Generalized Laguerre representation: Application to relativistic two-photon decay rates

Slater-type basis sets are commonly used to perform nonrelativistic and relativistic variational calculations. A major limitation of this method is that no more than about 14 powers per nonlinear parameter can be used while working in double-precision arithmetic without entering into numerical difficulties. Even for moderate-size basis sets, large numerical errors are introduced. We introduce a variational method based on the orthogonality properties of the Laguerre polynomials. All the matrix elements of the Dirac Hamiltonian are calculated in closed form, therefore avoiding numerical cancellations and vastly increasing the speed of the calculations. As a result, full double-precision accuracy is maintained with very large basis sets. Results for sets with 75 powers are given as examples. The method is applied to a new, very accurate relativistic calculation of the two-photon decay rates for hydrogenic ions. This calculation resolves a discrepancy between the previous calculation of Parpia and Johnson Phys. Rev. A 26, 1142 (1982) and that by Goldman and Drake Phys. Rev. A 24, 183 (1981) in favor of the latter.