Approximate electron density

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... 

The main advantage of using the ASA method to obtain approximate electron densities is the very important gain in computational efficiency to compute MQSM and thus perform similarity analysis among the molecules of some molecular set. This is important for those cases where either the molecules are very numerous, or very big or a combination of both. 

The most obvious drawback of Fourier space approaches is the computational cost of the Fourier transformation itself. However, this can be circumvented in some virtual screening applications. Gaussian functions are frequently used to approximate electron densities. Interestingly, the Fourier transform of a Gaussian function is again a Gaussian function and hence amenable to analytic transformation. 

Fig. 214. Sodium benzyl penicillin, (a) Approximate electron density map, b projection, with atomic coordinates appropriate to extended molecular configuration. (6) Error synthesis map. (c) The same electron density map as (a), with atomic coordinates appropriate to a curled configuration of the same molecule.

If the purpose of the analysis is the detection of shape changes, then these pseudo-densities and their complements are good diagnostic tools however, the pseudo-density scheme is not suitable to build approximate electron densities of large molecules from pseudo-density fragments obtained from smaller molecules. 

The case of 10 nuclei (dimension n=9) provides a large enough fragment size suitable for the rapid construction of approximate electron densities of large molecules. 

Fig. 3. Approximate electron density variation on jellium surfaces with periodic positive charge boundaries. The solid line gives the edge of the uniform ionic charge density. The dashed line indicates the contour where the electron density is equal to one-half its interior value.

However, the group gp is much too complicated for practical purposes of molecular shape characterization. Fortunately, the behavior of transformations t of family gp far away from the object p(r) is of little importance, and one can introduce some simplifications. Let us assume that the 3D function considered [e.g., an approximate electron density function p(r)], becomes identically zero outside a sphere S of a sufficiently large radius. As long as two symmorphy transformations tj and t2 have the same effect within this sphere, the differences between these transformations have no relevance to the shape of p(r), even if they have different effects in some domains outside the sphere. All such transformations t of equivalent effects within the relevant part of the 3D space can be collected into equivalence classes. In the symmorphy approach to the analysis of molecular shape, these classes are taken as the actual tools of shape characterization. 

Heavy-atom method Relative phases calculated for a heavy atom in a location determined from a Patterson map are used to calculate an approximate electron-density map. Further portions of the molecular structure may be identified in this map and used to calculate better relative phases, and therefore a more realistic electron-density map results. Several cycles of this process may be necessary in order to determine the entire crystal structure. 

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. 

The AFDF local molecular pieces can be used to build high quality approximate electron densities for large molecules, and also to study the local molecular subsystems themselves. 

In quantum mechanical calculations on atoms and molcules, we usually have only approximate electron density functions available. What is the relationship of hardness, 77, to p Will it increase to a maximum value as a set of trial basis functions becomes better and better, approaching the true electron density ... 

Bultinck, P., Carbo-Dorca, R. and Van Alsenoy, C. (2003) Quality of approximate electron densities and internal consistency of molecular alignment algorithms in molecular quantum similarity. 

In the preceding sections we have studied diatomic interactions via U(R). However, the study of diatomic interactions can also be carried out in terms of the force F(R) instead of the energy U(R), where R denotes the internuclear separation. Though there are several methods for the calculation of the force, the electrostatic theorem of Hellmann (1937) and Feynman (1939) is of particular interest in this section, since the theorem provides a simple and pictorial method for the analysis and interpretation of interatomic interactions based on the three-dimensional distribution of the electron density p(r). An important property of the Hellmann-Feyn-man (HF) theorem is that underlying concepts are common to both the exact and approximate electron densities (Epstein et al., 1967, and references therein). The force analysis of diatomic interactions is a useful semiclassical and therefore intuitively clear approach. And this results in the analysis of diatomic interactions via force functions instead of potential ones (Clinton and Hamilton, 1960 Goodisman, 1963). At the same time, in the authors opinion, it serves as a powerful additional instrument to reexamine model diatomic potential functions. 

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. 

The exact density functionals for arbitrary electron densities are not known. The simplest approximate density functionals are those which are exact for the uniform electron gas [16, 17] (the correlation energy functional is known exactly for the uniform electron gas only in the limits ofhigh and low electron densities the correlation energy for intermediate densities can be obtained by interpolation between these limits [14]). To improve upon the uniform electron gas functionals, Waldman and Gordon introduced scaling coefficients which depend on the number of electrons in the system [18]. Although these scaled functionals increase the accuracy of electron gas model calculations, this increased accuracy is due somewhat to a cancellation of the error due to the approximate electron density [19]. Accurate non-local density functionals have recently been developed [20, 21, 22], which lead to more accurate calculations of interaction energies [19,22]. 

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

Approximation Electron Density Fitting Algorithm Restricting Coefficients to Positive Values. 

Quality of Approximate Electron Densities and Internal Consistency of Molecular Alignment Algorithms in Molecular Quantum Similarity. 

To illustrate the magnitude of relativistic effects — kinematic as well as spin-orbit effects — electron density differences are depicted in Figure 16.4 for Ni(C2H2) and Pt(C2H2). For these plots various approximate electron densities have been subtracted from the four-component reference result. 

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