Promolecular density

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... 

Replacing ab initio densities with promolecular densities using the ASA expansion may seem a quite drastic approximation, but experience has shown that this is not the case [36 -0]. The reason is that the ASA method very well captures those areas where the density is the highest, namely near the cores of the atoms. On the other hand, the valence region is characterized by a much smaller density and thus has no big influence on the MQSM so that the ASA approach is certainly viable from a computational point of view. 

Grrones, X., Amat, L. and Carbo-Dorca, R. (2002) Modeling large macromolecular structures using promolecular densities./. Chem. Inf. Comput. Sci.,... 

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. 

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... 

The tVi are the expansion coefficients for the M s-type Gaussians, and we can see immediately the link between Eq. [42] and the wave function quadrature. So, for the calculation of ASA-based promolecular electron densities, we first need to develop a scheme for the fitting of the atomic densities. The exponents of the Gaussians may be chosen from, e.g., a well-tempered series.The coefficients may then be fitted against the true atomic ab initio electron density. Once these exponents and coefficients are set, these Gaussian exponents and coefficients are universally applicable. Promolecular densities p (r) can then be obtained quickly from Eq. [41]. 

This constraint automatically makes any promolecular density for some molecule, calculated with Eq. [41], fulfill the normalization constraint for that molecule. Such a constraint is most easily handled by a Lagrange multiplier. In ASA, normalized s-type Gaussians are applied, so the following constraint is introduced ... 

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... 

Boon et al. also studied several chiral molecules, which included again two amino acids (Ala and Leu) and CHFClBr, a prototype of chiral molecules. Ab initio total molecular electron densities yielded both Euclidean distances and Carbo indices between the enantiomers of these molecules. Molecular superposition was performed with, on the one hand, a manual alignment based on chemical intuition and the QSSA method, on the other hand. When analyzing the tables of the work by Boon et al. and comparing the results to the work by Mezey et al., similar values for the Euclidean distances between the two enantiomers appear for Ala and Leu, which once again illustrates the power of the ASA promolecular densities to yield quantum similarity measures in good agreement with those obtained from ab initio calculations. 

Modeling Large Macromolecular Structures Using Promolecular Densities. 

The topological analysis of the ELF provides a picture in which the electron density is distributed and localized in different volumes called basins, thus enabling one to discuss the reliability of simplified representations of electron densities in terms of superposition of promolecular densities or resonant Lewis structures. 

Densities are stable (see Sect. 18.3.2.2) to such an extent that NCI characteristics are already contained in the sum of atomic densities, pf [63, 64]. The resulting molecular density, also known as promolecular density, ff °, is then given by ... 

A promolecular density obtained from simple exponential atomic pieces is able to quantitatively predict low-density, low-reduced-gradient regions similar to density-functional results. The free atomic densities used in these calculations consist of one Slater-type function for each electron shell, lit to closely reproduce spherically-averaged, density-functional atomic densities (see Appendix, Fig. 18.19). 

Approximate promolecular densities were constructed by summing exponential atomic densities for bicyclo[2,2,2]octene, and the homomolecular dimers of methane and water. 

Promolecular densities obviously lack relaxation however, the promolecular densities are extremely useful in biomolecular systems, such as proteins or DNA. Because the calculation of the electron density in these systems becomes extremely computationally expensive, the promolecular density becomes an attractive option non-covalent interactions can be analyzed with only the molecular geometry required as input. 

The promolecular density, the denominator in Eq. (2), is defined as the sum of the densities of the isolated atoms p h positioned at the same coordinates as the atomic nuclei in the real molecule. Integration of the atomic density leads to the population of every atom ... 

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