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where γ is the Euler constant. The Bethe logarithm is defined as 

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The operator H (5) fs is the anomalous magnetic moment correction to the spin-dependent part of the Breit-Pauli Hamiltonian. H (5) fs does not contribute to the energies of the singlet states and to the spin-orbit averaged levels but it yields the mα 5 contribution to the fine structure splitting. It is given by 

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We note that despite the presence of terms with ln Z in Eq. (10), the correction E (5) ∞ (nlog) does not have loga- rithmic terms in its 1 /Z expansion. 

The recoil correction E (5) M consists of four parts [10], 

where 

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with ˜ Q$_{a}$ defined analogously to Eq. (11), and ( m/M ) E 3 is the first-order perturbation of E (5) ∞ due to the mass- polarization operator (6). The operator H (5) fs , rec yields a 

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nonvanishing contribution to the fine-structure splitting only. It is given by 

We note that the last term in Eq. (16) was omitted in the original derivation of Ref. [10]. 6 (6) 

The complete result for the m α correction E ∞ to the energy levels was derived by one of the authors (K.P.) in a series of papers [6, 7, 11, 12] 

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The first term in the above expression contains the com- plete logarithmic dependence of the m α 6 correction. The part of it proportional to ln α was first obtained in Ref. [13]. The remaining logarithmic part proportional to ln Z was implicitly present in formulas reported in Ref. [6, 7] (it originates from the expectation value of the operator 1 /r 3 $_{ab}$). In Eq. (19), we group all logarithmic terms together so that the remaining part does not have any logarithms in its 1 /Z expansion. 

The term E$_{sec}$ in Eq. (19) is the second-order perturba- tion correction induced by the Breit-Pauli Hamiltonian. (More specifically, it is the finite residual after separat- ing divergent contributions that cancel out in the sum with the expectation value of the effective mα 6 Hamilto- nian.) The first part of the effective Hamiltonian, H (6) $_{nrad}$, originates from the non-radiative part of the electron- nucleus and the electron-electron interaction. The next two terms, H (6) R 1 and H (6) R $_{2}$, are due to the one-loop and two-loop radiative effects, respectively. The last two parts H (6) fs and H (6) fs , amm are the spin-dependent opera- tors first derived by Douglas and Kroll [14]. They do not contribute to the energies of the singlet states and to the spin-orbit averaged levels. Expressions for these opera- tors are well known and are given, e.g., by Eqs. (3) and (7) of Ref. [15]. The non-radiative part of the m α 6 effec- tive Hamiltonian is rather complicated. For simplicity, we present it specifically for a two-electron atom. The corresponding expression reads [6, 7] 

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