  We give an analysis of the stability and displacement error for linear and
circular atomistic chains in the plane when the atomistic energy is
approximated by the Cauchy-Born continuum energy and by the quasi-nonlocal
atomistic-to-continuum coupling energy. We consider atomistic energies that
include Lennard-Jones type nearest neighbor and next nearest neighbor
pair-potential interactions. Previous analyses for linear chains have shown
that the Cauchy-Born and quasi-nonlocal approximations reproduce (up to the
order of the lattice spacing) the atomistic lattice stability for perturbations
that are constrained to the line of the chain. However, we show that the
Cauchy-Born and quasi-nonlocal approximations give a finite increase for the
lattice stability of a linear or circular chain under compression when general
perturbations in the plane are allowed. We also analyze the increase of the
lattice stability under compression when pair-potential energies are augmented
by bond-angle energies. Our estimates of the largest strain for lattice
stability (the critical strain) are sharp (exact up to the order of the lattice
scale). We then use these stability estimates and modeling error estimates for
the linearized Cauchy-Born and quasi-nonlocal energies to give an optimal order
(in the lattice scale) {\em a priori} error analysis for the approximation of
the atomistic strain in $\ell^2_\epsilon$ due to an external force.
