The positronic solutions of the Dirac Hamiltonian and approximate,
multi-electron Hamiltonians such as H
give rise to equations which
are more complicated to interpret physically and more difficult to implement
computationally.
Finite basis set expansion of the model space for such Hamiltonians requires
a basis set of at least twice the size of a non-relativistic basis set for
similar quality results.
Since quantum chemists are used to thinking in terms of purely electronic
wavefunctions, the introduction of the positronic terms
can serve to rob chemists of some of their intuition for
quantum chemical models. Additionally, the existance of the continuum of
negative energy eigenstates destroys the chemist's conception of a bound
chemical system.
The Foldy-Wouthoysen transformation reduces the coupling
between the postitronic and electronic solutions which arise from fully
relativistic Hamiltonians. The operators which result from such a
transformation can be very complicated, but provide solutions which fit more
comfortably into traditional concepts of electronic solutions.
The molecular Dirac-Coulomb hamiltonian contains only one electron operators
which couple the large and small components of the wavefunction. Therefore,
it is possible to transform 
to a two-component Hamiltonian by
simply transforming the one electron terms of the Hamiltonian.
The Dirac single-particle hamiltonian, 
, may be represented as
where the even portion of the Hamiltonian, which do not involve interactions between the large and small componenents, are given by
and the odd portions, which do describe the interaction between the large and small component, and are given by
A more mathematical definition of even and odd operators is given by the
requirement that an even operator, 
,
commutes with 
, and
that the odd operator, 
, anticommutes with 
.
It is important to note that
any operator, 
, may be represented as the sum of an even and odd
operator, 
and 
where
The objective of the Foldy-Wouthoysen transformation is to apply a unitary
transform to the relativistic, four-component wave equation in order to
decouple the large and small components by eradicating the odd portions of
the Hamiltonian to some order in an expansion parameter which will go to
zero in the non-relativistic limit. A good form of the unitary transform,
, turns out to be
where
This choice of 
will result in the transformed electronic wavefunction
where 
is, effectively, a two component wavefunction, since the
contribution of the small component
has been removed to some level of approximation.
Similarly, the transformed, two-component positronic wavefunction, 
may also be produced via the action of 
on the four-component
positronic wavefunction, 
, which will effectively remove the large
component contribution.
In order to achieve this
transformed wavefunction, we consider a somewhat altered version of our
original wave equation where the full, four-component wavefunction,
, has been
expressed as a product of the inverse transformation,
,
and the two component
wavefuction, and the entire equation has been pre-multiplied by 
to give
This gives us the definition of the transformed one-particle Hamiltonian,
Expanding 
and 
in terms of their power series
expansions, and collecting terms of similar order in 
gives
a series of terms which are most efficiently represented by nested
commutators
where the terms which have been omitted are of order 
.
The resultant Hamiltonian, 
,
may again be expressed as the sum of even and
odd operators, 
and
(and, now, an error
term of order 
) where the new even and odd terms are given by
Subsequently, the largest contributing odd term which remains may be removed through the use of the unitary transformation
to give new even and odd operators which are now higher order in the fine
structure constant, and a resudual error of the order 
.
Application of a third transform gives an even operator,
which is correct to
order 
. Truncation at this order represents a very useful
stopping point. The third order FW transformed Dirac one-particle Hamiltonian
for the electronic solutions is given by
In the absence of an external vector potential, 
, the two-component
Hamiltonian becomes somewhat simpler:
The first three terms of ( 4.120) and ( 4.119)
may be recognized as the
non-relativistic single-particle Hamiltonian. This decoupled, approximate
Hamiltonian, which is correct to order (mc
), may also be obtained
via the method of small components. In the method of small components, the
single particle wavefunction is replaced by
in order to remove the rest mass energy from the Hamiltonian to give the modified single particle equation
This relation may be expressed as a pair of coupled differential equations
in 
and 
, the large and small components.
Using the second differential equation to obtain an expression for 
in terms of 
gives
Substituting this result into the first differential equation gives
where
Using a rearrangement of ( 4.127)
the action of 
on 
is found to be
Because it is not possible to solve for 
analytically, and hence
achieve exact separation of 
and 
, except in the case of the
free particle, it is necessary to utilize the approximate form of 
given by
to give the approximate Hamiltonian
The last term in this Hamiltonian is somewhat disconcerting, since it is not
Hermitian. This Hamiltonian, however, was derived for the unnormalized
, and the renormalization will neccessarily be dependent upon the
magnitude of the coupling of large and small components, and hence on
. The Hamiltonian for the renormalized wavefunction,
, is given by
This transformed Hamiltonian is identical to the Foldy-Wouthoysen
transformed Hamiltonian given in ( 4.120), with the
exception that the rest mass energy of the electron, 
, has been removed.
