Constant electron density surfaces

FIGURE 2-8 Constant Electron Density Surfaces for Selected Atomic Orbitals. (a) (d) The cross-sectional plane is any plane containing the z axis, (e) The cross section is taken through the xz or yz plane, (f) The cross section is taken through the xy plane. (Figures (b)-(f) reproduced with permission from E. A. Orgyzlo and G. B. Porter, J. Chem. Educ., 1963, 40, 258.)... 

FIGURE 2.8 Constant Electron Density Surfaces for Selected Atomic Orbitals, (a)-(d) The cross-sectional plane is any plane containing the z axis. 

We emphasise again that the construction of an interatomic surface is parameter-free one could say that the surface draws itself. The use of molecular contour surfaces of constant electron density is artificial and serves here the purpose of visualisation. Here, the practical edge of the molecule, when in the gas phase, is set to /7 = 0.001 a.u., which is a typical value. Finally, we note that a molecule in the gas phase, i.e., alone in the universe, is a fiction sooner or later one will encounter another molecule, admittedly far away but one that still shares a topological boundary with the original isolated molecule. As a result, there is no need for a constant electron density surface to bound a molecule, that is, theoretically and philosophically in the gas phase and in the condensed phase, actually and practically. Any atom or collection thereof is always completely bounded by topological boundaries, which emerge naturally, without parameters. 

Electron Density Surface. A surface of constant Electron Density. 

A map of the electron density distribution around these atoms provides important information. It tells us to what distance from the adatom the surface is perturbed or, in catalytic terms, how many adsorption sites are promoted or poisoned by the adatom. The charge density contours in Fig. 6.27 are lines of constant electron density. 

Figure 3.10 Representations of the electron density ip2 of the Is orbital and the 2p orbital of the hydrogen atom. (b,e) Contour maps for the xe plane. (c,f) Surfaces of constant electron density. (a,d) Dot density diagrams the density of dots is proportional to the electron density. (Reproduced with permission from the Journal of Chemical Education 40, 256, 1963 and M. J. Winter, Chemical Bonding, 1994, Oxford University Press, Fig. 1.10 and Fig. 1.11.)...

Figure 3.15 An sp hybrid orbital, (a) left, radial functions for the 2s and 2p atomic orbitals right, radial function for the sp hybrid orbital (b) left, the shapes of the 2s and 2p atomic orbitals as indicated by a single contour value right, the shape of the sp hybrid orbital as indicated by the same contour, (c) The shape of a surface of constant electron density for the sp hybrid orbital, (d) Simplified representation of (c). (Reproduced with permission from R. J. Gillespie, D. A. Humphreys, N. C. Baird, and E. A. Robinson, Chemistry, 2nd Ed., 1989, Allyn and Bacon, Boston.)...

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 6.15 Three-dimensional representation of the sulfur atom in SC12. This atom is bounded by two interatomic surfaces (IAS) and one surface of constant electron density (p = 0.001 au). Topologically, an atom extends to infinity on its nonbonded side, but for practical reasons it is capped. Each interatomic surface contains a bond critical point (BCP).

Effect of diagonal-off-diagonal dynamic disorder (D-off-DDD). The polarization fluctuations and the local vibrations give rise to variation of the electron densities in the donor and the acceptor, i.e., they lead to a modulation of the electron wave functions A and B. This leads to a modulation of the overlapping of the electron clouds of the donor and the acceptor and hence to a different transmission coefficient from that calculated in the approximation of constant electron density (ACED). This modulation may change the path of transition on the potential energy surfaces. 

MolSurf parameters [33] are descriptors derived from quantum mechanical calculations. These descriptors are computed at a surface of constant electron density, with which a very fine description of the properties of a molecule at the Van der Waals surface can be obtained. They describe various electrostatic properties such as hydrogen-bonding strengths and polarizability, as well as Lewis base and acid strengths. MolSurf parameters are computed using the following protocol. 

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].

In fact, what is mapped is not the topography of the surface but contours of constant electron densities that is, the results are sensitive to the type of atom seen on the surface by... 

Contours of constant electron density for adsorbed atoms on aluminum. The straight line is the surface of the jellium. The center row shows contours of total density minus the superposition of frec-atom and bare metal densities. The bottom row is the profile of the bare metal density. [After Lang and William.s, 1976,1... 

The result of the calculations is the set of atomic orbitals familiar to all chemists. Figure 2-7 shows diagrams of s, p, and d orbitals and Figure 2-8 shows lines of constant electron density in several orbitals. The different signs on the wave functions are shown by different shadings of the orbital lobes in Figure 2-7, and the outer surfaces shown enclose 90% of the total electron density of the orbitals. The orbitals we use are the common ones used by chemists others that are also solutions of the Schrodinger equation can be chosen for special purposes. 

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.

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.)...

For an actual surface the positive charge is not uniformly distributed over a half-space. For steps at the surface or corrugated surfaces (on an atomic scale), the contours of constant electronic density have ridges and valleys corresponding to those of the distribution of positive ions at the surface but they are smoother and less bumpy. 

A number of studies have shown thatl(r) calculated on molecular surfaces defined by contours of constant electron density provide an effective tool for analysis of reactivity towards electrophiles [44-49]. The positions on a molecular surface where I(r) has its lowest values, the local surface minima (Is.min), are viewed as the locations of the least tightly bound electrons, and thus as the sites most likely to interact with an electrophile. However, in contrast to the electrostatic potential, I(r) reflects a molecule s ability to undergo charge transfer rather than its electrostatic interaction tendencies. The Is,min are therefore better suited than the Vmin for analyses of strong interactions that lead to the formation of covalent bonds. For example, it has been shown that the Is,min of aromatic systems can be used to identify and rank the sites most likely to undergo electrophilic attack that leads to electrophilic aromatic substitution [44]. The Vmin are not as successful in the same type of analysis. On the other hand, Vmin are much better suited than Is,min for characterization of hydrogen-bond-accepting sites [21]. 

Fig. 2.1. (a) and (b) schematically show the geometry of the sp, and sp orbital structures, (c) shows a calculated constant electron density probabihty surface for the constituent components of the sp orbital (one s and two of the three p orbitals) as well as one lobe of the hybrid orbital itself. 

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 3 Chemical symbols and schematic three-dimensional representations of the electron density distributions, for (a) a double bond system, ethylene, and (b) a conjugated double bond system, benzene. The shapes show the surfaces of constant electron density except for distortions necessary for clarity In depicting the overlapping or-bond and jr-bond electron clouds. 

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