Bonds critical points

A localized molecular orbital related to a certain type of bond. 

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Gradient vector paths around formamide. The paths terminate at atoms or at bond critical points (indicated by squares). 

Vector quantities, such as a magnetic field or the gradient of electron density, can be plotted as a series of arrows. Another technique is to create an animation showing how the path is followed by a hypothetical test particle. A third technique is to show flow lines, which are the path of steepest descent starting from one point. The flow lines from the bond critical points are used to partition regions of the molecule in the AIM population analysis scheme. 

Scheme 3 The electron density at the bond critical points and the center of 1 and 10...

The charge density study of benzoylacetone [8] revealed that the Laplacian at the bond critical points between the enol hydrogen and the oxygens has a negative value. This means that the bonds between that hydrogen and both the oxygens have covalent character. Furthermore the populations of the spherical valence parts of the multipole... 

We can measure the extent electronic charge is preferentially accumulated by a quantity called the ellipticity e. At the bond critical point it is defined in terms of the negative eigenvalues (or curvatures), Aj and A2 as e = (A1/A2) — I. As A1 < A2 < 0, we have that A ]/A2 > 1, and therefore the ellipticity is always positive. Tf e = 0 then we have a circularly symmetric electron density, which is typically found at bond critical points in linear molecules. 

Figure 6.13 Relief map of the electron density for methanal (formaldehyde) in the molecular plane. There is a bond critical point between the carbon and the oxygen nuclei, as well as between the carbon nucleus and each hydrogen nucleus. No gradient path or bond critical point can be seen between the two hydrogen nuclei because there is no point at which the gradient of the electron density vanishes. There is no bond between the hydrogen atoms consistent with the conventional picture of the bonding in this molecule.

Now we focus on the gradient paths, which do not terminate at a nucleus, but rather link two nuclei. For example, the bond critical point between C and H in Figure 6.14 is the origin of two gradient paths. One gradient path terminates at the hydrogen nucleus, the other at the carbon nucleus. This pair of gradient paths is called an atomic interaction line. It is found... 

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).

Figure 6.16 Molecular graphs for some molecules in their equilibrium geometries. A bond critical point is denoted by a black dot. The molecule HCCH is ethyne, H2CO is methanal, and H2CCH2 is ethene. [Adapted with permission from Bader [1990], Fig. 2.8.]...

The value of the electron density at this point, the bond critical point density pt,. 

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. 

The position of the bond critical point. The distance of this point from each of the nuclei is a measure of the size of each atom, that is, its bonding radius rb. 

The Diatomic Hydrides of Periods 2 and 3 6.9.1 Bond Critical Point Density... 

Figure 6.17 Contour map of p in the interatomic surface associated with the CC bond critical point in ethene. The plane of the plot is perpendicular to the molecular plane. The C and two H nuclei are projected onto the plane of the plot to indicate the orientation of the molecule. We see that electronic charge is preferentially accumulated in the direction perpendicular to the molecular plane, giving an elliptical shape to the electron density in this plane.

We assume that the bond critical point density is an approximate measure of the amount of density accumulated in the bonding region, that is, the amount of shared density. For bonds that are conventionally regarded as predominately covalent, pt, has a large value, and for bonds conventionally regarded as predominately ionic, pb has a low value. In a hypothetical purely ionic bond, the value of pb would be zero. 

As we have seen in earlier chapters, an important and much discussed bond property is the bond length. The length of a bond depends on its strength, and it therefore also depends on the bond critical point density and on the atomic charges. 

The strength of a bond increases with increasing bond critical point density pb and with increasing charges of the bonded atoms. 

The length of a bond decreases with increasing bond critical point density and increasing atomic charges, but increases with increasing coordination number of the atom to which it is bonded. 

Clearly not all these atomic and bond properties are independent of each other and it can be difficult to disentangle one from another. Nevertheless we will find these properties useful for discussing the properties of molecules, as we do for some typical molecules of the period 2 elements in this chapter. In particular, the amount of accumulated or shared density, which we assume is approximately measured by the bond critical point density, represents what is commonly called the covalent contribution to the bonding. The atomic charges represent what is commonly called the ionic contribution. 

Bonds between atoms with large charges and a small value of the bond critical point density are described as predominately ionic. 

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