Research: Science and Education
A Unified Approach to Electron Counting in Main-Group Clusters John E. McGrady Department of Chemistry, The University of York, Heslington, York YO10 5DD, United Kingdom;
[email protected] The family of main-group clusters encompasses a diverse spectrum of compounds, from highly electron-deficient species such as B6H62᎑ to the extremely electron-rich S8. As a result of this diversity, the clusters provide an ideal framework for introducing a wide range of chemical bond types, and a discussion of their properties is a fundamental component of many of undergraduate courses. In most textbook treatments, the topic is subdivided into three categories, electron-precise, electron-rich, and electron-deficient, depending on the ratio of valence orbitals to valence electrons (1, 2). The links between the three classes are, however, obscured by the different approaches to correlating structure with electron count adopted in the electron-rich and electron-deficient regimes. In the former, a reduction in the degree of electron deficiency is approached from the perspective of increasing the number of valence electrons while retaining a constant number of vertices (and hence valence orbitals). In contrast, in the electron-deficient domain, the Wade–Mingos rules (1, 3) approach a reduction in electron-deficiency not from the perspective of increasing the electron count, but rather by decreasing the number of valence orbitals through removal of one or more vertices. These divergent approaches encourage students to compartmentalize main-group cluster chemistry and, as a result, they often fail to appreciate that the three classes simply represent different regions in a continuum defined by two variables, electron count and vertex count. In this article, a unified approach to electron counting in main-group cluster chemistry is illustrated that emphasizes the links between the three classes, rather than the differences between them. Review of Electron Counting While it is not the purpose of this article to present an extensive review of traditional approaches to teaching electron counting, a brief overview serves to illustrate the key pedagogical issues involved.
Electron-Precise and Electron-Rich Clusters Of the three traditional classes of clusters, the electronprecise species, characterized by a total valence electron count
(TVEC) of 5n where n is the total number of vertices, generally receive the least attention owing to the relative paucity of well-characterized examples. The best known of these are the family of hydrocarbons, C4H4 (tetrahedrane), C6H6 (prismane) and C8H8 (cubane), which are thermodynamically unstable isomers of cyclobutadiene, benzene, and cyclooctatetraene, respectively (Figure 1). Tetrahedral P4, isoelectronic with C4H4, provides a more chemically familiar example from the realms of stable species. The electron-precise label reflects the fact that there are three electrons per vertex (3n in total) available for cluster bonding and, as each vertex is connected to three others, precisely two electrons are associated with each bond. In the CnHn family, the remaining two electrons per vertex are associated with terminal C⫺H bonds, while in naked clusters such as P4, the electrons form nonbonding lone pairs with similar directional properties. In the remainder of this paper, we will refer to these orbitals as the “exo” pairs to indicate that they are directed radially outwards from the cluster. The electron-precise clusters are usually taken as a reference point for rationalizing the structures of their electronrich counterparts, which are characterized by valence-electron counts greater than 5n. Given that each two-center, two-electron bond in the electron-precise structure has associated with it one occupied bonding orbital and one vacant antibonding orbital, it is clear that any increase in electron count can only be accommodated by occupation of antibonding orbitals, leading to bond cleavage. In this way, the structures of Te3S32+ and S6 (Figure 2) can be derived from the prismane structure (C6H6) by cleavage of two and three bonds, respectively. The addition of any further electrons to S6 would result in the formation of an open-chain species, so we can define an upper limit of 6n for the valence electron count of a “cluster”, which, in this context, includes ring structures.
Electron-Deficient Clusters The discussion so far has centered on the familiar concept of the two-center, two-electron bond and is therefore comfortable ground for most undergraduate chemists. Electron-deficient clusters (TVEC < 5n), in contrast, present more of a challenge because the clusters typically adopt deltahedral
S
Te Te
S Te
C 4H 4 (tetrahedrane) TVEC (= 5n)
20
C6H6 (prismane)
C8H8 (prismane)
30
40
Figure 1. Structures of 4-, 6-, and 8-vertex electron-precise clusters.
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S
S
S S
S S
C6H6
Te3S32+
S6
30
34
36
TVEC (> 5n)
S
Figure 2. Generation of the structures of electron-rich Te3S32+ and S6 starting from electron-precise C6H6.
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Continuum View What is often neglected in textbooks is the fact that, at some point, a reduction in electron deficiency along the closo–nido–arachno series must lead us back to electron-precise and even electron-rich counts. This link is, however, obscured by the radically different approaches to varying the degree of electron deficiency or electron richness adopted in the different domains. As shown above, in the Wade–Mingos
ⴚ
B6H62
B 5 H9
B4H10
TVEC
26
24
22
SEC
14
14
14
Figure 3. The Wade–Mingos approach to reducing the degree of electron deficiency in borane clusters.
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approach the degree of electron deficiency is reduced by decreasing the number of vertices (and hence the number of valence orbitals) within a constant skeletal (rather than total) electron count (Figure 3). For electron-rich clusters, in contrast, electron deficiency is reduced by increasing the total electron count while retaining a constant number of vertices (and hence valence orbitals) (see Figure 2 for an illustrative example). The absence of a common approach (or even a common reference point) means that the link between the different classes is rarely made. As a result, students often fail to appreciate the fact that electron-rich, electron-precise, and electron-deficient classes, as well as the hypoelectronic clusters that will be introduced later, simply occupy different domains in a continuum characterized by two variables, electron count and vertex count (Figure 4). This “continuum view”, illustrated in Figure 4, clearly illustrates the ambiguities that arise when main-group clusters are divided along traditional electron-rich, electron-pre-
electron-rich 5n
36
Valence Electron Count
structures (based on triangular faces), where it is not possible to assign a localized electron pair to each bond. The delocalized picture of bonding that lies at the heart of the Wade–Mingos rules takes as its reference point highly electron-deficient, approximately spherical clusters such as B6H62᎑. At each vertex, two electrons are involved in a localized B⫺H bond (the exo pairs introduced previously), leaving three orbitals but only two electrons to participate in the bonding of the cluster skeleton. One of these three orbitals is directed radially towards the center of the cluster, while the other two orbitals are oriented tangentially around the circumference. Occupation of the bonding combination of the radial orbitals, along with the n bonding orbitals derived from the tangential set yields the characteristic 2n + 2 skeletal electron count (SEC) of closo boranes, BnHn2᎑. Including the two electrons in each B⫺H bond yields a TVEC of 4n + 2, clearly lower than the electron-precise value of 5n. Within the Wade–Mingos approach, a reduction in the degree of electron deficiency can be achieved, at least conceptually, in a two-step process. Removal of a (B⫺H)2+ vertex leaves the skeletal electron count unchanged, but removes three orbitals from the skeletal framework. The addition of two protons then restores the charge, but reintroduces only two orbitals to the framework. The total number of skeletal electrons therefore remains unchanged, but the number of valence orbitals is reduced, and so the degree of electron deficiency is reduced. The classic series closo-B6H62᎑ → nidoB5H9 → arachno-B4H10 provides an illustrative example, where all three species share the same SEC, that is, 14 electrons, and the nido and arachno species are generated by successive removal of B⫺H vertices from the octahedral parent structure.
S6 4n + 8 Te3S32ⴙ
34 32 30 28 B5H11
26 24
P4Ph4 B5H9
22
B4H10
20
C4H4 4
electron-precise
C6H6
4n + 6 4n + 4 electron-deficient
B6H10
4n + 2
B6H62ⴚ
4n
TI66ⴚ
hypoelectronic
B5H52ⴚ
5
6
Vertex Count Figure 4. Vertex count and electron count in main-group clusters.
S6
S8
Figure 5. Typical electron-rich clusters showing location of bonding (bold lines), exo (shaded) and endo electron pairs.
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cise, or electron-deficient lines. Thus most students will confidently apply the Wade–Mingos rules to B4H10 (2n + 6 skeletal electrons) and conclude that its butterfly structure is generated by the loss of two vertices from the parent octahedron. They fail to note, however, that the total valence electron count of 22 (4n + 6) places the cluster firmly in the electron-rich category. In contrast, the butterfly structure of P2(PMe)2 is commonly rationalized in terms of addition of a pair of electrons to electron-precise P4 despite the fact that the total vertex and valence electron counts are identical to those in B4H10 (2). In simple algebraic terms, an arachno cluster (4n + 6 valence electrons) is only truly electron deficient for n > 6. For n = 6, it is electron precise, while for n < 6 it is actually electron rich! Similar ambiguities emerge in the nido series (4n + 4 electrons), which are electron deficient for n > 4 but electron precise for n = 4. Thus the classic electron-precise cluster, P4, can also be viewed as a nido cluster derived from a trigonal bipyramid by loss of one vertex.
antibonding
n
3n/2
endo
n
exo
n
n
bonding
n
3n/2
TVEC
2n − 1
2n
n
n–2
n+1
n+2
rich
precise
deficient
6n
5n
4n + 2
hypoelectronic 4n
Figure 6. Number of frontier orbitals of different types in electronrich, electron-precise, electron-deficient, and hypoelectronic clusters. Shaded areas indicate occupied orbitals.
b2 e b1 b1
a2
a1 a1
B4H10
C4H8
Figure 7. Destabilization of bonding orbitals by a square–diamond distortion.
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Unifying Principle: Destabilization of an Occupied MO The fundamentally different approaches adopted for electron-rich and electron-precise clusters versus electron-deficient clusters can be reconciled by noting one simple unifying principle: each reduction of two in the electron count must be associated with a structural deformation that destabilizes one of the occupied molecular orbitals. This simple concept is best illustrated by starting from typical electronrich cases (such as S8 or S6) and reducing the electron count, taking care to account for all valence electrons (not just the skeletal pairs) at each stage. At the electron-rich limit (6n valence electrons) the occupied manifold can be divided into two distinct subsets (Figure 5): one containing orbitals that are bonding between the clusters atoms (n orbitals) and the other containing orbitals that are nonbonding (2n orbitals) corresponding to the two lone pairs per vertex in a localized description of the bonding. The nonbonding orbitals can be further subdivided into two sets: the exo pairs, noted previously (shaded in Figure 5), that are directed radially out from the center and the endo pairs (shown unshaded in Figure 5) that are directed along the nonbonded edges of the cluster. In the hydrocarbon or borane clusters, the exo pairs are associated with C⫺H or B⫺H bonds. The relative energies of these three different types of occupied orbitals are shown schematically in the molecular orbital diagram in Figure 6. As noted above, each reduction of two in the electron count requires the destabilization of one orbital, and that orbital can, in principle, come from any one of the three occupied subsets (endo, exo, or bonding). Starting at the electron-rich limit, the simplest distortion that can destabilize a single occupied orbital is one that brings two nonbonded vertices together. This distortion destabilises the out-of-phase combination of two endo pairs, moving this orbital from the nonbonding set to the antibonding region. At the same time, its in-phase counterpart is stabilized and shifts into the bonding manifold. This pairwise bond formation can be repeated n兾2 times, after which all n endo lone pairs have been used, and the precise electron count (5n) is reached (Figure 6). At this point, the only remaining nonbonding orbitals are the exo lone pairs, which, by virtue of their orientation, are insensitive to pairwise motion of atoms (or indeed any other change in structure). As a result, any further reduction in electron count can only be accommodated by destabilizing an orbital from the bonding, rather than the nonbonding, manifold. The type of structural rearrangement required to bring this about is different from the pairwise motion characteristic of the transition from electron-rich to electron-precise counts, as is illustrated by the putative transformation from C4H8 (24 electrons) to B4H10 (22 electrons) in Figure 7. In C4H8 (D2d symmetry), the eight C⫺C bonding electrons are accommodated in orbitals with a1, b1, and e symmetry, all of which have dominant C⫺C bonding character. A contraction along one of the diagonals (a square–diamond distortion) then destabilizes one component of the formerly degenerate pair, precisely the result required to accommodate a reduction of two in the TVEC. The same type of square–diamond process is apparent in the transition from electron-precise C6H6, 30 valence electrons, to electron-deficient B6H62᎑, 26 valence
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a2"
This approach to the electron-deficient regime, starting from an electron-precise reference point and emphasizing the importance of a varying the total electron count (a vertical relationship in the context of Figure 4), is different from the normal Wade–Mingos approach, which would instead make the horizontal connection between C 6H 6 (formally an arachno cluster) and its electron-deficient closo “parent”, B8H82᎑. Developing the argument in this alternative way, however, clearly establishes the missing link between the electron-deficient versus the electron-rich and electron-precise domains.
a2u
e" eg
eu e' a1'
a1g
B6H62ⴚ
C6H6
Figure 8. Changes in the bonding manifold associated with the transition from trigonal prismatic (C6H6) to trigonal antiprismatic (B6H62᎑) geometry.
electrons (Figure 8), the former having a trigonal prismatic structure (D3h point symmetry) while the latter cluster is a trigonal antiprism (D3d, a subgroup of the full Oh point group of the octahedron). The two structures are related by contraction along the diagonals of all three square faces, or, equivalently, by the rotation of one of the triangular faces by 60⬚ relative to the other. In C6H6, the three occupied orbitals lying along the bonds linking the two triangular units separate into two groups, based on their symmetry properties with respect to the conserved 3-fold rotation axis. The totally symmetric (a1′) combination is largely unaffected by the rotation and remains bonding throughout. In contrast, the bonding interaction between the triangular faces in the doubly degenerate pair (e′) is reduced as soon as the two triangles move away from the totally eclipsed conformation, causing the two orbitals to rise in energy. The evolution of the orbital character is somewhat complicated in this case by an avoided crossing with the degenerate antibonding orbital, which is stabilized by the rotation. Nevertheless, Figure 8 clearly shows that the overall bonding character of the lowest degenerate orbital in the trigonal prism (e′) is substantially greater than that in its counterpart in the trigonal antiprism (eu ). The contraction along the diagonals of the square faces of the trigonal prism therefore brings about the destabilization of two orbitals, exactly as required to accommodate a decrease of four in the TVEC. The two examples shown in Figures 7 and 8 illustrate the general principle that an increase in the number of triangular faces in a structure (and hence an increase in the average number of nearest neighbors) destabilizes orbitals that already have bonding character, allowing the system to accommodate electron deficiency. The more complex nature of the distortion compared to the simple pairwise motions typical of the electron-rich domain should not, however, be allowed to obscure the simplicity of the underlying principle: each reduction of two in the electron count requires one orbital to be destabilized. 736
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Hypoelectronic Clusters The C6H6 → B6H62᎑ example shown in Figure 8 illustrates one further important feature of the electron counting procedure. While the transition from trigonal prism to trigonal antiprism destabilizes the two degenerate orbitals, it has little effect on the totally symmetric combination. As a result, the minimum electron count that can be attained by destabilizing members of the bonding subset is 4n + 2 (rather than 4n), the residual two electrons corresponding to the pair occupying the totally symmetric orbital. The closo electron count of 4n + 2 is usually taken as the ultimate limit of electron deficiency but, very recently, a new class of anionic clusters of the heavier group 13 elements (Tlnn᎑, n = 6, 7, 9, are representative examples; ref 4 ) has been identified where the electron count (4n) is even lower. These hypoelectronic species, have deltahedral structures similar to those adopted by the closo boranes, but two or more vertices are displaced towards the center of the cluster, effectively flattening the deltahedron. All of the endo pairs, along with the bonding orbitals, have been utilized to the maximum extent in reaching the 4n + 2 configuration, and thus only the exo pairs can be utilized to accommodate the further reduction in electron count. As noted previously, the orientation of these exo pairs should make them rather insensitive to any structural change. In the Tl clusters, however, the exo pairs are not strongly directional and the close approach of two vertices on opposite sides of a deltahedron therefore induces significant overlap, destabilizing the antisymmetric combination in precisely the same way as was observed for the endo analogues in the electron-rich regime. Indeed, the nature of the transannular bonding in Tl66᎑ and Tl77᎑ has been compared to the ‘inverted hybridization’ in tricyclo[1.1.1]pentane (5). Key Points The key ideas in this approach to electron counting are that all clusters can be considered as part of a continuum characterized by electron and vertex counts (Figure 4), and that the type of structural distortion characteristic of each regime is a direct consequence of the nature of orbitals involved (Figure 6). The utility of these concepts is illustrated in Figure 9, where the structural relationships within the 6vertex and 5-vertex families are summarized in detail. Starting from the cyclic S6 structure, the pairwise motion of two vertices destabilizes an antisymmetric combination of endo pairs, generating the transannular bond in [Te3S3]2+ (34 electrons). Two further distortions of a similar nature bring us to the electron-precise C6H6 (30 electrons), at which point the endo pairs have been fully utilized. The transition from electron-precise to electron-deficient regimes involves the
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TVEC
S6
[Te3S3] 2ⴙ
C6H6
B6H10
B6H62ⴚ
TI66ⴚ
36
34
30
28
26
24
electron-deficient
electron-rich
hypoelectronic
electron-precise
TVEC
S5
B5H11
B5H9
B5H5 2ⴚ
30
26
24
22
Figure 9. Structural relationships within the 6- and 5-vertex families.
destabilization of bonding pairs, and so a series of square– diamond processes is required to increase the number of triangular faces structure from two (C6H6) to five (B6H10) to eight (B6H62᎑). Finally, the transition from electron-deficient (closo-B6H62᎑) to hypoelectronic (Tl66᎑) is accommodated by a pairwise motion of two opposite vertices, destabilizing an antisymmetric combination of exo pairs. Thus far we have emphasized the central position of electron-precise species such as C6H6, but similar trends can be identified in odd-vertex families such as B5H11 → B5H9 → B5H52᎑, where no electron-precise species exists. The structure of the B5H11 species (26 electrons, an arachno borane in the Wade–Mingos treatment, but clearly electron-rich) can also be viewed as a derivative of a cyclic 30-electron species (such as the unknown S5 ), formed by the pairwise motion of two sets of vertices. A second pairwise motion yields B5H9, which sits on the electron-deficient side of the boundary so a further reduction in electron count must be accommodated by a square–diamond distortion, leading to trigonal bipyramidal B5H52᎑. Summary This article presents a unified approach to correlating structure and electron count across the entire spectrum of
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main-group clusters, from electron-rich species such as S6 all the way through to hypoelectronic Tl66᎑, via intermediate electron-precise (C6H6) and electron-deficient (B6H62᎑) cases. Most importantly, it establishes a link between the electrondeficient and electron-precise domains that is absent in standard treatments of the subject. The fundamental electronic principle linking all four classes is that, for each reduction of two in the electron count, a single occupied orbital must be destabilized. The classes are then seen to differ only in the nature of the orbital involved: nonbonding (endo), bonding and nonbonding (exo) in the electron-rich, electron-deficient and hypoelectronic domains, respectively. Literature Cited 1. Mingos, D. M. P. Introduction to Cluster Chemistry; PrenticeHall: Englewood Cliffs, NJ, 1990. Mingos, D. M. P. Acc. Chem. Res. 1984, 17, 311. 2. Woollins, J. D. Non-Metal Rings, Cages and Clusters; Wiley and Sons: Chichester, United Kingdom, 1988. 3. Wade, K. Electron-deficient compounds; Nelson and Sons, Ltd.: London, 1971. Wade, K. J. Chem. Soc., Chem. Commun. 1971, 792. 4. Corbett, J. D. Angew. Chem., Int. Ed. Engl. 2000, 39, 670. 5. King, R. B. Inorg. Chem. 2002, 41, 4722.
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