7 C. Total energies Δ E QED 2el = m α $^{2}$( Zα ) 3 [ a$_{31}$ ln( Zα ) − 2 + a$_{30}$ + ( Zα ) G ( Z ) ] , (35) E ( $^{12}$C$^{4+}$) = − 3 162 423 . 60(32) cm − $^{1}$, (36) The first two coefficients of the 1 /Z expansion of E $^{(6)}$(nlog) are listed in Table VI. A fit of our numeri- cal data agrees well with these coefficients. The agree- ment observed shows consistency of our numerical results with independent calculations at the level of the one- photon effects. We now turn to the contribution of order m α $^{2}$( Zα ) $^{4}$. This contribution is induced by nontrivial two-photon effects, so that a comparison drawn for this part will yield a much more stringent test of consistency of different approaches. Our total results for the ionization energy of the n = 1 and n = 2 states of helium-like atoms with the nuclear charge Z = 2 . . . 12 are listed in Table VIII. The fol- lowing values of fundamental constants were employed [19], R$_{∞}$ = 10 973 731 . 568 527(73) m − 1 and α − 1 = 137 . 0359 999 679(94). The atomic masses were taken from Ref. [30]. The part of E $^{(6)}$(nlog) that is of order m α $^{2}$( Zα ) 4 is implicitly present in the two-electron QED contribution calculated numerically in Ref. [8] to all orders in Zα . This contribution can be represented as (see Eq. (72) of Ref. [8]) The results for atomic helium presented in Table VIII differ from those reported previously only because of the different approximate treatment of the higher-order ( m α 7 and higher) contribution employed in this work. For the S states of helium, the present values are prac- tically equivalent to those of Refs. [6, 7]. (The difference is that now we include some contributions of order m α 8 and higher, which are negligible for helium but become noticeable for higher- Z ions.) However, for the helium P states, our present estimate of the higher-order contri- bution is by about 1 MHz higher than that of Ref. [7]. The reason is that the one-electron radiative correction of the p electron state was previously not included into the approximation (25). It is included now [see Eq. (29)] in order to recover the correct asymptotic behaviour of the radiative correction in the high- Z limit. where the remainder function G ( Z ) incorporates all higher orders in Zα . The two-electron QED correction comprises the so-called screened self-energy and vacuum- polarization contributions and the part of the two-photon exchange correction that is beyond the Breit approxima- tion. A selection of our theoretical results for transition en- ergies is compared with the theory by Drake [5, 31] and with experimental data in Table IX. Agreement between theory and experiment is excellent in all cases studied. We observe a distinct improvement in theoretical accu- racy as compared to the previous results by Drake. This improvement is due to the complete treatment of the cor- rections of order mα 6 and m $^{2}$/Mα5 accomplished in this work. The coefficients a$_{31}$ and a$_{30}$ in Eq. (35) correspond to the second term of the 1 /Z expansion of the leading QED correction E (5) ∞ . More specifically, a$_{31}$ corresponds to the coefficient c$_{1}$ from Table IV and a$_{30}$ , to that from Ta- ble V. The Z = 0 limit of the higher-order remainder function G ( Z ) corresponds to the third coefficient of the 1 /Z expansion of the correction E $^{(6)}$(nlog), G (0) = c$_{2}$ , for all states except 2 $^{1}$P$_{1}$ and 2 $^{3}$P$_{1}$ . The values of c$_{2}$ obtained by fitting our numerical data are listed in Ta- ble VI. For the 2 $^{1}$P$_{1}$ and 2 $^{3}$P$_{1}$ states, the coefficient c$_{2}$ is not directly comparable with the all-order results because of the mixing effects. Theoretical results for the fine structure splitting in- tervals 2 $^{3}$P$_{0}$ − 2 $^{3}$P$_{1}$ and 2 $^{3}$P$_{1}$ − 2 $^{3}$P$_{2}$ are not analyzed in the present work. This is because these intervals can nowdays be calculated more accurately (complete up to order mα $^{7}$), like it was recently done for helium [15]. We intend to perform such a calculation in a subsequent in- vestigation. The higher-order remainder function G ( Z ) inferred from the numerical results of Ref. [8] is plotted in Fig. 1, together with its limiting value at Z = 0 obtained by a fit of our numerical data. It should be stressed that the identification of the remainder implies a great deal of numerical cancellations, especially for the all-order re- sults. The comparison drawn in Fig. 1 provides a strin- gent cross-check of the two complementary approaches. The visual agreement between the results is very good for the S states, whereas for the P states a slight dis- agreement seems to be present. Among the results listed in Table VIII for helium-like ions, the ground-state energy of the carbon ion is of par- ticular importance, because it is used in the procedure of the adjustment of fundamental constants [32] for the determination of the mass of $^{12}$C4+ and, consequently, of the proton mass from the Penning trap measurement by Van Dyck et al. [33]. Our result for the ground-state ionization energy of helium-like carbon is It is tempting to merge the all-order and the Zα - expansion results by fitting the all-order data for G ( Z ) towards lower values of Z . However, we do not attempt to do this in the present work. The reasons are, first, that the numerical accuracy of the all-order results is apparently not high enough and, second, that the ex- pansion of the remainder function G ( Z ) contains terms ( Zα ) ln $^{2}$( Zα ) and ( Zα ) ln( Zα ), which cannot be reason- ably fitted with numerical data available in the high- Z region only. which is in agreement with the previous result by Drake [5] of − 3 162 423 . 34(15) cm − $^{1}$. We note that despite the fact that our calculation is by an additional order of α more complete than that by Drake, our estimate of un- certainty is more conservative. In summary, significant progress has been achieved during last decades both in experimental technique and