Several shells of equivalent electrons

A set of pairs of quantum numbers nf 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]  

In each product function, the same set of one-electron quantum numbers is arranged in the same order (usually in the standard order 1,2. N) but the electron coordinates ri,r2,r3. have been rearranged into some new order r. r. r, . The summation in (10.8) is over all N possible permutations P = jijih. .. jN of the normal coordinate ordering 123. ..N, and p is the parity of the permutation P (p = 0 if P is obtained from the normal ordering by an even number of interchanges, and p = 1 if an odd number of interchanges is involved). 

Antisymmetrized function (10.8) has the property that if any two one-electron functions are identical, then ip is identically zero (satisfying the Pauli exclusion principle). Its second very important property if any two electrons lie at the same position, e.g., ri = r2 (and they also have parallel spins Si = S2), then = 0. As the functions cp are continuous in the spatial variables (r, 0, tp), it follows that 1 1 must be unusually small whenever two electrons with parallel spin are close together. Thus, unlike the single product function, the antisymmetrized sum of product functions (10.8) shows a certain degree of electron correlation. This correlation is incomplete - it arises by virtue of the Pauli exclusion principle rather than as a result of electrostatic repulsion, and there is no correlation at all between two electrons with antiparallel spins [16]. 

Antisymmetrized wave function (10.8) may be written in the form of a determinant 

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The NRO approach is very efficient when accounting for correlation effects in the framework of the so-called extended method of calculation (see also Chapter 29) applied to electronic configurations having several shells of equivalent electrons with the same values of orbital quantum numbers. Its general theory is described in [199-204]. 

Part 2 is devoted to the foundations of the mathematical apparatus of the angular momentum and graphical methods, which, as it has turned out, are very efficient in the theory of complex atoms. Part 3 considers the non-relativistic and relativistic cases of complex electronic configurations (one and several open shells of equivalent electrons, coefficients of fractional parentage and optimization of coupling schemes). Part 4 deals with the second-quantization in a coupled tensorial form, quasispin and isospin techniques in atomic spectroscopy, leading to new very efficient versions of the Racah algebra. 

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. 

Deductions which are quite exact can be made from any properties of symmetry which the wave function must possess in virtue of the symmetrical character of the problem. The most important of these prop( rties of symmetry is the one which is involved in the complete equivalence of the electrons, and their consequent interchangeability the wave function must of course be the same, whether, say, the first electron is situatcKl in the K shell and the second in the L shell, or the second in the K shell and the first in the L shell. This leads to general rules for tlu>. tabulation of the terms in atoms with several radiating electrons. Still, the results thus obtained are not immediately com-paral)le with experinnvnt, since in wave mechanics, so far as developed above, an essential principle is lacking, which was discovered by Pauli, and which will come before our notice in next chapter. 

The structure of Cso has been proposed to be a truncated icosahedron with twenty six-membered rings and twelve five-membered rings. In this allotrope, the atoms are equivalent, giving a closed-shell electronic structure and a molecule that is unique in nature. While the existence of Ceo has been known for several years, moderately large-scale production and phase separation of Cao and other fullerenes were not possible before the work of Kratschmer et al. That breakthrough virtually assured rapid development in understanding the properties of these novel forms of matter. 

In this and the subsequent section we provide explicit expressions for the orbital and spin matrix elements, respectively, which describe the scattering of neutrons by equivalent, non-relativistic electrons in a single atomic shell. Derivations of the expressions are reported in several references. 

A special situation is encountered in the formation of a K-shell vacancy in systems with several equivalent corehole sites.Owing to the localization of the core orbitals in space, there will always exist several near-degenerate electronic states which can interact through vibrational modes of suitable symmetry. In this case, however, the vibronic Hamiltonian can be diagonalized by transforming to a suitable diabatic representation. These diabatic electronic states correspond to core holes localized on the equivalent sites. From the dynamical point of view, we are dealing here with a multidimensional weakly avoided crossing. From the structural... 

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