  Here we propose a new approach for performing a Taylor series expansion of
the first-principles computed energy of a crystal as a function of the nuclear
displacements. We enlarge the dimensionality of the existing displacement space
and form new variables (ie. slave modes) which transform like irreducible
representations of the space group and satisfy homogeneity of free space.
Standard group theoretical techniques can then be applied to deduce the
non-zero expansion coefficients a priori. At a given order, the translation
group can be used to contract the products and eliminate terms which are not
linearly independent, resulting in a final set of slave mode products. While
the expansion coefficients can be computed in a variety of ways, we demonstrate
that finite difference is effective up to fourth order. We demonstrate the
power of the method in the strongly anharmonic system PbTe. All anharmonic
terms within an octahedron are computed up to fourth order. A proper unitary
transformation demonstrates that the vast majority of the anharmonicity can be
attributed to just two terms, indicating that a minimal model of phonon
interactions is achievable. The ability to straightforwardly generate
polynomial potentials will allow precise simulations at length and time scales
which were previously unrealizable.
