Promolecular electron density

The optimum ( stockholder ) -electron fragments 0f(a,. ylr,..., r") are then uniquely determined by the molecular and promolecular -electron densities... 

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]. 

Identification of Local Maxima in Low-Resolution Promolecular Electron Density Distributions. 

Keywords Promolecular electron density distribution Poisson equation Coulomb potential Smoothing Molecular alignment Similarity index... 

Leherte L (2006) Similarity measures based on Gaussian-type promolecular electron density models alignment of small rigid molecules. J Comput Chem 27 1800-1816... 

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... 

Several approximate, semi-quantitative relations linking the above information-distance densities with the density difference function Ap(r) have been derived and numerically tested for selected linear molecules [29,30]. Since the molecular density is in general only slightly changed relative to the promolecular reference density, as a result of the mainly valence shell, minor reconstruction of the electron distribution,... 

All studied molecules in this work have been drawn and cleaned using WebLab ViewerPro [68], These initial molecular geometries have been optimized using Ampac 6.55 [69] at the AMI [70] semi-empirical level. Finally, molecular electronic density functions have been built using the Promolecular Atomic Shell Approximation [34-38], detailed below, using parameters fitted to the 6-311G basis set. 

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. 

When doing the molecular fitting of ASA coefficients compared with promolecular ASA, the ASA coefficients no longer sum to 1 for all atoms in the molecule, because the sum of the coefficients reflects the effect of polarization. In other words, rather than having Pa = Za in promolecular ASA, the values of Pa Za in fitted ASA reflect the internal electron density redistribution. In such a way, fitted ASA coefficients are reminiscent of the Stewart atoms idea, where atoms in molecules are considered as radially distorted atoms and are connected with so-called Stewart charges. Contrary to common promolecular ASA where the atomic densities of Eq. [41] integrate to Za ... 

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. 

For the set of steroid molecules illustrated in Figure 10, the geometries were obtained with the semi-empirical AMI method. Electron densities were then obtained with both B3LYP/6-31G single-point calculations and promolecular ASA densities. The latter are generated much faster than the ab initio densities. Once the electron density is known, the MQSM can be calculated with any of the positive definite operators mentioned earlier. 

In this section the density Ai (r) of the Kullback-Leibler functional for the information distance between molecular and promolecular electron distributions. 

Because of the nonquantitative nature of A/o(r) maps , the X-ray diffraction data recorded for coesite was used to genraate a total electron density distribution, p(r), for the mineral. In an analysis of the bond critical point properties of the distribution, Downs located the critical points along each of its Si—O bonds, determined the value of the electron density and the Laplacian of /o(r) at each of these critical points, V pfr, ), and mapped —V p(r) over the domain of each of its Si—O—Si skeletal units. A mapping of the total electron density distribution and its topological properties has a distinct advantage over a mapping of the deformation density in that The derivation of a unique and physically meaningful difference (deformation) electron density is a problem that cannot be solved since the choice of the promolecular reference density always implies some... 

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. 

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. 

Leherte L (2004) Hierarchical analysis of promolecular full electron-density distributions description of protein structure fragments. Acta Crystallogr Sect D 60 1254—1265... 

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