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It is shown that variational solutions corresponding to the bound states of the Dirac–Coulomb (DC) eigenvalue problem may be obtained using a complex-coordinate rotation method. The discrete eigenvalues of the DC Hamiltonian are treated as resonances coupled to the Brown–Ravenhall continuum (the superposition of the negative- and positive-energy one-electron continua). The method leads to a clean separation of the bound-state energies from the continuum. It is shown that the effects related to the resonance character of the bound states being an artefact of the DC approximation, in particular the instability of the ground state, are proportional to the third power of the fine structure constant α. Thus, the results correctly describe the physical reality up to the terms proportional to α2. The approach has been implemented for the case of the ground state of two-electron atoms described by the Dirac–Coulomb Hamiltonian in a space of the Hylleraas-type trial functions. The variational energies, calculated for Z = 2, 80, 90, correspond to the best available in the literature.

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.

Relations between matrices representing non-relativistic and relativistic N-electron Hamiltonians in N-electron model spaces are analyzed. The model spaces are defined as the antisymmetric parts of products of the N-th Kronecker power of either a two-dimensional (the non-relativistic case) or four-dimensional (the relativistic case) spinor space and of an orbital (or configurational) space. The explicit relation between the matrices corresponding to the relativistic and non-relativistic cases is derived and its practical implications are briefly discussed.

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