Hirshfeld promolecular electron density

Clearly, the Hirshfeld promolecular electron density is not likely to simplify the integrals in Eq. [39]. The essential difference between the Hirshfeld and ASA promolecular densities is that in the ITirshfeld method, the isolated atom electron densities pa(r) are obtained in the same basis set as the one in the ab initio calculation of the true molecular electron density, whereas in the ASA approach, the isolated atom densities are obtained in the way as described below. In the ASA method, we use a slightly different promolecular atomic shell approximation (PASA) electron density, where the number of electrons Pa attached to each atom a is introduced. The total promolecular electron density for an N-atom molecule is given by... 

This approach, also often called the stockholder scheme, was introduced in 1977 by Hirshfeld [22]. The central idea of the Hirshfeld method originates in x-ray crystallography. It proposes to divide the electron density among the atoms in a molecule, guided by a promolecular density. More precisely, once a molecular geometry is known, a promolecular density p°(r) is composed by simply summing the density of each atom A (denoted p°A(r)) in an isolated state ... 

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... 

Hirshfeld (or stockholder) charges are based on using atomic densities for partitioning the molecular electron density. The promolecular density is defined as the sum of atomic densities placed at the nuclear geometries in the molecule. The actual molecular electron density at each point in space is then partitioned by weighting factors according to the promolecular contributions. 

The Hirshfeld idea, already developed in 1977, calculates the so-called stockholder charges and is a popular method in conceptual DFT. It consists of the following rationale. First an electron density, represented as p", is obtained for a molecule A with some Hamiltonian and basis set. For every atom a, an isolated electron density p° is calculated within the same model. With the isolated atom electron densities for all N atoms comprising the molecule, a Hirshfeld promolecular density is obtained as... 

It has been amply demonstrated elsewhere [10-15] that many classical problems of theoretical chemistry can be approached afresh using this novel IT perspective. For example, displacements in information distribution in molecules relative to the promolecular reference consisting of nonbonded constituent atoms have been investigated [10-18], and the least biased partition of molecular electron distributions into subsystem contributions (e.g., densities of AIM) has been investigated as well [10,19-26], The IT approach has been shown to lead to the stockholder molecular fragments of Hirshfeld [27], These optimum density pieces have been derived from alternative global and local variational principles of IT. 

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