Electron density contour surfaces

Various choices have been proposed for atomic radii/ and the molecular surface obtained depends on this choice. 

Fused spheres VDWSs have been used extensively in computer-based molecular modeling, especially in the interpretation of biochemical processes and drug design. Besides their conceptual simplicity, their main advantage is that their computation and graphic display on a computer screen are relatively simple tasks, even for larger molecules. 

Note that the above representation of a formal molecular body by a level set F a) involves two simplifications  

It is assumed that a fixed nuclear configuration K adequately represents the actual nuclear arrangement. 

Those points of space where the electronic density is less than the threshold value a are not considered to belong to the molecule. 

---

An electron density contour surface piece with a larger Betti number usually (but not necessarily) has the larger surface area. If this is the case, then the sequence of Betti numbers in the above ordering is the same as the decreasing sequence of the Betti numbers. 

Electron density contour surfaces have the mathematical property of compactness, a generalization of the properties of closed and bounded . The Hausdorff distance h(A, B) between two (compact) subsets A and B of X is defined as the lowest upper bound h(A, B) = sup g gg(rf(a, B), d b, A) of distances between points a of A and the set B and distances between points b of B and the set A. In particular, the Hausdorff distance between two superimposed molecular contour surfaces (which are closed sets) is the minimum r value such that any point on either contour surface has at least one point of the other contour surface within a distance r. 

Fig. 31. Approximation of van der Waals cross-sections of inclusion channels in 1 alcohol clathrates21 (dimensions are in A hatched regions represent O atoms of the host matrix continous solid lines indicate surfaces of apolar attribute) (a) 1 MeOH (1 2) (approximately parallel to the 0(I -Cul vectors, cf. Fig. 17a) (b) 1 2-PrOH (1 2) (orientation as before) (c) 1 2-BuOH (1 1) (through a center of symmetry at 1,1/2,1/2, cf. Fig. 30c non-zero electron density contours) (d) 1 ethylene glycol (1 1) (in the plane of the C—C single bonds of a guest molecule, indicated by projected stick models non-zero electron density contours)...

Figure 2.8. Electron density contours for atomic chemisorption on jellium with electron density that corresponds to A1 metal. Upper row Contours of constant electron density in the plane normal to the surface. Center row Difference in charge density between isolated adatom and metal surface, full line gain and dashed line loss of charge density. Bottom row Bare metal electron density profile. Reproduced from [30].

Fig. 2.7 (a) Pictorial representation or the electron density in a hydrogen-like 2p orbital compared with lb) the electron density contours Tor the hydrogen-like 2pr orbital of carbon. Contour values are relative to the electron density maximum The xy plane is a nodal surface. The signs (+ and —) refer to those of the original wave function. [The contour diagram is from Ogryzlo, E. A4 Porter G. B J. Client. Ethic. 1963.40. 258. Reproduced with permission. ... 

The molecular volume descriptor, V, can be recognized as an important descriptor once one realizes that the free energy of solution is related in part to the size of the cavity that must be carved out of the solvent bath by the solute molecule during the solvation process. The surface area, A, of a molecule or a fragment of a molecule may be construed as a measure of the region available for interaction with another molecule. For computing V and A, one could use a particular electron density contour or a non-QM-derived measure of atomic size such as the van der Waals radii available from standard tables in physical chemistry textbooks. 

Electron density contour diagrams have been derived " (Figure 2) and potential energy curves and surfaces calculated In addition to this impressive quantity of accurate work on the parent compound a number of simplified schemes were proposed by Forster, Coulson and Moffit " and by Walsh these schemes have been widely employed for the interpretation of the photoelectron spectra and other properties of the derivatives of cyclopropane. 

Fig. S. (A) Emergent hybrid d orbitals at a metal surface (schematic). [After Bond 24).] (B) (Left) Electron-density contour map for the occupied a2, antibonding surface orbital of a cubooctahedral Ni,3 cluster, corresponding to the energy level —0.413 Ry, plotted in the plane of the square face containing atoms 1-4 of the cubooctahedron structure. (Right) Equivalent map but corresponding to the energy level -0.413 Ry plotted in the equatorial plane containing atoms 5-8 and 13 of the cubooctahedron structure.

Figure 2 Electron-density contours for chemisorption. Upper row contours of constant electron density in (any) plane normal to the metal surface containing the ad-atom nucleus (indicated by -f). The metal is to the left of the solid vertical line. Center row deformation charge density. The polarization of the core region, shown for Li, has been deleted for Si and Cl because of its complexity. Bottom row The bare-metal electron-density profile, shown to establish the distance scale. (From Ref. 38.)...

Fig. 5.3 Electron density contours lor an sp hybrid orbital. Note that the nodal surface does not pass through the nucleus.

Fig. 2.1. Electron density contour maps for the model MggOg of the MgO(OOl) surface in the plane containing the fourfold symmetry axis and the moiety O-Mg-O a) Mgpp +PC embedding b) PC only embedding c) electron density difference Ap = p(Mg909, Mg +PC) - p(Mg909,PC) with negative values indicated by dashed lines.

The critical point is the point at which the gradient vector field for the charge density is zero, that is, either a maximum or minimum along N. The condition Vp(r) N(r) = 0 applied to other paths between two atoms defines a unique surface that can represent the boundary of the atoms within the molecule. The electron density within these boundaries then gives the atomic charge. The combination of electron density contours, bond paths, and critical points defines the molecular graph. This analysis can be applied to electron density calculated by either MO or DFT methods. For a very simple molecule such as Hj, the bond path is a straight line between the nuclei. The... 

Figure 6.10. Electron density contours of CO chemisorbed on the Ni(lOO) surface [27. ...

Fig. 34a-c. Results of a model calculation of a Xe atom adsorbed on a high electron density jellium surface (e.g. aluminium), (a) Contours of constant electron density in a cut perpendicular to the surface through the center of the Xe atom, (b) Xe valence p-elechon density vs. distance (difference density between metal-adatom system and sum of clean metal plus single Xe atom except 5p level), (c) Effective single particle potential energy contributions due to electrostatic dipole, Ves, and the exchange-correlation interaction, Vxc, respectively, [82Lan],... 

Figure 1.27 Antibody combining site of 11C 10 in complex with hapten with electron density contoured at 1.5a. Representation of the electrostatic surface [152].

Fig.10 A local energy minimum of the water (darfc).. jnethanal (light) complex. Bond paths and interatomic surfaces (both in bold) are superimposed on an electron density contour plot...

Many functions, such as electron density, spin density, or the electrostatic potential of a molecule, have three coordinate dimensions and one data dimension. These functions are often plotted as the surface associated with a particular data value, called an isosurface plot (Figure 13.5). This is the three-dimensional analog of a contour plot. 

---