Representing the Electron Density

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. 

4 X 10-2, 8 X 10-2,. . . , au. Each subsequent contour value is then approximately twice the value of the preceding contour value. 

An early attempt to obtain insight from the molecular electron density was to subtract a reference density from it. The resulting difference density, Ap(r), introduced by Daudel and others is then simply  

The units we use in daily life, such as kilogram (or pound) and meter (or inch) are tailored to the human scale. In the world of quantum mechanics, however, these units would lead to inconvenient numbers. For example, the mass of the electron is 9.1095 X J0 31 kg and the radius of the first circular orbit of the hydrogen atom in Bohr s theory, the Bohr radius, is 5.2918 X 10 11 m. Atomic units, usually abbreviated as au, are introduced to eliminate the need to work with these awkward numbers, which result from the arbitrary units of our macroscopic world. The atomic unit of length is equal to the length of the Bohr radius, that is, 5.2918 X 10 n m, and is called the bohr. Thus 1 bohr = 5.2918 X 10 11 m. The atomic unit of mass is the rest mass of the electron, and the atomic unit of charge is the charge of an electron. Atomic units for these and some other quantities and their values in SI units are summarized in the accompanying table. 

There is another important reason for the existence of atomic units, namely, that quantum mechanical expressions, such as the Schrodinger equation, become simpler. When expressed in SI units, the Schrodinger equation for the hydrogen atom is 

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Atomic orbital Isa is associated with nucleus Ha and atomic orbital Isb with nucleus Hb. The first term in the electron density (is (r)) /(I -I-S ab) is taken to represent the amount of electron density associated with nucleus Ha- The corresponding term (ls (r)) /(I -I-S ab) represents the electron density associated with nucleus Hb and the remainder 2(lsA(r)lsB(r))/(l -t- AB) is taken to represent the amount shared by the two nuclei. Mulliken s first idea was to integrate these contributions, which gives the values 1/(1 -I-S ab), 1/(1 fi- AB) and (25ab)/(1 + ab). These values are assumed to contain some chemical information. Note that they sum to 2, the number of electrons in H2. 

According to the discussion in Section II, the quantity TT" represents the electron density about the nucleus in a hydrogen-like atom. The electron... 

Figs. 1-4 allow one to gain some further feeling about the quality of the approximation upon which our derivation of eq. (2.17) for n(x E) has been founded. Figs. 1 and 2 represent the electron density as a function of the coordinate x, in the absence of external electric field, for 2Ng=8 and 2Ne= 0 respectively. Excellent overall agreement between... 

It is possible, however, to avoid any violation of these fundamental properties, and derive a result on the local electron densities of non-zero volume subsystems of boundaryless electron densities of complete molecules [159-161]. A four-dimensional representation of molecular electron densities is constructed by taking the first three dimensions as those corresponding to the ordinary three-space E3 and the fourth dimension as that representing the electron density values p(r). Using a compactifi-cation method, all points of the ordinary three- dimensional space E3 can be mapped to a manifold S3 embedded in a four- dimensional Euclidean space E4, where the addition of a single point leads to a compact manifold representation of the entire, boundaryless molecular electron density. 

Another common method of representing the electron density distribution is as a contour map, just as we can use a topographic contour map to represent the relief of a part of the earth s surface. Figure 7a shows a contour map of the electron density of the SCI2 molecule in the Oh (xy) plane. The lines in which the interatomic surfaces, that are discussed later, cut this plane are also shown. Figure 7b shows a corresponding map for the H20 molecule. 

The quantity f n/(r) 2 represents the electron density as a function of r. The probability of finding the electron at a distance between r and r + dr, with no restrictions on 9 and cf> is obtained by integrating the probability density... 

The microscopic world of atoms is difficult to imagine, let alone visualize in detail. Chemists and chemical engineers employ different molecular modelling tools to study the structure, properties, and reactivity of atoms, and the way they bond to one another. Richard Bader, a chemistry professor at McMaster University, has invented an interpretative theory that is gaining acceptance as an accurate method to describe molecular behaviour and predict molecular properties. According to Dr. Bader, shown below, small molecules are best represented using topological maps, where contour lines (which are commonly used to represent elevation on maps) represent the electron density of molecules. 

The two-electron matrix G, the electron repulsion matrix (Eq. 5.104), is calculated from the two-electron integrals (Eqs. 5.110) and the density matrix elements (Eq. 5.81). This is intuitively plausible since each two-electron integral describes one interelectronic repulsion in terms of basis functions (Fig. 5.10) while each density matrix element represents the electron density on (the diagonal elements of P in Eq. 5.80) or between (the off-diagonal elements of P) basis functions. To calculate the matrix elements Grs (Eqs. 5.106-5.108) we need the appropriate integrals (Eqs. 5.110) and density matrix elements. These latter are calculated from... 

The Lande parameter ni in a system having spherical symmetry is essentially a one-electron operator quantity, as illustrated in Dr. Schmidtke s lecture (49). In a complex, the L. C. A. 0. M. O. description hence suggest that the new value of na (adapted to the lower Z due to central-field covalency) is multiplied by the square b representing the electronic density of the t2g sub-shell in MX6. Empirically, the dependence of ni is roughly a proportionality to Z, and we hence write... 

Figure 6-3. Representations of the hydrogen l v and 2pz orbitals (a) Plot of the angular wave function,. 4(0,

Here o represents the electron density per carbon-carbon bond, k is the strength of the induced magnetic dipoles in a given polyaromatic hydrocarbon relative to that in benzene, and / is a factor that corrects for the assumption that the electrons move in rectilinear segments, chosen by Pauling to be 1.23. With this approach, the magnetizability of almost any aromatic molecule may be calculated on the back of an envelope asf = oeW r nj ... 

This represents the electron density in a shell of radius r, including all values of the angular variables 6, [Pg.59]

We will now consider how to simulate this method of image formation in the X-ray diffraction experiment where we have to use a mathematical replacement for the objective lens. The studies by Porter are of great importance because they show how the Bragg reflections give the amplitude components of a Fourier series representing the electron density in the crystal (the electron-density map). In effect, Fourier analysis takes place in the diffraction experiment, so that the scattering of X rays by the electron density in the crystal produces Bragg reflections, each with a different amplitude F hkl) and relative phase Qhkl-... 

If a single point within the unit cell is chosen with fractional coordinates x,y,z (distance xa parallel to a, yb parallel to b, and zc parallel to c), the electron density p xyz) at that point is calculated by use of Equation 6.3. The right-hand side of this equation involves the summation of all measured Bragg reflections. If the intensities of 6,000 diffracted beams are measured, there will be 6,000 F hkl) values included in this summation for just one point. In practice, the electron density map is calculated by performing this summation at definite intervals of the asymmetric unit. We represent the electron density in electrons per A , and the summation is divided by the unit-cell volume, V. An example is given in Figure 6.9. 

Variations in F obs) maps are possible. The phases from direct methods can be combined with E hkl) values to calculate an E map, that is, an analogue of a normal electron-density map (which has F hkl) values as coefficients). The ideal E map represents the electron density of a structure composed of point atoms, rather than the smeared atoms found if values of F[hkl) are used. As a result, peaks are higher and sharper in an E map than in an F map. 

Up to this point, there are no approximations not even the central field approximation has been used, as is the case in all ab initio techniques in quantum chemistry. The procedure uses a set of molecular orbitals with the sole purpose of representing the electron density of a system of noninteracting electrons that happens to have the same density as the real system. All of this development comes out directly from the Schrodinger equation. All of the functionals or quantities in eq. (29) are known, except... 

According to the fundamental relations of crystallography (see e.g. refs. 1-5), appropriate Fourier transforms of the electron density p(r) define a set of structure factors, Fh. These quantities are the coefficients of a Fourier series representing the electronic density distribution p(r) in a crystal, where the three-dimensional periodicity of the crystal is exploited ... 

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