Atoms with More Than Two Electrons

The treatment of other atoms is similar to that of helium. In zero order we neglect electron-electron repulsions, and in higher-order calculations these repulsions are treated with the same approximation methods as used with the helium atom. From this point on in the chapter we will discuss only the results of such calculations. 

An application of the variation method to the lithium atom ground state uses an orbital wave function containing hydrogen-like orbitals with variable orbital exponents (effective nuclear charges) similar to that used with helium, except that different effective 

Find the value of (r) for a hydrogen-like Is orbital with Z = 2.686. Solution 

first ionization potential is defined as the energy required to remove one electron from an isolated neutral atom. If the orbitals for the other electrons are not significantly changed by the removal of one electron, the ionization potential is nearly equal to the magnitude of the energy of the orbital occupied by the outermost electrons. The ionization potential can therefore be used to obtain an estimate of the effective nuclear charge seen by the outer electrons. 

19 The Electronic States of Atoms. III. Higher-Order Approximations 

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The results in Table VI are of considerable interest also for atoms with more than two electrons, since they show the possibilities and limitations of the method of superposition of configurations/ when the latter are built up from one-electron functions... 

For atoms with more than two electrons, it is very difficult to obtain such a small absolute error in the energy as in the helium case, but, within an isoelectronic sequence, the relative error will, of course, go down rapidly with increasing atomic number Z. The method of superposition of configurations has been used successfully in a number of applications, particularly by Boys (1950-) and Jucys (1947-), and, for a more detailed survey of the work on atoms, we will refer to the special table on atomic calculations in the bibliography. This is a field of rapid development, where one can expect important new results within the next few years. 

However, we cannot handle atoms with more than two electrons in the same manner. For instance, for lithium, the wavefunction... 

Two conclusions from more advanced quantum mechanics guide us to a simplified approach for constructing MOs for atoms with more than two electrons. 

Up to the present time no method has been applied to atoms with more than two electrons which makes possible the computation of wave functions or energy levels as accurate as those for helium discussed in Section 29c. With the increasing complexity of the atom, the labor of making calculations similar to those used for the ground state of helium increases tremendously. Nevertheless, many calculations of an approximate nature have been carried out for larger atoms with results which have been of considerable value. We shall discuss some of these in this chapter.1... 

Atoms with more than two electrons atomic properties and periodic trends... 

Electronic structure calculations in atoms with more than two electrons require properly antisymmetrized trial functions. For a closed-shell atom with p electrons having paired spins, such a function can be written in the form of a normalized Slater determinant ... 

For atoms with more than two electrons, the wavefunctions must become more elaborate to satisfy the symmetrization principle. However, John Slater developed a general method for reliably generating many-electron spin-spatial wavefunctions, antisymmetric with respect to P21 (exchange of the electron labels 1 and 2), for any number of electrons. We call these wavefunctions Slater determinants, because they are obtained by taking the determinant of a matrix of possible one-electron wavefunctions. For example, for ground state He, there are two possible one-electron spin-spatial wavefunctions for each electron Isa and lsj8. We set up a 2 X 2 matrix in which each row corresponds to a different electron and each column to a different wavefunction ... 

Section 19.6 Atoms with More Than Two Electrons... 

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