Corrections to O(α⁷(lnα)mc²) fine-structure splittings and O(α⁶(lnα)mc²) energy levels in helium

Fully relativistic formulas for the energy-level shifts arising from no-pair exchange diagrams of two transverse photons plus an arbitrary number of Coulomb photons are derived in closed form within the external potential Bethe-Salpeter formalism. O(α⁷(lnα)mc²) corrections to the fine-structure splittings of helium are obtained and expressed in terms of expectation values of nonrelativistic operators. O(α⁷mc²) operators from exchange diagrams are found in nonrelativistic approximation. O(α⁶m²c²/M) nucleus-electron operators contributing to the fine-structure splittings are derived. Nonrelativistic operators of O(α⁶mc²) corrections to the triplet levels of helium are presented. Nonrelativistic operators of O(α⁶(lnα)mc²) corrections to the helium singlet levels and to positronium S levels are derived. O(α⁶m²c²/M) hydrogen and O(α⁶mc²) positronium P levels, and O(α⁶(lnα)mc²) corrections of first order to positronium S levels, are calculated using the derived operators for helium, in agreement with those obtained previously by others, except for one term in corrections to positronium P levels. In addition, the O(α⁶mc²) Dirac energies for hydrogenic non-S levels are exactly reproduced in a perturbative calculation. © 1996 The American Physical Society.