Hydrogen-like atom radial functions

Table Al.l Normalized radial functions / ,/(r) for hydrogen-like atoms...

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... 

Table 6.1. Radial functions R i for the hydrogen-like atom for n = to 6. The variable p is given by p = 2 Zrj na ...

Figure 6.4 The radial functions for the hydrogen-like atom.

As an example we apply the variational principle to the evaluation of the ground state energy of a hydrogen-like atom using a minimum basis set of two-component radial functions ... 

The correct limiting radial behavior of the hydrogen-like atom orbital is as a simple exponential, as in (A.62). Orbitals based on this radial dependence are called Slater-type orbitals (STOs). Gaussian functions are rounded at the nucleus and decrease faster than desirable (Figure 2.2b). Therefore, the actual basis functions are constructed by taking fixed linear combinations of the primitive Gaussian functions in such a way as to mimic exponential behavior, that is, resemble atomic orbitals. Thus... 

This is the most stable orbital of a hydrogen-like atom—that is, the orbital with the lowest energy. Since a Is orbital has no angular dependency, the probability density 2 is spherically symmetrical. Furthermore, this is true for all s orbitals. We depict the boundary surface for an electron in an s orbital as a sphere (Figure 1-2). The radial function ensures that the probability for finding the particle goes to zero for r — °°. 

The simplest analytical radial orbitals may be found by solving the radial Schrodinger equation for a one-electron hydrogen-like atom with arbitrary Z. They are usually called Coulomb functions and are expressed... 

Radial functions and energy eigenvalues for hydrogen-like atoms with this electron-nucleus potential are well-known in closed from, both in the non-relativistic and in the relativistic case. They can be found in every good textbook on quantum mechanics, for a compact reference see [48]. 

Due to the discontinuity of p r) at r = R, the second and higher derivar tives of V r) do not exist at this point. Only in the non-relativistic case the radial functions for hydrogen-like atoms with this electron-nucleus potential are known analytically in closed form. The corresponding energy eigenvalues must be determined iteratively [57-59]. 

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. 

Figure 6.1 Radial functions Pis (r) and Qij (r), in Hartree atomic units, of the ground state Dirac spinor of the hydrogen-like atoms with Z = 40,80,130, i.e., for Zr +, Hg +, and 130i +, With increasing nuclear charge the radial functions contract. The small component Qis (r) is small indeed for not too large nuclear charges, but its overall size increases with Z in such a way that Plsir) and Qis (r) are of similar size (but different sign) for Z = 130, i.e., if Z c in atomic units.

In the case of hydrogen-like atoms we have already noted that the number of nodes of the radial functions depends on the quantum numbers (see section 6.8.1). P ,. (r) always has (n — Zj — 1) radial nodes (as many as its nonrela-tivistic limit, the radial function P , , (r), has). Qn,K, (> ) has as many nodes as F ,k, (r) for negative values of K and one additional node for positive values... 

Table 17.2 gives the R i functions for n = 1,2, and 3. The entries in the table are for the hydrogen-like atom, which is a hydrogen atom with the nuclear charge equal to a number of protons denoted by Z. The He+ ion corresponds to Z = 2, the Li + ion corresponds to Z = 3, and so on. This modification to the R i functions will be useful when we discuss multielectron atoms in the next chapter. To obtain the radial factors and the energy levels of a hydrogen-like atom we need to replace the variable p by... 

The radial factors are a set of functions with two quantum numbers n, the principal quantum number, and /, the same quantum number as in the spherical harmonic functions. The hydrogen-like atom has a single electron and Z protons in the nucleus. The energy eigenvalues of the hydrogen-like atom depend only on the principal quantum number ... 

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. 

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

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. 

---