Leapfrog rule

The chemically significant fact about the leapfrog fullerenes is that, considered as neutral carbon frameworks Cy each has exactly 3 /2 bonding and 3n/2 antibonding 71 orbitals. As fullerenes exist for all n-20 + 2k(ki ), the leapfrog rule is thus ° ... 

The only closed-shell fullerenes predicted by simple Hiickel theory fall into two types the leapfrog fullerenes Ceo+en (n 1), and the cylinders 70+30 and 34+36 (m > 0) formed by stretching Dsh-Cio and Da-C 4 along then-respective 5 and 5 axes. The simplicity of the theory unfortunately prevents these rules having any general utility. Neither of the two IPR-isomers of 75 is predicted to be stable but Z>2- 76 has been isolated, whereas the expected fiillerene Ded-Cji has not, although its apparent low stability may be... 

For example, the second order (two-step) Leapfrog method frequently used in meteorology and oceanography can be deduced from the midpoint rule [66, 49, 158] ... 

A survey of the Hiickel spectra of the lower fullerenes rapidly reveals that properly closed n shells are very much the exception rather than the rule most fullerene isomers have pseudoclosed k configurations. The rare occurrences of properly closed shells can be almost entirely described by two magic number rules that are to the fullerenes what the Hiickel 4n + 2 is to monocyclic systems and the Wade n + 1 rule is to boranes. The two are the leapfrog and the carbon cylinder rules. 

Closed-shell fullerenes following the carbon cylinder mle are again rare (just 3 for ft < 100) and again an experimentally characterized fullerene is the parent of an infinite series of properly closed shells. For n < 112 the leapfrog and cylinder rules together exhaust the list of properly closed shells. They are not complete for all n, however, as the next section points out. 

Table 2. Enumeration of C Fullerene Isomers in the Range n < 140 with Sporadic Properly Closed Shells outside the Leapfrog and Carbon-Cylinder Magic-Number Rules...

The method is based on another time-marching scheme not mentioned in the above sections the leapfrog method [28, 29], also called the midpoint rule by Hairer and Wanner [6], using central differences. Equation (4.1) can be approximated as... 

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