It should be obvious that the formal CC
equations (3.7) and (3.8) are not in a form useful
for implementation on the computer. Before we may apply the CC method to
actual chemical problems, the equations must be re-written in terms of one-
and two-electron integrals and the cluster amplitudes (these amplitudes
are, in fact, the only unknown parts of the equations, other than the
energy itself). To that end, it is not convenient to deal with the
operator 
. Instead, with (a huge amount of)
mathematical foresight we use the effective Hamiltonian,
. The resulting equations
and
are equivalent to the CC equations (3.7) and (3.8) presented before[29], though it should be noted that the amplitude equations (3.17) now do not depend on the energy. At first glance, these equations seem even worse than our earlier set. However, there are a number of mathematical principles which allow us to use significant simplifications.
The first comes from the celebrated Campbell-Baker-Hausdorff formula, which is[39]
Inserting 
and 
into this equation, we obtain
This expansion is infinite in the general case, but conveniently truncates
itself for the Hamiltonian, , and excitation operators, .
The Hamiltonian (see appendix A) contains general-index creation and
annihilation operators, 
, which do not commute with the
excitation operators, 
in . However, it can be
shown that[27]
Hence, commutation of a general-index creation/annihilation pair with an
excitation pair serves to eliminate one of the general-index operators.
Since the Hamiltonian contains, at most, four general-index operators (in
the two-body terms), four such commutations will eliminate all of the
these terms from the expressions. The only terms left will be excitation
operators which will commute with , thereby eliminating all
higher-order commutators from (3.19). Therefore, the
Hausdorff expansion in and truncates exactly at the
quadruply-nested commutator
,
This expansion in commutators of with breaks the
problem into a number of much smaller ones: we must construct analytic
forms of the above commutators, their operation on the reference
, and finally the matrix elements after left projection by
the excited determinants 
and the reference
. The following section describes in detail how to
accomplish this task.
