Electron correlation methods other than CI may not be variational. For example, consider the coupled-cluster energy expression
If the operator 
is not trunctated, then we know that
. Generally, however, the
operator is truncated. Let us define 
for our truncated 
. Now define
. Note that in general
, which would have occured had
we used 
on the left. Then the
energy expression is
which, after expansion over the complete set of eigenvectors, becomes
This simplifies to
At this point we can go no farther, because the terms 
may
be negative, in contrast to the situation in equation ( 3.12).
For completeness, we also show that MBPT energies are not variational. The nth order MBPT wavefunction may be written [12] as
where the sum is over ``linked diagrams'' only. The nth order energy is then given by
Since this integral is not symmetric, the energy is not variational.
Only the first-order perturbation theory energy (which is also the
Hartree-Fock energy) is variational, since it uses
.
