Hydrogen-like Radial Wave Functions

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 forms of the radial wave function R r) and the angular wave function y (0,0) for a one-electron, hydrogen-like atom are shown in Table 8.2. The first thing to note is that the angular part of the wave function for an s orbital. 

The radial factors of the hydrogen-like atom total wave functions ip r, 0, tp) are related to the functions Sni(p) by equation (6.23). Thus, we have... 

The plots for hydrogen-like wave functions of radial function R(r) versus r, the distance from the nucleus and the probability distribution function 4jrr2[R(r) 2 versus r are shown... 

Fig. 3 shows a qualitative graphical representation of hydrogen-like wave functions for one-electron atoms which have to be replaced for many-electron atoms at least by Slater-type 107) analytical wave functions ifnlm (1) which are approximate as they contain no nodes in the radial part R ,. 

To conclude this review, we should like emphasize the fact that no serious argument can be presented for an exclusive use of pure atomic orbitals in quantum-chemical calculations, except that of the separation of the radial and angular parts of the wave function in the Hartree-Fock picture of the atoms themselves [80]. To the defenders of the traditional s, p, d. .. orbitals, we wish to reply that there are four coordinate systems for which the SchrQdinger equation of the hydrogenic atom can be solved [81], instead of eleven for a wave... 

We are also interested in knowing the total probability of finding the electron in the hydrogen atom at a particular distance from the nucleus. Imagine that the space around the hydrogen nucleus is made up of a series of thin spherical shells (rather like layers in an onion), as shown in Fig. 12.17(a). When the total probability of finding the electron in each spherical shell is plotted versus the distance from the nucleus, the plot in Fig. 12.17(b) is obtained. This graph is called the radial probability distribution, which is a plot of Atrr R versus r, where R represents the radial part of the wave function. 

As examples of radial and angular wave functions, those for values of the principal quantum number, , up to 3 are given, respectively, in Tables 2.3 and 2.4. Z represents the atomic number (1 for the hydrogen atom, but the formulae shown represent hydrogen-like atoms such as He for which Z = 2), and the term is the atomic unit of distance, explained below, and known as the Bohr radius. It has the value 52.9177 pm. 

Because the one-electron operators are identical in form to the one-electron operator in hydrogen-like systems, we use for the independent particle model of Eq. (8.96) for the basis of the many-electron wave function a product consisting of N such hydrogen-like spinors. This ansatz allows us to treat the nonradial part analytically. The radial functions remain unknown. In principle, they may be expanded into a set of known basis functions, but we focus in this chapter on numerical methods, which can be conveniently employed for the one-dimensional radial problem that arises after integration of all angular and spin degrees of freedom. 

TABLE 8.2 The Angular and Radial Parts of the Wave Functions for a Hydrogen-Like Atom... 

Since the interaction (4.304) is central, the associate wave equation may be separated in spherical polar coordinates to produce the normalized radial function. For the bound states hydrogenic atoms in the case of an infinitely heavy nucleus it looks like (Bransden Joachain, 1983) ... 

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