The Hirshfeld Approach

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 

The idea of Hirshfeld was to obtain atoms-in-molecules densities by defining the stock-amount or weight of an atom a in the electron density at r as 

If it is furthermore assumed that these weight coefficients remain valid for the true, ab initio density of the molecule. The Hirshfeld atomic electron density of the atom a in the molecule, denoted p (r), can be calculated as 

Application of Eq. [86] in the general definition of a molecular quantum similarity measure then gives  

Although this is the most obvious expression for defining the similarity between two atomic densities in the Hirshfeld way, a different expression was used in the study of molecular chirality. The latter approach will be discussed in some more detail in the application of molecular similarity to study molecular chirality. 

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There is a small number of reports of the application of the Hirshfeld approach to interionic interactions in the crystal structures of ionic liquids [52, 111, 124, 203, 574, 599]. However, some [600] have cautioned (because of weaknesses in charge partitioning) that its application should not necessarily lead to the setting aside of judgements based on the use of chemical intuition. 

All the different AIM methods that will be discussed below basically use this same approach but quite different in the nature of n (. Chronologically, we will discuss the Mulliken AIM, the Hirshfeld AIM, and the Bader AIM. This last approach will henceforth be called quantum chemical topology (QCT). There are more AIM methods, but most of them can be easily understood by the three selected emblematic approaches. 

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 ... 

Several forms of wf have already been used within the field of MQS. These methods include the Hirshfeld partitioning [30], Bader s partitioning based on the virial theorem within atomic domains in a molecule [64], and the Mulliken approach [65]. For more information on all the three methods, refer to Chapter 15. 

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]. 

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... 

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. 

In a related approach, Spackman has developed a force field [50] that includes an empirically derived part for the dispersion and repulsion terms, while coulombic energy terms are treated in the following maimer. The molecular electron distribution is divided into a promolecule term and a deformation term, as in the Hirshfeld definition. The product of the distributions in two interacting monomers A and B, as required in the evaluation of the coulombic energy, is then expanded as ... 

Hirshfeld and Schmidt (168) proposed that this problem might not arise in the polymerization of long bifunctional monomers in crystals of suitable structure, as in 97 — 98. They predicted that reaction in such systems would occur via a tilt of each molecule about its center of mass, so that little or no net displacement of the molecules would be involved. In fact, two successful realizations of this approach were subsequently reported, for the diacetylenes, which will be discussed in a later section, and for distyrylpyrazines and related compounds. Here we discuss the latter series. 

Fig. 9. Calculated total energies (upper row), molecular bond lengths (middle row), and Mulliken and Hirshfeld atomic charges (lower two rows) for the EE-type (left column) and N-type (right column) approaches of the H2 molecule to the Ni4 cluster, using a DFTprocedure.

The most widely used approach is Bader s quantum theory of atoms in molecules (QTAIM), which depicts AIM as nonoverlapping [3, 7-10], Other popular models, such as those of Stewart [4, 11] and Hirshfeld [5], depict AIM as overlapping spherical fuzzy electron densities. Both the advantages and disadvantages of these methods have been previously discussed [1]. 

Recently, Ayers et al. [24] have used sound mathematical arguments to recommend using Hirshfeld partitioning to compute the charges, and in our approach. 

Given the correlation between bond valence and electron density, it appears tempting to compare also what electron density maps and maps of the BVSE predict as ion transport pathways. Hirshfeld surface analysis has been explored to characterize intermolecular interactions in molecular crystals [47,48]. This analysis is based on the procrystal, which is obtained from superposition of spherical atomic electron densities placed at the crystal structure positions, a quantity that can readily be calculated from the structure using software tools such as CrystalExplorer [49]. The approach was also explored as a tool to map out voids in porous crystals such as metal organic framework materials and zeolites [50]. 

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