One-step formula for a well-defined intermediate state

2 One-step formula for a well-defined intermediate state 

If a well-defined hole-state with quantum numbers JpMp exists, for example, the 2p 1 2P3/2 hole-state in magnesium, the complex energy Ep(t) can be approximated by 

The next step concerns the evaluation of the involved matrix elements. Within the lowest approximation, final-state channel interactions are neglected, and the many-electron wavefunctions are expressed as superpositions of Slater determin-antal wavefunctions with the correct symmetry and parity. In the present case of 

In order to calculate the matrix elements with the Coulomb operator Vc, one again uses Slater determinantal wavefunctions, for the intermediate state xp(Mp, t) as well as for the complete final state which contains the doubly charged ion, f, and the two ejected electrons, x , (Ka, Kb). Assuming that there is no correlation between the two escaping electrons and that their common boundary condition applies separately to each single-particle function, the directional emission property is included in the factors f( ka) and f( kb), and one gets for this Coulomb matrix element C 

The Coulomb operator is a two-particle operator, i.e., it describes an interaction between at most two different orbitals on each side of its matrix element. Therefore, these matrix elements vanish unless the energies and spatial parts of the wave-function in the orbitals a or b coincide with t . This gives 

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