Normalization constant

Using Brillouin-Wigner perturbation theory we will proof that $Z=\langle N|N\rangle ^{-1}={\frac {\partial E_{n}}{\partial \epsilon _{n}}}$

In this theory, the exact state and exact energy can be written as follows:

$|N\rangle =|n\rangle +\sum _{k=1}^{+\infty }\lambda ^{k}\left(\sum _{m_{1}}'...\sum _{m_{k}}'|m_{1}\rangle {\frac {1}{E_{n}-\epsilon _{m_{1}}}}\langle m_{1}|H'|m_{2}\rangle .{\frac {1}{E_{n}-\epsilon _{m_{2}}}}\langle m_{2}|H'|m_{3}\rangle ...{\frac {1}{E_{n}-\epsilon _{m_{k-1}}}}\langle m_{k-1}|H'|m_{k}\rangle .{\frac {1}{E_{n}-\epsilon _{m_{k}}}}\langle m_{k}|H'|n\rangle \right)$

$E_{n}=\epsilon _{n}+\lambda \langle n|H'|n\rangle +\sum _{k=1}^{+\infty }\lambda ^{k+1}\left(\sum _{m_{1}}'...\sum _{m_{k}}'\langle n|H'|m_{1}\rangle {\frac {1}{E_{n}-\epsilon _{m_{1}}}}\langle m_{1}|H'|m_{2}\rangle .{\frac {1}{E_{n}-\epsilon _{m_{2}}}}\langle m_{2}|H'|m_{3}\rangle ...{\frac {1}{E_{n}-\epsilon _{m_{k-1}}}}\langle m_{k-1}|H'|m_{k}\rangle .{\frac {1}{E_{n}-\epsilon _{m_{k}}}}\langle m_{k}|H'|n\rangle \right)$