Constant electron density

If we consider the scattering from a general two-phase system (figure B 1.9.10) distinguished by indices 1 and 2) containing constant electron density in each phase, we can define an average electron density and a mean square density fluctuation as ... 

Electron Density Surface. A surface of constant Electron Density. 

Case 1 Atom on a Metal of Constant Electron Density... 

Let us describe the solid as having a constant electron density for all energies. Of course, such metals do not exist, but the situation gives us the simplest case. The matrix element Vat has also been assumed to be constant, meaning that A(e) is proportional only to the electron density of the metal. 

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. 

Hammett-Taft sigma constants Electron density TT-Bond reactivity Electron polarizability Dielectric constant Dipole moments Ionization potential Electron affinity... 

Figure 2-1. Representations of the electron density of the water molecule (a) relief map showing values of p(r) projected onto the plane, which contains the nuclei (large values near the oxygen atom are cut out) (b) three dimensional molecular shape represented by an envelope of constant electron density (0.001 a.u.). 

At the center of the approach taken by Thomas and Fermi is a quantum statistical model of electrons which, in its original formulation, takes into account only the kinetic energy while treating the nuclear-electron and electron-electron contributions in a completely classical way. In their model Thomas and Fermi arrive at the following, very simple expression for the kinetic energy based on the uniform electron gas, a fictitious model system of constant electron density (more information on the uniform electron gas will be given in Section 6.4) ... 

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.)...

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 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).

Note that in the reference model all the interactions of the electron with the medium polarization VeP are included in Eqs. (8) determining the electron states. The dependence of A and B on the polarization and intramolecular vibrations was entirely neglected in most calculations of the transition probability [the approximation of constant electron density (ACED)]. This approximation, together with Eqs. (4)-(7), resulted in the parabolic shape of the diabatic PES Ut and Uf. The latter differed only by the shift... 

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. 

Equations (50) and (51) show that for 0 < 6 < 1 the potential well for the electron near the donor site is more shallow than that in the initial equilibrium configuration. This leads to the fact that the radius of the electron density distribution in the transitional configuration is greater than in the initial equilibrium one (Fig. 3). A similar situation exists for the electron density distribution near the acceptor site. This leads to an increased transmission coefficient as compared to that calculated in the approximation of constant electron density (ACED). 

If the number Z/v of electrons in each particle is constant, electron density fluctuation... 

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. 

The key to investigating the topology of the electron density p is the gradient vector V p, which is perpendicular to a constant electron density snrface and points in the direction of steepest ascent. Then, a sequence of infinitesimal gradient vectors corresponds to a gradient path. Since gradient vectors are directed, gradient paths also have a direction They can go uphill or downhill. 

The derivative (dE/dZ)p at constant electron density p with respect to the nuclear charge of one of the nuclei is obtained with the help of the HeUmann-Feynman theorem [74]. So we get the potential at the center k, namely, VjJZk = dEjdZ )p, and the corresponding potential energy... 

Using Eq. (10.1) to obtain, with the help of the Hellmann-Feynman theorem [74], the derivative of AE at constant electron density p, taking the nuclear charge Z. as variable, namely... 

Summa summarum, Eq. (10.12) is certainly not the most efficient one for routine calculations of intrinsic bond energies. The reason is obvious—it lies with the partial derivatives (dski/dZk)p, which must be carried out at constant electron density p, meaning that this difficult calculation has to be made for each new s/ i, which is unpractical. 

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... 

Figure 8.3. The purple membrane. Contours of constant electron density are shown. (This is a simplified version of a figure which appears in Unwin, P.T.N. and Henderson, R. 1975 J. Mol. Biol. 94 425-40 and is reproduced by kind permission of the authors and Academic Press.) See the text for further discussion.

For a two-phase system having two components with volume fractions i and 2 and with corresponding constant electron densities pi and p2, the mean square of fluctuation in the electron density is given by ... 

The first term on the right hand side of Eq. (7.18) is the usual statistical shift determined at a constant electron density. However, the electron density is itself temperature dependent above the equilibration temperature because of the structural changes. The second term on the... 

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