Equivalent electrons terms

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. 

Table 7.2 Terms arising from some configurations of non-equivalent and equivalent electrons...

Of the terms arising from equivalent electrons those with the highest multiplicity lie lowest in energy. 

There are two further rules for ground terms which tell us whether a multiplet arising from equivalent electrons is normal or inverted. 

Let us now derive the Slater-Kirkwood38 formula in terms of our present quantities. A single subshell of equivalent electrons is assumed. Equation 29 may be rearranged to... 

The bond diagrams provide an obvious simple method of determining the allowed spectral terms for equivalent electrons with Russell-Saunders coupling, which may be convenient for the reason that it separates states of different multiplicity at the start. 

Rule 1 For terms resulting from equivalent electrons, those with the highest multiplicity will be the most stable. 

Spectroscopic terms arising from equivalent electronic configurations in L-S coupling... 

According to the vectorial model of an atom, the angular and spin momenta of equivalent electrons in LS coupling are coupled into total momenta L and S, respectively. Their numerical values L and S define the term and its multiplicity (2S+1L). The vectorial sum of L and S, caused by a spin-orbit interaction, makes the quantum number of total momentum J, describing the levels of the given term 2S+1Lj. In jj coupling we straightforwardly obtain J. 

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. 

Formula (12.19) for the particular case of the energy operator and of the transformation from the LS coupling to another one, in the general case of two shells of equivalent electrons, inside which there is LS coupling and their total momenta are LiSi and L2S2, respectively, will be of the form (for simplicity we skip the designations of the shells themselves and of the additional quantum numbers, distinguishing the terms of the shell with the same L,S,) ... 

Unit tensors are especially important for group-theoretical methods of studying the lN configuration. We can express the infinitesimal operators of the groups [10, 24, 98], the parameters of irreducible representations of which are applied to achieve an additional classification of states of a shell of equivalent electrons, in terms of them. 

The operator of the energy of electrostatic interaction of electrons in (14.65) is represented as a sum of second-quantization operators, and the appropriate submatrix element of each term is proportional to the energy of electrostatic interaction of a pair of equivalent electrons with orbital Lu and spin S12 angular momenta. The values of these submatrix elements are different for different pairing states, since, as follows from (14.66), the two-electron submatrix elements concerned are explicitly dependent on L12, and, hence, implicitly - on S12 (sum L12 + S12 is even). It is in this way that, in the second-quantization representation for the lN configuration, the dependence of the energy of electrostatic interaction on the angles between the particles shows up. This dependence violates the central field approximation. 

As has been shown, second-quantized operators can be expanded in terms of triple tensors in the spaces of orbital, spin and quasispin angular momenta. The wave functions of a shell of equivalent electrons (15.46) are also classified using the quantum numbers L, S, Q, Ml, Ms, Mq of the three commuting angular momenta. Therefore, we can apply the Wigner-Eckart theorem (5.15) in all three spaces to the matrix elements of any irreducible triple tensorial operator T(JC K) defined relative to wave functions (15.46)... 

Matrix elements of the operators of the interaction energy between two shells of equivalent electrons may be expressed, with the aid of the CFP, in terms of the corresponding two-electron quantities. Substituting in such formula the explicit expression for the two-electron matrix element, after a number of mathematical manipulations and using the definition of submatrix elements of operators composed of unit tensors, we get convenient expressions for the matrix elements in the case of two shells of equivalent electrons. The corresponding details may be found in [14], here we present only final results. 

Actually, only transitions described by one term in (25.23), take place. Formulas (25.22) and (25.23) are valid for the first and second forms of the Efc-transition operator (formulas (4.12) and (4.13)). The corresponding one-electron submatrix elements are given by (25.5) and (25.6). Analogous expressions for the third form of the /c-radiation operator are established in [77]. The appropriate selection and sum rules may be found in a similar way as was done for transitions between different configurations. It is interesting to mention that the non-zero conditions for submatrix elements for the operator Uk with regard to a seniority quantum number suggest new selection rules for the transitions in the shell of equivalent electrons v = v at odd and v = v, v 2 at even k values. 

Table 4 A list of atomic terms for dn configurations of equivalent electrons...

The terms arising from the electron configurations of equivalent electrons are listed in Table 4. 

Fig. 4 Genealogy of the d -terms. X is the term forbidden with equivalent electrons numbers in the circles represent the seniority.

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