  I show the infinite-layer rare-earth nickelates are near a structural quantum
critical point by mapping the energetics of their structural instabilities
using first priniciples calculations. I first confirm previous results that
show a phonon instability in the $P4/mmm$ phase leading to the $I4/mcm$
structure for $R$NiO$_2$ with $R$ = Sm--Lu. I then study the non-spin-polarized
phonon dispersions of the $I4/mcm$ phase and find that they exhibit rare-earth
size dependent instabilities at the $X$ and $M$ points for materials with $R$ =
Eu--Lu. Group-theoretical analysis was used to enumerate all the isotropy
subroups due to these instablities, and the distorted structures corresponding
to their order parameters were generated using the eigenvectors of the unstable
phonons. These structures were then fully relaxed by minimizing both the atomic
forces and lattice stresses. I was able to stabilize only five out of the
twelve possible distortions. The $Pbcn$ isotropy subgroup with the $M_5^+(a,a)$
order parameter shows noticeable energy gain relative to other distortions for
the compounds with late rare-earth ions. However, the order parameter of the
lowest-energy phase switches first to $X_2^- (0,a) + M_5^+ (b,0)$ and then to
$X_2^- (0,a)$ as the size of the rare-earth ion is progressively increased.
Additionally, several distorted structures lie close in energy for the early
members of this series. These features of the structural energetics persist
even when antiferromagnetism is allowed. Such a competition between different
order parameters that can be tuned by rare-earth ion substitution suggests that
any structural transition that could arise from the phonon instabilities
present in these materials can be suppressed to 0 K.
