Now we will construct explicit expressions for the commutators in the Hausdorff expansion using Wick's theorem and the normal-ordered form of the Hamiltonian. This will lead us to a complete algebraic construction of the CCD equations,
and
To begin with, we will construct 
, which appears as
the second term of the Hausdorff expansion (3.21). Recall that
the commutator is
The second-quantized definition of the cluster operator 
was
introduced in section 2.2
Note that this operator is already in normal-ordered form, so
We will deal with the left half of the one-electron commutator first. We may insert the second-quantized definitions of the two operators as
Now we must use the generalized Wick's theorem to re-write the operator strings in this half of the commutator.
which becomes, after evaluating each contraction
Once again, the signs on the terms arise due to the anti-commutation
relations for annihilation and creation operators. A simple rule that
helps us to evaluate the sign quickly is that the term is negative if there
is an odd number of operators between the contraction pair, and positive if
there is an even number. At this point we need to note several points:
first, by Wick's theorem, all contractions within a normal ordered string
are zero; second, all contractions between pairs of creation operators or
pairs of annihilation operators are zero; third, contractions in which a
creation operator is to the right are zero unless the index contracts to
the particle space; fourth, contractions in which an annihilation operator
is to the right are zero unless the index contracts to the hole space(see
Table 1). Finally, note that not all of the
operators can be contracted out of the list. Therefore, if we were to take
the reference expectation value of this operator,
, it must be zero; this
operator will not contribute terms to the CCD energy expression. This form
is still necessary, however, as we need to construct equations for the
double-excitation amplitudes. This half of the commutator simplifies to
We may re-index and combine terms to obtain
Now we will construct the second-quantized form for the right half of the commutator.
Now we must again use Wick's theorem to contract the operator strings in this half of the commutator.
Now, we make our lives much easier by recognizing that every contraction in this list is zero, by the rules stated before! (See Table 1.) Only the single term remains
Thus, the right half of the commutator is simply
The full commutator is the difference between equations (4.33) and (4.37)
The first terms (i.e. the non-contracted terms) in each half cancel each other out due to the negative sign on the second term. Therefore, the only terms which remain are those from the left half of the commutator for which contractions were possible. That is,
This leads us to an important generalization: the same simplifications are
possible for all commutators in the CC equations; since the Hamiltonian
operator fragments are always to the left in the commutators, the right
halves of each one will give only zero contractions with the 
operators, and the single non-contracted term will cancel with a similar
term in the left half of the commutator (this is because all operators in
this electron-conserving theory have even numbers of operators). This can
be summarized by saying that only terms which may be connected
(contracted) together remain in the final equations - this is closely
related to the linked diagram theorem of perturbation
theory. With this is mind, we may
conveniently write the CCD commutator equations in a mnemonic form
as[42]
and
where the C subscript means that only connected terms need be evaluated for the given equations. We will find this notation convenient not only for the construction of the algebraic equations, but also in our upcoming diagrammatic analysis and later in a term-by-term analysis of the CC equations including higher excitations.
Now we need to construct the contribution of 
to
the CCD amplitudes equation (4.41) to finish this term. This
requires that we evaluate
We have left-projected by 
rather than
because of the index definitions in (4.28).
Inserting the definition of into (4.39)
we obtain
The first term on the RHS of this equation will not contribute to the doubles equation since we cannot fully contract the two operator strings. The remaining two terms will survive, however. These give
and
The contractions are
and
respectively. These simplify (4.43) to
after re-indexing certain terms.
Now we wish to construct the remainder of the terms contributing to
in the energy and amplitude equations. We need
only the two-electron terms
Wick's theorem gives
In the final form for the commutator, however, the first term in this equation (i.e. the only term with no contractions) will not survive commutation, and may be ignored. Now, instead of completely evaluating these contractions and determining index pairings, let's take a more practical approach: We will only determine which terms contribute to specific equations; that is, we will choose one of the two CCD equations, and eliminate all of the terms from these contractions which cannot contribute.
We will deal with the energy equation first. Since the equation requires the reference expectation value, only the completely contracted terms will survive
Evaluation of the contractions gives
Note that the factor of 
has appeared in the final equation
because of the equivalence of the four individual terms.
Now consider which terms from equation (4.50) will survive in the doubles amplitudes equations. These will be the set of doubly-contracted terms only. Thus,
The contractions evaluate to
This equation can be simplified rather drastically by noticing that most of the terms are redundant. In fact, the first two terms are identical, the next two terms are identical, and finally the last 16 terms are identical. This gives
Now we must also contract each of the remaining normal-ordered strings with
to remove the final general-index
operators. We will do this term by term, noting that only the fully
contracted terms will contribute
Evaluating the contractions gives
Next we take
Evaluating the contractions gives
Finally we take
Evaluating the contractions gives
If we insert the contraction (4.61) into the first term on the RHS of equation (4.55), we find that all four terms arising from Wick's theorem are equivalent (after some index permutation). The same occurs for the contraction (4.61). Thus, these terms are
and
respectively. The situation for the remaining term in (4.55) is different. The four terms generated by Wick's theorem in (4.61), namely
are disctinct permutations. Finally, then, the simplified result for (4.55) is
Thus far we have completely evaluated the CCD energy equation (4.40), and have completely constructed one of three terms from the amplitude equation (4.41). The remaining terms in the CCD amplitude equation are
and
The first of these is trivial:
(Note that there are no contributions from 
since there is no
way to construct fully contracted terms between this operator and .) The final term, however, is rather
difficult. In order to evalutate this monstrosity we must evaluate the
full contractions in
While it is a useful exercise in masochism to evaluate this contraction using Wick's theorem, we will avoid it here. It should be painfully clear to the student by this point that the work involved would be tedious at best. The final results for this term are
The results of our extensive algebraic manipulations are the CCD
equations
and
In the next section, we will derive these same equations but from a far more elegant approach.
