# Advanced First-Principle Modeling of Relativistic Ruddlesden--Popper Strontium Iridates
## 2 Methodology
### The GW Method  
Thanks to the cRPA method, LSDA+\(U\) becomes de facto a parameter-free method in the sense that the \(U\) value is not taken as an adjustable parameter but calculated from first-principles. The LSDA+\(U\) method has thus been proven to be successful in describing the ground state properties for many correlated materials. However, the LSDA+\(U\) method is still limited to a one-electron single-determinant picture and cannot be expected to correct for inherent shortcomings of LSDA [65]. In particular, its descriptions on excited state properties such as band gaps and optical spectra are often not satisfactory. The \(GW\) approximation to the exchange correlation is derived systematically from many-body perturbation theory using the [PERSON]'s function formalism [66; 67] and provides a good approximation for the self-energy \(\Sigma\) of a many-body system of electrons by including many-body effects in electron-electron interaction. In this method, the self-energy in real space and imaginary time domain is approximated by \(\Sigma=-GW\)[66; 67; 68], where \(G\) is the single-particle Green's function and \(W\) is the fully screened Coulomb interaction calculated by Equation (11) within the RPA. The \(GW\) method has been widely and successfully applied to the calculations of quasiparticle energies for many types of systems (for reviews, see References [69; 70; 71]), and the resulting band gaps are often very close to the experimentally measured values [72; 73; 74; 75; 76; 77; 78; 79]. In practical \(GW\) calculations, the initial \(G\) and \(W\) are usually constructed using the DFT one-electron energies and orbtish. Among many existing different \(GW\) variants (e.g., in the order of increasing complexities, single-shot \(GW\) (\(G_{0}W_{0}\)) [75], eigenvalue self-consistent \(GW\) (\(\mathrm{ev}GW_{0}\)) [78], quasiparticle self-consistent \(GW\) (QSGW) [77; 80], fully self-consistent \(GW\) (\(\mathrm{sc}GW\)) [81; 82; 83; 84], and \(\mathrm{sc}GW\) including vertex corrections [85; 86]), the \(G_{0}W_{0}\) method is usually the most common choice due to its good compromise between accuracy and efficiency. To speed up the \(GW\) calculations and to extend their applications on large systems, several efforts and advances in different directions have been made, e.g., by lowering the number of unoccupied states [87], by completely eliminating the unoccupied states [88; 89; 90], by reducing the scaling with respect to system size [91; 92; 93; 94], and by optimizing existing implementations on current high-performance computing systems [95]. However, one should keep in mind that the reasonably satisfactory accuracy of the \(G_{0}W_{0}\) method mostly arises from error cancellation due to the lack of self-consistency and the absence of vertex corrections. To achieve a desired accuracy, the starting one-electron energies and orbtish become particularly important [96; 97]. A common practice is to choose a suitable exchange-correlation functional that can best represent the ground state of the system under study. For instance, for systems with localized \(d\) or \(f\) states, DFT+\(U\) obtained one-electron energies and orbitals are usually much closer to the ground state as compared to DFT and, hence, is a better starting point for subsequent \(G_{0}W_{0}\) calculations [98; 99]. Thisis the case for iridates where we have shown that DFT+\(U\) with the \(U\) calculated from cRPA predicts very similar band structures to that of \(G_{0}W_{0}\)[100]. In addition to good prediction of quasiparticle energies, in particular, band gaps, the \(GW\) method has also seen wide applications in other aspects. For instance, the \(GW\) method enables an elegant combination with DMFT, i.e., \(GW\)+DMFT [44; 101; 102; 103], since \(GW\) self-energy can be diagrammatically formulated in the same many-body framework as DMFT [48; 49]. Therefore, for \(GW\)+DMFT, one actually knows which Feynman diagrams are counted twice, and \(GW\)+DMFT thus overcomes the fundamental double counting problems in LDA+DMFT [52].