## 2 Theoretical Formulation
### Atomic Structure Calculation
We shall expand the two-electron eigenstates, \(\Psi_{EL}\) (_configuration interaction_, _CI basis_), on a known two-electron basis set (_configuration basis_), \(\Phi_{a}^{(L)}(\mathbf{r}_{1},\mathbf{r}_{2})\):  
\[\Psi_{EL}(\mathbf{r}_{1},\mathbf{r}_{2})=\sum_{a}C_{a}^{(EL)}\Phi_{a}^{(L)}( \mathbf{r}_{1},\mathbf{r}_{2}), \tag{4}\]  
with the intention to express Equation (3) as an alegbraic equation for the \(C_{a}^{EL}\) coefficients. We choose as configuration states, the eigenstates of the \(\hat{H}_{A}^{(0)},\mathbf{L}^{2},\hat{L}_{z},\mathbf{S}^{2},\hat{S}_{z}\), and \(\hat{\Gamma}\) (parity) operators, where \(\hat{H}_{A}^{(0)}=\hat{h}(\mathbf{r}_{1})+\hat{h}_{2}(\mathbf{r}_{2})\), \(\hat{L}=\hat{l}_{1}+\hat{l}_{2}\) and \(\hat{S}=\hat{s}_{1}+\hat{s}_{2}\). \(\hat{l}_{i}\) and \(\hat{s}_{i}\) are the \(i^{th}\) electron's angular and spin quantum operators, respectively. \(\hat{H}_{A}^{(0)}\) is the so-called zero-order two-electron Hamiltonian (of a physically fictitious system where the two-electrons are non-interacting), \(\hat{L}\) is the total angular momentum operator, \(\hat{L}_{z}\) is its projection onto the quantization axis (chosen to be the \(z\)-axis), and \(\hat{S},\hat{S}_{z}\) are the total spin quantum number and its \(z\)-axis projection, respectively. The CI states in Equation (4) are characterized only by their energy, \(E\), and total angular quantum number, \(L\), on the basis that the interaction with the field will not affect the total magnetic quantum number, \(M_{L}\), as well as the total spin, \(S\), and its \(z\)-axis projection, \(M_{S}\). These values will be equal to those of the initial state. For two electron systems, known to have a \({}^{1}\!S\) ground state, it is concluded that only states with \(M_{L}=0\), \(S=0\) and \(M_{S}=0\) are involved in the photoionization process. Accordingly, the zero-order states are fully determined if the set of \(L\) and \(a\equiv(n_{1}l_{1};n_{2}l_{2})\) parameters is given. The respective zero-order energy is equal to \(E_{0}=\epsilon_{1}+\epsilon_{2}\). Therefore, the configuration basis set is comprised of singlet \((S=0)\), spatially antisymmetric, angularly coupled, products of one-electron orbitals with \(m_{l}=0\):  
\[\Phi_{n_{1}l_{1}n_{2}l_{2}}^{(L)}(\mathbf{r}_{1},\mathbf{r}_{2})=C_{000}^{l_{ 1}l_{2}L}\hat{A}_{12}\left[\phi_{n_{1}l_{1}}(\mathbf{r}_{1})\phi_{n_{2}l_{2}}( \mathbf{r}_{2})\right], \tag{5}\]  
where \(\hat{A}_{12}\) is the antisymmetrization operator and \(C_{000}^{l_{1}l_{2}L}\) is the Clebsch-Gordan coefficient that ensures \(M_{L}=0\)[26]. Projection of the above zero-order basis states onto Equation (3) leads again to a matrix diagonalization problem, the solution of which provides the CI energies, \(E\), and CI coefficients, \(C_{a}^{(EL)}\). The physical interpretation of the coefficient is that \(|C_{a}^{(EL)}|^{2}\) represents the contribution of the (un-correlated) configuration, \(\Phi_{a}^{L}\) (characterized by the set \({}^{\prime}a=(n_{1}l_{1};n_{2}l_{2})\)\({}^{\prime}\)), in the formation of the CI state, \(\Psi_{EL}\), with energy \(E\) and angular momentum number \(L\).