Bader partitioning

The topological analysis of the total density, developed by Bader and coworkers, leads to a scheme of natural partitioning into atomic basins which each obey the virial theorem. The sum of the energies of the individual atoms defined in this way equals the total energy of the system. While the Bader partitioning was initially developed for the analysis of theoretical densities, it is equally applicable to model densities based on the experimental data. The density obtained from the Fourier transform of the structure factors is generally not suitable for this purpose, because of experimental noise, truncation effects, and thermal smearing. 

Relative to these deviations, we have also calculated other parameters regarding the Si atoms with the Bader partition scheme of the electron density for both (110) and (100) planes (Table 11.3). The obtained Bader charges for the Si(sp ) or Si(sp ) atoms possess opposite signs for both (110) and (100) planes. The different charges are coherent with the proposed origin of the band gap of 0.5 eV for a partially ionic Si-Si bond [45], which explains the non-metallic character of the Si (100) surface [51]. Small differences are observed between the Si(ip ) Bader charges for the (110) and (100) surface but the Si sp ) charges coincide for both planes. 

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... 

Bader et al. have developed a theory of molecular structure [8], based on the topological properties of the electron density p(r). In this theory, a molecule may be partitioned into atoms or fragments by using zero-flux surfaces that satisfy the condition... 

Several forms of wf have already been used within the field of MQS. These methods include the Hirshfeld partitioning [30], Bader s partitioning based on the virial theorem within atomic domains in a molecule [64], and the Mulliken approach [65]. For more information on all the three methods, refer to Chapter 15. 

One way of getting rid of distortions and basis set dependence could be that one switches to the formalism developed by Bader [12] according to which the three-dimensional physical space can be partitioned into domains belonging to individual atoms (called atomic basins). In the definition of bond order and valence indices according to this scheme, the summation over atomic orbitals will be replaced by integration over atomic domains [13]. This topological scheme can be called physical space analysis. Table 22.3 shows some examples of bond order indices obtained with this method. Experience shows that the bond order indices obtained via Hilbert space and physical space analysis are reasonably close, and also that the basis set dependence is not removed by the physical space analysis. 

When space is partitioned with discrete boundaries, as in Eq. (6.7) and in the Bader virial partitioning method, the moments can be derived directly from the structure factors by a modified Fourier summation, as described for the net charge in chapter 6. 

A promising simplification has been proposed by Bader (1990) who has shown that the electron density in a molecule can be uniquely partitioned into atomic fragments that behave as open quantum systems. Using a topological analysis of the electron density, he has been able to trace the paths of chemical bonds. This approach has recently been applied to the electron density in inorganic crystals by Pendas et al. (1997, 1998) and Luana et al. (1997). While this analysis holds great promise, the bond paths of the electron density in inorganic solids are not the same as the more traditional chemical bonds and, for reasons discussed in Section 14.8, the electron density model is difficult to compare with the traditional chemical bond models. 

An alternative physical observable that has been used to define partial atomic charges is the electron density. In X-ray crystallography, the electron density is direedy measured, and by comparison to, say, spherically symmetric neutral atoms, atomic partial charges may be defined experimentally, following some decisions about what to do with respect to partitioning space between the atoms (Coppens 1992). Bader and co-workers have adopted a particular partitioning scheme for use with electronic structure calculations that defines the atoms-in-molecules (AIM) method (Bader 1990). In particular, an atomic volume is... 

Since difference electron densities, deformation densities or valence electron densities are not observable quantities, and since the Hohenberg-Kohn theorem64 applies only to the total electron density, much work has concentrated on the analysis of p(r). The accepted analysis method today is the virial partitioning method by Bader and coworkers67, which is based on a quantum mechanically well-founded partitioning of the molecular... 

R. F. W. Bader and P. M. Beddall, /. Chem. Phys., 56, 3320 (1972). Virial Field Relationship for Molecular Charge Distributions and the Spatial Partitioning of Molecular Properties. 

Finally, we mention briefly an interesting series of papers by Bader and coworkers,215 who have partitioned the charge distribution in a variety of hydrides in a particular way. Details and references to this series of papers are given in ref. 215. 

It seems clear that in molecules with one type of atom only, say ozone, one should have a similar sort of decomposition. In independent work and from a different direction, Bader and his co-workers have likewise insisted on the importance of Vp in dividing a molecule into fragments which are localized and transferable. As one example, the work of Bader and Beddall58 effects such a partitioning into fragments based on the virial theorem, and the interested reader is referred to this and various other papers which throw very considerable light on electron densities in specific molecules and on partitioning the electron cloud.59-85... 

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],... 

The postulates of VSEPR theory are consistent with the partitioning of electron density according to Bader s atoms-in-molecules method [173], in which the electron pairs return as the valence shell charge concentration. 

The alternative approach is to count the number of electrons in an atom s space. The question is how to define the volume an individual atom occupies within a molecule. The topological electron density analysis (sometimes referred to as atoms-in-molecules or AIM) developed by Bader uses the electron density itself to partition molecular space into atomic volumes. 

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