  We use first-principles theory to show that the ingredients assumed to be
essential to the occurrence of negative thermal expansion (NTE) -- rigid unit
phonon modes with negative Gr\"{u}neisen parameters -- are neither sufficient
nor necessary for a material to undergo NTE. Instead, we find that NTE in
PbTiO$_3$ involves a delicate interplay between the phonon properties of a
material (Gr\"{u}neisen parameters) and its anisotropic elasticity. These
unique insights open new avenues in our fundamental understanding of the
thermal properties of materials, and in the search for NTE in new materials
classes.
