Topological atom critical points

H-bonds intermolecular H-bonds Tr-electron delocalization topological parameters critical points QTAM (Quantum Theory Atoms in Molecules method) covalent hydrogen bonds interaction energy decomposition scheme. 

Points on the zero-flux surfaces that are saddle points in the density are passes or pales. Should the critical point be located on a path between bonded atoms along which the density is a maximum with respect to lateral displacement, it is known as a pass. Nuclei behave topologically as peaks and all of the gradient paths of the density in the neighborhood of a particular peak terminate at that peak. Thus, the peaks act as attractors in the gradient vector field of the density. Passes are located between neighboring attractors which are linked by a unique pair of trajectories associated with the passes. Cao et al. [11] pointed out that it is through the attractor behavior of nuclei that distinct atomic forms are created in the density. In the theory of molecular structure, therefore, peaks and passes play a crucial role. 

Figure 6.15 Three-dimensional representation of the sulfur atom in SC12. This atom is bounded by two interatomic surfaces (IAS) and one surface of constant electron density (p = 0.001 au). Topologically, an atom extends to infinity on its nonbonded side, but for practical reasons it is capped. Each interatomic surface contains a bond critical point (BCP).

The topological characteristics of CPs on bonds gives a quantitative explanation of the known effect that the formation of a Ge crystal is accompanied by shifting the electron density towards the Ge-Ge bonding line. This is can be seen by comparing the parameters of the curvature of the electron density at the critical point (3,-1) with analogous parameters for a procrystal (a set of noninteracting spherical atoms placed at the same... 

Bond critical points represent extremes of electronic density. For this reason, these points are located in space where the gradient vector V p vanishes. Then the two gradient paths, each of which starts at the bond critical point and ends at a nucleus, will be the atomic interaction line. When all the forces on all the nuclei vanish, the atomic interaction line represents a bond path. In practice, this line connects two nuclei which can consequently be called bonded [5]. In terms of topological analysis of the electron density, these critical points and paths of maximum electron density (atomic interaction lines) yield a molecular graph, which is a good representation of the bonding interactions. 

What makes this bond so unusual is that the proton-accepting atom is a hydrogen atom, although the topological properties of the electron density at the bond s critical points allow us to distinguish these two types of bonding. 

The promolecule density shows (3, — 1) critical points along the bond paths, just like the molecule density. But, as the promolecule is hypothetical and violates the exclusion principle, it would be incorrect to infer that the atoms in the promolecule are chemically bonded. In a series of topological analyses, Stewart (1991) has compared the model densities and promolecule densities of urea,... 

The electron density is a continuous function that is experimentally observable, hence uniquely defined, at all points in space. Its topology can be described in terms of the distribution of its critical points, i.e. the points at which the electron density has a zero gradient in all directions. There are four kinds of critical point which include maxima (A) usually found near the centres of atoms, and minima (D) found in the cavities or cages that lie between the atoms. In addition there are two types of saddle point. The first (B) represents a saddle point that is a maximum in two directions and a minimum in the third, the second (C) represents a saddle point that is a minimum in two direction and a maximum in the third. One can draw lines of steepest descent connecting the maxima (A) to the minima (D), lines whose direction indicates the direction in which the electron density falls off most rapidly. Of the infinite number of lines of steepest descent that can be drawn there exists a unique set that has the property that, in passing from the maximum to the minimum, each line passes successively through a B and a C critical point. This set forms a network whose nodes are the critical points and whose links are the lines of steepest descent connecting them. 

Then, analyzing the electron density topology requires the calculation of Vp and of the hessian matrix. After diagonalization one can find the critical points in a covalent bond characterized by a (3, -1) critical point, the positive curvature X3 is associated with the direction joining the two atoms covalently bonded, and the X2, curvatures characterize the ellipticity of the bond by ... 

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. 

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