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Correlation Effects on Magnetic Structure and Lattice Dynamics of LaMn$_7$O$_{12}$: A First-Principles Study

LaMn$_7$O$_{12}$, a quadruple perovskite oxide (AA'$_3$B$_4$O$_{12}$-type), has attracted attention for its notable bifunctional activity in oxygen evolution and reduction reactions. Here, we systematically investigate the magnetic phase diagram and lattice dynamics of LaMn$_7$O$_{12}$ using two density functional theory plus Hubbard U (DFT + U) approaches: the spin-density and the charge-only-density formalism. Phase diagram analysis as a function of U and J shows that both methods stabilize the experimentally observed antiferromagnetic (AFM) configuration (C-type AFM at the B-site and ferrimagnetic structure at the A'-site Mn ions) at U = 3.5 eV and J = 0.8 eV. These U and J values are consistent with those obtained from the constrained random phase approximation. Furthermore, we observe the dynamical stability of the AFM phase through phonon dispersion curves and analyze the Raman-active phonon modes. These results highlight the critical role of appropriate U and J parameters in accurately describing the properties of LaMn$_7$O$_{12}$.