Aromaticity is among the most frequently used concepts in the chemical literature. Nevertheless, it has to be stressed that aromaticity is not an observable that could be directly obtained from molecular wave function. So, it remains as a controversial concept and its physical origin is still being debated. This has lead to multiple ways for assessing the aromatic or anti-aromatic character of molecules. Thus, several indices of aromaticity have been proposed throughout the literature. It is well-known that aromaticity is strongly related to the electron delocalisation of the electrons along all the centres of the ring. The introduction of the n-order density matrix, and development of n-centre delocalisation indices (n-DIs), has provided a quantitative tool to measure electron delocalisation within a certain ring. Therefore, n-DIs can be regarded as a promising way to determine the actual role played by electron delocalisation in aromaticity. n-DIs are found to be in line with most of the qualitative trends predicted by the harmonic oscillator model of aromaticity indices, HOMAs, aromaticity indices derived from the Chemical Graph Theory, properties of the ring critical points, and circuit resonance energies. It has to be noticed that the direct comparison of n-DIs computed for rings with a different number of centers does not provide a picture of relative aromatic stabilisation. In fact, aromatic stabilisation does not only depend on the extension of the electron delocalisation over the n centres, but also on the number of centres. n-DIs dependence on the number of centres and electrons in a ring prevents to use their direct comparison as a measure of aromatic stabilisation unless an appropriate scaling is applied. Effective Scaled Electron Delocalisation indices, ESED, have been developed for comparing the total aromaticity of polyaromatic hydrocarbons. n-DIs exhibit very different behaviour for aromatic and antiaromatic hydrocarbons. Thus, an aromatic compound is one whose multicenter electron delocalisation is larger than that of the corresponding linear structure, whereas an antiaromatic compound is one where the multi-centre electron delocalisation is smaller than that of the linear molecule. Those compounds that display very small differences between linear and cyclic structures may be called non-aromatic. n-DIs are also found as a promising tool to study the aromaticity of transition states in concerted processes. This book also reports on its application to the study of the aromaticity in all metal compounds.
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