  By using the concept of the quantum discord (QD), we study the spin-1/2
antiferromagnetic Heisenberg chain with next-nearest-neighbor interaction. Due
to the SU(2) symmetry and $Z_{2}$ symmetry in this system, we obtain the
analytical result of the QD and its geometric measure (GMQD), which is
determined by the two-site correlators. For the 4-site and 6-site cases, the
connection between GMQD (QD) and the eigenenergies was revealed. From the
analytical and numerical results, we find GMQD (QD) is an effective tool in
detecting both the first-order and the infinite-order quantum-phase-transition
points for the finite-size systems. Moreover, by using the entanglement
excitation energy and a universal frustration measure we consider the
frustration properties of the system and find a nonlinear dependence of the
GMQD on the frustration.
