Electron density distribution critical point

The shape of the electron density distribution in a plane through the bond critical point and perpendicular to the bond as measured by its ellipticity e. 

FIGURE 9. Perspective drawing of the calculated electron density distribution p (r) in the plane of the cyclopropane ring [HF/6-31 G(d,p) calculations]. Point p denotes the position of the bond critical point between two neighbouring C atoms. For better presentation, density values above 14 e A 3 are cut off... 

FIGURE 11. Gradient vectorfield of the HF/6-31 G(d,p) electron density distribution p (r) calculated for the plane of the cyclopropane ring. Bond critical points p are denoted by dots. There are three different types of trajectories type 1 trajectories start at infinity or the centre of the ring and end at a carbon nucleus type II trajectories (heavy lines) define the bond path linking two neighbouring carbon atoms type III trajectories form the three zero-flux surfaces between the C atoms (in the two-dimensional display only their traces can be seen). They terminate at the bond critical points... 

Another insight into the nature of a covalent bond is provided by analysing the anisotropy of the electron density distribution p (r) at the bond critical point p. For the CC double bond, the electron density extends more into space in the direction of the n orbitals than perpendicular to them. This is reflected by the eigenvalues 2, and k2 of the Hessian matrix, which give the curvatures of p (r) perpendicular to the bond axis. The ratio 2, to /.2 has been used to define the bond ellipticity e according to equation 8S0 ... 

FIGURE 13. Topological CC bond orders n of homocyclopropenium (44) and homotropenylium cation (45) calculated from the MP2 electron density distribution p(r) at the bond critical points. Note that n values for C1,C3 of 44 and C1,C7 of 45 correspond to interaction indices. MP2 bond lengths (in A) are also given44,56... 

IV. BOND CRITICAL POINT PROPERTIES OF THE ELECTRON DENSITY DISTRIBUTION OF THE SKELETAL Si-O-Si UNIT... 

The same data collection and reduction techniques are commonly used by the same workers for many different polymers. Therefore, data for these other polymers may contain errors on a similar scale, but that the errors have usually, but not always, gone undetected (8). If more than 500 reflections are observed, from single crystals of simple molecules, recognizable electron-density distributions have been derived from visually estimated data classified only a "weak", "medium" or "strong". The calculation of the structure becomes more sensitive to the accuracy of the intensity data as the number of data points approaches the number of variables in the structure. One problem encountered in crystal structure analyses of fibrous polymers is that of a very limited number of reflections (low data to parameter ratio). In addition, fibrous polymers usually scatter x-rays too weakly to be accurately measured by ionization or scintillation counter techniques. Therefore, the need for a critical study of the photographic techniques of obtaining accurate diffraction intensities is paramount. 

Fig. 5.42 Contour lines for p, the electron density distribution, in a homonuclear diatomic molecule X2. The lines originating at infinity and terminating at the nuclei and at the bond critical point C are trajectories of the gradient vector field (the lines of steepest increase in p two trajectories also originate at C). The line S represents the dividing surface between the two atoms (the line is where the plane of the paper cuts this surface). S passes through the bond critical point and is not crossed by any trajectories...

The quantum theory of atoms in molecules (QTAIM) [25, 26] is based on analyses of the electron density distribution. The electron density of such systems such as simple molecules or ions, and also complexes, complex molecular and ionic aggregates, as well as crystals may be analyzed using this approach. QTAIM is a powerful tool that allows characterizing of various interactions covalent bonds, ionic bonds, van der Waals interactions and, what is the most important for this review, also HBs. The analysis of critical points of the electron density is very useful. For the critical points (CPs), the gradient of electron density, p(r), vanishes ... 

On the basis of these definitions one can describe chemical bonding in molecules containing noble gas elements with the aid of the properties of p(r). One starts by searching for the bond paths 2uid their associated bond critical points Tg in the molecular electron density distribution. If all bond paths are found, then the properties of p(r) along the bond paths will be used to characterize the chemical bonds. For example, the value of can be used to determine a bond order, the anisotropy of Pp can be related to the n character of a bond, the position of the bond critical point is a measure of the bond polarity and the curvature of the bond path reveals the bent-bond character of a bond [17, 19]. 

Figure 6 shows a contour line diagram of the electron density distribution of the van der Waals complex Hcj (He,He distance 2.74 A). The electron density at the midpoint between the two He atoms is just 0.008 e/A, quite different from the values found for a typical covalent bond between first-row elements (1-5 e/A ). Despite the smallness of p(r) in the internuclear region, the He nuclei are linked by a MED path and the midpoint is the position of a (3, — 1) critical point (Fig. 6). As pointed out above this does not imply the existence of a chemical bond. The energy density H(r) is positive at the (3, — 1) critical point, which means that the kinetic energy rather than the potential energy dominates in the internuclear region. There is no chemical bond between the He atoms.

Fig. 6a and b.. Perspective drawing (a) and contour line diagram (b) of the electron density distribution p(r) of He2 at its van dcr Waals distance (2.74 A) shown with regard to a plane that rantains the nuclei. MED path and (3, — 1) critical point are indicated in the contour line diagram. (HF/6-31G(p) calculations)... 

A description of the electronic structure and the nature of bonding in polyatomic noble gas ions follows exactly the same lines pursued in the case of diatomic noble gas molecules (Sect. 4.5). Once the electronic wave function has been determined by ab initio calculations, the electron density distribution can be calculated, the MED path and (3, — 1) critical points can be found, and the properties of p(r) and H(r) can be evaluated at the critical points of p(r). This has been done for NgjX (X = C, N, O), NgCCNg +, NgCCH+, and NgCN+ ions. We will review in this section the results of these investigations [7, 13]. 

The theory of AIM allows one to study the concept of chemical bond and the bond strength in terms of electron density distribution function [6, 193]. It exploits the topological features of electron density and thereby a definition of chemical bonding through bond path and bond critical point (BCP). A BCP (it is a point at which gradient vector vanishes, Vp(r) = 0) is found between the... 

Figure 1 Ground state electron density distributions of LiF, CO and N2 in their equilibrium geometries, superposed with bond paths and the intersection of the interatomic surfaces. The surface in LiF is characteristic of an ionic interaction, that in CO of a polar interaction and that in N2 of a shared interaction. An interatomic surface intersects the bond path at the position of the bond critical point. The outer contour value is 0.001 au and the remaining contours increase in value in the order 2 x 10", 4 x 10", 8 x 10" au with n beginning at -3 and increasing in steps of unity.

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