  The spin-1/2 chain with isotropic Heisenberg exchange $J_1$, $J_2 > 0$
between first and second neighbors is frustrated for either sign of J1. Its
quantum phase diagram has critical points at fixed $J_1/J_2$ between gapless
phases with nondegenerate ground state (GS) and quasi-long-range order (QLRO)
and gapped phases with doubly degenerate GS and spin correlation functions of
finite range. In finite chains, exact diagonalization (ED) estimates critical
points as level crossing of excited states. GS spin correlations enter in the
spin structure factor $S(q)$ that diverges at wave vector $q_m$ in QLRO($q_m$)
phases with periodicity $2\pi/q_m$ but remains finite in gapped phases.
$S(q_m)$ is evaluated using ED and density matrix renormalization group (DMRG)
calculations. Level crossing and the magnitude of $S(q_m)$ are independent and
complementary probes of quantum phases, based respectively on excited and
ground states. Both indicate a gapless QLRO($\pi/2$) phase between $-1.2 <
J_1/|J_2| < 0.45$. Numerical results and field theory agree well for quantum
critical points at small frustration $J_2$ but disagree in the sector of weak
exchange $J_1$ between Heisenberg antiferromagnetic chains on sublattices of
odd and even-numbered sites.
