Normalization constant

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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: