# Advanced First-Principle Modeling of Relativistic Ruddlesden--Popper Strontium Iridates
## 2 Methodology
### Constrained Random Phase Approximation  
Initially, the cRPA method [44; 45; 46] was designed to calculate the \(U\) value in order to merge density functional theory (DFT) with dynamical mean field theory (DMFT) [47; 48; 49] for the low energy part of the effective electronic Hamiltonian [50; 51; 52]. Since the DFT+\(U\) method can be regarded as the static limit of DFT+DMFT in terms of self-energy, the cRPA method has also been used in the DFT+\(U\) method as a way to reduce dependence on the parameter \(U\) in routine DFT+\(U\) calculations. Within the framework of cRPA [44; 45; 46; 53], the electronic system was subdivided into two subspaces. One subspace describes the correlated degrees of freedom, which are dominant close to the Fermi level (called target correlated subspace), and the remaining part acts as an effective medium, which screens the interaction between the correlated electrons in the correlated subspace. To avoid double counting, all polarization effects between the correlated electrons \(\chi^{c}\) that have already been accounted for by the \(+U\) term in DFT+DMFT (or DFT+\(U\)) were removed from the total irreducible polarizability \(\chi\), leading to the rest polarizability \(\chi^{r}\)[45; 46; 53]  
\[\chi^{r}=\chi-\chi^{c}. \tag{9}\]  
The partially screened Coulomb interaction \(U\) was then calculated in the random phase approximation (RPA) but using a \"constrained\" polarizability, i.e., the rest polarizability \(\chi^{r}\)[45; 46; 53]  
\[U^{-1}=V^{-1}-\chi^{r}, \tag{10}\]  
where \(V\) is the bare (unscreened) Coulomb interaction. From Equations (9) and (10), it is trivial to show that the fully screened interaction \(W\) can be recovered by  
\[W^{-1}=V^{-1}-\chi=U^{-1}-\chi^{c}. \tag{11}\]  
Finally, the matrix elements of \(U\) were evaluated by [45; 46; 53]  
\[U_{ijkl}(\omega)=\iint d\mathbf{r}d\mathbf{r}^{\prime}w_{i}^{*}(\mathbf{r})w _{j}^{*}(\mathbf{r}^{\prime})U(\mathbf{r},\mathbf{r}^{\prime},\omega)w_{k}( \mathbf{r})w_{l}(\mathbf{r}^{\prime}), \tag{12}\]  
where \(w(\mathbf{r})\) is the atomic-like local orbtials such as the maximally localized Wannier functions [54; 55]. Clearly, the partially screened interaction is essentially frequency-dependent. Its dynamical effects have been discussed in the DFT+DMFT and \(GW\)+DMFT methods [56; 57]. For the DFT+\(U\) method, normally, the orbital-averaged value of static \(U\) is adopted [53]:  
\[U=\frac{1}{N^{2}}\sum_{i}^{N}\sum_{j}^{N}U_{ijij}(\omega=0), \tag{13}\]  
where \(N\) is the number of local orbtials within the correlated subspace. A final note on cRPA is in place here. If the target correlated states form an isolated manifold around the Fermi level, the removal of \(\chi^{c}\) from \(\chi\) in Equation (9) can be performed straightforwardly either in the plane wave basis [46] or in the Wannier function basis [58; 59]. However, when the correlated states are strongly entangled with those non-correlated(usually \(s\) or \(p\)) states of the system, the target correlated subspace becomes not trivially defined and evaluation of \(\chi^{c}\) must be treated carefully. To address this problem, [PERSON] and [PERSON] [60] proposed a method that disentangles the states around the Fermi level such that a minimal basis set in the Bloch domain within a given energy window is developed. In this method, the calculated effective interactions depend strongly on the choice of the energy window [60]. In order to achieve a good representation of the entangled bands, additional non-correlated \(s/p\)-like states were suggested to be included in the Wannier projection and the effective interaction was calculated using a weighted polarizability, where the weights were defined as probabilities for Bloch states being correlated [61; 62; 63].