Hydrogenlike radial wave functions

Table 21-3.—The Hydrogenlike Radial Wave Functions n = 1, K shell ...

In the table of hydrogenlike radial wave functions the polynomial contained in parentheses represents for each function the associated Laguerre polynomial as defined by... 

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. 

TABLE 6.1 Radial Factors in the Hydrogenlike-Atom Wave Functions... 

Normalized Radial Wave Functions R(r) for Hydrogenlike Atoms... 

Fig. 10. Radial distribution of 2s, 3s, and 3d orbitals in the ion, calculated using hydrogenlike wave functions and Burns rules for estimating the screening constants (US).---, 2s ----, 3s and.., 3d.

Ground-State Wave Function and Energy. For the ground state of the hydrogenlike atom, we have = 1, / = 0, and m = 0. The radial factor (6.100) is... 

Table 6.1 lists some of the normalized radial factors in the hydrogenlike wave functions. Figure 6.8 graphs some of the radial functions. The r factor makes the radial functions zero at r = 0, except for s states. 

For a many-electron atom, the self-consistent-field (SCF) method is used to construct an approximate wave function as a Slater determinant of (one-electron) spin-orbitals. The one-electron spatial part of a spin-orbital is an atomic orbital (AO). We took each AO as a product of a spherical harmonic and a radial factor. As an initial approximation to the radial factors, we can use hydrogenlike radial functions with effective nuclear charges. 

For the real hydrogenhke functions (a) What is the shape of the n — / — 1 nodal surfaces for which the radial factor is zero (b) The nodal surfaces for which the

same plane.) (c) It can be shown that there are Z — m surfaces on which the 0 factor vanishes. What is the shape of these surfaces (d) How many nodal surfaces are there for the real hydrogenlike wave functions ... 

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