One particularly nice feature of the CI method is that the calculated
lowest energy eigenvalue is always an upper bound to the exact ground
state energy. Our approximate wavefunction 
can always be expressed as a
linear combination of the exact nonrelativistic
eigenvectors 
, which span the entire
N-electron space.
and the energy is given by
Now if is normalized, and we substitue the expansion over
exact eigenfunctions (equation 3.9) into the equation above,
we obtain
where 
is the ith energy eigenvalue, i.e.\
.
Now subtract 
,
the exact nonrelativistic ground state energy, from both sides to obtain
or
since is normalized and 
.
Since are greater than or equal to for all
values of i and the coefficients 
are necessarily non-negative,
the right hand side of equation (3.12) is non-negative, so that
. We should also point out that the variational theorem
holds for the Hartree-Fock method as well as for CI, since equation
(3.10) is valid for the Hartree-Fock energy--for a given set of
MO's, the HF energy can be formulated as a (trivial) 1 x 1 CI eigenvalue
problem. In a similar manner, the MCSCF method (where MO's and
CI coefficients are optimized) is also ``variational.''
It should be clear that instead of using the exact nonrelativistic
eigenfunctions in equation (3.9), we could also
have used an expansion over the exact eigenfunctions within the
one-electron space spanned by (i.e. we could expand the
approximate CI wavefunction
in terms of the full CI wavefunctions).
This means that not only is the approximate energy an upper bound to the
exact nonrelativistic ground-state energy, but it is also an upper bound
to the full CI energy in the given one-electron basis.