  By using the coupled cluster method, the numerical exact diagonalization
method, and the numerical density matrix renormalization group method, we
investigated the properties of the one-dimensional Heisenberg chain with
alternating antiferromagnetic and ferromagnetic next nearest neighbor
interactions. In the classical limit, the ground state is in the collinear Neel
state if a<1/2, while for a>1/2, there is an noncollinear canted state. For the
quantum case, we found that, although the classical Neel state is absent, the
canted state exists if the frustration parameter a exceeds a critical point
ac1. The precise critical point ac1 can be determined by using the coupled
cluster method and the numerical exact diagonalization method separately. The
results of the coupled cluster method and the exact diagonalization method both
disclose that the type of phase transition occurring at ac1 changes from a
classical second-order transition to a quantum first-order transition due to
quantum fluctuation. Although there is another critical point ac2 in a finite
system at which the ground state evolves from the canted state to the collinear
Neel plus ferromagnetic state, that state is absent because ac2 tends to
infinity in the thermodynamic limit.
