Springer: New York, 2014. (28) Tsukuda, T.; Häkkinen, H. Protected Metal Clusters: From Fundamentals to Applications; Elsevier: New York, 2015. (29) Davenas, J.; Rabette, P. Contribution of clusters physics to materials science and technology. Proceedings of the NATO Advanced Study Institute on Impact of Clusters Physics in Materials Science and Technology, Cap d’Agde, France, 1982. (30) Jena, P.; Rao, B.; Khanna, S. Physics and Chemistry of Small Clusters; Virginia Commonwealth University: Richmond, VA, 1987.

medical sciences, for imaging, drug design, and drug delivery. We look forward to an exciting future when cluster science will be an integral part of materials science.

AUTHOR INFORMATION Corresponding Author

*E-mail: [email protected]. ORCID

Puru Jena: 0000-0002-2316-859X Notes

The authors declare no competing financial interest. Biographies Puru Jena, Distinguished Professor of Physics at Virginia Commonwealth University, received his Ph.D. at the University of California. He joined Virginia Commonwealth University in 1980 and has been there ever since, with the exception of two years when he was a program director at the National Science Foundation and Jefferson Science Fellow and senior science adviser at the U.S. Department of State. Dr. Jena’s research covers a wide range of topics in condensed matter physics and nanoscience dealing with the structure−property relationships of matter, such as atomic clusters, cluster-assembled materials, superion inspired materials for energy storage and conversion, superatoms as the building block of a three-dimensional periodic table, and hydrogen storage materials. He is the author of nearly 600 papers and 13 edited books. Dr. Jena’s honors include Outstanding Scientist of Virginia, Presidential Medallion from Virginia Commonwealth University, Fellow of the American Physical Society, Outstanding Faculty Award from the State Council of Higher Education of Virginia, and Award of Excellence and Outstanding Scholar Award from Virginia Commonwealth University, and cochair of the Presidential Commission on bilateral scientific collaboration between the U.S. and Russia. Qiang Sun, Professor at Peking University and visiting professor at Virginia Commonwealth University, received his Bachelor degree in physics from Southwest University in 1984, Master degree in theoretical physics from Sichuan University in 1987, and Ph.D. degree in condensed matter physics from Nanjing University in 1996. His current research is focused on nanostructure physics (clusters, 2D materials) and physics of energy materials (hydrogen storage, ion battery, thermoelectrics, and CO2 conversion). He served as an Associated Editor of Journal of Renewable and Sustainable Energy during 2010−2015. Currently he is a member of the editorial committee of Materials Transactions, Annals of Materials Science & Engineering, and Materials Science and Nanotechnology.

ACKNOWLEDGMENTS The U.S. Department of Energy, Office of Basic Energy Sciences, Division of Materials Sciences and Engineering, supported this work in part under award no. DE-FG0296ER45579. We are grateful to Yawei Li, Jiabing Yu, and Junyi Liu, Shuo Wang, Wei Wu, and in particular to Tiashan Zhao, for helping with the references. We also thank Dr. Jian Zhou, Dr. Hong Fang, and Mr. Dillon Hensley for a critical reading of the manuscript. REFERENCES (1) Blakemore, R. P. Magnetotactic Bacteria. Annu. Rev. Microbiol. 1982, 36, 217−238. (2) Mie, G. Beiträge zur Optik trüber Medien, speziell kolloidaler Metallösungen. Ann. Phys. 1908, 330, 377−445. CR


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