Since the normalization of \( \Psi_0 \) is at our disposal and since \( C_0 \) is by hypothesis non-zero, we may arbitrarily set \( C_0=1 \) with corresponding proportional changes in all other coefficients. Using this so-called intermediate normalization we have
$$ \langle \Psi_0 | \Phi_0 \rangle = \langle \Phi_0 | \Phi_0 \rangle = 1, $$resulting in
$$ |\Psi_0\rangle=(1+\hat{C})|\Phi_0\rangle. $$