  In this article we investigate the linear-chain spin-half $J_1$--$J_2$ model
by using high-order coupled cluster method (CCM) calculations. We employ three
model states, namely, a nearest-neighbour (n.n.) N\'eel model state in which
neighbouring spins are antiparallel, a next-nearest-neighbour (n.n.n.) N\'eel
model state in which next-neighbouring spins are antiparallel, and finally a
type of "double spiral" model state with {\it two} pitch angles. For $J_2/J_1
\le \frac 12$, we find that the n.n. N\'eel model state produces the lowest
energies. For $J_2/J_1 > \frac 12$, we find that the stable states for the
quantum system are those for the "traditional" spiral state in which the two
pitch angles are identical and the collinear n.n.n. N\'eel model state. We show
that ground-state energies for the collinear n.n.n. model state are lower than
those of the spiral state in a finite region for $J_2/J_1>\frac 12$. We produce
results for the dimer order parameter by using the n.n.n. N\'eel model state
and these results are in reasonable agreement with previous results of DMRG.
Again for the n.n.n. N\'eel model state, it is shown that the correlation
length increases with increasing $J_2/J_1$ above $J_2/J_1=\frac 12$ using this
model state and that shift in the peak of the structure function occurs from
$q=\pi$ to $q=\pi/2$ as one increases $J_2/J_1$. Although these results are not
conclusive, they are intriguing because they suggest that a ground state
exhibiting collinear order might occur for some region for $J_2/J_1 \ge \frac
12$.
