Hirshfeld population analysis

The first term in this expression is an entropy of mixing term related to electron transfer the second term is the information loss due to polarization of the AIM. Minimizing the information loss per atom results in the Hirshfeld population analysis [64,65] and many other results in the broad field of chemical information theory [26,66-75]. Zeroing the entropy of mixing term by choosing a reference ion that has the same number of electrons as the AIM, one obtains the Hirshfeld-I population analysis [76,77],... 

It is well-known that a superposition of isolated atomic densities looks remarkably much like the total electron density. Such a superposition of atomic densities is best known as a promolecular density, like it has been used by Hirshfeld [30] (see also the chapter on atoms in molecules and population analysis). Carbo-Dorca and coworkers derived a special scheme to obtain approximate electron densities via the so-called atomic shell approximation (ASA) [31-35]. Generally, for a molecule A with atoms N, a promolecular density is defined as... 

In these examples, the condensed Fukui functions were computed using Hirsh-feld population analysis [26], which is unique among the commonly employed population analysis methods, because the same results are obtained from the response of molecular fragment and the fragment of molecular response approaches [24]. There are other arguments in favor of the Hirshfeld scheme too [27,28], many of them based on the tendency for the atom-condensed Hirshfeld Fukui functions to be nonnegative [25,29,30]. Nonetheless, condensed Fukui functions maybe computed using any population analysis method common methods... 

Details on the numerical evaluation of the descriptors will be given in the individual cases but in most cases a computational DFT approach is used, with a hybrid functional of the B3LYP type [32]. Condensation of f(r) or sir) is done with conventional population analysis techniques (Mulliken [33], Natural Population Analysis (NPA) [34]) or with the Hirshfeld technique [35], often used by our group [36]. 

Saha, S., Roy, R.K. and Ayers, P.W., Are the Hirshfeld and Mulliken population analysis schemes consistent with chemical intuition Int. J. Quantum Chem. 109 (9), 1790-1806 (2009). 

For a different population analysis model, the constraint takes a different form. For example, in the Hirshfeld method [32], the atomic population takes the form ... 

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