Complex coordinate rotation and relativistic Hylleraas-CI: helium isoelectronic series

A combination of the Hylleraas-CI (Hy-CI) and complex coordinate rotation (CCR) methods has been applied to study the nuclear charge dependence of the eigenvalues of the Dirac-Coulomb (DC) Hamiltonian corresponding to the ground states of helium isoelectronic series atoms. It has been shown that the CCR, due to the separation of the localized states from the unphysical Brown-Ravenhall continuum, removes the instabilities of the bound-state eigenvalues observed in large-basis set Hy-CI results. The Hy-CI-CCR results are in very good agreement with the most accurate ones available in the literature. Surprisingly, the difference between the DC Hy-CI-CCR eigenvalues and the eigenvalues of the positive-energy projected no-pair Hamiltonian is equal, up to the numerical accuracy of the results, to (Zα)3/6π, i.e. to (Zα)3 relativistic many-body perturbation theory contribution for electron-electron Coulomb interaction operator. An excellent agreement between the Hy-CI-CCR eigenvalues shifted by (Zα)3/6π and the no-pair ones confirms the very high accuracy achieved in both approaches. The numerical accuracy of the Hy-CI-CCR DC eigenvalues is estimated to eight significant figures