Relativistic and nonrelativistic finite-basis-set calculations of low-lying levels of hydrogenic atoms in intense magnetic fields

A finite-basis-set method is used to calculate relativistic and nonrelativistic binding energies of an electron in a static Coulomb field and in magnetic fields of arbitrary strength (0B\ensuremath{\le}${10}^{13}$G). The basis set is composed of products of Slater- and Landau-type functions, and it contains the exact solutions at both the Coulomb limit (B=0) and the Landau limit (Z=0). Relativistic variational collapse is avoided and highly accurate results are obtained with the basis set. The relativistic corrections obtained for intense magnetic fields (B\ensuremath{\gtrsim}${10}^{9}$G) differ from the previous relativistic calculations based on the adiabatic approximations. It is found that the sign of the relativistic correction changes from negative to positive near B\ensuremath{\approxeq}${10}^{11}$ G for the ground state and near B\ensuremath{\approxeq}${10}^{10}$ G for the 2${\mathit{p}}_{3/2}$(\ensuremath{\mu}=-3/2) excited state of hydrogen. The method is checked to be very accurate by means of the virial theorem, sum rules, and the relativistic low-B limit where comparison can be made with perturbation results. In the nonrelativistic limit of the Dirac equation, our results agree with other accurate nonrelativistic calculations available and with our own calculations based on the Schr\"odinger equation, which converge to more significant digits than previous calculations for the whole range of magnetic fields.