Before we can cast many-electron operators into a normal-ordered form, we must state the generalized Wick's theorem[40, 41]. In mnemonic form, this theorem is
where the 
represents a contraction between annihilation and
creation operators which will be defined below. In short, this version of
Wick's theorem says that given a string of annihilation and creation
operators, this may be re-written as the normal-ordered product of the
string, plus the normal-ordered product after all single-contractions among
operator pairs, plus all double contractions, etc., plus all full
contractions.
A contraction of two annihilation/creation operators allows for the elimination of the operator pair. The general definition of a contraction is
where A and B are some creation/annihilation operators. Contractions of annihilation operators together or of creation operators together such as
are zero since annihilation and creation operators anti-commute. (Show this.) A contraction in which the left operator is of creation type is
If the string is already normal-ordered, which is the case if the right-hand operator is a particle index, then this contraction is zero, since the normal-ordering does nothing to the last term on the RHS. However, if the string isn't normal-ordered, which is the case if the left-hand index is a hole index, then the order of the operators must be reversed, thereby introducing a minus sign
By the relations (A.6), this anti-commutator is zero unless i=q. Therefore, this contraction is
Now consider the case in which the left operator is of annihilation type is
By the same arguments as before, we find that the only non-zero possibility is
These results are summarized in Table 1.
Tabla: Results of all possible pairwise contractions resulting from the
anticommutation relations for annihilation and creation operators
(A.6). Only two non-zero contractions exist.
One implication of Wick's theorem is that the reference expectation value of an operator written as a string of annihilation and creation operators is zero unless all operators in the string may be eliminated via contractions:
An important extension of this theorem allows us to define contractions between normal-ordered strings of operators. This ``Generalized Wick's Theorem'' is
where the contraction sum runs over only those contractions between operators in different normal-ordered products, since contractions within a normal-ordered string are, by definition, zero. This form of Wick's Theorem will be particularly useful in the construction of the coupled-cluster equations.