Bader electron density contour

R F W Bader s theory of atoms in molecules [Bader 1985] provides an alternative way to partition the electrons between the atoms in a molecule. Bader s theory has been applied to many different problems, but for the purposes of our present discussion we will concentrate on its use in partitioning electron density. The Bader approach is based upon the concept of a gradient vector path, which is a cuiwe around the molecule such that it is always perpendicular to the electron density contours. A set of gradient paths is drawn in Figure 2.14 for formamide. As can be seen, some of the gradient paths terminate at the atomic nuclei. Other gradient paths are attracted to points (called critical points) that are... 

Fig. 5.18 (a) Total electron density contours for the carbon monoxide molecule. The carbon atom is on the left (b) Total electron density contours for the dinitrogen molecule. [From Bader, R. F. W. Bandrauk, A. D. J. Chem. Phys. 1968,49, 1653. Reproduced with permission.]... 

Fig. 1.3. Electron density contours in the water molecule, as calculated by Bader. Densities are given in atomic units (i a.u. = = 6-74geA ).

Figure 6.18 Contour maps of the ground state electronic charge distributions for the period 2 diatomic hydrides (including H2) showing the positions of the interatomic surfaces. The outer density contour in these plots is 0.001 au. (Reproduced with permission front Bader [1990].)...

Here we see clearly the large concentration of density around the oxygen nucleus, and the very small concentration around each hydrogen nucleus. The outer contour is an arbitrary choice because the density of a hypothetical isolated molecule extends to infinity. However, it has been found that the O.OOlau contour corresponds rather well to the size of the molecule in the gas phase, as measured by its van der Waal s radius, and the corresponding isodensity surface in three dimensions usually encloses more than 98% of the total electron population of the molecule (Bader, 1990). Thus this outer contour shows the shape of the molecule in the chosen plane. In a condensed phase the effective size of a molecule is a little smaller. Contour maps of some period 2 and 3 chlorides are shown in Figure 8. We see that the electron densities of the atoms in the LiCl molecule are only very little distorted from the spherical shape of free ions consistent with the large ionic character of this molecule. In... 

The first step in our procedure is to compute an optimized structure for each molecule and then to use this geometry to compute the electronic density and the electrostatic potential. A large portion of our work in this area has been carried out at the SCF/STO-5G //SCF/STO-3G level, although some other basis sets have also been used. We then compute V(r) on 0.28 bohr grids over molecular surfaces defined as the 0.001 au contour of the electronic density (Bader et al. 1987). The numbers of points on these grids are converted to surface areas (A2), and the and Fs min are determined. Our statistically based interaction in-... 

An effective approach is to compute V(r) on an appropriately-defined molecular surface, because this is what is seen or felt by the other reactant. Such a surface is of course arbitrary, because there is no rigorous basis for it. A common procedure has been to use a set of fused spheres centered on the individual nuclei, with van der Waals or other suitable radii [34—37]. We prefer, however, to follow the suggestion of Bader et al. [38] and take the molecular surface to correspond to an outer contour of the electronic density. This has the advantage of reflecting features such... 

The microscopic world of atoms is difficult to imagine, let alone visualize in detail. Chemists and chemical engineers employ different molecular modelling tools to study the structure, properties, and reactivity of atoms, and the way they bond to one another. Richard Bader, a chemistry professor at McMaster University, has invented an interpretative theory that is gaining acceptance as an accurate method to describe molecular behaviour and predict molecular properties. According to Dr. Bader, shown below, small molecules are best represented using topological maps, where contour lines (which are commonly used to represent elevation on maps) represent the electron density of molecules. 

Figure 8.IB shows an experimental contour map of electron density for the H2O molecule in plane y-z, after Bader and Jones (1963). The electron density is higher around the nuclei and along the bond directrix. The experimental electron density map conforms quite well to the hybrid orbital model of Duncan and Pople (1953) with the LCAO approximation.

The qualitative study of electronic structure through the electron (number) density p(r) relies heavily on linecut diagrams, contour plots, perspective plots, and other representations of the density and density differences. There is a review article by Smith and coworkers [302] devoted entirely to classifying and explaining the different techniques available for the pictorial representation of electron densities. Beautiful examples of this type of analysis can be seen in the work of Bader, Coppens, and others [303,304]. 

For analyzing noncovalent interactions, we typically evaluate V(r) on a three-dimensional surface of the molecule. For this purpose, we use the 0.001 au (electrons/bohr3) contour of its electronic density, as suggested by Bader et al. [51]. This surface encompasses at least 96% of the electronic charge of the molecule, and reflects its specific features, such as lone pairs, n electrons, strained bonds, etc., which is not true of surfaces created by overlapping atomic spheres. 

We have computed V(r) on nanotube surfaces, both inner and outer, which we define as the 0.001 electrons/bohr3 contours of the electronic density, as proposed by Bader et al.45 The resulting surface potentials are labeled Vs(r). 

We have shown in earlier work that it is possible to quantitatively relate a variety of liquid, solid and solution phase properties to the electrostatic potential patterns on the surfaces of the individual molecules [64-66]. Among these properties are pKa, boiling points and critical constants, enthalpies of fusion, vaporization and sublimation, solubilities, partition coefficients, diffusion constants and viscosities. For these purposes, we take the molecular surface to be the 0.001 au contour of the molecular electronic density p(r), following the suggestion of Bader et al [67],... 

A more realistic outline of a molecular surface can be defined in terms of the outer contours of electron density according to Bader [173]. To avoid excessive computation the densities of large molecules may be built up from previously calculated densities of smaller fragments [213]. The most obvious approach, to approximate molecular density by the sum of atomic densities over the promolecule has also been explored [214]. This approach works well since the deformation density associated with bond formation is small compared to the total density [215]. The total density may therefore be represented by a sum over spherically averaged atomic densities, p(r) = Pa(t)-... 

As mentioned in Chapter 8 (page 172), the double bond is associated with an elliptical distribution of electronic charge in the plane perpendicular to the CC nuclear axis and containing its mid-point where the electronic density has a local maximum (critical point in the theory of Bader). The relief diagram and the contour plots of Fig. 9.7 taken from the work of Bader et al. (ref. 92) show the distribution of the electronic charge density in the nuclear plane of the molecule. 

However, our preference is to follow the suggestion of Bader et al. [86] and take the surface to be an outer contour of the molecular electronic density. This has the important advantage that it reflects features specific to the particular molecule, such as lone pairs or strained bonds. We normally choose p(r) = 0.001 electrons/bohr3, but we have confirmed that other low values of p(r), e.g., 0.002 electrons/bohr3, would serve equally well [87]. 

We will look in particular at the detailed features of V(r) computed on the surfaces of energetic molecules, which is designated Vs(r). This raises the question of how to define a molecular surface. In the past, this has often been done by means of fused spheres centered at the nuclear positions and having, for example, the corresponding van der Waals atomic radii [112,113]. More recently, however, there has been an increasing tendency to follow Bader et al [WA] and take the surface to be some outer contour of the electronic density. This has the advantage that the surface then reflects the specific features of that molecule, such as lone pairs and strained bonds. We use this approach, with p(r) = 0.001 electrons/bohr other outer contours, e.g. p(r) = 0.002 electrons/bohr, would serve just as well [115]. 

Fig. 1.2. (a) Contour map of electron density in the plane of the ethene molecule, (b) Contour map of electron density perpendicular to the plane of the ethene molecule at the midpoint of the C=C bond. Reproduced with permission from R. F. W. Bader, T. T. Nguyen-Dang, and Y. Tal, Rep. Prog. Phys., 44, 893 (1981). 

R. F. W. Bader and associates at Canada s McMaster University have derived a means of describing the electron distribution associated with specific atoms in a molecule, called the atoms in molecules (AIM) method. The foundation of this approach is derived from quantum mechanics and principles of physics. It uses the methods of topology to identify atoms within molecules. The electron density of a molecule is depicted by a series of contours. Bond paths are the paths of maximum electron density between any two atoms. The critical point is a point on the bond path where the electron density is a maximum or a minimum with respect to dislocation in any direction. The bond critical point is defined by the equation... 

In closing, we should also include the molecular graphs introduced by Bader and co-workers.These graphs are derived from the entire electron density function p(r, K) and not from one molecular contour surface. In this sense, these are 2D descriptors of molecular 3D models. 

As is evident already in Sect. 16.1, V(r) provides an effective basis for interpreting and predicting noncovalent interactions [41,42], It has in fact been shown that condensed phase physical properties that depend upon noncovalent interactions can be expressed analytically in terms of the features of V(r). For these purposes, V(r) is commonly computed on the surface of the molecule, taking this to be the 0.001 au (electrons/bohr ) contour of its electronic density, as proposed by Bader et al. [44]. Representing the surface as a contour of p(r) has the advantage that it is specific to that particular molecule, and reflects its lone pairs, it electrons, etc. V(r) computed on a molecular surface is labeled Vs(r) its local most positive and most negative values, of which there may be several, are designated Vs,max and Vs min, respectively. 

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