Relativistic electronic structure calculations with positive-energy and negative-energy orbitals

Abstract

We give an overview of a new approach to the theory and practice of relativistic electronic structure calculations. The self-consistent field (SCF) approaches and the no-pair formulation are seen as particular cases of a rigorous general framework in which both positive-energy and negative-energy orbitals are incorporated on an equal footing from the outset. Starting from any reasonable relativistic Hamiltonian, we construct its complete representation H in an antisymmetrized N-particle basis made by a fixed set of orbitals. Since H is not bounded from below, a direct variational approach is doubtful. It is shown that the N eigenvalues of H split into two sets: the lowest N − correspond to a mathematically well-defined problem, but upon optimization of the orbital basis in every case they fall towards minus infinity (this is not the variational collapse, which refers to another problem). The remaining N + eigenvalues are upper bounds to the eigenvalues of the no-pair Hamiltonian, and acquire physical meaning upon energy-optimization through a variational principle. After optimization, they represent bonafide bound states approximating the ground state, the first-excited state, and so on. Convergence towards definite eigenvalues, as the size of the orbital basis is increased, is not necessarily from above, as in the variational nonrelativistic case, but may be from below or show an oscillatory behaviour. The solution of the SCF equations for model wavefunctions with vanishing singly-excited configuration interaction (CI) coefficients are shown to be identical with the eigenfunctions of the singly-excited CI representation of H with eigenvalue N −+1, thus justifying the standard SCF approach. Also, distinct from traditional SCF, higher excited states can be readily calculated in our formulation, and for any combination of open shells. A variety of atomic calculations illustrating the features and scope of our approach are presented. The corresponding documented computer programs are freely available.