The zero-order contribution Δ E (0) $_{scrSE}$, depicted on Fig. 2(a), is the difference between the SE corrections calculated with and without the screening potential Here, the subscript “C” labels the energies and wave functions calculated with the Coulomb potential of the nucleus only, while Σ( ε ) denotes the unrenormalized self-energy operator. The contribution of the diagrams depicted in Fig. 2(b) is conveniently divided into irreducible and reducible parts [63]. The irreducible part is represented by the expression where the sum over b runs over all core electron states, P is the permutation operator, giving rise to the sign ( − 1) P of the permutation, Δ = ε$_{P a}$ − ε$_{a}$ , and I ( ω ) is the interelectronic- interaction operator defined in the Coulomb gauge by Eqs. (8) and (9). The expression for the reducible part is given by The vertex part, corresponding to Fig. 2(c), is given by where u = 1 − i 0 preserves the proper treatment of poles of the electron propagators. Expres- sions (18)-(21) suffer from ultraviolet divergences. To cancel these divergences explicitly we have employed the renormalization scheme presented in details in Refs. [4, 64]. The infrared divergences which occur in some terms of the expressions (20) and (21) are regularized by introducing a nonzero photon mass and canceled analytically. 25 Δ E (0) scrSE = 〈 a | Σ( ε$_{a}$ ) | a 〉 − 〈 a$_{C}$ | Σ( ε$_{a}$$_{C}$ ) | a$_{C}$ 〉 . (18) Δ E (1 , irr) scrSE = 2 ∑ b ∑ P ( − 1) P [ ε n = ε a ∑ n 〈 P aP b | I (Δ) | nb 〉〈 n | Σ( ε$_{a}$ ) | a 〉 ε$_{a}$ − ε$_{n}$ + ε n = ε b ∑ n 〈 P aP b | I (Δ) | an 〉〈 n | Σ( ε$_{b}$ ) | b 〉 ε$_{b}$ − ε$_{n}$ ] − 2$^{ε}$$^{n}$$^{̸}$$^{=}$$^{ε}$$^{a}$ ∑ n 〈 a | V$_{scr}$ | n 〉〈 n | Σ( ε$_{a}$ ) | a 〉 ε$_{a}$ − ε$_{n}$ , (19) Δ E (1 , red) scrSE = ∑ b ∑ P ( − 1) P [ 〈 P aP b | I (Δ) | ab 〉 ( 〈 a | Σ $^{′}$( ε$_{a}$ ) | a 〉 + 〈 b | Σ $^{′}$( ε$_{b}$ ) | b 〉 ) − 〈 P aP b | I $^{′}$(Δ) | ab 〉 ( 〈 a | Σ( ε$_{a}$ ) | a 〉 − 〈 b | Σ( ε$_{b}$ ) | b 〉 ) ] −〈 a | V$_{scr}$ | a 〉〈 a | Σ $^{′}$( ε$_{a}$ ) | a 〉 . (20) Δ E (1 , ver) scrSE = ∑ b ∑ P ( − 1) P i 2 π ∫ ∞ −∞ d ω ∑ n 1 , n $_{2}$[ 〈 P bn$_{1}$ | I (Δ) | bn$_{2}$ 〉〈 P an$_{2}$ | I ( ω ) | n$_{1}$a 〉 ( ε$_{P a}$ − ω − uε$_{n}$$_{1}$ )( ε$_{a}$ − ω − uε$_{n}$$_{2}$ ) + 〈 P an$_{1}$ | I (Δ) | an$_{2}$ 〉〈 P bn$_{2}$ | I ( ω ) | n$_{1}$b 〉 ( ε$_{P b}$ − ω − uε$_{n}$$_{1}$ )( ε$_{b}$ − ω − uε$_{n}$$_{2}$ ) ] − i 2 π ∫ ∞ −∞ d ω ∑ n 1 , n 2 〈 n$_{1}$ | V$_{scr}$ | n$_{2}$ 〉〈 an$_{2}$ | I ( ω ) | n$_{1}$a 〉 ( ε$_{a}$ − ω − uε$_{n}$$_{1}$ )( ε$_{a}$ − ω − uε$_{n}$$_{2}$ ) , (21)