Application of the complex-coordinate rotation to the relativistic Hylleraas-CI method: a case study

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