This journal is c The Royal Society of Chemistry 2013 Chem. Commun., 2013, 49, 3615--3616 3615

Cite this: Chem. Commun.,2013,49, 3615

The significance and impact of Wade’s rules

Alan J. Welch*

The emergence of a set of simple yet powerful electron counting rules following a classic paper by

Wade published in 1971 in J. Chem. Soc. D has transformed the way chemists think about the structures

of clusters with delocalised skeletal bonding.

In 1971 a landmark paper in cluster chemistry ‘‘The StructuralSignificance of the Number of Skeletal Bonding Electron-pairs inCarboranes, the Higher Boranes and Borane Anions, and VariousTransition-metal Carbonyl Cluster Compounds’’ was publishedin J. Chem. Soc. D,1 the forerunner of Chemical Communications.The principles outlined in this communication were to becomeknown as Wade’s rules and they provide a straightforward andeloquent rationalisation of the shapes of ‘‘electron-deficient’’†cluster compounds in terms of the number of skeletal electronpairs (SEPs) these molecules possess. The communicationfocused on boranes, carboranes and other heteroboranes, low-valent transition-metal clusters whose structures could not beexplained by 2c–2e bonding, carbido transition-metal clustersand mixed transition-metal–main group element clusters.Shortly after Wade’s initial communication, Mingos extendedthe principle of counting SEPs to electron-precise and electron-rich clusters, provided a generalised method of calculating theSE contribution of a wide variety of transition-metal fragmentsincluding recognising that it was not necessary to know whetherCO ligands were terminal or bridging, and introduced theprinciple that excess electrons resulted in bond breaking.2

Thereafter the rules became popularly known as the Wade–Mingos rules or, more formally, the Polyhedral Skeletal ElectronPair theory.

In whatever form they are known the greatest impact of ‘‘therules’’ has been in providing a simple and elegant explanationof the shapes of ‘‘electron-deficient’’ clusters, since prior tothem the main approach to the structures of the boranes andrelated species was Lipscomb’s topological model involvingstyx numbers and rules,3 which was somewhat limited inapplicability. The stimulus for Wade’s paper was the earlierrecognition by Williams4 that the structures of open boranesand carboranes were not (as had been previously assumed)simply fragments of an icosahedron. Rather, once they werearranged into three families (closo, nido and arachno) it wasclear that, structurally, a nido (n � 1)-vertex polyhedron and anarachno (n � 2)-vertex polyhedron were fragments of theappropriate parent closo n-vertex polyhedron, the structures ofthese parent polyhedra being exemplified by dianionic borates[BnHn]2� and by the carboranes C2B(n�2)Hn. The usual representa-tion of these structural relationships is that popularised byRudolph5 and reproduced in Fig. 1.

Wade’s outstanding contribution was to recognise whythese structural patterns existed, i.e. that closo n-vertex, nido(n � 1)-vertex and arachno (n � 2)-vertex polyhedra all sharedthe same number (n + 1) of skeletal bonding molecular orbitalsand hence SEPs. Alternatively, expressing the rules in theirmore usual form, a closo cluster has (n + 1), a nido cluster has(n + 2) and an arachno cluster has (n + 3) SEPs where n is simplythe number of cluster vertices. These empirical rules workbecause they are underpinned by the results of molecularorbital analyses6 and more generally by Tensor Surface HarmonicTheory.7

Skeletal electron counting rules are now very much part ofthe fabric of modern cluster chemistry and are routinely taughtin all undergraduate inorganic chemistry courses. For thoseactive in cluster research they provide an immediate and veryclear starting point from which to think about and describecluster structures. The impact of the rules on the researchcommunity is clearly evidenced by a current total citation count

Institute of Chemical Sciences, Heriot-Watt University, Edinburgh, EH14 4AS, UK.

E-mail: a.j.welch@hw.ac.uk

Received 4th January 2013,Accepted 13th March 2013

DOI: 10.1039/c3cc00069a

www.rsc.org/chemcomm

† This term, which originates from the fact that such clusters contain moreconnections between adjacent, covalently-bonded, atoms than skeletal electronpairs, should be used cautiously. Whilst the fragments of which such clusters arecomposed are electron deficient, e.g. {BH} has 3 valence orbitals but only 2valence electrons, in terms of molecular orbitals the clusters themselves are not,e.g. in [B12H12]2� all of the skeletal bonding molecular orbitals, and non of theantibonding molecular orbitals, are occupied. A more suitable descriptor of thenature of the clusters that Wade’s rules rationalise might therefore be ‘‘clusterswith delocalised skeletal bonding’’ (we thank a referee for this suggestion).

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3616 Chem. Commun., 2013, 49, 3615--3616 This journal is c The Royal Society of Chemistry 2013

in excess of 2000 for Wade’s initial J. Chem. Soc. D communicationand his subsequent full paper8 emphasising the importance of theskeletal electron count in understanding the structures of a widevariety of cluster types.

Wade’s rules have not only rationalised the structures ofa vast number of cluster compounds but they have alsostimulated much further research, two modest examples ofwhich involve our own work. Several years ago we recognisedthat although {B10H12}2� and {B10H12}4� fragments of metalla-boranes were topologically exactly the same they were, in termsof Wade’s rules, structurally different, the former being a nidofragment of the 11-vertex parent polyhedron (the octadecahedron)and the latter being an arachno fragment of the 12-vertex parent(the icosahedron). By geometrically analysing the structures of{B10} fragments of MB10H12 metallaboranes studied crystallo-graphically and referencing these against unequivocal standards,we were able to establish the presence of measurably different{B10H12}2� and {B10H12}4� units in a variety of known MB10H12

species and were able to validate these results by independentconfirmation of the metal oxidation states that followednaturally from them.9 In a later study we analysed the pentagonalpyramidal {B6} fragments of rhodathiaboranes and carbar-hodaboranes that had been previously regarded as ‘‘rule

breakers’’,10 having nido 11-vertex cages but, seemingly, only12 SEPs. After establishing that the cages really were nido (andnot distorted closo) we were able to recognise the presence oftwo 1-e agostic interactions that afforded the metal centre (andhence the cluster) an additional electron pair, confirming thatthese species conform to Wade’s rules after all.11 Supportingevidence was furnished by simple deprotonation whichswitched off the agostic interactions and resulted in closooctadecahedral structures (this structural change was fullyreversible on reprotonation). Had Wade’s rules not existed wewould almost certainly not been stimulated to undertake thesestudies, and even if we had we would not have had a suitableframework within which to understand the results.

It is indisputable that the emergence of simple but powerfulelectron counting rules in the early 1970s has had a vital andlasting influence on cluster chemistry. Crucially, both Wadeand Mingos perceptively recognised the importance of afragment approach to the bonding within clusters, i.e. thatthe precise nature of the constituents of the cluster wasarguably less important than the skeletal electron countthese constituent fragments provided. Thus, e.g., both a {BH}fragment and a {Ru(CO)3} fragment each provide 3 clusterbonding orbitals and 2 skeletal electrons. They therefore havethe same bonding capability and consequently both [B6H6]2�

and [Ru6(CO)18]2� are structurally related, both closo octahedralclusters with (n + 1) SEPs. It would not be for another four orfive years that the idea of a common bonding capability forsuperficially very different fragments was crystallised inthe isolobal analogy,12 which remains a vital concept bothwithin and between modern inorganic/organometallic/organicchemistry today.

Notes and references1 K. Wade, J. Chem. Soc. D, 1971, 792.2 D. M. P. Mingos, Nature (London) Phys. Sci., 1972, 236, 99.3 W. N. Lipscomb, Boron Hydrides, Benjamin, New York, 1963.4 R. E. Williams, Inorg. Chem., 1971, 10, 210.5 R. W. Rudolph, Acc. Chem. Res., 1976, 9, 446.6 e.g. R. W. Rudolph and W. R. Pretzer, Inorg. Chem., 1972, 11, 1974;

however, molecular orbital calculations on polyhedral boronhydrides predate Wade’s rules, e.g. H. C. Longuet-Higgins andM. de V. Roberts, Proc. R. Soc. London, Ser. A, 1955, 230, 110.

7 A. J. Stone, Mol. Phys., 1980, 41, 1339.8 K. Wade, Adv. Inorg. Chem. Radiochem., 1976, 18, 1.9 S. A. Macgregor, A. J. Wynd, N. Moulden, R. O. Gould, P. Taylor,

L. J. Yellowlees and A. J. Welch, J. Chem. Soc., Dalton Trans., 1991, 3317.10 (a) G. Ferguson, M. C. Jennings, A. J. Lough, S. Coughlan,

T. R. Spalding, J. D. Kennedy, X. L. R. Fontaine and B. Stıbr,J. Chem. Soc., Chem. Commun., 1990, 891; (b) S. Coughlan,T. R. Spalding, G. Ferguson, J. F. Gallagher, A. J. Lough, X. L. R.Fontaine, J. D. Kennedy and B. Stıbr, J. Chem. Soc., Dalton Trans.,1992, 2865.

11 (a) K. J. Adams, T. D. McGrath, Rh. Ll. Thomas, A. S. Weller andA. J. Welch, J. Organomet. Chem., 1997, 527, 283; (b) K. J. Adams,T. D. McGrath, G. M. Rosair, A. S. Weller and A. J. Welch,J. Organomet. Chem., 1998, 550, 315.

12 The first reference to the word isolobal appears in M. Elian, M. M. L.Chen, D. M. P. Mingos and R. Hoffmann, Inorg. Chem., 1976,15, 1148.

Fig. 1 The structural relationships between closo (left column), nido (centre)and arachno (right column) polyhedra noted by Williams4 which were thestimulus for the development of Wade’s rules.1 Adapted from ref. 5.

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