in the Coulomb gauge. Here x 12 = x 1 − x $_{2}$, ˜ ω = √$_{ω}$ 2 + i 0 . The branch of the square root is fixed by the condition Im ˜ ω > 0 . In order to handle the infrared divergencies it is convenient to introduce the photon mass µ . In the Feynman gauge, it results in the replacement √$_{ω}$ 2 + i 0 → √ ω 2 − µ 2 + i 0 . The limit µ → 0 should be taken after removing the divergencies. The operator I ( ω ) has the following symmetry properties: I ( ω ) = I ( − ω ) , I $^{′}$( ω ) ≡ d d ω I ( ω ) = − I $^{′}$( − ω ) , I $^{′′}$( ω ) ≡ d 2 d $^{2}$ω I ( ω ) = I $^{′′}$( − ω ) . (13) The interaction with the external magnetic field can be repre sented by an operator T$_{0}$ . For both cases under consideration ( g factor and hyperfine splitting) it is proportional to [ r × α ]$_{z}$ , what defines the angular momentum structure. Explicit formulas f or T$_{0}$ will be given in Sec. V. III. SCREENED SELF-ENERGY The diagrams of the screened self-energy correction, corresponding to the first term in Eq. (9), are shown in Fig. 1. We divide the total contribution of these diagrams into the reducible and irreducible parts. The irreducible part is the sum of the terms where the energies of the intermediate states are different from the energy of the initial state. The reducible part is the remainder. We denote the irreducible parts of each diagram A – F by the same letter: Δ E SE( A − F ) SQED . It is convenient to divide them into three groups, according to the number of the ω -dependent denominators ( ω is the virtual photon energy, over which the integration is performed). The terms with only one denominator ( A , B and E ) are referred to as a ”modified self-energy” terms, since all of them have the form of a matrix element of the self-energy operator, 〈 X | Σ( ε ) | Y 〉 . The terms with two denominators ( C and F ) are denoted as a ”modified vertex”, and the diagram D with three denominators is denoted as a ”double-vertex”. The red ucible parts are considered together with the related contributions that arise from the second and the third terms in Eq. (9). Their sum is divided into three parts: G , H , and I , according to the number of the ω -dependent denominators. These three parts are considered together with the ”modified self-energy”, ”modified vertex”, and ”double-vertex” terms, respectively. Finally, the total c ontribution of the two-electron self-energy 6