Relativistic two-photon decay rates of $2s_{\frac{1}{2}}$ hydrogenic ions

Rates are calculated for the decay of metastable $2s_{\frac{1}{2}}$ ions to the ground state by the simultaneous emission of two photons. The calculation includes all relativistic and retardation effects, and all combinations of photon multipoles which make significant contributions up to $Z=100\mathrm{}$. Summations over intermediate states are performed by constructing a finite-basis-set representation of the Dirac Green's function. The estimated accuracy of the results is ± 10 ppm for all $Z$ up to 100. The decay rates are about $20 {\left(\alpha Z\right)}^2$% larger than an earlier calculation by Johnson owing to the inclusion of higher-order retardation effects. The general question of gauge invariance in two-photon transitions is discussed.