Based on a special variant of the plaquette expansion, an operator is constructed whose eigenvalues give the low-energy singlet spectrum of a spin-1/2 Heisenberg antiferromagnet on a square lattice with nearest-heighbor and frustrating next-nearest-neighbor exchange couplings J 1 and J 2. It is well known that a nonmagnetic phase arises in this model for 0.4 ≲ J 2/ J 1 ≲ 0.6, sandwiched by two Néel ordered phases. In agreement with previous results, we observe a first-order quantum phase transition (QPT) at J 2 ≈ 0.64 J 1 from the non-magnetic phase to the Néel one. A large gap (≳ 0.4 J 1) is found in the singlet spectrum for J 2 < 0.64 J 1, which excludes a gapless spin-liquid state for 0.4 ≲ J 2/ J 1 ≲ 0.6 and the deconfined quantum criticality scenario for the QPT to another Néel phase. We observe a first-order QPT at J 2 ≈ 0.55 J 1, presumably between two nonmagnetic phases.