  The development of patch test consistent quasicontinuum energies for
multi-dimensional crystalline solids modeled by many-body potentials remains a
challenge. The original quasicontinuum energy (QCE) has been implemented for
many-body potentials in two and three space dimensions, but it is not patch
test consistent. We propose that by blending the atomistic and corresponding
Cauchy-Born continuum models of QCE in an interfacial region with thickness of
a small number $k$ of blended atoms, a general quasicontinuum energy (BQCE) can
be developed with the potential to significantly improve the accuracy of QCE
near lattice instabilities such as dislocation formation and motion. In this
paper, we give an error analysis of the blended quasicontinuum energy (BQCE)
for a periodic one-dimensional chain of atoms with next-nearest neighbor
interactions. Our analysis includes the optimization of the blending function
for an improved convergence rate. We show that the $\ell^2$ strain error for
the non-blended QCE energy (QCE), which has low order
$\text{O}(\epsilon^{1/2})$ where $\epsilon$ is the atomistic length scale, can
be reduced by a factor of $k^{3/2}$ for an optimized blending function where
$k$ is the number of atoms in the blending region. The QCE energy has been
further shown to suffer from a O$(1)$ error in the critical strain at which the
lattice loses stability. We prove that the error in the critical strain of BQCE
can be reduced by a factor of $k^2$ for an optimized blending function, thus
demonstrating that the BQCE energy for an optimized blending function has the
potential to give an accurate approximation of the deformation near lattice
instabilities such as crack growth.
