Coulomb interactions three-electron atoms

Coulomb-Coulomb interaction for three-electron atom is represented by two-electron Feynman diagrams Fig. la, b and three-electron diagrams Fig.2a. 

Besides taking into account the two-electron diagrams Fig.lc, d, e, f, Coulomb-Breit interaction for three-electron atom represented by the three-electron diagrams Fig.2b, c (as it is mentioned above the contribution of the diagrams Fig.2b, c is considered as doubled contribution of the diagram Fig.2b). The formulas for irreducible part of the diagram Fig.2b is (see the Appendix)... 

All three forms of the dipole matrix element are equivalent because they can be transformed into each other. However, this equivalence is valid only for exact initial- and final-state wavefunctions. Since the Coulomb interaction between the electrons is responsible for many-body effects (except in the hydrogen atom), and the many-body problem can only be solved approximately, the three different forms of the matrix element will, in general, yield different results. The reason for this can be seen by comparing for the individual matrix elements how the transition operator weights the radial parts R r) and Rf(r) of the single-particle wavefunction differently ... 

Nonbonding interactions play a major role in determining the three-dimensional structure of a molecule. Such interactions are composed of repulsive and attractive contributions, such as van der Waals repulsion, London dispersion forces, coulombic interactions, and delocalizations of electrons due to the through-space interactions between atomic orbitals. 

When, however, the atom C is close to a—b, the interaction of the three electrons and of the three nuclei enter into the Coulomb energy and... 

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