Topological atom

In addition to looking for data trends in physical property space using PCA and PLS, trends in chemical structure space can be delineated by viewing nonlinear maps (NLM) of two-dimensional structure descriptors such as Unity Fingerprints or topological atom pairs using tools such as Benchware DataMiner [42]. Two-dimensional NLM plots provide an overview of chemical structure space and biological activity/molecular properties are mapped in a 3rd and/or 4th dimension to look for trends in the dataset. 

Till now, the AIM in QCT comes from an entirely topological origin. The single most important step forward in the theory was the realization that a quantum mechanical AIM coincides exactly with this topological atom [53]. The definition of an interatomic surface is given by... 

Scheme 3.12. The topological atom connectivity of several MCRs Ugi-, Passerini-, Asinger- and Cewald variants.

The coincidence of the topological and quantum definitions of an atom means that the topological atom is an open quantum subsystem, free to exchange charge and momentum with its environment across boundaries which are defined in real space and which, in general, change with time. It should be emphasized that the zero-flux surface condition is universal— it applies equally to an isolated atom or to an atom bound in a molecule. The approach of two initially free atoms causes a portion of their surfaces to be shared in the creation of an interatomic surface. Atomic surfaces undergo continuous deformations as atoms move relative to one another. They are, however, not destroyed as atoms separate. 

Molecular orbital theory has played the central role in the definition and understanding of problems of electronic structure. The charge density plays the corresponding role in the definition and understanding of the concepts associated with molecular structure. The previous chapters have shown that atoms, bonds, and structure are indeed consequences of the dominant topological property exhibited by a molecular charge distribution. What remains to be done is to demonstrate that the topological atom and its properties have a basis in quantum mechanics. 

A theory is only justified by its ability to account for observed behaviour. It is important, therefore, to note that the theory of atoms in molecules is a result of observations made on the properties of the charge density. These observations give rise to the realization that a quantum mechanical description of the properties of the topological atom is not only possible but is also necessary, for the observations are explicable only if the virial theorem applies to an atom in a molecule. The original observations are among the most important of the properties exhibited by the atoms of theory (Bader and Beddall 1972). For this reason and for the purpose of emphasizing the observational basis of the theory, these original observations are now summarized. They provide an introduction to the consequences of a quantum mechanical description of an atom in a molecule. 

The weights w can be any chemical or topological atomic properties. Examples of chemical -> atomic properties are - van der Waals volume, atomic mass, - polarizability examples of -> local vertex invariants are - vertex degree, - path degree, - walk degree. 

Topological atomic charges provide a canonical numbering of graph vertices, each vertex being ordered according to its branching, centrality and cyclicity. 

Moreover, topological atomic valencies TAVs, or corrected second moments) were derived from fractional atomic charges by rescaling the values with respect to the 2 -order molecular self-returning walk count srw ... 

Analogous to the topological atomic charge definition is that of topological bond orders, defined as the limit of... 

The topological bond order thus defined can be interpreted as the one-electron distribution over the bonds in a molecule, analogous to the one-electron distribution over all atoms given by the topological atomic charges ... 

Bonchev, D. and Kier, L.B. (1992). Topological Atomic Indices and the Electronic Charges in Alkanes. J.Math.Chem., 9,75-85. 

Bonchev, D. and Gordeeva, E.V. (1995). Topological Atomic Charges, Valencies, and Bond Orders. J.Chem.Inf.Comput.Sci.,35,383-395. 

D. M. M. Jaradat, S. Mebs, L. Chgcihska, and P. Luger, Experimental charge density of sucrose at 20 K Bond topological, atomic, and intermolecular quantitative properties, Carbohydr. Res., 342 (2007) 1480-1489. 

P. L. A. Popelier and P. J. Smith, Quantum topological atoms. In Chemical Modelling Applications and Theory, edited by A. Hinchlifle (The Royal Society of Chemistry, London, 2002), Vol. 2, Chapter 8, pp. 391 48. 

A (3,—3)CP, where p exhibits a local maximum (typically occurring at nuclear positions), is an attractor in the gradient field, and the topological atom is defined as the union of the attractor and its associated basin. The... 

While topological atoms in an isolated molecule are open domains with infinite boundaries, those in a perfect periodic crystal are defined by closed surfaces. In other words, the atomic basins in a crystal are formed by gradient paths originating from a true minimum ((3,- -3)CP) and terminating at a true... 

These descriptors are conceptually the same as the topological atom pairs, the difference is that S E-vectors are based on simple atom types and between any two atom types all the paths are calculated instead of the shortest one. 

The two considered atoms need not be directly connected and the separation can be the topological distance between them [Carhart, Smith et al., 1985] these descriptors are usually called topological atom pairs being based on the topological representation of the molecules. Atom type is defined by the element itself, the number of heavy-atom cormections and number of 7t electron pairs on each atom. 

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