Inorganica Chimica Acta 363 (2010) 4392–4398

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Inorganica Chimica Acta

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How icosahedral are icosahedral clusters?

Santiago AlvarezDepartament de Química Inorgànica and Institut de Química, Teòrica i Computacional, Universitat de Barcelona, Martí i Franqués 1-11, 08028 Barcelona, Spain

a r t i c l e i n f o a b s t r a c t

Article history:Available online 15 September 2010

Dedicated to Achim Muller

Keywords:IcosahedraContinuous shape measuresSymmetryDodecaboratesClusters

0020-1693/$ - see front matter � 2010 Elsevier B.V. Adoi:10.1016/j.ica.2010.06.057

E-mail address: santiago@qi.ub.es

Icosahedral symmetry, not contemplated within the crystallographic space groups, is nevertheless pres-ent to a high degree of perfection in a variety of clusters, in molecular, ionic, covalent or metallic struc-tures. The application of continuous shape measures to those structures allows for a quantification of thedegree of icosahedral symmetry present in each case, a first necessary step for a deeper discussion of thefactors that favor the adoption of the icosahedron as a stable structural motif. Examples analyzed includeboranes and carbaboranes, main group, rare earth or transition metal clusters, ligand-bridged polynuclearcomplexes and sets of donor atoms in mono- or polynuclear complexes. Specific examples are found ofstructures that appear along the minimal distortion pathways from the icosahedron to the cuboctahedronor the anticuboctahedron.

� 2010 Elsevier B.V. All rights reserved.

1. Introduction

But then there is the icosaspherein which at last we have steel-cutting at its summit of economysince twenty triangles conjoined, can wrap oneball or double-rounded shellwith almost no waste, so geometricallyneat, it’s an icosahedron.

Marianne Moore, The Icosasphere.

The icosahedron, one of the five Platonic solids is character-ized by 12 vertices related through five and threefold symmetryrotations and belongs to the icosahedral (Ih) point group. How-ever, there is no room in crystallographic space groups for five-fold symmetry, to the point that it was only after Caspar andKlug developed the structural principles of icosahedral virusesthat previous observations from other researchers were under-stood and accepted [1]. Contemporaneously, Mackay proposed[2] nesting of icosahedral shells as a non-crystallographic densepacking principle, and the impact that those ideas had on struc-tural studies of nanometer-sized icosa-twins, atomic clusters,intermetallics and quasicrystals has been the object of a recentreview. The discovery of quasicrystals with pentagonal diffrac-tion patterns two decades later was still received with disbeliefand rejection by skeptical colleagues who thought the crystals

ll rights reserved.

analyzed were twinned [3,4]. Although presently the existenceof both molecules with icosahedral symmetry and pentagonalcrystals without translational symmetry are well accepted, thefact that the icosahedral symmetry is not contemplated in thecurrent crystallographic approach, leaves us with little informa-tion about the degree of perfection with which groups of atomsform icosahedra.

In this paper I wish to show how the analysis of crystal struc-tures by means of continuous shape and symmetry measures [5–7] allows us to quantify the degree of icosahedricity of 12-atomgroups in a wide variety of chemical compounds. In addition,structures distorted from the regular icosahedron can in many in-stances be described as being along a minimal distortion path to-ward an alternative regular polyhedron (e.g., the cuboctahedron),calibrated by the value of the ‘‘path deviation function” [8].Furthermore, one can determine the exact position of a givenstructure along such a pathway by means of the ‘‘generalizedpolyhedral interconversion coordinate” that can adopt values from0% to 100% [9].

The type of information provided by the continuous shapemeasures methodology should represent a starting point for abetter understanding of the bonding relationships that holdgroups of atoms together in icosahedral molecules or clusters.The icosahedral groups to be analyzed here include boranesand carbaboranes, transition metal or main group metal clusterslinked by metal–metal bonds, polynuclear metal complexes sup-ported by bridging ligands, arrangements of ligand atoms aroundor within shells of metal atoms (as in Keplerates [10]), or clus-ters of metal atoms in extended structures of intermetallicphases.

Fig. 1. Distribution of the icosahedral shape measures in the families of dodecaborates, carbadodecaborates and dicarbadodecaboranes, obtained from the CSD.

1 Cristal y acero. Prisma fucsia que al sol de la tarde arde como una estratosféricabotella de granadina gigante, como un icosaedro de granates furiosos.

S. Alvarez / Inorganica Chimica Acta 363 (2010) 4392–4398 4393

2. Icosahedral boranes

I carry the Platonic solids in my neurological kit and decided tostart with these.‘‘What is this?” I asked, drawing out the first one.‘‘A cube, of course.”‘‘Now this?” I asked, brandishing another.He asked if he might examine it, which he did swiftly and systemat-ically: ‘‘A dodecahedron, of course. And don’t bother with the others– I’ll get the icosahedron, too.”Abstract shapes clearly presented no problems.

Oliver Sacks, The Man Who Mistook His Wife for a Hat.

Boron loves the icosahedron. B12 icosahedra are found in its ele-mental structures and in a wide variety of binary and ternary bor-on compounds. [11] Here we look at three related skeletons: thedodecaborate anions (B12R12

2�, 129 structures analyzed), the car-badodecaborate anions (CB11R12

�, 175 structures), and the dicarb-adodecaboranes (C2B10R12, 1180 structures), whose structureshave been retrieved from the Cambridge and Karlsruhe structuraldatabases. No restriction was made as to the number and natureof the substituents, and for the anionic dodecaborates we searchedfor structures of salts with either inorganic or organic cations. Thedegree of icosahedricity of those skeletons has been evaluatedthrough their icosahedral shape measures [7,12]. In short, a zerovalue of the shape measure is indicative of a perfect icosahedralshape, and increasing values indicate increasing degrees of distor-tion. The distribution of the icosahedral shape measures amongthose molecules is shown in Fig. 1, where we can see that the dode-caborates are very close to the regular icosahedron, in spite of thevariety of substitution patterns presented by them. In contrast,when one or two atoms of the boron skeleton are substituted bya carbon atom, only one example is found with an icosahedralshape measure of less than 0.02.

The high degree of icosahedricity of the dodecaborates must bestressed, given the variety of counterions and the presence ofasymmetrical substitution patterns. Another interesting observa-tion can be made in the symmetrically substituted dodecaboratesB12X12

2� (X = Me, F, Cl, Br or I; 13 structures). In those cases, theboron skeletons appear to be highly symmetric (icosahedral shapemeasures of less than 0.02), and the X atoms form also nearly per-fect icosahedra, if with slightly larger distortions (shape measuresof up to 0.04). The hydroxo- and alkoxo-substituted analogues (20structures analyzed) show a different distortive behavior, with lar-ger distortions from icosahedricity of both the O12 and B12 shells(S(icosahedron) 6 0.14) which are, however, uncorrelated. It thusseems that the B12 framework is in general highly symmetric andperturbations resulting from the introduction of a carbon atom inthe icosahedral skeleton has a significant influence on the loss oficosahedral symmetry. On the other hand, the polyhedra of theX12 substituents retain most of the icosahedricity but are some-what more deformable than the B12 group, whereas perturbationsdue to intermolecular ionic interactions have a minor effect on thesymmetry of the inner B12 icosahedron.

3. Main group M12 icosahedra

Glass and steel. Fuchsia prism that burns under theafternoon sun like a stratospheric giant bottle of pome-granate juice, like an icosahedron of frantic garnets.1

Francisco Antonio Ruiz Caballero, El rascacielos rosa.

Main group metal clusters, either appearing as independent

Icosahedron CuboctahedronMn12

Fig. 2. Relationship between the icosahedron, the cuboctahedron and the middle point of the minimal distortion path represented by the Mn12 group in a Mn21 compound. The upperpart shows the sets of atoms related through the threefold rotation that is preserved along the path (outlined in red), while the lower view is a projection down the same axis.

4394 S. Alvarez / Inorganica Chimica Acta 363 (2010) 4392–4398

molecules or ions and present in extended structures, have beenanalyzed. Among them, an Al12(i-Bu)12 dianion, [13] and theM @ Pb12 centered dianionic clusters (M = Pd [14] or Pt [15]) haveall icosahedral measures smaller than 0.01. Also minor distortions(shape measures smaller than 0.10) are present in the Ni @ Pb12

dianion [14] and in Ga12 clusters [16].A Ga24 cluster [17] is formed by two nested Ga12 icosahedra.

While the inner shell has a high degree of icosahedricity, accordingto the corresponding shape measures (0.10), the outer shell isstrongly distorted (1.22). Another highly distorted Al12 icosahe-dron is found within an Al77 cluster [18], but that distortion willbe discussed in more detail below.

MAl12 clusters (M = Mo, W, Tc, Re) in compounds with theWAl12 structure [19–21] show a constant degree of icosahedricity(0.03). Such a small distortion could probably be attributed tothe symmetry loss due to the interaction with a cube of eightneighboring clusters. The Tl13 centered clusters present in com-pounds of the type AxByTl13 (where A and B are alkaline elements;5 structures analyzed) show deviations from the icosahedron cali-brated by shape measures between 0.03 and 0.15, and the In12

groups [22,23] in K39In80 and K17In41 are still more distorted(0.20 and 0.22). The most distorted icosahedra found in this familycorrespond to Na12 and Cs12 groups in Na8Si46 and Cs8Sn46, respec-tively [24,25] (icosahedral measure of 0.81 in both cases), quitehigh compared to other clusters, but still sufficiently small to war-rant them being named icosahedra.

4. Transition metal M12 icosahedra

In the vast Eastern fringesOf blue the planets fade,The alchemist thinks on the secretLaws that link planets and metals.2

Jorge Luis Borges, El Alquimista.

2 En los vastos confines orientales / del azul palidecen los planetas, / el alquimistapiensa en las secretas / leyes que unen planetas y metales.

We have already noted in a previous work [26] the perfect icos-ahedricity of Pd12 and Mo12 groups in the impressive clusters Pd145

and Mo132 [27,28]. A variety of other groups of transition metalatoms have been found with only minute deviations from the ico-sahedron (shape measures smaller than 0.08), both as metal–metalbonded clusters or as ligand-bridged polynuclear complexes: Ni12

[29], Mo12 in Mo78Fe20 [30], Au12 [31–33], Pt12 [34], Au6Ag6

[35,36]. Also a Sm12 compound [37] and an Ag12 cluster in [38]Ag13OsO6 have been found to be highly icosahedral. Other groupswere found to present slightly greater distortions from the icosahe-dron (shape measures between 0.10 and 0.22): Au8Ag4 [39],Au8Cu4 [39], Rh12 [40], and a Cd12 group in the solid state com-pound Na13Cd1�xTlx)27 [41].

Finally, there are some strongly distorted icosahedra, character-ized by shape measures between 0.80 and 1.80: Mn12 [42], Pt12

[43], and Cu12 [44]. When the shape measures indicate importantdistortions, it is reasonable to investigate whether an alternativepolyhedron may describe better the geometry of the group ofatoms under consideration. Also, one should search for thepossibility of a more accurate structural description as being alongsome specific distortion pathway that interconverts two idealpolyhedra.

An intriguing possibility in the present case consists in the trans-formation of the icosahedron into a cuboctahedron. Such a processwould convert an object with icosahedral symmetry into one ofoctahedral symmetry, thus replacing fivefold symmetry operationsby fourfold ones. By using a continuous shape measures analysis,we have been able to detect a beautiful example of a molecularstructure intermediate between the icosahedron and the cuboctahe-dron. It corresponds to a group of 12 MnIII ions held together by oxobridges within a Mn21 polynuclear complex [42]. Even if such a Mn12

has been described as a ‘‘slightly distorted icosahedron”, its shapemeasures relative to the icosahedron and the cuboctahedron are1.13 and 1.60, respectively, with a deviation of only 0.3% from theminimal distortion path and a 46% conversion to the cuboctahedron.Three snapshots along this pathway are schematically shown inFig. 2.

A similar description applies to the structure of the inner Al13

shell of the Al77 cluster reported by Ecker et al. [18]. Its relativelyhigh icosahedral shape measure (0.89), together with a not too

Table 1Shape measures and positions relative to polyhedral interconversion pathwaysa forseveral M12 transition metal clusters.

Cluster Ref. D u S(ICO) S(ACOC)

Icosahedron-anticuboctahedron pathwayRh12 [40] 9.2 18 0.22 5.32Rh12 [40] 10.5 16 0.17 5.77Cu12 [44] 10.4 52 1.74 2.25Au8Ag4 [39] 4.7 13 0.12 5.39Au12 [45] 11.5 15 0.16 5.94Cd12 [41] 13.2 16 0.16 6.12

Icosahedron-cuboctahedron pathwayS(COC)

Mn12 [42] 0.3 46 1.11 1.60Al77 [18] 9.1 41 0.89 2.49

a D and u are the path deviation function and the generalized interconversioncoordinate from the icosahedron to the anticuboctahedron or the cuboctahedron (inpercentage). S(ICO), S(ACOC) and S(COC) stand for the shape measures relative tothe icosahedron, anticuboctahedron and cuboctahedron, respectively.

Fig. 3. Comparison of the structure of the VCu12 group [44] (left, the central V atomis omitted for clarity) with that of an ideal structure with a 45% distortion along theminimal distortion path from the icosahedron to the anticuboctahedron (right). Thestructures are shown in a projection down the threefold rotation axis common tothe icosahedron and the cuboctahedron.

S. Alvarez / Inorganica Chimica Acta 363 (2010) 4392–4398 4395

high value of its cuboctahedral measure (2.49) tell us that thiscluster is close to the interconversion path between the two idealpolyhedra (path deviation function of 9%), with an approximate de-gree of conversion to the cuboctahedron of 41%.

A different distortion is found in a VCu12 group [44] heldthrough oxo and alkoxo bridges. In this case, the icosahedron is sig-nificantly distorted toward the anticuboctahedron, within a 10% ofthe minimal distortion path. The analysis of that path indicatesthat it consists in a concerted rotation and planarization of sixequatorial atoms around a threefold symmetry axis of the icosahe-dron, with a D3 symmetry preserved throughout that path. Thismeans that the geometries along that path are chiral, as can beseen in Fig. 3, where the experimental and ideal twisted structuresappear as mirror images. However, the reported crystal structurecorresponds to a non enantiomorphic space group and it is there-fore racemic.

After seeing that some structures of severely distorted icosahedra,with S(icosahedron) values between 0.89 and 1.74, can be describedas icosahedra twisted toward either the cuboctahedron or the antic-uboctahedron, we looked in more detail to less distorted structures,i.e., those having S(icosahedron) values between 0.10 and 0.22(Table 1), and found them to be closer to the path to the anticubocta-hedron than to that leading to the cuboctahedron. Their positionalong that path, can be seen in the shape map presented in Fig. 4.

5. Icosahedral ligand polyhedra

–there, in is out,everywhere and nowhere,things are themselves or different,encaged in an icosahedronthere is an insect knitting musicwhile another insect unweavessyllogisms woven by a spiderhanging on the moon threads . . .3

Octavio Paz, Pasado en claro.

There are two different ways in which donor atoms of electro-negative elements can be icosahedrally arranged. First, when 12donor atoms coordinate a metal atom, the icosahedron is the

3 –allá dentro es afuera, / es todas partes y ninguna parte, / las cosas son las mismas yson otras, / encarcelado en un icosaedro / hay un insecto tejedor de música / y hay otroinsecto que desteje / los silogismos que la araña teje / colgada de los hilos de la luna ...

most spherical arrangement, so there may be a possibility of find-ing icosahedral coordination spheres. However, since coordinationnumber 12 is rarely found for transition metals, we will turn oureyes to rare earths and to alkaline or alkaline-earth metals. A sec-ond way to obtain a set of icosahedral donor atoms is as anarrangement of bridging ligands around a polynuclear skeletonof metal atoms.

Let us look first at compounds with 12 donor atoms coordi-nated to a single metal atom. The nicest coordination icosahedrafound (shape measures between 0.02 and 0.07) correspond to lan-thanide cations (La, Pr, Sm or Gd) in aminocarboxylato complexes[46–50]. In those systems, each donor group acts as a bis(mono-dentate) bridging ligand that links the rare earth coordination ico-sahedron with one of the six octahedrally coordinated transitionmetal ions, which display an octahedral arrangement throughedge sharing with the icosahedron (Fig. 5). Notice that the combi-nation of the coordination icosahedron of the rare earth and theoctahedron formed by the six transition metal atoms provides an-other beautiful example of the close relationship between icosahe-dral and octahedral symmetries discussed by us elsewhere [26].The same construction principle is found in MNi6 (M = Sr, Ba, La,Ce, Pr, Nd) compounds [51,52], with the multidentate ligand 3,7-diazanona-2,7-dien-2,8-dicarboxylato, all having icosahedralshape measures of less than 0.07. Other examples of icosahedralcoordination spheres are found around the Ba2+ and Ca2+ ions inthe solid state compounds Ba3Ga6Cl24�2C6H6 and CaCu3Ti4O12,respectively [43,53].

Two tridentate (1 and 2) and one hexadentate (3) ligands seemto be well adapted to an icosahedral coordination sphere, even ifwith non-negligible distortions (shape measures between 0.12and 0.43). These have been found in complexes [Ba(1)4]2+ [54],[La(2)4]3+ [55], and [R(3)2]+, where R = U [56], Nd [57,58], Eu [59]or Sm [56]. While the construction of an icosahedron by appendingfour tridentate ligands such as 1 and 2 to span trigonal faces seemsstraightforward, we have found no additional examples of 12-coor-dinated complexes with such ligands. In contrast, the structures ofthe various [R(3)2]+ compounds all present icosahedral coordina-tion spheres with quite similar shape measures, of about 0.4, whichare likely a signature of the ligand’s geometry. It results from ashortening of the edges occupied by chelate rings, a distortionopposite to that leading to the cuboctahedron (Fig. 2). In such a dis-tortion, two opposing triangular faces of the icosahedron rotate inopposing directions relative to the central hexagon that is plana-rized, finally adopting the eclipsed conformation of theanticuboctahedron.

Fig. 5. Structure of a SmNi6 prolinato complex [50], showing the central SmO12

coordination sphere and the octahedrally arranged Ni atoms with octahedralcoordination.

Fig. 4. Shape maps showing the minimal distortion paths (continuous lines) between the icosahedron and the cuboctahedron (a) and between the icosahedron and theanticuboctahedron (b), as well as the position of several M12 distorted icosahedra discussed in the text that are along those paths (see Table 1).

4396 S. Alvarez / Inorganica Chimica Acta 363 (2010) 4392–4398

NH

OH

OHOH

NH

NH

N

NB

N

NN

N

H

NN

N

1

2

3

The second way through which 12 donor atoms can be set intoan icosahedron consists in arranging them on a polyhedral scaffoldof metal atoms. The most elegant choice in this case consists incapping the pentagonal faces of an M20 dodecahedron with 12 li-gands, thereby forming the dual polyhedron, the icosahedron. Thisarchitecture, that requires the ligands to act as pentuple bridges, is

realized in an As12 icosahedron (with a shape measure of 0.05) sup-ported by a Li20 dodecahedron [60], and in nested P12 (icosahedralshape measure of 0.19) and Li16Al4 polyhedra [61].

A second, less obvious alternative, consists in arranging l2

bridging ligands on each edge of an octametallic cube, in such away that each face of the cube is decorated with two donor atoms[62]. However, that arrangement results in a regular icosahedrononly when the position of the donor atoms is such that the distancebetween donors on one face of the cube is the same as that be-tween those in neighboring faces. As a result, the icosahedra ob-tained in this way present in general sizeable distortions fromthe ideal shape, as in the family of Se12-bridged Ag8 cubes (shapemeasures in excess of 0.55) [63–65]. Other examples of this typeof architecture have been identified by us in a previous communi-cation [26] including the almost ideal case of the [Cu8(dtsq)6]4- an-ion (dtsq = dithiosquarate) [66], for which the Cu8 set presents acubic shape measure of 0.03 and the S12 group has a correspond-ingly small icosahedral shape measure of 0.04.

6. Concluding remarks

The professional crystallographers said it wasimpossible; you could not have icosahedralsymmetry. Today, of course, nobody doubts this.

Donald Caspar3

The varied examples examined here tell us that there is roomfor perfect icosahedral symmetry within crystals of different sorts,despite the absence of fivefold axes in crystallographic symmetryspace groups. The aim of this work, rather than providing explana-tion for the stability of the icosahedral structures, was to test thetools that can tell us how perfectly icosahedral a given cluster ofatoms is, and to furnish a data set for future detailed studies.

Some preliminary conclusions that stem out from the presentstudy are:

(a) In molecular crystals, covalent bonding (even if with somedegree of delocalization, as in the case of dodecaboratesand isolobal transition metal clusters) may be responsiblefor an ideal icosahedral symmetry. While chemical substitu-tions within the icosahedron may result in significant distor-tions from the ideal geometry, different externalsubstitution patterns and intermolecular interactions inthe solid state have in general a minor distortive effect.

(b) Scaffolding with nested cubic or dodecahedral layers ofatoms provides an efficient way of building up icosahedra.

S. Alvarez / Inorganica Chimica Acta 363 (2010) 4392–4398 4397

(c) The present results advocate for a more systematic use of acombination of concurrent local and global symmetries[67] to describe molecular structures and to attempt to cor-relate properties and symmetry.

I wish to call also the attention on the important fact that icosa-hedral molecules have latent octahedral symmetry that is instru-mental in building up cubic crystals. Even when the space groupis not cubic, we have shown that the deviations from a cubic struc-ture are rather small [26]. Paradigmatic examples are provided bya Mo132 compound prepared by Müller and coworkers, that show-cases three nested icosahedral polyhedra [68], the Ag188 compoundreported by Fenske and coworkers [69], and the Fe30Mo72 capsulebuilt around a PMo12 Keggin unit [70]. Those compounds crystal-lize in non cubic space groups, but very close to a perfect fcc struc-ture [26]. It is also worth mentioning that it is not rare to findicosahedral viruses, such as the turnip yellow mosaic virus [71],the simian virus 40 [72] or the polyoma virus [73] crystallizingin cubic space groups.

We have previously noted [26] that in some structures withseveral concentric icosahedral clusters, such as Li13Cu6Ga21, theoutermost icosahedron looses a significant part of its fivefold sym-metry, probably due to the onset of interactions with the surround-ings of cubic symmetry, whereas the inner polyhedra arepractically unaffected by the cubic packing and retain the full ico-sahedral symmetry. A similar situation, but to a lesser degree, isfound in the cesium salt of the dodecaborate anion, in which theouter polyhedron of hydrogen atoms involved in intermolecularinteractions is slightly distorted from the regular icosahedron,whereas the inner boron cluster forms practically a perfect icosa-hedron. In the potassium salt of the same borane anion, though,the icosahedral symmetry of the hydrogen atoms seems to beunperturbed by intermolecular interactions.

7. Computational details

For the calculation of the shape measures reported in this paperwe have used the SHAPE program [74], that can be obtained from theauthor upon request. The structural data analyzed was retrievedfrom the Cambridge Structural Database [75,76], version 5.31,and from the ICSD [77].

Acknowledgments

Support from the Ministerio de Investigación, Ciencia e Innovación(MICINN, Project CTQ2008-06670-C02-01-BQU), and from theComissió Interdepartamental de Ciència i Tecnologia (CIRIT, Grants2009SGR-1459 and XRQTC) is gratefully acknowledged.

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