Wave functions, approximate hydrogenlike

The operation of the selection rule for l for hydrogen and hydrogenlike ions can be seen by the study of the fine structure of the lines. The phenomena are complicated, however, by the influence of electron spin.1 In alkali atoms the levels with given n and varying l are widely separated, and the selection rule for l plays an important part in determining the nature of their spectra. Theoretical calculations have also been made of the intensities of lines in these spectra with the use of wave functions such as those described in Chapter IX, leading to results in approximate agreement with experiment. 

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. 

Secondary Electron Emission from the Li.a and M2,3 Core Levels. In calculating the emission of the secondary electrons from the Is hydrogenlike core level, we succeeded in making analytical estimates and obtaining simple approximating expressions for the amplitude [Eq. (61)], the intensity of emission [Eq. (62)], and the angular correlation function [Eqs. (52), (53), (56)]. However in the study of the 3d metal SEFS it is essential to describe the emission of the secondary electrons from L2,3 and M2,3 core levels. Consider the ionization process of the L2,3 and M2,3 core level of the atom with the wave functions of the core level electron taken as... 

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. 

The orbital concept and the Pauli exclusion principle allow us to understand the periodic table of the elements. An orbital is a one-electron spatial wave function. We have used orbiteils to obteiin approximate wave functions for many-electron atoms, writing the wave function as a Slater determinant of one-electron spin-orbitals. In the crudest approximation, we neglect all interelectronic repulsions and obtain hydrogenlike orbitals. The best possible orbitals are the Heu tree-Fock SCF functions. We build up the periodic table by feeding electrons into these orbitals, each of which can hold a pair of electrons with opposite spin. 

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. 

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 Is function in this wave function is a helium-atom Is function, which ideally is an SCF atomic function but can be approximated by a hydrogenlike function with an... 

To get approximations to higher MOs, we can use the linear-variation-function method. We saw that it was natural to take variation functions for Hj as linear combinations of hydrogenlike atomic-orbital functions, giving LCAO-MOs. To get approximate MOs for higher states, we add in more AOs to the linear combination. Thus, to get approximate wave functions for the six lowest linear combination of the three lowest m = 0 hydrogenlike functions on each atom ... 

The subscripts a and b refer to the two atoms, and the bar indicates spin function j8. The 1 function in this wave function is a helium-atom Is function, which ideally is an SCF atomic function but can be approximated by a hydrogenlike function with an effective nuclear charge. The VB wave function for Hc2 has each electron paired with another electron in an orbital on the same atom and so predicts no bonding. 

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