The Radial Wave Functions

The radial wave functions all show an exponential decay as the radius increases. The exponential decay is slower with increasing n because the denominator in the exponential term contains a factor of hoq. Thus, the average radius (or size) of an orbital also increases with n. For n I, the radial functions all have at least one radial 

TABLE 4.1 Mathematical forms of the radial wave functions, R(i), and angular wave functions, Y 0, f ), for the hydrogen atom for the first few sets of allowed quantum numbers, where = 52.9 pm, the Bohr radius. 

When a wave function or a product of wave functions is integrated over all space, the volume element in Cartesian coordinates is dr = dx dy dz. In polar coordinates. 

Definition of the volume element dr in polar coordinates 6x=6V=r sin0 drd 0 d . [ University Science Books, Mill Valley, CA. Used with permission. All rights reserved. McQuarrie, 

Example 4-1. Use Equation (3.18) to prove that the Bohr radius Oq has a value of 52.9 pm when n = I. 

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Solution of the Schrodinger equation for R i r), known as the radial wave functions since they are functions only of r, follows a well-known mathematical procedure to produce the solutions known as the associated Laguerre functions, of which a few are given in Table 1.2. The radius of the Bohr orbit for n = 1 is given by... 

Figure 1.7 Plots of (a) the radial wave function (b) the radial probability distribution function and (c) the radial charge density function 4nr Rl( against p...

The metal cluster will be modeled as an infinitely deep spherical potential well with the represented by an infinitely high spherical barrier. Let us place this barrier in the center of the spherical cluster to simplify the calculations. The simple Schrodinger equation, containing only the interaction of the electrons with the static potential and the kinetic energy term and neglecting any electron-electron interaction, can then be solved analytically, the solutions for the radial wave functions being linear combinations of spherical Bessel and Neumann functions. 

Consider now the solutions of the spherical potential well with a barrier at the center. Figure 14 shows how the energies of the subshells vary as a function of the ratio between the radius of the C o barrier Rc and the outer radius of the metal layer R ui- The subshells are labeled with n and /, where n is the principal quantum number used in nuclear physics denoting the number of extrema in the radial wave function, and / is the angular momentum quantum number. 

The radial wave functions used are thus the hydrogen-like 2p and 3d functions, J ai(r) and J 32-(r), for all orbitals of the L and M shells, respectively the symbols pts, and i 3j, piP, p3d represent these multiplied by the angular parts 1 (for s), /3 cos 8 (for p), and /5/4 (3 cos2 0-1) (for d), rather than the usual hydrogen-like orbitals. The 2-axis for each atom points along the internuclear axis toward the other atom. 

The effective atomic numbers in the radial wave functions cannot be evaluated by minimizing the energy integral, because of neglect of inner shells. In all the calculations reported the effective atomic numbers were given the value 1. 

The Relation between the Shell Model and Layers of Spherons.—In the customary nomenclature for nucleon orbitals the principal quantum number n is taken to be nr + 1, where nr> the radial quantum number, is the number of nodes in the radial wave function. (For electrons n is taken to be nT + l + 1.) The nucleon distribution function for n = 1 corresponds to a single shell (for Is a ball) about the origin. For n = 2 the wave function has a small negative value inside the nodal surface, that is, in the region where the wave function for n = 1 and the same value of l is large, and a large value in the region just beyond this surface. 

The nature of the radial wave functions thus leads us to the following interpretation1 of the subshells of the shell model ... 

The Slater—Condon integrals Ft(ff), Ft(fd), and Gj-(fd), which represent the static electron correlation within the 4f" and 4f 15d1 configurations. They are obtained from the radial wave functions R, of the 4f and 5d Kohn—Sham orbitals of the lanthanide ions.23,31... 

The bound-state energies and eigenfunctions can be obtained by solving the Schrodinger equation with boundary conditions that the radial wave function vanishes at both ends... 

In the rigid rotor approximation, the radial wave function is independent of /. The total wave function is... 

The typical behavior of M0 v is shown in Figure 1.6. One should note that, for the Morse potential, and in lowest approximation, the radial wave functions and thus v are independent of /. This is no longer the case for more general potentials and for the exact solution of the Morse problem. 

In the Clementi and Roetti tables, the radial wave function of all orbitals in each electron subshell j is described as a sum of Slater-type functions ... 

The radial wave functions depend on radius only via the combination p = rrUr-Za and it is convenient to write it explicitly as a function of this dimensionless variable... 

The gamma functions Ak(p) and Bj(pt) may be obtained by the use of recursion formulas an extensive tabulation is due to Flodmark (141). In the case of Slater orbitals of principal quantum number 4 or 6, application of Slater s rules leads to nonintegral powers of r in the radial wave function consequently, changing to spheroidal coordinates introduces A and B functions of nonintegral k values, that is, incomplete gamma functions. These functions can, however, be computed (56, 57) and the overlap... 

Line shapes. Because of the energy normalization of the radial wave-functions, Eq. 5.63, the summations over the free-state energies, Eq. 5.62, become integrations,... 

Because no symmetry operation can alter the value of R(n, r), we need not consider the radial wave functions any further. Symmetry operations do alter the angular wave functions, however, and so we shall now examine them in more detail. It should be noted that, since A(0, 0) does not depend on n, the angular wave functions for all s, all / , all d, and so on, orbitals of a given type are the same regardless of the principal quantum number of the shell to which they belong. Table 8.1 lists the angular wave functions for sy p, d, and / orbitals. 

The most apparent feature of the radial wave functions is that tbey all represent an exponential "decay", and that for n 2 the decay is slower than for m — I. Tlus may be generalized for all radial functions They decay as For this rea-... 

Usually, for both theoretical and semi-empirical determination of energy spectra, radial integrals that do not depend on term energy of the configuration are used. More exact values of the energy levels are obtained while utilizing the radial wave functions, which depend on term. Therefore, there have been attempts to account for this dependence in semi-empirical calculations. Usually the Slater parameters are multiplied by the energy dependent coefficient... 

FIGURE 1.25 The radial wave-functions of the first three s-orbitals of a hydrogenlike atom. Note that the number of radial nodes increases (as n — l), as does the average distance of the electron from the nucleus. Because the probability density is given by ijr2, all s-orbitals correspond to a nonzero probability density at the nucleus. 

The anisotropic coupling of the halogen nuclei arises essentially from the p-character of the wave-function. The axial symmetry of p-electrons then implies that the maximum value of the anisotropic coupling is observed when the magnetic field is parallel to the direction of the p-orbital. It depends on the average value of the inverse cube of the radial wave-function and can be shown to have the value ... 

The radial wave function has (n — l+l) nodes, where n and l are the quantum numbers. To solve the radial atomic wave equation above, the Herman-Skillman method [12] is usually used. The equation above may be rewritten in a logarithmic coordinate of radius. The radial wave equation is first expressed in terms of low-power polynomials near the origin at the nucleus [13]. With the help of the derived polynomials near the origin, the equation is then numerically solved step by step outward from the origin to satisfy the required node number. At the same time, the radial wave equation is solved numerically from a point far away from the origin, where the radial wave function decays exponentially. The inner and outer solutions are required to be connected smoothly including derivative at a connecting point. 

The im (9) functions are related to the associated Legendre polynomials, and the first few are listed in table 6.1. The R i(r) are the radial wave functions, known as associated Laguerre functions, the first few of which are listed in table 6.2. The quantities n, l and m in (6.8) are known as quantum numbers, and have the following allowed values ... 

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