2 H (4) = ∑ a [ − ⃗ p 4 a 8 + πZ 2 δ $^{3}$( r$_{a}$ ) + Z 4 ⃗ σ$_{a}$ · ⃗ r$_{a}$ r 3 a × ⃗ p$_{a}$ ] + a<b { − π δ $^{3}$( r$_{ab}$ ) − 1 2 p i a ( δ ij r$_{ab}$ + r i ab r j ab r 3 ab ) p j b − 2 π 3 ⃗ σ$_{a}$ · ⃗ σ$_{b}$ δ $^{3}$( r$_{ab}$ ) + σ i a σ j b 4 r 3 ab ( δ ij − 3 r i ab r j ab r 2 ab ) + 1 4 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$_{a}$ )] } . (7) E ( n ) = E ( n ) ∞ + E ( n ) M + E ( n ) M 2 + . . . , (2) H (4) rec = Zm 2 M ∑ ab [ ⃗ r$_{a}$ r 3 a × ⃗ p$_{b}$ · ⃗ σ$_{a}$ − p i a ( δ ij r$_{a}$ + r i $_{a}$r j a r 3 a ) p j b ] . (8) H (2) ≡ H$_{0}$ = ∑ a ( ⃗ p 2 a 2 − Z r$_{a}$ ) + ∑ a<b 1 r$_{ab}$ . (3) E (5) ∞ (log) = 14 3 ln( Zα ) ∑ $_{a<b}$〈 δ $^{3}$( r$_{ab}$ ) 〉 + 4 Z 3 ln [ ( Zα ) − 2 ] ∑ a 〈 δ $^{3}$( r$_{a}$ ) 〉 , (9) E (5) ∞ (nlog) = 164 15 ∑ $_{a<b}$〈 δ $^{3}$( r$_{ab}$ ) 〉 − 14 3 ∑ a<b ˜ Q$_{ab}$ + [ 19 30 − ln ( k$_{0}$ Z 2 )] 4 Z 3 a 〈 δ $^{3}$( r$_{a}$ ) 〉 + 〈 H (5) fs 〉 , (10) E (2) M = − m M E (2) ∞ + 〈 H (2) rec 〉 , (4) E (2) M 2 = ( m M ) 2 E (2) ∞ − 2 m M 〈 H (2) rec 〉 + 〈 H (2) rec 1 ( E 0 − H$_{0}$ ) $_{′}$H (2) rec 〉 , (5) ˜ Q$_{ab}$ = 〈 1 4 πr 3 ab + δ $^{3}$( r$_{ab}$ ) ln Z 〉 , (11) − 3 〈 1 r 3 〉 ≡ lim a → 0 ∫ d $^{3}$r φ $^{∗}$( ⃗ r ) φ ( ⃗ r ) × [ 1 r 3 Θ( r − a ) + 4 π δ $^{3}$( r ) ( γ + ln a ) ] , (12) H (2) rec = m M ∑ a<b ⃗ p$_{a}$ · ⃗ p$_{b}$ (6) [9], where E ( n ) ≡ mα $^{n}$E ( n ) is a contribution of order α n and may include powers of ln α . Each of E ( n ) is in turn ex- panded in powers of the electron-to-nucleus mass ratio m/M where E ( n ) M denotes the correction of first order in m/M and E ( n ) M 2 is the second-order correction. To note, for the nonrelativistic energy, it is more natural to expand in m$_{r}$/M (where m$_{r}$ is the reduced mass) rather than in m/M , since such expansion has smaller coefficients. For the relativistic corrections, however, the natural recoil expansion parameter is m/M , so for consistency we use it for the nonrelativistic energy as well. The finite nuclear mass correction to the Breit contri- bution E (4) M is conveniently separated into the mass scal- ing, the mass polarization, and the operator parts. The mass scaling prefactor is ( m$_{r}$/m ) 4 for the first term in Eq. (7) and ( m$_{r}$/m ) $^{3}$, for all the others. The mass po- larization part represents the first-order perturbation of E (4) ∞ by the mass-polarization operator (6). The opera- tor part is given by the expectation value of the recoil addition to the Breit-Pauli Hamiltonian, The terms of the double perturbation expansion (1) and (2) are expressed as expectation values of some effec- tive Hamiltonians (in some cases, of nonlocal operators) and as second- and higher-order perturbation corrections induced by these Hamiltonians (operators). It is note- worthy that the expansion (1) is employed also for the states that are mixed by the relativistic effects, namely 2 $^{1}$P$_{1}$ and 2 $^{3}$P$_{1}$ . The mixing effects are treated pertur- batively. (So, the leading effect due to the 2 $^{1}$P$_{1}$ − 2 $^{3}$P$_{1}$ mixing appears naturally as the second-order mα 6 cor- rection, together with contributions from other interme- diate states.) This differs from the approach used, e.g., in Ref. [5], where a two-by-two matrix was constructed for this pair of states and the energies were obtained by a diagonalization. E (5) ∞ is the leading QED correction [2, 3]. We divide it into the logarithmic and the nonlogarithmic parts, E (5) ∞ = E (5) ∞ (log) + E (5) ∞ (nlog), which are given by The leading contribution to the energy E (2) ∞ ≡ E 0 is the eigenvalue of the nonrelativistic Hamiltonian, and where The first- and second-order recoil corrections to the non- relativistic energy are given by where and the singular operator r is defined by is the mass polarization operator. The leading relativistic correction E (4) ∞ is given by the expectation value of the Breit-Pauli Hamiltonian H (4)