Hirsch rule

There are similarities and differences between the Hirsch rule and those developed by physicists, originally to rationalize the magic numbers observed experimentally for simple metal clusters. The reader is referred to the excellent review by de Heer for more information [71]. The magic numbers for sodium clusters, 8, 20,... 

Hirsch and co-workers calculated NICS values for tetrahedral clusters of N, P, As, Sb, and Bi, as well as for the corresponding tetra-anions composed of Si, Ge, Sn, or Pb atoms, finding diatropic values for 2,n(n 1) jr-systems.296a b It was postulated by Hirsch, Schleyer, and their co-workers that for icosahedral fullerenes and their hetero-analogues the Hiickel rule, involving 4/2+2 //-electrons, should be replaced by the 2(/2+l)2 electron rule.296... 

Hirsch s rule has more limited applicability than the Hiickel rule. However, the 2(n +1)2 concept has been used very successfully to interpret relative fullerene stabilities [66], and to suggest new systems, including neutral and charged non-fullerene carbon [67] and homoaromatic cages [68]. All these species have large NICS values in their centers and satisfy other criteria of aromaticity. 

While Hirsch conceived his 2(n + l)2 electron rule for spherical aromatics, subsets of three-dimensionally aromatic molecules having very high symmetries ( Ti, Oj, h, etc.), it can be applied to lower symmetry clusters such as the nine-vertex examples above. In cluster molecules the highest degeneracy MOs of a spherically harmonic atom set split into related, but lower degeneracy (or even non-degenerate) components. 

More advanced mathematical aspects of aromaticity are given in other references [33, 34]. Some alternative methods beyond the scope of this chapter for the study of aromaticity in deltahedral molecules include tensor surface harmonic theory [35-38] and the related Hirsch 2 N -b 1) electron-counting rule for spherical aromaticity [39]. The topological solitons of nonlinear field theory related to the Skyrmions of nuclear physics have also been used to describe aromatic cluster molecules [40]. 

Nonplanar situations arise in fullerenes where a description of ji electrons in three dimensions is necessary. Haddon argued that orbital orthogonality is the key to the o—n separability, and he presents a recipe on how to conserve orbital orthogonality in three dimensions. Hirsch et al. showed that a new electron counting rule, different from the Hiickel An + 2 rule, can be used to describe the spherical aromaticity of fullerenes of symmetry. This interesting work may provide a stimulus to take a closer look into the a—n separation of fullerenes. 

Today, it is clear that aromaticity is possible in 3-dimensional, as well as planar, systems, such as quasi-spherical cages of fullerenes [43,44] and polyhedral boranes (e.g. Bi2Hi2 ) [44, 45], in carbon nanotubes and in some metal clusters (e.g., AusZn+, Au2o) [46], The 2( - -1) rule proposed by Hirsch [47] and successfully applied to design various novel aromatic compounds, serves as the 3-dimensional... 

In 2000, Hirsch s 2( +1) rule of aromaticity for spherical compounds [18,46,47] was introduced as the spherical analog of Hiickers 4n + 2 mle. Hirsch s mle is based on the fact that the Ji-electron system of an icosahedral fullerene can be, in a first approximation, considered as a spherical electron gas surrounding the surface of a sphere. The corresponding wave functions of this electron gas are characterized by file angular momentum quantum number 1(1 = 0,1,2,...), with each energy level 2 Z + 1 times degenerated, and thus all jt-sheUs are completely filled when we have 2 ( + 1) electrons. For such reason, spherical species with 2(n +1) Ji-electrons are aromatic, like icosahedral 20, or Cgo ". ... 

Hirsch A, Chen Z, Jiao H (2000) Spherical aromaticity in icosahedral fullerenes the 2(N + 1) rule. Angew Chem Int Ed 39 3915-3917... 

Chen Z, Jiao H, Hirsch A, Thiel W (2001) The 2(N -i-1) rule for spherical aromaticity further validation. J Mol Model 7 161-163... 

Aromaticity is useful to rationalize and understand the structure and reactivity of many organic molecules. In 1971, Wade proposed a similar concept to describe delocalized a-bonding in closed-shell boron deltahedra, which follow a 2n-t-2 skeletal electron rule [34]. This concept has been extended by Hirsch to treat spherical clusters by his 2(n + 1) rule, and various applications to organic and inorganic clusters have been reported [35]. However, stability based on aromaticity had not been confirmed for any metallic moiety until Li et al. [36] published their seminal paper Observation of All-metal Aromatic Molecules. ... 

As in the Hiickel model, only the connection between the atoms is taken into account, the 4n and 4n + 2 rules remain valid in the case of rings where s electrons are itinerant. For example, Lij has a perfect symmetry and two itinerant s electrons. Hence it is expected to be aromatic. The Hiickel model can also be applied to three-dimensional systems, such as fullerenes [4], where electron count rules can also be formulated. In the case of icosahedral fullerenes, Hirsch et al. have formulated such a rule [5], but other electron counting rules also exist [6]. As in fullerenes the itinerant electrons are delocalized over a nearly spherical surface, these rules account for spherical aromaticity [7]. 

Some examples for 3D spherical aromaticity can be fonnd in different fullerenes. It is well known that the Ji orbitals of the carbon atoms in fullerenes are perpendicular to the surface and these electrons are itinerant, similarly to the benzene. Hence, the electronic structure of these compounds can be modeled using a Hiickel-type model, and the condition to obtain a closed electronic structure and aromaticity can also be formulated as previously done by Hirsch et al. Their rule is valid for icosahe-dral fullerenes. We will see that spherical aromaticity is a more common feature of metal clusters. 

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