is the unperturbed wave function. The time-Fourier transform of the Green function is defined as The Feynman rules for the Green function are given in Ref. [49]. The energy shift Δ E$_{a}$ and the Green function g ( E ) are to be expanded as a power series in α , Then from Eq. (3) we find the first-, second- and third-order te rms of the energy shift, Equation (9) and the Feynman rules for g ( E ) yield the formal expressions for the total contribution of the two-loop diagrams presented in Fig. 3. Consequently replacing each electron propagator according to Eq. (2) we obtain the formal expressions for the two-electron one-loop diagrams displayed in Figs. 1 and 2. For brevity we introduce the operator where α µ = (1 , α ) are the Dirac matrices, and D$_{µν}$ is the photon propagator. It is given by in the Feynman gauge and by g ( E, x $^{′}$, x ) δ ( E − E $^{′}$) = 1 2 πi ∫ + ∞ −∞ d t d t ′ exp( iE $^{′}$t′ − iEt ) 〈 0 | T ψ ( t $^{′}$, x $^{′}$) ψ $^{†}$( t, x ) | 0 〉 . (4) Δ E$_{a}$ = Δ E (1) a + Δ E (2) a + Δ E (3) a + . . . , (5) Δ g$_{aa}$ ( E ) = Δ g (1) aa ( E ) + Δ g (2) aa ( E ) + Δ g (3) aa ( E ) + . . . . (6) Δ E (1) = 1 2 πi ∮ Γ d E ( E − E (0) a )Δ g (1) $_{aa}$( E ) , (7) Δ E (2) = 1 2 πi Γ d E ( E − E (0) a )Δ g (2) $_{aa}$( E ) − 1 2 πi ∮ Γ d E ( E − E (0) a )Δ g (1) $_{aa}$( E ) 1 2 πi ∮ Γ d E $^{′}$Δ g (1) aa ( E $^{′}$) , (8) Δ E (3) = 1 2 πi Γ d E ( E − E (0) a )Δ g (3) $_{aa}$( E ) − 1 2 πi ∮ Γ d E ( E − E (0) a )Δ g (2) $_{aa}$( E ) 1 2 πi ∮ Γ d E $^{′}$Δ g (1) aa ( E $^{′}$) − 1 2 πi Γ d E ( E − E (0) a )Δ g (1) $_{aa}$( E ) × [ 1 2 πi ∮ Γ d E $^{′}$Δ g (2) aa ( E $^{′}$) − ( 1 2 πi ∮ Γ d E $^{′}$Δ g (1) aa ( E $^{′}$) ) 2 ] . (9) I ( ω ) = e $^{2}$α $^{µ}$α$^{ν}$D$_{µν}$ ( ω ) , (10) D$_{µν}$ ( ω, x $_{12}$) = g$_{µν}$ exp( i ˜ ω | x 12 | ) 4 π | x 12 | , (11) D$_{00}$ ( ω, x $_{12}$) = 1 4 π | x 12 | , D$_{i}$$_{0}$ ( ω, x $_{12}$) = D$_{0}$$_{k}$ ( ω, x $_{12}$) = 0 , D$_{ik}$ ( ω, x $_{12}$) = δ$_{ik}$ exp( i ˜ ω | x 12 | ) 4 π | x 12 | + ∇ 1 i ∇ 2 k 1 − exp( i ˜ ω | x 12 | ) 4 πω $^{2}$| x 12 | , ( i, k = 1 , 2 , 3) , (12) 5