We need to say something about so-called core-polarization effects. To do this, we have to introduce elements from many-body perturbation theory.

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 $\langle \Phi_0\vert \Phi_0\rangle=1$.