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The atomic and magnetic structures of (Cu$X$)LaNb$_2$O$_7$ ($X$=Cl and Br)are investigated using the density-functional calculations. Among severaldozens of examined structures, an orthorhombic distorted $2\times 2$ structure,in which the displacement pattern of $X$ halogens resembles the modelconjectured previously based on the empirical information is identified as themost stable one. The displacements of $X$ halogens, together with those of Cuions, result in the formation of $X$-Cu-$X$-Cu-$X$ zigzag chains in the twomaterials. The nearest-neighbor interaction within the zigzag chains aredetermined to be antiferromagnetic (AFM) for (CuCl)LaNb$_2$O$_7$ butferromagnetic (FM) for (CuBr)LaNb$_2$O$_7$. On the other hand, the first twoneighboring interactions between the Cu cations from adjacent chains are foundto be AFM and FM respectively for both compounds. The magnitudes of all thesein-plane exchange couplings in (CuBr)LaNb$_2$O$_7$ are evaluated to be aboutthree times those in (CuCl)LaNb$_2$O$_7$. In addition, a sizable AFMinter-plane interaction is found between the Cu ions separated by two NbO$_6$octahedra. The present study strongly suggests the necessity to go beyond thesquare $J_1-J_2$ model in order to correctly account for the magnetic propertyof (Cu$X)$LaNb$_2$O$_7$.

Atomic and magnetic structures of (CuCl)LaNb$_2$O$_7$ and (CuBr)LaNb$_2$O$_7$: Density functional calculations