Corrections to O(${\mathrm{\alpha }}^7$(lnα)${\mathit{mc}}^2$) fine-structure splittings and O(${\mathrm{\alpha }}^6$(lnα)${\mathit{mc}}^2$) 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(${\mathrm{\alpha }}^7$(lnα)${\mathit{mc}}^2$) corrections to the fine-structure splittings of helium are obtained and expressed in terms of expectation values of nonrelativistic operators. O(${\mathrm{\alpha }}^7$ ${\mathit{mc}}^2$) operators from exchange diagrams are found in nonrelativistic approximation. O(${\mathrm{\alpha }}^6$ ${\mathit{m}}^2$ ${\mathit{c}}^2$/M) nucleus-electron operators contributing to the fine-structure splittings are derived. Nonrelativistic operators of O(${\mathrm{\alpha }}^6$ ${\mathit{mc}}^2$) corrections to the triplet levels of helium are presented. Nonrelativistic operators of O(${\mathrm{\alpha }}^6$(lnα)${\mathit{mc}}^2$) corrections to the helium singlet levels and to positronium S levels are derived. O(${\mathrm{\alpha }}^6$ ${\mathit{m}}^2$ ${\mathit{c}}^2$/M) hydrogen and O(${\mathrm{\alpha }}^6$ ${\mathit{mc}}^2$) positronium P levels, and O(${\mathrm{\alpha }}^6$(lnα)${\mathit{mc}}^2$) 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(${\mathrm{\alpha }}^6$ ${\mathit{mc}}^2$) Dirac energies for hydrogenic non-S levels are exactly reproduced in a perturbative calculation. © 1996 The American Physical Society.