Platonic and Archimedean solids in discrete metal-containing clusters

Abstract

Metal-containing clusters have attracted increasing attention over the past 2–3 decades. This intense interest can be attributed to the fact that these discrete metal aggregates, whose atomically precise structures are resolved by single-crystal X-ray diffraction (SCXRD), often possess intriguing geometrical features (high symmetry, aesthetically pleasing shapes and architectures) and fascinating physical properties, providing invaluable opportunities for the intersection of different disciplines including chemistry, physics, mathematical geometry and materials science. In this review, we attempt to reinterpret and connect these fascinating clusters from the perspective of Platonic and Archimedean solid characteristics, focusing on highly symmetrical and complex metal-containing (metal = Al, Ti, V, Mo, W, U, Mn, Fe, Co, Ni, Pd, Pt, Cu, Ag, Au, lanthanoids (Ln), and actinoids) high-nuclearity clusters, including metal-oxo/hydroxide/chalcogenide clusters and metal clusters (with metal–metal binding) protected by surface organic ligands, such as thiolate, phosphine, alkynyl, carbonyl and nitrogen/oxygen donor ligands. Furthermore, we present the symmetrical beauty of metal cluster structures and the geometrical similarity of different types of clusters and provide a large number of examples to show how to accurately describe the metal clusters from the perspective of highly symmetrical polyhedra. Finally, knowledge and further insights into the design and synthesis of unknown metal clusters are put forward by summarizing these “star” molecules.

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Xi-Ming Luo

Xi-Ming Luo received his MS Degree from the College of Chemical Engineering, Nanjing Tech University in 2019. Currently, he is a PhD candidate at Zhengzhou University under the supervision of Prof. Shuang-Quan Zang. His research focuses on the synthesis of atomically precise silver clusters, their growth mechanisms and structure-property relationship.

Ya-Ke Li

Ya-Ke Li received her PhD Degree in Physical Chemistry from the Institute of Chemistry, Chinese Academy of Sciences in 2018 under the supervision of Prof. Sheng-Gui He. After 3 years’ postdoctoral research with Prof. Knut R. Asmis at Leipzig University in Germany, she joined the College of Chemistry in Zhengzhou University as an Associate Research Professor. Her research interest focuses on the structure and reactivity of gas-phase metal clusters toward small molecules such as methane and carbon dioxide.

Xi-Yan Dong

Xi-Yan Dong is an Associate Professor in the College of Chemistry and Chemical Engineering, Henan Polytechnic University. She pursued her postgraduate study at Zhengzhou University and received her PhD Degree in Chemistry under the supervision of Prof. Thomas C. W. Mak and Prof. Shuang-Quan Zang. Her research focuses on silver clusters and luminescent crystalline materials.

Shuang-Quan Zang

Shuang-Quan Zang received his PhD Degree in Chemistry from Nanjing University in 2006 under the supervision of Prof. Qingjin Meng. After his postdoctoral research with Prof. Thomas C. W. Mak at The Chinese University of Hong Kong, he joined the College of Chemistry of Zhengzhou University. He received The National Science Fund for Distinguished Young Scholars in 2018. He is serving as the Dean of the College of Chemistry and Green Catalysis Center at Zhengzhou University. His current scientific interests focus on atomically-precise metal clusters, cluster-assembled materials, and functional metal–organic frameworks.

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1. Introduction

Metal-containing clusters whose main core or skeleton originates from the coordination-driven self-assembly of basic units such as metal atoms M^0/n+, metal-oxo MO_x^m−, and secondary building blocks (SBBs, Fig. 1) are emerging as unique crystalline materials, and thus have attracted significant interest owing to their fascinating structures and great potential in numerous applications.^1–23 Among the many types of metal-containing clusters, the highly symmetric structures are considered exceptional due to their complexity and beauty, which is inspired by the structure of the ‘star’ molecule, i.e., buckminsterfullerene (C₆₀) discovered in 1985.²⁴ The poly-nuclearity and high-nuclearity (with nuclearity of 30 or greater) metal clusters with orderly arrangements of metal atoms, especially those meeting well-known regular Platonic and/or Archimedean solids (Fig. 2 and 3), present aesthetically pleasing structures and intriguing physical properties.^25–30 However, the synthesis of highly symmetric metal clusters has opened up a challenging field for chemists and materials scientists. These outcomes are usually serendipitous discoveries.

Fig. 1 Regular packing of basic units (metal atoms M^0/n+, metal-oxo MO_x^m−, and SBBs) to form highly symmetric metal clusters protected by additional ligands. For clarity, ligands were omitted.

Fig. 2 All Platonic solids and their corresponding point group.

Fig. 3 All thirteen Archimedean solids and their corresponding point group.

Over the last two decades, due to their continuing enthusiasm and efforts, synthetic chemists have proposed and designed multiple feasible synthesis strategies, developed new synthesis methods, and improved technology for the growth of single-crystals, resulting in several very exciting achievements in the synthesis and applications of metal clusters with different structures (especially high-nuclearity metal nanoclusters), such as cubic-box-shaped {Mo₆₄Ni₈Ln₆},³¹ dodecahedra-shaped {Mo₂₄₀},³² hedgehog-shaped {Mo₃₆₈},³³ tetrahedral {Dy₃₀Co₈W₁₀₈},³⁴ dodecameric {(Ta/Nb)₃P₂W₁₅O₆₂}₁₂,³⁵ protein-sized {Nb₂₈₈},³⁶ C₆₀-shaped {U₆₀},³⁷ fullerene-like {Ti₄₂},³⁸ wheel {Mn₈₄},³⁹ {Fe₁₆₈},⁴⁰ {Co₃₂},⁴¹ ligand-free {As@Ni₁₂@As₂₀},⁴² {Pt₆Ni₃₈},^43–45 ring {Pd₈₄},⁴⁶ {(PtPd)₁₆₅},⁴⁷ {Zn₁₄},⁴⁸ {Cd₆₆},⁴⁹ {Hg₃₂},⁵⁰ {Ce₁₀₀},⁵¹ {Gd₁₄₀},⁵² cubic {Ln₉₆Ni₆₄} (Ln = Gd, Dy, Y),⁵³ hexagon-shaped {Ni₃₆Gd₁₀₂},⁵⁴ spherical {Ln₁₀₄} (Ln = Gd, Nd),⁵⁵ {Gd₁₅₈Co₃₈},⁵⁶ {Cu₁₇₉},⁵⁷ {Ag₃₇₄},⁵⁸ {Ag₄₉₀},⁵⁹ {Au₂₇₉},⁶⁰ {Al₇₇},⁶¹ {Ga₈₄},⁶² {In₆₈},⁶³ {Sn₃₄},⁶⁴ {Pb₁₈I₄₄},⁶⁵ and {Bi₅₀}.⁶⁶ Among them, highly symmetrical polyhedral structures (metal atoms or SBBs as vertices), ranging from simple tetrahedral ones to complex frameworks with multiple shells (Fig. 1), are always the most impressive structures, causing us to marvel at their nature.

The synthesis and structural chemistry of different giant metal-containing clusters and their related properties and applications have been previously reviewed.⁶⁷ For example, Tasiopoulos and Christou et al. reviewed the synthesis, structures and magnetic properties of giant 3d (transition metal) and 3d–4f (lanthanide) paramagnetic metal clusters (including Mn, Fe, Co, Ni, Cu, and Ln) in moderate oxidation states.⁶ Long and Zheng et al. surveyed a wide variety of fullerene-like metal-containing molecular clusters.⁸

Recently, Scheer reviewed giant non-biological discrete metal-based species (molecules and ions) with nanometer size, including metal clusters, ligand-based metal–organic cages and spacer-based supramolecules.²¹

However, most of the metal cluster-related reviews focused on discussing metal clusters according to the metal type (Ti,^2,68–70 3d metal,⁶ 3d–4f metal,^6,71–73 Ln,⁵ Cu,^11,74 Ag,^75,76 Au,^12,15,77 polyoxometalates(POMs),^4,78–80 Al, Ga,^81,82etc.), the chemical properties (magnetism,^6,73,83 optics,⁷⁵ catalysis,^1,13 biology,⁸⁴etc.) of these metal clusters and the types of shell ligands (thiolate,⁸⁵ phosphine,^86–88 alkynyl,^89,90 carbonyl,²² nitrogen⁹¹/oxygen⁹² donor ligands, etc.), with few summarizing their structural features. In 2005, Alvarez systematically summarized and described the chemistry of polyhedra (inorganic) and highlighted the geometrical relationships between different polyhedra.²⁹ Zaworotko and co-workers summarized the design and synthesis of metal–organic frameworks (MOFs) using metal–organic polyhedra (Platonic and Archimedean solids) as super-molecular building blocks.⁹³ Recently, Choe and Lah reviewed the topologies and the classification of hollow metal–organic polyhedra/cages.⁹⁴ However, over the past two decades, different from MOFs and metal–organic polyhedra/cages, these exciting ‘star’ metal clusters have not been linked from a polyhedron perspective, in which the ordered arrangement of atoms or metal-cluster SBBs satisfy the regular polyhedron feature of Platonic and Archimedean solids. Thus, from the perspective of regular polyhedral geometry, taking highly symmetric crystal structures as examples, a systematic description of the recent development of poly-/high-nuclearity metal clusters in different fields of inorganic chemistry is urgently required. This will provide a highly valuable overview of what is possible and what approaches are successful for the formation of these poly-/high-nuclearity metal clusters. Meanwhile, a summary on the different properties resulting from different metal-type clusters with similar metal arrangements will help us reasonably predict the unknown structures of related metal clusters based on the existing results and obtain more excellent properties.

In this review, we summarize and introduce atomically precise discrete metal aggregates with structural characteristics of Platonic and Archimedean solids created in the field of chemistry (mainly inorganic chemistry), including metal clusters with metal–anionic cores (anionic: oxo/hydroxide/chalcogenide/halide, etc.), metal clusters with metal cores (with metal–metal binding) protected by thiolate, phosphine, alkynyl, carbonyl and nitrogen/oxygen donor ligands, and metal clusters with metal–anionic cores and metal–metal bonds. Herein, the polyhedral structural features of these metal clusters (special attention to complex high-nuclearity ones) will be discussed in the different chapters according to the metal type (polyoxometalates, group IV(Ti/Zr/Hf)/VII(Mn)/VIII(Fe/Co/Ni/Pd/Pt) metal, lanthanide, coinage metals (Cu/Ag/Au), and others), while their corresponding synthetic strategies and methods will be briefly introduced. By analyzing the examples from exciting achievements in the field of metal clusters over the past few decades, we show how different sets of atoms can be organized into different polyhedra, which can further combine into more complex or nested structures. In the following section, we discuss the synthesis (mainly from the perspective of shell ligands and templated anions) and structural prediction (from the perspective of the close packing of the underlying cells (metal atoms M^0/n+, metal-oxo MO_x^m−, and SBBs) and the geometrical relationship between different Platonic and Archimedean polyhedra) of the giant ‘star’ clusters. The current research status including the major challenges will be elaborated at the end, together with some personal perspectives for future development.

All the metal clusters mentioned in this review were isolated as single-crystalline states, enabling their precise structural characterization by single-crystal X-ray diffraction. However, due to our limited knowledge, some metal clusters featuring Platonic and Archimedean solids, especially heterometallic clusters, may not be included in this review. Specifically, we searched the literature up to October 2022. It should be noted that metal–organic cages (metal atoms or metal clusters as vertexes and polydentate organic ligands as edges) with regular polyhedral features are beyond the scope of this review.

2. Platonic and Archimedean solids

2.1 Platonic solids

The simplest but best-known family of polyhedra, i.e., Platonic solids, contain five common polyhedra (tetrahedron, hexahedron (cube), octahedron, dodecahedron and icosahedron, Fig. 2 and Table 1), all of which are constructed from a single type of regular polygon (triangle, square or pentagon with equivalent vertices and equivalent edges). These five Platonic solids are regular polyhedra and belong to one of the highly symmetric point groups, i.e., tetrahedral (T_d), octahedral (O_h), and icosahedral (I_h).

Table 1 Characteristics of Platonic and Archimedean solids

Platonic polyhedra	Vertices (V)^a		Faces (F)^b	Edges (E)
a Where the numbers 3, 4, 5, 6, 8 and 10 represent the faces around that vertex. For instance, (3,6,6) represents one triangle, one hexagon, and another hexagon, respectively. b The index indicates the number and type of the polygon faces. For example, 43 + 46 represents four tetragonal and four hexagonal faces. c There are two enantiomeric versions of this polyhedron. Note: The listed polyhedron conforms to Euler's rule of V + F − E = 2.
Tetrahedron	4	(3,3,3)	43	6
Cube	8	(4,4,4)	64	12
Octahedron	6	(3,3,3,3)	83	12
Dodecahedron	20	(5,5,5)	125	30
Icosahedron	12	(3,3,3,3,3)	203	30

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Archimedean polyhedra	Vertices (V)		Faces (F)	Edges (E)
Truncated tetrahedron	12	(3,6,6)	43 + 46 = 8	18
Cuboctahedron	12	(3,4,3,4)	83 + 64 = 14	24
Truncated cube	24	(3,8,8)	83 + 68 = 14	36
Truncated octahedron	24	(4,6,6)	64 + 86 = 14	36
Rhombicuboctahedron	24	(3,4,4,4)	83 + 64 + 124 = 26	48
Truncated cuboctahedron	48	(4,6,8)	124 + 86 + 68 = 26	72
Snub cube^c	24	(3,3,3,3,4)	243 + 83 + 64 = 38	60
Icosidodecahedron	30	(3,5,3,5)	203 + 125 = 32	60
Truncated dodecahedron	60	(3,10,10)	203 + 1210 = 32	90
Truncated icosahedron	60	(5,6,6)	125 + 206 = 32	90
Rhombicosidodecahedron	60	(3,4,5,4)	203 + 304 + 125 = 62	120
Truncated icosidodecahedron	120	(4,6,10)	304 + 206 + 1210 = 62	180
Snub dodecahedron^c	60	(3,3,3,3,5)	603 + 203 + 125 = 92	150

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Numerous examples of discrete metal clusters featuring Platonic solids (tetrahedron, cube, octahedron, dodecahedron, and icosahedron) have been found in the literature (discussed in detail below).⁹⁵ Among them, simple tetrahedral (M₄) and hexahedral (M₆) clusters are the most common ones and will not be discussed in detail in the subsequent chapters. In tetrahedral clusters, cubane-like (Fig. 4a–c) and adamantane-like (Fig. 4d–f) cores often appear, which can also be used as building blocks to build larger cluster structures.

Fig. 4 (a) Molecular structure of [Ti₄(μ₃-O)₄(OⁱPr)₄(Ph₂PO₂)₄].⁹⁶ (b) Cubane-like M₄(μ₃-O)₄ core. (c) Cubane molecule. (d) Ball-and-stick representations of [(η-C₅Me₅)₄V₄(μ₂-O)₆].⁹⁷ (e) Adamantane-like M₄(μ₂-O)₆ core. (f) Adamantane molecule.

2.2 Archimedean solids

Archimedean solids are the second family of high symmetry polyhedrons, which are often called semiregular solids. There are 15 Archimedean solids (Fig. 3), two of which, the snub cube and snub dodecahedron, are chiral and have two pairs of enantiomers.²⁹ Similar to Platonic solids, Archimedean solids feature a single type of vertex, namely, equivalent vertices surrounded by regular polygons (Table 1). All 13 Archimedean polyhedra can be obtained by simple manipulation of the Platonic polyhedrons.²⁹ In addition to chiral Archimedean solids, other centrosymmetric ones have been found in atomically precise metal-containing clusters, mainly in truncated tetrahedron, cuboctahedron, truncated cube, truncated octahedron and rhombicuboctahedron. The specific cases are described in detail in the following sections.

The term “keplerate” refers to a type of concentric nested structure that contains two or multi-Platonic and Archimedean solids, which was first proposed by Müller's group because of its similarity to Kepler's early cosmos model.^25,98

The term “Mackay structure” refers a type of dense packing of equal-sized spheres, which can form concentric multi-shells of either regular icosahedra (Platonic solid) or cuboctahedra (Archimedean solid).³⁰ The nth shell conforming to Mackay rules consists of (10n² + 2) spheres. For example, the first shell can be a 12-atom icosahedron (Platonic solid) or cuboctahedron, either surrounding a central atom or a cavity, whereas the second shell, which is either a v₂-icosahedron or v₂-cuboctahedron (where ν_n denotes (n + 1) atoms along each edge of the polyhedron), consists of 42 atoms and encapsulates the 1st shell.

3. Platonic and Archimedean solids in POMs

Polyoxometalates (POMs) are a class of metal–oxygen clusters that have achieved rapid development due to their intriguing structural features and wide potential applications in the past two decades.^{3,4,78,79,99–109} These clusters contain highly symmetrical core assemblies of MO_x units, in which M is the early transition metal with the highest oxidation state, such as V, Mo, W, Nb and Ta. POMs often adopt quasi-spherical structures based on Archimedean and Platonic solids of considerable topological interest, such as the α-Keggin cuboctahedral structure of {XM₁₂O₄₀} (X = P, Si, As; M = V, Mo, W, Nb, etc.) with T_d symmetry (Fig. 5a and b) and Lindqvist-type octahedral structure of {M₆O₁₉} (M = V, Mo, W, Ta, Nb, etc.) with O_h symmetry (Fig. 5c and d). The related high-nuclearity and/or giant POMs with beautiful and complex polyhedral structures will be described below according to different metal elements (polyoxovanadates, polyoxomolybdates, polyoxotungstates, polyoxoniobates, and others).

Fig. 5 Polyhedral representation and corresponding polyhedral arrangement of the metal atoms of the α-Keggin {XM₁₂O₄₀} (a and b) and Lindqvist {M₆O₁₉} POMs (c and d). Structures shown correspond to K₁₂[Ti₂O₂][SiNb₁₂O₄₀] (a and b)⁹⁹ and Na₇[HNb₆O₁₉] (c and d).^110,111

3.1 Polyoxovanadates

In the small polyoxovanadates (POVs), in addition to the classic inorganic clusters (e.g., Lindqvist-type octahedron), some V–O clusters featuring Platonic and Archimedean solids are constructed by introducing organic composites.^80,112,113 Furthermore, when other metals are present in the reaction system, unexpected structures can be obtained. For example, the reaction of [V₂(μ-Cl)₃(THF)₆]₂[Zn₂Cl₆] precursor with PhCOONa in THF solution yielded an octa-nuclear heterometal cluster [V^IIIZnO(O₂CC₆H₅)₃(THF)]₄, which contained a remarkable array of two concentric tetrahedra, i.e., V₄ within Zn₄ (V₄Zn₄, Fig. 6a),¹¹⁴ while the reaction of V₂O₅ with [(η-C₅R₅)Rh(OH)₂]₂ yielded a decanuclear heterometallic cluster [(C₅Me₅)Rh]₄(V₆O₁₉) (V₆Rh₄, Fig. 6b), containing a remarkable array of two concentric polyhedra, i.e., V₆ (octahedron) within Rh₄ (tetrahedron).¹¹⁵

Fig. 6 Ball-and-stick representation and nested metal skeletons of (a) [V^IIIZnO(O₂CC₆H₅)₃(THF)]₄,¹¹⁴ (b) [(C₅Me₅)Rh]₄(V₆O₁₉),¹¹⁵ (c) [H₈(V^VO₄)V^IV₁₈O₄₂]⁷⁻,¹¹⁶ (d) {PNb₁₂O₄₀(V^IVO)₆}³⁻,¹¹⁷ and (e) [Ce^IV₁₂(V^VO)₆O₂₄(H₂O)₂₄(SO₄)₈]²⁺.¹¹⁸

In 2019, Das reported the preparation of two high-nuclearity POVs, i.e., [V^VO₄@V^IV₁₈O₄₂] (V₁₉, Fig. 6c) and [Cl@V^IV₁₈O₄₂], exhibiting proton conductivity property.¹¹⁶ The spherical V₁₉ is a classical “Keplerate” cluster, with the metal atoms in a polyhedral nested arrangement, V^VO₄(center)@V₁₂(cuboctahedron)@V₆(octahedron). Hu's group reported the preparation of a hexa-capped Keggin-type PONb, i.e., {PNb₁₂O₄₀(V^IVO)₆}³⁻ (V₆Nb₁₂, Fig. 6d).¹¹⁷ The metal atoms in V₆Nb₁₂ can be simplified as a highly symmetrical core–shell structure, which consists of Nb₁₂ (cuboctahedron) and V₆ (octahedron), enveloping the center {PO₄} group.

In 2022, Nyman and Farha synthesized three Ce^IV/V^V oxo clusters with a [Ce₁₂(VO)₆O₂₄]¹⁸⁺ cubane core capped by different ligands (SO₄²⁻, Ce₁₂V₆, Fig. 6e; toluenesulfonic acid; nitro-benzenesulfonic acid).¹¹⁸ In the Ce₁₂V₆ cube, the Ce^IV ions are located in the center of the 12 edges, while the vanadyl (V^VO₅) groups cap the six faces and SO₄²⁻ anions are situated on the eight corners. The arrangement of the heterometallic atoms is similar to the above-mentioned V₁₉, which is a nested structure (Fig. 6e), i.e., V₆(octahedron)@Ce₁₂(cuboctahedron). Interestingly, the functionalization of the [Ce₁₂(VO)₆O₂₄]¹⁸⁺ core with organic ligands achieved adjustable catalytic reactivity.

Zhang and Schmitt prepared four high-nuclearity POV cages stabilized by organoarsonate ligands and provided a highly effective method using the template effect and modular self-assembly to realize the successive construction of nanoscopic symmetrical polyoxovanadate cages.¹¹⁹ In the biggest one of the four POV cages reported, i.e., {Cl₆@[(V^VO)₁₆(V^IVO)₈O₂₄(O₃AsC₆H₅)₈]}⁶⁻ (Cl₆@V₂₄As₈, Fig. 7a), the V atoms as the vertices of the hollow structure adopt a topology of truncated octahedron. The formation of the 24-metal Archimedean arrangement is the result of the assembly of 6 square tetramers {V₄O₈} induced by 6 Cl⁻ ions as templates, in which the tetramers cap the vertices of the encapsulated Platonic {Cl₆} octahedron. This cluster can be classified as a keplerate, with concentric arrangements of both Platonic and Archimedean solids (Cl₆(octahedron)@V₂₄(truncated octahedron)@As₈(hexahedron), Fig. 7b).

Fig. 7 (a) Structure of Cl₆@V₂₄As₈. (b) {V₂₄As₈} cluster topology with the six Cl⁻ as anion templates highlighted using the space-filling representation.¹¹⁹

Besides, Streb reported the preparation of a tetrahedron-shaped 58-nuclear Ba–V–O cluster, H₅[Br@Ba₁₀(NMP)₁₄(H₂O)₈[V₁₂O₃₃]₄] (Ba₁₀V₄₈, NMP = N-methyl-2-pyrrolidone).¹²⁰ In Ba₁₀V₄₈, the octahedral Ba₁₀ shell (tetrahedron(Ba₄)@octahedron(Ba₆)) templated by a Br⁻ anion is capped by four [V₁₂O₃₃]⁶⁻ SBBs in a tetrahedral fashion.

3.2 Polyoxomolybdates

By introducing reducing agents, synthetic chemists achieved the preparation and isolation of many Mo^V-containing polyoxomolybdates (POMos, Fig. 8) and the self-assembly of metal-oxide building blocks (POM subunits, such as triangular {Mo₃}, pentagonal {(Mo)Mo₅}, and tripod-shaped {Mo₆} groups generally), crossing the boundary of the small molecule world, as well as constructing multi-functional and highly complex molecular nano-objects.⁴ The main types of related giant polyoxomolybdates include (Fig. 9–11) core–shell-type {Mo₆₀} (Fig. 9),¹²¹ cubic-box-shaped {Mo₆₄Ni₈Ln₆} (Fig. 10),³¹ {Mo₈₄} (Fig. 11a and b),¹²² spherical keplerates {M₇₂M′₃₀} (M = Mo, M′ = V^IV, Cr^III, Fe^III, and Mo^V, Fig. 11c and d),^123–125 {M₇₂Mo₆₀} (M = Mo, and W, Fig. 11e and f),^126–128 dodecahedra-shaped {Mo₂₄₀} (Fig. 11g and h),³² and hedgehog-shaped {Mo₃₆₈}³³ nanoclusters.

Fig. 8 (a and b) Combined polyhedral/ball-and-stick representation of Mo₁₆.¹²⁹ (c and d) Ball-and-stick representation and nested structure of ε-La₄PMo₁₂, respectively.¹³⁰ (e and f) Ball-and-stick and polyhedral representations of Mo₂₄, and rhombicuboctahedral arrangement of 24 metal atoms, respectively.¹³¹

Fig. 9 (a) Ball-and-stick representation of Zn₈Mo₂₈ and simplified 36-metal-based skeleton with four regular polyhedrons nested together (Mo₄(tetrahedron)@Zn₄(tetrahedron)@Mo₂₄(truncated octahedron)@Zn₄(tetrahedron)).¹³² (b) Ball-and-stick representation of Mo₆₀. (c) {Mo^V₆O₂₅}-based assembly of Mo₆₀ from the inside out, an ε-Mo₁₂ core (truncated tetrahedron), an Mo₂₄ cage (truncated tetrahedron) and four Mo₆ arranged in a tetrahedron. (d) Successive Archimedean polyhedral description featuring only equivalent vertex metal atoms of Mo₆₀.¹²¹

Fig. 10 Ball-and-stick (a) and polyhedral (b) representations of Mo₆₄Ni₈La₆. (c) Simplified 88-metal-based skeleton with five regular polyhedra nested together.³¹

Fig. 11 (a and b) Structure of Mo₈₄ and the cubic arrangement with 8 {Mo₆} (green) as vertexes and 12 {Mo₃} (bright green) as linkers, respectively.¹²² (c and d) Structure of Mo₁₀₂ and the icosahedral arrangement with 12 {(Mo)Mo₅} as vertexes and 30 O = Mo(H₂O) (bright green) as linkers, respectively.¹²⁵ (e and f) Structure of Mo₁₃₂ and the icosahedral arrangement with 12 {Mo₆} (blue and green) as vertexes and 30 {Mo^V₂} (bright green) as linkers, respectively.¹²⁶ (g and h) Structure of Mo₂₄₀ and the dodecahedral arrangement with 20 {Mo₆} (green) as vertexes and 30 {Mo₄} (bright green) as linkers, respectively.³²

In 1993, Müller and co-workers reported the preparation of two 16-metal polyoxomolybdates with the same polyanion, i.e., [(Mo^VIO₃)₄Mo^V₁₂O₂₈ (OH)₁₂]⁸⁻ (Mo₁₆, Fig. 8a).¹²⁹Mo₁₆ in full tetrahedral T_d symmetry contains a truncated tetrahedral core (Mo^V₁₂O₂₈(OH)₁₂) capped with four {Mo^VIO₃} motifs with a tetrahedral arrangement, that is, Mo₁₂(truncated tetrahedron)@Mo₄(tetrahedron, Fig. 8b). In 2002, Sécheresse's group reported the first ε-Keggin cation (ε-PMo₁₂O₄₀) with a central {PO₄}, stabilized by four {La(H₂O)₄}³⁻ capping groups with a tetrahedral fashion, which has the detailed formula {ε-PMo^V₈Mo^VI₄O₃₆(OH)₄{La(H₂O)₄}₄]⁵⁻ (ε-La₄PMo₁₂, Fig. 8c and d).¹³⁰ The incorporation of four metal ions on the skeleton of ε-PMo₁₂O₄₀ (especially zinc-modified, ε-Zn₄PMo₁₂) provided the possibility to connect the ε-POM clusters into extended frameworks (1D, 2D and 3D) through coordination interactions between the metal ions and organic ligands.^133,134

In 2022, Wang and co-workers reported a co-crystal example, (ⁿBu₄N)₆H₂[{Mo^V₂₄O₄₈(OCH₃)₃₂}{Mo^V/VI₂₄O₅₂(OCH₃)₂₈}₂], assembled with an anionic [{Mo^V₂₄O₄₈(OCH₃)₃₂}]⁸⁻ (Mo₂₄, Fig. 8e) and two neutral [{Mo^V/VI₂₄O₅₂(OCH₃)₂₈}] high-nuclear iso-polymolybdates.¹³¹ The two iso-polymolybdates have the same metal frame, protected by different numbers of O²⁻ linkers and CH₃O⁻ groups (Fig. 8f). Notably, this is the first 24-metal cluster with a rhombicuboctahedral character (Fig. 8e), although a rhombicuboctahedron can be observed as part of or a shell in many complex cluster structures. Recently, Zhang and Wang reported the preparation of three T_d symmetry Mo₄O₄-cubane-templated heteropolyoxometalates, i.e., M₈Mo₂₈ (M = Zn or Co), M₈Mo^V₈Mo^VI₂₀O₈₈(H₂O)₄ (Fig. 9a), and Zn₈Mo^V₈Mo^VI₂₀O₈₈(H₂O)(NH(CH₃)₂)₃.¹³² In M₈Mo₂₈, the metal-oxo Mo₄O₄ core (Mo₄: tetrahedron) acts as a template to form the larger truncated octahedral Mo₂₄ shell, which is composed of six tetragons and eight 8-ring windows. Two additional sets of tetrahedrally arranged heterometal atoms (Zn or Co) are filled inside and outside the center of eight six-membered ring windows (Fig. 9a).

In the same year, under hydrothermal conditions, Wang obtained the largest fully reduced Na₈[Mo^V₆₀O₁₄₀(OH)₂₈] (Mo₆₀, Fig. 9b) polyoxomolybdate to date, which is a good all-inorganic light-absorbing material with efficient improved photo-electronic performance.¹²¹Mo₆₀ with T_d symmetry can be described as one ε-Keggin Mo₁₂ (truncated tetrahedron) encapsulated in an unprecedented Mo₂₄ cage, which is further face-capped and surrounded by four tripod-shaped {Mo₆} motifs. In addition, the whole structure of Mo₆₀ (Fig. 9c) can be regarded as the splicing of tripod-shaped {Mo₆} secondary building blocks (SBB), where ε-Mo₁₂ can be viewed as consisting of four {Mo₆} SBBs fused together. From the inside to the outside, the topologies of the 60 metals in the triple nesting structure contain five T_d symmetrical Archimedean solids (three truncated tetrahedrons and two cuboctahedrons), abbreviated as Mo₁₂@Mo₁₂@Mo₁₂@Mo₁₂@Mo₁₂ (Fig. 9d).

Besides, Cronin engineered highly reduced polyoxometalates with a super-cube structure by incorporating Ni^II and Ln^III ions, i.e., Na₉[Mo^V₅₂Mo^VI₁₂Ni₈Ln₆H₂₆O₂₀₀(H₂O)₃₀](ClO₄) (Mo₆₄Ni₈Ln₆, Ln = La, Fig. 10a; Ce, Pr).³¹ The super-cubic structure of Mo₆₄Ni₈Ln₆ (Fig. 10b) is comprised of 5 different overlayed and co-centred Platonic or Archimedean structures, i.e., a primary truncated cuboctahedron embedding a {Mo^VI₁₂} cuboctahedron and capped by different metal ions on its different faces to form an {Ni^II₈} cube, an {Ln^III₆} octahedron, and a disordered {Mo^V₄} tetrahedron, respectively (Fig. 10c).

In 2012, Cadot reported the preparation of a high-nuclearity polyoxothiomolybdate featuring a cubic box shape, i.e., [H₈Mo₈₄S₄₈O₁₈₈(H₂O)₁₂(CH₃COO)₂₄]³²⁻ (Mo₈₄, Fig. 11a).¹²² The Mo₈₄ cage is composed of 8 triangular {Mo₆S₆O₆(OH)}⁵⁺ as 3 connected vertices and 12 {Mo^VI₃O₁₁}⁴⁻ as edges, featuring a large inner void (diameter up to 9 Å) and 6 opening square windows (Fig. 11b).

In 1998, Müller's group prepared an inorganic giant fullerene-like 132-metal cluster based on {MoO_x}, i.e., (NH₄)₄₂[Mo^VI₇₂Mo^V₆₀O₃₇₂(CH₃COO)₃₀(H₂O)₇₂] (Mo₁₃₂, Fig. 11e), and first mentioned the term “Keplerate”.^{126–128,135}Mo₁₃₂ can be regarded as the product of the assembly of 12 {Mo₁₁} subunits ({(Mo^VI)(Mo^VI₅)(Mo^V₅)}) according to the regular icosahedron, wherein (Mo^VI)(Mo^VI₅) are vertexes and {Mo^V₂} are edges (Fig. 11f). The Mo atoms of {Mo₁₁} with C₅ symmetry can be divided into three parts (types I–III, Fig. 11f), including a central seven-coordinated Mo atom (type I, blue; 12 Mo: icosahedron) with pentagonal-bipyramid geometry, sharing its five equatorial edges with five {Mo^VIO₆} octahedra (type II, green; 60 Mo (12 {Mo₅}): rhombicosidodecahedron), and the other five {Mo^VO₆} octahedra (type III, bright green; 60 Mo (30 {Mo₂}): truncated icosahedron) share corners with that of the type-II {Mo^VIO₆}. When red-brown solution of Mo₁₃₂ is oxidized, the deep-blue rhombohedral crystals of smaller spherical Mo-clusters, i.e., [{(Mo)Mo₅O₂₁(H₂O)₄(CH₃COO)}₁₂{MoO(H₂O)}₃₀] (Mo₁₀₂, Fig. 11c), were isolated after a few days.¹²⁵ Similar to Mo₁₃₂, Mo₁₀₂ is composed of 12 {(Mo)Mo₅} pentagonal units, but they are partially reduced. The units follow the same arrangement rule (icosahedron) as Mo₁₃₂, which are interlinked through 30 O = Mo(H₂O) species. Twelve five-membered {Mo₅} rings ignore the center and build a twisted rhombicosidodecahedron skeleton (Mo atoms as equal (3,4,5,4) vertices), while 30 linkers as vertices (3,5,3,5) form another Archimedean solid (icosidodecahedron, Fig. 11d).

In 2020, Lan reported the preparation of a giant polyoxomolybdate featuring hollow opening dodecahedra, [Mo₂₄₀(OH)₆₀O_620−x(SO₃)_20−x(SO₄)_x]^(80−2x)− (Mo₂₄₀, Fig. 11g).³²Mo₂₄₀ is the highest-nuclearity metal cage with hollow opening dodecahedra reported to date. Mo₂₄₀ consists of 20 tripodal {Mo₆O₂₁(SO₄)}/{Mo₆O₂₂(SO₃)} as three connected vertices and 30 {Mo₄O₁₆} as edges, featuring a large inner cavity (diameter of up to 18 Å, Fig. 11h). Due to the presence of rich terminal oxygen atoms, water molecules, and counter cations (H⁺ and NH₄⁺), Mo₂₄₀ exhibits ultrahigh proton conductivity (1.03 × 10⁻¹ S cm⁻¹ at 80 °C and 98% relative humidity (RH)), which is the highest among POM-based crystalline materials.

3.3 Polyoxotungstates

Metal-substituted polyoxometalates, especially metal-substituted polyoxotungstates (POWs) constitute one of the most exciting developments in POM chemistry.^{68,106,136,137} The motivation comes not only from their intriguing structural diversity but also their special properties, which are applicable to magnetism, catalysis, medicine, and materials science. Here, the metal includes transition-metals (TMs = Ti, Fe, Mn, Co, Ni, and Cu), lanthanides (Ln) and main group metals. Among the various structures, some exhibit regular polyhedral geometric arrangements based on metal atoms and SBBs, forming super-tetrahedral, cubic, octahedral, and icosahedral structures.

In 2003, Kortz's group reported for the first time the structural characterization of Ti^IV-substituted polyoxoanions based on Wells–Dawson type and prepared the first tetrameric Ti^IV-substituted POMs, [{Ti₃P₂W₁₅O_57.5(OH)₃}₄]²⁴⁻ (Ti₁₂P₂W₁₅, Fig. 12a).¹³⁸ The 12-Ti-substituted polyoxotungstate is composed of four {Ti₃P₂W₁₅} equivalent fragments, connected by terminal Ti–O–Ti bonds, leading to a tetrahedral POM cluster with T_d symmetry. The central Ti₁₂O₄₆ core of Ti₁₂P₂W₁₅ consists of four {Ti₃O₁₃} subgroups constructed by three edge-shared {TiO₆} octahedra. With Ti metal as the vertices, 12 atoms can form a slightly distorted truncated tetrahedron (Fig. 12b), wherein the Ti⋯Ti separations in the four {Ti₃P₂W₁₅} groups and between adjacent groups are 3.129(6)–3.208(6) Å and 3.488(6)–3.535(6) Å, respectively. In 2014, Kortz's group again reported the preparation of a tetrahedral Ti-substituted polyoxotungstate, [Ti₆(TiO₆)(AsW₉O₃₃)₄]²⁰⁻ (Ti₇AsW₉, Fig. 12c).¹³⁹Ti₇AsW₉ contains a novel Ti-center octahedral core, which consists of a central TiO₆ (octahedron geometry) surrounded by six TiO₅ (square-pyramid geometry) and capped by four {AsW₉} tri-lacunary fragments (Fig. 12d). In 2018, Yang's group reported the synthesis of an Sb(V)-analogue of Ti₇AsW₉, [Ti₇O₆(SbW₉O₃₃)₄]²⁰⁻ (Ti₇SbW₉).¹⁴⁰Ti₇SbW₉ is a good heterogeneous catalyst for the catalytic oxidation of multiple thioethers, possessing high conversion and remarkable selectivity and exhibiting good stability and recyclability. Nomiya's group also reported the preparation of two types of Ti^IV-substituted polyoxotungstates, [(P₂W₁₅Ti₃O_60.5)₄X]³⁷⁻ (X = Cl⁻,¹⁴¹ NH₄⁺,¹⁴² similar to Ti₁₂P₂W₁₅) and [Cl@(P₂W₁₅Ti₃O₆₂)₄{μ₃-Ti(OH)₃}₄]⁴⁵⁻ (Ti₁₆P₂W₁₅, Fig. 12e).¹⁴³Ti₁₆P₂W₁₅ consists of four equivalent substructures of tri-Ti^IV-substituted POM {Ti₃P₂W₁₅}, similar to Ti₁₂P₂W₁₅, four extra bridging groups (Ti(OH)₃), and one Cl⁻ ion encapsulated in the center of the cluster. The four {Ti₃} fragments of Ti₁₆P₂W₁₅ form a twisted cuboctahedron geometry, where four Ti(OH)₃ motifs capped at the triangles in a tetrahedral fashion (Fig. 12f). Some TM-substituted (TM = Fe,^144–149 Mn,¹⁵⁰ and Cr¹⁵¹) polyoxotungstate super-tetrahedral clusters with a truncated tetrahedral or cuboctahedron core have been published.

Fig. 12 Combined polyhedral/ball-and-stick representation of Ti-substituted polyoxotungstates and their central Ti–O core: (a and b) [{Ti₃P₂W₁₅O_57.5(OH)₃}₄]²⁴⁻;¹³⁸ (c and d) [Ti₆(TiO₆)(AsW₉O₃₃)₄]²⁰⁻;¹³⁹ and (e and f) [Cl@(P₂W₁₅Ti₃O₆₂)₄{μ₃-Ti(OH)₃}₄]⁴⁵⁻.¹⁴³

In 2016, Kortz's group reported the synthesis of two mixed-valent Mn₁₆-substituted polyoxotungstate super-tetrahedra, [Mn^III₁₀Mn^II₆O₆(OH)₆(PO₄)₄(SiW₉O₃₄)₄]²⁸⁻ (Mn₁₆P₄SiW₉, Fig. 13a) and [Mn^III₄Mn^II₁₂(OH)₁₂(PO₄)₄(SiW₉O₃₄)₄]²⁸⁻.¹⁵² The mixed-valent cationic {Mn₁₆} cores are encapsulated by four {μ₆-PO₄} groups and four trivacant [SiW₉O₃₄]⁹⁻ units in a tetrahedral fashion. Mn₁₆P₄SiW₉ can also be regarded as four {Mn₃SiW₉} fragments connected by four {μ₆-PO₄} motifs, with an encapsulated {Mn₄O₄} cubane unit. The arrangement of the metal and bridged groups of Mn₁₆P₄SiW₉ can be considered a complex nested structure (abbreviated as Mn₄@P₄@Mn₁₂@{SiW₉}₄), contained in an innermost {Mn₄O₄} core, and moving outward, there are five regular polyhedrons, a tetrahedron, an elongated cuboctahedron and a tetrahedron (Fig. 13b). The synthesis of Co and Ni analogues with 16-metal-substituted polyoxotungstate was also successful.^153,154

Fig. 13 (a) Combined polyhedral/ball-and-stick representation of total structure and (b) central core of Mn-substituted polyoxotungstates [Mn^III₁₀Mn^II₆O₆(OH)₆(PO₄)₄(SiW₉O₃₄)₄]²⁸⁻.¹⁵² (c) Combined polyhedral/ball-and-stick representation of super-octahedron {Mn^III₁₂Mn^II₃Na(H₂O)₄₀(OH)₁₂(P₂W₁₂Nb₆ O₆₂)₆}⁴¹⁻ and its simplified polyhedral model, Mn^III₁₂@Mn^II₃Na@(P₂W₁₂Nb₆)₆.¹⁵⁵

In addition to the above-mentioned TM₁₆-substituted POM-based tetrahedral clusters, which are connected and stabilized by {XO₄} (X = P, As, Si and Ge) groups in a tetrahedral fashion, there are TM₁₆-containing POM clusters constructed by one central tetranuclear cluster unit and four tri-substituted POMs. In 1999, Hill's group reported the synthesis of an Nb₁₆-substituted polyoxotungstate with an [Nb₄O₆]⁸⁻ tetrahedral core, [Nb₄O₆(Nb₃SiW₉O₄₀)₄]²⁰⁻.¹⁵⁶ Four [Nb₃SiW₉O₄₀]⁷⁻ Keggin units are linked together by Nb–O–Nb–O–Nb bonds. This work realized the shape heredity of [Nb₄O₆]⁸⁻ in overall tetrahedral POMs. In 2014, Wang's group reported two polytantalotungstate tetrahedrons with different cations, [Ta₄O₆(SiW₉Ta₃O₄₀)₄]²⁰⁻.¹⁵⁷

By introducing Mn^II in the metastable {(NbO₂)₆P₂W₁₂O₅₆} in an acid aqueous solution of H₂O₂, Niu et al. synthesized a {P₂W₁₂Nb₆}-based mixed-valence Mn^II/III cluster, K₁₀{H₃₁Mn^III₁₂Mn^II₃Na(H₂O)₄₀(OH)₁₂(P₂W₁₂Nb₆O₆₂)₆} (Mn₁₅Na(P₂W₁₂Nb₆)₆, Fig. 13c), exhibiting single-molecule magnet property.¹⁵⁵ In the case of hexameric Mn₁₅Na(P₂W₁₂Nb₆)₆, it is effective to clarify the complex architecture from a topological viewpoint involving Platonic and Archimedean solids. Four [Mn^III₃(OH)₃(H₂O)₆]⁶⁺ motifs make up the basic Mn₁₂ core, in which the distribution of 12 Mn^III can be seen as an elongated cuboctahedron. Besides, four hinges (Mn^II and Na^I) are distributed in a tetrahedral fashion and six [P₂W₁₂Nb₆O₆₂]⁹⁻ SBBs arranged in an octahedral mode capping the outside of the Mn₁₂ cuboctahedron.

In 2020, Chen and co-workers reported the organoboron-functionalization of polyoxotungstates ([M₃P₂W₁₅O₆₂]⁹⁻ (M = Ta^V and Nb^V)), resulting in the hierarchical assembly of two tetrameric aggregates, {Na(3-PyB)₄[M₃P₂W₁₅O₆₂]₄}²⁷⁻ (3-PyBH₂ = 3-pyridineboronic acid), and two giant polyoxometalate-based dodecameric nanocapsules containing up to twelve Dawson-type POM units, {K₄(5-PymB)₃(5-PymBH)₁₂[M₃P₂W₁₅O₆₂]₁₂}⁸⁶⁻ (5-PymBH₂ = 5-pyrimidinylboronic acid, Fig. 14).³⁵ In the super-tetrahedron, four Ta₃-substituted Dawson-type building blocks are linked by four μ₃-bridging organoboron (3-PyB) ligands into a tetrahedral aggregate (Ta₁₂: cuboctahedron), similar to a tetrahedral tetramer through the inorganic unit (PO₄³⁻) mentioned above. Unexpectedly, when the organic linker was changed from 3-PyB to 5-PymB, a giant super-icosahedral based on {M₃P₂W₁₅O₆₂} was obtained. Twelve {M₃P₂W₁₅O₆₂} building blocks are linked by 12 μ₂-5-PymBH and three μ₃-5-PymB to form four trimers (Fig. 14a), resulting in the formation of a giant heteropolyoxotungstate icosahedral cage through 4 K⁺ ions arranged in tetrahedral mode (Fig. 14b).

Fig. 14 Combined polyhedral/ball-and-stick representation of trimer {(5-PymB)(5-PymBH)₃[M₃P₂W₁₅O₆₂]₃}³²⁻ (a) and nano-capsule {K₄(5-PymB)₃(5-PymBH)₁₂[M₃P₂W₁₅O₆₂]₁₂}⁸⁶⁻ (b).³⁵

Main group metals also connected lacunary POM motifs to construct highly symmetric polyhedral structures. In 2005, Kortz's group reported two cases of Sn-containing heteropolytungstates in different works, i.e., tetrahedron [{Sn(CH₃)₂(H₂O)}₂{Sn(CH₃)₂}As₃(AsW₉O₃₃)₄]²¹⁻ (Sn₃As₃AsW₉) ¹⁵⁸ and icosahedron [{Sn(CH₃)₂(H₂O)}₂₄{Sn(CH₃)₂}₁₂(XW₉O₃₄)₁₂]³⁶⁻ (X = P, Sn₃₆(PW₉)₁₂ (Fig. 15) and As).¹⁵⁹Sn₃As₃AsW₉ is composed of four {AsW₉O₃₃} POM units linked together by three As(III) atoms and two {Sn(CH₃)₂(H₂O)} and one {Sn(CH₃)₂} group, resulting in a chiral polyoxoanion with C₁ symmetry. Sn₃₆(PW₉)₁₂ is composed of twelve tri-lacunary [PW₉O₃₄]⁹⁻ Keggin units, which are linked by 12 inner {Sn(CH₃)₂} and 24 outer {Sn(CH₃)₂(H₂O)} groups, resulting in a more 100-metal POMs with T_h symmetry. If the [{Sn(CH₃)₂(H₂O)}₂{Sn(CH₃)₂}(XW₉O₃₄)]³⁻ fragment is a vertex, Sn₃₆(PW₉)₁₂ can be described as an almost perfect icosahedron (Fig. 15). The 12 inner and 24 outer Sn atoms also form a highly symmetrical arrangement, respectively.

Fig. 15 Combined polyhedral/ball-and-stick representation of [{Sn(CH₃)₂(H₂O)}₂₄{Sn(CH₃)₂}₁₂(XW₉O₃₄)₁₂]³⁶⁻ and its simplified icosahedron skeleton.¹⁵⁹

In 2018, Zheng's group reported the synthesis of a giant ionic spherical alkali-metal-POM composite cluster, [K₄₂Ge₈W₇₂O₂₇₂(H₂O)₆₀]³⁸⁻ (K₄₂Ge₈W₇₂, Fig. 16a).¹⁶⁰K₄₂Ge₈W₇₂ is not only a rare high-nuclearity K-H₂O cluster {K₄₂(H₂O)₆₀} (Fig. 16b), but also a new-type all-inorganic ionic porous material in which the cluster can be linked together through Na⁺, to construct a 3D framework possessing versatile properties (high stability, proton conductivity (6.8 × 10⁻² S cm⁻¹ at 98% RH and 85 °C), and water vapor adsorption 52 cm³ g⁻¹). The rare {K₄₂(H₂O)₆₀} cluster in K₄₂Ge₈W₇₂ exhibits a core–shell framework (Fig. 16b) of K₆(octahedron)@K₂₄(truncated octahedron)@K₁₂ (cuboctahedron), stabilized by eight C_3v-symmetry tri-vacant Keggin-type [GeW₉O₃₄]¹⁰⁻ with a cubic arrangement (Fig. 16c).

Fig. 16 (a) Polyhedral representation of [K₄₂Ge₈W₇₂O₂₇₂(H₂O)₆₀]³⁸⁻. (b) Core–shell {K₄₂(H₂O)₆₀} cluster, K₆(octahedron)@K₂₄(truncated octahedron)@K₁₂(cuboctahedron). (c) Eight C_3v-symmetry tri-vacant [GeW₉O₃₄]¹⁰⁻ with a cubic arrangement.¹⁶⁰

Due to their luminescent and magnetic properties, the introduction of lanthanide (Ln) ions into POM systems to construct Ln-containing POM-based clusters and materials is an important research field in POM chemistry.^109,161,162 In 2015, Powell and co-workers described the preparation of a giant tetrahedral 3d–4f heterometallic POM [{(GeW₉O₃₄)₂Dy₃(OH)₃(H₂O)}₆{Co^II₂Dy₃(OH)₆(H₂O)₆}₄]⁵⁶⁻ (Dy₃₀Co₈W₁₀₈, Fig. 17a), which shows single-molecule magnet (SMM) behaviour.³⁴Dy₃₀Co₈W₁₀₈ (Fig. 17b) is composed of six sandwich-type building blocks ({(GeW₉O₃₄)₂Dy₃(OH)₃(H₂O)}¹⁴⁻), where a planar-triangular trinuclear cluster, {Dy₃(OH)₃(H₂O)}, is sandwiched by a pair of tri-lacunary Keggin-type [GeW₉O₃₄]¹⁰⁻ and four trigonal-bipyramidal 3d–4f five-metal clusters {Co₂Dy₃(μ₃-OH)₆(H₂O)₆}⁷⁺. Each dimeric POM {(GeW₉O₃₄)₂Dy₃(OH)₃(H₂O)} links two 3d–4f clusters, whilst each {Co₂Dy₃(μ₃-OH)₆(H₂O)₆} is coordinated by three dimeric moieties, resulting in an assembly (dimeric POMs as linkers and 3d–4f clusters as nodes) with idealized T_d symmetry (Fig. 17b). In 2021, the analogues [{(GeW₉O₃₄)₂Ln₃ (CO₃)(OH₂)₃}₆{M₂Ln₃(OH)₆(H₂O)₆}₄]⁵⁰⁻, where Ln  =  Gd and Y and M  =  Zn, Mn, and Co, were reported by the same research group.¹⁶³

Fig. 17 (a) Ball-and-stick representation of [Dy₃₀Co₈Ge₁₂W₁₀₈O₄₀₈(OH)₄₂(H₂O)₃₀]⁵⁶⁻. (b) Tetrahedral geometry of Dy₃₀Co₈W₁₀₈, based on 6 {(GeW₉O₃₄)₂Dy₃(OH)₃ (H₂O)}¹⁴⁻ and 4 {Co₂Dy₃(μ₃-OH)₆(H₂O)₆}⁷⁺ building blocks.³⁴

In 2021, Kong et al. reported the preparation of a giant 3d–4f polyoxometalate inorganic super-tetrahedron {[La₆(SbO₃)₄(H₂O)₂₀]@[LaNi₁₂W₃₅Sb₃P₃O₁₃₉(OH)₆]₄}⁸⁶⁻ (La₆@(LaNi₁₂)₄, Fig. 18a) with high proton conductivity (6.8 × 10⁻² S cm⁻¹ at 100% RH and 295 K).¹⁶⁴ The La₆@(LaNi₁₂)₄ aggregate features a super-tetrahedral cluster@cluster framework (Fig. 18b), which is assembled from four [LaNi₁₂W₃₅Sb₃P₃O₁₃₉(OH)₆]²³⁻ SBBs (Fig. 18c) encapsulating a central [La₆(SbO₃)₄(H₂O)₂₀] core. The SBB can be viewed as one WO₄Ni₁₂(OH)₆ cluster stabilized by three tri-lacunary Keggin [PW₉O₃₄] and one [LaW₇O₂₄] anion.

Fig. 18 (a) Inorganic super-tetrahedron {[La₆(SbO₃)₄(H₂O)₂₀]@[LaNi₁₂W₃₅Sb₃P₃O₁₃₉(OH)₆]₄}⁸⁶⁻. (b) Simplified nested skeleton Sb₄@La₆@(LaNi₁₂)₄ of La₆@(LaNi₁₂)₄. (c) [LaNi₁₂W₃₅Sb₃P₃O₁₃₉(OH)₆] SBB (LaNi₁₂), containing one [LaW₇O₂₄(SbO₃)₃], one [WO₄Ni₁₂(OH)₆] and three [PW₉O₃₄].¹⁶⁴

In 2015, Kortz prepared three hexameric super-octahedron, Ln₁₂-containing 60-tungstogermanate-based six {GeW₁₀O₃₈}¹²⁻ SBBs, [Na(H₂O)₆@Eu₁₂(OH)₁₂(H₂O)₁₈Ge₂(GeW₁₀O₃₈)₆]³⁹⁻ (Fig. 19), [Na(H₂O)₆@Gd₁₂(OH)₆(H₂O)₂₄Ge(GeW₁₀O₃₈)₆]³⁷⁻, and [(H₂O)₆@Dy₁₂(H₂O)₂₄(GeW₁₀O₃₈)₆]³⁶⁻.¹⁶⁵ The complex structures of these polyanions can be clearly elucidated from a topological polyhedral viewpoint. As shown in Fig. 19, six H₂O coordinated with the central Na⁺ ion, 12 Eu^III ions, and six {GeW₁₀O₃₈}¹²⁻ SBBs are simplified as vertexes, which give rise to three Platonic solids arranged one inside the other, Na(H₂O)₆(octahedron)@Eu₁₂(icosahedron)@(GeW₁₀O₃₈)₆(octahedron). In 2019, Zheng and co-workers synthesized a series of octahedron-shaped Ln₁₄-substituted polyoxotungstates, H₂₇Na₁₆[(Ln₁₄(H₂O)W₄(OH)O₁₄)(WO₄)₄ (GeW₁₀O₃₈)₆] (Ln₁₄W₆₈, Ln = Eu, Gd, Tb, Dy, Ho, Er).¹⁶⁶Ln₁₄W₆₈ possesses a core–shell structure, which is made up of a W₄O₁₅ core (tetrahedron), Ln₁₄ tetrahedron (shell 1), W₄ tetrahedron (shell 2) and Ge₆W₆₀ octahedron (shell 3). The Ln–O–W core is stabilized by 6 bi-vacant Keggin-type [GeW₁₀O₃₈]¹²⁻ with an octahedral arrangement.

Fig. 19 Combined polyhedral/ball-and-stick representation and Platonic solid-based shell structure of super-octahedron [Na(H₂O)₆@Eu₁₂(OH)₁₂(H₂O)₁₈Ge₂(GeW₁₀O₃₈)₆]³⁹⁻.¹⁶⁵

3.4 Polyoxoniobates

In contrast to the above-mentioned V/Mo/W-containing POMs, the development of polyoxoniobates (PONbs) is still in its infancy, owing to their very difficult synthesis. Besides the classical [Nb₆O₁₉]⁸⁻ (octahedra)^110,111 and [XNb₁₂O₄₀]¹⁰⁻ (X = Si and Ge; cuboctahedra),¹⁶⁷ PONbs with a regular polyhedron feature are rare. Great advancements in giant iso-polyoxoniobates and hetero-polyoxoniobates were achieved by Zheng et al., who synthesized a series of nanoscale discrete aggregates, such as dimer [H₄Nb₅₂O₁₅₀]³⁶⁻, trimer [Li₃KNb₈₁O₂₂₅]⁴¹⁻, and tetramer [Li₈Nb₁₁₄O₃₁₆]⁵⁴⁻ (Nb₁₁₄, Fig. 20) based on {Nb₂₆} or {Nb₂₇} SBBs,¹⁶⁸ as well as the protein-sized macromolecule [Nb₂₈₈O₇₆₈(OH)₄₈(CO₃)₁₂]¹⁸⁰⁻.³⁶ The giant super-tetrahedral PONb Nb₁₁₄ consists of two parts (Fig. 20), i.e., four 27-nuclear {LiNb₂₇} SBBs with a tetrahedral arrangement and a slightly twisted tetrahedral kernel Li₄Nb₆(μ₃-O)₄ (Li₄Nb₂@Nb₄).

Fig. 20 Combined polyhedral/ball-and-stick representation and simplified nested metal-SBB skeleton of super-tetrahedron [Li₈Nb₁₁₄O₃₁₆]⁵⁴⁻.¹⁶⁸

The introduction of different metals into PONbs greatly enriched this family. In 2008, Casey's group reported a new v₂-octahedral homoleptic polyoxotitanoniobate, [N(CH₃)₄]₁₀[Ti₁₂Nb₆O₄₄] (Ti₁₂Nb₆, Fig. 21), where ν_n denotes (n + 1) atoms along each edge.¹⁶⁹Ti₁₂Nb₆ consists of a cuboctahedral Ti₁₂-shell [Ti₁₂O₁₄]²⁰⁺, of which six Ti₄-square faces are capped by six {NbO₅} units with an octahedral fashion. The nested type (or Keplerate type) makes the overall cluster hexagonal, and thus Ti₁₂Nb₆ is a type of “super-Lindqvist” ion (Fig. 21). Its Ta(V)-analogue, TMA₁₀Ti₁₂Ta₆O₄₄ (TBA = tetramethylammonium), was also successfully isolated in 2016.¹⁷⁰

Fig. 21 Ball-and-stick and polyhedron representation of v₂-octahedral polyoxotitanoniobate cluster [Ti₁₂Nb₆O₄₄]¹⁰⁻ and Keplerate-type arrangement of the 12 Ti atoms (cuboctahedron) and 6 Nb atoms (octahedron).¹⁶⁹

In 2015, Wang's group reported a Co₈-containing PONb, Na₆K₁₂[H₂Co₈O₄(Nb₆O₁₉)₄] (Co₈Nb₂₄, Fig. 22a).¹⁷¹ The Lindqvist-type PONb [Nb₆O₁₉]⁸⁻ fragments of Co₈Nb₂₄ show a tetrahedral arrangement and envelope an octa-nuclear cluster, {Co^II₄Co^III₄}. The mixed-value cluster {Co^II₄Co^III₄} is comprised of a central {Co^III₄O₄} cubane core, surrounded by four Co^II ions in a tetrahedral fashion. The anion cluster of Co₈Nb₂₄ can be seen as three nested tetrahedrons of different sizes from the inside to the outside, denoted as Co^III₄@Co^II₄@{Nb₆}₄. In 2017, this group also reported the preparation of a tetrahedral PONb with a larger kernel (Cu₁₆Nb₄) based on four [Nb₆O₁₉]⁸⁻ SBBs, {Cu₁₆Nb₄O₁₇ (H₂O)₁₂(AsO₄)₄(Nb₆O₁₉)₄}²⁶⁻ (Cu₁₆Nb₂₈, Fig. 22b).¹⁷²

Fig. 22 (a) Polyhedral/ball-and-stick representation of [H₂Co₈O₄(Nb₆O₁₉)₄]¹⁸⁻ (Co₈Nb₂₄) with a triple tetrahedron nested framework, {Co^III₄O₄}@Co^II₄@{Nb₆}₄.¹⁷¹ (b) Polyhedral/ball-and-stick representation of {Cu₁₆Nb₄O₁₇(H₂O)₁₂(AsO₄)₄(Nb₆O₁₉)₄}²⁶⁻ (Cu₁₆Nb₂₈) and its simplified polyhedral architecture.¹⁷² (c) Polyhedral/ball-and-stick representation of [NbO₈Cu₂₄(Nb₇O₂₂)₈]³⁶⁻ and simplified nested structure of Cu₂₄Nb₅₇.¹⁷³ (d) Polyhedral/ball-and-stick representation of K₁₂[Nb₂₄O₇₂H₂₁]₄.¹⁷⁴ (e) Polyhedral/ball-and-stick representation of {Eu₁₂W₁₂O₃₆(H₂O)₂₄(Nb₆O₁₉)₁₂}.¹⁷⁵

In 2010, Wang's group reported a cubic PONb, K₁₂Na[H₂₃NbO₈Cu₂₄(Nb₇O₂₂)₈] (Cu₂₄Nb₅₇) with eight [Nb₇O₂₂] units enveloping a big regular core [NbO₈Cu₂₄] (Fig. 22c).¹⁷³Cu₂₄Nb₅₇ with O point symmetry embodies the beauty of symmetry and can be regarded as a combination of multiple O-symmetry polyhedra from the inside to outside, i.e., NbO₈(center)@Cu₁₂(cuboctahedron)@Nb₈(cube)@Cu₁₂(cuboctahedron)@(Nb₆)₈(cube). In 2012, their group again reported the preparation of three new PONbs, KNa₂[Nb₂₄O₇₂H₂₁], K₂Na₂[Nb₃₂O₉₆H₂₈], and K₁₂[Nb₂₄O₇₂H₂₁]₄ (K₁₂Nb₉₆, Fig. 22d), showing triangle, molecular square, and cuboctahedral geometries, respectively.¹⁷⁴ These PONbs can be used as catalysts for hydrogen evolution upon UV-light. Twelve Lindqvist-type PONb [Nb₆O₁₉]⁸⁻ fragments of K₁₂Nb₉₆ show a slightly distorted cuboctahedral arrangement. K₁₂Nb₉₆ is assembled by four {KNb₂₄O₇₂} subunits joined by eight K ions. The {KNb₂₄O₇₂} cluster has three Lindqvist-type PONb [Nb₆O₁₉]⁸⁻ fragments with a triangle distribution, held together by an Nb₆ ring of six corner-sharing {NbO₆} or {NbO₅(H₂O)}, in which one K⁺ ion is trapped in the center. Thus, K₁₂Nb₉₆ can be viewed as a huge tetrahedral cage.

In 2016, Zheng's group synthesized a giant hollow heterometallic PONb with sodalite-type 24-metal Ln₁₂W₁₂ cages, {Ln₁₂W₁₂O₃₆(H₂O)₂₄(Nb₆O₁₉)₁₂} (Ln₁₂W₁₂Nb₇₂, Ln = Y, La, Pr, Nd, Sm, Eu, Gd, Tb, Dy, Ho, Er, and Yb, Fig. 22e).¹⁷⁵ The overall structures of Ln₁₂W₁₂Nb₇₂ can be viewed as bi-layered self-assembly, Ln₁₂W₁₂@(Nb₆)₁₂. The hollow structure of 24-metal Ln₁₂W₁₂ adopts a topology of the truncated octahedron, and the twelve Lindqvist-type PONb [Nb₆O₁₉]⁸⁻ fragments in the outer layer form an icosahedral arrangement. Interestingly, Ln₁₂W₁₂Nb₇₂ acts as a building block, bridged by [Cu(en)₂]²⁺, resulting in a 3D framework, while Y₁₂W₁₂Nb₇₂ is a discrete cluster. In 2019, this group first reported the preparation of two inorganic-organic hybrid polyoxoniobate cages, K₄@{[Cu₂₉(OH)₇(H₂O)₂(en)₈(trz)₂₁][Nb₂₄O₆₇(OH)₂(H₂O)₃]₄} (K₄Cu₂₉Nb₉₆, en = ethanediamine and Htrz = 1,2,4-triazole) and [Cu(en)₂]@{[Cu₂(en)₂(trz)₂]₆(Nb₆₈O₁₈₈)} (Cu₁₃Nb₆₈).¹⁷⁶ The cage materials exhibited high vapor adsorption capacity and hydrogen evolution activity upon visible light. K₄Cu₂₉Nb₉₆ shows a giant tetrahedral nanocage, in which four Nb₂₄ clusters ([H₂Nb₂₄O₆₉(H₂O)₃]¹⁶⁻, like {KNb₂₄O₇₂} in K₁₂Nb₉₆) are assembled by a metal–organic polyhedron of K₄@Cu₂₉ ([K₄Cu₂₉(OH)₇(H₂O)₂(en)₈(trz)₂₁]³⁰⁺). Cu₁₃Nb₆₈ is a giant cubic cluster, in which eight Lindqvist-type PONb [Nb₆O₁₉]⁸⁻ fragments with a cubic fashion are gathered by a spherical Nb₂₀ polyoxoniobate cage ({Nb₂₀O₇₂N₂₄}), resulting in a nanoscale high-nuclear PONb cage encapsulating an [Cu(en)₂] complex and stabilized by six bi-Cu-complexes ([Cu₂(trz)₂(en)₂]²⁺).

3.5 Other POMs

Since the beginning of this century, actinides have also been incorporated in polyoxometalates of transition metals,^177–181 and the first uranium (U) derivative (U₆O₁₃(2,2′-bipyridine)₂ (1,2,4-^tBu₃C₅H₂)₂) with an octahedron [U₆O₁₃] core was prepared in 2001.¹⁸⁰ Actinide-containing polyoxometalate clusters, which are mainly divided into two parts (U-containing heteropolyoxometalates and U-oxo-clusters),¹⁸² exhibit versatile and stunning structures, such as tetrameric, hexameric U-containing heteropolyoxometalates (Na₃₂[(UO₂)₁₂O₄(H₂O)₁₂(P₂W₁₅O₅₆)₄],¹⁸³ Na₂₄[U₆(OH)₄O₄(AsW₉O₃₃)₄],¹⁸⁴ and (NH₄)₃₀[(UO₂(H₂O))₆(UO₂(H₂O)₆(AsW₉O₃₄)₆]¹⁸⁵), Keggin-type {U₁₃},¹⁸⁶ and cage-like U-oxo-clusters.

The preparation of a series of cage-like structures (Fig. 23) has greatly enriched the large family of cage-like clusters.^{37,180,181,187–196} Burns’ group reported a series of U clusters with oxide (O²⁻), peroxide (O₂²⁻), and hydroxide (OH⁻) bridges, consisting of 20,¹⁹⁴ 24, 28, 32,¹⁸¹ 36,³⁷ 40,¹⁹⁶ 44,³⁷ 50,¹⁹⁶ 60,³⁷ 84,¹⁹⁷ and more metal atoms.¹⁷⁸ In these U(IV) clusters, the U(IV) unit (usually hexagonal bipyramid) shares equatorial edges with its surroundings to form four-, five-, and six-membered ring-like subunits, corresponding to topological squares, pentagons, and hexagons, respectively. Further these metal polygons are integrated into regular structures, such as dodecahedron-shape U₂₀ (Fig. 23a),¹⁹⁴ truncated-octahedron-shape U₂₄ (Fig. 23b),¹⁸¹ and truncated-icosahedron-shape U₆₀ (like buckminsterfullerene, C₆₀; Fig. 23c).³⁷ By introducing additional inorganic and organic ligands in the U clusters, more cage aggregates with polyhedral characteristics can be constructed.^{187–189,198}

Fig. 23 Polyhedral representation of (a) [(UO₂)₂₀(O₂)₃₀]²⁰⁻, 20-metal framework presenting a dodecahedral feature.¹⁹⁴ (b) [(UO₂)₂₄(O₂)₂₄(OH)₂₄]²⁴⁻, 24-metal framework, presenting a truncated octahedral feature.¹⁸¹ (c) [(UO₂)₆₀(O₂)₆₀(OH)₆₀]⁶⁰⁻, 60-metal framework, presenting a truncated icosahedral feature.³⁷

In addition to the U-based clusters, Burns’ team also extensively studied other actinide elements, for example, plutonium(IV) oxide nanocluster [Pu^IV₃₈O₅₆Cl₅₄(H₂O)₈]¹⁴⁻ (Pu₃₈, Fig. 24a).¹⁹⁵ A detailed structural analysis showed that the metal arrangement of the [Pu₃₈O₅₆] core classifies it as typical Keplerate, i.e., a μ₆-O²⁻-centered octahedron {Pu₆}, which is encapsulated by a {Pu₈} cube, and further encapsulated by a truncated octahedral {Pu₂₄} shell (Fig. 24b). By controlling the H₂O content and in the presence of PhCOOH in tetrahydrofuran (THF), Thierry Loiseau solvothermally synthesized an analogue of uranium, U₃₈O₅₆Cl₁₈(THF)₈(PhCOO)₂₄·8THF.¹⁹⁹ Subsequently, a series of U₃₈ with the same core but different protection shells was synthesized and the cluster growth mechanism was deeply explored.^200–203 In 2006, two research groups simultaneously published similar 38-nuclear Bi₃₈ clusters, [Bi₃₈O₄₄(HSal)₂₆(Me₂CO)₁₆(H₂O)₂]⁴⁺ (H₂sal = (2-HO)C₆H₄COOH)²⁰⁴ and [Bi₃₈O₄₅(hfac)₂₄] (hfac = hexafluoroacetylacetone)²⁰⁵ stabilized with different carboxylic acid ligands. Similar Keplerate Bi₃₈ clusters can also be obtained with other carboxylic acid ligands.^206–211 In 2022, Zhang et al. prepared a series of Ti_x-coated Bi₃₈ heterometallic clusters, Bi₃₈@Ti_x (x = 14, 16, 18 and 20). The sensitized outer shell (Ti_x) of Bi₃₈@Ti_x clusters can regulate and influence the photocurrent response and CO₂ cycloaddition reaction.²¹²

Fig. 24 (a) Ball-and-stick representation of [Pu₃₈O₅₆Cl₅₄(H₂O)₈]¹⁴⁻. Color codes: Pu, yellow, blue or orange; O, red; and Cl, green. (b) [Pu₃₈O₅₆] core exhibiting a nested structure, Pu₆(octahedron)@Pu₈(cube)@Pu₂₄(truncated icosahedron).¹⁹⁵

4. Platonic and Archimedean solids in clusters of group IV metals (Ti, Zr, and Hf)

Group IV metal (Ti, Zr, and Hf)-oxo clusters with unique structural stability, reactivity and electronic properties represent a diverse class of compounds with rapidly growing applications in catalysis, materials chemistry, electronics and optics.^{2,68–70,213}

4.1 Ti clusters

As one of the most widely applied photocatalytic materials, titanium dioxide (TiO₂) and its molecular model compounds, titanium-oxo (Ti-O) clusters, are attracting increasing attention.^{2,68–70,214} Most Ti–O clusters are passivated with organic ligands or lacunary polyoxometalate fragments (discussed in detail in Section 3). Some regular Ti–O clusters have an intimate structural relationship with Platonic solids, such as four-metal tetrahedra (Ti₄),^215,216 six-metal octahedra (Ti₆),²¹⁷ eight-metal cube (Ti₈),^218,219 and 12-metal icosahedra (Ti₁₂).²²⁰

Recently, Jian Zhang and Lei Zhang team applied an effective coordination-delayed-hydrolysis strategy for the synthesis and structural modulation of atomically precise titanium-oxo clusters.^2,68,221 Although there have been many examples of the construction of giant Ti–O clusters (with nuclearity of 30 or greater), with up to 52 metal-core,²²² there is only one Ti–O cluster (Ti₄₂) with polyhedral feature. Through the reaction of Ti(OⁱPr)₄ with formic acid under solvothermal conditions, this group isolated the first fullerene-like Ti–O cage, H₆[Ti₄₂(μ₃-O)₆₀(OⁱPr)₄₂(OH)₁₂)] (Ti₄₂, Fig. 25a and b) with an outer diameter of 1.53 nm, wherein the {Ti₄₂O₆₀} core exemplifies the highest possible symmetry for molecules (icosahedral symmetry, I_h, like C₆₀).³⁸ The structure of the 42-titanium framework, abbreviated as {Ti₁₂}@{Ti₃₀} (Fig. 25c), features a nested two-shell arrangement of 42 metal ions, comprised of one icosahedron (Platonic solid) and one icosidodecahedron (Archimedean solid), constructed by 12 seven-coordinated Ti atoms and 30 five-coordinated Ti atoms (Fig. 25b), respectively. Interestingly, the highly symmetrical arrangement of the 60 μ₃-O units linking 42 Ti atoms forms a rare rhombicosidodecahedral cage, resulting from the polyhedral-like arrangement of 42 metals and the stabilization of a square({O₄})/pentagon({O₅}) O-ring by each six/seven-coordinate Ti atom (Fig. 25d).

Fig. 25 (a) Molecular structure of fullerene-like Ti₄₂ cluster. (b) Polyhedral presentation of Ti₄₂. Color codes: {TiO₇}, green and {TiO₅}, orange. (c) Nested two-shell arrangement of the 42 metal ions, {Ti₁₂}@{Ti₃₀} (icosahedron@icosidodecahedron). (d) Rhombicosidodecahedral cage {O₆₀} of Ti₄₂.³⁸

4.2 Zr and Hf clusters

Compared with Ti–O clusters, the synthesis and studies of Zr–O and Hf–O clusters are still very scarce, especially with polyhedral characteristics, wherein mainly M₄ tetrahedra,²²³ M₆ octahedra,^215,224 and simple M₁₀ super-tetrahedra (M₆(octahedron)@M₄(tetrahedron), like V₆Rh₄ in Fig. 6b).^225,226

By mixing NaOH in methanol and the alkoxide of zirconium (Zr(OCH₃)₄), an unstable 13-metal cluster Zr₁₃(μ₄-O)₈(OCH₃)₃₆ (Zr₁₃) was obtained, which is similar to the α-Keggin POM structure (Fig. 5).²²⁷ The Hf-analogue Hf₁₃(μ₄-O)₈(OCH₃)₃₆ (Hf₁₃, Fig. 26a) was also prepared.²²⁸Hf₁₃ features a highly symmetric molecule (O_h) similar to Keggin-type of POMs (Fig. 26b), comprised of an {Hf(μ₄-O)₈} cubic core and a cuboctahedral shell {Hf₁₂(μ₂-OCH₃)₂₄(μ-OCH₃)₁₂} (Hf⋯Hf distances: 3.527(2)–3.586(2) Å). In addition, the inorganic (8 μ₄-O²⁻) and organic (24 μ₂-OCH₃⁻ and 12 μ₁-OCH₃⁻) ligands that act as linkers and caps also exhibit highly ordered arrangements (Fig. 26c) and form a Russian nesting doll-like structure (Hf@O₈@Hf₁₂@O₂₄@O₁₂). Unexpectedly, Hf₁₃ features chemical ultra-stability, keeping the original color, transparency, and morphology of the crystals without significant change in extremely harsh environments (both alkali (20 M boiling NaOH) and acid (10 M HNO₃) aqueous solution), in sharp contrast to the isomorphic Zr₁₃. Due to the excellent stability in acid-base aqueous solution, Hf₁₃ can be used as a luminescence probe for the detection of high-concentration acid-based solution, exhibiting high selectivity and repeatability.

Fig. 26 (a) Molecular structure of Hf₁₃(μ₄-O)₈(OCH₃)₃₆ (Hf₁₃). (b) Polyhedral presentation of Keggin-type Hf₁₃. (c) Three-layer Matryoshka structure of 8 μ₄-O²⁻ (cube), 24 μ₂-OCH₃⁻ (rhombicuboctahedron) and 12 μ₁-OCH₃⁻ (cuboctahedron).²²⁸

5. Platonic and Archimedean solids in clusters of group VII metal (Mn)

In the past two decades, intense research efforts have been concentrated on manganese (Mn) cluster chemistry owing to their interesting magnetic properties, especially high-spin (S) ground state,^229–232 magnetocaloric effects (MCE)^83,233,234 and single-molecule magnets (SMMs).^6,71,73,235 Mn-containing clusters have proven to be one of the most effective sources of SMMs, including the first one, [Mn₁₂O₁₂(O₂CCH₃)₁₆(H₂O)₄].²³⁶ Meanwhile, the Mn atoms in multi-/high-nuclearity Mn-containing clusters can be stabilized in multiple oxidation states (such as Mn^II, Mn^III, and Mn^IV), which are conducive to the formation and coordination of the bridging unit μ_3/4/5/6-O²⁻, further promoting the construction of various multi-/high-nuclearity clusters.⁶

Thus far, besides tetra-nuclear clusters with regular tetrahedra^237,238 and hexa-nuclear clusters with regular octahedral arrangements,^239–241 no metal skeleton of Mn clusters with only one of the other Platonic solids (cube, dodecahedron, and icosahedron) features has been found. In small Mn clusters, some simple nested structures were reported, such as Mn₈ (Mn₄@Mn₄)^242–249 and Mn₁₀ (Mn₆@Mn₄).^{229,230,233,234,250–258}

In addition to the simple clusters with Platonic solid feature described above, the Mn clusters exhibiting an Archimedean solid-type (mainly truncated tetrahedron and cuboctahedron) metal arrangement can also be found, such as Mn₁₂ (truncated tetrahedral geometry),²⁵⁹ Mn₁₃ with super-cubane {Mn₁₃O₁₄} (Mn@O₆(octahedron)@Mn₁₂(cuboctahedron)@O₈(cube)),^260–264 and α-Keggin-type Mn₁₃ (Mn@Mn₁₂(cuboctahedron)).²⁶⁵ The first Mn-based Keplerate cluster with high symmetry (O_h) and high-spin (S = 35/2) ([Mn^IIMn^III₁₂(μ₄-O)₈(μ₄-Cl)₆(tert-butyl-PO₃)₈], Mn₁₃, Fig. 27a) was formed by the reaction of MnCl₂·4H₂O, KMnO₄ and tert-butylPO₃H₂ in the presence of Et₃N.²³¹ Keplerate Mn₁₃ consists of concentric nested arrangements of both Platonic ({(μ₄-O)₈} cube, {P₈} cube, and {Cl₆} octahedron) and Archimedean ({Mn₁₂} cuboctahedron) solids (Fig. 27b).

Fig. 27 (a) Ball-and-stick representation of cluster [Mn^IIMn^III₁₂(μ₄-O)₈(μ₄-Cl)₆(tert-butyl-PO₃)₈]. (b) Polyhedron arrangement of Mn-based keplerate.²³¹

Through the reaction of Mn(NO₃)₂·4H₂O, with triethanolamine (teaH₃) and 6-chloro-2-hydroxypyridine (Hchp) in MeOH, an icosanuclear super-tetrahedral(v₃) Mn^II/III cluster [Mn^II₄Mn^III₁₆O₁₂(OH)₄(tea)₈(chp)₄] (Mn₂₀, Fig. 28a) was obtained by Murray et al.²⁶⁶ The twenty Mn ions in Mn₂₀ are primarily connected by ten μ₄-O²⁻ groups, giving rise to a distorted super-tetrahedral core {Mn^II₄Mn^III₁₆O₁₀}. Specifically, ten corner-sharing μ₄-O-centered {(μ₄-O)Mn₄} tetrahedra self-organize to construct a larger v₃-tetrahedron, consisting of a v₁-tetrahedral Mn₄ core encapsulated by a v₃-tetrahedral Mn₁₆ shell (Mn₁₂(truncated tetrahedron) + Mn₄(tetrahedron)). This Mn₂₀v₃-super-tetrahedron as an inner core also appeared in larger Mn₂₆ clusters, [Mn^II₄Mn^III₂₂(pdol)₁₂(μ₃-CH₃O)₁₂(μ₃-O)₆(μ₄-O)₁₀(N₃)₆] (Mn₂₆, pdol²⁻ = dipyridylketonediolate ion, Fig. 28b), which is identified as a single-molecule magnet according to low-temperature ac susceptibility data.²⁶⁷ The core of the Mn₂₆ cluster consists of an ideal v₃-tetrahedron Mn₂₀ (Fig. 28c) and six external Mn ions in an octahedral manner, which are connected by six μ₃-O²⁻ groups. If the complex v₃-tetrahedron is regarded as a combination of a tetrahedron and truncated tetrahedron featuring only equivalent vertex Mn atoms, then the metal core of Mn₂₆ can be simplified as a combination of four Platonic/Archimedean polyhedrons, Mn₄(tetrahedron)@Mn₁₂(truncated tetrahedron)@Mn₄(tetrahedron)@Mn₆(octahedron) (Fig. 28d). Two similar 26-metal manganese clusters have also been prepared.^235,268

Fig. 28 (a) Ball-and-stick representation of cluster [Mn^II₄Mn^III₁₆O₁₂(OH)₄(tea)₈(chp)₄] (Mn₂₀) with a distorted v₃-tetrahedral {Mn₂₀(μ₄-O)₁₀} core.²⁶⁶ (b) Ball-and-stick representation of cluster [Mn^II₄Mn^III₂₂(pdol)₁₂(μ₃-OCH₃)₁₂(μ₃-O)₆(μ₄-O)₁₀(N₃)₆] (Mn₂₆) with a v₃-tetrahedral {Mn₂₀(μ₄-O)₁₀(μ₃-O)₁₈} core capped by six external Mn ions. (c) v₃-Tetrahedral {Mn₂₀(μ₄-O)₁₀} framework, Mn₄@O₆@Mn₁₂@O₄@Mn₄. (d) Arrangement of 26 Mn atoms and complementary successive Platonic/Archimedean solid description of Mn₂₆ featuring only equivalent vertex metal atoms.²⁶⁷

In 2021, Huang and Li reported the preparation of a 38-metal high-nuclearity Mn^II cluster, [Mn^II₃₈(CO₃)₉O₆Cl₂₄(bmpbt)₁₂(H₂bmpbt)₆]²⁺ (Mn₃₈, H₂bmpbt = 5,5′-di(4-methyl-pyridin-2-yl)-3,3′-bi(1,2,4-triazole), Fig. 29a) with an inner inorganic core [Mn₁₈(CO₃)₉] and metal–organic periphery.²⁶⁹ The core–shell is viewed as an interesting “Matryoshka doll”: Mn₆(octahedron)@Mn₁₂(truncated tetrahedron)@Mn₈(cube)@{Mn₂}₆(octahedron) (Fig. 29b). Note that the vertices of the outermost octahedral shell are [Mn₂OCl₄] subunits rather than single metal atoms.

Fig. 29 (a) Ball-and-stick representation of [Mn₃₈(CO₃)₉O₆Cl₂₄(bmpbt)₁₂(H₂bmpbt)₆]²⁺ (Mn₃₈). (b) Matryoshka doll-like metal skeleton of Mn₃₈.²⁶⁹

In 2016, Tasiopoulos's group reported the preparation of a magnetic nanosized cluster, [Mn^III₃₆Mn^II₁₃(μ₄-O)₃₂(μ₃-OCH₃)₈(μ₃-hp)₂₄(HCOO)₆(DMF)₁₂](OH)₈ (Mn₄₉, H₂hp = 2-(hydroxymethyl)phenol, DMF = dimethylformamide, Fig. 30a), consisting of eight high-spin and high-symmetry (T_d) [Mn^III₆Mn^II₄(μ₄-O)₄] (S = 22) super-tetrahedral building blocks.²³² The Mn–O core {Mn^III₃₆Mn^II₁₃(μ₄-O)₃₂} of Mn₄₉ (Fig. 30b) consists of an octa-coordinated Mn^II-centered v₁-cuboctahedron {Mn^IIMn^III₁₂}, encapsulated by an outer v₂-cuboctahedral shell {Mn^II₁₂Mn^III₂₄} (Fig. 30c). The giant mixed-valence Mn₄₉ cluster features high virtual O_h symmetry and represents unique examples of metal-oxo clusters based on the assembly of a large number of edge-sharing high-nuclearity magnetic building blocks. Besides, Mn₄₉ possesses high-spin ground-state values (S = 61/2) and displays SMM behavior.

Fig. 30 (a) Ball-and-stick representation of cationic cluster [Mn₄₉O₃₂(OCH₃)₈(hp)₂₄(O₂CH)₆(DMF)₁₂]⁸⁺. (b) Metal core {Mn^III₃₆Mn^II₁₃(μ₄-O)₃₂} consisting of a centered v₁-cuboctahedron {Mn^IIMn^III₁₂(μ₄-O)₈} encapsulated by a larger v₂-cuboctahedral shell {Mn^II₁₂Mn^III₂₄(μ₄-O)₂₄}. (c) Arrangement of 49 Mn atoms and complementary successive Archimedean solid description of Mn₄₉ featuring only equivalent vertex metal atoms, Mn@Mn₁₂(cuboctahedron)@Mn₂₄(rhombicuboctahedron)@Mn₁₂(cuboctahedron).²³²

In 2007, Tong and co-workers reported the synthesis of a giant heterometallic neutral cluster with T_d symmetry and unusual magnetic properties, [Cu^I₄Cu^II₁₃Mn^II₄Mn^III₁₂Mn^IV₁₂(μ₃-O)₁₂(μ₄-O)₂₈(tea)₁₂(HCO₂)₆(H₂O)₄] (Cu₁₇Mn₂₈, Fig. 31a).²⁷⁰ Unprecedently, five metal oxidation states (Cu^I, Cu^II, Mn^II, Mn^III, and Mn^IV) coexist in the Cu₁₇Mn₂₈ cluster (Fig. 31b). The most striking structural feature is that all the six-coordinated 28 Mn atoms are joined by forty bridged oxygen groups (12 μ₃-O²⁻ and 28 μ₄-O²⁻) into four {Mn^IIMn^IV₃O₄} and six {Mn^III₂Mn^IV₂O₄} cubanes, which are further bridged to be an adamantane-like Mn cage by sharing the inner 12 Mn^IV ions (Fig. 31c). Finally, the metallic arrangement of the metal core can be a complex Matryoshka structure (abbreviated as μ₄-Cu^II@Mn^IV₁₂@Cu^I₄@Mn^III₁₂@Cu^II₁₂@Mn^II₄), contained in an innermost {Cu^II(μ₄-O)₄} core and moving outward, five regular polyhedra, i.e., a truncated tetrahedron, a tetrahedron, a distorted truncated tetrahedron, a cuboctahedron and a tetrahedron (Fig. 31d).

Fig. 31 (a) Ball-and-stick of Cu₁₇Mn₂₈. (b) Metal-oxo core {Cu₁₇Mn₂₈(μ₄-O)₂₈} of Cu₁₇Mn₂₈. (c) Adamantane-like Mn^II₄Mn^III₁₂Mn^IV₁₂O₂₈ cluster containing ten sharing-vertices cubane-like units (four {Mn^IIMn^IV₃O₄} and six {Mn^III₂Mn^IV₁₂O₄}). (d) Complementary successive Platonic/Archimedean polyhedral description of Cu₁₇Mn₂₈ core-geometry featuring only equivalent vertex metal atoms: {Cu^II(μ₄-O)₄}(tetrahedron)@Mn^IV₁₂(truncated tetrahedron)@Cu^I₄(tetrahedron)@Mn^III₁₂(distorted truncated tetrahedron)@Cu^II₁₂(cuboctahedron)@Mn^II₄(tetrahedron).²⁷⁰

6. Platonic and Archimedean solids in clusters of group VIII metals (Fe, Co, Ni, Pd, and Pt)

6.1 Fe clusters

Iron (Fe) clusters, especially high-nuclearity Fe aggregates, continue to be synthetic targets for synthetic chemists and materials scientists worldwide, not only because of their interesting structures and magnetic and catalytic properties but also their relevance to the Fe-storage ferritin proteins.^271,272 Tetrahedral Fe₄,²⁷³ octahedral Fe₆,^274,275 and cubic Fe₈ clusters^276,277 have been reported but are not discussed in detail in this review. In addition to cubic Fe₈ clusters, some octa-nuclear Fe–O clusters with an inner Fe₄(μ-O)₄-cubane and four peripheral tetrahedral arranged Fe centers in an Fe₄@Fe₄ manner were reported.^{243,278–286}

The first open-shell Keggin-type 13-metal cluster, [Fe^III₁₃(μ₄-O)₄F₂₄(μ₂-OMe)₁₂]⁵⁻ (Fe₁₃, Fig. 32a) containing 13 high-spin d⁵ Fe^III atoms, was reported by Bino and co-workers.²⁸⁷ The anion of Fe₁₃ has an ideal α-Keggin-type structure with T_d symmetry (Fig. 32b), wherein the 12 surrounding Fe^III atoms exhibit a cuboctahedron arrangement and a central tetrahedral {Fe^III(μ₄-O)₄} core (Fig. 32c). Polyfluorometalate (Fe₁₃) demonstrated for the first time the combination of transition-metal (TM) atoms with an open-shell configuration and F⁻ ligands, leading to the isolation of a new Keggin-type cluster. The Keggin Fe₁₃-oxo core is regarded as a basic structural unit for iron oxyhydroxide [FeO(OH)] cluster materials, such as ferrihydrite (Fe₅HO₈·4H₂O) and magnetite (Fe₃O₄). Since then, with the efforts of synthetic chemists, not only a series of Fe–O/OH clusters containing α- or ε-Keggin-type Fe₁₃-oxo cores (such as Fe₁₇ (Fig. 32d–f),^288,289Fe₁₉ (Fig. 32j–l),²⁹⁰ Fe₃₀,²⁹¹ and Fe₃₄ (Fig. 32m–o)²⁹²) were prepared, but also heterometal ions were used to capture and stabilize the basic motifs (like Bi³⁺-stabilized Bi₆Fe₁₃ (Fig. 32g–i)^293–295 and Ln³⁺-shell-protected La₆Fe₁₃,²⁹⁶ Ln₁₆Fe₂₉ (Ln = Y, Gd),²⁹⁷ Ln₁₂Fe₃₃ (Ln = Y, Gd, Pr, and Dy).^297,298 The synthesis and preparation of these metal oxyhydroxide clusters of different sizes can help us understand the formation mechanism of related natural minerals.

Fig. 32 Ball-and-stick, polyhedral and metal arrangement representations of (a–c) [Fe₁₃O₄F₂₄(OMe)₁₂]⁵⁻ (Fe₁₃),²⁸⁷ (d–f) [Fe₁₇O₁₆(OH)₁₂(py)₁₂Cl₄]³⁺ (Fe₁₇),²⁸⁸ (g–i) Bi₆[Fe₁₃O₁₆(OH)₁₂(O₂C(CCl₃)₁₂]⁺ (Bi₆Fe₁₃),²⁹³ (j–l) [Fe₁₉O₁₄(OEt)₃₀]⁻ (Fe₁₉),²⁹⁰ and (m, n and o) [Fe₃₄O₃₈(μ₂-OH)₁₂Br₁₂(py)₁₈] (Fe₃₄).²⁹²

By the simple reaction of anhydrous FeCl₃ in pyridine (py), a 17-metal Fe^III-oxo/hydroxy cluster [Fe₁₇O₁₆(μ₄-OH)₁₂(py)₁₂Cl₄]³⁺ (Fe₁₇, Fig. 32d–f) was prepared by Heath et al.^288,289 The core of Fe₁₇ also has a central tetrahedral {Fe^III(μ₄-O)₄} motif, then bridging twelve outer octahedral {Fe^III(μ₂-OH)₂(μ₃-O)(μ₄-O)₂(py)}. In contrast to Fe₁₃, the twelve metal atoms form a truncated tetrahedron, further assembled with a central tetrahedron into a POM-like ε-Keggin architecture (Fig. 8 and 32e). Four hexagons of this truncated tetrahedron are capped by four additional {Fe^III(μ₃-O)₃Cl} motifs with a tetrahedral arrangement (Fig. 32f). Using pyridine to control the slow hydrolysis of Fe^III ions, Fe₃₀²⁹¹ and Fe₃₄ (Fig. 32m–o)²⁹² were successively synthesized.

In the presence of hexamethylenetetramine, the dissolution and reaction of anhydrous FeBr₃ in a mixture solvents (py and CH₃CN) resulted in the formation of a giant Fe^III-oxyhydroxide cluster, [Fe^III₃₄(μ₄-O)₄(μ₃-O)₃₄(μ₂-OH)₁₂Br₁₂(py)₁₈]Br₂ (Fe₃₄, Fig. 32m).²⁹² From the inside to the outside, there are alternating layers of tetrahedral (green) and hexahedral (brown) Fe^III ions by oxide (μ₃-O²⁻ and μ₄-O²⁻) and hydroxide (μ₂-OH⁻) ions in Fe₃₄ (Fig. 32n), similar to the enlarged version of Fe₁₇. The metallic skeleton (Fig. 32o) described an adamantane-like {Fe^III₄(μ₃-O)₆} tetrahedron encapsulated within an elongated truncated tetrahedron, whose hexagons are capped by four {Fe^III₃(μ₃-O)Br₃} triangles. Interestingly, there is a regular octahedron inside the elongated truncated tetrahedron. Thus, the metallic skeleton of Fe₃₄ can be viewed as the combination of concentric polyhedrons, Fe₄(tetrahedron)@Fe₆(octahedron)@Fe₁₂(elongated truncated tetrahedron)@(Fe₃)₄(tetrahedron).

In 2015, Nyman's group isolated a discrete Bi₆-coated Fe₁₃-oxo/hydroxy cluster Bi₆[FeO₄Fe₁₂O₁₂(OH)₁₂(O₂C(CCl₃)₁₂]⁺ (Bi₆Fe₁₃, Fig. 32g), which has the same structural features as ferrihydrite.^293–295Bi₆Fe₁₃ has an α-Keggin-type Fe–O core similar to Fe₁₃ (Fig. 32h), which is comprised of a central {Fe^IIIO₄} tetrahedron and a cuboctahedra shell {Fe₁₂}, capped by six Bi atoms in an octahedral fashion (Fig. 32i). The Fe^III-oxy/hydroxy aggregate of Bi₆Fe₁₃ may be an intermediate state from ion to iron-oxy/hydroxide precipitates or nanoparticles.

Besides, Spandl's group reported the preparation of a similar 19-metal v₂-octahedral “Super-Lindqvist” Fe₁₉ cluster, [HFe₁₉O₁₄(OEt)₃₀] (Fe₁₉, Fig. 32j).²⁹⁰ The Fe–O skeleton of Fe₁₉ (without C atoms) exhibits an approximate O_h symmetry. In the center of Fe₁₉, there are six μ₆-O in a nearly perfect octahedral manner, surrounding a central six-coordinated Fe^III atom. {Fe(μ₆-O)₆} is enveloped by an octahedral nested shell, {Fe₁₈(μ₃-O)₈(OEt)₆(μ₂-OEt)₂₄} (Fig. 32l, inner cuboctahedron {Fe₁₂} and outer octahedron {Fe₆}, like Ti₁₂Nb₆, Fig. 21). Although the 12-metals are arranged in a polyhedron (cuboctahedron) as in the α-Keggin structure, the spatial layout of the coordination atoms oxygen varies greatly (Fig. 32b, h and k). Fe₁₉ features a core–shell arrangement of metal atoms and O bridges/ligands (μ₃-O, μ₆-O, μ₂-OEt, and μ₁-OEt) contained in an innermost Fe^III atom and moving outward, six Platonic or Archimedean solids (μ₆-Fe@(μ₆-O)₆@Fe₁₂@(μ₃-O)₈@(μ₂-OEt)₂₄@Fe₆@(μ-OEt)₆), i.e., an octahedron, a cuboctahedron, a cube, a cuboctahedron and two octahedrons.

From Fe₁₃, Fe₁₇, and Fe₁₉ to the most recently reported Fe₃₄, exhibiting the structural similarity, these clusters are prepared by a simple bottom-up synthetic methodology. Thus, larger Fe^III–O/OH clusters with high symmetry are likely to be successfully prepared soon.

Besides metal–O²⁻/OH⁻ clusters with core–shell frameworks, some interesting hollow cluster-based cages were isolated and characterized.^40,299–302 For example, a 64-nuclear cubic hollow cage, {[Fe^III₈O₃(tea)(teaH)₃(HCOO)₆]₈ (HCOO)₁₂}¹²⁺ (Fe₆₄).³⁰¹ The cubic Fe₆₄ consists of 8 corners of octanuclear [Fe^III₈O₃(tea)(teaH)₃(HCOO)₆] subunits with a propeller-like Fe^III₈O₃ core and 12 edges of anti–anti HCOO⁻ ligands.

6.2 Co clusters

In numerous works, Co and Ni clusters are often studied together, owing to their same or similar skeleton protected by the same inorganic and organic ligands. The employment of organic oxygen-containing ligands has been proven to be a successful strategy for the synthesis of multi-/high-nuclearity Co and Ni clusters. Small Co/Ni-containing clusters featuring simple Platonic solids (tetrahedron,^{273,303–305} octahedron,^{241,306–308} and cube^309–315), Archimedean solid (truncated tetrahedron),^316–318 and common nested structure (M₄(tetrahedron)@M₄(tetrahedron),^319–322 M₆(octahedron)@M₄(tetrahedron),^323–325 and M₆(octahedron)@M₈(cube),³²⁶ M = Co or Ni) are not discussed here.

In 2017, Bai and co-workers reported the preparation of a homochiral mixed-valence v₃-supertetrahedral cobalt cluster from a racemic ligand, [Co^III₄Co^II₁₆(μ₆-O)₄(μ₃-OH)₁₂(S-bme)₁₂(OAc)₆]⁶⁺ (Co₂₀-T, Hbme = 1H-(benzimidazol-2-yl)ethanol, Fig. 33a).³²⁷Co₂₀-T presents a beautiful mixed-valence Co-oxo supertetrahedral core, {Co^III₄Co^II₁₆(μ₆-O)₄(μ₃-OH)₁₂} (Fig. 33b), constructed by an inner tetrahedron {Co^III₄} and outer large v₃-tetrahedron {Co^II₁₆} (Co₁₂(truncated tetrahedron)@Co₄(tetrahedron)). Alternatively, its 20 Co atoms are organized into 11 cubane-like units in three forms (one {Co^III₄(μ₆-O)₄}, six {Co^III₂Co^II₂(μ₆-O)₂(μ₃-OH)₂}, and four {Co^IIICo^II₃(μ₆-O)(μ₃-OH)₃}). Ten cubane groups on the periphery are connected by sharing corners, and the central {Co^III₄(μ₆-O)₄} motif is encapsulated in the center by sharing edges with six neighbors {Co^III₂Co^II₂(μ₆-O)₂(μ₃-OH)₂} (Fig. 33c). The chiral Co₂₀-T is the largest super-tetrahedral Co cluster reported to date, which exhibits high photocatalytic activity and ferromagnetic behavior.

Fig. 33 (a) Ball-and-stick representation of cationic cluster [Co₂₀O₄(OH)₁₂(S-bme)₁₂(OAc)₆]⁶⁺. (b) Co–O core structure, {Co^III₄Co^II₁₆(μ₆-O)₄(μ₃-OH)₁₂}. (c) Illustration of the assembly of 11 vertex-sharing or edge-sharing cubane units ({Co^III₄(μ₆-O)₄} + 6{Co^III₂Co^II₂(μ₆-O)₂(μ₃-OH)₂} + 4{Co^IIICo^II₃(μ₆-O)(μ₃-OH)₃}) into a v₃-tetrahedron.³²⁷

In 2012, Huang's group reported the preparation of an eight-cobalt-capped 20-metal isopolyoxometalate cluster with an α-Keggin-type structure, [Co^II₂₀(μ₃-OH)₂₄(MMT)₁₂(SO₄)]²⁺ (Co₂₀-1, MMT = 2-mercapto-5-methyl-1,3,4-thiadiazole, Fig. 34a).³²⁸ A 20-metal Ni analogue has also been reported.³²⁹Co₂₀-1 has a non-crystallographic T_d symmetry and possesses an open-shell SO₄²⁻-centered α-Keggin core [SO₄@Co₁₂(μ₃-OH)₂₄N₁₂]²⁺, with a cuboctahedral arrangement. The α-Keggin-like core was capped by eight metal atoms in a cubic fashion (Fig. 34b). Clusters (Mn₁₃Cu₈,³³⁰ Cu₂₀,^331,332 Cu₁₂Zn₈,³³² and Cu₁₂Mg₈³³¹) similar to the metal framework (cuboctahedron(M@M₁₂ or M₁₂)@cube(M₈)) have also been reported.

Fig. 34 Ball-and-stick representation and metal arrangement of (a and b) [Co₂₀(OH)₂₄(MMT)₁₂(SO₄)]²⁺ (Co₂₀-1)³²⁸ and (c and d) [Co₂₀(OH)₁₂(maleate)₁₂(Me₂NH)₁₂]⁴⁺ (Co₂₀-2).³³³

Liang and co-workers also reported two isostructural 20-metal clusters, [M₂₀(μ₃-OH)₁₂(maleate)₁₂(Me₂NH)₁₂](BF₄)₃(OH) (M = Co^II (Co₂₀-2) or Ni^II; H₂maleate = fumaric acid, Fig. 34c),³³³ different from Co₂₀-1. The spherical Co₂₀-2 with T_h symmetry possesses an ideal cubic {M₈(μ₃-OH)₁₂} core, where the M atoms as vertices are bridged by μ₃-OH⁻ groups as the edges, while these μ₃-OH⁻ groups are coordinated further by outer identical 12 metal atoms, which build a distorted M₁₂ icosahedral shell to encapsulate the cubic core (Fig. 34d).

In 2011, Liao's group obtained p-tert-butylthiacalix[4]arene(TC₄A)-capped high-nuclearity bi-metallic {Co^II₂₄M₈} (M = Mo or W) nanospheres, [Co^II₂₄(TC₄A)₆(MO₄)₈Cl₆]²⁺ (Co₂₄, Fig. 35a).³³⁴ In Co₂₄, each Cl⁻ anion is encapsulated by a different metal–organic super calix, resulting in the constructing of six square-like Co₄Cl-TC₄A SBBs. Each Co₄Cl-TC₄A linked with four MoO₄²⁻ or WO₄²⁻ anions, and each MoO₄²⁻ or WO₄²⁻ as a template connects with three Co₄Cl-TC₄A motifs, forming a flat hexagonal arrangement (Fig. 35b). Consistent with Cl₆@V₂₄As₈ described above (Fig. 7), Co₂₄ with O_h symmetry has the same topology (keplerate structure: Cl₆(octahedron)@Co₂₄(truncated octahedron)@Mo₈/W₈(cube)).

Fig. 35 Ball-and-stick representation and metal frameworks of (a and b) [Co₂₄(TC₄A)₆(MoO₄)₈Cl₆]²⁺ (Co₂₄)³³⁴ and (c and d) {Co₃₂O₂₄(H₂O)₂₄(TC₄A)₆} (Co₃₂).⁴¹

In 2009, Liao and co-workers reported the preparation of a giant spherical Co cluster capped by TC₄A ligands, {Co^II₂₄Co^III₈(μ₃-O)₂₄(H₂O)₂₄(TC₄A)₆} (Co₃₂, Fig. 35c).⁴¹ A 32-metal Ni analogue had also been reported.³³⁵Co₃₂ is also a nested structure, possessing a Co^III₈ cubic core encapsulated by a 24-metal shell with a truncated octahedral arrangement (Fig. 35d). Interestingly, the SBBs (M₄-TC₄A) of Co₂₄ and Co₃₂ all exhibit a regular octahedral arrangement. By introducing suitable polydentate ligands to assemble thiacalixarene-based SBBs, a series of metal cluster–organic polyhedra with a regular arrangement (such as tetrahedra, octahedra, and icosahedra) were obtained.^336–342

In 2013, the dimethylmalonate(Me₂Mal²⁻)-protected anion cage cluster [Co₃₆(H₂O)₁₂(μ₃-OH)₂₀(HMe₂Mal)₂(Me₂Mal)₂₈]⁶⁻ (Co₃₆) was prepared.³⁴³ The isostructural 36-Ni cluster was also obtained.³⁴⁴ The metal skeleton of Co₃₆ can be seen as an arrangement of two shells, in which the 24 metals inside form eight triangles with a cubic fashion or a distorted truncated cube, and the other twelve metals on the outside can form a slightly distorted cuboctahedron.

6.3 Ni clusters

Given that the coordination forms of Ni and Co are similar, the Ni–O/OH clusters featuring metal frameworks of Platonic and Archimedean solids are highly similar with the above-mentioned cobalt clusters. The related structures have already been discussed and will not be repeated here.

Due to the existence of the original state of the metal bonds, atomically precise Ni-containing clusters with direct metal–metal bonding have been widely synthesized and studied. Nuckolls and co-workers described a series of solid-state materials constructed from the binary assembly of atomically precise molecular clusters featuring direct metal–metal bonding and activated electronic transport, [Co₆Se₈(PEt₃)₆][C₆₀]₂ (Co₆Se₈P₆), [Cr₆Te₈(PEt₃)₆][C₆₀]₂ (Cr₆Te₈P₆), and [Ni₉Te₆(PEt₃)₈][C₆₀] (Ni₉Te₆P₈, Fig. 36a).³⁰⁷ These clusters with O_h symmetry show the classic Keplerate structure, as follows: (i) Co₆Se₈P₆ and Cr₆Te₈P₆ have three-shell topology, i.e., Co₆/Cr₆(octahedron)@Se₈/Te₈(cube)@P₆(octahedron) and (ii) Ni₉Te₆P₈ (Fig. 36b) exhibits a core–shell structure, i.e., (μ₈-Ni)@Ni₈(cube)@Te₆(octahedron)@P₈(cube).

Fig. 36 Ball-and-stick and atomic arrangement representation of (a and b) [Ni₉Te₆(PEt₃)₈] with Ni@Ni₈@Te₆@P₈³⁰⁷ and (c and d) [As@Ni₁₂@As₂₀]³⁻.⁴²

In 2003, Eichhorn's group prepared a ligand-free intermetalloid cluster, [As@Ni₁₂@As₂₀]³⁻ (Fig. 36c).⁴² This cluster exhibits interpenetrating reciprocal Platonic polyhedrons and contains an icosahedral Ni₁₂ fragment with a central μ₁₂-As atom, encapsulated by an As₂₀ dodecahedral (fullerene-like) shell, resulting in the formation of an onion-skin-like [As@Ni₁₂@As₂₀]³⁻ with nearly perfect I_h point symmetry (Fig. 36d). The structures of the same ligand-free core–shell-type intermetalloid cluster built from other elements A@B₁₂@A₂₀ (A = As, Sn, Pb, Sb; B = Ni, Mg, Zn, Cd, Mn, and Pd) have been synthesized.^345–348

Recently, Zacchini and Berben reviewed the advances in metal carbonyl clusters involving groups VIII (Fe, Co, Ni, Pd, Pt, etc.) elements, with emphasis on their synthesis, electronic structures and catalysis.²² Nickel carbonyl clusters featuring Platonic and Archimedean solids are summarized and discussed here. In 2015, Zacchini's group reported the preparation of some Ni poly-carbide carbonyl nanoclusters.³⁴⁹ During the optimization experiment, a by-product ([Ni₃₂C₆(CO)₃₆]⁶⁻, Ni₃₂C₆, Fig. 37a) with low yield was isolated. The hexacarbide Ni₃₂C₆ exhibits the classical Keplerate-type structure and can be viewed as a combination of three Platonic or Archimedean solids (Fig. 37b), abbreviated as Ni₈(cube)@C₆(octahedron)@Ni₂₄(truncated octahedron).

Fig. 37 (a) Ball-and-stick representation of [Ni₃₂C₆(CO)₃₆]⁶⁻. (b) Nested structure of [Ni₃₂C₆(CO)₃₆]⁶⁻, Ni₈(octahedron)@C₆(hexahedron)@Ni₂₄(truncated octahedron).³⁴⁹

In addition to the preparation of homopolymetallic carbonyl clusters, Ni-containing heterometallic clusters have also been extensively studied, such as Au₆Ni₁₂,³⁵⁰Ag₁₆Ni₂₄,³⁵¹Pt₁₄Ni₂₄,³⁵² and super-octahedron Pt₆Ni₃₈.^43–45 Dahl's group reported the synthesis of the first bimetallic Au-Ni carbonyl cluster, [Au₆Ni₁₂(CO)₂₄]²⁻ (Au₆Ni₁₂).³⁵⁰Au₆Ni₁₂ contained an ideal T_d configuration and its metal core can be regarded as an Au₆ octahedron capped by four tetrahedral {Ni₃(CO)₈} triangular units (12 Ni atoms: elongated cuboctahedron). The first high-nuclearity bimetallic Ag–Ni carbonyl cluster [Ag₁₆Ni₂₄(CO)₄₀]⁴⁻ (Ag₁₆Ni₂₄, Fig. 38a) was also reported by Dahl's group.³⁵¹Ag₁₆Ni₂₄ possesses a 40-metal cubic pseudo-T_d closed-packed geometry (Fig. 38b), which can be described as a central Ag₁₆ kernel (v₂-truncated tetrahedron: Ag₄(tetrahedron)@Ag₁₂(truncated tetrahedron)) linked by direct Ag–Ni bonding to four triangular {Ni₆(CO)₁₀} units featuring a v₂-cuboctahedron, i.e., Ni₁₂(truncated tetrahedron)@Ni₁₂(cuboctahedron). Each triangular {Ni₆(CO)₁₀} can be viewed as a hexagonal close-packing (hcp) extension of the inner Ag₁₆ core along one of the four symmetry-equivalent 3-fold axes, causing each of the Ag atoms of Ag₄(tetrahedron) to be encapsulated by 12 hcp metal atoms.

Fig. 38 (a) Ball-and-stick representation of [Ag₁₆Ni₂₄(CO)₄₀]⁴⁻ (Ag₁₆Ni₂₄). (b) Four-layered nested 40-metal core Ag₄(tetrahedron)@Ag₁₂(truncated tetrahedron)@Ni₁₂(truncated tetrahedron)@Ni₁₂(cuboctahedron) of Ag₁₆Ni₂₄.³⁵¹

With the reaction of [NⁿBu₄]₂[Ni₆(CO)₁₂] in THF solution with K₂PtCl₄, Giuliano Longoni and co-workers obtained hexagonal black crystals of the heterometal Pt–Ni truncated octahedral cluster [Pt₁₄Ni₂₄(CO)₄₄]⁴⁻ (Pt₁₄Ni₂₄, Fig. 39a).³⁵²Pt₁₄Ni₂₄ with a core–shell structure is comprised of a face-centre cubic (fcc) Pt₁₄ core and Ni₂₄ truncated octahedral shell (Fig. 39b). The fcc-Pt₁₄ core geometry can be viewed as the combination of an octahedron (Pt₆) and cube (Pt₈). The structural polyhedron of the 38-metal Pt–Ni core can be abbreviated as Pt₆@Pt₈@Ni₂₄, featuring a core–shell arrangement of metal atoms contained in an octahedron and moving outward, a hexahedron and a truncated octahedron.

Fig. 39 (a) Ball-and-stick representation of [Pt₁₄Ni₂₄(CO)₄₄]⁴⁻ (Pt₁₄Ni₂₄). (b) Core–shell structure of Pt₁₄Ni₂₄, Pt₆(octahedron)@Pt₈(cube)@Ni₂₄(truncated octahedron).³⁵² (c) Ball-and-stick representation of [Pt₆Ni₃₈(CO)₄₈]⁶⁻ (Pt₆Ni₃₈). (d) Pt₆ octahedral core and v₃-octahedral Ni₃₈ shell.⁴⁴ (e) 38-metal-based v₃-octahedron consists of three components, cube, truncated octahedron and octahedron.

Manassero and co-workers first synthesized four v₃-octahedral heterometal Pt–Ni clusters with 44 metal atoms, [Pt₆Ni₃₈(CO)₄₈]⁶⁻ (Pt₆Ni₃₈, Fig. 39c).^43–45 The 44-metal v₃-octahedral structure is comprised of an octahedral Pt₆ core and a v₃-octahedral Ni₃₈ shell (Fig. 39d). The v₃-octahedral Ni₃₈ shell can be viewed as the combination of three polyhedra (cube Ni₈ + truncated octahedron Ni₂₄ + octahedron Ni₆, Fig. 39e) with the metal as the vertex. Thus, the metal core of the carbonyl cluster can be viewed as the assembly of four Platonic and Archimedean solids (Pt₆@Ni₈@Ni₂₄@Ni₆).

6.4 Pd clusters

In the field of the metal-based catalysts, palladium-containing nanoparticles and materials occupy a privileged position, mainly due to their excellent catalytic performance, such as catalysts in coupling and hydrogenation reactions, oxidation catalysts in automobile emission-control systems, and reforming catalysts in the production of high-octane gasoline.^353–356 As ideal models, the precise structure of polyoxopalladates (POPs)³⁵⁶ and Pd clusters with direct metal–metal bonding¹⁵ can help researchers understand the underlying catalytic mechanisms.

Controlled hydrolysis-condensation processes of {Pd^IIO₄} square-planar units in the presence of oxyacid groups (such as PO₄³⁻, AsO₄³⁻, and SeO₃²⁻) induce the self-assembly of these discrete units to obtain polynuclear Pd^II-oxo clusters.³⁵⁶ Since the first cubic POPs [H₆Pd^II₁₃O₈(AsO₄)₈]⁸⁻ were reported by Kortz and co-workers in 2008,³⁵⁷ more than seventy POPs have been obtained, which encompass a large structural variety. Among them, only in the family of cubic clusters, with the general formula of MPd₁₂L₈ (M = Na^I, Ca^II, Sc^III, Cr^III, Mn^II, Fe^III, Co^II, Ni^II, Cu^II, Zn^II, Ga^III, Sr^II, Y^III, Pd^II, In^III, and Ln^III; L = PO₄³⁻, AsO₄³⁻, SeO₃²⁻, PhPO₃³⁻, and PhAsO₃³⁻), the twelve peripheral Pd-metal atoms are arranged in the form of a cuboctahedron (an Archimedean solid), in which the degree of deviation from the ideal polyhedron is related to the nature of the selected capping group (such as charge, X–O bond length (X = P, As, Se), steric hindrance, and geometry).³⁵⁶ Here, we choose [Pd^II₁₃(As^VPh)₈O₃₂]⁶⁻ with O_h symmetry (without phenyl substituents) to introduce the cuboid-shaped POP cluster.³⁵⁷ The structure of {Pd₁₃As₈O₃₂} can be described as Keplerate type, featuring Platonic and Archimedean solids of different sizes, similar to a complicated “Matryoshka doll sequence”, i.e., Pd@O₈(cube)@Pd₁₂(cuboctahedron)@As₈(cube)@O₂₄(truncated cube).

In addition to POPs, palladium can also form an interesting family of highly condensed carbonyl/phosphine-protected high-nuclearity homopalladium/heteropalladium clusters.^22,45 Lawrence F. Dahl and co-workers reported the preparation of two Pd- and one Pd–Pt-carbonyl cluster with the icosahedral Mackay hard-sphere model, Pd₅₅(P(ⁱPr)₃)₁₂(μ₃-CO)₂₀ (Pd₅₅, Fig. 40a),³⁵⁸ Pd₁₄₅(CO)_x(PEt₃)₃₀ (x ≈ 60, Pd₁₄₅, Fig. 40b),³⁵⁹ and (μ₁₂-Pt)Pd_164−xPt_x(CO)₇₂(PPh₃)₂₀ (x ≈ 7, (PtPd)₁₆₅, Fig. 40c).⁴⁷Pd₅₅ is the first crystallographically documented example of a 55-metal cluster with Mackay two-shell icosahedron. Pd₅₅ features pseudo-I_h symmetry (without isopropyl groups), and the 55-Pd core (ideally I_h symmetry) can be viewed as simply a “Matryoshka doll sequence”. The center μ₁₂-Pd atom (color code: black) is encapsulated by the icosahedron in shell 1 (Pd₁₂, Fig. 40d), which in turn is encapsulated by the icosahedron in shell 2 (Pd₄₂, Fig. 40e). The 42-metal shell can be regarded as a combination of an icosidodecahedron and icosahedron (Fig. 40h). Meanwhile, the peripheral ligands (12 P(ⁱPr)₃ and 20 μ₃-CO) are arranged in the form of icosahedron and pentagonal dodecahedron, respectively. In addition to the same interior core (Pd₅₅), Pd₁₄₅ has a third-shell (shell 3) that possesses 60 equivalent vertices with the arrangement of rhombicosidodecahedra (Fig. 40f). 30 additional Pd atoms with the icosidodecahedral arrangement cap the thirty square faces of shell 3 and each Pd is attached by phosphine ligands (PEt₃). The structure of the 145-palladium core, which can be abbreviated as μ₁₂-Pd@Pd₁₂@Pd₄₂(Pd₁₂ + Pd₃₀)@Pd₆₀@Pd₃₀, features a core–shell arrangement of metal atoms contained in the innermost μ₁₂-Pd atom, and moving outward, five Platonic or Archimedean solids, i.e., an icosahedron, icosidodecahedron, icosahedron, rhombicosidodecahedron, and icosidodecahedron. The (PtPd)₁₆₅ cluster features a Pt-centered four-shell 165-atom Pt-Pd core with interesting intershell bridging carbonyl ligands (Fig. 40c). The same interior 115-metal-atom three-shell (Pd_center@Pd₁₂@Pd₄₂@Pd₆₀) polyhedral geometries of Pd₁₄₅ can be observed in the 165-metal-atom four-shell Pt-centered (PtPd)₁₆₅ nanoclusters (Pt_center@Pd_12−xPt_x (x ≈ 1.2)@Pd_42−xPt_x (x ≈ 3.5)@Pd_60−xPt_x (x ≈ 2.2)@Pd₅₀). In (PtPd)₁₆₅, the outermost 50 Pd–metal atoms construct shell 4, featuring an arrangement of pentagonal dodecahedra (Fig. 40g). Shell 4 can be regarded as a combination of Pd₃₀(icosidodecahedron) and Pd₂₀(pentagonal dodecahedron) if it is described with only equivalent vertex metal atoms (Fig. 40i). Thus, the structure of the 165-atom Pt–Pd core, which can be abbreviated as μ₁₂-Pt@(PtPd)₁₂@(PtPd)₄₂((PtPd)₁₂ + (PtPd)₃₀)@(PtPd)₆₀@Pd₅₀(Pd₂₀ + Pd₃₀), features a core–shell arrangement of metal atoms contained in the innermost μ₁₂-Pt atom, and moving outward, six Platonic or Archimedean solids, i.e., an icosahedron, icosidodecahedron, icosahedron, rhombicosidodecahedron, icosidodecahedron, and pentagonal dodecahedron.

Fig. 40 (a) Two-shell structure of Pd₅₅(P(ⁱPr)₃)₁₂(μ₃-CO)₂₀ containing a Mackay icosahedral cluster core.³⁵⁸ (b) Three-shell structure of Pd₁₄₅(PEt₃)₃₀.³⁵⁹ (c) Four-shell structure of (μ₁₂-Pt)Pd_164−xPt_x(CO)₇₂(PPh₃)₂₀ (x ≈ 7). (d–g) Shells 1–4 of (PtPd)₁₆₅: (PtPd)₁₂ icosahedron with a central (μ₁₂-Pt) atom, (PtPd)₄₂ icosahedron, (PtPd)₆₀ rhombicosidodecahedron, and Pd₅₀ pentagonal dodecahedron. (h) (PtPd)₄₂(icosahedron) is transformed into (PtPd)₁₂(icosahedron) and (PtPd)₃₀(icosidodecahedron). (i) Pd₅₀(pentagonal dodecahedron) is transformed into Pd₂₀(pentagonal dodecahedron) and Pd₃₀(icosidodecahedron).⁴⁷ For clarity, the C, O, and H atoms are omitted. Color codes: Pd(Pt), black, indigo, red, green, yellow; and P, pink.

In addition to the above-mentioned Pd clusters protected by carbonyl and phosphine ligands (Fig. 40), through the reduction of an organometallic complex [Pd₃(μ₃-C₇H₇)₂]²⁺ with tetraarylborate, Murahashi afforded an interesting cationic organometallic cluster, [Pd₁₃(μ₄-Tr)₆]²⁺ (Tr = cycloheptatrienyl), containing a Pd-centered cuboctahedral Pd₁₃ core protected by an octahedral ligand–shell (μ₄-Tr)₆.³⁶⁰

6.5 Pt clusters

No platinum (Pt) analogues featuring Platonic and/or Archimedean solids of polyoxopalladates have been reported to date. Wickleder and Pley discovered the first noble metal-based POMs (NH₄)₄[Pt^III₁₂O₈(SO₄)₁₂] in 2004.³⁶¹ In the anion cluster, the 12 Pt^III ions form a {Pt^III₁₂} skeleton with a remarkably distorted icosahedral arrangement, owing to the shorter metal–metal bond (Pt⋯Pt: ca. 2.53 Å) in the six {Pt₂} groups and longer Pt⋯Pt separations (ca. 3.45 Å) between the adjacent {Pt₂} groups.

Besides Pt–O clusters, platinum carbonyl (Pt_x(CO)_y) clusters have been reported to be an ideal model to understand the CO adsorption on surfaces of ultrasmall metal crystallites and nanoparticles.^22,362 Researchers synthesized a series of carbonyl-protected homoplatinum/heteroplatinum clusters with direct metal–metal bonding, and some Pt_x(CO)_y clusters featuring classical Platonic or Archimedean solids.^{43,44,47,352,363} Masciocchi and co-workers first reported the single-crystal structure of a cubic close-packed (ccp) Pt₃₈ cluster, [N(PPh₃)₂]₂[Pt₃₈(CO)₄₄] (Pt₃₈), which disclosed the exact stereochemistry of 44 carbonyl ligands that had never been determined before.³⁶⁴Pt₃₈ possessed a ccp metal core of idealized O_h symmetry. The 38 platinum atoms constitute a truncated octahedron (like Pt₁₄Ni₂₄, Fig. 39) with three chemically distinct types of metal atoms, exhibiting different polyhedron geometries, Pt₆(hexahedron)@Pt₈(octahedron)@Pt₂₄(truncated octahedron). In addition, Pt₃₈ bears 32 terminal carbonyl groups and 12 edge-bridging CO ligands. Thus, the overall ideal symmetry is lowered to D_2d.^365,366

7. Platonic and Archimedean solids in lanthanide-containing clusters

The chemistry of lanthanide clusters has become one of the most interesting research frontiers, and lanthanide-containing clusters (including lanthanide-exclusive clusters and heterometallic clusters containing lanthanide elements) are attracting widespread current interest due to their fantastic architecture (especially high-nuclearity lanthanide clusters,^5,6,8 such as wheel Ce₇₀,³⁶⁷ Gd₁₄₀,⁵² cubic Ln₉₆Ni₆₄,⁵³ hexagon-shaped Ni₃₆Gd₁₀₂,⁵⁴ tubular Dy₇₂,³⁶⁸ spherical Ln₁₀₄,⁵⁵ and Gd₁₅₈Co₃₈⁵⁶), and intriguing properties in the fields of single-molecular magnets (SMMs),⁷³ magnetocaloric effects (MCEs),^7,72,83 and catalysis.^369,370

7.1 Lanthanide-exclusive clusters

Among the reported lanthanide-exclusive clusters, a series of tetrahedral Ln₄,^71,371–377 octahedral Ln₆,^378–381 cubic Ln₈,³⁸² and Keggin-type Eu₁₃ (cuboctahedron)³⁸³ clusters has been obtained, while tetrahedral, octahedral and cubic cores have been widely used as building blocks in high-nuclearity Ln clusters.³⁸⁴

Utilizing linear-like multi-coordinated 1-methyl-3,5-bis[3-(pyrid-2-yl)-1,2,4-triazolyl]pyridine (H₂MebptpO) ligands reacting with Ln^III salt under solvothermal reaction, Tong's group obtained a series of supertetrahedral clusters, [Ln₂₀(μ₄-O)₁₁(μ₃-OMe)₁₂(μ₂-OMe)₈(MebptpO)₄(PhCOO)₈(H₂O)₄]²⁺ (Ln₂₀, Ln = Tb, Dy, Ho, and Er, Fig. 41a).³⁷³ In the super-tetrahedral Ln₂₀ cluster, the two perpendicular edges opposite the super-tetrahedron are bridged by two pairs of MebptpO²⁻ ligands (Fig. 41b) and the remaining four edges are protected by eight PhCOO⁻ and eight μ₂-OMe⁻ ligands. Additionally, each face and vertex of the v₃-tetrahedron is capped by three μ₃-OMe⁻ groups and one H₂O molecule, respectively. Eleven μ₄-O²⁻ are all located at the center of the vertex-sharing {Dy₄(μ₄-O)} units and the outer ten μ₄-O²⁻ atoms form a v₂-tetrahedron, encapsulating an inner μ₄-O²⁻ (Fig. 41c). The metal arrangement of the v₃-tetrahedral cluster can be described as a large Dy₁₆ tetrahedral shell (Dy₁₂(truncated tetrahedron)@Dy₄(tetrahedron)) wrapping a small Dy₄ tetrahedral core.

Fig. 41 (a) Structure of [Dy₂₀(μ₄-O)₁₁(μ₃-OMe)₁₂(μ₂-OMe)₈(MebptpO)₄(PhCOO)₈(H₂O)₄]²⁺. (b) Coordination mode of linear-like MebptpO²⁻ of Ln₂₀. (c) {Dy₂₀(μ₄-O)₁₁} super-tetrahedral core with nested arrangement, Dy₄(tetrahedron)@Dy₁₆(tetrahedron).³⁷³

Using different hydroxyl-containing small organic molecules (Hchp (2-chloro-6-hydroxypyridine), mdaH₂ (N-methyl-diethanolamine), and Hdmp (2,2-dimethylol propionic acid)), Zheng and co-workers synthesized a series of highly symmetric spherical gadolinium(Gd^III) clusters with polyhedral characteristics ranging from small to large (Fig. 42), [Gd₉(μ₄-OH)₂(μ₃-OH)₈(mda)₄(mdaH)₂(mdaH₂)₂(NO₃)₇(CH₃OH)₄] (Gd₉), [Gd₁₅(μ₅-Cl)(μ₃-OH)₂₀(dmp)₁₀(H₂O)₁₀](ClO₄)₈Cl₆ (Gd₁₅), [Gd₂₀(μ₅-CO₃)₁₂(chp)₃₀(NO₃)₆(H₂O)₆] (Gd₂₀), [Gd₃₂(μ₄-OH)₆(μ₃-OH)₄₈(mda)₁₂(NO₃)₁₂(H₂O)₂₄]⁶⁺ (Gd₃₂), [Gd₅₀(μ₅-Cl)₁₂(μ₃-OH)₈₀(dmp)₃₀(H₂O)]²⁸⁺ (Gd₅₀), and [Gd₆₀(μ₆-CO₃)₈(μ₃-OH)₉₆(dmp)₂₈(NO₃)₈(H₂O)₈]³²⁺ (Gd₆₀).³⁸⁵ Magnetization analyses revealed the magneto-structural correlation in the Gd^III clusters and this correlation can be used as a “fingerprint” to identify these aggregates. The structural analysis revealed that four high-symmetry spherical Gd^III clusters can be viewed as the self-assembly of polymetallic lanthanide fragments that contain different topological polygons and polyhedrons, such as {Gd₃(μ₃-OH)} triangle, {Gd₄(μ₄-OH)} square, {Gd₄(μ₃-OH)₄} tetrahedron, {Gd₅(μ₅-CO₃)} pentagon, {Gd₅(μ₅-Cl)} pentagon, and {Gd₆(μ₆-CO₃)} hexagon (Fig. 42e). Simple anion templates (such as OH⁻, Cl⁻, and CO₃²⁻) facilitate the formation of these fragments and complete spherical clusters.^386,387

Fig. 42 Core structures and topological representation of the four spherical Gd^III clusters Gd₂₀ (a), Gd₃₂ (b), Gd₅₀ (c), and Gd₆₀ (d). (e) Synthetic strategy of the four spherical Gd^III clusters and two intermediates (Gd₉ and Gd₁₅).³⁸⁵ Color codes: Gd, dark green; Cl, yellow; O, pink; and C, gray.

Gd₂₀ with approximate I_h symmetry, resembles a pentagonal dodecahedron with the Gd^III atoms lying on the vertices and twelve μ₅-CO₃²⁻ anions on the slightly distorted hexagonal faces {Gd₅(μ₅-CO₃)} (Fig. 42a). Similar dodecahedral Ln₂₀ clusters were also reported.^388,389Gd₃₂ polyhedron features six {Gd₄(μ₄-OH)} squares and 32 {Gd₃(μ₃-OH)} triangles and its 24 Gd^III sites construct a truncated octahedron with each of regular hexagonal faces capped by a Gd^III atom (Fig. 42b). Gd₃₂ can be viewed as a Keplerate, Gd₂₄(truncated octahedron)@Gd₈(octahedron), similar to Co₂₄ (Co₂₄@Mo₈/W₈, Fig. 35). Interestingly, Gd₃₂ is the first giant Gd^III cluster core with O_h symmetry. Gd₅₀ displays an unprecedented core–shell Keplerate structure with approximate I_h symmetry, in which an icosidodecahedral core Gd₃₀ (12 {Gd₅(μ₅-Cl)} pentagons and 20 {Gd₃(μ₃-OH)} triangles) is encapsulated by an outer dodecahedral shell Gd₂₀ (Fig. 42c). The 30 Gd^III atoms inside Gd₅₀ cluster construct a topology of Archimedean solid (icosidodecahedron) with each of the regular triangle faces capped by a Gd^III atom, resulting in 20 vertex-sharing {Gd₄(μ₃-OH)₄} tetrahedrons/cubanes with a dodecahedral arrangement.

Gd₆₀ also displays a core–shell structure and its metal core exhibits approximately O_h symmetry (Fig. 42d). The 36 Gd^III atoms inside the Gd₆₀ cluster construct a truncated topology of truncated octahedra, containing twenty-four {Gd₃(μ₃-OH)} triangles, six {Gd₄(μ₃-OH)₈} squares, and eight {Gd₆(μ₆-CO₃)} hexagons. Each of regular triangle faces of the doubly truncated octahedron is capped by a Gd^III atom, resulting in 24 vertex-sharing {Gd₄(μ₃-OH)₄} tetrahedra/cubanes with a truncated octahedral arrangement. Similar 60-metal (Er, Ho, and Y) core–shell clusters were first reported by Zhang and co-workers in 2009.³⁹⁰ Subsequently, other Ln₆₀ clusters with a similar metal framework were also reported by Xu's group.^391,392 In addition, two different research groups revealed two different ways to generate similar Ln₆₀ clusters.^393,394

By controlling the hydrolysis of Ln(ClO₄)₃ (Ln = Nd and Gd) in the presence of CH₃COO⁻ and N-acetyl-D-glucosamine, Kong and co-workers prepared three beautiful highly symmetric cage-like clusters containing 104 lanthanide atoms, i.e., [Nd₁₀₄(μ₄-O)₃₀(μ₃-OH)₁₆₈(CH₃COO)₆₀(H₂O)₁₁₂(ClO₄)₆](ClO₄)₁₈ (Nd₁₀₄-1, Fig. 43) and [Ln₁₀₄(μ₄-O)₃₀(μ₃-OH)₁₆₈(CH₃COO)₅₆ (H₂O)₁₁₂(ClO₄)₆]·(ClO₄)₂₂ (Ln = Nd, Nd₁₀₄-2; Gd, Gd₁₀₄).⁵⁵ These clusters represent the second largest lanthanide-exclusive cluster to date, while the largest is the {Gd₁₄₀} wheel.⁵² These huge Ln oxyhydroxide clusters (Fig. 43a) can be seen as being assembled from square-pyramidal [Nd₅(μ₄-O)(μ₃-OH)₄]⁹⁺ units (Fig. 43b) and [Nd(μ₃-OH)₆]³⁻ linkers (Fig. 43d). Four square-pyramidal SBBs are clustered together by corner sharing to generate a 16-metal tetramer centered on a μ₄-O²⁻ group, [Nd₁₆(μ₄-O)₅(μ₃-OH)₂₀]¹⁸⁺ (Fig. 43c). Six of these tetrameric assemblies are arranged as vertices in an octahedral form, and eight [Nd(μ₃-OH)₆]³⁻ linkers further fill the center of the eight faces of the octahedron, resulting in the formation of the giant hollow structure. It is a typical Keplerate structure with its 104 Ln³⁺ ions organized into four distinct shells (Fig. 43e). From the inside to the outside, the topologies of the 104 metal core contain an inner Platonic solid (octahedron) and outer three Archimedean solids (a truncated cuboctahedron, truncated octahedron, and rhombicuboctahedron), abbreviated as Ln₈@Ln₄₈@Ln₂₄@Ln₂₄. Gd₁₀₄ possesses the second largest magnetic entropy change (−ΔS_m = 46.9 J kg⁻¹ K⁻¹ at 2 K for ΔH = 7 T) among the known Ln-exclusive clusters.

Fig. 43 (a) Molecule structure of cationic cluster Nd₁₀₄-1. (b) Square-pyramidal [Nd₅(μ₄-O)(μ₃-OH)₄]⁹⁺ unit. (c) Tetramer [Nd₁₆(μ₄-O)₅(μ₃-OH)₂₀]¹⁸⁺. (d) Linker [Nd(μ₃-OH)₆]³⁻. (e) Arrangement of the 104 Nd atoms in the cluster core of Nd₁₀₄-1 with nested four-shell structure.⁵⁵

The research group led by Professor George Christou developed a simple bottom-up method, where Ce^III/V salt and small organic acid molecules react in solutions containing pyridine or its derivatives, to prepare ultra-small CeO₂ nanoparticles in Ce-oxo cluster form (from Ce₆,^395,396 Ce₁₀,³⁹⁶ Ce₁₆, Ce₁₉,³⁹⁷ Ce₂₀,³⁹⁵ Ce₂₄, Ce₃₈, and Ce₄₀,³⁹⁸ to the largest Ce₁₀₀ with a size of 2.4 nm⁵¹). These Ce–O clusters possess a regular metal core with the fluorite structure of bulk CeO₂ and reflect critical information at atomic resolution of important surface features, helping CeO₂ nanoparticles to be better used in catalysis and other fields. Among them, the metal skeletons of Ce₆ (octahedron), Ce₁₀ (Ce₆@Ce₄, octahedron@tetrahedron), Ce₁₆ (Ce₄@Ce₁₂, tetrahedron@truncated tetrahedron), and Ce₃₈ (octahedron@cube@truncated icosahedron, such as Pu₃₈, Fig. 24) feature Platonic and Archimedean solids.

7.2 Heterometallic clusters containing lanthanide

Owing to the diverse aesthetic structures, existence of significant d–d, d–f, and f–f magnetic exchange, and intriguing magnetocaloric effect (MCE) and single-molecule magnet (SMM) properties of d–f high-nuclearity heterometallic clusters containing lanthanide (Ln) and other metal ions, especially transition metals (TM), are of continuing interest.⁷² In particular, Tong et al. summarized the recent advances in heterometallic clusters based on cubane {TM_xLn_4−x} subunits, focusing on their structure and magnetic properties.⁷¹ Thus, here we will not discuss simple heterometallic clusters containing lanthanides, only a few examples of typical high-nuclearity clusters. For example, the v₂-tetrahedral cluster M₁₀ with an {Fe₄Ln₆(μ₄-O)₄} (Ln = Dy and Ho) core,³⁹⁹v₂-octahedral cluster (M₁₈:M₁₂@M₆, [(ClO₄)@Cu₁₂Ln₆(OH)₂₄(H₂O)₁₈(Pyb)₁₂]¹⁷⁺ (Ln₆Cu₁₂, Ln = Y, Nb, and Gd, and Pyb = pyridine betaine)) with an α-Keggin-type Cu–O core),^331,400 and truncated tetrahedron ([Ln₁₂(MoO₄)₄(HL)₆(μ₃-OH)₄(OOCCH₃)₁₂] (Ln₁₂Mo₄, Ln = Gd, Eu, and Sm; H₃L = (E)-2-(2,3-dihydroxypropylimino)methyl)-phenol)⁴⁰¹ and (2-MepyH)₅[(μ₄-Cl)Pr₄Sb₁₂(μ₃-O)₁₂(μ₄-O)₆Cl₁₆] (Pr₄Sb₁₂, 2-MepyH⁺ = monoprotonated 2-methylpyridine)).⁴⁰²

With the development and improvement of synthetic strategies, e.g., controlling the hydrolysis of metal ions in the presence of small organic molecules containing carboxyl and/or hydroxyl groups, significant progress in the preparation of high-nuclearity Ln-containing clusters has been achieved, including the above-mentioned Ln clusters (Gd₉, Gd₁₅, Gd₂₀, Gd₃₂, Gd₅₀, Gd₆₀ (Fig. 42), and Gd₁₀₄ (Fig. 43)) and various heterometallic (Ln–Ti, Ln–Fe, Ln–Co, Ln–Ni, Ln–Cu, Ln–Zn, etc.) clusters. Long and Kong's team made a great contribution in this regard, preparing a series of representative clusters, many of which fit the theme of this review, such as Ln₂₄Zn₄ (Fig. 44), Gd₃₀Co₁₂, Ln₁₂Fe₃₃ (Fig. 45), and La₂₀Ni₃₀ (Fig. 45).

Fig. 44 (a) Molecule structure of [Gd₂₄Zn₄O₅(OH)₄₄(CH₃COO)₁₂ (H₂O)₄₈]¹⁴⁺ (Gd₂₄Zn₄). (b) Metal core Zn₄@{Gd₆}₄, wherein four octahedrons encapsulate a tetrahedron. (c) Nested arrangement of 28 metals, Zn₄(tetrahedron)@Gd₁₂(truncated tetrahedron)@Gd₁₂(elongated cuboctahedron).⁴⁰³

Fig. 45 (a) Ball-and-stick representation of [Pr₁₂Fe₃₃(NO₃)₆(L-van)₄(D-van)₅(TEOA)₁₂(μ₃-OH)₁₂(μ₄-OH)₁₂(μ₄-O)₂₈(H₂O)₄]⁴⁺ (Pr₁₂Fe₃₃). (b) ε-Keggin-type Fe₁₃-oxo core encapsulated by Pr₁₂Fe₂₀ shell (Fe₄@Pr₁₂@Fe₄@Fe₁₂). (c) [Fe(μ₄-OH)₃(μ₄-O)₃]⁶⁻ linking the inner Fe₁₃-oxo core and outer Pr^III shell. (d) Hexagon [Pr₆Fe₄(TEOA)₃]²¹⁺.²⁹⁸ (e) Molecule structure of [La₂₀Ni₃₀(IDA)₃₀(CO₃)₆(NO₃)₆(OH)₃₀(H₂O)₁₂]¹²⁺ (La₂₀Ni₃₀). (f) La₂₀ core enveloped by an outer Ni-IDA shell, in which 50 metal atoms present a nested arrangement, La₂₀(dodecahedron)@Ni₃₀(icosidodecahedron).⁴⁰⁴

Through the self-assembly of Ln(ClO₄)₃ (Ln = Gd and Sm) and Zn(OAc)₂ in a mixed solution containing NaOH, two rare high-nuclearity heterometallic Ln–Zn clusters [Ln₂₄Zn₄(μ₆-O)₄(μ₄-O)(μ₃-OH)₄₄(CH₃COO)₁₂(H₂O)₄₈] (ClO₄)₁₄ (Ln₂₄Zn₄, Ln = Gd and Sm, Fig. 44a) were obtained,⁴⁰³ where the Gd-containing cluster exhibited a good magnetocaloric effect (−ΔS_m = 31.4 J kg⁻¹ K⁻¹ at 2 K for ΔH = 7 T). Ln₂₄Zn₄ with ideal T_d symmetry features four octahedra {(μ₆-O)@Ln₆(μ₃-OH)₈} encapsulating a {(μ₄-O)@Zn₄} tetrahedron (Fig. 44b). Alternatively, the metal core of Ln₂₄Zn₄ can be viewed as a three-shell Keplerate, as shown in Fig. 44c. From the inside to the outside, the topologies of the 28-metal core possess an innermost tetrahedron, followed by a truncated tetrahedron and elongated cuboctahedron with four small triangles, four big triangles and six rectangle faces.

By replacing Zn(OAc)₂ with other transition metal salts and adjusting the types of Ln^III salts, other high-symmetry clusters ([Gd₃₀Co^II/III₁₂(OH)₅₆(NO₃)₁₂(CH₃COO)₃₀(H₂O)₃₀]·(NO₃)₂₂ (Gd₃₀Co₁₂)⁴⁰⁵ and [Pr₁₂Fe₃₃(NO₃)₆(L-van)₄(D-van)₅(TEOA)₁₂(μ₃-OH)₁₂(μ₄-OH)₁₂(μ₄-O)₂₈(H₂O)₄]·(ClO₄)₃·(NO₃)·10H₂O (Pr₁₂Fe₃₃, L/D-van = L/D-valine, TEOA = triethanolamine, Fig. 45a)²⁹⁸) were obtained in a similar manner. Gd₃₀Co₁₂ features a double-shell structure, possessing a twisted icosahedron (Co₁₂) encapsulating a slightly distorted icosidodecahedron (Gd₃₀).⁴⁰⁵ Meanwhile, Gd₃₀Co₁₂ exhibits a good magnetocaloric effect, i.e., −ΔS_m = 44.7 J kg⁻¹ K⁻¹ at 2 K for ΔH = 7 T. High-symmetry Pr₁₂Fe₃₃ (Fig. 45a) is a typical example of capturing Keggin Fe₁₃-oxo clusters by using Ln^III ions, and their Ln–organic shells make the functionalization of Fe₁₃-oxo clusters convenient (Fig. 45b).²⁹⁸ The cationic core of Pr₁₂Fe₃₃ can be described from the inside to outside as follows: (1) the ε-Keggin Fe₁₃-oxo capped by four [Fe(μ₄-OH)₃(μ₄-O)₃]⁶⁻ (Fig. 45c) in the hexagonal “window” of the truncated tetrahedron generates a Fe₁₇-oxo core (Fe@Fe₁₂@Fe₄) and (2) the Fe₁₇-oxo unit further capped by 12 Pr^III and 4 Fe^III forms a Fe₂₁Pr₁₂-oxo core (Fe@Fe₁₂@Fe₄@Pr₁₂@Fe₄), which is further capped by four [Fe^III₄(TEOA)₃]³⁺ motifs (Fig. 45d), producing a total lanthanide-Fe oxyhydroxide core (Fe(center)@Fe₁₂(truncated tetrahedron)@Fe₄(tetrahedron)@Pr₁₂(truncated tetrahedron)@Fe₄(tetrahedron)@Fe₁₂(cuboctahedron)) of Pr₁₂Fe₃₃. These lanthanide-containing clusters stabilized and constructed by small ligands possess beautiful symmetry and high magnetocaloric properties.⁷

Iminodiacetate (IDA), possessing five coordinating atoms (one N and four O atoms), has achieved major success in the construction and assembly of heterometallic 3d–4f nanoclusters. A series of high-nuclear Ln–Ni/Co clusters with core–shell or giant hollow structures has been synthesized using IDA and its derivatives, as well as an IDA-containing multi-ligand strategy, such as Ln₁₈Ni_24(23.5) (Ln = Gd and Eu),⁴⁰⁶ Ln₂₀Ni₂₁ (Ln = Gd,⁴⁰⁷ Pr and Nd⁴⁰⁸), Gd₂₂Ni₂₁,⁴⁰⁹ Ln₂₃Ni₂₀ (Ln = Gd and Eu),⁴¹⁰ La₂₀Ni₃₀,^404,408 Nd₃₈Ni_42.5,⁴¹¹ Ln₄₀Ni₄₄ (Ln = Gd and Eu),⁴¹² Pr₄₂Ni₃₈,⁴¹¹ Ln₅₂Ni₅₂ (Ln = Gd and Dy),⁴¹³ Ln₅₂Ni₅₆ (Ln = Eu, Pr, Nd and Gd),^414,415 Gd₅₄Ni₅₄,⁴¹⁶ La₆₀Ni₇₆,⁴¹⁷ La₆₈Ni₉₀, La₇₆Ni₈₈,⁴¹⁸ Gd₇₈Ni₆₄, Eu₇₈Ni₆₂,⁴¹⁹ Ln₉₆Ni₆₄ (Ln = Gd, Dy, Y),⁵³ Gd₄₄Co₂₈, Gd₉₅Co₆₀,⁴²⁰ and Gd₁₅₈Co₃₈.⁵⁶ In these Ln–Ni/Co clusters, simple 3d/3d–4f complexes with IDA ligands assemble into a peripheral shell connected by carboxylate O atoms, which greatly control the degree of lanthanide hydrolysis. This controlled hydrolysis and additional anions (such as NO₃⁻, CO₃²⁻, Cl⁻, Br⁻ and I⁻) afford polyhedral Ln–OH or Ln–OH–Ln/TM cores of the eventual complex heterometallic clusters.

The first high-nuclearity heterometallic Ln–Ni cluster protected by IDA ligands is [La₂₀(OH)₃₀(CO₃)₆(NiIDA)₃₀(NO₃)₆ (H₂O)₁₂](CO₃)₆ (La₂₀Ni₃₀, Fig. 45e).⁴⁰⁴ Keplerate La₂₀Ni₃₀ features a double-sphere structure with an outer spherical shell constructed by 30 Ni-IDA metal-complexes, encapsulating the inner one of 20 La atoms. The topologies of the double-shell structure can be viewed as an inner dodecahedron encapsulated by an icosidodecahedron, La₂₀@Ni₃₀ (Fig. 45f). The nested arrangement of the two distinct sets of metal atoms exhibits symmetrical beauty as both ideally possess I_h symmetry (dodecahedron and icosidodecahedron).

Based on a mixed-ligand (IDA and 2,2-dimethylol propionic acid (DMPA)) approach, Zheng's group isolated three isostructural gigantic Ln–Ni clusters, [Ln₉₆Ni₆₄(μ₃-OH)₁₅₆(IDA)₆₆ (DMPA)₁₂(CH₃COO)₄₈(NO₃)₂₄(H₂O)₆₄]Cl₂₄ (Ln = Gd, Gd₉₆Ni₆₄; Dy, Y).⁵³Gd₉₆Ni₆₄ with a unique porous cube-like structure represents the second highest nuclearity heterometallic Ln-TM clusters. The cation of Gd₉₆Ni₆₄ is a cube-like aggregate of [Gd₉₆Ni₆₀(μ₃-OH)₁₅₆(IDA)₆₀(DMPA)₁₂(CH₃COO)₄₈(NO₃)₂₄(H₂O)₅₂]²⁴⁺ (abbreviated as {Gd₉₆Ni₆₀}) with four add-on {Ni(IDA)(H₂O)₃} fragments. {Gd₉₆Ni₆₀} consists of eight propeller-like {Ni₆Gd₉} fragments on each vertex of its cubic framework and twelve triangular {NiGd₂} units as edges. These hollow structures exhibit highly selective adsorption for CO₂ over N₂ or CH₄ at ambient temperature and Gd₉₆Ni₆₄ exhibits a large magnetocaloric effect (−ΔS_m = 42.8 J kg⁻¹ K⁻¹ at 3 K for ΔH = 7 T).

Through the ‘multi-anions-template’ strategy and controlling the hydrolysis of metal ions (Gd and Co) in the presence of the IDA-derivative N-methyliminodiacetic acid (H₂MIDA), the largest 3d–4f heterometallic cluster [Gd₁₅₈Co₃₈Cl₁₂(μ₃-OH)₂₃₆(CO₃)₉₀(MIDA)₄₂(Sar)₆(CH₃COO)₁₈(H₂O)₈₄]·Cl₂₄·144H₂O (Gd₁₅₈Co₃₈Cl₁₂, HSar = sarcosine, Fig. 46a) to date was obtained, which exhibits a large magnetocaloric effect (−ΔS_m = 46.95 J kg⁻¹ K⁻¹ at 2 K for ΔH = 7 T).⁵⁶ Different from the reported giant hollow Ln-containing clusters (wheel Ce₇₀,³⁶⁷ Gd₁₄₀,⁵² cubic Ln₉₆Ni₆₄,⁵³ cage-like Gd₉₅Co₆₀,⁴²⁰ hexagon-shaped Ni₃₆Gd₁₀₂,⁵⁴ tubular Dy₇₂,³⁶⁸ and spherical Ln₁₀₄⁵⁵), the giant Gd₁₅₈(μ₃-OH)₂₃₆(CO₃)₉₀ templated by twelve halide ions is a solid assembly, and 38 Co complexes (30 Co(μ₃-OH)₃(MIDA) and 8 Co(μ₃-OH)₃(H₂O)₃) cap its surface. As shown in Fig. 46b, the cationic Gd₁₅₈Co₃₈Cl₁₂(μ₃-OH)₂₃₆(CO₃)₉₀ core protected by multiple organic small molecules and water molecules can be described (from the inside out, Fig. 46b) as follows: (1) the 20 Gd^III ions in the center form a dodecahedron framework by 12 CO₃²⁻ and 36 μ₃-OH⁻ groups (like Gd₂₀ in Fig. 42a), whose 12 pentagonal “windows” are capped by 12 templates (Cl⁻) in an icosahedron fashion. (2) Gd₂₀@Cl₁₂ with I_h symmetry further is encapsulated by a cuboctahedron-like heterometallic framework (Gd₆Co₄)₈, built by eight propeller-like Gd₆Co₄ SBBs with a cubic arrangement. (3) Twelve Gd^III are packed outside the junction between the Gd₆Co₄ SBBs and present an icosahedral arrangement. (4) Eight saddle-like Gd₁₂ SBBs are capped on the hexagonal “windows” of the cuboctahedron-like (Gd₆Co₄)₈ framework, forming the cubic structure of the main body. (5) In addition, there are 6 Gd^III (near-octahedral configuration) and 6 Co^II/III (hexagonal geometry) at the periphery of the cubic Gd₁₅₂Co₃₂Cl₁₂ cluster.

Fig. 46 (a) Ball-and-stick representation of cationic cluster of Gd₁₅₈Co₃₈Cl₁₂. (b) Metal (152 Gd and 32 Co) framework templated by 12 Cl⁻ in Gd₁₅₈Co₃₈Cl₁₂.⁵⁶

8. Platonic and Archimedean solids in coin metals (Cu, Ag, and Au) clusters

Atomically precise coinage metal (Cu, Ag, and Au) cluster crystalline materials are new categories and have been comprehensively studied owing to their aesthetically beautiful molecular structures and fascinating properties such as quantized electronic absorption, fluorescence, and catalysis.^{1,11,12,15,21,75–77,85,89–92,421} With the diversification of synthetic methods and ligands, especially the introduction of numerous reducing agents, an increasing number of coinage metal clusters have been obtained. Due to the unique metal–metal bonds, polyhedral cores with Platonic and Archimedean solids can commonly appear in these clusters, such as regular tetrahedron, cube, octahedron, and icosahedron.

8.1 Cu clusters

8.1.1 Cu^II-containing clusters. Because of its single electron, the bi-valent Cu^II ion can be employed to construct interesting paramagnetic metal clusters, which has triggered extensive research.^6,21,422 Monakhov and Kondinski summarized the copper(II)-oxo/hydro-oxo clusters with polyoxometalate-like architectures.⁴²² Many of the examples of Cu^II-containing clusters have some polyhedral features, such as v₂-tetrahedron M₁₀ (M₆(cube)@M₄(tetrahedron), [Cu^II₁₀(μ₄-O)₄(MeCN)₄{(OPh₂Si)₂O}₆];⁴²³ [Cu^II₆Cd^II₄{(CH₂O)₃C(CH₂OH)}₄(piv)₈(H₂O)₆] (Cu₆^IICd₄^II, C(CH₂OH)₄ = pentaerythritol, Hpiv = pivalic acid)⁴²⁴), M₁₄(M₈@M₆) (such as Tl₅[Cl@Cu^I₈Cu^II₆(D-pen)₁₂] (Cu₈^ICu₆^II, D-pen = D-penicillamine)^425–428) and the above-mentioned complex clusters (Cu₁₆Nb₂₈ (Fig. 22b), Cu₂₄Nb₅₇ (Fig. 22c), Cu₁₇Mn₂₈ (Fig. 31a)).

In 2016, Xu's group reported the preparation of three cubic 20-metal clusters containing Cu^II, [(CO₃)@Cu^II₂₀(μ₃-OH)₂₄(CH₃COO)₈(n-propylamine)₁₀](CH₃COO)₆, [(CO₃)@Cu^II₂₀(μ₃-OH)₂₄(CH₃COO)₆(iso-propylamine)₈](CH₃COO)₈, and [(CO₃)@Cu^II₁₂Zn^II₈(μ₃-OH)₂₄(CH₃COO)₁₂](CH₃COO)₂.³³² Although there are some differences in their peripheral ligands, the core of these clusters is the same, and a double-shell metal skeleton with carbonate as the anion template, {(CO₃)@Cu₁₂M₈(μ₃-OH)₂₄} (M = Cu and Zn), is similar to the above-mentioned Co₂₀-1 (Fig. 34a and b). The {Cu₁₂M₈} core can be further viewed as a cuboctahedron shell {Cu₁₂} with each triangular face capped by Cu^II or Zn^II ions from an outer cube. It should be noted that two other analogous 20-metal clusters, {Cu₂₀} and {Cu₁₂Mg₈}, were revealed by Winpenny et al.³³¹

8.1.2 Cu^I/0 clusters. Due to their low coordination number and the possible presence of metal–metal bonds, the structures of mono-valent copper(I) and copper(I/0) clusters and di-valent copper(II) clusters are significantly different.⁷⁴ The classical polyhedral configurations can be found here (Fig. 47), such as tetrahedron, octahedron,⁴²⁹ cube,^430–444 cuboctahedron,^445–452 and truncated octahedron (Fig. 47f).⁴⁵³ In addition, here the classic cases are introduced, such as simple nested clusters, M₈ (M₄(tetrahedron)@M₄(tetrahedron)),^454–457 M₁₄ (M₈(cube)@M₆(octahedron) or M₆(octahedron)@M₈(cube), Fig. 47a and b),^458–463 and M₁₆ (M₁₂(cuboctahedron)@M₄(tetrahedron), Fig. 47c and d).^464,465

Fig. 47 Ball-and-stick representation of (a) [Au@Cu₁₄(SPh^tBu)₁₂(PPh(C₂H₄CN)₂)₆]⁺ with an Au@Cu₁₄ (Cu₈@Cu₆: cube@octahedron) framework,⁴⁶⁶ (b) [Cl@Cu₆Ag₈(C≡CFc)₁₂]⁺ with a Cl@Cu₆(octahedron)@Ag₈(cube) framework,⁴⁶² (c) [I@{TpMoS₄Cu₃}₄]⁻,⁴⁶⁴ (d) [Ag@Cu₁₂S₈@Cu₄(dpph)₆]⁺,⁴⁶⁵ (e) [Cu₁₃(S₂CNⁿBu₂)₆(C≡CC(O)OMe)₄]⁺,⁴⁴⁶ and (f) Cu₂₄(O₃Si(2,6-ⁱPr₂C₆H₃)N(SiMe₃))₈.⁴⁵³

In 2021, Zhu's group reported the preparation of an ultrabright Au–Cu cluster with two free valence electrons (2e), [Au@Cu₁₄(SPh^tBu)₁₂(PPh(C₂H₄CN)₂)₆]⁺ (Au@Cu₁₄, Fig. 47a), with the record phosphorescence quantum yield (71.3%) in non-degassed solution at room temperature.⁴⁶⁶Au@Cu₁₄ has a single-Au-atom center encapsulated by a rigid Cu(I) complex cage (Cu₈@Cu₆: cube@octahedron). Numerous clusters of core–shell structures (M(center)@M₈(cube)@M₆(octahedron)) protected by other ligands have also been reported.^467,468

Anion-templated 14-metal clusters with a rhombic dodecahedral metal framework M^I₆Ag^I₈ (M = Cu, Ag, and Au) fall in the theme of our review. M^I₆Ag^I₈ (M = Cu, Ag, and Au) clusters protected by different ligands have been continuously reported, such as [Cl@M^I₆Ag^I₈(C≡CFc)₁₂]⁺ (≡CFc⁻ = ferrocenylacetylide, Fig. 47b).⁴⁶² The M^I₆Ag^I₈ cluster forms a rhombic dodecahedron made up of 12 M^I₂Ag^I₂ quadrangles. Alternatively, six M^I atoms occupy the corners of a regular octahedron, whereas eight Ag^I atoms are located at the corners of a regular cube. Interestingly, each of the triangular faces in the M^I₆ octahedron is capped with an Ag^I atom and each of the square faces in the Ag^I₈ cube is capped with an M^I atom. In 2020, Zang's group showed that o-carboranealkynyl-protected metal clusters, Cu₆Ag₈ and Ag₁₄ (octahedron@cube), can serve as catalysts, enabling spontaneous ignition to achieve one-off combustion reactions.⁴⁶³

Lang's group reported the preparation of a unique tetracubane cluster, [PyH][(μ₁₂-I){TpMo(μ₃-S)₄Cu₃}₄] ({MoCu₃}₄, Tp = hydridotris(pyrazol-1-yl)borate, Py = pyridine, Fig. 47c).⁴⁶⁴ The [I@{TpMo(μ₃-S)₄Cu₃}₄]⁻ anion can be viewed as a μ₁₂-I⁻ (as anionic template)-centred tetrahedral cage, in which four corners are occupied by four incomplete cubane-like [TpMo(μ₃-S)₄Cu₃}₄] fragments. The twelve Cu^I atoms form a slightly distorted cuboctahedron cage. Sun isolated two cluster-in-cage structures with an Ag@Cu₁₂ cuboctahedron (Archimedean solid) in a Cu₄(dpph)₆ tetrahedral (Platonic solid) cage, [Ag@Cu^I₁₂S₈@Cu₄(dpph)₆]X (X = OH or PF₆, dpph = bis(diphenylphosphino)hexane, Fig. 47d).⁴⁶⁵ Xin and co-workers synthesized a bi-metallic Cu^I–Mo^VI–S cluster, [ⁿBu₄N]₄[Cu₁₂Mo₈S₃₂] (Cu₁₂Mo₈).⁴⁶⁹ The anionic icosanuclear cluster [Cu₁₂Mo₈S₃₂]⁴⁻ of Cu₁₂Mo₈ exhibits an octameric supra-cubane-like architecture. In the supra-cubane-like/cubic cage, eight Mo^VI atoms are located at the corners, whereas the midpoints of the twelve edges are occupied by Cu^I atoms, exhibiting a cuboctahedral arrangement. The anion cluster [Cu₁₂Mo₈S₃₂]⁴⁻ stabilized by different counter cations was also prepared.^470–472

In addition to the above-mentioned monovalent copper cluster, Liu and co-workers reported the first structurally characterized two-electron superatom Cu^0/I cluster, [Cu₁₃(S₂CNⁿBu₂)₆(C≡CR)₄](PF₆) (Cu₁₃, R = C(O)OMe, or C₆H₄F; Fig. 47e), with a centered cuboctahedral arrangement Cu₁₃ (as an ideal model of the bulk copper fcc structure).⁴⁴⁶ Besides, Liu and co-workers also reported the synthesis of the first rhombicuboctahedral Cu^I–H clusters, [H₁₅Cu₂₈{S₂CNR}₁₂]PF₆ (H₁₅Cu₂₈, NR = NⁿPr₂ or aza-15-crown-5, Fig. 48a).⁴⁷³ The H₁₅Cu₂₈ cluster is air- and moisture-stable both in the solid state and in solution. The cation cluster of H₁₅Cu₂₈ with D_2d symmetry contains a central irregular Cu₄ tetrahedron with an interstitial hydride and an outer slightly distorted Cu₂₄ rhombicuboctahedron (Fig. 48b), in which the Cu⋯Cu separation is 2.603(1)–2.824(1) Å. The outer shell Cu₂₄ is further encapsulated in an array of twelve {S₂CNPr₂}⁻ (dtc = di-n-propyldithiocarbamate) ligands. Each dtc ligand in a μ₄-η²,η² binding mode bridges four Cu^I atoms (Fig. 48c) and the 24 S atoms of the twelve ligands are arranged in a truncated octahedral cage (Fig. 48d). The hydrides of H₁₅Cu₂₈ are distributed in three locations, playing an important role in the following connections: (i) one central hydride; (ii) six hydrides with an octahedral array, located inside square faces of Cu₂₄ rhombicuboctahedron and bridging between the inner and outer Cu^I polyhedrons (Fig. 48f); and (iii) eight hydrides with a cubic manner, located outside the triangular faces of the Cu₂₄ rhombicuboctahedron (Fig. 48e). The aesthetically pleasing nested-structure of H₁₅Cu₂₈ can be expressed as μ₄-H@Cu₄(tetrahedron)@H₆(octahedron)@Cu₂₄(rhombicuboctahedron)@H₈(cube)@S₂₄(truncated octahedron), and the geometric entity is reminiscent of a Chinese puzzle ball (Fig. 48g). Intriguingly, limited H₂ evolution was observed upon the exposure of H₁₅Cu₂₈ solutions to mild conditions at room temperature.

Fig. 48 (a) Ball-and-stick representation of [H₁₅Cu₂₈{S₂CNPr₂}₁₂]⁺ cation. (b) Rhombicuboctahedral arrangement of 24 Cu^I atoms. (c) Each of the 12 square faces of the Cu₂₄ shell is capped by an {S₂CNPr₂}⁻ ligand. (d) Cu₂₄ shell is enclosed by a truncated octahedron of 24 S atoms from twelve {S₂CNPr₂}⁻ ligands. (e) Eight hydrides μ₃-H⁻ on triangular faces of the Cu₂₄ shell. (f) Six hydrides with an octahedron array inside square faces of the Cu₂₄ shell. (g) Core assembly: μ₄-H@Cu₄(tetrahedron)@H₆(octahedron)@Cu₂₄(rhombicuboctahedron)@H₈(cube)@S₂₄(truncated octahedron).⁴⁷³ Colour codes: C, grey; Cu, cyan; H, red; N, pink; and S, yellow.

In 2018, Fischer and co-workers prepared a paramagnetic and all-hydrocarbon ligand-stabilized heterometallic cluster, [Cu₄₃Al₁₂](Cp*)₁₂ (Cu₄₃Al₁₂, Fig. 49a, Cp* = η⁵-C₅Me₅), obtained from the reaction of [AlCp*]₄ and [CuMes]₅ (Mes = mesityl).⁴⁷⁴Cu₄₃Al₁₂ possesses a magic atom-number (55) and features a Mackay-type nested icosahedral structure, similar to Pd₅₅ (Fig. 40a), and a unique open-shell 67 electron super-atom configuration. The Mackay-type structure of Cu₄₃Al₁₂ is composed of a μ₁₂-Cu-centered Cu₁₃ icosahedral core and a heterometallic Cu₃₀Al₁₂v₂-icosahedral shell. The 44-metallic shell can also be divided into two parts, i.e., icosidodecahedron Cu₃₀ (Fig. 49b) and (AlCp*)₁₂ icosahedral shell (Fig. 49c). The complete structure of Cu₄₃Al₁₂ can be described by a concentric core–shell, Cu@Cu₁₂@Cu₃₀@(AlCp*)₁₂. A similar M₄₃ metal kernel containing Cu was also observed in the bimetallic Au₁₉Cu₃₀ nanocluster,⁴⁷⁵ featuring a core–shell structure of Au@Au₁₂@Cu₃₀@Au₆ and consisting of an Au-centered icosahedral core Au₁₃ encapsulated by an icosidodecahedral shell Cu₃₀ and an outmost hexagonal Au₆. Zhu and Wang et al. also reported a bimetallic Ag₆₁Cu₃₀ nanocluster, which is composed of an Ag@Ag₁₂@Cu₃₀ kernel, capped by a peripheral Ag₄₈(SAdm)₃₈S₃ shell.⁴⁷⁶

Fig. 49 (a) Ball-and-stick representation of [Cu₄₃Al₁₂](Cp*)₁₂. (b) Core–shell arrangement of 43 Cu atoms, μ₁₂-Cu@Cu₁₂(icosahedron)@Cu₃₀(icosidodecahedron). (c) (AlCp*)₁₂ icosahedral shell.⁴⁷⁴ (d) Ball-and-stick representation of [(IDipp)₆Cu₅₅].⁵⁷

Accidentally, two unprecedented low-valent copper clusters [(IDipp)₆Cu₅₅] (Cu₅₅, Fig. 49d) and [(IDipp)₁₂Cu₁₇₉] were obtained as minor products from the reductive decomposition of the unstable N-heterocyclic carbine (NHC) copper-boryl complex [(IDipp)Cu-Bneop] (IDipp = 1,3-bis(2,6-diisopropylphenyl)imidazol-2-ylidene, neop = (OCH₂)₂CMe₂)) by Kleeberg's team.⁵⁷ The copper atoms in the remarkable metalloid Cu₅₅ cluster adopt approximately icosahedral symmetry and a Cu@Cu₁₂@Cu₃₀@Cu₁₂ arrangement similar to Cu₄₃Al₁₂ (Fig. 49a) and Pd₅₅ (Fig. 40a).

8.2 Ag clusters

Similar to Cu^I-containing clusters, a large number of multi-/high-nuclearity clusters with metal cores featuring Platonic and Archimedean solids (such as tetrahedron,⁴⁷⁷ cube,^478–480 octahedron,⁴⁸¹ tetra-capped tetrahedron (M₄@M₄),^455,482,483 cuboctahedron,^478,484,485 and v₂-tetrahedron (M₆@M₄)⁴⁸⁶) has been obtained in atomically precise silver clusters but will not be discussed in detail here. A distinct feature has been demonstrated in Cu^I and Cu^I/0 clusters, which is more vividly manifested in the silver cluster, where the anions play an important role ((i) template and directing agent to control the shape and size of the structures; (ii) stabilizing and balancing the local positive charges from metal ions; and (iii) improving the cluster functionality) in the formation of the structure (Fig. 47b and c) in Cu^I clusters, while that (the metal occupies the position of the anion, Fig. 47a and e) in the Cu^I/0 clusters requires additional free valence electrons.⁷⁶ For example, hexa-capped body-centered cubic (bcc) silver clusters (X/Ag@Ag₈(cube)@Ag₆(octahedron)): [X@Ag^I₁₄(C≡C^tBu)₁₂]Y (X = F⁻, Cl⁻, Br⁻; Y = OH⁻, BF₄⁻)^487,488 and [Ag₁₅(Ntriphos)₄(Cl₄)](NO₃)₃ (Ntriphos = tris((diphenylphosphino)methyl)amine) with 8 free electrons.⁴⁸⁹ Interestingly, a homoleptic alkynyl-protected 2e Ag₁₅ cluster [Ag@Ag₁₄(C≡C^tBu)₁₂](CH₃O) with almost the same composition and structure as [X@Ag^I₁₄(C≡C^tBu)₁₂]⁺ was also successfully prepared, which exhibited high activity for the CO₂ reduction reaction (FE(CO) = 95% at −0.6 V and TOF_max = 6.37 s⁻¹).⁴⁹⁰ A series of similar alloy M₁₅⁺ (Ag₉Cu₆, Au₇Ag₈, and Au₂Ag₈Cu₅) is also available.^491–493

8.2.1 Ag(I) clusters. Firstly, some complex structures with T symmetry in silver(I) clusters are introduced. In 2015, Sun and co-workers reported the preparation of a T_d symmetric spherical molecule, {(HNEt₃)[Ag₃₇S₄(SC₆H₄^tBu)₂₄(CF₃COO)₆(H₂O)₁₂]} (Ag₃₇S₄, Fig. 50a).⁴⁹⁴ The skeleton of the Ag₃₇S₄ cluster is comprised of an Ag₃₆ shell, featuring a slightly distorted truncated tetrahedral shape and enveloping an inner AgS₄ tetrahedron. By careful disassembly, the Ag₃₆ shell can be seen as a combination of three slightly distorted Archimedean polyhedra (two truncated tetrahedra originating from 12 vertices or 6 middle Ag₂ fragments of edges, and one cuboctahedron made up of 4 Ag₃ faces). Since then, this research group successively reported several cases of nearly T-symmetric clusters. In 2019, they again successfully prepared chiral Ag₆Z₄@Ag₃₆ ([Ag₆@Ag₃₆S₄(SC₆H₄^tBu)₂₄(CF₃COO)₆(H₂O)₁₂]·4CF₃COO, Z = S (Ag₄₂S₄, Fig. 50b) or Se) nanoclusters featuring the same truncated tetrahedral shell (Ag₃₆) and enlarged tetrahedral core Ag₆Z₄ with an Ag₆ octahedron.⁴⁹⁵ Due to the tetrahedral Ag–S core expansion (AgS₄ in Ag₃₇S₄ → Ag₆S₄ in Ag₄₂S₄), the outer Ag₃₆ shell is also slightly enlarged and its degree of distortion is enhanced. The expansion of the inner core (AgS₄ → Ag₆S₄) and change in the outer shell (Ag₃₆) promote the reduction of molecular symmetry and realize homo-chiral crystallization. In 2021, this group again reported a 58-nuclei Ag supertetrahedral cage, capturing a supramolecular aggregate ([CH₃OH@(SO₄)₄ (H₂O)₆]⁸⁻), [CH₃OH@(SO₄)₄(H₂O)₆@Ag₅₈(ⁱPrC₆H₄S)₄₀(CF₃COO)₆(H₂O)₁₀]·4CF₃COO.⁴⁹⁶

Fig. 50 Crystallographic structures of (a) [Ag₃₇S₄(SC₆H₄^tBu)₂₄(CF₃COO)₆(H₂O)₁₂]}^3− 494 and (b) [Ag₆@Ag₃₆S₄(SC₆H₄^tBu)₂₄(CF₃COO)₆(H₂O)₁₂]⁴⁺.⁴⁹⁵

In 2020, Zhang prepared a series of silver clusters with various combinations of Platonic and/or Archimedean solid arrangements, induced by a flexible trifurcate metallo-ligand Ti(Sal/5-FSal)₃ (SalH₂ = salicylic acid, 5-FSalH₂ = 5-fluorosalicylic acid): Ag₈{Ti(Sal)₃)}₄ (Ag₈Ti₄, Ag₄@Ag₄@Ti₄, Fig. 51a), H₂Ag₁₂(SⁱPr)₆{Ti(Sal)₃)}₄ (Ag₁₂Ti₄, Ag₁₂@Ti₄, Fig. 51b), SO₄@Ag₂₂(SⁱPr)₁₂{Ti(Sal)₃)}₄ (Ag₂₂Ti₄, Ag₄@Ag₆@Ag₁₂@Ti₄, Fig. 51c), (SO₄)₂@Ag₃₆(SⁱPr)₂₄{Ti(5-FSal)₃)}₄ (Ag₃₆Ti₄, Ag₁₂@Ag₁₂@Ag₁₂@Ti₄, Fig. 51d), and Ag₄₂S₄(SⁱPr)₁₈(SO₄)₄{Ti(Sal)₃)}₄ (Ag₄₂Ti₄, Ag₆@Ag₂₄@Ag₁₂@Ti₄, Fig. 51e).⁴⁸³ The Ag/Ag–S clusters (number of metal atoms ranging from 8 to 42) are each capped by four metallo-ligands Ti(Sal/5-FSal)₃, thereby forming supertetrahedra {Ti₄} with different edge lengths (1–2 nm). These supertetrahedra {Ti₄} further induce a tetrahedral geometry (approximate T symmetry) in the protected Ag/Ag–S aggregates. Owing to the chiral feature of the propeller-shaped metallo-ligand Ti(Sal/5-FSal)₃, the Ag/Ag–S cores in these silver clusters exhibit distinctly distorted polyhedral structures. In Ag₈Ti₄, there is one tetrahedral Ag₄ inside and 4 Ag atoms in the tetrahedral pattern outside (Fig. 51a). The Ag₁₂ core in Ag₁₂Ti₄ forms a distorted cuboctahedron (Fig. 51b), often as nodes in Ag cluster-based assembled materials.^497–499 In the limited space of {Ti(Sal)₃)}₄ in Ag₂₂Ti₄, a 22-metal truncated tetrahedral cage templated by SO₄²⁻ exists. In this Archimedean solid (Ag₁₂, truncated tetrahedron), two Platonic solids exist (Fig. 51c), an Ag₆ octahedron (metal atoms located on edges of truncated tetrahedron Ag₂₂) and an Ag₄ tetrahedron (metal atoms from the center of four faces of Ag₂₂). The larger SO₄²⁻-templated truncated tetrahedral metal cage in Ag₃₆Ti₄ is comprised of three Archimedean solids, two Ag₁₂ truncated tetrahedra and one Ag₁₂ cuboctahedron (Fig. 51d), similar to the outer Ag₃₆ shell in Ag₃₇S₄ and Ag₄₂S₄ (Fig. 50). The largest silver cluster Ag₄₂Ti₄ in this system (Fig. 51e) is similar to Ag₄₂S₄ (Fig. 50b) in terms of metal number and core–shell (Ag₆S₄@Ag₃₆) structure. Comparing the two 42-metal Ag–S structures, they basically have the same tetrahedral core capping an Ag₆ octahedron, but different truncated tetrahedral Ag₃₆ shells (Fig. 50b and Fig. 51e), which may have originated from the template effect of SO₄²⁻ in Ag₄₂Ti₄. From the inside to the outside, the topologies of the 46-metal framework contain an inner Platonic solid (octahedron) and outer three Platonic and/or Archimedean solids (a truncated octahedron, truncated tetrahedron, and tetrahedron), abbreviated as Ag₆@Ag₂₄@Ag₁₂@Ti₄. Subsequently, Zang et al. achieved restricted growth from single Ag(I) atoms (Ag@Ti₅) to all-carboxylate-protected superatomic Ag cluster (Ag₆@Ti₆) by controlling the size of titanium–organic cages.⁴⁸²

Fig. 51 Crystallographic structures and metal arrangements of (a) Ag₈{Ti(Sal)₃)}₄, (b) Ag₁₂(SⁱPr)₆{Ti(Sal)₃)}₄²⁻, (c) SO₄@Ag₂₂(SⁱPr)₁₂{Ti(Sal)₃)}₄, (d) Ag₃₆(SⁱPr)₂₄{Ti(5-FSal)₃)}₄, and (e) Ag₄₂S₄(SⁱPr)₁₈(SO₄)₄{Ti(Sal)₃)}₄.⁴⁸³

Next is the Ag^I cluster with higher symmetry. In 2016, Zang and co-workers reported the preparation of a novel spherical Ag^I–S cluster, [Ag₃₀(^tBuS)₂₀V^V₁₀V^IV₂O₃₄] (V₁₂@Ag₃₀, Fig. 52a), featuring a dodecahedrane-like shell {Ag₃₀(^tBuS)₂₀}¹⁰⁺ with approximately O_h symmetry (Fig. 52b), which tightly wraps an unusual C_2h polyoxovanadate (POV) anion core {V^V₁₀V^IV₂O₃₄}¹⁰⁻.⁵⁰⁰ In the Ag–S shell of V₁₂@Ag₃₀, each ^tBuS⁻ ligand adopts the μ₃-η¹,η¹,η¹ coordination mode to cap an Ag triangle. Each Ag^I atom connects two neighboring S atoms to construct an almost linear S–Ag–S motif. Five vertex-sharing motifs form a quasi-regular pentagon, and then twelve vertex-sharing pentagons come together to constitute a Platonic polyhedron (regular dodecahedron, Fig. 52b). Twenty silver triangles capped by ^tBuS⁻ ligands share Ag vertices to form a slightly distorted icosidodecahedron. It is worth noting that V₁₂@Ag₃₀ is the first reported spherical cluster featuring an Ag^I–organic dodecahedral nano-cage and an icosidodecahedra silver skeleton. In addition to the semiregular Archimedean icosidodecahedral metal arrangement found in single-shell silver clusters, a similar Ag₃₀ metal framework was also observed in multi-shell silver clusters. In 2012, Mak's group constructed two high-nuclearity silver ethynide clusters, [Cl₆Ag₈@Ag₃₀(^tBuC≡C)₂₀(ClO₄)₁₂] (Ag₃₈Cl₆) and [Cl₆Ag₈@Ag₃₀(chxC≡C)₂₀(ClO₄)₁₀](ClO₄)₂ (chx = cyclohexyl), bearing the same novel central core Cl₆Ag₈ (octahedron@cube, like Cu₆Ag₈ in Fig. 47b) and a similar shell Ag₃₀ (icosidodecahedra).⁵⁰¹ In the synthesis of Ag₃₈Cl₆, a rhombohedral silver ethynide cluster [Cl@Ag₁₄(^tBuC≡C)₁₂]OH⁴⁸⁷ was used as the Ag precursor to react with other silver salts (AgClO₄) in CH₂Cl₂, resulting in a structural transformation and increase in nuclearity from 14 to 38. In 2017, Shen and Mizuta reported the preparation of an alkynyl-protected Pt-doped 8-electron Ag superatomic nanocluster, [PtAg₄₂(C≡CC₆H₄CH₃)₂₈](SbF₆)₆.⁵⁰² In 2020, Zhang et al. synthesized and characterized a trifluoromethanesulfonate and tert-butylacetylene co-protected Ag₄₃ nanocluster, Ag₄₃(^tBuC≡C)₂₄(CF₃SO₃)₈ (Ag₄₃, Fig. 52c).⁵⁰³ The 43-metal kernel can be viewed as a concentric core–shell structure (Fig. 52d), which is abbreviated as Pt/Ag@Ag₁₂(icosahedron)@Ag₃₀(slightly distorted icosidodecahedron).

Fig. 52 (a) Crystallographic structure of [Ag₃₀(^tBuS)₂₀V₁₂O₃₄] cluster. (b) Dodecahedrane-like shell {Ag₃₀(^tBuS)₂₀}¹⁰⁺.⁵⁰⁰ (c and d) Crystallographic structure and concentric metal core–shell framework of Ag₄₃(^tBuC≡C)₂₄(CF₃SO₃)₈, respectively.⁵⁰³

In 2015, Xie and co-workers reported the preparation of an MoO₄²⁻-templating high-nuclearity silver(I) cluster co-protected by phosphonate and thiolate ligands, [CH₃OH₂]₆{MoO₄@Ag₁₂@(ⁿBuPO₃)₈S₆@Ag₃₆(^tBuS)₂₄} (Ag₄₈, Fig. 53), featuring a phosphonate-functionalized cubic silver(I) cluster core (Ag₁₂@(ⁿBuPO₃)₈S₆, Fig. 53c) encapsulated by a silver(I)-thiolate shell with a truncated octahedral geometry (Ag₃₆(^tBuS)₂₄, Fig. 53d).⁵⁰⁴ The template anion MoO₄²⁻ in μ₁₂-η³,η³,η³,η³ ligation mode is bound to its 12 neighboring silver atoms, resulting in the formation of an MoO₄²⁻-centered Ag₁₂ core with slightly distorted cuboctahedron (Fig. 53b). In the Ag₁₂ cuboctahedron, the six square faces are capped by six S²⁻ ions in an octahedral fashion, and the eight triangular faces are stabilized by eight cubic arranged ⁿBuPO₃²⁻ ligands (Fig. 53c). The MoO₄²⁻-centered Ag₁₂ core is further attached to the outer truncated v₃-octahedron shaped silver(I)-thiolate shell by the S²⁻ ions and the phosphonate ligands (Fig. 53d). Meanwhile, the S²⁻ ions and the phosphonate ligands are located inside the center of the six square and eight hexagon faces of the truncated octahedron, respectively. Ag₄₈ can be described as an arrangement of multiple concentric shells with polyhedral features, MoO₄(tetrahedron)@Ag₁₂(cuboctahedron)@(ⁿBuPO₃)₈(cube)@S₆(octahedron)@Ag₃₆(^tBuS)₂₄(truncated octahedron). In 2017, this group again prepared an S²⁻-templated high-nuclearity silver(I) cluster co-protected by phosphonate, ethynide and thiolate ligands, {S@Ag₁₂S₆@Ag₃₆(^tBuC≡C)₁₂(^tBuS)₁₂(ⁿBuPO₃)₂(ⁿBuPO₃H)₆}, possessing a nested polyhedron framework similar to Ag₄₈.⁵⁰⁵

Fig. 53 (a) Crystallographic structure of anionic cluster {MoO₄@Ag₁₂@(ⁿBuPO₃)₈S₆@Ag₃₆(^tBuS)₂₄}⁶⁻ (Ag₄₈). (b) Cuboctahedral Ag₁₂ core encapsulating an MoO₄²⁻ anion as a template. (c) Phosphonate-functionalized cubic silver(I) cluster core, Ag₁₂@(ⁿBuPO₃)₈S₆. (d) Truncated v₃-octahedron-shaped silver(I)–thiolate shell, Ag₃₆(^tBuS)₂₄ (without C atoms), where its eight hexagonal centers are occupied by ⁿBuPO₃²⁻ ligands (without C and O atoms).⁵⁰⁴

In 2020, Sun and co-workers reported the preparation of two giant globular Ag^I nanoclusters with a sodalite-type silver orthophosphate skeleton {(Ag₃PO₄)₈}, [(PO₄)₈@Ag₉₀S₆(^tBuS)₂₄(PhPO₃)₁₂ (PhPO₃H)₆] (Ag₉₀, Fig. 54a–c) in pseudo-T_h symmetry⁵⁰⁶ and [(PO₄)₈@Ag₁₀₂S₆(CyS)₃₀(H₂PO₄)₆(HPO₄)₆(CF₃COO)₁₈] (Ag₁₀₂, Cy = cyclohexyl, Fig. 54d–f) in D_3h symmetry.⁵⁰⁷ The ball-shaped Ag₉₀ (Fig. 54a) contains three concentric polyhedra (octahedron(Ag₆, O_h)@truncated octahedron(Ag₂₄, O_h)@rhombicosidodecahedron(Ag₆₀, I_h)) with apparently incompatible symmetry (O_h and I_h, Fig. 54b). The arrangement of 90 metals is similar to that in the model devised by the mathematician John Conway called Kepler's Kosmos. The polyhedra of the inner (Ag₆, octahedron) and middle (Ag₂₄, truncated octahedron) shells have been observed in many structures. A rhombicosidodecahedron is rarely seen, where this polyhedral feature was previously observed in the third shell in Pd₁₄₅ and (PtPd)₁₆₅ (Fig. 40), mainly because its formation requires more atoms (60) and can only appear in huge complex nanoclusters or nanocages. This multi-shell structure (Ag₆@Ag₂₄@Ag₆₀, Fig. 54b) with polyhedral features but distinctly different symmetries (symmetry raises from the inside out) can be attributed to the role of the small anions inside. These anions (six S²⁻ and eight PO₄³⁻, Fig. 54c) are important templates, charge balancers and structural support materials. The S²⁻ anions are located between the middle layer (Ag₂₄) and the outermost layer (Ag₆₀), presenting an octahedral arrangement (Fig. 54c). Six S²⁻ play an important role in the formation and stabilization of 12 Ag₄ quadrilaterals in the truncated octahedron and rhombicosidodecahedron (Fig. 54c). The PO₄³⁻ tetrahedra in a cubic fashion penetrate the hexagonal Ag₆ windows of the middle truncated octahedral layer, connecting the inner octahedron and the outer rhombicosidodecahedron (Fig. 54c). PO₄³⁻ is essential for the formation of the hexagon of the truncated octahedron (Fig. 54c), which is observed in many truncated octahedral structures (Fig. 35 and 53). Ag₁₀₂ (Fig. 54d) features an analogous multi-shell inner core of Ag₆@Ag₂₄@Ag₆₀. Furthermore, six linear Ag₂ staples in octahedral mode further surround this core (Fig. 54e). The difference in the outer shells of the structures in Ag₉₀ and Ag₁₀₂ comes from the difference in their protecting ligands (^tBuS⁻, PhPO₃²⁻, and PhPO₃H⁻ in Ag₉₀ and CyS⁻, H₂PO₄⁻, HPO₄²⁻, and CF₃COO⁻ in Ag₁₀₂). Interestingly, the high symmetry of the overall structure makes the arrangement of peripheral ligands also have polyhedral characteristics, i.e., 12 PhPO₃²⁻ with icosahedral arrangement and 6 PhPO₃H⁻ exhibiting an octahedral framework in Ag₉₀ and 6 H₂PO₄⁻ and 6 HPO₄²⁻ with icosahedral mode in Ag₁₀₂ (Fig. 54f).

Fig. 54 (a) Crystallographic structure of [(PO₄)₈@Ag₉₀S₆(^tBuS)₂₄(PhPO₃)₁₂(PhPO₃H)₆]. (b) Topologies of 90-metal framework of Ag₉₀. (c) Distribution of eight PO₄³⁻ and six S²⁻ anions as templates in the cluster and their detailed coordination towards Ag atoms in different shells.⁵⁰⁶ (d) Crystallographic structure of [(PO₄)₈@Ag₁₀₂S₆(CyS)₃₀(H₂PO₄)₆(HPO₄)₆(CF₃COO)₁₈]. (e) 102 metal framework topologies of Ag₁₀₂. (f) Distribution of eight PO₄³⁻ anions as templates with cubic fashion and 12 H_xPO₄^(3−x)− anions (6 H₂PO₄⁻ and 6 (HPO₄)₆²⁻) as peripheral protection ligands in icosahedral mode.⁵⁰⁷

In 2021, Sun's group prepared the largest and highly symmetric silver chalcogenide cluster reported thus far, (H₃O)₂[S₂₂(SO₄)₃₆@Ag₁₉₂(CyS)₆₆(NO₃)₁₂]·4CH₃OH (Ag₁₉₂, Fig. 55a) with 6 silver (Ag₆@Ag₃₀@Ag₁₂@Ag₁₂@Ag₆₀@Ag₇₂) and 14 chalcogenide ((SO₄)₈((SO₄)₄@(SO₄)₄)@S₆@S₄@S₁₂@(SO₄)₁₂@(SO₄)₁₂@(SO₄)₄@(CyS)₁₂@(CyS)₆@(CyS)₂₄@(CyS)₁₂@(CyS)₁₂@(NO₃)₁₂) octahedral and tetrahedral shells, which are assigned to Platonic or Archimedean solids.⁵⁰⁸ The metal framework was analyzed first. From the inside out, the 192 silver atoms can be divided into several parts (Fig. 55b and c), as follows: v₁-octahedron (turquoise), Ag₆ (shell 1); v₃-octahedron (blue, truncated octahedron@octahedron), Ag₃₀ (Ag₂₄@Ag₆, shell 2); truncated tetrahedron (pink), Ag₁₂ (shell 3); cuboctahedron or doubly truncated tetrahedron (dark blue),⁵⁰⁹ Ag₁₂ (shell 4); spherical hollow cage (orange), Ag₆₀ (shell 5); and four bowl-like subunits (Ag₁₈, green) with tetrahedral arrangement, Ag₇₂ (shell 6). Shell 5 containing 60 metal atoms can be subdivided into several non-standard Archimedean solids (Fig. 55c), i.e., two truncated octahedra with 24 (4,6,6) vertices and one cuboctahedron with 12 (3,4,3,4) vertices. In shell 6, the bowl-like subunit (Ag₁₈) can be seen as a part of the rhombicosidodecahedron. The 72-Ag atoms can be split into three truncated octahedra with 24 (4,6,6) vertices (Fig. 55c). Next, the arrangement of a large number of anions (22 S²⁻ and 36 SO₄²⁻) inside Ag₁₉₂ is discussed. Six μ₆-S²⁻ (octahedral arrangement, Fig. 55d) and eight SO₄²⁻ in cubic mode (penetrated in the center of the v₃-triangle in shell 2, Fig. 55e) connect adjacent shells, forming an {Ag₆@(Ag₃SO₄)₈@S₆} core, similar to {Ag₆@(Ag₃PO₄)₈@S₆} in Ag₉₀ and Ag₁₀₂ (Fig. 54c). However, the difference in the position of the S²⁻ atoms (on the quadrilateral face for Ag₁₉₂ and outside the quadrilateral face for Ag₉₀ and Ag₁₀₂) and more anion intercalation make the arrangement of the outer silver atoms very different. The remaining 16 S²⁻ anions are distributed between shell 2 (or 3) and shell 5 (Fig. 55d), of which four μ₆-S²⁻ templates with a tetrahedral arrangement connect the two triangles (from shell 3 and 5) and the other 12 (truncated tetrahedral arrangement) connect shells 2, 3, and 5. The remaining 28 SO₄²⁻ anions are all in four Ag₁₈ bowls (Fig. 55e) and function as glue to connect and consolidate the outer silver shells 2, 4, 5, and 6. 28 SO₄²⁻ can also be arranged to form three Platonic or Archimedean solids, i.e., a truncated tetrahedron, cuboctahedron and tetrahedron (Fig. 55e). In addition, the distribution of the peripheral protective ligands (66 CyS⁻ and 12 NO₃⁻) also exhibits polyhedral characteristics (Fig. 55a), i.e., truncated tetrahedron (12 CyS⁻), octahedron (6 CyS⁻), truncated octahedron (24 CyS⁻), cuboctahedron (12 CyS⁻), truncated tetrahedron (12 CyS⁻), and cuboctahedron (12 NO₃⁻).

Fig. 55 (a) Crystallographic structure of [S₂₂(SO₄)₃₆@Ag₁₉₂(CyS)₆₆(NO₃)₁₂]²⁻ (Ag₁₉₂). (b) Six Ag shells with T_d or O_h symmetry in Ag₁₉₂. (c) Successive Platonic/Archimedean polyhedral description featuring only equivalent vertex metal atoms. (d) Distribution of 22 S²⁻ anions in three shells. (e) Distribution of 36 SO₄²⁻ anions in four shells.⁵⁰⁸

In addition to these silver clusters with inner cores, Sun et al. prepared an unprecedented buckyball-like Ag₁₈₀ nanocage ([Ag₁₈₀(ⁱPrS)₉₀(CH₃SO₃)₄₄]·(CH₃SO₃)₄₆·34CH₃OH) with icosahedral (I_h) topology via the solvothermal reaction of [Ag(ⁱPrS)]_n and CH₃SO₃Ag in methanol.⁵¹⁰ 180 Ag atoms can be divided into two types of 4-valent ((3,4,5,4) or (3,4,6,4)) vertices to constitute 360 edges and 182 faces (sixty trigons, ninety tetragons, twelve pentagons, and twenty hexagons). Ag₁₈₀ can be regarded as the product of the perfect combination of chemistry and geometric mathematics, which can be viewed as an assembly composed of 60 equivalent silver trigons or 12 equivalent pentagons and 20 equivalent hexagons. If the silver trigons are viewed as vertices, the topology (truncated icosahedron) will be similar to C₆₀. Excitingly, a careful split reveals that Ag₁₈₀ can be understood as a combination of two Archimedean solids (truncated icosahedron built by 12 silver pentagons and truncated icosidodecahedron constructed by 20 silver hexagons) with icosahedral (I_h) point-group symmetry.

8.2.2 Ag(I/0) clusters. With the continuous innovation in the development of reducing agents and synthesis methods, a large number of silver clusters containing the reduced state has emerged. In this section, we present these remarkable results in order of symmetry from low (tetrahedral (T)) to high (octahedral (O) and icosahedral (I)).

In 2011, Huang et al. reported the first reductive Ag^I/0 cage-like cluster, [Ag₃₄(CO₃)₁₂Cl₄(dppm)₁₂] (Ag₃₄, dppm = 1,1′-bis(diphenylphosphine)-methane, Fig. 56a), with perfect T_h symmetry.⁵¹¹ All the Ag atoms are connected by twelve CO₃²⁻ anions originating from atmospheric CO₂. The 34 Ag atoms of Ag₃₄ can be divided into two parts (Ag₁₀@Ag₂₄, Fig. 56b), as follows: (i) the 24 Ag^I atoms on the surface construct a distorted truncated tetrahedral framework, in which four triangles are capped by μ₃-Cl⁻ ions and (ii) the ten Ag^I/0 atoms in the interior form an v₂-tetrahedron core with an octahedral Ag₆. In 2020, Xie and co-workers reported the preparation of an array of phosphine and oxometallate ligand co-protected superatomic Ag nanoclusters, Ag₂₈(dppb)₆(MoO₄)₄ (Ag₂₈Mo₄, dppb = 1,4-bis(diphenylphosphino)butane, Fig. 56c), Ag₂₈(dppb)₆(WO₄)₄ and Ag₃₂(dppb)₁₂(MoO₄)₄(NO₃)₄, which all feature a core–shell structure of Ag₄@Ag₂₄ (Fig. 56d) protected by four oxometallate ligands (MoO₄²⁻ or WO₄²⁻) and six phosphine ligands (dppb).⁵¹² The Ag₄@Ag₂₄ metal skeleton of Ag₂₈Mo₄ can be described as an ideal ν₁-tetrahedral Ag₄ core capped on all its triangle faces by four ν₂-triangular Ag₆ fragments. Alternatively, the twenty-four-metal silver shell of Ag₂₈Mo₄ can also be viewed as the fusion of two distorted truncated tetrahedrons, including an inner Ag₁₂ skeleton capped by four tetrahedral-arranged MoO₄²⁻ anions, and an outer Ag₁₂ framework, which is consolidated by peripheral phosphine ligands (Fig. 56d).

Fig. 56 Crystallographic and core–shell structures of (a and b) [Ag₃₄(CO₃)₁₂Cl₄(dppm)₁₂] with Ag₁₀@Ag₂₄,⁵¹¹ (c and d) Ag₂₈(dppb)₆(MoO₄)₄ with Ag₄@Ag₁₂@Ag₁₂,⁵¹² (e and f) [Cu₁₂Ag₂₈(2,4-SPhCl₂)₂₄]⁴⁻ with Ag₄@Ag₁₂@Ag₁₂@Cu₁₂,⁵¹³ and (g and h) Cd₁₂Ag₃₂(SePh)₃₆ with Ag₄@Ag₁₂@Ag₁₂@Cd₁₂@Ag₄, respectively.⁵¹⁴

A similar core–shell structure of Ag₄@Ag₂₄ was observed in the heterometallic [Cu₁₂Ag₂₈(2,4-SPhCl₂)₂₄]⁴⁻ (Cu₁₂Ag₂₈, 2,4-SPhCl₂ = 2,4-dichlorobenzenethiol, Fig. 56e and f)⁵¹³ and Cd₁₂Ag₃₂(SePh)₃₆ (Cd₁₂Ag₃₂, Fig. 56g and h)⁵¹⁴ clusters. Different from Ag₂₈Mo₄ co-protected by phosphine and oxometallate ligands, the Ag₄@Ag₂₄ skeleton of Cu₁₂Ag₂₈ and Cd₁₂Ag₃₂ is capped by four Cu₃(SPhCl₂)₆ (Cu₁₂: distorted cuboctahedron, Fig. 56f) and Cd₃Ag(SePh)₉ (Cd₁₂(distorted cuboctahedron)@Ag₄(tetrahedron), Fig. 56h) fragments, respectively. Revisiting the geometry of the core–shell structure containing an Ag₄@Ag₂₄ skeleton (Fig. 56c–h), a series of distortions from the core to the shells lowers the point-group symmetry (from centrosymmetric T_d to chiral T), turning them into chiral clusters. Due to the strong ion–pair interactions between the [Cu₁₂Ag₂₈(2,4-SPhCl₂)₂₄]⁴⁻ anion clusters and counter-cations (TBA⁺ = tetrabutylammonium), chiral quaternary ammonium cations can be used to successfully separate the enantiomers.⁵¹³ In 2021, Zhu and co-workers reported the preparation of a 44-metal racemic nanocluster with the same metal arrangement (Ag₄@Ag₁₂@Ag₁₂@Cu₁₂@Ag₄) as Cd₁₂Ag₃₂, [Ag₃₂Cu₁₂(CH₃COO)₁₂(SAdm)₁₂(P(CH₃OPh)₃)₄] (HSAdm = 1-adamantanethiol).⁵¹⁵ The deracemization process was achieved by ligand exchange (CH₃COO⁻ → 2-chloropropionic acid). The optical activity was conserved after further exchange (2-chloropropionic acid → Br⁻), to give two new optically active nanoclusters, R/S-[Ag₂₈Cu₁₆Br₁₂(SAdm)₁₂(P(CH₃OPh)₃)₄] with Ag₄@Ag₁₂@Ag₁₂@Cu₁₂@Cu₄ core.

Giant T-symmetric reducing Ag nanoclusters with a dense metal core (>50 metal atoms) have not been reported to date, whereas two larger ones were successfully constructed by introducing additional anionic templates. In 2022, Zang presented a nearly perfectly-symmetrical doubly truncated tetrahedral 70-Ag nanocluster with pseudo-T_d symmetry, {Ag₇₀S₄(SⁱPr)₂₄(CF₃COO)₂₀(DMF)₃}²⁻ (Ag₇₀, Fig. 57a), containing a very rare geometric configuration among atomically-precise clusters.^509,516 The core of this cluster consists of an unprecedented fcc-based (fcc = face centered cubic) Ag₄@Ag₁₂S₄ with tetrahedron symmetry, while all-tetrahedral symmetries Ag₄@Ag₁₂S₄@Ag₁₂@Ag₄₂ develop its multi-shells (Fig. 57b). Overall, high-symmetry makes the arrangement of atoms in Ag₇₀ more regular and attractive, and the overall skeleton (including metal, anions, organic ligands) can be transformed into a combination of several Platonic or Archimedean solids (Fig. 57b), which is denoted as Ag₄@Ag₁₂@S₄@Ag₁₂@Ag₆@Ag₂₄@Ag₁₂@(SⁱPr)₁₂@(SⁱPr)₁₂, i.e., from the inside to the outside, a tetrahedron (Ag₄), truncated tetrahedron (Ag₁₂), tetrahedron (S₄), truncated tetrahedron (Ag₁₂), octahedron (Ag₆, located in the center of the rectangular face of outer Ag₄₂ shell), truncated octahedron (Ag₂₄, consisting of four six-membered rings from the Ag₄₂ shell), cuboctahedron (or say doubly truncated tetrahedron, Ag₁₂), truncated tetrahedron ((SⁱPr)₁₂), and cuboctahedron ((SⁱPr)₁₂).

Fig. 57 (a) Crystallographic structure of {Ag₇₀S₄(SⁱPr)₂₄(CF₃COO)₂₀(DMF)₃}²⁻ (Ag₇₀). (b) Successive Platonic/Archimedean polyhedral description featuring only equivalent vertex metal atoms.⁵⁰⁹

In 2021, Sun successfully isolated a reductive silver thiolate nanocluster, [Ag₈₉S₁₆(CyS)₄₂(p-NH₂PhAsO₃)₄](NO₃)₃ (Ag₈₉, Fig. 58a), containing an icosahedral μ₁₂-Ag@Ag₁₂ kernel passivated by 4 tetrahedra {AgS₄} and 4 ligands (p-NH₂PhAsO₃²⁻), and further encapsulated by an outer Ag₇₂ shell (Fig. 58b).⁵¹⁷ Comparing the two pseudo-T_d-symmetric reductive Ag–S nanoclusters (Ag₇₀ and Ag₈₉, Fig. 57 and 58), the interior (Ag₄@Ag₁₂@S₄ in Ag₇₀ and Ag@Ag₁₂@(AgS₄)₄ in Ag₈₉) expands due to the change in the templated anions (S²⁻ in Ag₇₀ and AgS₄⁷⁻ in Ag₈₉), and thus the outer shell also expands (Ag₁₂@Ag₄₂ in Ag₇₀ and Ag₂₄@Ag₄₈ in Ag₈₉). Similar to Ag₇₀, the overall Ag–S skeleton can be transformed into a combination of multiple Platonic or Archimedean solids (Fig. 58c), denoted as Ag@Ag₁₂@S₁₂@Ag₄@S₄@Ag₂₄@Ag₁₂@Ag₂₄@Ag₁₂, including one icosahedron, two tetrahedra, three truncated tetrahedra, and two truncated octahedra.

Fig. 58 (a and b) Crystallographic and core–shell structure of [Ag₈₉S₁₆(CyS)₄₂(p-NH₂PhAsO₃)₄]³⁺, respectively. (c) Successive Platonic/Archimedean polyhedral description featuring only equivalent vertex metal atoms.⁵¹⁷

In addition to the above-mentioned T-symmetric multi-shell structures, there is a striking family of polyhedral with octahedral (O_h) topology, i.e., box-shaped fcc Ag nanoclusters with the same vertex ([Ag(SR)₃(PR′₃)], as shown in Fig. 59).^518–531 In 2013, Zheng and co-workers reported the preparation of a thiolate and phosphine co-protected Ag₁₄ cluster containing two zero-valent Ag atoms, [Ag₁₄(3,4-SPhF₂)₁₂(PPh₃)₈] (Ag₁₄, 3,4-SPhF₂ = 3,4-difluorothiophenol, Fig. 59a), exhibiting excellent yellow luminescence.⁵¹⁸Ag₁₄ is a neutral cluster possessing a nearly cubic shape cluster and it had two important structural features, significantly different from that of the reported thiolate-protected Au clusters, as follows: (i) all thiolate (SPhF₂⁻) ligands bond to three Ag atoms, and thus no classical staple motifs are found in Ag₁₄ and (ii) the metal core of Ag₁₄ is an Ag₆⁴⁺ octahedron with two zero-valent Ag atoms, representing one of the basic structural fragments of single-crystalline face-centered-cubic (fcc) bulk metals (e.g., Au and Ag). The Ag₆⁴⁺ octahedral core is encapsulated by a cubic {Ag₈(3,4-SPhF₂)₁₂(PPh₃)₈} cage, consisting of eight vertex-sharing [Ag(3,4-SPhF₂)₃(PPh₃)] tetrahedra with a cubic arrangement (Fig. 59a). Alternatively, Ag₁₄ can be described as an octa-capped octahedron, Ag₆@Ag₈. In 2018, Zang's group reported the preparation of an analogous Ag₁₄ superatomic silver cluster, [Ag₁₄(C₂B₁₀H₁₀S₂)₆(CH₃CN)₈] (NC-1, C₂B₁₀H₁₀S₂ = 1,2-dithiolate-o-carborane).⁵²⁵ Given that NC-1 with labile CH₃CN is unstable, monodentate pyridine ligands (such as pyridine (Py) and p-methylpyridine (p-MePy)) were used to site-specifically replace these solvent molecules, obtaining two new Ag₁₄ clusters [Ag₁₄(C₂B₁₀H₁₀S₂)₆(Py/p-MePy)₈] (NCs-2,3), which exhibited ultrastability up to 150 °C in air. Moreover, by introducing bidentate N-containing ligands to bridge the Ag₁₄ building blocks, a hierarchical series of 1D-to-3D superatomic Ag cluster-assembled materials (SCAM-1,2,3,4) was successfully obtained, which exhibited tunable dual emission with wide-range thermochromism. Thereafter, a series of similar Ag₁₄ (Ag₆@Ag₈ with octahedron@cube) superatomic clusters was prepared.^519–524

Fig. 59 Crystallographic structures of cubic box-like Ag nanoclusters: (a) [Ag₁₄(3,4-SPhF₂)₁₂(PPh₃)₈] (Ag₁₄);⁵¹⁸ (b) [Ag₄₀(2,4-SPh(CH₃)₂)₂₄(PPh₃)₈] (Ag₄₀);⁵²⁹ (c) [Ag₄₆(2,5-SPh(CH₃)₂)₂₄(PPh₃)₈]²⁺ (Ag₄₆);⁵²⁹ (d) [Ag₆₃(3,4-SPhF₂)₃₆(PⁿBu₃)₈]⁺ (Ag₆₃);⁵²⁴ (e) [Ag₁₀₀(3,4-SPhF₂)₄₈(PPh₃)₈]⁻ (Ag₁₀₀);⁵³⁰ (f) outer eight [Ag(SR)₃(PR′₃)] tetrahedron motifs with cubic fashion; and (g) inner cores with nested structures: octahedron (Ag₆ core in Ag₁₄), cube@truncated octahedron (Ag₈@Ag₂₄ core in Ag₄₀), octahedron@cube@truncated octahedron (Ag₆@Ag₈@Ag₂₄ core in Ag₄₆), v₁-cuboctahedron@v₂-cuboctahedron (Ag@Ag₁₂@Ag₄₂ core in Ag₆₃), and v₁-octahedron@v₃-octahedron@rhombicuboctahedron (Ag₆@Ag₃₈@Ag₄₈ core in Ag₁₀₀), respectively.

In 2018, utilizing a common mixed-ligand strategy, Zhu and co-workers synthesized a unique pair of new cube-shaped superatomic nanoclusters, [Ag₄₀(2,4-SPh(CH₃)₂)₂₄(PPh₃)₈] (Ag₄₀, 2,4-SPh(CH₃)₂ = 2,4-dimethylbenzenethiolate, Fig. 59b) and [Ag₄₆(2,5-SPh(CH₃)₂)₂₄(PPh₃)₈]²⁺ (Ag₄₆, 2,5-SPh(CH₃)₂ = 2,5-dimethylbenzenethiolate, Fig. 59c), representing two new members of the high-nuclearity cubic cluster family.⁵²⁹ The crystal structures of Ag₄₀ and Ag₄₆ show that the skeleton of both clusters have a common Ag₃₂S₂₄P₈ protecting framework, in which a unique Ag₂₄ shell with a truncated octahedron arrangement encapsulated by eight cubic-arranged tetrahedron units [Ag(SPh(CH₃)₂)₃(PPh₃)]. Differently, Ag₄₀ has an unprecedented simple-cubic Ag₈ core, whereas Ag₄₆ has a common face-centered cubic Ag₁₄ core. The metal skeleton of Ag₄₀ and Ag₄₆ can be described by the form of nested polyhedrons, abbreviated as Ag₈(cube)@Ag₂₄(truncated octahedron)@Ag₈(cube) for Ag₄₀ and Ag₆(octahedron)@Ag₈(cube)@Ag₂₄(truncated octahedron)@Ag₈(cube) for Ag₄₆. Simultaneously, Pradeep reported two similar superatomic clusters, [Ag₄₀(2,4-SPh(CH₃)₂)₂₄(PPh₃)₈](NO₃)₂ and [Ag₄₆(2,4-SPh(CH₃)₂)₂₄(PPh₃)₈](NO₃)₂, co-crystallized in the same crystal.⁵²⁸ In 2019, Zheng and co-workers reported the preparation of a hydride-rich Ag–S–P nanocluster, [Ag₄₀(2,4-SPh(CH₃)₂)₂₄(PPh₃)₈H₁₂]²⁺ (Ag₄₀H₁₂).⁵²⁶ Similarly, the metal framework of Ag₄₀H₁₂ consists of three concentric shells (abbreviated as Ag₈@Ag₂₄@Ag₈). The presence of 12 hydrides (H⁻) in Ag₄₀H₁₂ was systematically identified using various characterization techniques and DFT calculations, with H⁻ located outside the midpoints of the 12 edges of the inner Ag₈ cube.

Through the optimization of experiments for the synthesis of Ag₁₄, Zheng's group successfully prepared a pair of nearly perfect half-cubic-shape superatomic clusters ([Ag₃₈(3,4-SPhF₂)₂₆(PR₃)₈], R = Ph or ⁿBu) and a pair of larger nearly perfect cubic superatomic clusters ([Ag₆₃(3,4-SPhF₂)₃₆(PR₃)₈]⁺X⁻, R = Ph, X = Br; or R = ⁿBu, X = BPh₄, denoted as Ag₆₃, Fig. 59d) in 2017.⁵²⁴ The metal framework of Ag₆₃ is formed by eight face-sharing fcc Ag₁₄ units arranged to form a larger cube and made up of a v₂-cuboctahedral Ag₅₅ core and cubic [Ag(3,4-SPhF₂)₃(PR₃)] (R = Ph or ⁿBu) shell. Its Ag₅₅ metal core consists of a central μ₁₂-Ag atom surrounded by twelve Ag atoms with a cuboctahedral arrangement (Ag₁₂), which in turn are encapsulated by 42 Ag atoms with a distorted v₂-cuboctahedral fashion (Ag₄₂). The v₂-cuboctahedron Ag₄₂ can be viewed as a combination of Ag₆ (octahedron, made up of 6 Ag atoms from the centers of the six rectangles Ag₉), Ag₂₄ (rhombicuboctahedron, consisting of 8 Ag₃ triangles from the centers of the eight v₂-triangles Ag₆) and Ag₁₂ (cuboctahedron, constituted by 12 Ag atoms from vertices). Thus, the metal framework of Ag₆₃ can be abbreviated as μ₁₂-Ag@Ag₁₂(v₁-cuboctahedron)@Ag₄₂(v₂-cuboctahedron, Ag₆(octahedron) + Ag₂₄ (rhombicuboctahedron) + Ag₁₂(cuboctahedron))@Ag₈(cube). In 2020, Zhu and Yu reported the largest known fcc Ag nanocluster, [Ag₁₀₀(3,4-SPhF₂)₄₈(PPh₃)₈] (Ag₁₀₀, Fig. 59e).⁵³⁰Ag₁₀₀ is the first all-octahedral symmetric four-layered nesting structure, similar to a Russian doll. From the inside to the outside, the topologies of the 100 Ag atoms contain an inner Platonic solid (v₁-octahedron, Ag₆) enveloped by outer three Platonic/Archimedean solids (v₃-octahedron, Ag₃₈; rhombicuboctahedron, Ag₄₈; and cube, Ag₈), abbreviated as Ag₆@Ag₃₈@Ag₄₈@Ag₈. The v₃-octahedron Ag₃₈ can be viewed as a combination of Ag₈ (cube, made up of 8 Ag atoms from the centers of the eight v₃-triangles Ag₁₀), Ag₂₄ (truncated octahedron, consisting of 8 edge-shared Ag₅ triangles from the centers of the eight v₃-triangles Ag₁₀) and Ag₆ (octahedron, constituted by 6 Ag atoms from vertices). The rhombicuboctahedron shell (Ag₄₈) is made up of eight v₂-triangles Ag₆ with a cubic arrangement, which can be split into a truncated cube (Ag₂₄) constituted by 8 Ag₃ triangles from the centers of the eight v₂-triangles Ag₆ and a rhombicuboctahedron (Ag₂₄). Interestingly, the surface ligands of all the fcc silver nanoclusters share similar structural features, in which the phosphines serve as terminal ligands to bridge the eight vertices of the polyhedron (Fig. 59f), while the thiolate ligands occupy the twelve edges and eight faces of half-cube or cube. The cubic nanoclusters with the same Ag₈S₂₄P₈ surface but different cores (Fig. 59g) provide an ideal platform for the study of structure–property correlation.

Bigioni et al. reported an ultra-stable silver nanocluster with I_h-symmetrical Ag₃₂ (icosahedron@dodecahedron, Ag₁₂@Ag₂₀) kernel (Ag₄₄(p-MBA)₃₀⁴⁻, p-MBA = p-mercaptobenzoic acid, Fig. 60a; denoted as Ag₄₄) produced by a simple synthetic protocol, which is a single size product without further size sorting.⁵³² The purity, yield, and stability of Ag₄₄ were substantially better than that of other gold or metal nanoparticles at that time, owing to its ultra-stable 32-silver-atom dodecahedral core (Fig. 60b), suitable silver-sulfur protective shell and unique electronic structure (closed-shell 18 free valence electrons with a large energy gap). Ag₄₄ consists of a hollow icosahedron Ag₁₂ encapsulated by a dodecahedron Ag₂₂, and further stabilized by six {Ag₂(p-MBA)₅} units with an octahedral arrangement. During the same period, Zheng reported the preparation of similar structures (Ag₄₄ and Au₁₂Ag₃₂).⁵³³ After extensive literature research, we found that numerous high-nuclearity silver clusters have similar icosahedral cores, but the outer metal shells of most of them do not have polyhedral features, such as Ag₂₀,⁵³⁴ Ag₂₁,^535–537 Ag₂₂,⁵³⁶ Ag₂₅,^538–541 Ag₃₄,⁵⁴² Ag₃₅,⁵⁴³ Ag₆₁,⁵⁴⁴ and Ag₁₁₂.⁵⁴⁵ However, there are some examples of icosahedral cores encapsulated by shells with a regular polyhedral arrangement in Ag-containing clusters. It should be noted that 12 or 13-metal silver clusters with icosahedral configuration have not been successfully isolated to date.

Fig. 60 Crystallographic structure and core–shell framework of Ag₄₄(p-MBA)₃₀⁴⁻.⁵³²

In 2017, Zhu and co-workers reported the preparation of a novel 8-electron Au-Ag alloy cluster, Au₄Ag₁₃(DPPM)₃(2.5-SPhMe₂)₉ (Au₄Ag₁₃, DPPM = bis(diphenylphosphino)methane, Fig. 61a), exhibiting interesting crystallization-induced emission enhancement.⁵⁴⁶Au₄Ag₁₃ consists of four-Au-doped icosahedral Au₄Ag₉ kernel protected by three AgS₂P and one AgS₃ motif with a tetrahedral arrangement (Au@(AuAg)₁₂@Ag₄, Fig. 61b). In the same year, Liu and co-workers isolated the first atomically precise superatomic Ag nanoclusters stabilized by Se-donor ligands, [Ag₂₀{Se₂P(OⁱPr)₂}₁₂] and [Ag₂₁{Se₂P(OEt)₂}₁₂]⁺ (Ag₂₁, Fig. 61c), through a common ligand-exchange strategy.⁵⁴⁷ Furthermore, the reaction of Ag₂₁ with Au precursor resulted in the formation of a heteroatom-doped analogue [AuAg₂₀{Se₂P(OEt)₂}₁₂]⁺. The Ag skeleton of Ag₂₁, possessing an idealized T symmetry, can be described as a μ₁₂-Ag-centered icosahedron Ag₁₃ inscribed in a large cube formed by eight Ag atoms (Ag@Ag₁₂@Ag₈, Fig. 61d).

Fig. 61 (a) Crystallographic structure of Au₄Ag₁₃(DPPM)₃(2,5-SPhMe₂)₉. (b) Four-Au-doped icosahedral Au@Au₃Ag₉ kernel enveloped by outer Ag₄ tetrahedron.⁵⁴⁶ (c) Crystallographic structure of [Ag₂₁{Se₂P(OEt)₂}₁₂]⁺. (d) Centered icosahedral Ag@Ag₁₂ unit inscribed in an outer Ag₈ cube.⁵⁴⁷

In 2015, Bakr's group reported the synthesis of an atomically precise 8-electron superatomic Ag nanocluster, [Ag₂₉(BDT)₁₂(TPP)₄]³⁻ (Ag₂₉, BDT = 1,3-benzenedithiol and TPP = triphenylphosphine, Fig. 62a), featuring four tetrahedrally symmetrical sites formed by the supramolecular self-assembly of twelve wide footprint bidentate thiolate ligands (BDT) on an Ag₁₃ core.⁵⁴⁸ The overall core–shell structure of Ag₂₉ consists of a common center icosahedral 13-metal core inscribed in an exterior shell Ag₁₆S₂₄P₄ (Fig. 62b). In the Ag₁₆S₂₄P₄ shell, the remaining 16 Ag atoms can be divided into two categories, as follows: (i) 12 Ag atoms cap all 12 Ag atoms of the icosahedral core, leading to four silver triangles in a distorted tetrahedral arrangement and (ii) each of the additional four Ag atoms is protected and stabilized by one P atom and three S atoms, resulting in the formation of four Ag₁S₃P₁ motifs in a tetrahedral fashion (Fig. 62b). Thus, the Ag₁₆S₂₄P₄ shell is composed of four pairs of S atom-sharing Ag₃S₆ and Ag₁S₃P₁ motifs, providing the complete passivation of Ag₂₉. Alloy nanoclusters with the same structure have also been prepared, such as [AuAg₂₈(BDT)₁₂(TPP)₄],⁵⁴⁹ [PtAg₂₈(BDT)₁₂(TPP)₄],⁵⁵⁰ [Ag₁₇Cu₁₂(BDT)₁₂(TPP)₄], and [AuAg₁₆Cu₁₂(BDT)₁₂(TPP)₄].⁵⁵¹ These M₂₉ clusters have high stability and excellent optical and catalytic properties.

Fig. 62 Crystallographic structures and core–shell frameworks of (a and b) [Ag₂₉(BDT)₁₂(TPP)₄]³⁻,⁵⁴⁸ (c and d) [Ag₂₅Cu₄(PhC≡C)₁₂(PPh₃)₁₂Cl₆]¹¹⁺,⁴⁶⁷ and (e and f) PtAg₂₈(S-Adm)₁₈(PPh₃)₄, respectively.⁵⁵²

In 2021, Zheng reported the synthesis of a hydride-rich Ag–Cu nanocluster, [Ag₂₅Cu₄(PhC≡C)₁₂(PPh₃)₁₂Cl₆H₈]³⁺ (Ag₂₅Cu₄, Fig. 62c), using (PPh₃)₂CuBH₄ as a new reducing agent.⁴⁶⁷Ag₂₅Cu₄ is made up of an icosahedral Ag₁₃ core capped by a mixed Ag–Cu–Cl shell, exhibiting a tetra-stratified configuration (Fig. 62d), i.e., μ₁₂-Ag(center)@Ag₁₂(icosahedron)@Cu₄(tetrahedron)@Cl₄(tetrahedron)@Ag₁₂(truncated tetrahedron). Ag₂₅Cu₄ exhibited high stability and no decomposition was observed after exposure to CH₂Cl₂ for 7 days by UV-Vis detection. In addition, Zhu reported a case of M₂₉ nanocluster (PtAg₂₈(S-Adm)₁₈(PPh₃)₄, HS-Adm = 1-adamantanethiol, PtAg₂₈, Fig. 62e) with different core–shell configurations in 2017.⁵⁵² The 29 metal atoms of PtAg₂₈ present a typical v₃-supertetrahedral configuration, containing a tetra-stratified arrangement (Fig. 62f), i.e., μ₁₂-Pt(center)@Ag₁₂(cuboctahedron)@Ag₁₂(truncated tetrahedron)@Ag₄(tetrahedron). Since then, this research group conducted in-depth research on related scientific issues such as alloying and functionalization based on the M₂₉ structure.⁵⁵³ It should be noted that these three types of M₂₉ clusters have similarities, but are not identical (Fig. 62), and it can be seen that shell engineering (i.e., peripheral protection ligands) has a great influence on their structure.

In addition to being encapsulated in simple low-symmetric Platonic and Archimedean solids (such as tetrahedra (Fig. 61a and b), cubes (Fig. 61c and d), and truncated tetrahedra (Fig. 62)), Ag-centered icosahedra (Ag@Ag₁₂) can also appear in high-symmetric dodecahedra (I_h). In 2021, Zang reported the preparation of a fullerene-like Ag nanocluster Ag₃₃(C₂B₁₀H₁₀S₂)₁₂ (Ag₃₃, C₂B₁₀H₁₀S₂H₂ = 9,12-dimercapto-1,2-closo-carborane, Fig. 63a) with an Ag@Ag₁₂(icosahedron)@Ag₂₀(dodecahedron) core–shell framework (Fig. 63b).⁵⁵⁴ Interestingly, continuous Cu-doping in the outer Ag₂₀ dodecahedral shell first induced lower symmetry to generate an Ag@Ag₁₂@Ag_20−nCu_n (6 ≥ n ≥ 2) chiral analog, which shows clearly chiroptical activity, and then produced a highly symmetrical achiral cluster, Ag@Ag₁₂@Ag_20−nCu_n (20 ≥ n ≥ 13).

Fig. 63 (a) Crystallographic structure of Ag₃₃(C₂B₁₀H₁₀S₂)₁₂. (b) Tri-stratified configuration of Ag₃₃: Ag(center)@Ag₁₂(icosahedron)@Ag₂₀(dodecahedron).⁵⁵⁴ (c) Crystallographic structure of Au₂₄Ag₂₀(2-SPy)₄(PhC≡C)₂₀Cl₂. (d) Hollow concentric three-shell of Au₂₄Ag₂₀: Au₁₂(icosahedron)@Ag₂₀(dodecahedron)@Au₁₂(icosahedron).⁵⁵⁵

In 2015, Zheng and co-workers reported the synthesis of a bi-metallic 18-electron superatomic nanocluster, Au₂₄Ag₂₀(2-SPy)₄(PhC≡C)₂₀Cl₂ (Au₂₄Ag₂₀, 2-SPy = 2-pyridylthiolate, Fig. 63c), and it was found for the first time that three different ligands (2-SPy⁻, PhC≡C⁻, Cl⁻) coexist on the surface of Au₂₄Ag₂₀.⁵⁵⁵ The metal skeleton of Au₂₄Ag₂₀ can be described as a Keplerate (Fig. 63d), featuring a concentric three-shell Au₁₂(icosahedron)@Ag₂₀(dodecahedron)@Au₁₂(icosahedron). Due to the synergy of multiple ligands, the outer metal–ligand shell has the characteristics of regular polyhedra (icosahedrons), which cannot be formed in 44-/45-metal clusters (Fig. 60) protected by single ligands.⁵⁵⁶

In 2018, Zheng and co-workers reported the co-crystallization of two atomically precise, different-size metal aggregates in a 1 : 1 ratio. One is an atomically closed-shell intermetallic spherical nanoparticle (AuAg)₂₆₇(2,4-SPhMe₂)₈₀ ((AuAg)₂₆₇, Fig. 64a) and the other is a smaller electronically closed-shell trigonal-prismatic (AuAg)₄₅(2,4-SPhMe₂)₂₇(PPh₃)₆ nanocluster.⁵⁵⁷ The larger (AuAg)₂₆₇ has four notable features, as follows: (i) a concentric four-shell icosahedral structure of Ag@M₁₂@M₄₂@M₉₂@Ag₁₂₀(2,4-SPhMe₂)₈₀ (M = Au or Ag) with an inner-core M₁₄₇v₃-icosahedron (Fig. 64b–f); (ii) an open electron shell possessing 187 delocalized-electrons; (iii) fully metallic, plasmonic behavior; and (iv) zero HOMO–LUMO energy gap. The 147-metal inner-core M₁₄₇ (M= Au or Ag) exhibits an unprecedented high-symmetry structure of three-layer Mackay icosahedron mode (Fig. 64c–e), encapsulated by a metal Ag₁₂₀ shell in anti-Mackay configuration (Fig. 64f) and an outmost fullerene-like thiolate-ligand (2,4-SPhMe₂)₈₀ shell. The anti-Mackay fourth shell Ag₁₂₀ consists of twenty v₂-triangles Ag₆ (Fig. 64g), sixty squares Ag₄ (Fig. 64h), and twelve pentagons Ag₅, and these polyhedra form a semi-regular polyhedron by sharing edges. The center of each Ag₆ triangular and Ag₄ square face is capped by a 2,4-SPhMe₂⁻ ligand. Interestingly, the 60 S atoms from the peripheral protective ligands can be arranged as a slightly distorted fullerene C₆₀, and 12 additional S atoms are located at the center of twenty hexagons of the fullerene ball, building a dodecahedral framework.

Fig. 64 (a) Ball-and-stick representation of spherical (AuAg)₂₆₇(2,4-SPhMe₂)₈₀. (b) First Mackay ν₁-icosahedral (Au/Ag)₁₂ shell with a central Ag atom. (c) Second Mackay ν₂-icosahedral (Au/Ag)₄₂ shell. (d) Third Mackay v₃-icosahedral (Au/Ag)₉₂ shell. (e) Fourth anti-Mackay Ag₁₂₀ shell. (f) v₃-icosahedron (Au/Ag)₉₂ including three polyhedrons, where the metal atoms are only at the vertices (icosahedron, truncated icosahedron, and dodecahedron). (g) Anti-Mackay Ag₁₂₀ with spherical geometry including two polyhedra (rhombicosidodecahedron and truncated dodecahedron).⁵⁵⁷ Color codes: Au/Ag, green, blue, red, and dark yellow; S, yellow; and C, gray.

In the past few decades, two-layer M₅₅ Mackay icosahedra have been found in a diversity of atomically precise nanosized metal clusters, characterized fully by X-ray crystallographic analysis. For example, two-shell clusters Pd₅₅ (Fig. 40a) and Cu₄₃Al₁₂ (Fig. 49), three-shell cluster Pd₁₄₅ (Fig. 40b), four-shell clusters (PtPd)₁₆₅ (Fig. 40c); and some clusters (such as Au₁₃₃,^558,559 and Ag₁₁₂⁵⁴⁵) without a complete regular polyhedral shell. In (AuAg)₂₆₇, the next Mackay overlayer based on the M₅₅ icosahedron is the first to be crystallographically observed, giving rise to three well-known complete Mackay metal shells of 147 atoms in contrast to the embryonic growth of decahedral (Au₁₀₂,⁵⁶⁰ Au₁₃₀,⁵⁶¹ Au₂₄₆,⁵⁶² Ag₁₁₂,⁵⁴⁵ Ag₁₃₆,⁵⁸ Ag₃₀₇,⁵⁶³ and Ag₃₇₄⁵⁸) and fcc cuboctahedral Au₁₄₆⁵⁶⁴ and Au₂₇₉⁶⁰ clusters possessing a partial irregular shell without polyhedral feature.

Among the numerous silver-containing metal carbonyl clusters, there are also examples of polyhedral features, such as [Ag₁₆Ni₂₄(CO)₄₀]⁴⁻ (Ag₁₆Ni₂₄, Fig. 38d–f). In 1992, Longoni's group reported the synthesis of a paramagnetic tetra-anion, [(μ₁₂-Ag)Ag₁₂{Fe(CO)₄}₈]⁴⁻ (Ag₁₃Fe₈, Fig. 65), behaving as a 4-electron donor.^565–567 The molecular structure of Ag₁₃Fe₈ consists of an (μ₁₂-Ag)Ag₁₂-centered cuboctahedron capped on eight triangular faces by Fe(CO)₄ motifs, displaying C_3v symmetry, and the eight Fe atoms form a slightly distorted cube. Also, its suggested molecular symmetry is O_h.

Fig. 65 Ball-and-stick representation and metal framework of [(μ₁₂-Ag)Ag₁₂{Fe(CO)₄}₈]⁴⁻ (Ag₁₃Fe₈).⁵⁶⁵

8.3 Au clusters

Although Cu, Ag and Au belong to the same group and their mono-value ions possess a similar valence-electron configuration, the structures of their corresponding clusters exhibit significant differences.^12,15,77,421

As mentioned above, cubic copper or silver clusters can be easily built through suitable ligands, but similar cubic gold clusters constructed with similar ligands have not be observed to date. Recently, Zheng and co-workers obtained a metastable nona-nuclear superatomic cubic Au cluster, [Au₉(PPh₃)₈]⁺ (Fig. 66a), featuring a rare, nearly perfect body-centered cubic (bcc) metal core, capped by eight cubic-arranged phosphine ligands.⁸⁷ Thereafter, Schnepf and Clayborne reported a 20-electron superatomic gold cluster Au₂₀(^tBu₃P)₈ (Au₂₀, Fig. 66b) in the oxidation state ±0, resembling the Au bulk structure.⁸⁸ The metal atoms of Au₂₀ show a highly symmetric hollow cubic arrangement, Au₁₂(cuboctahedron)@Au₈(cube).

Fig. 66 (a) Crystallographic structure of [Au₉(PPh₃)₈]⁺ with cubic core Au@Au₈.⁸⁷ (b) Crystallographic structure of [Au₂₀(^tBu₃P)₈] with bigger cubic core Au₁₂@Au₈.⁸⁸ (c) Computed structure of Au₂₀(PPh₃)₄. Inset: Tetrahedral structure of the Au₂₀⁻ cluster.^568,569 (d) Crystallographic structure of [Au₂₀(P(P(CH₂)₂Ph₂)₃)₄]⁴⁺ with distorted icosahedral unit Au@Au₁₂.⁵⁷⁰ (e and f) Crystallographic structure of [Au₂₀(PPhpy₂)₁₀Cl₄]²⁺ with Au₂₀ kernel (e), generated from the fusion of two tri-vacant icosahedral Au@Au₉ subunits (f).⁵⁷¹

The preparation and successful structure elucidation of Au₂₀ remind people of the ligand-free T_d-Au₂₀ pyramidal cluster found by Wang et al. in 2003, which is highly stable in the gas phase (Fig. 66c).⁵⁶⁸ The geometries of the ligand-free metal clusters experimentally found in the gas phase were assigned by combining experimental data (such as ion-mobility spectrometry, trapped ion electron diffraction photoelectron spectroscopy, and infrared photodissociation spectroscopy), with related quantum chemical calculations. These structures have been extensively studied by M. M. Kappesm,⁵⁷² L.-S. Wang,^568,573,574 and others.^19,575–577 Comparing the structure of the ligand-free systems with that of the corresponding ligand-protected atomically precise metal clusters will help shed light on the influence of ligands on the formation of cluster structures. This is clearly reflected in the efforts to synthesize T_d-Au₂₀ clusters in single crystal solids.^{568–571,573,578–581} To maintain the integrity of the pyramidal structure, phosphine ligands were selected to synthesize T_d-Au₂₀ (Fig. 66c).⁵⁷³ Wang synthesized a 20-metal cluster using tetraphosphine ligands (tris[2-(diphenylphosphino)ethyl]phosphine = P(P(CH₂)₂Ph₂)₃), [Au₂₀(P(P(CH₂)₂Ph₂)₃)₄]Cl₄, with a distorted icosahedral kernel Au@Au₁₂ and Au(Au₂)₃ motif (Fig. 66d);⁵⁷⁰ as well as a new Au₂₂ nanocluster protected by diphosphine ligands (1,8-bis(diphenylphosphino)octane).⁵⁷⁹ Besides, Wang reported a phosphine-protected Au₂₀ nanocluster (Fig. 66e) in which its core can be viewed as being generated from the fusion of two tri-vacant icosahedral Au@Au₉ subunits (Fig. 66f).⁵⁷¹ Unfortunately, the crystal structures of these Au₂₀ cores did not reveal the expected pyramid feature, possibly reflecting that the introduction of protected ligands dramatically altered the electronic and geometrical structure of the metal clusters.

In the reported 12- or 13-nuclear gold clusters, a cuboctahedron arrangement similar to that of copper and silver clusters was found, such as [Ph₄As]₄[Au₁₂S₈]⁵⁸² and [NaAu₁₂Se₈].⁵⁸³ More interestingly, an individual icosahedral metal framework was never found in copper and silver clusters, while the first atomically precise centered icosahedral gold cluster, [Au₁₃(PMe₂Ph)₁₀Cl₂](PF₆)₃ (Fig. 67a), was reported by Mingos et al. in 1981.⁵⁸⁴ Some 13-nuclear centered icosahedral gold clusters protected by different ligands have been reported.⁵⁸⁵ Commonly, Au₁₃ icosahedron as part of the metal core often appears in larger metal clusters (such as Au₁₈, Au₂₀, Au₂₅, Au₃₆, Au₃₇, Au₃₈, Au₅₅, and Au₆₀), which is critical for cluster growth and formation.^77,421 In 2013, Zheng and co-workers reported the preparation of a set of mixed-ligand-stabilized Au₁₃Cu_x (x = 2, 4, and 8) bimetallic 8-electron superatomic nanoclusters, featuring centered icosahedral Au₁₃ core and parts of triangle faces capped by 2, 4, or 8 Cu atoms.⁵⁸⁶ Specifically, the four Cu atoms of [Au₁₃Cu₄(PPh₂Py)₄(SC₆H₄-tert-C₄H₉)₈]⁺ (Au₁₃Cu₄) exhibit a slightly distorted tetrahedral arrangement.

Fig. 67 (a) Crystallographic structure of the cationic cluster [Au₁₃(PMe₂Ph)₁₀Cl₂]³⁺.⁵⁸⁴ (b) Crystallographic structure of [Au₃₂(Ph₃P)₈(dpa)₆]²⁺ cationic cluster.⁵⁸⁷ (c) Crystallographic structure of Au₃₂(Et₃P)₁₂Cl₈. (e) Au₁₂@Au₂₀ two-shell structure.⁵⁸⁸

In addition to these single-shell gold clusters with Platonic/Archimedean polyhedral features, multi-shell gold clusters with similar characteristics have been also prepared. Wang's group prepared a golden fullerene Au₃₂ cluster employing a bi-ligand strategy, [Au₃₂(Ph₃P)₈(dpa)₆](SbF₆)₂ (Hdpa = 2,2′-dipyridylamine, Fig. 67b),⁵⁸⁷ consisting of a hollow Au₁₂@Au₂₀ Keplerate cage (two concentric Platonic solids: icosahedron@dodecahedron; Fig. 67d) co-protected by six amido and eight phosphine ligands, similar to the above-mentioned Ag₃₃(C₂B₁₀H₁₀S₂)₁₂ (Ag₃₃, Fig. 62a). Interestingly, the arrangement of six amido (dpa) ligands can be viewed as an octahedron, while eight phosphine (Ph₃P) ligands bridging at eight vertexes of the dodecahedral Au₂₀ shell form a cubic coordination polyhedron. Simultaneously, Schnepf et al. reported the preparation of three 32-metal gold clusters with the same Au₁₂@Au₂₀ kernel but different protective ligands, Au₃₂(R₃P)₁₂Cl₈ (R = Et, Fig. 67c; ⁿPr, ⁿBu).⁵⁸⁸

The reduction of the Au precursor ((Ph₃P)AuCl) with NaBH₄ in the presence of a sulfur (S²⁻) source (HSC(SiMe₃)₃), resulting in the formation of one of the largest metalloid Au clusters, Au₁₀₈S₂₄(PPh₃)₁₆ (Au₁₀₈, Fig. 68), was reported by Schnepf et al. In Au₁₀₈ (Fig. 68a), the v₃-octahedral Au₄₄ (Au₆@Au₃₈(Au₈@Au₂₄@Au₆), like Pt₆Ni₃₈ in Fig. 39) core is surrounded by a rhombicuboctahedron-shaped Au₄₈S₂₄ shell, where the “hollow window” of the Au₄₈S₂₄ shell (Au₂₄(rhombicuboctahedron)@S₂₄(rhombicuboctahedron)@Au₂₄(truncated octahedron), Fig. 68c) is capped by 16 additional Au(PPh₃) units (Au₁₂(cuboctahedron)@Au₄(tetrahedron), Fig. 68d), leading to the complete protection (Au₆₄S₂₄(PPh₃)₁₆ shell) of the Au₄₄ core (Fig. 68b).

Fig. 68 (a) Molecular structure of Au₁₀₈S₂₄(PPh₃)₁₆ (Au₁₀₈). (b) v₃-octahedron Au₄₄ (Au₆@Au₃₈) core and spherical Au₆₄S₂₄(PPh₃)₁₆ shell. (c) Rhombicuboctahedron-shaped Au₄₈S₂₄ framework containing six Au₈S₄ motifs (assembled by four corner-sharing Au₃S). (d) Outer PPh₃-coordinated 16 Au atoms.⁸⁶

In 2018, Wu and Jin et al. reported the preparation of an all-thiolate ligand-protected gold nanocluster Au₁₄₄(SCH₂Ph)₆₀ (Fig. 69a),⁵⁸⁹ while Wang's group isolated an all alkynyl protected Au₁₄₄(2-FC₆H₄C≡C)₆₀ nanocluster.⁵⁹⁰ Two 144-Au clusters exhibit the same spherical four-shell metal core architectures, abbreviated as Au₁₂@Au₄₂(Au₃₀@Au₁₂)@Au₆₀@Au₃₀. Taking Au₁₄₄(SCH₂Ph)₆₀ as an example for structural analysis, the innermost shell (shell 1, Fig. 69b) is a hollow 12-atom Mackay v₁-icosahedron, observed in thiolate-protected gold nanoclusters for the first time. Shell 1 is further enclosed by shell 2 of an Au₄₂ Mackay v₂-icosahedron (Fig. 69c), comprising a hollow complete two-shell Mackay icosahedron Au₅₄ (Au₁₂@Au₄₂). The two-shell Mackay icosahedron is encapsulated by an Au₆₀ rhombicosidodecahedral shell, also known as an anti-Mackay icosahedron (Fig. 69d). The three-shell metal kernel is wrapped by the outmost metal-ligand shell (Fig. 69f), consisting of thirty linear staple motifs (Fig. 69e). Also, 30 additional Au atoms in Shell 4 are arranged in an icosidodecahedral manner.

Fig. 69 (a) Crystallographic structure of Au₁₄₄(SCH₂Ph)₈₀. (b) Shell 1: hollow Au₁₂ icosahedron. (c) Shell 2: Au₄₂ Mackay v₂-icosahedron (icosahedron + icosidodecahedron). (d) Shell 3: Au₆₀ rhombicosidodecahedron. (e) S–Au–S linear staple motif. (f) Shell 4: icosidodecahedral arrangement of 30 Au atoms of linear S–Au–S staple motifs.⁵⁸⁹

9. Platonic and Archimedean solids in other metal clusters

Compared with transition metal clusters, the main group metal (Al,^{61,81,82,591,592} Ga,^62,81,82,593 In,⁶³ Sn,^64,594 Pb,⁶⁵ Sb,^595,596 Bi,^66,204–212etc.) clusters that meet the theme of this review are relatively scarce. Herein, we briefly summarize the aluminium (Al) clusters with polyhedral characteristics and some T-symmetry metal-chalcogenide clusters.

9.1 Al clusters

In 1988, Werner Uhl reported the first bi-nuclear complex, [Al₂{CH(SiMe₃)₂}₄], containing Al–Al (metal–metal) bonding.⁵⁹⁷ Subsequently, Al clusters with metal–metal bonding have attracted widespread attention due to their unique structures.^81,82,591 A series of Al clusters with Platonic polyhedral characteristics has been reported, such as Al₄ tetrahedra,^598–600 Al₆ octahedra,^593,601,602 Al₈ cubes (Fig. 70a),^603–609 and Al₁₂ icosahedra (Fig. 70b).^610,611

Fig. 70 (a) Cubic Al₈ cluster, [(AlH)₆(AlNMe₃)₂(CCH₂Ph)₆].⁶⁰³ (b) Icosahedral Al₁₂ cluster, [Al₁₂ⁱBu₁₂]²⁻.⁶¹⁰ (c) Face-centered cubic Al₁₄ cluster, Si@Al₈(AlCp*)₆.⁶¹² (d) ε-Keggin-type Al₁₃ cluster, [AlO₄Al(OH)₁₂(OC₂H₄OH)₁₂], with a truncated tetrahedral metal skeleton.⁶¹³ (e) Crystallographic structure of cationic core–shell Al₂₄ cluster, [Al₂₄(OH)₃₂(PhCOO)₁₂(EtO)₂₄]⁴⁺, with inorganic truncated cubic Al-hydroxy core and calixarene-like organic shell.⁶¹⁴

Great advancements in giant Al clusters were reported by Schnöckel and co-workers. Their works on core–shell structures that satisfy the subject of this review are listed here, including Al₁₄ (Si@Al₈@Al₆, Fig. 70c),^612,615 fullerene-like Al₃₈(AlCp*)₁₂ (Al₅₀),⁶¹⁶ Si@Al₅₆[N(2,6-ⁱPr₂C₆H₃)SiMe₃]₁₂ (Al₅₆, Fig. 71a)⁵⁹² and [Al₇₇(N(SiMe₃)₂)₂₀]²⁻. Al₅₀ has an irregular Al₈ core and a slightly distorted icosidodecahedral Al₃₀ shell, where 12 pentagons on the shell are covered by 12 AlCp* units with a regular icosahedral arrangement, similar to the peripheral framework of Cu₄₃Al₁₂ (Fig. 49).

Fig. 71 (a) Ball-and-stick representation of Si@Al₅₆[N(2,6-ⁱPr₂C₆H₃)SiMe₃]₁₂. (b) Si@Al₃₂ core (Si@Al₄@Al₄@Al₂₄) enveloped in the {Al₆[N(2,6-ⁱPr₂C₆H₃)SiMe₃]₃}₄ (Al₁₂(truncated tetrahedron)@Al₁₂(cuboctahedron)) shell.⁵⁹²

In the center of the Al₅₆ cluster (Fig. 71a), a μ₈-silicon atom is surrounded by 8 Al atoms, forming two distinct Al₄ tetrahedra (Fig. 71b). Its core is surrounded by a truncated octahedral shell containing 24 Al atoms. These 33 atoms (1 Si + 32Al) are all members of the kernel (Si@Al₄@Al₄@Al₂₄, Fig. 71b). Four no-adjacent pentagons of the Al₂₄ truncated octahedron are capped by four triangles {Al₆[N(2,6-ⁱPr₂C₆H₃)SiMe₃]₃} in a tetrahedral manner (Fig. 71b). The four Al₃ triangles in the center of {Al₆[N(2,6-ⁱPr₂C₆H₃)SiMe₃]₃} can form a truncated tetrahedron, while 12 additional periphery Al atoms bound by organic ligands can construct a distorted cuboctahedron (Fig. 71b).

In addition to metal clusters with metal–metal bonding, the synthesis and properties of Al–O clusters are also flourishing.^617–619 Since the discovery of the first polyoxyaluminum cluster, Al₁₃, a series of Al–O clusters has been produced via the hydrolysis of Al salt, which is widely used in catalysis, water treatment and other fields.⁵⁹¹ There are five isomers (α, β, γ, δ and ε) in Keggin Al₁₃ clusters.^591,620 In 2018, He and Sun et al. reported the preparation of a glycol-coordinated ε-Keggin-type Al₁₃ cluster, [AlO₄Al(OH)₁₂(OC₂H₄OH)₁₂]Cl₇ (Fig. 70d).⁶¹³

In recent years, Zhang and Fang's team has made a series of groundbreaking advances in the study of Al-oxo clusters, including the synthesis high-nuclearity Al–O clusters, Al₂₄ (Fig. 70e)⁶¹⁴ and Al₃₂⁶¹⁸ and the discovery, size expansion, supramolecular assembly and the application of iodine capture of wheel-like clusters.^{617,619,621–623} Recently, this group reported the preparation of a series of water-stable porous Al₂₄ crystalline materials ([Al₂₄(OH)₃₂(PhCOO)₁₂(EtO)₂₄]⁴⁺, Fig. 70e), which remove trace iodine from I₂/KI aqueous solution effectively (down to 4 ppm concentration). The structure of Al₂₄ shows an inorganic and organic “core–shell” feature and two-tier cavity arrangement. The inorganic “core” Al₂₄(OH)₃₂ is a rare Archimedean solid (truncated cube), consisting of eight Al₃ trigons and six Al₈ octagons. The organic “shell” is a large calixarene-like annular cavity with 6 face centers. Although there are reports of 24 metal-centered truncated cube-like metal–organic cages,^624–631 this is the first metal-oxo cluster possessing a pure inorganic metal framework.

9.2 Supertetrahedrons in metal chalcogenides

Among the clusters with polyhedral features introduced above, tetrahedral or supertetrahedral clusters (M₄, M₁₀, and M₂₀) appear frequently, which exist in various metal clusters.^632–640 They can all be represented by T_n, where T represents a tetrahedron and n is the number of metal layers in the cluster or the number of {ME₄} (E = group 16 element, O, S and Se) tetrahedra along each edge.^{445,632–650} The main-group metal (Ga, In, Ge, and Sn) chalcogenides and their heterometallic doped counterparts based on supertetrahedral clusters have been widely explored due to their interesting properties, such as ion exchange and fast-ion conductivity.⁶³⁵ The supertetrahedral T₆ clusters are the largest reported in this category.⁶⁵¹ A T₁ unit is a single tetrahedron like InS₄⁵⁻, whilst the T₂ cluster possesses an adamantane-like core, whose metal corners are capped by four non-metal atoms (Fig. 72a).⁶⁴¹ Alternatively, T_n (n = 3–6) can be described as the self-assembly of corner-sharing tetrahedra (Fig. 72a–d, T₂: 4 {ME₄};⁶⁴¹ T₃: 10 {ME₄};^{642,647,648,652} T₄: 20 {ME₄};⁴⁴⁵ T₅: 35 {ME₄}^643,644,648 and T₆: 56 {ME₄}⁶⁵¹). These supertetrahedral clusters can be assembled into a network structure or larger clusters (Fig. 72e).⁶⁴⁵

Fig. 72 (a) T₂ supertetrahedron, In₄S₁₀⁸⁻.⁶⁴¹ (b) T₃ supertetrahedron, In₁₀S₂₀¹⁰⁻.⁶⁴² (c) T₄ supertetrahedron, Mn₄In₁₆S₃₅¹⁴⁻.⁶⁴³ (d) T₅ supertetrahedron, Cu₅In₃₀S₅₄Cl₂¹⁵⁻.⁶⁴⁴ (e) Cd₁₆In₆₄S₁₃₄⁴⁴⁻ supertetrahedron with a large cavity.⁶⁴⁵

10. Synthesis and structural prediction of giant ‘star’ clusters with Platonic and Archimedean solids

10.1 Summary and consideration for the synthesis of giant ‘star’ clusters

Most of the highly symmetrical metal clusters described in this review were prepared via wet chemistry and their synthetic methods vary widely due to the variety of different types of metal clusters involved (I: metal clusters with metal–anionic cores; II: metal clusters with metal cores (with metal–metal binding); III: metal clusters with metal–anionic cores and metal–metal bonds). Thus, the deliberation and selection of the organic/inorganic ligands, metal sources, and reaction conditions (metal ion/ligand ratio, pH, solvents, anionic templates, reducing agents, etc.) play crucial roles in the synthesis of metal clusters. Due to the unpredictability of the inner metal core, based on the relevant typical cases mentioned in this review, here we mainly discuss the effect of ligands and anionic templates on the construction of high-symmetry metal clusters with Platonic and Archimedean solid features. In this section, we attempt to summarize the common structural features and sources of high symmetry in metal clusters, especially high-nuclearity metal clusters with core–shell structures. It should be noted that we will not discuss the synthesis of SBB-based multimeric structures here, mainly giant inorganic POMs.

The formation of metal clusters (especially high-nuclearity) involves the organization of a large number of disordered metal ions into a complex aggregate in an ordered manner. Undoubtedly, a suitable shell-ligand is crucial, or even the most important, for the construction of high-symmetry metal clusters with a metal core or metal–anionic core. This directly affects the arrangement of the outermost metal atoms. In many cases where the metal cores exhibit polyhedral characteristics, the framework of the outermost metal atoms loses its regularity and symmetry due to the surface ligands (such as Ag₄₄ (Fig. 60), and Au₂₇₉(SPh^tBu)₈₄ with a cuboctahedron Au₁₄₇ kernel in an fcc arrangement (Au@Au₁₂(v₁-cuboctahedron)@Au₄₂(v₂-cuboctahedron)@Au₉₂(v₃-cuboctahedron)), and irregular Au₁₃₂(SPh^tBu)₈₄ shell). The ligands (mainly organic) distributed on the surface of the high-nuclearity metal clusters listed in the text are mainly divided into two categories, as follows:

(1) Simple or highly symmetric ligands with little or no geometrical restrictions in the inner metal cores or metal–anionic cores, e.g., X⁻ (X = F (Fe₁₃, Fig. 32a), Cl (Pu₃₈, Fig. 24a), Br (Fe₃₄, Fig. 32), I (super-octahedron Pb₁₈I₄₄⁸⁻)⁶⁵); E²⁻ (E = O, S or Se); CO (metal carbonyl clusters, such as Ni₃₂C₆ (Fig. 37), Ag₁₆Ni₂₄ (Fig. 38), Pt₁₄Ni₂₄, Pt₆Ni₃₈ (Fig. 39), Pd₅₅, Pd₁₄₅, (PtPd)₁₆₅ (Fig. 40), Ag₁₃Fe₈ (Fig. 65)); H₂O (Mo₆₄Ni₈Ln₆ (Fig. 10)); RO⁻ (R groups mainly are simple alkyl, such as –CH₃ (Mo₂₄ (Fig. 8e), Zr₁₃, Hf₁₃ (Fig. 26)), –CH₂CH₃ (Fe₁₉ (Fig. 32)), and –CH(CH₃)₂ (Ti₄₂, Fig. 25)); R′COO⁻, where the R′ groups can be simple (–H, Fe₆₄; –CH₃ (Nd₁₀₄ (Fig. 43), Ln₂₄Zn₄ (Fig. 44), Gd₃₀Co₁₂) or phenyl moieties; R′′SO₃⁻/R′′PO₃²⁻/R′′AsO₃²⁻ inducing the formation of a regular M₃ triangle (Ce₁₂V₆ (Fig. 6e), Mn₁₃ (Fig. 27)) or M₆ hexagon motifs (Cl₆@V₂₄As₈ (Fig. 7), Ag₈₉ (Fig. 58)); cyclopentadiene and its derivatives (Cp* in Cu₄₃Al₁₂ (Fig. 49), Al₁₄ (Fig. 70)); pyridine (Fe₁₇, Fe₃₄ (Fig. 32); Ce₃₈) and its derivatives (pdol²⁻ in Mn₂₆ (Fig. 28b)); simple polycarboxylic acid (such as IDA²⁻ in La₂₀Ni₃₀ (Fig. 45) and Gd₉₆Ni₆₄, MIDA²⁻ in Gd₁₅₈Co₃₈Cl₁₂ (Fig. 46)), polyol ligands (such as triethanolamine in Mn₂₀ (Fig. 28a) and Cu₁₇Mn₂₈ (Fig. 31a); 2-(hydroxymethyl)phenol in Mn₄₉ (Fig. 30)), and oxygen-rich ligands containing carboxylic acids and hydroxyls (dmp⁻ in Gd₆₀ (Fig. 42)); simple thiolate (ⁱPrS⁻ (Ag_xTi₄ (x = 12, 24, 36, 42, Fig. 51), Ag₇₀ (Fig. 57)), ^tBuS⁻ (V₁₂@Ag₃₀, Fig. 52; Ag₄₈, Fig. 53; Ag₉₀, Fig. 54), CyS⁻ (Ag₁₉₂, Fig. 55; Ag₈₉, Fig. 58), thiophenol and its derivatives (^tBuPhS⁻: Ag₃₇S₄, Ag₄₂S₄ in Fig. 50; 2,4-SPhMe₂: Ag₄₀ in Fig. 59b, (AuAg)₂₆₇ in Fig. 64; 3,4-SPhF₂: Ag₁₄, Ag₆₃ and Ag₁₀₀ in Fig. 59); phosphine (PEt₃ in Ni₉Te₆P₈ (Fig. 36a); P^tBu₃ in Au₂₀ (Fig. 66); PPh₃ in Ag₂₉ (Fig. 62), Ag_x (x = 14, 40, 46, 63, 100, Fig. 59), Au₉ (Fig. 66), Au₁₀₈ (Fig. 68)); and alkynyl ligands (^tBuC≡C⁻ in Ag₃₈Cl₆ and Ag₄₃ (Fig. 52c)).

(2) Ligands inducing specific structures. For example, the linear-like multidentate organic ligand MebptpO²⁻ with seven coordination atoms on the same side in Ln₂₀ (Fig. 41), thiacalix[4]arene and its derivatives (p-tert-butylthiacalix[4]arene (H₄TCA), inducing the formation of regular M₄ square modules in Co₂₄, Co₃₂ (Fig. 35) and other giant clusters).³³⁶

In addition to shell-ligands, the important role of the anions in the formation, passivation and protection of the metal core cannot be ignored.⁷⁶ According to most of the examples in this review, it is not difficult to find that anions as templates exhibit unique advantages in the control of the structure of highly symmetrical nanoclusters and the preparation of giant metal nanoclusters.

The anion templates can be roughly divided into four categories according to their shape including simple spherical (like halogen anions, C⁴⁻, Si⁴⁻, S²⁻, and Se²⁻), triangular (NO₃⁻ and CO₃²⁻), tetrahedral-shaped (ClO₄⁻, SO₄²⁻, CrO₄²⁻, MoO₄²⁻, WO₄²⁻, PO₄³⁻, AsO₄³⁻, SiO₄⁴⁻, etc.) and complex POM anions ({V₁₂O₃₄}¹⁰⁻ in V₁₂@Ag₃₀ (Fig. 52)).⁶⁵³ The role of anions is mainly reflected in three aspects, as follows: (1) anions as structure-directing agents can influence and control the shape and size of the structures, especially in small clusters (α-Keggin POMs (Fig. 5), Cl@Cu₆Ag₈ (Fig. 47b)) and POM-templated high-nuclear Ag clusters.⁶⁵³ (2) Anions can effectively stabilize the metal core and balance the local positive charges from the metal cations. (3) Anions with intrinsic properties can improve or endow cluster functionality.

The progress over the past few decades has witnessed the transition from the initial single anion template strategy for the preparation of small metal clusters (XO₄-templated α-Keggin POMs, Cl@Cu₆Ag₈ (Fig. 47b), Cl@Ag₁₄, SO₄-templated Co₂₀-1 (Fig. 34), etc.) to the now multiple homo- (e.g., 6Cl⁻ in Cl₆@V₂₄As₈ (Fig. 7) and Mn₁₃ (Fig. 27); 12Cl⁻ in Gd₅₀ (Fig. 42); 4S²⁻ in Ag₃₇S₄, Ag₄₂S₄ (Fig. 50), and Ag₇₀ (Fig. 57); 16S²⁻ in Ag₈₉ (Fig. 58); 6C⁴⁻ in Ni₃₂C₆ (Fig. 37), etc.) or hetero-anion templates (e.g., 12Cl⁻ + 90CO₃²⁻ in Gd₁₅₈Co₃₈Cl₁₂ (Fig. 46), MoO₄²⁻ + 6S²⁻ in Ag₄₈ (Fig. 53), 8PO₄³⁻ + 6S²⁻ in Ag₉₀ (Fig. 54), and 22SO₄²⁻ + 36S²⁻ in Ag₁₉₂ (Fig. 55), etc.) approach to achieve the preparation of high-nuclearity metal clusters. Here, we mainly discuss the role of the anions in the construction of regular metallic polygons and polyhedrons in local space (Fig. 73), and the size expansion of metal-oxo/hydroxide/chalcogenide clusters. In the introduction of Ln (Fig. 42) and Ag clusters (Fig. 54), the construction of topological polygons and polyhedra based on anionic templates is briefly discussed. The polygons and polyhedra constructed with different shapes of anions (sphere, triangle, and tetrahedron) as templates (Fig. 73) are summarized in detail here. With the exception of octagons and decagons, almost all the faces (trigon, square, pentagon, and hexagon) used to form a Platonic or Archimedean solid are covered here (Table 1). In addition to the polygons directly induced by anions or surface ligands, more are formed during the assembly of multiple small modules in high-nuclearity metal clusters.

Fig. 73 Polygonal motifs (trigon, square, pentagon, and hexagon) or polyhedral modules (tetrahedron, octahedron, cube, and cuboctahedron) constructed from different types of anion templates. Initial examples: O@Mo₃, Mo₆₀; O@Co₄, [Co₈(μ₄-O)(8-hydroxyquinoline)₁₂]²⁺;³²⁰ O@Nd₄, Nd₁₀₄; O@V₆, V₆Rh₄; Cl@V₄, Cl₆@V₂₄As₈; Cl@Gd₅, Gd₅₀; S@Ag₆, Ag₁₉₂; S@Ag₈, Ag₉₀; Si@Al₈, Al₁₄; CO₃@Gd₅, Gd₂₀; CO₃@Gd₆, Gd₆₀; SO₄@Ce₃, Ce₁₂V₆; MoO₄@Co₆, Co₂₄; SO₄@Ag₉, Ag₁₉₂; SO₄@Ag₁₂, Ag₁₉₂; and SO₄@Ag₁₂, Ag₉₀.

Notably, most of the anion-templated metal-containing clusters surveyed in this review are kinetic products rather than thermodynamic ones, and the incorporation of anions in many cases is inadvertently an accidental result rather than rationally designed.^654–656 In recent years, with some exploration on the role of various anion templates in directing the multi-step assembly process of high-nuclearity metal clusters, the guiding role of the homo-/hetero-anions in the “dynamic” process becomes clearer, which is not limited to the somewhat “static” role for directing structure or supporting construction.^{387,393,394,501,516,657} Recently, the study by Zheng and Feng provided a rare groundbreaking example, i.e., how to generate a large target–lanthanide hydroxide cluster (Ln₆₀ cluster with an individual sodalite cage structure, Fig. 42d) through hetero-anion-guided rational stepwise assembly (Fig. 74).³⁹³ A reasonable “dynamic” route starting from the Er(ClO₄)₃ metal salt to Er₆₀ ([Er₆₀(μ₆-CO₃)₈(μ₄-I)₇(μ₂-NO₃)₁₂(μ₃-OH)₉₆(His)₂₄(H₂O)₃₆](ClO₄)₅I₄-(NO₃)₁₆, H-His = L-histidine) was outlined (Fig. 74a) by capturing three key intermediates (Er₁₂ ([Er₁₂(μ₄-I)₂(μ₃-OH)₁₆(His)₈(H₂O)₂₀](ClO₄)₁₀), Er₃₄ ([Er₃₄(μ₆-CO₃)₂(μ₅-CO₃)₂(μ₄-I)₂(μ₃-OH)₄₈(His)₂₀(H-His)₈(H₂O)₂₄](ClO₄)₁₅I₉), and Er₄₈ ([Er₄₈(μ₆-CO₃)₄(μ₅-CO₃)₃(μ₄-I)₄(μ₂-NO₃)₃(μ₃-OH)₇₂(His)₂₄(H-His)₆(H₂O)₂₄](ClO₄)₁₁I₆(NO₃)₁₀) with the guidance of I⁻, I⁻/CO₃²⁻, I⁻/CO₃²⁻/NO₃⁻, respectively, Fig. 74b) in this process. The simplified geometries based on the cubane-like [Er₄(μ₃-OH)₄] subunits of the lower-degree Er₁₂, Er₃₄, and Er₄₈ represent 1/6, 1/2, and 3/4 of the complete sodalite solid of Er₆₀, respectively.

Fig. 74 (a) Synthetic strategies (stepwise assembly (I → II → III → IV) and direct synthesis (I, V, VI, VII)) of the target Er₆₀ cluster and three intermediates (Er₁₂, Er₃₄ and Er₄₈). (b) Positions of anion templates (I⁻ and CO₃²⁻) in the lanthanide hydroxide Ln_n(μ₃-OH)_m frameworks of Er₁₂, Er₃₄, Er₄₈ and Er₆₀.³⁹³

In addition to the structural template effect, the strategy of intercalating a large number of anions has shown increasing advantages in the construction of high-nuclearity clusters. The typical examples include the largest atomically precise metal cluster [Ag₄₉₀S₁₈₈(SC₅H₁₁)₁₁₄]⁵⁹ and the largest 3d–4f heterometallic cluster Gd₁₅₈Co₃₈Cl₁₂ (Fig. 46). The progress of the rich structural variety of Cu and Ag chalcogenide clusters was reviewed by Fenske,¹¹ and the preparation of clusters from polynuclear to nanosized molecules consisting of hundreds of metal atoms can be achieved with the introduction of S²⁻ or Se²⁻ anions.

Furthermore, two representative examples are given to introduce the role of anions in the structural expansion (Fig. 75). Gd₂₀, La₂₀Ni₃₀ and Gd₁₅₈Co₃₈Cl₁₂ have similar Ln₂₀(μ₅-NO₃/CO₃)₁₂ kernels, while the number of metal atoms and size of their clusters differ greatly (Fig. 75a–c). In contrast to Gd₂₀ (Fig. 75a), the expansion (20 → 50) of La₂₀Ni₃₀ (Fig. 75b) originates from the introduction of Ni-IDA metallo-ligands (marked in blue). Gd₁₅₈Co₃₈Cl₁₂ with nuclearity of 196 (Fig. 75c) is a perfect product of the ‘metallo-ligand-controlled hydrolysis’ and ‘multi-anions-template’ strategies. The chloride ions outside the Gd₂₀(μ₅-CO₃)₁₂ kernel construct 12 anion (OH⁻ and CO₃²⁻)-rich cavities, which greatly balance the charges from the Gd^III cations and grow a Gd–OH–CO₃ layer outward (marked in sky blue, Fig. 75c). Meanwhile, the metallo-ligands (Co-MIDA and Gd-MIDA) well-stabilize the total cluster structure. Ag₄₈, Ag₉₀, and Ag₁₉₂ demonstrate the role of anions in the core expansion and epitaxial growth of Ag clusters (Fig. 75d–f), respectively. With the same octahedral arrangement of six S²⁻ anions, the expansion of the core and increase in the size of the peripheral silver shell are realized by the insertion of more tetrahedral-shaped anions (from one MoO₄²⁻ in Ag₄₈ (Fig. 75d) to eight PO₄³⁻ in Ag₉₀ (Fig. 75e)). In contrast to Ag₉₀ with a similar core ([Ag₃₀(PO₄)₈S₆], Fig. 75e), the expansion (90 → 192) of Ag₁₉₂ (Fig. 75f) is attributed to the intermediate anionic layer [Ag₃₀(SO₄)₂₈S₁₆] located between the [Ag₃₀(SO₄)₈S₆] core and surface shell of [Ag₁₃₂(CyS)₆₆(NO₃)₁₂].

Fig. 75 (a–c) 4f and 3d–4f (transition-lanthanide) metal clusters (a, Gd₂₀; b, La₂₀Ni₃₀; and c, Gd₁₅₈Co₃₈Cl₁₂) containing an Ln₂₀(NO₃/CO₃)₁₂ inner core. With the introduction of anions and peripheral metallo-ligands, the size of the clusters expanded successively. (d–f) Silver clusters templated by spherical and tetrahedral-shaped anions together (d, Ag₄₈; e, Ag₉₀; and f, Ag₁₉₂). Organic ligands and the central part (Ag₆@(SO₄)₈@S₆) in Ag₁₉₂ (f) have been omitted for clarity.

The specific classical and various innovative synthesis methods (for example, ‘ligand-controlled hydrolysis’ strategy for metal-oxo/hydroxide clusters; ‘multi-anion-template’ strategy; ‘building-block’ method; ‘reduction of metal precursor’ for metalloid clusters; ‘disproportionation reaction’ for main group metal clusters such as Al and Ga; and hetero-metal strategy) of metal clusters were summarized and discussed in detail in the above main text and relevant reviews, and will not be repeated here.

10.2 Structural prediction of new ‘star’ clusters with Platonic and Archimedean solids

Based on the close-packing principle (such as face-centre cubic (fcc) close packing, hexagonal close packing (hcp) and body-centered cubic (bcc) close packing) of the equal sphere-like elements (metal atoms M^0/n+, metal-oxo MO_x^m−, or SBBs, Fig. 1), and the relevant examples mentioned herein, and the geometric relationships between different Platonic and Archimedean solids (Fig. 2 and 3), we attempt to make some predictions for non-synthesized ‘star’ molecules that satisfy the topological features of Platonic and Archimedean solids. There are mainly two cases, as follows: (1) the existing metal frameworks of metal clusters may also appear in other systems in the future and (2) some reasonable speculations about unknown structures with polyhedron features are made based on the current examples.

The first type of structure prediction is based on the large number of successful examples, where the family of α-Keggin-type molecules (XM₁₂, M = V, Mo, W, Nb, etc.Fig. 76) is a typical representative. A variety of elements as the central atoms (X), from the original main group atoms (B, Al, Si, Ge, P, and As) to the later transition metals (Fe, Co, and Cu), and more recently the alkali/alkaline earth metals (Na, Ca), was successfully encapsulated in inorganic structures. Besides traditional inorganic POMs, similar α-Keggin structures have been found in metal clusters protected by organic ligands (U₁₃,¹⁸⁶Fe₁₃,²⁸⁷Zr₁₃,²²⁷Hf₁₃,²²⁸ Mn₁₃,²⁶⁵ and Eu₁₃,³⁸³Fig. 76) and Keggin-type cores also appear in larger metal clusters (such as Co₂₀-1, Fig. 34). Other isomers (like γ-Keggin AgSn₁₂)⁶⁵⁸ have also been reported. Under suitable synthetic conditions and the protection of ligand, more Keggin and existing polyhedral structures based on other metals will be synthesized in the near future. Similarly, more polyhedral cores (such as v₁-icosahedron Ag₁₂/Ag₁₃, Cu₁₂/Cu₁₃, and v₂-icosahedron Ag₅₅ and Au₅₅) appearing in large metalloid clusters can theoretically be isolated separately based on the effective protection of ligands, and there are many examples to prove this (such as Pd₅₅/Pd₁₄₅/(PdPt)₁₆₅ (Fig. 40) and icosahedron Au₁₃ (Fig. 67 and 69)).

Fig. 76 Polyhedral mode of α-Keggin-type metal clusters: classical inorganic POM, {Fe₁₃(μ₄-O)₄F₂₄(μ₂-OMe)₁₂} (Fe₁₃), Zr₁₃(μ₄-O)₈(OCH₃)₃₆ (Zr₁₃), Hf₁₃(μ₄-O)₈(OCH₃)₃₆ (Hf₁₃), {Mn₁₃O₆(OH)₂(OMe)₄(bemp)₆} (H₃bemp = 2,6-bis[N-(2-hydroxyethyl)iminomethyl]-4-methylphenol),²⁶⁵ and {Eu₁₃O₈(Ph₄Si₄O₈)₆(DMF)₁₂}.³⁸³

The second category is to predict unknown structures based on existing structural features, which are undoubtedly quite difficult, and only a general trend is discussed here. As the number of metal atoms increases, to adjust the charge balance inside the metal-containing aggregates, the structures generally have the following three forms: (1) the presence of a large number of anion-rich holes (such as Gd₁₅₈Co₃₈Cl₁₂, Fig. 46 and 74; Ag₁₉₂, Fig. 55 and 74); (2) hollow structures with more surfaces, which facilitate the coordination of more negatively charged ligands (such as Mo₂₄, Fig. 8; Mo₈₄, Mo₁₀₂, Mo₁₃₂, Mo₂₄₀, Fig. 11; Ti₄₂, Fig. 25; and Ag₁₈₀⁵¹⁰); and (3) the internal metal atoms are reduced to form a metalloid core. The first approach is based on the use of multi-anions, and anions with high symmetry help promote the local symmetry, thereby affecting the molecular symmetry.

The second case is especially common in aggregates containing more than 50 metal atoms, and in addition to cage-like structures, there are also wheel-like (such as Ce₇₀,³⁶⁷ Mn₈₄,³⁹ and Gd₁₄₀⁵²), tubular (such as Dy₇₂),³⁶⁸ and various hollow structures (such as Ni₃₆Gd₁₀₂).⁵⁴ In the case of cage-like structures based on M-O/OH/S (non-metal organic cage), their constituent units have certain regularity. For example, U₂₀ (Fig. 23a), Gd₂₀ (Fig. 42a), Ti₄₂ (Fig. 77a), U₆₀ (Fig. 23c), Mo₁₀₂ (Fig. 77b), and Mo₁₃₂ (Fig. 77c) can be regarded as different forms of an I_h symmetry assembly of 12 pentagon basic units (M@M₅ or M₅). We believe that these pentagon-motifs can also be used to construct cage structures with hitherto unknown metal frameworks featuring Platonic and Archimedean solids and may be a species (M₆₀ (12 M₅) or M₇₂ (12 M@M₅) with rhombicosidodecahedron or snub dodecahedron feature) between Ti₄₂ (Fig. 77a) and Mo₁₀₂ (Fig. 77b) or a larger hollow monolayer structure (Fig. 77d).

Fig. 77 (a–c) {M@M₅}-based metal-oxo cages (a, Ti₄₂; b, Mo₁₀₂; and c, Mo₁₃₂). (d) Platonic and Archimedean solids with 12 pentagon faces. From top to bottom: dodecahedron, icosidodecahedron, truncated icosahedron, rhombicosidodecahedron, and snub dodecahedron.

The third way is fully reflected in the synthesis of giant gold and silver clusters in recent years. Compared with small clusters, surface ligands have less effect on the arrangement of the metal atoms inside high-nuclearity clusters with multi-shell structures, showing the same packing form (fcc) as bulk materials. In many cases, the metalloid kernels have high symmetry, but the metal-ligand layer symmetry on the surface may be reduced. Among the many reported structures, there are several families of Ag/Au clusters with regular size expansion, such as tetrahedral (Fig. 50, 51, 57 and 58) and cubic shape (Fig. 59 and 78). There are obviously many missing members of these metal cluster families, and the cubic silver clusters with O_h-symmetry metal cores (Fig. 59 and 78) are taken as an example for structural analysis and prediction. Ag₁₄ (Ag₆@Ag₈), Ag₄₆ (Ag₆@Ag₈@Ag₂₄@Ag₈), and Ag₁₀₀ (Ag₆@Ag₈@Ag₂₄@Ag₆@Ag₂₄@Ag₂₄@Ag₈) with the same Ag₆ in their center clearly show the manner and regularity of the fcc Ag core growth. The appearance of Ag₄₀ (Ag₈@Ag₂₄@Ag₈) suggests that the internal metal atomic arrangement can be tuned under the same symmetry. The successful synthesis of Ag₆₃ (Ag@Ag₁₂@Ag₆@Ag₂₄@Ag₁₂@Ag₈) proves that regular growth with the cuboctahedron Ag₁₃ (Ag@Ag₁₂) as the core can also be achieved. In the reported Au clusters, some structures (Au₂₀ (Au₁₂@Au₈, Fig. 66b) and Au₂₇₉(SPh^tBu)₈₄ (Au@Au₁₂@Au₄₂@Au₉₂@Au₅₄@Au₄₈@Au₃₀)) with cuboctahedral kernels were observed. Thus, we expect that Ag clusters with similar cuboctahedral kernels will be successfully synthesized.

Fig. 78 Structure (analogous [Ag(SR)₃(PR′₃)] motifs) of equal vertices of cubic box-like Ag nanoclusters: (a) [Ag₁₄(3,4-SPhF₂)₁₂(PPh₃)₈] (Ag₁₄); (b) [Ag₄₀(2,4-SPh(CH₃)₂)₂₄(PPh₃)₈] (Ag₄₀) and [Ag₄₆(2,5-SPh(CH₃)₂)₂₄(PPh₃)₈]²⁺ (Ag₄₆); (c) [Ag₆₃(3,4-SPhF₂)₃₆(PⁿBu₃)₈]⁺ (Ag₆₃); and (d) [Ag₁₀₀(3,4-SPhF₂)₄₈(PPh₃)₈]⁻ (Ag₁₀₀). Metal-framework with eight [Ag(S/SeR)₃] motifs arranged in cubes around the periphery: (e) [Ag₂₁{Se₂P(OEt)₂}₁₂]⁺ (Ag@Ag₁₂@Ag₈@(Se₃)₈, Ag₂₁); (f) Ag₃₃(C₂B₁₀H₁₀S₂)₁₂ (Ag@Ag₁₂@Ag₂₀@(S₃)₈, Ag₃₃) and (g) [Ag₃₉(SPhF₅)₂₄(PPh₃)₈]²⁻ (Ag@Ag₁₂@Ag₁₈@(Ag(SPhF₅)₃(PPh₃))₈, Ag₃₉). Some Ag⋯Ag interactions and organic ligands have been omitted for clarity. (h) Intermediate vertex-deficient dodecahedral Ag₁₈ shell of Ag₃₉.⁶⁵⁹

In addition to the regularity of the inner metal kernels, they also have similar eight [Ag(SR)₃(PR′₃)] vertices at the periphery (Fig. 78a–d), which are also observed in some non-O_h-symmetry clusters (Fig. 78e–g). We present the bold speculation that in the future, Platonic and Archimedean polyhedra belonging to different symmetry groups will exist in the same multi-shell Ag cluster, which may be the following structures: Ag₂₈ (Ag₁₂(icosahedron)@Ag₈(cube)@Ag₈(cube)), Ag₂₉ (Ag(center)@Ag₁₂(icosahedron)@Ag₈(cube)@Ag₈(cube)), Ag₄₀(Ag₁₂(icosahedron)@Ag₂₀(dodecahedron)@Ag₈(cube)) and Ag₄₁(Ag(center)@Ag₁₂(icosahedron)@Ag₂₀(dodecahedron)@Ag₈(cube)). Some recent experimental results also confirm the rationality of our conjecture. Zang's group synthesized a distorted cubic silver cluster, Na₂[Ag₃₉(SPhF₅)₂₄(PPh₃)₈] (Ag₃₉, Fig. 78g), co-protected by phosphine (PPh₃) and thiolate (HSPhF₅ = pentafluorobenzenethiol).⁶⁵⁹ The metal arrangement (Ag(center)@Ag₁₂(icosahedron)@Ag₁₈@Ag₈(cube)) of Ag₃₉ is consistent with our speculation, but the Ag₁₈ shell reveals vertex vacancy defects, deviating from the dodecahedron constructed by 20 vertices (Fig. 78h).

11. Conclusions and future prospective

In this review, we discussed more than 100 examples of high-nuclearity metal clusters with metal frameworks featuring Platonic and Archimedean solids. It is not difficult to recognize that some polyhedral arrangements, for example, five Platonic solids (tetrahedron, cube, octahedron, dodecahedron, and icosahedron) and most parts of 13 Archimedean solids (truncated tetrahedron, cuboctahedron, truncated cube, truncated octahedron, rhombicuboctahedron, icosidodecahedron, and rhombicosidodecahedron), appear in these clusters. The discovery of these highly symmetric metal clusters will undoubtedly guide synthetic chemists to explore new structures and giant molecules with similar characteristics, not only to pursue structural aesthetics, but more importantly to create materials with unique properties and useful applications.

However, it should be noted that the characteristics (including nucleus, shape, and configuration) of the crystal structures of metal clusters, especially giant metal clusters are mostly unpredictable and frequently the result of serendipitous discovery. Their acquisitions are affected by a large number of factors, which may depend on the specific structural characteristics of the surface ligands, reaction conditions and other factors. Thus, high-throughput synthetic workstations combined with machine learning and the application of artificial intelligence techniques in cluster synthesis will be helpful for the rapid development of this field. Although there are great difficulties in the synthesis process, the research on these clusters has provided an in-depth understanding of the possible mechanisms that cause their assembly, the establishment of bridges from atoms to nanoparticles, and new systems to study new physical and chemical properties. These interesting polyhedral cluster structures have shown excellent properties and applications in many fields, but whether similar polyhedral structures have parallel properties and the relationships between different polyhedral structures are not clear. Many studies have shown that small cubane-like M₄ clusters have similar properties, but other high-nuclearity polyhedral metal clusters have not been studied to date.

We summarized and analyzed the metal clusters from the perspective of polyhedra (Platonic and Archimedean solids), hoping that more researchers will pay attention to these fascinating molecules with high symmetry. Simultaneously, it is hoped that the polyhedral approach for describing and understanding the complex structures can bring some new insights into related research, and stimulate research on higher-level relationships, such as the influence of the electronic properties on the geometric structure and pointing out the possible systematics in the dependence of the structure on metal and ligand.

Conflicts of interest

There are no conflicts to declare.

Acknowledgements

This work was supported by the National Natural Science Foundation of China (No. 92061201, 21825106, U21A20277, 21975065) and Zhengzhou University.

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