Topological atom definition

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... 

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. 

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

The topological definition of an atom and the associated ideas of structure and structural stability are introduced in Section 2. These ideas can be presented in a pictorial and qualitative manner. The quantum definition of an open system and its identification with the topological atom are presented in Section 3. The consequences of this identification are explored without presenting its derivation, the approach that must necessarily be followed in this general introduction to the theory. 

Figure 3 Dual-topology definition for ZMP (1) and AMP (2). Dashed structures incorporate dummy atoms (D).

Figure 3. Single topology definition for compound JG365 and JG365A. The chemical symbols with the D prefix indicate dummy atoms and X=Ac-Ser-Leu-Asn-(Phe-Hea-Pro)...

Quantum mechanics applies to a segment of a system, that is, to an open system, if the segment is bounded by a surface of zero flux in the gradient vector field of the density. Thus the quantum mechanical and topological definitions of an atom coincide [1]. The quantum mechanical rules for determining the average value of a property for a molecule, as the expectation value of an associated operator, apply equally to each of its constituent atoms. 

One way of getting rid of distortions and basis set dependence could be that one switches to the formalism developed by Bader [12] according to which the three-dimensional physical space can be partitioned into domains belonging to individual atoms (called atomic basins). In the definition of bond order and valence indices according to this scheme, the summation over atomic orbitals will be replaced by integration over atomic domains [13]. This topological scheme can be called physical space analysis. Table 22.3 shows some examples of bond order indices obtained with this method. Experience shows that the bond order indices obtained via Hilbert space and physical space analysis are reasonably close, and also that the basis set dependence is not removed by the physical space analysis. 

A modem description of a conventional hydrogen bond as well as its older, more accurate definition are based on Bader s theory of atoms in molecules (AIM theory) [4]. Bader considers matter a distribution of charge in real space of point-like nuclei embedded in the diffuse density of electron charge, p(r). All the properties of matter are manifested in the charge distribution and the topology... 

Definition) All atoms of the ligands to be compared arc ordered according to their topologic distance (number of bonds to the stereogenic center). All atoms of equal topologic distance are said to form a sphere, The atoms in the first sphere are called proximal atoms (p). The atoms of spheres II, III,. . N.. are bound to the stereogenic center via 2, 3. , N.. bonds. 

This unique labeling problem 14) originally required the definition of multiple equilibrium positions and redundant atom labelling schemes... resulting in a tedious over-definition of the molecular topology 14). 

From their definitions, one may admit that topological indices may code two structural factors, namely, the size and the shape of the molecule. This assumption is proved by calculating 94) the schemes (B) in Figure 2 for the 36 polyalkylcyclohexanes with 6-10 carbon atoms, and for a subseries of 22 polyalkylcyclohexanes with 10 carbon atoms (resulting in scheme (C)). The schemes (A)-(C) corroborated with the... 

For a limited discussion of fractal geometry, some simple descriptive definitions should suffice. Self-similarity is a characteristic of basic fractal objects. As described by Mandelbrot 58 When each piece of a shape is geometrically similar to the whole, both the shape and the cascade that generate it are called self-similar. Another term that is synonymous with self-similarity is scale-invariance, which also describes shapes that remain constant regardless of the scale of observation. Thus, the self-similar or scale-invariant macromolecular assembly possesses the same topology, or pattern of atomic connectivity, 62 in small as well as large segments. Self-similar objects are thus said to be invariant under dilation. 

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