Topological properties, principles

If the topological property which defines an atom is also one of physical significance, then it should be possible to obtain from quantum mechanics an equivalent mechanical definition. As demonstrated in Chapters 5 and 8, this can be accomplished through a generalization of the quantum action principle to obtain a statement of this principle which applies equally to the total system or to an atom within the system. The result is a single variational principle which defines the observables, their equations of motion, and their average values for the total system or for an atom within the system. 

An understanding of chemistry requires a regional description of a system. The notion that a molecule can be viewed as a collection of atoms linked by a network of bonds, a notion that has already been shown to be rooted in the topological properties of the charge distribution, is the operational principle... 

An additional limitation of the index theorem is that in higher dimensions, there may often be different time scales corresponding to fast and slow reactions, so that the dynamics rapidly relax to a manifold that is embedded in the N-sphere. In this case, the Euler-Poincare characteristic of the submanifold may be different from that of the iV-sphere and (14) would have to be modified appropriately. A very general mathematical approach in which the geometric and topological properties chemical kinetic equations is treated has recently been developed. Although the formulation is in principle capable... 

In this approach, connectivity indices were used as the principle descriptor of the topology of the repeat unit of a polymer. The connectivity indices of various polymers were first correlated directly with the experimental data for six different physical properties. The six properties were Van der Waals volume (Vw), molar volume (V), heat capacity (Cp), solubility parameter (5), glass transition temperature Tfj, and cohesive energies ( coh) for the 45 different polymers. Available data were used to establish the dependence of these properties on the topological indices. All the experimental data for these properties were trained simultaneously in the proposed neural network model in order to develop an overall cause-effect relationship for all six properties. 

For over a decade, the topological analysis of the ELF has been extensively used for the analysis of chemical bonding and chemical reactivity. Indeed, the Lewis pair concept can be interpreted using the Pauli Exclusion Principle which introduces an effective repulsion between same spin electrons in the wavefunction. Consequently, bonds and lone pairs correspond to area of space where the electron density generated by valence electrons is associated to a weak Pauli repulsion. Such a property was noticed by Becke and Edgecombe [28] who proposed an expression of ELF based on the laplacian of conditional probability of finding one electron of spin a at t2, knowing that another reference same spin electron is present at ri. Such a function... 

Consequently, it must be emphasized that precautions have to be taken with the conventional rough description of molecules based on the chemical bond pattern. In a molecule that contains at least two atoms which do not belong to the first row of the periodic table, the energy and all the monoelectronic properties are literally spread out over the whole molecule. Obviously, the concept of chemical bond, based as it is on the principle of topological proximity, is inadequate on its own for a correct description of the chemical and physical behavior of such a molecule. 

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