Using Brillouin-Wigner perturbation theory we will proof that
In this theory, the exact state and exact energy can be written as follows:
where
does not allow the running indexes equal to n.
Taking the derivative of
with respect
to, using the chain rule ,we get:
From this we can solve for
Now let's evaluate
from
We have
, therefore the summing over
is equivalent to setting
. We get:
Let's define
and exchange the indexes as follows:
Doing so we can see that
exactly equals to
given in (2). Therefore: