Atomic eigenvalue solver

The JASMINE component of iQIST software package is used to solve a local atomic Hamiltonian defined as,

\[\begin{equation} \hat{H}_{\text{atom}}=\hat{H}_{0}+\hat{H}_{\text{Coulomb}}, \end{equation}\]

where, $\hat{H}_{0}$ is the on-site term, and $\hat{H}_{\text{Coulomb}}$ is the Coulomb interaction term.

The $\hat{H}_{0}$ term includes the crystal field (CF) splitting term $\hat{H}_{\text{CF}}$, spin-orbit coupling (SOC) term $\hat{H}_{\text{SOC}}$, and other terms such as a Zeeman term in case of the presence of external magnetic field,

\[\begin{equation} \hat{H}_{0}=\hat{H}_{\text{CF}}+\hat{H}_{\text{SOC}}+\text{other on-site terms}, \end{equation}\]

We can re-write the $\hat{H}_{\text{atom}}$ in second quantization form,

\[\begin{equation} \hat{H}_{\text{atom}}=\sum_{\alpha\beta}E_{\alpha\beta}\hat{f}_{\alpha}^{\dagger}\hat{f}_{\beta} +\sum_{\alpha\beta\gamma\delta}U_{\alpha\beta\gamma\delta}\hat{f}_{\alpha}^{\dagger} \hat{f}_{\beta}^{\dagger}\hat{f}_{\delta}\hat{f}_{\gamma}, \end{equation}\]

where, $\alpha,\beta,\gamma,\delta$ is the single particle orbital-spin index, the first term is the on-site term, the second one is the Coulomb interaction term.