Electrons non-equivalent

3(a) Non-equivalent electrons Non-equivalent electrons are those that have different values of either n or 1 so that, for example, those in a 3p 3d or 3p 4p configuration are non-equivalent whereas those in a 2p configuration are equivalent. Coupling of angular momenta of non-equivalent electrons is rather more straightforward than for equivalent electrons. 

First we consider, for non-equivalent electrons, the strong coupling between orbital angular momenta, referred to as it coupling, using a particular example. 

Flowever, the values of the total orbital angular momentum quantum number, L, are limited or, in other words, the relative orientations of f j and 2 are limited. The orientations which they can take up are governed by the values that the quantum number L can take. L is associated with the total orbital angular momentum for the two electrons and is restricted to the values 

It can be shown quite easily that, for a filled sub-shell such as 2p or L = 0. Space quantization of the total orbital angular momentum produces 2L - - 1 components with M] = L, L —, —L, analogous to space quantization of f. In a filled sub-shell 

The coupling between the spin momenta is referred to as xx coupling. The results of coupling of the s vectors can be obtained in a similar way to U coupling with the difference that, since x is always 4, the vector for each electron is always of magnitude 3 ft/2 

The labels for the terms indicate the value of 5 by having 25 +1 as a pre-superscript to the S, P,D. label. The value of 25 + 1 is known as the multiplicity and is the number of values that Ms can take these are 

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In a similar way the coupling of a third vector to any of the L in Figure 7.4(a) will give the terms arising from three non-equivalent electrons, and so on. 

Table 7.2 lists the terms that arise from various combinations of two non-equivalent electrons. 

Again, for the filled orbitals L = 0 and 5 = 0, so we have to consider only the 2p electrons. Since n = 2 and f = 1 for both electrons the Pauli exclusion principle is in danger of being violated unless the two electrons have different values of either or m. For non-equivalent electrons we do not have to consider the values of these two quantum numbers because, as either n or f is different for the electrons, there is no danger of violation. 

In the excited electron configuration given, there are two electrons in partly filled orbitals, a Ad (electron 1) and a 5/ electron (electron 2). These are non-equivalent electrons (Section 7.1.2.3a) and we need consider only the coupling of the orbital angular momenta, fj andf2> and the spin angular momenta, Sj and S2-... 

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 ... 

As was already mentioned, due to the Pauli exclusion principle, which states that no two electrons can have the same wave functions, a wave function of an atom must be antisymmetric upon interchange of any two electron coordinates. For a shell of equivalent electrons this requirement is satisfied with the help of the usual coefficients of fractional parentage. However, for non-equivalent electrons the antisymmetrization procedure is different. If we have N non-equivalent electrons, then a wave function that is antisymmetric upon interchange of any two electron coordinates can be formed by taking the following linear combination of products of one-electron functions [16] ... 

Let us illustrate this method in the special case of two non-equivalent electrons ni/ir fe, described by four momenta two orbital li, I2 and two spin si, S2- Having in mind the commutativity of the addition as well as the fact that the interaction of the orbital momentum of a given electron with its own spin momentum is much stronger than that with... 

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. 

In order to indicate the parity, defined here as (—l),1+ 2, we have to add to the term a special symbol (e.g. for odd configurations the small letter o). Then, for example, the levels of the configuration nsn p will be lP[, 3 0,1,2- Thus, the spectra of two non-equivalent electrons will consist of singlets and triplets. 

Let us notice that momenta of each shell may be coupled into total momenta by various coupling schemes. Therefore, here, as in the case of two non-equivalent electrons, coupling schemes (11.2)—(11.5) are possible, only instead of one-electronic momenta there will be the total momenta of separate shells. To indicate this we shall use the notation LS, LK, JK and JJ. Some peculiarities of their usage were discussed in Chapters 11 and 12 and will be additionally considered in Chapter 30. Therefore, here we shall restrict ourselves to the case of LS coupling for non-relativistic and JJ (or jj) coupling for relativistic wave functions. We shall not indicate explicitly the parity of the configuration, consisting of several shells, because it is simply equal to the sum of parities of all shells. 

The extension of a given determinantal wavefunction (called the parent wave-function, which in the simplest case can be just a one-electron spin-orbital) to include another non-equivalent electron (or even a group of non-equivalent electrons) is made with the help of vector-coupling or Clebsch-Gordan coefficients... 

From the known values of the Clebsch-Gordan coefficients, determinantal wavefunctions with selected angular momenta can easily be evaluated by adding to the parent function the new non-equivalent electron orbital and taking into account the Clebsch-Gordan coefficient ... 

In an external field there are accordingly 36 different energy levels in aU, for the case of two non-equivalent -electrons. 

Here (in the case of equivalent p-electrons), therefore, there are only" five terms in the absence of a magnetic field, as compared with ten terms in the case of non-equivalent electrons. The terms P, and found above fall out here (owing to the exclusion principle). 

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