The Dirac equation has been shown to successfully treat the interactions of
electrons with nuclei in a special-relativistic manner, and so represents a
good starting point for a special-relativistic many-electron Hamiltonian.
What is needed next is a description of the electron-electron interaction,
and an associated quantum mechanical operator, 
.
If the electron-electron interaction is not
altered significantly by the introduction of special relativity, then the
standard, non-relativistic coulomb operator, 
, might not be a bad guess. With
the coulomb interaction, the Hamiltonian for an n electron system would be
given by
This is known as the Dirac-Coulomb Hamiltonian, 
.
The associated wave equation, 
, is not Lorentz
invariant, and so 
does not represent a proper special
relativistic Hamiltonian.
In order to obtain a two-electron interaction term which is consistent with
special relativity, it is necessary to turn to quantum electrodynamics (QED).
In order to cast the QED electron-electron interaction term into a
reasonable form, it is necessary to expand it in perturbative series in
orders of the fine structure constant, 
.
Retaining only the terms which contribute up to order 
, gives the
Coulomb operator plus the Breit interaction term
In practice, however, the full Breit magnetic interaction term is often cumbersome to implement, and so an approximation of the Breit operator known as the Gaunt operator, may be used
The Gaunt operator is not gauge invariant, but it avoids having to solve
integrals over operators more complicated than 
, and
includes the largest contributions of the Breit interaction.
More rigorous forms of the molecular Hamiltonian have been suggested, but
they are, at best, only approximately Lorentz covariant to some order in
. Because the two-electron
magnetic interaction terms are typically small, the
contributions beyond second order in 
are typically chemically
unimportant. Because of this the more rigorous Hamiltonians, which
typically involve more complicated operators than even the Breit
interaction, have received far less attention from quantum chemical
investigators, and so the Dirac-Coulomb, Dirac-Coulomb-Gaunt, and
Dirac-Coulomb-Breit Hamiltonians are the most widely used
four-component special relativistic molecular Hamiltonians.