6 The value for the ground state is given only for com- pleteness, since much more accurate numerical results are available in the literature [22, 23]. The numerical results for the leading relativistic correction E (4) are summarized in Table II. the lines described in Refs. [6, 7]. Here we only note that calculations for higher values of Z often exhibit a slower numerical convergence (and numerical stability) than for helium, especially so for the second-order corrections in- volving singular operators. The variational optimization of nonlinear parameters for the symmetric second-order corrections was performed in several steps with increas- ing the number of basis functions on each step up to N = 1000 or 1200. The final values were obtained with merging several basis sets and enlarging the number of functions up to N = 5000 − 7000. The nonsymmetric second-order corrections were evaluated as described in Ref. [15]. The calculations were performed in the quadru- ple, sixtuple, and octuple arithmetics implemented in Fortran 95 by V. Korobov [27]. The leading QED correction E (5) is given by Eqs. (9), (10), and (15). Computationally the most problematic part of it is represented by the Bethe logarithm ln( k$_{0}$ ) and its mass-polarization correction ln( k$_{0}$ )$_{M}$. Accurate calculations of ln( k$_{0}$ ) were performed by Drake and Gold- man [24] for helium-like like atoms with Z ≤ 6 and by Korobov [25] for Z = 2. Calculations of the recoil correc- tion to the Bethe logarithm were reported by Pachucki and Sapirstein [10] for Z = 2 and by Drake and Gold- man [24] for Z ≤ 6. In the present investigation, we per- form accurate evaluations of the Bethe logarithm ln( k$_{0}$ ) and its recoil correction ln( k$_{0}$ )$_{M}$ for helium-like ions with Z ≤ 12. The calculational approach is described in Ap- pendix A. Table VII presents the results for the finite nuclear size correction and approximate values of the higher-order ( mα 7 and higher) correction to the ionization energy . The uncertainty of the total theoretical prediction origi- nates from the higher-order radiative effects; it was esti- mated by dividing the absolute value of this correction by Z . The values for the root-mean-square radius of nuclei were taken from Ref. [28]. Table III summarizes the numerical results obtained and gives a comparison with the previous calculations. Numerical values for the Bethe logarithm are presented for the difference ln( k$_{0}$ ) − ln( Z $^{2}$) since this difference has a weak Z -dependence and does not contain any logarithms in its 1 /Z expansion. The table also lists the coefficients of the 1 /Z expansion of ln( k$_{0}$/Z $^{2}$). The leading coef- ficient c$_{0}$ is known from the hydrogen theory; accurate numerical values can be found in Ref. [26]. The higher- order coefficients were obtained by fitting our numerical data. It is interesting to compare them with the analo- gous results reported previously by Drake and Goldman [24]. For the next-to-the-leading coefficient c$_{1}$ , the re- sults agree up to about 4-5 digits for S states and about 3-4 digits for P states. For the higher-order coefficients, the agreement gradually deteriorates. However, the re- sults for the sum of the two expansions agree very well with each other. More specifically, the maximal absolute deviation between the values of the Bethe logarithms for Z > 12 obtained with our 1 /Z -expansion coefficients and with those by Drake and Goldman is 1 × 10 − 8 for the 1 $^{1}$S state, 3 × 10 − 8 for the 2 $^{1}$S state, 6 10 − 9 for the 2 $^{3}$S state, 1 10 − 7 for the 2 $^{1}$P state, and 6 × 10 − 8 for the 2 $^{3}$P state. So, the accuracy of these expansions is sufficient for most practical purposes. In this subsection we discuss the calculational results obtained for the mα 6 correction in more detail and make a comparison with the results obtained previously within the all-order, 1 /Z -expansion approach. The log- arithmic part of the correction, E $^{(6)}$(log), behaves as mα $^{3}$( Zα ) 3 for large Z and thus corresponds to diagrams with three photon exchanges that have not yet been ad- dressed within the all-order approach. The nonlogarith- mic part E $^{(6)}$(nlog), however, contains some pieces that are known and identified below. B. Comparison with the all-order approach For all states except 2 $^{1}$P$_{1}$ and 2 $^{3}$P$_{1}$ , the leading term of the 1 /Z expansion of E $^{(6)}$(nlog) is of order m ( Zα ) 6 and comes from the Zα expansion of the one-electron Dirac energy. For the 2 $^{1}$P$_{1}$ and 2 $^{3}$P$_{1}$ states, the leading term is of the previous order in 1 /Z , m ( Zα ) $^{6}$Z , and is due to the mixing of these levels. More specifically, the mixing con- tribution is δE$_{mix}$ = ∣ 〈 2 $^{1}$P$_{1}$ | H $^{(4)}$| 2 $^{3}$P$_{1}$ 〉 ∣ $^{2}$/ [ E$_{0}$ (2 $^{1}$P$_{1}$ ) − E$_{0}$ (2 $^{3}$P$_{1}$ )] for the 2 $^{1}$P$_{1}$ state and that with the opposite sign, for 2 $^{3}$P$_{1}$ . The contribution of order m ( Zα ) 6 for the mixing states comes from the expansion of the one- electron Dirac energy and from the expansion of δE$_{mix}$ . The next term of the 1 /Z expansion is of order mα ( Zα ) 5 and comes from the one-electron one-loop ra- diative correction and from the one-photon exchange cor- rection. The radiative part is well known [19]. The part due to the one-photon exchange was obtained for the 1 $^{1}$S , 2 $^{3}$S , 2 $^{3}$P$_{0}$ , and 2 $^{3}$P$_{2}$ states analytically in Ref. [29] and for the other states numerically in this work. For the 2 $^{1}$P$_{1}$ and 2 $^{3}$P$_{1}$ states, there is a small additional mixing contribution, which we were unable to determine unam- biguously. Another part of the calculation of E (5) that needs a separate discussion is the evaluation of the expectation value of the singular operator 1 /r $^{3}$, which is defined by Eq. (12). The calculational approach is described in Ap- pendix B. Total results for the logarithmic and the non- logarithmic part of the leading QED correction are sum- marized in Tables IV and V, respectively. The results are in good agreement with the previous calculations [5]. Table VI presents the numerical values of the mα 6 cor- rection, the main result of this investigation. The corre- sponding calculations for atomic helium were reported in Refs. [6, 7]; our present numerical values agree with the ones obtained previously. Calculations performed in this work for helium-like ions were accomplished along