Molecular isodensity contours

One approach to the approximate representation of molecular bodies is based on molecular isodensity contours, MIDCOs, defined with respect to some fixed nuclear configuration K and some electron density threshold a. A MIDCO G(a,K) is defined (in the fixed nuclear configuration approximation) as the collection of all those points r of the three-dimensional space where the electronic density is equal to the threshold a ... 

The fundamental principle we shall follow in the local shape analysis of functional groups and local molecular moieties is a strict analogy with the shape analysis of complete molecules. Accordingly, instead of molecular isodensity contour (MIDCO) surfaces, the main tool of analysis will be the fragment isodensity contour (FIDCO) surfaces. Some of the ideas and concepts described in this section are illustrated in Figure 1. 

FIGURE 1 The fuzzy body of the electron density of a bovine insulin molecule is represented by three molecular isodensity contour surfaces (MlDCOs), for the density thresholds of 0.1, 0.01, and 0.001 a.u. (atomic unit), respectively, as computed using the MEDLA method. Bovine insulin was among the proteins selected for the first ab initio quality electron density computations for macromolecules. ... 

Figure 1.2 The three-dimensional, fuzzy "body" of the charge density distribution of allyl alcohol can be represented by a series of "nested" molecular isodensity contours (MIDCO s). Along each MIDCO the electronic density is a constant value. Three such MIDCO s are shown for the constant electron density values of 0.2, 0.1, and 0.01 (in atomic units), respectively. A contour surface of lower density encloses surfaces of higher density. These MIDCO s are analogous to a series of Russian wooden dolls, each larger doll enclosing a smaller one. These ab initio MIDCO s have been calculated for the minimum energy conformation of allyl alcohol using a 6-31C basis set.

Level sets F(a) [as well as the closely related density domains DD(a), as we shall see in the next section] provide a representation of formal molecular bodies. A similar definition gives a useful concept of a formal molecular surface the concept molecular isodensity contour surface (MIDCO). For any formal nuclear configuration K, it is possible to define a surface by choosing a small value a for the electronic density, and by selecting all those points r in the 3D space where the density p(r) happens to be equal to this value a, that is, where equation (2.3) is fulfilled. For an appropriate small value a, this contour surface may be regarded as the surface of the essential part of the molecule and, in short, it may be referred to as the molecular surface. These surfaces, the molecular isodensity contour surfaces, or MIDCO s, are denoted by G(a) and are defined as... 

If molecular shape is represented by the electronic density, then the shape analysis can be performed on the molecular isodensity contours (MIDCO s) of the calculated density. Some of the elementary properties of MIDCO s have been... 

The concept of density domains can be introduced in the context of isodensity contour surfaces. A molecular isodensity contour surface, MIDCO G(K,a)y of nuclear configuration K and density threshold a is defined as the collection of all points r of the 3D space where the electronic density is equal to the threshold value a. In the notation for electron density it is useful to specify the nuclear configuration K of the molecule and in the forthcoming discussion the notation p(K, r) will be used for the electron density of nuclear configuration K. Accordingly, the MIDCO G(K,a) is defined as... 

According to a recent proposal [4,8,9], chemical bonding within formal molecular bodies can be described by molecular isodensity contour (MIDCO) surfaces and by density domains (DD) that are the formal bodies enclosed by... 

A formal definition of a molecular isodensity contour surface, MIDCO G(K, a) of nuclear configuration K and density threshold a is given as the collection of points of the 3D space where the electronic density p(K,r) is equal to the... 

An important fact has been pointed out by Parr and Berk the bare nuclear potential Vn(r) shows many similarities with the electronic density function p(r). The computed isopotential contours of the composite nuclearpotential VnC lwere remarkably similar to some of the molecular isodensity contours (MIDCOs) of the electronic ground states in several simple molecules. One may regard the composite nuclear potential as the harbinger of electronic density, and isopotential contours of the composite nuclear potential V (r) can serve as surprisingly good approximations of MIDCOs. The nuclear potential contours (NUPCOs) are suitable for an inexpensive, approximate shape representation of molecules. 

The shape of molecules is the shape of their three-dimensional 3D electron density clouds. In Figure 1, nine molecular isodensity contour surfaces (MIDCOs) of the electron density cloud of the water molecule are shown, at density levels ranging from 0.001 au to 0.45 au, where the atomic unit of charge density is used, 1 au = e bohr e is the electronic charge). Molecular shape has a fundamental effect on both physical and chemical properties of molecules. The development and application of computational approaches to molecular shape analysis are among the main roles of computational chemistry. 

Figure 1 Molecular isodensity contour surfaces, MIDCOs, of water shown at nine density levels ranging from 0.001 au to 0.45 au (electron density in atomic units, 1 au = e bohr where e is the electronic charge)...

A typical discrete set of chemical importance is a nuclear arrangement in the clamped nucleus version of the Bom-Opp-enheimer approximation. A single molecular isodensity contour (MIDCO) surface is a crisp continuum set. A simple generalization of the CSM approach from finite, discrete point sets to continua is provided by the crisp average of sets. 

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