Two and more shells of equivalent electrons

In Chapter 9 we discussed the classification of the terms and energy levels of a shell of equivalent electrons using the LS coupling scheme. Here we shall consider the case of two non-equivalent electrons. As we shall see later on, generalization of the results for two non-equivalent electrons to the case of two or more shells of equivalent electrons is straightforward. 

A set of pairs of quantum numbers n,7, with the indicated number of electrons having these quantum numbers, is called an electronic configuration of the atom (ion). Thus, we have already discussed the cases of two non-equivalent electrons and a shell of equivalent electrons. If there is more than one electron with the same nf, then the configuration may look like this ... 

With jj coupling, the spin-angular part of the one-electron wave function (2.15) is obtained by vectorial coupling of the orbital and spin-angular momenta of the electron. Then the total angular momenta of individual electrons are added up. In this approach a shell of equivalent electrons is split into two subshells with j = l 1/2. The shell structure of electronic configurations in jj coupling becomes more complex, but is compensated for by a reduction in the number of electrons in individual subshells. 

The most widely used qualitative model for the explanation of the shapes of molecules is the Valence Shell Electron Pair Repulsion (VSEPR) model of Gillespie and Nyholm (25). The orbital correlation diagrams of Walsh (26) are also used for simple systems for which the qualitative form of the MOs may be deduced from symmetry considerations. Attempts have been made to prove that these two approaches are equivalent (27). But this is impossible since Walsh s Rules refer explicitly to (and only have meaning within) the MO model while the VSEPR method does not refer to (is not confined by) any explicitly-stated model of molecular electronic structure. Thus, any proof that the two approaches are equivalent can only prove, at best, that the two are equivalent at the MO level i.e. that Walsh s Rules are contained in the VSEPR model. Of course, the transformation to localised orbitals of an MO determinant provides a convenient picture of VSEPR rules but the VSEPR method itself depends not on the independent-particle model but on the possibility of separating the total electronic structure of a molecule into more or less autonomous electron pairs which interact as separate entities (28). The localised MO description is merely the simplest such separation the general case is our Eq. (6)... 

Valence shell electrons of the atoms in a molecule are either shared or unshared. The shared electrons are found in either o ox it bonds. Unshared electrons are found in AOs (usually hybrid AOs for first-row elements). Lewis structures provide a way to indicate the shared and unshared pairs of electrons in molecules. Sometimes, however, it is possible to indicate the electron distribution in molecules by more than one Lewis structure. For example, a car-boxylate anion can be represented by two equivalent but different Lewis structures. 

Figure 2.2 Demonstration of the two equivalent nomenclatures used for the description of inner-shell levels and X-ray transitions (also Auger transitions, see below). The vertical direction is regarded as the energy axis (but is not to scale here). On the left-hand side is given the notation which is frequently used in inner-shell spectroscopy, on the right-hand side the corresponding single-orbital quantum numbers with n, t and j being, respectively, the principal quantum number, the orbital angular momentum and the total angular momentum which includes the spin of the electron. Also shown are the main X-ray transitions with their spectroscopic notation (for a more complete plot which includes...

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