# Advanced First-Principle Modeling of Relativistic Ruddlesden--Popper Strontium Iridates
## 2 Methodology
### The Bethe-Salpeter Equation  
From an ab initio perspective, the established state-of-the-art method for calculating neutral excitation energies and optical spectra is to solve the Bethe-Salpeter equation (BSE) for full reducible two-particle polarizability \(\Pi\) by solving a four-point Dyson-like equation [70; 106; 107; 108; 110; 111; 112]:  
\[\Pi^{-1}=\chi^{-1}-K, \tag{14}\]  
where \(\chi\) is the independent irreducible two-particle polarizability and \(K\) is the interaction kernel, which in the \(W\) approximation (\(\delta\Sigma/\delta G\approx-W\)) can be expressed as [70; 106; 107; 108; 110; 111; 112]:  
\[K=V-W, \tag{15}\]  
where \(V\) is the unscreened Coulomb interaction and \(W\) is the screened exchange interaction. Diagrammatically, if the ladder diagrams caused by \(W\) are not considered, then the BSE in Equation (14) reduces to the RPA. By taking into account both the bubble and ladder diagrams, the BSE explicitly accounts for the excitonic effects resulting from electron-hole interactions. For reviews on BSE, we refer to References [70; 113; 114]. In order to obtain precise peak positions in optical spectra, accurate calculations of quasiparticle energies are important, in particular, the size of the band gap. In addition, the electron-hole interactions have to be properly described. Hence, the natural procedure to calculate the optical spectra consists in the solution of the BSE using the \(GW\) quasiparticle (QP) energies, adopting the static screening \(W\) calculated from the RPA for the electron-hole interactions [70]. This approach is referred to as \(GW\)+BSE and has been widely used to predict the optical spectra of various systems, e.g., molecules, clusters, semiconductors, and insulators [115; 116; 117; 118; 119; 120; 121; 122]. In practice, the BSE is recast into the following generalized eigenvalue problem [70; 123]:  
\[ \begin{pmatrix}\mathbf{A}&\mathbf{B}\\ \mathbf{B}^{*}&\mathbf{A}^{*}\end{pmatrix}\begin{pmatrix}\mathbf{X}\\ \mathbf{Y}\end{pmatrix} =\Omega \begin{pmatrix}\mathbf{I}&0\\ 0&-\mathbf{I}\end{pmatrix}\begin{pmatrix}\mathbf{X}\\ \mathbf{Y}\end{pmatrix} , \tag{16}\]  
where matrices \(\mathbf{A}\) and \(\mathbf{B}\) read [123]  
\[A_{ai,bj}=(\epsilon_{a}-\epsilon_{i})\delta_{ij}\delta_{a,b}+\langle ib\mid K \mid aj\rangle, \tag{17}\]  
\[B_{ai,bj}=\langle ij\mid K\mid ab\rangle. \tag{18}\]  
where indices \(i,j\) and \(a,b\) represent occupied and unoccupied states, respectively. \(\Omega\) is the eigenvalues of the BSE, i.e., the excitation energies of the system. \(X\) and \(Y\) are the two-particle electron-hole eigenstates in the product basis for the electron-hole pairs: \(\psi_{i}(\mathbf{r})\psi_{a}^{*}(\mathbf{r}^{\prime})\) and \(\psi_{a}(\mathbf{r})\psi_{i}^{*}(\mathbf{r}^{\prime})\)  
\[\langle(\mathbf{r},\mathbf{r}^{\prime})|X,Y\rangle=\sum_{i,a}[X_{ia}\psi_{i}( \mathbf{r})\psi_{a}^{*}(\mathbf{r}^{\prime})+Y_{ia}\psi_{a}(\mathbf{r})\psi_{i }^{*}(\mathbf{r}^{\prime})]. \tag{19}\]  
Obviously, Equation (16) is a complex non-Hermitian eigenvalue problem that is difficult to solve using standard eigenvalue solvers. A common practice to avoid this difficulty in solid-state physics is to neglect the resonant-antiresonant coupling \(\mathbf{B}\), thereby reducing the problem to a Hermitian eigenvalue problem. This is known as the Tamm-Dancoff approximation (TDA). Although the full solution in Equation (16) that is beyond the TDA can be recast into a solvable quadratic Hermitian eigenvalue problem by using the time-inversion symmetry [123], it was shown that the TDA shows tiny differences in the optical spectra of solids compared to the full solution of the BSE [123]. In addition, going beyond TDA is technically intractable, when the spin-orbit coupling is considered [123]. Hence, for the iridates under study where the SOC plays important roles, the TDA has to be employed [100].