In this section, we parallel the arguments of Pauling and Wilson
[14], p. 186.
So far, we have shown only that the energy calculated as the expectation
value of some trial function must be an upper bound to the true ground
state energy 
. In certain cases, we may derive a similar
result for other states. If we take a trial function 
such
that the first k coefficients in equation (3.9) are zero, then
we may subtract 
from equation (3.11) to obtain
Now since we've assumed 
, this simplifies to
Once again, we can see that every term on the right side is nonnegative,
so 
.
There are any number of cases in which we have a trial function of the
form just described. Consider, for example, a calculation on a triplet
state for a molecule which has a singlet ground state. If our trial
function is constrained to be a triplet, then all singlet eigenfunctions
will have zero coefficients in the expansion of the
trial function. In this case, the energy we minimize from the triplet
trial function will be an upper bound to the lowest triplet energy,
even though there is a lower-lying singlet state. Similar arguments can
be made for spatial symmetry.