Atomic basins, physical space analysis

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. 

The disadvantage of the physical space analysis is that the calculation of atomic basins and the subsequent integration is not always straightforward, and definitely requires much more time than the Hilbert space analysis (recall the latter is instantaneous). Our experience shows that the latter analysis does provide satisfactory information so that it is not necessary to perform the physical space analysis. 

The AIM approach partitions the physical space into atomic basins based on a topological analysis of the electron density itself, but several other methods have been proposed for dividing the space into atomic contributions. 

The main requirement in the determination of bond orders is to derive rules on how to measure the number of electrons shared between two atoms. For this purpose, a definition of an atom in a molecule is required, which, however, cannot be formulated in a unique and unambiguous way [169]. Quantum chemical calculations are typically performed in the Hilhert-space analysis, where atoms are defined by their basis orbitals. Such an analysis, however, strongly depends on both the atomic basis set chosen and the type of wave function used. The position-space representation, on the other hand, where atoms are defined as basins in three-dimensional physical space does not suffer from these insufficiencies. In this chapter, we present one option for a three-dimensional atomic decomposition scheme and the reader is referred to Refs. [170-173] for further examples. 

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