Non-Bonding Orbitals in Hydrocarbons

A conjugated organic system is described as alternant if the atoms in it can be divided into two groups starred and unstarred in such a way that no two atoms are of like parity i.e., which belong to the same group, are directly linked. Some typical examples of even and 

Therefore, the coefficients of the paired molecular orbitals are numerically the same, but the signs of one set of MO s are reversed in going from one MO to the other. 

These conclusions can also be interpreted in terms of the m - n rule developed above and are a direct consequence of the properties of the Eq. (1) matrix. The Pairing Theorem does not completely define the spectrum of molecular orbitals in an alternant hydrocarbon, because (1) is consistent with the presence of an even number of molecular orbitals. Whilst this might generally mean that the number of non-bonding molecular orbitals is 0, there are instances when there are 2, 4, etc. For example, both butadiene and cyclo-butadiene are even alternant hydrocarbons, but the former has no non-bonding molecular orbitals and the latter 2 (see Fig. 13). 

Group theory can be used to resolve this distinction, because the sets of paired molecular orbitals defined above are related by specific symmetry relationships. If ]  

For example, for cir-butadiene (C2v point group) Fa corresponds to the bi representation (the carbon atoms lie in the xz plane) and the paired molecular orbitals shown in Fig. 13 are related in the following manner  

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