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Summary The design principle of supramolecular architectures through a bottom-up strategy has always been guided by polyhedra such as Platonic solids, Archimedean solids, and prisms (antiprisms) with high symmetry. Non-uniform convex polyhedra with regular faces, i.e., Johnson solids, are rarely constructed experimentally although their topologies have been predicted through mathematical theory. Herein, self-assembly of a giant triangular orthobicupola [Zn24612], one of 92 Johnson solids J27, has been achieved from the coordination between 24 Zn2+ and 12 tetrakis-terpyridine building blocks 6 in DMF (N,N-Dimethylformamide). Different from classical Oh or Td symmetric cuboctahedron, the structure of this triangular orthobicupola revealed D3 symmetry which has been unambiguously determined by synchrotron X-ray analysis, NMR, and mass spectrometry. Surprisingly, the self-assembly process has been found to be solvent dependent. The same Zn2+ and tetrakis-terpyridine ligand 6 assembled into another pentagonal prism with the composition of [Zn20610] when using methanol as the reaction solvent.

The synthesis of molecular cages consisting of fully fused, π-conjugated rings is rare due to synthetic challenges including preorganization, large strain, and poor solubility. Herein, we report such an example in which a tris-2-aminobenzophenone precursor undergoes acid-mediated self-condensation to form a truncated tetrahedron, one of the 13 Archimedean solids. Formation of eight-membered [1,5]diazocine rings provides preorganization and releases the strain while still maintains weak π-conjugation of the backbone. Thorough characterizations were performed by X-ray, NMR, and UV-vis analysis, assisted by theoretical calculations. The cage exhibits a rigid backbone structure with a well-defined cavity that confines a magnetically shielded environment. The solvent molecule, o-dichlorobenzene, is precisely encapsulated in the cavity at a 1:1 ratio with multiple π···π, C-H···π, and halogen···π interactions with the cage skeleton, implying its template effect for the cage closing reaction. Our synthetic strategy opens the opportunity to access more complex, fully fused, three-dimensional π-conjugated cages.

Silver clusters with more than 2 concentric silver shells are scarce. Here, we enable self-assembly and crystallize SD/Ag192a, a highly symmetric silver chalcogenide cluster (SCC) with 192 silver cations in 6 shells and 136 anionic groups in 14 shells. All but 1 of these 20 concentric shells are Platonic or Archimedean solids. All have octahedral or tetrahedral symmetry and align the maximum number of their 2-, 3-, and 4-fold axes of rotational symmetry, thus identifying the cluster as a Keplerate. A rhombic dodecahedron supershell, formed from the first 3 anionic shells, is the keystone for the entire structure. But, nearly all of the edges in these polyhedral shells are too long to represent bonds. What mechanism of coordination chemistry holds the shells together? Like Na+ ions held electrostatically inside adjacent cube-shaped anionic compartments in a crystal of NaCl, individual Ag+ ions sit inside adjacent octahedron-shaped anionic compartments that fill space. Similarly, like Cl- ions in NaCl, individual anionic groups sit inside adjacent cationic (Ag+) compartments, mostly uniform polyhedra, that also fill space.

Rational control of the self-assembly of large structures is one of the key challenges in chemistry, and is believed to become increasingly difficult and ultimately impossible as the number of components involved increases. So far, it has not been possible to design a self-assembled discrete molecule made up of more than 100 components. Such molecules-for example, spherical virus capsids-are prevalent in nature, which suggests that the difficulty in designing these very large self-assembled molecules is due to a lack of understanding of the underlying design principles. For example, the targeted assembly of a series of large spherical structures containing up to 30 palladium ions coordinated by up to 60 bent organic ligands was achieved by considering their topologies. Here we report the self-assembly of a spherical structure that also contains 30 palladium ions and 60 bent ligands, but belongs to a shape family that has not previously been observed experimentally. The new structure consists of a combination of 8 triangles and 24 squares, and has the symmetry of a tetravalent Goldberg polyhedron. Platonic and Archimedean solids have previously been prepared through self-assembly, as have trivalent Goldberg polyhedra, which occur naturally in the form of virus capsids and fullerenes. But tetravalent Goldberg polyhedra have not previously been reported at the molecular level, although their topologies have been predicted using graph theory. We use graph theory to predict the self-assembly of even larger tetravalent Goldberg polyhedra, which should be more stable, enabling another member of this polyhedron family to be assembled from 144 components: 48 palladium ions and 96 bent ligands.

The spontaneous formation of discrete spherical nanosized molecules is prevalent in nature, but the authentic structural mimicry of such highly symmetric polyhedra from edge sharing of regular polygons has remained elusive. Here we present a novel ball-shaped {(HNEt3)[Ag37S4(SC6H4(t)Bu)24(CF3COO)6(H2O)12]} cluster () that is assembled via a one-pot process from polymeric {(HNEt3)2[Ag10(SC6H4(t)Bu)12]}n and CF3COOAg. Single crystal X-ray analysis confirmed that is a Td symmetric spherical molecule with a [Ag36(SC6H4(t)Bu)24] anion shell enwrapping a AgS4 tetrahedron. The shell topology of belongs to one of 13 Archimedean solids, a truncated tetrahedron with four edge-shared hexagons and trigons, which are supported by a AgS4 Platonic solid in the core. Interestingly, the cluster emits green luminescence centered at 515 nm at room temperature. Our investigations have provided a promising synthetic protocol for a high-nuclearity silver cluster based on underlying geometrical principles.

Buckminsterfullerene (C60) represents a perfect combination of geometry and molecular structural chemistry. It has inspired many creative ideas for building fullerene-like nanopolyhedra. These include other fullerenes, virus capsids, polyhedra based on DNA, and synthetic polynuclear metal clusters and cages. Indeed, the regular organization of large numbers of metal atoms into one highly complex structure remains one of the foremost challenges in supramolecular chemistry. Here we describe the design, synthesis, and characterization of a Ag180 nanocage with 180 Ag atoms as 4-valent vertices (V), 360 edges (E), and 182 faces (F)––sixty 3-gons, ninety 4-gons, twelve 5-gons, and twenty 6-gons––in agreement with Euler’s rule V − E + F = 2. If each 3-gon (or silver Trigon) were replaced with a carbon atom linked by edges along the 4-gons, the result would be like C60, topologically a truncated icosahedron, an Archimedean solid with icosahedral (Ih) point-group symmetry. If C60 can be described mathematically as a curling up of a 6.6.6 Platonic tiling, the Ag180 cage can be described as a curling up of a 3.4.6.4 Archimedean tiling. High-resolution electrospray ionization mass spectrometry reveals that {Ag3}n subunits coexist with the Ag180 species in the assembly system before the final crystallization of Ag180, suggesting that the silver Trigon is the smallest building block in assembly of the final cage. Thus, we assign the underlying growth mechanism of Ag180 to the Silver-Trigon Assembly Road (STAR), an assembly path that might be further employed to fabricate larger, elegant silver cages.

Sulfonates were incorporated into six novel high-nuclearity silver(I) thiolate clusters under the guidance of anion templates varied from S2- , SO42- , α-[Mo5 O18 ]6- , β-[Mo5 O18 ]6- , [Mo2 O8 ]4- , to [Mo4 O14 (SO4 )]6- . Single crystal X-ray analysis revealed that SD/Ag1, SD/Ag3, SD/Ag5, and SD/Ag6 are discrete [S@Ag60 ], [α-Mo5 O18 @Ag36 ], [Mo2 O8 @Ag30 ]2 , and [Mo4 O14 (SO4 )@Ag73 ] clusters, respectively, whereas SD/Ag2 and SD/Ag4 are one-dimensional (1D) chains based on the [SO4 @Ag20 ] and [β-Mo5 O18 @Ag36 ] cluster subunits, respectively. Their silver skeletons are protected exteriorly by thiolates and sulfonates and interiorly supported by diverse anions as templates. Structurally, cluster SD/Ag1 is a typical core-shell structure comprised of an inner Ag12 cuboctahedron and an outer Ag48 shell. The sulfate-templated drum-like Ag20 cluster subunits are bridged by PhSO3- to give a 1D chain of SD/Ag2. Complex SD/Ag3 and SD/Ag4 are spindle-like Ag36 clusters with isomeric [Mo5 O18 ]6- inside, and the latter is further extended to a 1D chain through PhSO3- bridges. A pair of [Mo2 O8 ]4- templated gourd-like Ag30 clusters are dimerized in a head-to-head fashion to form SD/Ag5. Complex SD/Ag6 is the largest cluster in this family and doubly templated by unprecedented [Mo4 O14 (SO4 )]6- anions. Geometrically, the silver shells of SD/Ag1-SD/Ag5 show the polyhedral features of Johnson solids, instead of the usual Platonic or Archimedean solids. Solution behaviors and luminescent properties were also investigated in detail.

Biochemists have coaxed a protein to assemble into a snub-cube-shaped cage. Snub cubes, which have 38 faces—6 squares and 32 equilateral triangles—are one of the 13 Archimedean solids, first descri...

Understanding the self-assembly of small structural units into large supramolecular assemblies remains one of the great challenges in structural chemistry. We have discovered that tetrahedral supramolecular cages, exhibiting the shapes of Archimedean solids, can be self-assembled by hydrogen bonding interactions using tricationic N-donors (1 or 2) in cooperation with water (W). Single crystal X-ray analysis shows that cage (2)4(W)6, assembled in an aqueous solution of cation 2 and KPF6, consists of four tripodal trications linked by six water monomers and resembles the shape of a truncated tetrahedron. Similarly, cage (1)4(W6)4 spontaneously self-assembles in an aqueous solution of cation 1 and NH4PF6 and consists of four tripodal cations and four water hexamers. Here, each of the four (H2O)6 units act as tritopic nodes between three distinct tripodal cations forming a polyhedron similar to the cantellated tetrahedron. These two well-defined cages are assembled via total of 12 and 36 hydrogen bonds, respe...

Supramolecular coordination cages provide unique restricted inner cavities that can be exploited for molecular recognition purposes and catalysis. Their syntheses often involve complex self-organization processes and rely on the identification of preorganized, kinetically stable building units that provide ligand-accessible coordination sites. Here we report a highly effective protocol for the successive buildup of symmetrical nanoscopic polyoxometalate (POM) cages. Our methodology takes advantage of a supramolecular templating effect and utilizes the structure-directing influence of octahedral {X(x)(H(2)O)(6-x)} (X = Br(-), Cl(-); x = 2, 4, 6) assemblies that reside inside the hollow cluster shells and determine the arrangement of di- and tetranuclear vanadate units. The approach allows the preparation of a series of high-nuclearity POM cages that are characterized by {V(16)As(8)}, {V(16)As(10)}, {V(20)As(8)}, and {V(24)As(8)} core structures. In the latter cluster cage, the vanadium centers adopt a truncated octahedral topology. The formation of this Archimedean body is the direct result of the assembly of six square {V(4)O(8)} units that cap the vertices of the encapsulated Platonic {Cl(6)} octahedron. To the best of our knowledge, this {V(24)As(8)} cage is the largest hybrid vanadate cluster reported to date.

We develop an analytical model to predict equilibrium shapes of two-component heterogeneous vesicles or capsules. Using a free energy functional including the bending energies of the two components and line tension contributions, the model describes shape transitions between spherical and polyhedral (faceted) states, complementing and extending results of previous numerical simulations. In the parameter space of relative area fraction, bending modulus ratio, and line tension, a region of polyhedral shapes occurs for weak line tension and large bending modulus ratio and is very robust towards changes in the modeling assumptions. At large enough line tension, the spherical shape fragments into two components. Within the polyhedral region, we compare the energies of all regular and semiregular polyhedra, as well as those of arbitrary prismatic shapes. We find that the largest bending modulus contrasts together with larger line tension favor polyhedra with small face number as optimal shapes. In this region, we also demonstrate the counter-intuitive result that the most symmetric polyhedra are not energetically optimal, with specific Archimedean solids and specific prismatic shapes beating more isotropic (e.g. Platonic) polyhedra. Furthermore, all polyhedra of lowest energy are found to be three-fold coordinated. The shape transition boundary for polyhedra can be computed analytically. The model can be utilized to predict heterogeneous vesicle shapes and to estimate physical properties of the components constituting observed vesicles.

We report star polyhedral gold nanocrystals synthesized by colloidal reduction with ascorbic acid in water at ambient conditions. We identify two distinct classes of star nanocrystals: multiple-twinned crystals with fivefold symmetry, and monocrystals. These respective classes correspond to icosahedra and cuboctahedra, two Archimedian solids, with preferential growth of their {1 1 1} surfaces. Due to this preferential growth, the {1 1 1} faces of the original Archimedean solids grow to become tetrahedral pyramids, the base of each pyramid being the original polyhedral face. By assuming a star morphology, gold nanocrystals increase their proportion of exposed {1 1 1} surfaces, which possess the lowest surface energy among low-index crystallographic planes for FCC crystals. Thus, we propose that the driving force for star nanocrystal formation could be the reduction in surface energy that the crystals experience. Interestingly, icosahedrally derived star nanocrystals possess a geometric morphology closely resembling the great stellated dodecahedron, a Kepler-Poinsot solid. r 2005 Elsevier B.V. All rights reserved. PACS: 61.46.+w; 81.07.Bc; 68.37.Lp; 68.37.Hk

The superconducting (SC) phase in phase-separated (PS) ${\mathrm{K}}_{0.8}\mathrm{F}{\mathrm{e}}_{1.6+x}\mathrm{S}{\mathrm{e}}_{2}\phantom{\rule{0.16em}{0ex}}(0\ensuremath{\le}x\ensuremath{\le}0.15)$ materials is found to crystallize on Archimedean solidlike frameworks, and this structural feature originates from a spinodal phase separation (SPS) at around ${T}_{s}\ensuremath{\approx}540\phantom{\rule{0.28em}{0ex}}\mathrm{K}$, depending slightly on the Fe concentration, as revealed by in situ heating TEM observations and shown in a phase diagram. Two stable phases in ${\mathrm{K}}_{0.8}\mathrm{F}{\mathrm{e}}_{1.6+x}\mathrm{S}{\mathrm{e}}_{2}$ are demonstrated to be the SC ${\mathrm{K}}_{0.5}\mathrm{F}{\mathrm{e}}_{2}\mathrm{S}{\mathrm{e}}_{2}$ and antiferromagnetic (AFM) ${\mathrm{K}}_{0.8}\mathrm{F}{\mathrm{e}}_{1.6}\mathrm{S}{\mathrm{e}}_{2}$. The spinodal waves go along the systematic $[113]$ direction and result in notable lamellar structure as illustrated by strain-field theoretical simulations. The three-dimensional SC framework is constructed with hollow truncated octahedron similar to that discussed for Archimedean solids. Based on this structural model, we can efficiently calculate the volume fraction of the SC phase in this type of PS SC material.

The hydrolysis of Ln(ClO4)3 in the presence of acetate leads to the assembly of the three largest known lanthanide-exclusive cluster complexes, [Nd104(ClO4)6(CH3COO)60(μ3-OH)168(μ4-O)30(H2O)112]·(ClO4)18·(CH3CH2OH)8·xH2O (1, x ≈ 158) and [Ln104(ClO4)6(CH3COO)56(μ3-OH)168(μ4-O)30(H2O)112]·(ClO4)22·(CH3CH2OH)2·xH2O (2, Ln = Nd; 3, Ln = Gd; x ≈ 140). The structure of the common 104-lanthanide core, abbreviated as Ln8@Ln48@Ln24@Ln24, features a four-shell arrangement of the metal atoms contained in an innermost cube (a Platonic solid) and, moving outward, three Archimedean solids: a truncated cuboctahedron, a truncated octahedron, and a rhombicuboctahedron. The magnetic entropy change of ΔS(m) = 46.9 J kg(-1) K(-1) at 2 K for ΔH = 7 T in the case of the Gd104 cluster is the largest among previously known lanthanide-exclusive cluster compounds.

The use of the novel pro-ligand H4L combining the complimentary phenolic oxime and diethanolamine moieties in one organic framework, results in the formation of the first example of a [MnIII 12] truncated tetrahedron and an extremely rare example of a Mn cage conforming to an Archimedean solid.

The spherical capsule skeleton of the host–guest system [{Mo6O19}2−⊂{MoVI72FeIII30O252(CH3COO)20(H2O)92}]4−1a—built up by 12 {(MoVI)MoVI5} type pentagonal units linked by 30 FeIII centers which span the unique icosahedral Archimedean solid, the icosidodecahedron—can now be constructed deliberately and with a simpler composition than before from an acidified aqueous molybdate solution containing the mentioned (virtual) pentagonal units; the encapsulated hexamolybdate—normally not formed in water—is built up in an unprecedented way concomitant with capsule growth, while being directed by the corresponding internal electrophilic surface functionalities.

A covalently closed molecular complex whose double-helical edges have the connectivity of a truncated octahedron has been assembled from DNA on a solid support. This three-connected Archimedean solid contains six squares and eight hexagons, formed from 36 edges arranged about 24 vertices. The vertices are the branch points of four-arm DNA junctions, so each vertex has an extra exocyclic arm associated with it. The construct contains six single-stranded cyclic DNA molecules that form the squares and the extra arms; in addition, there are eight cyclic strands that correspond to the eight hexagons. The molecule contains 1440 nucleotides in the edges and 1110 in the extra arms; the estimated molecular weight for the 2550 nucleotides in the construct is 790 kDa

We consider the possibility that the stability of the Archimedean structure C60 (a truncated icosahedron) is a consequence of the aromaticity of this nonalternant molecule. As the model, we used the structure of C12 in the form of a truncated tetrahedron — the first of the Archimedean solids, having triangular faces in addition to hexagonal faces (rather than the pentagonal faces in the analogous C60 molecule of symmetry Ih). On the basis of calculation of the topological resonance energy of the C12 molecule of Td symmetry, we conclude that the structure with different bond lengths is antiaromatic, while the C12 molecule (whose hexagonal faces are benzene rings of the Kekule type) has a slight degree of aromatic character. MNDO and AMl calculations have shown that these structures for C12 correspond to minima on the potential energy surface, but the structure with different bond lengths has lower energy. We also observe an isomer of C12 having benzene rings of the quinoid type, which corresponds to a minimum on the potential energy surface but is less stable than the structure with equal bonds. Aromatic stabilization is expected, as shown by the calculation for the tetracation C4+ 12 of Td symmetry with virtually equal lengths of all the carbon-carbon bonds.

Modern research in the field of small metallic systems has confirmed that many nanoparticles take Platonic and Archimedean solids related shapes. A Platonic solid looks the same from any vertex, and intuitively they appear as good candidates for atomic equilibrium shapes. A good example is the icosahedral (Ih) particle that only shows {111} faces that produce a more rounded structure. Indeed, many studies report the Ih as the most stable particle at the size range r≤20 A for noble gases and for some metals. In this chapter, we discuss the structure and shape of mono- and bimetallic nanoparticles in the size range from 1–300 nm. First, AuPd nanoparticles (1–2 nm) that show dodecahedral atomic growth packing. Next, in the range of 2–5 nm, we discuss a surface reconstruction phenomenon observed also on AuPd and AuCu nanoparticles. These binary alloy nanoparticles show the fivefold edges truncated, resulting in {100} faces on decahedral structures, an effect largely envisioned and reported theoretically, with no experimental evidence in the literature before. Next, we review a monometallic system (≈5 nm) that we termed the decmon. Finally, we present icosahedrally derived star gold nanocrystals (100–300 nm) which resemble the great stellated dodechaedron, a Kepler-Poisont solid. We conclude that the shape or morphology of some mono- and bimetallic particles evolves with size following the sequence from atoms to the Platonic solids. As the size increases, they tend to adopt Archimedean related shapes and then beyond the Archimedean (Kepler-Poisont) solids, up to the bulk structure of solids.

Cubane (C8H8) and fullerene (C60) are famous cage molecules with shapes of platonic or archimedean solids. Their remarkable chemical and solid-state properties have induced great scientific interest. Both materials form polymorphic crystals of molecules with variable orientational ordering. The idea of intercalating fullerene with cubane was raised several years ago but no attempts at preparation have been reported. Here we show that C60 and similarly C70 form high-symmetry molecular crystals with cubane owing to topological molecular recognition between the convex surface of fullerenes and the concave cubane. Static cubane occupies the octahedral voids of the face-centred-cubic structures and acts as a bearing between the rotating fullerene molecules. The smooth contact of the rotor and stator molecules decreases significantly the temperature of orientational ordering. These materials have great topochemical importance: at elevated temperatures they transform to high-stability covalent derivatives although preserving their crystalline appearance. The size-dependent molecular recognition promises selective formation of related structures with higher fullerenes and/or substituted cubanes.

We provide a summary of our results in three-dimensional, coordination-driven self-assembly based on the directional-bonding methodology, in which the stoichiometric mixing of complementary building blocks, with appropriate, predefined geometries, leads to targeted, nanoscopic cages. Using this motif, we have synthesized high-symmetry ensembles resembling the Platonic solids, such as dodecahedra, and the Archimedean solids, such as truncated tetrahedra and cuboctahedra, as well as other cages, like trigonal bipyramids, adamantanoids, and trigonal prisms. The synthesis and characterization of these compounds is discussed, as is some host-guest chemistry.

We provide a summary of our results in three-dimensional, coordination-driven self-assembly based on the directional-bonding methodology, in which the stoichiometric mixing of complementary building blocks, with appropriate, predefined geometries, leads to targeted, nanoscopic cages. Using this motif, we have synthesized high-symmetry ensembles resembling the Platonic solids, such as dodecahedra, and the Archimedean solids, such as truncated tetrahedra and cuboctahedra, as well as other cages, like trigonal bipyramids, adamantanoids, and trigonal prisms. The synthesis and characterization of these compounds is discussed, as is some host-guest chemistry.

A family of nanoscale-sized supramolecular cage compounds with a polyhedral framework is prepared by self-assembly from tritopic building blocks and rectangular corner units via noncovalent coordination interactions. These highly symmetrical cage compounds are described as face-directed, self-assembled truncated tetrahedra with Tdsymmetry.

One of the goals of supramolecular chemistry is the construction of very large assemblies, assemblies which may shed light on those found in the life processes. Herein we describe two quite large assemblies, but those which are amenable to characterization in solution by NMR techniques and in the solid state by X-ray crystallography. This relative ease of characterization is due to the extraordinally high symmetry, particularly of the larger assembly.

The cover picture shows a novel hierarchic endohedral clusterization of H2O molecules in the form of a dodecahedron ((H2O)20, red), a further “mounted” dodecahedron (green), and a rhombicosidodecahedron ((H2O)60, yellow). The shell/host (Mo atoms blue, O red) is built up by 12 pentagonal {(Mo)Mo5} type building blocks (one is highlighted as a blue ring),which are connected by 30 MoV2 linkers with the consequence that 20 nanosized Mo9O9 pores/rings of classical crown ether quality are formed in which 20 guanidinium cations are encapsulated (C black, N green). The MoV2 type linkers are stabilized by PO2H2−/SO42− ligands (P/S purple). As the clusterization in the cavity takes place after filling the receptors/pores at the cluster surface with guests, a process is modeled by which a cell converts an extracellular molecular signal into a response. The representative red ring below the blue ring is marked out by the 60 H2O ligands coordinated to the pentagonal Mo units, which altogether form a rhombicosidodecahedron (not shown completely). As the geometric forms described are those of Platonic and Archimedean solids Plato and Archimedes feel involved. Further details are reported by A. Muller et al. on p. 3604 ff.

Mass spectra of carbon \({{\rm{c}}^{\rm{ + }}}_{\rm{n}}{\rm{,}}{{\rm{c}}^{\rm{ - }}}_{\rm{n}}{\rm{,}}{{\rm{c}}^{{\rm{2 + }}}}_{\rm{n}}\) aggregates produced by various techniques show three different n regions. For n ≲ 10, an alternation behaviour appears with aggregates more abundant for a given parity of n. For 10 ≲ n ≲ 30, the mass spectra exhibit a modulo-4 periodicity. For n ≳ 40, an alternation behaviour appears again and some species as \({{\rm{c}}^{\rm{ + }}}_{60}\) are very intense. These results which are related to the stabilities of the aggregates are discussed.