We assume here that we are only interested in the ground state of the system and expand the exact wave function in term of a series of Slater determinants
$$ \vert \Psi_0\rangle = \vert \Phi_0\rangle + \sum_{m=1}^{\infty}C_m\vert \Phi_m\rangle, $$where we have assumed that the true ground state is dominated by the solution of the unperturbed problem, that is
$$ \hat{H}_0\vert \Phi_0\rangle= W_0\vert \Phi_0\rangle. $$The state \( \vert \Psi_0\rangle \) is not normalized, rather we have used an intermediate normalization \( \langle \Phi_0 \vert \Psi_0\rangle=1 \) since we have \( \langle \Phi_0\vert \Phi_0\rangle=1 \).