Convergence of approximate two-component Hamiltonians: how far is the Dirac limit.

A systematic elimination of the off-diagonal parts of the Dirac Hamiltonian is carried out in the spirit of the Douglas-Kroll [Ann. Phys. 82, 87 1974] approach and the recently proposed infinite-order two-component method. The present approach leads to a series of approximate two-component Hamiltonians which are exact through a certain order in the external potential. These Hamiltonians are used to study the convergence pattern of approximate two-component theories. It is shown that to achieve an acceptably high accuracy for low-lying one-electron levels in heavy and superheavy systems one needs to use approximate Hamiltonians of prohibitively high order in the external potential. One can conclude that the finite-order two-component Hamiltonians are of limited usefulness in accurate relativistic calculations for heavy and superheavy systems.