Negative-energy contributions to transition amplitudes in heliumlike ions

We derive the leading term in an $\alpha Z$ expansion for the negative-energy (virtual electron-positron pair) contributions to the transition amplitudes of heliumlike ions. The resulting expressions allow us to perform a general analysis of the negative-energy contributions to electric- and magnetic-multipole transition amplitudes. We observe a strong dependence on the choice of the zeroth-order Hamiltonian, which defines the negative-energy spectrum. We show that for transitions between states with different values of total spin, the negative-energy contributions calculated in a Coulomb basis vanish in the leading order while they remain finite in a Hartree basis. The ratio of negative-energy contributions to the total transition amplitudes for some of nonrelativistically forbidden transitions is shown to be of order $1/Z.$ In the particular case of the magnetic-dipole transition $3{}^3{S}_1\to 2{}^3{S}_1,$ we demonstrate that the neglect of negative-energy contributions, in an otherwise exact no-pair calculation, would lead one to underestimate the decay rate in helium by a factor of 1.5 in calculations using a Hartree basis and by a factor of 2.9 using a Coulomb basis. Finally, we tabulate revised values of the line strength S for the magnetic-quadrupole ${(M}_2)$ transition $2{}^3{P}_2\to 1{}^1{S}_0.$ These values include negative-energy contributions from higher partial waves, which were neglected in our previous calculations.