Topologies definitions

Tsirelson, V.G., Zou, P.F. and Bader, R.F.W. (1995) Topological definition of crystal structure determination of the bonded interactions in solid molecular chlorine, Ada Cryst., A51, 143-153. 

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. 

Bond Order Indices Calculated Using the Topological Definition... 

Cyclopropyl homoconjugation can be easily detected and described as long as one retains its topological definition. This also holds to some extent in the case of no-bond homoconjugation. However, as soon as one has to assess the chemical relevance of homoconjugation and to determine a homoconjugative bond (electron) delocalization energy, one needs, as mentioned in Section I, suitable reference compounds for comparison. 

Topological Definition of Atoms, Bonds, and Structure.—The definitions of an isolated atom, of an atom in a molecule, of a chemical bond, and of molecular structure derive from the properties of critical points of the charge density. 

Because of the dominant topological property exhibited by a molecular charge distribution—that it exhibits local maxima at the positions of the nuclei—the imposition of the quantum boundary condition of zero flux leads directly to the topological definition of an atom. Indeed, the interatomic... 

Recalling that the topological definition of an atom implies the zero-flux surface condition, a region of space bounded by a surface satisfying eqn (5.86) at the point of variation is henceforth called an atom. It is required that, as tends to ip, 2(( ) is continuously deformable into the region flfi/r) as.sociated with the atom. The region 2(< ) thus represents the atom in the varied total system, which is described by the trial function <, just as 1( ) represents the atom when the total system is in the state described by kp. 

The condition, started in eqn (8.113), that the atomic action integral vanish for all time intervals may be taken as the quantum definition of an atom. It is a direct consequence of the topological definition of an atom as the union of... 

It is the use of zero-flux surfaces for the topological definition of atoms or functional grouping of atoms in molecules that maximizes the extent of transferability of their properties between systems. 

Gatti, C., Saunders, V.R. and Roetti, G. (1994) Crystal field effects on the topological properties of the electron density in molecular crystals the case of urea, /. Chem. Phys., 101, 10686-10696. Tsirelson, V.G., Zou, P.F. and Bader, R.F.W. (1995) Topological definition of crystal structure determination of the bonded interactions in solid molecular chlorine. Cryst., A51, 143-153. Platts, J.A. and Howard, S.T. (1996) Periodic Hartree-Fock calculations on crystalline HCN,/. Chem. Phys., 105,4668-4674,... 

The beauty of the above topological definition of the atom in a molecule lies in the fact that it coincides with the rigorous quantum mechanical definition of an open subsystem [27, 33, 34]. In particular, the atomic action integral, which is defined through the atomic one-particle Lagrangian density, is zero within the atomic volume ... 

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