Topological equivalent state

These topological indexes, based on the molecular connectivity approach, include three types the ""Xr molecular connectivity chi indexes that characterize the structural attributes of molecules, the ""k kappa indexes of molecular shape, and the topological equivalence state T values that individually characterize atoms and groups in the molecular skeleton and are used primarily to determine chemically equivalent atoms within a molecule. A further development of this approach has led to the electrotopological state atom indexes, which will not be discussed here but will be presented elsewhere. Molecular connectivity chi indexes are discussed in the first part of this paper along with illustrative applications. Then kappa shape indexes are discussed. The topological state index is discussed in the final section. 

The importance of the arguments we have outlined lies in the fact that they provide a theoretical foundation both for aromaticity-antiaromaticity and for pericyclic selection rules. They furthermore demonstrate the relationship between the two The topological equivalence between an array of p orbitals in a w system of a carbon chain or ring and a pericyclic transition state, composed of an... 

MOLCONN-Z EduSoft, LC www.eslc.vabiotech.com/molconn/ Molecular connectivity, molecular connectivity difference, and kappa shape indices, E-state indices, atom-type and group-type E-state indices, topological equivalence classification of atoms, other topological indices, counts of subgraphs paths, rings, clusters, etc. vertex eccentricities... 

The imaginary connection of overlapping lobes of 2p-AO transition state results in a curve which is topologically equivalent (homeomorphic) to the curve obtained in the similar analysis of a benzene molecule. The difference between a and ji overlaps of AOs is not taken into consideration. 

The antiaromatic geometry found along the concerted path of ground-state-forbidden pericyclic reactions, which is topologically equivalent to an antiaromatic Hiickel [4n]annulene or MObius [An + 2]annulene, is a particularly interesting type of biradicaloid geometry. (Cf. Section 4.4.) Other biradicaloid geometries and combinations of those mentioned are equally possible. 

Hall, L.H. and Kier, L.B. (1990). Determination of Topological Equivalence in Molecular Graphs from the Topological State. Quant.Struct.-Act.Relat., 9,115-131. 

These ideas also generalize neatly to higher-order systems. A fixed point of an th-order system is hyperbolic if all the eigenvalues of the linearization lie off the imaginary axis, i.e., Re(Aj iO for / = ,. . ., . The important Hartman-Grobman theorem states that the local phase portrait near a hyperbolic fixed point is topologically equivalent to the phase portrait of the linearization in particular, the stability type of the fixed point is faithfully captured by the linearization. Here topologically equivalent means that there i s a homeomorphism (a continuous deformation with a continuous inverse) that maps one local phase portrait onto the other, such that trajectories map onto trajectories and the sense of time (the direction of the arrows) is preserved. 

The topological state index (T ) of an atom represents the position of the atom in the scaffold of the molecular structure in relation to aU other atoms of the molecule (hut based on topology, i.e., on the connectivity and not on the three-dimensional structure) [54], Chemically and topologically equivalent atoms have identical indices. 

Hall LH, Kier LB. Determination of topological equivalence in molecular graphs from the topological state. Quant Struct-Act Relat 1990 9 115-131. 

This pair of delta values is seen as a characterization of the atom in its valence state. The simple delta, 5, describes the role of the atom in the skeleton in terms of its connectedness and count of sigma electrons it could be called the sigma electron descriptor. The valence delta, 8, encodes the electronic identity of the atom in terms of both valence electron count and core electron count. It could be called the valence electron descriptor. The isolated, unbonded atom may be thought of as characterized by its atomic number, Z, and the number of valence electrons, Z. In its valence state, the bonded atom is characterized by 8 and 8. Embedded in the molecular skeleton, the full characterization of the atom in the environment of the whole molecule is given by the topological equivalence value, described in a later section, and the electrotopological state value, presented separately.A representation of the whole molecule is accomplished by the combination of chi, kappa, and topological state indexes. 

As a further development, Hall and Kier introduced the idea of the topological state of the graph vertex (which represents the skeletal atom). The topological state of the skeletal atom is the basis of both the determination of topological equivalence as well as computation of graphical indexes which can be related to the properties of the atoms in the molecule, The atom is viewed in the context of the full topology of the whole molecule. 

The key to useful topological state values is an appropriate form for the r, values. Hall and Kier have shown that simple forms, such as the graph distance d,j, are not useful because they fail to indicate proper topological equivalence. To ensure representation of topological equivalence, two features of the paths must be encoded (1) atomic identity and (2) the sequence of atoms in each path. It has been shown that both these characteristics can be encoded as follows. Atomic identity can be encoded using the molecular connectivity valence delta value, 8. The discussions concerning chi indexes and related quantities have shown the validity of the valence delta value as a characterization of atoms. 

Figure 13.18 The Evans principle applied to the Diels-Alder reaction and the [2-l-2]cycloaddition. The delocalized system of electrons in the transition state of the Diels-Alder reaction is topologically equivalent to the 3t system of benzene. The transition state is stabilized by aromaticity, and therefore the reaction is thermochemically...

Figure 13.26 The orbital overlap of the orbital basis of the coarctate transition state of the cyclopropyl carbene fragmentation is topologically equivalent to the % system of...

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