Relativistic Variational Calculations For Helium-like Systems

Ab initio relativistic variational calculations are performed to obtain the 1{dollar}\\sp1{dollar}S{dollar}\\sb0{dollar}, 2{dollar}\\sp1{dollar}S{dollar}\\sb0{dollar}, 2{dollar}\\sp1{dollar}P{dollar}\\sb0{dollar}, 2{dollar}\\sp3{dollar}S{dollar}\\sb1{dollar}, 2{dollar}\\sp3{dollar}P{dollar}\\sb0{dollar}, 2{dollar}\\sp3{dollar}P{dollar}\\sb1{dollar} and 2{dollar}\\sp3{dollar}P{dollar}\\sb2{dollar} terms of helium and helium-like argon, krypton, xenon and mercury. The variational approach is formulated for the stationary problem for the Dirac-Coulomb-Breit Hamiltonian. The approximate solutions to the problem are sought in the form of variational expansions in terms of products of normalizable variational one-electron bi-spinors. Then the variational approach based on the minimax formulation of Talman generalized to the two-electron system is applied. The minimax formulation is discussed in the context of the variational approach to the one-electron Dirac-Coulomb problem and the two-electron relativistic problem. In the application to the two-electron problem there are two types of the finite-basis-set expansions considered: (a) a configuration interaction type basis; and (b) an expansion in terms of explicitly correlated coordinate functions. In both cases trial functions are chosen from the class of the Dirac-Slater type products having proper behaviour at the nucleus. A general optimization procedure with respect to scaling parameters of the Rayleigh-Ritz expansion was developed. The method, which involves analytical computation of the gradient and Hessian of variational eigenvalues, has applications in a number of problems employing finite-basis-set expansions. It is found that for the light ions, {dollar}\\alpha{dollar} Z {dollar}\\ll{dollar}1, the results obtained with the present variational method for the Dirac-Coulomb Hamiltonian coincide with the results obtained from an approximate non-relativistic Schrodinger Hamiltonian. For heavy ions, the results of the proposed variational approach coincide with the results obtained within the framework of the no-pair Hamiltonian from quantum electrodynamics. In the absence of the electron-electron interactions in the Hamiltonian the computed variational spectrum is reduced to the sum of the one-electron Dirac-Coulomb energies. The results obtained in the proposed variational approach lead to theoretical transition frequencies that compare with experimental transition frequencies. It is concluded that the proposed variational method can be considered as a legitimate approach to the determination of the bound state spectrum of relativistic helium-like ions.