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 Dirac Hamiltonian. Expansion of the normalization terms leads immediately to the zeroth-order regular approximation (ZORA) and first-order regular approximation (FORA) Hamiltonians as the zeroth- and first-order terms of the expansion. The expansion may be taken to infinite order by using an un-normalized Foldy–Wouthuysen transformation, which results in the ZORA Hamiltonian and a nonunit metric. This infinite-order regular approximation, IORA, has eigenvalues which differ from the Dirac eigenvalues by order E3/c4 for a hydrogen-like system, which is a considerable improvement over the ZORA eigenvalues, and similar to the nonvariational FORA energies. A further perturbation analysis yields a third-order correction to the IORA energies, TIORA. Results are presented for several systems including the neutral U atom. The IORA eigenvalues for all but the 1s spinor of the neutral system are superior even to the scaled ZORA energies, which are exact for the hydrogenic system. The third-order correction reduces the IORA error for the inner orbitals to a very small fraction of the Dirac eigenvalue.