Calculation of the biquadratic spin interactions based on the spin cluster expansion for \textit{ab initio} tight-binding models

We develop a calculation scheme using \textit{ab initio} tight-binding Hamiltonians to evaluate biquadratic magnetic interactions. This approach relies on the spin cluster expansion combined with the disordered local moment (DLM) method, originally developed within the multiple scattering Korringa-Kohn-Rostoker method. Applying it to a single-orbital Hubbard model with two sublattices, we show that the evaluated DLM biquadratic interactions are in good agreement with those obtained from the strongly correlated limit, demonstrating the wide applicability of the method to various magnetic systems with large local moments. We then apply it to the \textit{ab initio} tight-binding models for elemental magnetic metals; the resulting magnetic interactions align well with previous literature. Finally, we explore its performance in more complex compounds, such as transition metal dichalcogenides with intercalation of 3\textit{d} transition metals and potassium electrosodalite. The obtained results for both compounds show good agreement with experiments. The present approach offers a convenient \textit{ab initio} path for evaluating biquadratic interactions and understanding the electronic mechanisms controlling them.