The corresponding contributions of the screened VP diagrams  depicted in Fig. 2(d)-(f)  are where U$_{VP}$ denotes the VP potential  and I$_{VP}$ (Δ) is the interelectronic-interaction operator modified by the electron-loop. For the renormalization of the expressions (23)-(26) we refer to the works [5  65]. Accordingly  these contributions are divided into the Uehling and Wichmann-Kroll parts. The renormalized Uehling parts of the VP operators U$_{VP}$ and I$_{VP}$ (Δ) are given by the expressions (see  e.g.  Ref. [5]) where the density ρ$_{eff}$ is related to the nuclear binding and local screening potentials via the Poisson equation Δ V$_{nuc}$ ( r )+Δ V$_{scr}$ ( r ) = 4 παZρ$_{eff}$ ( r ). U scr VP differs from U$_{VP}$ only by replacing ρ$_{eff}$ with ρ$_{scr}$   where the density ρ$_{scr}$ is related to the screening potential V$_{scr}$ . The Wichmann- Kroll parts of the expressions (23)-(25) are evaluated employing the approximate formula for the Wichmann-Kroll potential [66]. The Wichmann-Kroll contribution to Eq. (26) is relatively small [5] and is neglected in the present consideration. The numerical evaluation is based on the wave functions constructed from B-splines employing the dual-kinetic-balance finite basis set method [67]. The sphere model for the 26 U$_{VP}$ ( r ) = − 2 α $^{2}$Z 3 r ∫ ∞ 1 d t √$_{t}$ 2 − 1 t 3 ( 1 + 1 2 t 2 ) ∫ ∞ 0 d r $^{′}$r $^{′}$ρ$_{eff}$ ( r $^{′}$) × [exp ( − 2 | r − r $^{′}$| t ) − exp ( − 2 | r + r $^{′}$| t )]   (27) I$_{VP}$ (Δ   r$_{12}$ ) = α α$_{1}$$_{µ}$α µ 2 r$_{12}$ 2 α 3 π ∫ ∞ 1 d t √$_{t}$ 2 − 1 t 2 ( 1 + 1 2 t 2 ) exp ( − √$_{4}$$_{t}$ 2 − Δ $^{2}$r$_{12}$ )   (28) Δ E ( b ) a scrVP = Δ E (0) scrVP + Δ E (1   irr) scrVP + Δ E (1   red) scrVP + Δ E (1   b ) $_{scrVP}$  (22) Δ E (0) scrVP = 〈 a | U$_{VP}$ | a 〉 − 〈 a$_{C}$ | U$_{VP}$ | a$_{C}$ 〉   (23) Δ E (1   irr) scrVP = 2 ∑ b ∑ P ( − 1) P [ ε n = ε a ∑ n 〈 P aP b | I (Δ) | nb 〉〈 n | U$_{VP}$ | a 〉 ε$_{a}$ − ε$_{n}$ + ε n = ε b ∑ n 〈 P aP b | I (Δ) | an 〉〈 n | U$_{VP}$ | b 〉 ε$_{b}$ − ε$_{n}$ ] − 2$^{ε}$$^{n}$$^{̸}$$^{=}$$^{ε}$$^{a}$ ∑ n 〈 a | V$_{scr}$ | n 〉〈 n | U$_{VP}$ | a 〉 ε$_{a}$ − ε$_{n}$   (24) Δ E (1   red) scrVP = − ∑ b ∑ P ( − 1) $^{P}$〈 P aP b | I $^{′}$(Δ) | ab 〉 ( 〈 a | U$_{VP}$ | a 〉 − 〈 b | U$_{VP}$ | b 〉 )   (25) Δ E (1   b ) scrVP = ∑ b ∑ P ( − 1) $^{P}$〈 P aP b | I$_{VP}$ (Δ) | ab 〉 − 〈 a | U scr VP | a 〉   (26)