Another method of decoupling the large and small components of the DHF wavefunctions is given by the Douglas-Kroll method[22,23]. First presented by Douglas and Kroll in 1974, this method begins with a first order Foldy-Wouthoysen transformation performed in momentum space. The momentum space Hamiltonian is given by
where the action of the external potential, 
, on a momentum space
state vector, 
, is given by the expression
The unitary operator used in the first-order transformation is constructed using those free-particle momentum-space eigensolutions of the Dirac Hamiltonian associated with the positive energy eigenvalues
The resultant unitary transformation operator is given by
When this unitary operator is applied to the single particle Hamiltonian, the result is given by
with 
and 
given by
Use of the definitions
provides more compact expressions for the even and odd operators
From this point, Douglas and Kroll proposed the use of a second unitary transformation, of the form
where 
is anti-Hermitian, and is selected such that, in the
resulting transformed Hamiltonian
the 
term is exactly canceled by the
term.
This condition is fufilled if 
obeys the equation
Because the operator 
is linear in 
, an
integral operator, 
must be expressed in terms of its kernel
Using the functional form of 
, a more explicit expression for
the anti-Hermitian operator, 
may be obtained
This transformation removes all coupling of the upper and lower halves of
the transformed wavefunction to first order in 
. Douglas and Kroll
suggested that further separation, to arbitrary orders in the
external potential were possible through the repeated application of the
n
order transformation
where 
is, again, an anti-Hermetian operator linear in the
external potential.
Separation to second order in the external potential represents a practical
stopping point, since the higher order transformations become extremely
complicated. Furthermore, higher order contributions
should only be important for the proper treatment of the lowest lying
electrons, and so should not substantially affect the description of
the valence electronic structure.
The large component block of the single-particle Hamiltonian which has been
decoupled to second order in 
is given by
The resultant single-particle Hamiltonian may be employed used in conjunction with the coulomb inter-electron repulsion operator to define a many-electron hamiltonian:
This Hamiltonian is frequently known as the no-pair Douglas-Kroll Hamiltonian, and it has been extensively discussed and tested by Hess and coworkers[24,23,25]. Molecular wavefunctions may be obtained with the no-pair DK Hamiltonian in much the same fashion that solutions are obtained in the non-relativistic case, and investigations of systems containing heavy atoms by means of DK based electronic structure methods have given very encouraging comparison to fully relativistic results.[4]