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:

Taking the derivative of with respect to, using the chain rule ,we get:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle + \frac {\partial E_{n}}{\partial \epsilon _{n}} \sum_{k=1}^{+\infty}\lambda ^{k+1} \frac {\partial}{\partial E_{n}} \left ( \sum_{m_{1}}'...\sum_{m_{k}}' \langle n|H'|m_ {1} \rangle \frac {1}{E_{n}- \epsilon _{m_{1}}}\langle m_{1}|H'|m_{2} \rangle.\frac {1}{E_{n}- \epsilon _{m_{2}}}\langle m_{2}|H'|m_{3} \rangle...\frac {1}{E_{n}- \epsilon _{m_{k-1}}}\langle m_{k-1}|H'|m_{k} \rangle.\frac {1}{E_{n}- \epsilon _{m_{k}}}\langle m_{k}|H'|n \rangle \right ) }

From this we can solve for

Now let's evaluate from