  The nature of the lattice instability connected to the structural transition
and superconductivity of (Sr,Ca)$_3$Ir$_4$Sn$_{13}$ is not yet fully
understood. In this work density functional theory (DFT) calculations of the
phonon instabilities as a function of chemical and hydrostatic pressure show
that the primary lattice instabilities in Sr$_3$Ir$_4$Sn$_{13}$ lie at phonon
modes of wavevectors $\mathbf{q}=(0.5,0,0)$ and $\mathbf{q}=(0.5,0.5,0)$.
Following these modes by calculating the energy of supercells incorporating the
mode distortion results in an energy advantage of -14.1 meV and -9.0 meV per
formula unit respectively. However, the application of chemical pressure to
form Ca$_3$Ir$_4$Sn$_{13}$ reduces the energetic advantage of these
instabilities, which is completely removed by the application of a hydrostatic
pressure of 35 kbar to Ca$_3$Ir$_4$Sn$_{13}$. The evolution of these lattice
instabilities is consistent with experimental phase diagram. The structural
distortion associated with the mode at $\mathbf{q}=(0.5,0.5,0)$ produces a
distorted cell with the same space group symmetry as the experimentally refined
low temperature structure. Furthermore, calculation of the deformation
potential due to these modes quantitatively demonstrates a strong
electron-phonon coupling. Therefore, these modes are likely to be implicated in
the structural transition and superconductivity of this system.
