Now we wish to apply these concepts to the second-quantized form of the Hamiltonian
which is simply the sum of equations (A.15) and
(A.23)
We begin by re-writing the pair of operators
in the one-electron part of the Hamiltonian using Wick's
theorem
The contraction rules we examined before state that the contraction is zero
unless 
acts in the hole space, i.e. unless 
is occupied
in the reference wavefunction. In addition, p and q must be the
same. This simplifies the one-electron part of the equation to
Now we re-write the string of annihilation and creation operators from the two-electron part of 
as
Again, all of these contractions are zero unless the leftmost operator of the contraction acts in the hole space. This leads to the simplified form
where the notation 
means that p acts within the hole space.
Again, the signs on the terms arise from the anti-commutation relations for
annihilation and creation operators (A.6) which must be
used in order to place operators adjacent to one another prior to
evaluation of the contraction. Inserting this expression back into the
equation for the two-electron part of the Hamiltonian we obtain
Remembering that 
we may re-index terms and
combine where appropriate to obtain
The complete Hamiltonian is therefore
Note that the first and second terms on the RHS of this equation are simply the spin-orbital Fock operator (in normal-ordered form) and that the last two terms are the HF energy! Thus, we may write
or
Therefore, the normal-ordered Hamiltonian is
This result is easily generalized -- the normal-ordered form of an operator is simply the operator less the reference expectation value of itself. For the example of the Hamiltonian, above, the normal-ordered Hamiltonian is just the Hamiltonian minus the SCF energy.
