# Advanced First-Principle Modeling of Relativistic Ruddlesden--Popper Strontium Iridates
## 2 Methodology
### The Bethe-Salpeter Equation
With the BSE eigenvectors \(X^{\Lambda}\) and eigenvalues \(\Omega_{\Lambda}\) in the TDA in hand, the frequency-dependent macroscopic dielectric function can be calculated as [100; 123]  
\[ \begin{split}\varepsilon(\omega)=1-&\lim_{\mathbf{q }\to 0}V(\mathbf{q})\sum_{\Lambda}\biggl{(}\frac{1}{\omega-\Omega_{ \Lambda}+\mathrm{i}\eta}-\frac{1}{\omega+\Omega_{\Lambda}-\mathrm{i}\eta} \biggr{)}\\ &\times\Bigg{\{}\sum_{\mathbf{k}}w_{\mathbf{k}}\sum_{\
u,\ell} \langle\psi_{\ell\mathbf{k}}|e^{\mathrm{i}\mathbf{q}\cdot\mathbf{r}}|\psi_{ \
u\mathbf{k}}\rangle X^{\Lambda}_{c\
u\mathbf{k}}\Bigg{\}}\times\Bigg{\{}c.c. \Bigg{\}},\end{split}  \tag{20}\]  
with the oscillator strengths \(S_{\Lambda}\) associated with the optical transitions defined by [100]  
\[S_{\Lambda}=\mathbf{Tr}\Bigg{[}\biggl{\{}\sum_{\mathbf{k}}w_{\mathbf{k}}\sum _{\
u,\ell}\langle\psi_{\ell\mathbf{k}}|e^{\mathrm{i}\mathbf{q}\cdot\mathbf{ r}}|\psi_{\
u\mathbf{k}}\rangle X^{\Lambda}_{c\
u\mathbf{k}}\biggr{\}}\times \bigg{\{}c.c.\Bigg{\}}\Bigg{]}. \tag{21}\]  
where \(V\) is the bare Coulomb interaction, \(\eta\) is a positive infinitesimal, and \(w_{\mathbf{k}}\) is the \(k\)-point weights. \(\psi_{\
u\mathbf{k}}\) and \(\psi_{\ell\mathbf{k}}\) refer to occupied and unoccupied one-electron wave functions, respectively. From \(\varepsilon(\omega)\), the real part of the optical conductivity is then derived by  
\[ \begin{split}\mathrm{Re}[\sigma(\omega)]&=\frac{ \omega}{4\pi}\mathrm{Im}[\varepsilon(\omega)].\end{split}  \tag{22}\]  
Despite of the success of the \(GW\)+BSE approach, its high computational cost due to unfavorable scaling has limited its use to small or moderate systems. Time-dependent density functional theory (TDDFT) [70; 124; 125; 126] provides an alternative approach that has been widely used. In this approach, the interacting polarizability \(\Pi\) is obtained by solving the following two-point Dyson-like equation [70]  
\[\Pi^{-1}=\chi^{-1}-(V+f_{\mathrm{xc}}), \tag{23}\]  
where \(f_{\mathrm{xc}}\) is the exchange-correlation kernel, which is defined as the derivative of the time-dependent exchange-correlation potential \(v_{\mathrm{xc}}\) with respect to the time-dependent density [70]. Comparing Equations (14) and (15), one clearly sees that the difference between TDDFT and BSE lies in the interaction kernels, though the two methods are formulated in different notations (two-point vs. four-point). However, like \(v_{\mathrm{xc}}\), the exact \(f_{\mathrm{xc}}\) is unknown, and hence, how to explore an approximate but appropriate form for \(f_{\mathrm{xc}}\) becomes a central challenging task in the community of TDDFT. Among the existing approximate exchange-correlation kernels, the nanoquanta kernel [127; 128; 129; 130] and bootstrap kernel of [PERSON] [131] are particularly successful in yielding a satisfactory description of the excitonic effects. Nevertheless, the nanoquanta kernel in TDDFT is computationally as expensive as solving the BSE, since the kernel is essentially derived from the BSE and explicit calculations of the two-electron four-orbital integrals are needed. The bootstrap kernel is not derived from first-principles and its description of bound excitons is not satisfactory [132]. To overcome the slow performances of the \(GW\)+BSE while maintaining its accuracy to a large extent, several approaches have been proposed focusing on improving the band gaps by using hybrid-functional calculations and/or improving the screening effects by introducing a material-specific model screening-dependent dielectric function [133; 134; 135; 136; 137; 138].