Operators involving systems of identical particles may be represented using annihilation and creation operators. As an example, we consider the one-electron hamiltonian
where the summation runs over all N electrons. We may represent this operator in the basis of spin-orbitals as
where the closure relationship for the set of orbitals has been used. When
operates on a determinantal function
where 
is the usual permutation operator. The fact that
is invariant under permutation of electrons has been used to simplify the
equation. If we now expand each term on the RHS using expression
A.10, we obtain
This equation may be simplified using annihilation and creation operators by realizing that
for example. (Why is this true, based on the definition of annihilation and creation operators?) So
Thus, may be written as
where the sum over q technically runs over only states occupied in the determinant upon which h acts. However, since q will give a zero result if q is not occupied in this determinant anyway, we may allow the sum to run over all orbitals without error.
Now examine the two-electron operator
where the restricted summation runs over all N electrons. Represent this operator in the spin-orbital basis as
When 
operates on a determinantal wavefunction
The expansion (A.17) may be inserted above to obtain
Again we may simplify things by realizing that
Thus
We can write compactly as
where we have extended the sums over r and s to include all orbitals,
and the factor of 
has been inserted to account for
redundancies in r and s. Note that for the electronic Hamiltonian
operator, 
. An equivalent expression for is
where 
is the standard antisymmetrized two-electron
integrals and another factor of has been inserted to account
for the additional redundancies introduced by this notation. Note that the
form of equations (A.15) and (A.23)
contains one- and two-electron integrals explicitly and neither depends on
the number of electrons. These expressions will be invaluable in the
evaluation of perturbation-theory and coupled-cluster theory equations,
among others.
