Radial distribution of the electron density

Flo. 19. Radial distributions of the electron density of a-K2CrO( (85). (a) Observed... 

Fig. 4. Radial distribution of the electron density in equlatomic mai nese compounds MnBi (1), MnSb (2), and MnAs (3).

The existence of some overlap of the 3d shells in mai anese arsenide is particularly noticeable in curves showing the radial distribution of the electron density U(r) = 4irr p(r) (see Fig. 4). The maximum of the radial distribution, the position of which may be associated with the radius of the 3d shell, moves to higher values of r on reducli the atomic number of the anion. The followii values of rjjj jj were obtained for the compounds studied for MnBi 0.48, MnSb 0.50, and MnAs 0.56 A. 

Fig. 2.20 Radial distributions of the electron density in the atomic spheres of UC, UN and UO and in free U, C, N and O atoms. Dashed lines - free atoms chain lines - atoms in solids. Solid lines are the differences between the electron densities in the atomic spheres of the crystals and in free atoms.

Often, it is more meaningful physically to make plots of the radial distribution function, P(r), of an atomic orbital, since this display emphasizes the spatial reality of the probability distribution of the electron density, as shell structure about the nucleus. To establish the radial distribution function we need to calculate the probability of an electron, in a particular orbital, exhibiting coordinates on a thin shell of width, Ar, between r and r - - Ar about the nucleus, i.e. within the volume element defined in Figure 1.6. 

Figpre 3.3 presents the probability density distribution for the one-electron atom. The distribution of the electron density is shown for the principal quantum numbers n = 1,2,3 and the I = 0,1,2. It is obvious that the radial probabihty density has significant values in restricted ranges of the radial coordinate. When the atom is in one of its quantum states the electron can probably be found within a certain so-called electronic sheh. The characteristic radius of this sheU is generally determined by the principle quantum number n. One can also see from Figure 3.3 that the quantum number I affects the radial density distribution. Additional maxima appear for n = 2,1 = 0 = 3,1 = 0 and 1. However, there are no such maxima when I takes its largest possible value. 

Covalent chemical bonds between atoms of the same or a different species rely on the interaction of the outermost — or valence — electrons. Even though one speaks of electrons one should rather think of electron clouds, i.e. of electronic density distributions. The radial and angular distribution of the electron density is described by one electron wave functions — also called atomic orbitals — which are derived as a solution of the quantum mechanical Schrodinger equation ... 

A simple modification of the IAM model, referred to as the K-formalism, makes it possible to allow for charge transfer between atoms. By separating the scattering of the valence electrons from that of the inner shells, it becomes possible to adjust the population and radial dependence of the valence shell. In practice, two charge-density variables, P , the valence shell population parameter, and k, a parameter which allows expansion and contraction of the valence shell, are added to the conventional parameters of structure analysis (Coppens et al. 1979). For consistency, Pv and k must be introduced simultaneously, as a change in the number of electrons affects the electron-electron repulsions, and therefore the radial dependence of the electron distribution (Coulson 1961). 

Since an a priori definition of the effective region is hardly possible, each atomic region is usually approximated by a spherical region around the atom, where the radius is taken as its ionic, atomic, or covalent bond radius. The radial distribution of electron density around an atom is also useful to estimate the effective radius of an atom, particularly in ionic crystals. In an ionic crystal, the distance from the metal nucleus to the minimum in the radial distribution curve generally corresponds to the ionic radius. As an example, the radial distribution curves around K in o-KvCrO., (85) are shown in Fig. 19a. The radial distributions of valence electrons (2p electrons) exhibit a minimum at 1.60 A for K(l) and 1.52 A for K(2), respectively. These distances correspond to the ionic radii in crystals (1.52-1.65 A)... 

Fie. 21. Radial distribution of the valence electron density around the Co atom in... 

The radial distribution of electron probability density for the sodium atom. The shaded area represents the 10 core electrons. The radial distributions of the 3s, Ip, and 3d orbitals are also shown. Note the difference in the penetration effects of an electron in these thiee orbitals. 

Fig. 5.3 Comparison of the experimental and SCF radial distributions of the total electron density for the argon atom (reprinted with permission from lef. 44 copyright 1953, American Physics Society).

Without even solving the equation in the radial variable, just knowmg the spherical harmonics gives us a lot of information about the structure of the atom. The probability distribution of the electron is usually pictured as a cloud, where the density of the cloud denotes greater probability. Nodes of the spherical functions are choices of 9 and where the solution is zero. For example,... 

If the radial distributions of the inner s orbitals are now compared with those for the 3d orbital (Fig 10), it is seen that the Is and 2s orbitals lie almost entirely inside the 3d orbital and polarization will produce a negative spin density at the nucleus In contrast, the 3s orbital is slightly more diffuse than the 3d, thus producing a positive spin density at the nucleus, i.e., it polarizes the electron nearer the nucleus thus giving it the same spin as itself. The net effect is the sum of these... 

The above cited methods use point multipolar representation of the electron density associated with atoms or localized orbitals. Since the radial part of the electronic charge distribution is neglected, the electrostatic potentials become very poor in the close proximity of the molecule, e.g. in the lone pair regions, including contacts with hydrogen bonded substrates or solvents. 

A schematic representation of the electron densities (radial distribution functions, RDFs) of the 4/, Sd, and 6s orbitals. The 6s electron has five small maxima of electron density (or probability) mostly within the 4/ and 5d RDFs. These enable the 6s orbital to penetrate through the filled 4/and 5d electron clouds and therefore experience a greater-than-expected effective nuclear charge. The 6p electron (not shown for reasons of clarity) is similar to the 6s but has one fewer small maximum of electron density. 

New figure (Figure 14.4) showing a schematic representation of the electron densities (radial distribution functions, RDFs) of the 4f, 5d, and 6s orbitals... 

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