Relativistic regular approximations revisited: An infinite-order relativistic approximation

The concept of the regular approximation is presented as the neglect of the energy dependence of the exact Foldy-Wouthuysen transformation of the Dira c Hamiltonian. Expansion of the normalization terms leads immediately to th e zeroth-order regular approximation (ZORA) and first-order regular approxi mation (FORA) Hamiltonians as the zeroth- and first-order terms of the expa nsion. The expansion may be taken to infinite order by using an un-normaliz ed Foldy-Wouthuysen transformation, which results in the ZORA Hamiltonian a nd a nonunit metric. This infinite-order regular approximation, IORA, has e igenvalues which differ from the Dirac eigenvalues by order E-3/c(4) for a hydrogen-like system, which is a considerable improvement over the ZORA eig envalues, and similar to the nonvariational FORA energies. A further pertur bation analysis yields a third-order correction to the IORA energies, TIORA . Results are presented for several systems including the neutral U atom. T he IORA eigenvalues for all but the 1s spinor of the neutral system are sup erior even to the scaled ZORA energies, which are exact for the hydrogenic system. The third-order correction reduces the IORA error for the inner orb itals to a very small fraction of the Dirac eigenvalue. (C) 1999 American I nstitute of Physics. [S0021-9606(99)30128-8].