Hydrogenlike wave functions

In Tables III-l, III-2, and III-3 there are given the expressions for the three component parts of the hydrogenlike wave functions for all values of the quantum numbers that relate to the normal states of atoms. The expressions for m(0) are given in both the complex form and the real form. 

Following Mulliken, we shall occasionally refer to one-electron orbital wave functions such as the hydrogenlike wave functions of this chapter as orbitals. In accordance with spectroscopic practice, we shall also use the symbols s, p, d, /, g, to refer to states characterized by the values 0, 1, 2, 3, 4, , respectively, of the azimuthal quantum number l, speaking, for example, of an s orbital to mean an orbital with 1 = 0. 

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.

Estimation of the relative intensity of the autoionization process [27] has shown that the second-order process intensity constitutes 10-15% of the first-order process intensity. Based on the results obtained and the dipole approximation for electron transitions [28], the authors of [27] draw the conclusion that the SEFS structure is formed in the process of coherent scattering of a secondary electron emitted from the valence band as a result of excitation by an incident electron (the first-order process). By contrast, the intensity of autoionization, i.e., the second-order process, was estimated [29-31] with hydrogenlike wave functions. The autoionization intensity in the region of the existence of the SEFS spectrum was shown to be comparable to the intensity of the first-order processes. 

Atomic Emission of Electrons from the Valence State in the Second-Order Process. Consider the atomic process of electron emission from the valence band on exciting the atom core level by an incident electron. As previously, the angular dependence of the intensity of emission of the final electron will be considered isotropic. To describe the emission of electrons from the valence band in the second-order process we will use hydrogenlike wave functions [Eqs. (40), (64), (65)] of an electron of the core level la), and the wave function of the valence-state electron will be given by... 

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. 

The hydrogenlike wave functions are one-electron spatial wave functions and so are hydrogenlike orbitals (Section 6.5). These functions have been derived for a one-electron atom, and we cannot expect to use them to get a truly accurate representation of the wave function of a many-electron atom. The use of the orbital concept to approximate many-electron atomic wave functions is discussed in Chapter 11. For now we restrict ourselves to one-electron atoms. 

In general we have two HJ states correlating with each separated-atoms state, and rough approximations to the wave functions of these two states will be the LCAO functions and - f, where / is a hydrogenlike wave function. The functions... 

The Schrodinger equation for the D-dimensional analogue of hydrogen (equation (88)) can be solved exactly, both in direct space and in reciprocal space and in both cases the solutions involve hyperspheri-cal harmonics. In this section we shall discuss the close relationship between hyperspherical harmonics, harmonic polynomials, and exact D-dimensional hydrogenlike wave functions. We shall also discuss the importance of these functions in dimensional scaling and in the hyperspherical method. 

The familiar 3-dimensional hydrogenhke wave functions in direct space can be expressed as confluent hypergeometric functions multiplied by spherical harmonics. We shall see that the jD-dimensional hydrogenlike wave functions can also be expressed as confluent hypergeometric functions, but in this case they axe multipUed by hyper-spherical harmonics. 

S.P. Alliluev [29,30] was able to obtain exact D-dimensional hydrogenlike wave functions in momentum space by a generalization of Fock s method. In Alliluev s treatment, Fock s transformation was generalized in such a way as to project D-dimensional momentum space onto the surface of a (DH-l)-dimensional hypersphere [24]. The momentum-space hydrogenlike wave functions could then be shown to be proportional to (D-f-l)-dimensional hyperspherical harmonics. 

From the above discussion, we can see that, both in direct space and in reciprocal space, the D-dimensional hydrogenlike wave functions involve hyperspherical harmonics and we shall therefore devote a little space to discussing these functions. Hyperspherical harmonics are closely related to harmonic polynomials [24]. In fact, hyperspherical harmonics are nothing but harmonic polynomials, orthonormal-ized in an appropriate way, and divided by appropriate powers of the hyperradius. Let us therefore begin by looking briefly at the theory of harmonic polynomials. 

The orthonormality properties of the hyperspheiical harmonics can then be used to show that the Fourier-transformed P-dimensional hydrogenlike wave functions in equation (90) are properly normalized. 

E. For the reciprocal-space solutions, ko represents the radius of the hypersphere onto which momentum-space is mapped by the generalized Fock transformation (equation (89)). As we saw above, the momentum-space wave functions are proportional to hyperspheri-cal harmonics on the surface of this hypersphere. The hyperspherical harmonics form a complete set, in the sense that any well-behaved function of the hyperangles can be expanded in terms of them. A set of hydrogenlike wave functions, all corresponding to the same value of ko (but with variable Z) is called a Sturmian basis [33-38,24] and such a basis set has the degree of completeness just mentioned. However, if Z is held constant while ko is variable within the set, then the continumn functions are required for completeness. 

In the case of the ground-state hydrogenlike wave function, V o> we have ... 

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 ... 

In Table 6 2 we give the complete normalized hydrogenlike wave functions for n = 1, 2, 3, using the above form for the angular part of the wave functions. 

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