On the difference between the pyroxenes LiFeSi$_{2}$O$_{6}$ and LiFeGe$_{2}$O$_{6}$ in their magnetic structures and spin orientations

The clinopyroxenes LiFeSi$_{2}$O$_{6}$ and LiFeGe$_{2}$O$_{6}$, crystallizing in a monoclinic space group P2$_{1}$/c, are isostructural and isoelectronic Their crystal structures are made up of zigzag chains of edge-sharing FeO$_{6}$ octahedra containing high-spin Fe3$+$ ions, which run along the c direction. Despite this structural similarity, the two have quite different magnetic structures and spin orientations. In LiFeSi$_{2}$O$_{6}$ the Fe spins have a ferromagnetic coupling within the zigzag chains along $c$ and such FM chains have an antiferromagnetic coupling along $a$. In contrast, in LiFeGe$_{2}$O$_{6}$, the spins have an AFM coupling within the zigzag chains along $c$ and such FM chains have an $\uparrow \uparrow \downarrow \downarrow $ coupling along $a$. In addition, the spin orientation is parallel to $c$ in LiFeSi$_{2}$O$_{6}$, but is perpendicular to $c$ in LiFeGe$_{2}$O$_{6}$. To explain these differences in the magnetic structure and spin orientation, we evaluated the spin exchange parameters by performing energy mapping analysis based on LDA$+$U and GGA$+$U calculations and also by evaluating the magnetocrystalline anisotropy energies in terms of GGA$+$U$+$SOC and LDA$+$U$+$SOC calculations. Our study show that the magnetic structures and spin orientations of LiFeSi$_{2}$O$_{6}$ and LiFeGe$_{2}$O$_{6}$ are better described by LDA$+$U and LDA$+$U$+$SOC calculations.