The approximate Hamiltonian which arises from the Foldy-Wouthoysen
transformation and the method of small components is frequently known as the
Pauli Hamiltonian, and methods which utilize this Hamiltonian as a starting
point are said to stem from the Pauli formulation.
In contrast to the Dirac formulation,
the Pauli formulation strongly suggests
that relativistic effects may be viewed as a perturbation to the
non-relativistic case. Most of the terms retained in ( 4.120)
and ( 4.119)
have direct physical interpretations. The term on the second line of
( 4.119) corresponds
to the interaction of the electron spin magnetic
moment with an external magnetic field, and is generally known as the Zeeman
term. The Zeeman term is zeroth order in the fine structure constant,
, and may be obtained directly from classical magnetostatics.
The first relativistic correction to this term, roughly second order in the
fine structure constant, is given by the fifth line in
( 4.119).
The term on the second line of ( 4.120)
corresponds to the interaction of the electron
spin magnetic moment with the magnetic field produced by the electron's
motion within an electric field.
In the case that the electric feild is that of the nucleus,
this term represents spin-orbit coupling.
The fourth term represents the relativistic correction to the kinetic energy
term. This term is commonly called the mass-velocity (MV) term.
The final term of ( 4.120) gives what is known as the Darwin operator.
This operator has no simple classical analogue.
In order to get a feel for the physical origin of this term, it is useful to
investigate the relativistic velocity operator, 
.
In non-relativistic quantum mechanics, the velocity operator is given simply
by the classically intuitive 
. The relativistic analog,
on the other hand, is given by
Since the eigenvalues of any single component of 
may only take
the value 
, this gives the somwhat disturbing result that each
component of the velocity of an electron has a magnitude c. Further
insight may be divined by inspection of the time derivative of 
.
Such an analysis reveals that even slow moving electrons appear to undergo an
extremely small amplitude, highly oscillatory motion about its average
postition
which has been termed Zitterbewegung, or ``trembling motion''.
The Darwin term represents the first order correction to the ``average
position'' model of non-relativistic quantum mechanics.
One of the most popular methods which employ the Pauli Hamiltonian is known as the MVD or mass-velocity plus Darwin method. In the MVD method, only the j-independent terms of the Pauli Hamiltonian are retained, to give
Once a converged HF wavefunction has been obtained, the MVD Hamiltonian may be employed in order to obtain a estimate of the first order relativistic correction to the electronic energy. Unfortunately, because evaluation of the Darwin term leads to singularities in the region of the nucleus, and the fact that the contributions from the mass-velocity term diverge with higher order applications, the perturbative expansion of the energy may not be extended beyond first-order for molecular systems. These effects are a result of the finite basis set expansion of the electronic wavefunction. Because only the first-order correction to the electronic energy may be obtained, no estimate of the relativistic effects on the wavefunction may be obtained.
For atomic systems, the singularities associated with the
MVD Hamiltonian may be avoided by solving for the atomic radial
wavefunctions numerically. Cowan and Griffin first
proposed the application of the MVD Hamiltonian to atomic systems
in 1976[21].
By incorporating the mass-velocity and Darwin terms in the
variational optimization of the radial wavefunctions in what they called the
HFR method, Cowan and Griffin were
able to produce atomic spin orbitals which
exhibited orbital energies and 
values which closely matched those of
the l-averaged atomic orbitals produced by the DHF method for atoms as heavy
as uranium.