Combined cluster and atomic displacement expansion for solid solutions and magnetism

Finite temperature disordered solid solutions and magnetic materials are difficult to study directly using first principles calculations, due to the large unit cells and many independent samples that are required. In this work, we develop a combined cluster expansion and atomic displacement expansion, which we fit to first principles energies, forces, and stresses. We then use the expansion to calculate thermodynamic quantities at nearly first principles levels of accuracy. We demonstrate that by treating all the relevant degrees of freedom (DOF) explicitly, we can in some cases achieve better convergence than a simple cluster expansion, and we can naturally treat coupling between structural DOF and chemical or magnetic DOF. As examples, we use our expansion to calculate properties of Si_1-xGe_x, magnetic MnO, Al with vacancies, and K_xBi_1-xTiO₃.