Application of distance and similarity measures. The comparison of molecular electronic structures in arbitrary electronic states

Covalent bond indices and ionicities from similarity measures

“…As will be shown in next section, an analysis of the domain-restricted reduced density matrices may be carried out by means of the mathematical concepts of distance [23] and similarity measures [24,25]. The distance measure, d, between two n-dimensional elements x and y in a metric space M is defined as a real valued function fulfilling the following properties [15,24,25]: dðx; yÞ P 0; dðx; yÞ ¼ 0 () x ¼ y; dðx; yÞ ¼ dðy; xÞ ðsymmetryÞ; dðx; zÞ 6 dðx; yÞ þ dðy; zÞ ðtriangle inequalityÞ:…”

“…Particular examples [24,25] of distance measure d in different metric spaces M are those of Bray-Curtis (BC)…”

“…Its value is given by a real valued function s fulfilling the properties 0 6 sðx; yÞ 6 1; sðx; xÞ ¼ 1; sðx; yÞ ¼ sðy; xÞ; ð7Þ that quantifies the number of matches between some features of two objects (quantitative or qualitative), whereas the distance measures the 'difference' between them. The use of similarities instead of distances requires to consider some specific functional relationships between s and d. In this work we will use s = 1 À d due to the physical nature of the quantities involved [24,25]. Note that in this framework the distance function is bounded to unity, i.e., it is normalized [24,25].…”

Similarity ideas in the theory of pericyclic reactivity

“…32, No. 6, 1992 697 the whole two-step process (eq 18) can be expressed in terms R -I -P (18) of similarity indices of individual reaction steps by the relation ístepwise = O ~rRl) + rR,(l ~ ) = U _ rRIrIp) (19) Since the extent of reorganization in the one-step concerted process is given by eq 16, it is possible to expect that the relative ease of concerted and stepwise reaction mechanisms can be estimated on the basis of comparison of the similarity index trp with the product rRinp. If the value of trp is higher than the value of the product, the concerted reaction mechanism can be expected to be preferred.…”

A classification of polyenes into 4L + 2‐ and 4L‐classes on the basis of Coulson's bond orders and information theory and its application to the interpretation of electrocyclic reactions

“…The conservation of the individuality of a fragment in a definite electronic state of a complex molecule can be estimated rather correctly in the framework of an arbitrary MO-LCAO approximation using the partial similarity measure between the electronic structure of the fragment and of the corresponding isolated reference molecule [42].…”