Natural population analysis spin density

Spin density calculations analyze the difference between each atom s population of a- and [3-electrons using various techniques. Mulliken population analysis140 is still often used for determining spin densities despite its deficiencies experienced sometimes when large basis sets are used. The Lowdin population analysis Scheme141 tries to circumvent some of the unreasonable orbital populations predicted by the Mulliken scheme however it is somewhat still basis set dependent. The natural bond orbital (NBO) methodology142 is a whole set of analysis techniques, including natural population analysis (NPA), which is less basis set dependent than the Mulliken scheme and often considered one of the most reliable methods for population analysis. 

In natural population analysis (NPA) of open-shell species (see, for example, 1/0-3.10), the natural spin density is evaluated for each NAO, then summed over NAOs on each atom to give and finally over all atoms to give net overall spin density of the species (as measured hy ESR spectroscopy see Chapter 7). This provides a very detailed picture of spin charge and spin polarization distributions throughout the molecule, allowing one to quantify (or rationally design) specific magnetic properties of interest. 

So far we have presented cDFT as a means to impose a charge or a spin density on a group of atoms. However, we have not specified how such charges are calculated in practice. It is well known that atomic charges cannot be strictly defined in quantum mechanics and, as a corollary, that multiple population analysis (PA) approaches can be developed to reach this goal. A central point PAs have to address is to define a criterion for assigning each fraction of the electronic density of every point in space to the individual atoms. Some PAs realize this partition with criteria relying on the KS molecular orbital coefficients. The MuUiken [227], Lowdin [228], and the more sophisticated Natural Bond Orbital [229,230] approaches are examples of... 

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