Atomic shell approximation

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... 

Constans, P. and Carbo, R. (1995). Atomic Shell Approximation Electron Density Fitting Algorithm Restricting Coefficients to Positive Values. JChem.Inf.Comput.Sci., 35,1046-1053. 

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. 

Practical computation of the integral (1) becomes computationally expensive when the involved density functions correspond to large molecules or have been calculated at high computational levels. Even concrete applications of MQSM have been carried out at the ab initio level, when several molecules are studied simultaneously, as in QSAR studies, MQSM need to be computed several times, preventing their usage at these stages. In order to overcome this problem, the promolecular atomic shell approximation (PASA) [34-38] has been defined as a model of the true ab initio density, devised as a linear combination of 15 functions, and mathematically expressed as ... 

Constans P, Carbo R. Atomic shell approximation electron density fitting algorithm restricting coefficients to positive values. J Chem Inf Comput Sci 1995 35 1046-1053. 

Constans P, Amat L, Fradera X, Carbo-Dorca R. Quantum Molecular Similarity Measures (QMSM) and the Atomic Shell Approximation (ASA). In Carbo-Dorca R, Mezey PG, eds. Advances in Molecular Similarity. Vol. 1. London JAI Press, 1996 187-211. 

Amat L, Carbo-Dorca R. Quantum similarity measures under atomic shell approximation first-order density fitting using elementary Jacobi rotations. J Comput Chem 1997 18 2023-2039. 

It is well known that the eDF can be used for computing molecular expectation values. However, from the point of view of the development of practical applications, first order DF beginning with ab initio functions and ending as well into Atomic Shell Approximation (ASA) forms [39], are accepted work candidates for QSM applications too [40]. General QSM definition proves to contain expectation values as a particular case [41], some summarised details can be found in Appendix A. First-order eDF are customarily used in QSM molecular comparisons, since the initial description of the original concept [16]. Higher order eDF can be employed as well in QSM calculations [31,33,35]. Problems consisting in systems other than molecules can be studied in the same way [41], and the QSM applications can be extended to the area of statistical mechanics distributions [43] as well. 

When the practical implementation of QSM has been considered in this laboratory, a simplified manner to construct the first-order eDF form [39] has also been proposed and named Atomic Shell Approximation (ASA) eDF. A procedure has been recently described [39e),f)], bearing the correct necessary conditions to obtain positive definite ASA eDF, possessing appropriate probability distribution properties. 

A typical dendrogram obtained from overlap MQSM using atomic shell approximation electron densities is shown in Figure 2. 

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... 

Measures Under Atomic Shell Approximation First Order Density Fitting Using Elementary Jacobi Rotations. 

In their work related to the Promolecular Atom Shell Approximation (PASA), Amat and Carbo-Dorca used atomic Gaussian ED functions that were fitted on 6-31IG atomic basis set results [35]. In the PASA approach that is considered in the present work, a promolecular ED distribution Pa is represented analytically as a weighted summation over the nat atomic ED distributions p, which are described in terms of series of three squared Is Gaussian functions fitted from atomic basis set representations [36] ... 

All molecules were described using each of the four following smoothed molecular fields, that is, the promo-lecular atomic shell approximation (PASA) of the full electron density (ED) [35], a charge density (CD) calculated using the Poisson equation [33], the Coulomb electrostatic potential [34], and the Atomic Property Fields (APF) described by Totrov [15]. 

Amat L, Caibo-Dorca R (2000) Molecular electronic density fitting using elementary Jacobi rotations under atomic shell approximation. J Chem Inf Comput Sci 40 1188-1198... 

The computational background where the present CQS4 chapter will be developed does not differ abruptly from the previous CQS papers and is basically grounded in the well-known atomic shell approximation (ASA) [30-34] in order to express, within an advanced polarized promolecular framework, the quantum mechanical molecular density fimctions (DF). To obtain additional information on recent ASA application development, see for example the previous CQS chapters [12-14] and some application to describe DF for large macromolecular structures [35, 36] to see the adequate behavior for ASA DF. 

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