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In this section we discuss some general considerations concerning which
N-electron basis functions should be included in the CI space (given that,
in the general case, there are too many for us to include all
of them).
Certainly
if we can find a class of N-electron functions which rigorously have zero
Hamiltonian matrix elements with the desired CI wavefunction, then none of
these basis functions will contribute at all to our approximate wavefunction
and they should not be included in the CI space; the Hamiltonian
will be block diagonal, and all the functions in this class will contribute
to the wrong block. We will now prove the following: consider an operator
which commutes with 
. If
and
then
First we show that 
is an eigenfunction of
with eigenvalue 
. Define
Now apply to 
Where we have used the given that 
.
We may now write
Now consider again equation (4.1). If we take the adjoint of
this equation we obtain
Now use the fact that 
(we assumed was
Hermitian) and that the eigenvalues of a Hermitian operator are real. This
yields
Multiply on the right by
Now multiply equation (4.6) on the left by 
to
obtain
If we subtract equation (4.10) from equation (4.9)
we arrive at
Since we assumed 
, then 
.
Recalling the definition of , we have
which was to be proven. Thus if our desired wavefunction is an eigenfunction
of some Hermitian operator that commutes with the Hamiltonian, our CI space
need not include those N-electron functions which are eigenfunctions of
this operator with different eigenvalues.
As an example, consider the spin angular momentum operator 
.
If we want to solve for a state 
of spin S, then we know
and any basis function of a different spin can be excluded from the CI.
Slater determinants are generally not eigenfunctions of .
However, we can take linear combinations of Slater determinants so that we
do have eigenfunctions of ; such functions are generally called
configuration state functions, or CSF's. The advantage of using CSF's is
that we can throw out all functions with the wrong eigenvalue S--they
contribute to another, noninteracting block of the Hamiltonian matrix.
This reasoning also applies to symmetry operations of point groups, such
that we can throw away any N-electron basis functions (whether
determinants or CSF's) which have the wrong irreducible representation. We
can also restrict the basis functions according to their eigenvalues with
respect to the operator 
. For a triplet state, we can perform the
calculation using basis functions which have 
or 1. If we
were to include basis functions of all these values of 
, we
would obtain a triply-degenerate answer--as one should
expect!