Mulliken populations description

There is some ambiguity about Mulliken population analysis in the literature. This is because various software packages print different portions of the analysis and may name them slightly differently. The description here follows some of the more common conventions. 

As with other schemes of partitioning the electron density in molecules, Mulliken population analysis is arbitrary and is strongly dependent on the particular basis set employed. However, the comparison of population analyses for a series of molecules is useful for a quantitative description of intramolecular interactions, chemical reactivity and structural information. In another approach, the Lowdin population analysis, the atomic orbitals are first transformed to an orthogonal set, as are the molecular orbital coefficients [Lowdin, 1970]. 

Although the Mulliken population analysis is useful for the discussion of two-center bonding, it is not sufficient for the description of the three- or four-center bonds that may exist in metal clusters. In order to elucidate the multicenter bonds, we examine contour maps of the charge densities of Na4 and Mg4 as typical tetramers of AM and AE, respectively. We use the differential charge density, Ap defined by eq. (3)., to express the charge redistribution on the formation of chemical bonds. 

Reed, Weinstock, and Wienhold compared the basis set dependencies of Mulliken populations, NPA, and IPP (Table 4).i Whereas the Mulliken populations again vary wildly, the NPA and IPP charges are relatively stable when split-valence basis sets are employed. One should keep in mind that variations in the populations with increasing basis set size can result from two factors— an inherent basis set dependency in the methodology or a significant change in the electronic distribution itself due to the improved basis set. The population differences between the STO-3G and 4-3IG results reflect the inadequate description of the density with the small single-zeta basis. 

Similar concerns apply to molecular orbitals. One constructs molecular orbitals and populates them with electrons is a manner analogous to an individual atom by adopting the linear combination of atomic orbitals (LCAO) approximation. While this might lend the impression that molecular orbitals are merely an extension of atomic orbitals, they are conceptually distinct. An atomic orbital is a description of the state of motion of an electron subject to the influence of a single nucleus plus other electrons. But molecular orbitals describe electron motions in the field of two or more nuclei plus the other electrons and the use of the LCAO method is merely a matter of mathematical convenience (Gavroglu and Simoes 2012, p. 83). The delocalized character of molecular orbitals is conceptually quite distinct from the idea of atomic orbitals, and Mulliken - one of the originators of the molecular orbital approach - was at pains to distinguish his conceptual scheme from the methods employed to compute them (ibid, pp. 84—85). 

Finally, in order to give a chemical sounding description of the electronic structure of these molecules the natural bond orbital (NBO) approach and the related natural population analysis (NPA) - have been computed and discussed. The NPA approach is particularly effective in the case of inorganic complexes, since it gives a description of the electronic distribution less sensitive to the computational parameters (e.g. basis set) with respect to other more commonly used population analysis such as the Mulliken one [70]. 

Related with the chemieal bonding description, there is for first interest also a method for determining the atomic partition function in molecule here the Mulliken picture of the population analysis, a., will be presented yet modified, in the way of assuming the overlapping of charge unequally distributed between the atoms involved in the bond. Considering the linear combination of atomic orbitals (LCAO) expansion of the molecular function, the density assoeiated to the atomic orbitals F. is written (Rychlewski Parr, 1986 Li Parr, 1986) ... 

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