HMO Criteria for Aromaticity in Cyclic pp Systems

A comparison of various properties of the HMO model for linear and mono-cyclic conjugated (pp) systems made it possible to derive numerical values for a set of upper or lower bounds differentiating theoretically between typical aromatic or olefinic behaviour in cyclic tt systems U3f The most important of these are collected in (3). 

The notations and definitions used here are given in Table 3. 

Condition (3a) is necessary to ensure a closed-shell ground-state configuration, whereas conditions (3b) and (3c) are minimum conditions for aromatic thermodynamic ground-state stability of 7r systems differing in their number and topological connection of pn orbitals, as well as in the number of ir electrons present. Condition (3d) is necessary to specify low aromatic reactivity. 

But the striking excess property aromaticity might be even better characterized by values close to that of the standard aromatic compound, benzene, which are given in (3) for comparison. 

In this section no distinction is made between outer or inner d orbitals, and for simplicity it is assumed that the ri-orbital center may be described by the 

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