From Density Functional Theory to Spin Hamiltonians: Magnetism in $d^5$ Honeycomb Compound OsCl$_3$

Magnetism in strongly correlated honeycomb systems with $d^5$ electronic configuration has garnered significant attention due to its potential to realize the Kitaev spin liquid state, characterized by exotic properties. However, real materials exhibit not only Kitaev exchange interactions but also other magnetic exchanges, which may drive the transition from a spin liquid phase to a long-range ordered ground state. This work focuses on modelling the effective spin Hamiltonian for two-dimensional (2D) honeycomb magnetic systems with $d^5$ electronic configurations. The Hubbard-Kanamori (HK) Hamiltonian equipped with spin-orbit coupling and electron correlations is considered where onsite energies and hopping parameters, preserving the crystal symmetry, are extracted from the first principles Density functional theory (DFT) calculations. Exact diagonalization (ED) calculations for the HK Hamiltonian on a two-site cluster are performed to construct the effective magnetic Hamiltonian. The ground-state magnetic properties are explored using the semi-classical Luttinger-Tisza approach. As a representative case, the magnetic ground state of the $d^5$ honeycomb system OsCl$_3$ is investigated, and the variation of magnetic exchange parameters with respect to the correlation strength (U) and Hund's coupling ($J_H$) is analyzed. The magnetic ground state exhibits zigzag antiferromagnetic ordering for a chosen value of $U$ and $J_H$, consistent with DFT results. This study provides insight into the magnetism of OsCl$_3$ and offers a computationally efficient alternative to traditional energy-based methods for calculating exchange interactions for strongly correlated systems.