The relativistic interaction of two electrons in the self-consistent field method

The extension of the self-consistent field method for the relativistic case has been discussed in a previous paper, There no attempt was made to consider the interaction energy of two electrons to a greater degree of accuracy than that given by the Coulomb energy. In the present paper the interaction of the spins and the effect of retardation is introduced. The method is then applied to the evaluation of the separations of the components of the 23 P term of helium. The relativistic expression for the interaction of two electrons has been discussed by several authors. The expressions they obtain may be shown to agree as far as terms of the first order ine 2 and the square of the fine structure constant. We shall follow the discussion of Bethe and Fermi since this seems to be most suited for application to the self-consistent field method. We require the matrix elements of the interaction energyI of two electrons 1 and 2, corresponding to given transitions of the two electrons. We denote the states of electron 1 by N1 , N'1 , N"1 , ..., and those of electron 2 by N2 , N'2 , N"2 , ..., where each N stands for the set of four quantum numbers specifying a state of the electron. Then the matrix element (N1 , N2 |I| N'1 , N'2 ) corresponding to a transition N1 → N'1 for electron 1 and N2 → N'2 for electron 2 is found as follows. We form the charge and current density corresponding to a transition N1 → N'1 of electron 1. In Hartree atomic units these are (N1 |ρ| N'1 ) = Ψ* (N'1 |1) Ψ (N1 |1)e i (E'1 -E1 )t (1)