# Advanced First-Principle Modeling of Relativistic Ruddlesden--Popper Strontium Iridates
## 2 Methodology  
### Noncollinear LSDA+\(U\) Model  
For a detailed derivation of noncollinear LSDA+\(U\) model, we refer to Reference [37]. Here, we just recapitulate the most important aspects of the underlying algorithm--the proper model Hamiltonian on which the noncollinear LSDA+\(U\) model is based. To this end, we start from a general on-site Hamiltonian describing the interaction \(V_{ij,lk}\) between electrons occupying orbitals \(i,j,k_{i}\):  
\[\hat{\mathcal{H}}=\frac{1}{2}\sum_{i,j,k,l}\sum_{\sigma^{\dagger}_{\xi}}V_{ij, lk}\hat{c}^{\dagger}_{i,\sigma}\hat{c}^{\dagger}_{j,\sigma}\hat{c}_{k,\sigma}\hat{c}_{l, \sigma}, \tag{1}\]  
where \(\hat{c}_{l,\sigma}\) and \(\hat{c}^{\dagger}_{l,\sigma}\), respectively, represent annihilation and creation operators for the orbital \(l\) and the spin \(\sigma\). Using an analogy with the theory of isotropic elasticity [38], where the four-index matrix of elastic constants \(C_{ijkl}\) has the same symmetry as \(V_{ij,lk}\), the Hamiltonian (1) for the \(p\)-electron cubic harmonic orbital case can be written exactly as [39]  
\[\hat{\mathcal{H}}=\frac{1}{2}\bigg{(}U-\frac{J}{2}\bigg{)}:\hat{N}^{2}:-\frac{ J}{4}:\hat{\mathbf{M}}^{2}:+\frac{J}{2}\sum_{i,j}:(\hat{n}_{ij})^{2}:. \tag{2}\]  
where \(\hat{N}=\sum_{m,\sigma}\hat{c}^{\dagger}_{m,\sigma}\hat{c}_{m,\sigma}\) is the operator of the total number of electrons on a site; \(\hat{n}_{kl}=\sum_{\sigma}\hat{c}^{\dagger}_{k,\sigma}\hat{c}_{l,\sigma}\); and  
\[\hat{\mathbf{M}}=\sum_{m,\hat{c},\hat{c}^{\dagger}}\hat{c}^{\dagger}_{m, \sigma}\hat{c}_{\xi\xi^{\prime}}\hat{c}_{m,\xi^{\prime}}\]is the total magnetic moment vector operator associated with a site. :: denotes normal ordering of creation and annihilation operators, and \(\sigma_{\xi\xi^{\prime}}\) are the Pauli matrices. The normally ordered terms in Equation (2) have the following form [37]:  
\[:\hat{N}^{2}: = \hat{N}^{2}-\hat{N},\] \[:\hat{\mathbf{M}}^{2}: = \hat{\mathbf{M}}^{2}-3\hat{N},\] \[:\hat{M}_{2}^{2}: = \hat{M}_{2}^{2}-\hat{N},\] \[:(\hat{n}_{kl})^{2}: = \sum_{\sigma_{\xi}^{\prime}}\xi_{k,\sigma}^{\dagger}\xi_{k,\xi} \xi_{l,\xi}\xi_{l,\sigma}. \tag{3}\]  
The first two terms in Equation (2) are essentially the Hamiltonian of the collinear Stoner model on which the collinear LSDA+\(U\) models [40; 41; 42] are based  
\[\hat{\mathcal{H}}=\frac{U}{2}\left(\hat{N}^{2}-\hat{N}\right)-\frac{I}{4} \left(\hat{N}^{2}-2\hat{N}\right)-\frac{I}{4}\hat{M}^{2}, \tag{4}\]  
where \(\hat{N}_{\sigma}=\sum_{l}\hat{n}_{l,\sigma}\), \(\hat{N}=\hat{N}_{\uparrow}+\hat{N}_{\downarrow}\), and \(\hat{M}=\hat{N}_{\uparrow}-\hat{N}_{\downarrow}\), with the exception that, in Equation (2), the magnetic moment operator is a vector quantity. However, the Hamiltonian (2) includes an extra third term, required by symmetry and absent in Equation (4). This term is related to the orbital moment of electrons on a lattice site [39]. Following the derivation given in References [40; 42], the noncollinear LSDA+\(U\) model correction from Hamiltonian (2) equals the difference between the exact and mean-field expectation values of these terms, resulting in [37]  
\[E_{\text{LSDA+}U}-E_{\text{LSDA}}=\] \[\left[\frac{1}{2}\left(U-\frac{I}{2}\right)-\frac{I}{4}+\frac{I}{ 2}\right]\sum_{m,\sigma}\left(n_{m,\sigma}-n_{m,\sigma}^{2}\right), \tag{5}\]  
where each term in square brackets corresponds to a respective term in Hamiltonian (2) and \(n_{m\sigma}\) is the electron occupation number of an orbital state \(m\) with spin index \(\sigma\). The last term \((J/2)\sum_{m,\sigma}\left(n_{m,\sigma}-n_{m,\sigma}^{2}\right)\), missing in the derivations given in References [40; 42], results from the last term in Equation (2). It is clear from Equation (5) that the terms containing parameter \(J\) cancel each other exactly and that only the term proportional to parameter \(U\) remains.