H (5) fs , rec = m M Z 4 π ∑ ab ⃗ r$_{a}$ r 3 a × ⃗ p$_{b}$ · ⃗ σ$_{a}$ . (18) ln( k$_{0}$ ) = 〈 ∑ a ⃗ p$_{a}$ ( H$_{0}$ − E $_{0}$) ln [ 2 ( H$_{0}$ − E $_{0}$) ] ∑ b ⃗ p$_{b}$ 〉 2 π Z 〈 ∑ c δ $^{3}$( r$_{c}$ ) 〉 . (13) E (6) ∞ = − ln( Zα ) π ∑ $_{a<b}$〈 δ $^{3}$( r$_{ab}$ ) 〉 + E$_{sec}$ + 〈 H (6) nrad + H (6) R 1 + H (6) R 2 + H (6) fs + H (6) fs , amm 〉 . (19) H (5) fs = Z 4 π ∑ a ⃗ σ$_{a}$ · ⃗ r$_{a}$ r 3 a × ⃗ p$_{a}$ + ∑ a<b { 1 4 π σ i a σ j b r 3 ab ( δ ij − 3 r i ab r j ab r 2 ab ) + 1 4 π r 3 ab [ 2 ( ⃗ σ$_{a}$ · ⃗ r$_{ab}$ × ⃗ p$_{b}$ − ⃗ σ$_{b}$ · ⃗ r$_{ab}$ × ⃗ p$_{a}$ ) + ( ⃗ σ$_{b}$ · ⃗ r$_{ab}$ × ⃗ p$_{b}$ − ⃗ σ$_{a}$ · ⃗ r$_{ab}$ × ⃗ p$_{a}$ )] } . (14) E (5) M = m M ( E 1 + E 2 + E 3 ) + 〈 H (5) fs , rec 〉 , (15) E 1 = − 3 E (5) ∞ + 4 Z 3 ∑ a 〈 δ $^{3}$( r$_{a}$ ) 〉 − 14 3 ∑ $_{a<b}$〈 δ $^{3}$( r$_{ab}$ ) 〉 , (16) E 2 = Z 2 [ − 2 3 ln( Zα ) + 62 9 − 8 3 ln ( k$_{0}$ Z 2 ) ] ∑ a 〈 δ $^{3}$( r$_{a}$ ) 〉 − 14 Z 2 3 ∑ a ˜ Q$_{a}$ , (17) 3 nonvanishing contribution to the fine-structure splitting only. It is given by where γ is the Euler constant. The Bethe logarithm is defined as We note that the last term in Eq. (16) was omitted in the original derivation of Ref. [10]. The complete result for the m α 6 correction E (6) ∞ to the energy levels was derived by one of the authors (K.P.) in a series of papers [6, 7, 11, 12] 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 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], 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. where 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] 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