Single Subshell Approximation

Show that the width of the number operator Ngp = Yls satisfies 

The many-electron hamiltonian can be partitioned into an unperturbed part consisting of the Fock operator in diagonal form and a perturbation consisting of modified interaction terms. One can write 

When the discussion is limited to a single open subshell and to perturbations within such a shell, an interesting formulation of the many-electron atomic problem can be achieved that exhibits useful particle-hole symmetry. The summations in the perturbation term of the hamiltonian then run only over electron states nlmu) of the open subshell (nQ. This means that the electron repulsion integrals can be expressed as 

Electron pair annihilation, n LMi,SMs), and creation, n ( LMiSMs), operators can be defined such that tt creates a normalized two-electron state with the specified orbital and spin angular momenta when applied to the vacuum 

It follows from the symmetry properties of the 3 symbols and from the anticommutation relations of the electron field operators that the triplet operators do not exist for L even, while the singlet operators exist only for such L. 

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The quality of this approximation, which was introduced by Heisenberg to describe the deep core-hole excitations of X-ray spectroscopy, varies according to the element and also according to the excitation energy. This aspect will be discussed in chapter 7 (see in particular fig. 7.1 and the related discussion in section 7.2). The fact that holes are stable results in some extremely useful simplifications of atomic spectra for example, if a vacancy is created in the fu subshell, i.e. if we remove one electron to create the /13 hole, then this hole can behave like a single particle, i.e. a closed subshell minus one electron, despite the fact that it is made up from thirteen electrons. 

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