Incorporating Dirac negative-energy solutions into calculations of many-electron systems

Attempts to incorporate negative-energy solutions of the Dirac equation into calculations of many-electron systems have presented difficulties. While excluding negative-energy solutions completely from the conventional many-electron Dirac Hamiltonian using positive-energy projection operators under the "no-virtual pair approximation"(NVPA) is an approach often used, the NVPA necessarily provides an incomplete description of the many-electron system. Rather than excluding Dirac negative-energy solutions under the NVPA, this paper provides an approach that establishes Dirac negative-energy solutions and positive-energy solutions as degenerate eigenfunctions of a common Hamiltonian. It is shown how to use the probability densities arising from positive energy functions and Dirac negative-energy functions to balance the relative contributions from these eigenfunctions, and further establish that the only function associated with an observable state is the "large" component of the positive-energy eigenfunction. The use of the positive- and Dirac negative-energy eigenstates is illustrated in various scenarios, including central-field systems.