Density envelope

We can conveniently think of p as a gas with a nonuniform density, which is more compressed and therefore more dense in some regions, and less compressed or less dense in other regions. Since the electron density p(x, y, z) of a molecule varies in three dimensions, we need a fourth dimension to represent it completely. Nevertheless we can get a good idea of the behavior of p by plotting constant electron density envelopes. 

Figure 6.3 Constant electron density envelope maps for SCI2 for three different contour values (a) p = 0.001 au, (b) p = 0.200 au and (c) p = 0.133 au. (a) This constant density envelope shows the practical outer boundary of the molecule broadly corresponding to the van der Waals envelope, (b) This constant density envelope demonstrates that for higher p values the envelope becomes disconnected into three surfaces each encompassing a nucleus, (c) This constant density envlope is plotted at the highest p value for which the molecular envelope is still connected or encompasses the whole molecule.

Figure 5. Three-dimensional isodensity envelopes of (a) SCI2, (b) H2O, and (c) Cl2. The outer envelope has the value of 0.001 au, the van der Waals envelope the inner one is the bond critical point density envelope (pb-envelope).

We begin with a comparison of the various DO models to each other. Based on a parametric procedure that takes account of the molecular volume encompassed by the 0.001 a.u. electron density envelope, Wong et al. [297] suggested that an appropriate spherical cavity radius is 3.8 A. Szafran et al. [157]... 

The three-way helical junctions all play key roles in directing the architecture of the ribozyme. The low-resolution electron density envelope is good enough to fit cylindrical representations of helical segments and has provided a starting point for more detailed structural modeling of the VS ribozyme to include the junctions and tertiary interactions. 

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. 

The distributions of charge for molecules in this series are illustrated in Fig. 1.1 in terms of an outer envelope of the charge density p and, specifically for the five- and six-carbon members, in Fig. 6.9 in the form of contour maps of p. The latter maps show the bond paths linking the nuclei and indicate the intersection of the interatomic surfaces with the plane of the diagram. The intersection of these same surfaces with the density envelopes are shown in Fig. 1.1 and they define the methyl and methylene groups as envisaged by chemists and as defined by theory. The diagrams show qualitatively what the atomic properties will demonstrate quantitatively, that the methyl and... 

The atoms of the adamantane cage are not totally enclosed. Their volumes v(C) and u (H), necessary for the calculation of the atomic pressures, are determined by the intersection of the 0.001 au density envelope with the interatomic surfaces bounding the atom. This volume contains 99.991% of the density of a methine carbon atom, for example. Contributions to the surface integrals vs X), beyond their intersection with the 0.001 envelope, should make negligibly small contributions to the atomic pressures. 

The electron distribution varies with basis set, thereby also changing the molecular size. The molecular volume was calculated for the molecule in the gas phase for each basis set. Wong, Wiberg and Frisch[49] found that there was a strong correlation between the calculated 0.001 electrons/Bohr density envelope (DE) and the experimentally measured molar volume (V ), in that the ratio where Voe is the volume enclosed by the den-... 

In the Isodensity Polarised Continuum Model the cavity is based on an isosurface of the total electron density calculated at the level of theory being employed. Only the electron density describing the isosurface is required to fully specify this model instead of the set of radii for the spheres used in the Polarised Continuum Model. The cavity is derived entirely from the electronic environment. In the present study, the isosurface W is defined by the 0.001 electrons/Bohr density envelope. 

Figure 15. The 0.001 au outer density envelope for the hydrocarbons ethane to pentane showing the group boundaries. The numbers shown are the transfers of charge induced between equivalent groups per unit field, by fields applied in the directions indicated by the arrows on the left-hand side of the figure...

The 0.001 au density envelopes of methane, ethane, propane, butane, pentane, and hexane. (Adapted from reference 216.)... 

Figure 13.1 Molecular graph of a drug with generic name acetaminophen (brand name Tylenol) supplemented with finite volumes of some topological atoms (O, N, and H). The outer boundary of each atom consists of the 0.001 a.u. constant electron density envelope.

Density The mass of an object divided by its volume. Variations are created by defining the volume measuring technique. These different techniques of measuring or calculating volume give rise to green bulk density, fired bulk density, apparent density, true density, envelope density and theoretical density. 

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