  The relativistic Hartree-Fock and electron correlation methods without the
negative-energy orbital problem are examined on the basis of the quantum
electrodynamics (QED) Hamiltonian. First, several QED Hamiltonians previously
proposed are sifted by the orbital rotation invariance, the charge conjugation
and time reversal invariance, and the nonrelativistic limit. A new total energy
expression is then proposed, in which a counter term corresponding to the
energy of the polarized vacuum is subtracted from the total energy. This
expression prevents the possibility of total energy divergence due to electron
correlations, stemming from the fact that the QED Hamiltonian does not conserve
the number of particles. Finally, based on the Hamiltonian and energy
expression, the Dirac-Hartree-Fock (DHF) and electron correlation methods are
reintroduced. The resulting QED-based DHF equation has the same form as the
conventional DHF equation, but also formally describes systems specific to QED,
such as the virtual positrons in the hydride ion and the positron in
positronium. Three electron correlation methods are derived: the QED-based
configuration interactions and single- and multireference perturbation methods.
Numerical calculations show that the total energy of the QED Hamiltonian indeed
diverges and that the counter term is effective in avoiding the divergence. The
theoretical examinations in the present article suggest that the molecular
orbital (MO) methods based on the QED Hamiltonian not only solve the problem of
the negative-energy solutions of the relativistic MO method, but also provide a
relativistic formalism to treat systems containing positrons.
