As discussed in appendix A many-electron operators are often conveniently expressed in terms of annihilation and creation operators -- that is, in second quantization. We can go further with this, however, by introducing the concept of a normal-ordered second-quantized operator in the particle-hole formalism. Normal-ordered forms of the Hamiltonian and the cluster operators make algebraic (and diagrammatic) analysis of the coupled-cluster equations a much simpler task.
A normal-ordered product of annihilation and creation operators is one in
which all annihilation operators lie to the right of all creation
operators; any such product may be expressed in normal-ordered form using
the anti-commutation relations given in appendix A.
This is a useful form because a
normal-ordered product gives a zero result when applied to the true vacuum
state, 
(i.e. a state containing no electrons), if the string
contains any annihilation operators. However, we can see from the CC
equations (3.7) and (3.8) that the operators used in
CC theory ( 
and 
) are applied, not to the true vacuum,
but to the reference wavefunction, 
.
Thus, a zero result is obtained only when those
annihilation operators corresponding to states unoccupied in the reference
(which we will hereafter refer to as ``particle'' states) or creation
operators corresponding to states occupied in the reference (which we will
hereafter refer to as ``hole'' states) are applied to .
Therefore, in the ``particle-hole'' formalism, normal-ordering of a string of annihilation and creation
operators means that all particle-creation (a) and all
hole-annihilation ( 
) operators lie to the right of (i.e. are
applied before) all particle-annihilation ( 
) and hole-creation
(i) operators. For example, given the string of operators
the normal-ordered form of this string is
where the 
denote the normal-ordering. The sign arises due to the
anti-commutation relations among annihilation and creation operators
(A.6). The reason for this definition will become
apparent in subsequent discussions.