Study of the magnetic structures of the atacamite and botallackite forms of Cu$_{2}$(OH)$_{3}$Cl in terms of first principles DFT calculations

The atacamite and botallackite forms of Cu$_{2}$Cl(OH)$_{3}$ are made up of edge- and corner-sharing Cu(OH)$_{4}$Cl$_{2}$ octahedra. Each polymorph consists of two slightly different types of Cu(OH)$_{4}$Cl$_{2}$ octahedra, with their Cu x$^{2}$-y$^{2}$ magnetic orbitals contained in the Cu(OH)$_{4}$ square planes. Atacamite and botallackite are different in the way the Cu(OH)$_{4}$ square planes are connected, but their magnetic properties of atacamite and botallackite are quite similar. To explain these observations, the crystal structures of the two polymorphs were optimized by first principles DFT calculations. We then evaluated the spin exchange interactions of the two polymorphs using the optimized structures on the basis of DFT calculations. To a first approximation, both polymorphs were found to be described by a uniform 1D antiferromagetic chain model with spin frustration arising from the next-nearest-neighbor interactions. Implications of these observations were explored.