Coupled cluster (CC) methods begin with an approximate wavefunction which is
somewhat more complicated than those proposed for the CI and MP2 methods.
Instead of casting the wavefunction as a simple linear combination of
excited determinants, CC methods employ excitation operators,
,
whose action on the DHF reference wavefunction is given by
where the scalar quantity 
is known as an
amplitude. These operators are classified according to the number of
spinors which their application swaps. The sum of all operators of a
particular class forms a single 
operator, for example
the sum of all 
forms the 
operator.
These operators are then employed in an exponential fashion to form a
coupled-cluster wavefunction
In the case that all the 
operators up to 
are included in
the exponential operator, the resultant coupled-cluster wavefunction is
equivalent to the FCI wavefunction.
Usually, however,
only the 
and 
operators are employed to give
what is known as the CCSD wavefunction. Left projection of 3.104 by
the reference state, a singly excited state vector, and a doubly excited
state vector gives three sets of equations which subsequently allow for the
iterative solution of 
, 
, and 
.
The non-relativistic
CCSD method, and the associated CCSD(T) method[16,17],
which attempts to
estimate the effects of the inclusion of the 
operator in the
exponential excitation operator, have proved to be extremely successful in
providing chemically accurate predictions for a large number of difficult
chemical systems. Practical implementations of the DHF based versions of
these methods have been reported by Lindgren[18]
as well as Dyall, Lee, and Visscher[19,20].
Dyall et al. claim that their
implementation of CCSD,
which employs double group symmetry and Kramer's restricted
single particle functions, should employ 32 times the number of floating
point operations required by a similar implementation of the
equivalent non-relativistic CCSD.[19]
The success of the non-relativistic CC methods, the rapid development of
modern workstation and supercomputer capabilities, and the existence of
highly optimized implementations of these methods in such program packages
as MOLFDIR[11]
all indicate that, in the future, relativistic coupled cluster
methods will play an important role in the treatment of systems which exhibit
significant relativistic and correlation effects.
