Hydrogenlike atom wave functions

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

For very large values of rAB we know that in its normal state the system consists of two normal hydrogen atoms. Its wave functions (the state having two-fold degeneracy) are then Ui,x(l) Ui,s(2) and i,B(l) Wi ,(2) or any two independent linear combinations of these two (the wave function ulwave function for electron 1 about nucleus A,... 

It was mentioned by Heitler and London in their first paper1 that rough theoretical considerations show that two normal helium atoms repel each other at all distances. The evaluation of the energy for the wave function of Equation 44-3 with uA and uR hydrogenlike Is wave functions with effective atomic number Z = was carried out by Gentile.2 A more... 

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. 

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 wave functions for a state of a hydrogenlike atom described by the quantum numbers n (total quantum number), l (azimuthal quantum number), and m (magnetic quantum number) are usually expressed in terms of the polar coordinates r, 8, and . The orbital wave function is a product of three functions, each depending on one of the coordinates ... 

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. 

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. 

Concerning molecules, the wave function (molecular orbital) for a hydrogenlike molecule, for instance, is expanded in terms of hydrogen-like atomic orbitals Xaj(f) belonging to hydrogen-like atoms / = 1,2, respectively, as... 

The problem of the structure of the hydrogen atom is the most important problem in the field of atomic and molecular structure, not only because the theoretical treatment of this atom is simpler than that of other atoms and of molecules, but also because it forms the basis for the discussion of more complex atomic systems. The wave-mechanical treatment of polyelectronic atoms and of molecules is usually closely related in procedure to that of the hydrogen atom, often being based on the use of hydrogenlike or closely related wave functions. Moreover, almost without exception the applications of qualitative and semiquantitative wave-mechanical arguments to chemistry involve the functions which occur in the treatment of the hydrogen atom. 

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. 

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... 

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... 

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... 

In dealing with molecules, the real hydrogenlike orbitals are more useful than the complex ones. For example, we shall see in Section 15.6 that the real atomic orbitals 2p, 2py, and 2p of the oxygen atom have the proper symmetry to be used in constructing a wave function for the H2O molecule, whereas the complex 2p orbitals do not. 

Suppose we take the interelectronic repulsions in the li atom as a perturbation on the remaining terms in the Hamiltonian. By the same steps used in the treatment of helium, the unperturbed wave functions are products of three hydrogenlike functions. For the ground state,... 

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