  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 explicitly, we can achieve improved convergence properties as compared
to a simple cluster expansion, and our model naturally includes both
configurational and vibrational entropy. In addition, we can treat coupling
between structural and chemical or magnetic degrees of freedom. As examples, we
use our expansion to calculate properties of Si$_{1-x}$Ge$_x$, magnetic MnO, Al
with vacancies, and Ba$_x$Sr$_{1-x}$TiO$_3$.
