content_image/381850.jpg content_image/381852.jpg content_image/381855.jpg content_image/381856.jpg content_image/381859.jpg content_image/381849.jpg content_image/381851.jpg content_image/381853.jpg content_image/381854.jpg content_image/381858.jpg content_image/381857.jpg
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 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. 

The leading contribution to the energy E (2) ∞ ≡ E 0 is the eigenvalue of the nonrelativistic Hamiltonian, 

The first- and second-order recoil corrections to the non- relativistic energy are given by 

where 

is the mass polarization operator. 

The leading relativistic correction E (4) ∞ is given by the expectation value of the Breit-Pauli Hamiltonian H (4) 

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[9], 

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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, 

<img src='content_image/381851.jpg'>

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 

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and 

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where 

and the singular operator r − 3 is defined by 

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