Relation between different variants of the generalized Douglas–Kroll transformation through sixth order

Wolf et al. have recently investigated a generalized Douglas-Kroll transformation. From a general class of unitary transformations that can be used in the Douglas-Kroll transformation, they pick one which is supposed to give, at a given order, an optimal transformed Dirac Hamiltonian. Results were presented through the fifth order. However, no data were given to demonstrate to which extent the so-called "optimal" Douglas-Kroll transformation is superior to other choices. In this work, the Douglas-Kroll transformation is extended to the sixth order for the first time, using computer algebra algorithms to obtain the working equations. It is shown how, at a given order, different variants of the Douglas-Kroll Hamiltonians are related. Various choices of the generalized transformation are examined numerically for the ground states of the one-electron atomic ions with nuclear charges Z=20, 40, 60, 80, 100, and 120. It is shown that compared to the improvement obtained by including the next order, the differences between various choices for the generalized Douglas-Kroll transformation are almost negligible. Results closest to the Dirac eigenvalues are not obtained with the optimal Douglas-Kroll transformation given by Wolf et al., but with the parametrization originally suggested by Douglas and Kroll.