A Keplerian Ag90 nest of Platonic and Archimedean polyhedra in different symmetry groups

Polyhedra are ubiquitous in chemistry, biology, mathematics and other disciplines. Coordination-driven self-assembly has created molecules mimicking Platonic, Archimedean and even Goldberg polyhedra, however, nesting multiple polyhedra in one cluster is challenging, not only for synthesis but also for determining the alignment of the polyhedra. Here, we synthesize a nested Ag90 nanocluster under solvothermal condition. This pseudo-Th symmetric Ag90 ball contains three concentric Ag polyhedra with apparently incompatible symmetry. Specifically, the inner (Ag6) and middle (Ag24) shells are octahedral (Oh), an octahedron (a Platonic solid with six 3.3.3.3 vertices) and a truncated octahedron (an Archimedean solid with twenty-four 4.6.6 vertices), whereas the outer (Ag60) shell is icosahedral (Ih), a rhombicosidodecahedron (an Archimedean solid with sixty 3.4.5.4 vertices). The Ag90 nanocluster solves the apparent incompatibility with the most symmetric arrangement of 2- and 3-fold rotational axes, similar to the arrangement in the model called Kepler’s Kosmos, devised by the mathematician John Conway.


Results
Structures of SD/Ag90a and SD/Ag90b. With a combination of anion-template and geometrical-polyhedron strategies [31][32][33][34][35][36][37] and with careful choice of organic ligands ( t BuSH and PhPO 3 H 2 ) and anion templates (S 2− and PO 4 3− ), we one-pot synthesized Ag 90 nanoclusters under the solvothermal condition as dark brown or red rhombic crystals, depending on the polymorphs. They are stable under ambient conditions because the bulk sample of them can keep the color and morphology unchanged for at least one month. The PO 4 3− ion was considered as anion-template mainly due to its high negative charge and T d symmetry. The former feature can aggregate more Ag + ions to form high-nuclearity cluster through electrostatic attraction, whereas the latter can shape cluster with a T-related symmetry. The Ag 90 nanoclusters can be crystallized into monoclinic P2 1 /n or triclinic P-1 phases dictated by the silver salts used (Fig. 1), hereafter denoted as SD/ Ag90a and SD/Ag90b, respectively. Although the different anions, PhCOO − and CH 3 SO 3 − , did not participate in the final structures of the Ag 90 clusters, they may have influenced the crystallization process through supramolecular interactions such as hydrogen bonds, which causes the formation of the ultimate crystalline phases. Other common silver salts, such as AgBF 4 , CF 3 COOAg, CF 3 SO 3 Ag, and AgNO 3 , were tried in the above synthesis experiments, but none produced desired clusters. A series of characterization techniques such as single-crystal X-ray crystallography, infrared spectroscopy, 31 P nuclear magnetic resonance, ultraviolet-visible absorption spectroscopy, fluorescence spectroscopy, and thermogravimetric analysis were used on these two nests (Supplementary Figs. 1-5, 13-17 and Supplementary Tables 1-3). The electrospray ionization mass spectrometry (ESI-MS) of SD/Ag90a dissolved in CH 2 Cl 2 or CH 3 OH did not give useful data, which indicates that either (i) SD/Ag90a is fragmented during the ionization process or (ii) it is neutral and is hard to ionize under mass spectrometer conditions -even when we added CsOAc to aid in ionization 38 .
As deduced from crystallography, SD/Ag90a (Fig. 2a) and SD/ Ag90b ( Supplementary Fig. 6a) 9,10). Due to the similarity of their molecular structures, we take SD/Ag90a as representative for discussions in detail below. SD/Ag90a crystallized into the monoclinic P2 1 /n space group and conformed to pseudo-T h symmetry. SD/Ag90a is a neutral cluster with all Ag(I) atoms in 3-or 4-coordination with S and/or O atoms. The all-silver framework (Fig. 2b) is composed of three concentric nested polyhedra, an outer Ag 60 rhombicosidodecahedron with 60 3.4.5.4 vertices (Fig. 2c), a middle Ag 24 truncated octahedron with 24 4.6.6 vertices (Fig. 2d), and an inner Ag 6 octahedron with 6 3.3.3.3 vertices (Fig. 2e), where the numbers 5, 4, and 3 represent faces around that vertex, respectively, 5-gons, 4-gons, and 3-gons. Of note, the outer Ag 60 shell is geometrically reminiscent of the third shell (Pd 60 ) in Dahl's Pd 145 cluster 15 . All vertices on these three polyhedra are Ag(I) atoms, and all edges are built from the connection of adjacent two Ag atoms. The Ag-Ag edge lengths in outer, middle, and inner shells range from 2.96 to 4.03, 3.02 to 3.48, and 3.51-3.61 Å, respectively (Supplementary Fig. 11 and Supplementary Table 3). Some of these Ag···Ag edge lengths, shorter than 3.44 Å, twice the Van der Waals radius of silver(I) ion, can be deemed as argentophilic interactions that contribute to the stability of the silver shells. Of note, the long Ag···Ag edges in the Ag 6 octahedron also rule out a subvalent nature, which usually produces short Ag···Ag distances approximating to 2.88 Å 39 . By measuring distances between inversion-related pairs of Ag atoms in the same shell, the diameters of outer, middle and inner shells are determined to be 1.5, 1.0, and 0.5 nm, respectively.
Each μ 12 -κ 3 :κ 3 :κ 3 :κ 3 PO 4 3− ion (as anion-template) penetrates the hexagonal windows of the Ag 24 shell to connect all three silver shells (Ag-O: 2.3-2.6 Å) by linking two silver 3-gons of the Ag 6 and Ag 60 shells and one 6-gon of the Ag 24 shell (Fig. 2f, g). Six μ 8 -S 2− ions from in situ decomposition of t BuSH intercalate the aperture between Ag 24 and Ag 60 shells (Ag-S: 2.43-2.89 Å) by linking two 4-gons up and down from these shells, respectively (Fig. 2h) 28 . Based on the above analysis, we found that the tetrahedral PO 4 3− ion has a special role in shaping silver 3-gons and 6-gons, essential elements to construct the rhombicosidodecahedron and the truncated octahedron, respectively. As for the spherical S 2− ion, it assists in fabricating the silver 4-gon, an essential element for both the rhombicosidodecahedron and the truncated octahedron. Both inorganic anions act not only as templates to shape the silver polyhedra by defining the essential c Arrangement of the tetrahedron with its self-dual, another tetrahedron, with alignment along threefold axes with a face in front, threefold axes with a vertex in front, and twofold axes.
polygon elements but also function as glue to consolidate the overall nested silver shells. The ligand coverage on the surface of the outer Ag 60 shell is polygon selective with 24 t BuS − and 6 PhPO 3 H − on thirty 4-gons and 12 PhPO 3 2− on twelve 5-gons. There are no ligands capping the twenty 3-gons. The PhPO 3 H 2 ligand exhibits two kinds of deprotonated forms, PhPO 3 2− and PhPO 3 H − , that respectively coordinate with twelve 5-gons (μ 5 -κ 2 :κ 2 :κ 1 ) and six 4-gons (μ 4 -κ 2 : κ 2 ) on the surface of the Ag 60 shell (Ag-O: 2.2-2.6 Å). The 31 P NMR (nuclear magnetic resonance) of the digestion solution of SD/Ag90a shows two sharp peaks with chemical shifts at δ = −1.07 and 15.62 ppm ( Supplementary Fig. 2), corresponding to H 3 PO 4 and PhPO 3 H 2 , respectively, which clearly verify the existence of two different P-containing chemicals in SD/Ag90a. From the ligation modes of each coordinative component, we suggest that all of them play roles to shape different silver polygons, paving the way for further construction of polyhedral silver nanoclusters. The overall structure is reinforced by a combination of argentophilic interactions (<3.44 Å) and the scaffolding provided by all other coordination bonds. The remarkable structure of SD/Ag90a has not been previously observed in the family of silver nanoclusters.

Alignment of shells with compatible point-group symmetry.
We now ask about the alignment of the icosahedral and octahedral cages in the SD/Ag90a nest. For the dodecahedron and its dual (the icosahedron), both Platonic solids with icosahedral (I h ) symmetry and thus "compatible", nesting may be based on alignment of all of the five-, three-and twofold axes of rotational symmetry ( Fig. 3a; Supplementary Table 4). The same full alignment could obtain for nests of any icosahedral structures, including the six icosahedral Archimedean solids and an infinity of other icosahedral structures. Likewise, nesting of octahedral shells like the cube and its dual (the octahedron)-both Platonic solids-with each other (Fig. 3b) and of tetrahedral shells like the tetrahedron and its self-dual (the tetrahedron) ( Fig. 3c) with each other may be based on alignment of all rotational axes (Supplementary Table 4). Indeed, the Ag 73 cited above is just such a symmetry-compatible nesting of octahedral silver cages 30 .
Alignment of shells with incompatible point-group symmetry.
Although it might be assumed that only cages with compatible symmetry (e.g., icosahedral with icosahedral) could be nested, for the Zometool toy the mathematician John Conway created a model called "Kepler's Kosmos", a model that aligns the five Platonic solids, the two with icosahedral symmetry, the two with octahedral, and the one with tetrahedral 40  Kepler's Kosmos provides one possible answer to the question of how to align icosahedral polyhedra with fivefold rotational axes but no fourfold, octahedral polyhedra with fourfold axes but no fivefold, and tetrahedral polyhedra with neither. Conway's  Table 4) has these properties: (i) None of the six fivefold axes of the dodecahedron (or icosahedron) is aligned with any rotational axis of an octahedral or tetrahedral polyhedron. (ii) Four of the ten threefold axes of the dodecahedron (or icosahedron) are aligned with all four threefold axes of the cube (or octahedron) and all four threefold axes of the tetrahedron (Fig. 4b-e). (iii) Three of the fifteen twofold axes of the dodecahedron (or icosahedron) are aligned with all three (orthogonal) fourfold axes of the cube (or octahedron) and all three (orthogonal) twofold axes of the tetrahedron (Fig. 4f-i). This quintuple nest itself has tetrahedral (T d ) symmetry. However, other alignments are possible. Here, we devise an alternative alignment with three different combinations of fourand twofold axes to produce a nest with just three different, orthogonal, twofold axes ( Fig. 5; Supplementary Fig. 12b, Supplementary Table 4, and Supplementary Movie 1), producing a quintuple nest with lower (D 2 ) symmetry. Of course, it is also possible to align none of the rotational axes, producing a quintuple nest with trivial (C 1 ) symmetry.
Alignment of shells in SD/Ag90a. Given different symmetric arrangements of the three shells-as in the Kepler's Kosmos with point-group T d (Fig. 4), alignment of only three different orthogonal twofold axes with point-group D 2 (Fig. 5) and no alignment of rotational axes (thus C 1 )-we ask how the shells in SD/Ag90a align. The icosahedral outer shell of SD/Ag90a has fivefold axes, but these are absent in the octahedral middle and inner shells (Fig. 2b). The octahedral inner and middle shells compatibly align all of their four-, three-and twofold axes, as in Fig. 3b. The icosahedral outer shell aligns four of its threefold axes with all four of the threefold axes of the octahedral shells (Fig. 6a) and three of its twofold axes with all three fourfold axes of the octahedral shells (Fig. 6b). Of note, the interstitial anions of S 2− and PO 4 3− also have important influences on aligning the three shells. Specifically, the threefold axes of the Ag 6 , Ag 24 , and Ag 60 shells pass through the PO 4 3− ions, and the twofold axis of Ag 60 shell and fourfold axis of Ag 24 shell pass through the S 2− ions, thus dictating the alignment of three different shells in the unique fashion discussed above. The alignment of these approximate polyhedra is nearly as good as in the same nest with regular polyhedra (Fig. 6c, d). Thus, the arrangement of the icosahedral and octahedral shells in the Ag 90 triple nest is the same as in Kepler's Kosmos (Fig. 4).
However, without a tetrahedral shell, SD/Ag90a is a subset of Kepler's Kosmos, with just I h (3.4.5.4) and O h (4.6.6 and 3.3.3.3) shells. As both I h and O h structures have inversion symmetry, their combination in SD/Ag90a also has inversion symmetry. Thus, the regular version of SD/Ag90a (Fig. 6c, d), with four threefold axes, three twofold axes, mirrors, inversion, and a symmetry order of 24, has T h symmetry (Supplementary Table 4).
Optical properties of SD/Ag90a. The solid-state ultravioletvisible (UV/Vis) absorption spectra of SD/Ag90a and [ t BuSAg] n were measured at room temperature. As shown in Fig. 7a, SD/ Ag90a exhibits a wide absorption range spanning UV and Vis regions with an absorption maximum at 419 nm. Compared with the absorption of [ t BuSAg] n at 280 nm, the absorption edge is  significantly red-shifted in SD/Ag90a, which should be caused by the ligand-to-metal charge transfer (LMCT) or/and clustercentered (CC) transitions. Based on the Kubelka-Munk function of (αhυ) 1/2 = κ(hυ − E g ) (E g is the band gap (eV), h is Planck's constant (J·s), υ is the light frequency (s −1 ), κ is the absorption constant, and α is the absorption coefficient) 45 , the band gaps of SD/Ag90a and [ t BuSAg] n precursor were determined as 0.69 and 2.34 eV, respectively ( Supplementary Fig. 3), which indicates that the aggregation of multiple silver atoms into a cluster structure has an important influence on the HOMO-LUMO gap.
We further studied the photocurrent responses of SD/Ag90a and [ t BuSAg] n driven by visible-light in a typical three-electrode system (ITO glass as the working electrode, platinum wire as the assisting electrode and Ag/AgCl as the reference electrode) and keeping the bias voltage at 0.6 V. Upon on-off cycling irradiation with LED light (λ = 420 nm; 50 W; intervals of 10 s), clear photocurrent responses were observed for SD/Ag90a and [ t BuSAg] n (Fig. 7b). The photocurrent density of SD/Ag90a (0.9 μA cm −2 ) was five times larger than that of [ t BuSAg] n (0.17 μA cm −2 ), indicating that SD/Ag90a has better generation and separation efficiency of photoinduced electrons/holes pairs in ITO electrodes 46 . The generation of photocurrent may involve photoinduced charge migration from S 3p to the Ag 5s orbits.
The solid state emission spectra of SD/Ag90a were recorded as a function of temperature from 293 to 93 K with 40 K as an interval, showing luminescence thermochromic behavior (Fig. 8). The luminescence of SD/Ag90a originates from the ligand-tometal charge transfer transition with a charge transfer from S 3p to Ag 5s orbitals and/or mixed with a cluster-centered transition related to Ag···Ag interactions 47 . SD/Ag90a shows gradually blueshifted emissions from 700 to 684 nm (λ ex = 468 nm) upon cooling, possibly related to enhanced molecular rigidity at lower temperatures. During the cooling process, the emission intensity shows a nearly tenfold increase from 293 to 93 K, which is attributed to reduction of the nonradiative decay at low temperature [48][49][50] . The emission lifetime of SD/Ag90a (Supplementary Fig. 4), falling in the microsecond scale (τ SD/Ag90a = 17.80 μs) at 93 K, suggests a triplet phosphorescence origin.

Discussion
We have successfully synthesized an Ag 90 nanocluster with overall pseudo-T h symmetry. This silver nanocluster is divided into three shells as Ag 6 @Ag 24 @Ag 60 from inner to outer. The Ag 6 inner shell is an octahedron (a Platonic solid with 6 3.3.3.3 vertices), the Ag 24 middle shell is a truncated octahedron (an Archimedean solid with 24 4.6.6 vertices), and both have octahedral (O h ) symmetry. However, the Ag 60 outer shell is a rhombicosidodecahedron (an Archimedean solid with 60 3.4.5.4 vertices and icosahedral (I h ) symmetry). The SD/Ag90a nanocluster solves the apparent incompatibility among their symmetry groups with the most symmetric arrangement of two-and threefold axes. Creation of a nest with all three of the polyhedral symmetries, icosahedral, octahedral and tetrahedral-and resembling Kepler's Kosmos even more closely-remains an exciting challenge.

Data availability
The X-ray crystallographic coordinates for structures reported in this article have been deposited at the Cambridge Crystallographic Data Centre, under deposition numbers CCDC: 1913186-1913187. These data can be obtained free of charge from the Cambridge Crystallographic Data Centre via www.ccdc.cam.ac.uk/data_request/cif. Received: 7 January 2020; Accepted: 8 June 2020;