  We develop a relativistic model to describe the bound states of positive
energy and negative energy in finite nuclei at the same time. Instead of
searching for the negative-energy solution of the nucleon's Dirac equation, we
solve the Dirac equations for the nucleon and the anti-nucleon simultaneously.
The single-particle energies of negative-energy nucleons are obtained through
changing the sign of the single-particle energies of positive-energy
anti-nucleons. The contributions of the Dirac sea to the source terms of the
meson fields are evaluated by means of the derivative expansion up to the
leading derivative order for the one-meson loop and one-nucleon loop. After
refitting the parameters of the model to the properties of spherical nuclei,
the results of positive-energy sector are similar to that calculated within the
commonly used relativistic mean field theory under the no-sea approximation.
However, the bound levels of negative-energy nucleons vary drastically when the
vacuum contributions are taken into account. It implies that the
negative-energy spectra deserve a sensitive probe to the effective interactions
in addition to the positive-energy spectra.
