Isodensity contours

Figure 1. Relief maps of the electron density of (a) SCI2 and (b) H2O in the plane of the nuclei (density and distances from the origin of the coordinate system in au). Isodensity contour lines are shown in the order 0.001,0.002,0.004,0.008 (four outermost contours) 0.02,0.04,0.08 (next three) 0.2,0.4,0.8 (next three). The density is truncated at 2.00 au (innermost contour). These contours are shown in blue, violet, magenta, and green, respectively, on the figure in the table of contents (p 1028).

A more practical representation of the electron distribution in a molecule can be obtained from the probability density contour maps. Isodensity contours in the molecular plane and in a plane parallel to the molecular plane at an altitude of 0.8 atomic unit have been calculated230 for three nucleic acid bases (adenine, thymine, and cytosine) from non-empirical wave functions. The first type of contour gives an overall picture of cr-bonding in the molecule, and the second characterizes the 77-electron density. 

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. 

In Fig. 4, isodensity contour plots of the Kohn-Sham LUMO are shown. The familiar dumbbell shape of the p-like orbital is not... 

Fig. 4. The anatomy of a p-like state . Two isodensity contour maps (+ 0.01 and + 0.03 a.u." ) of the same LUMO orbital are shown side by side. Unlike the p-Uke orbitals in one-electron models, LUMO states in MQC MD-DFT and CIS models have the lobes pushed outwards between the first and the second solvation shells, with < 20% of the spin density residing inside the cavity. This results in considerable firagmentation of the diffuse part of the wavefunction. The O 2p orbitals are strongly polarized, with opposite signs of the orbitals attained by water molecules on the opposite sides of the cavity in the direction of transition dipole moment.

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... 

The concept of density domains is related to the concept of MIDCO in a simple way. A maximum connected part of an isodensity contour surface G(a) and the corresponding part of the level set F(a) enclosed by it is called a density domain, DDj(a) [109]. Below we shall give a more formal definition and describe the most essential properties of density domains. 

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... 

Parr and Berk [309] have found that the isopotential contours of the nuclear potential V (r) of simple molecules show a remarkable similarity to the actual isodensity contours of the electronic ground states of these molecules. 

For example, if the shape domains are defined in terms of local convexity, and if we select the locally convex domains, then the shape groups of G(a) are the homology groups of the truncated isodensity contour surface G(a,2), obtained from the molecular contour surface G(a) by eliminating all domains of index p = 2. This family of shape groups, obtained by cutting out all locally convex domains of G(a), has been studied in most detail for several molecules [192,262,263,342]. 

The points located by this procedure along the line segments generate a dot representation of the electronic isodensity contour (MIDCO) surface G(a) for any value a of the contour density, as long as a < a < ao, where a , is the maximum density value for points along the low density envelope F in the union of all the P") sets, and ao is the minimum density value for points along the high density envelope Fq in the union of all the pOj sets. 

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