  It is shown that four-component (4C), quasi-four-component (Q4C), and exact
two-component (X2C) relativistic Hartree-Fock (HF) equations can be implemented
in an unified manner, by making use of the atomic nature of the small
components of molecular 4-spinors. A model density matrix approximation can
first be invoked for the small-component charge/current density functions,
which gives rise to a static, pre-molecular mean field (pmf) to be combined
with the one-electron term. As a result, only the nonrelativistic-like
two-electron term of the 4C/Q4C/X2C Fock matrix needs to be updated during the
iterations. A `one-center small-component' approximation can then be invoked in
the evaluation of relativistic integrals. That is, all atom-centered
small-component basis functions are regarded as extremely localized nearby the
position of the atom to which they belong, such that they have vanishing
overlaps with all small- or large-component functions centered at other nuclei.
Under these approximations, the 4C, Q4C, and X2C mean-field and many-electron
Hamiltonians share precisely the same structure and accuracy. Beyond these is
the effective quantum electrodynamics Hamiltonian that can be constructed in
the same way. Such approximations lead to errors that are orders of magnitude
smaller than other sources of errors (e.g., truncation errors in the one- and
many-particle bases as well as uncertainties of experimental measurements) and
are hence safe to use for whatever purposes. The quaternion forms of the 4C,
Q4C, and X2C equations are also presented in the most general way, based on
which the corresponding Kramers-restricted open-shell (RKOHF) variants are
formulated for `high-spin' open-shell systems.
