Analytic energy gradients for the orbital-optimized second-order Møller-Plesset perturbation theory (OMP2) are presented. The OMP2 method is applied to difficult chemical systems, including those where spatial or spin symmetry-breaking instabilities are observed. The performance of the OMP2 method is compared with that of second-order Møller-Plesset perturbation theory (MP2) for investigating geometries and vibrational frequencies of the cis-HOOH(+), trans-HOOH(+), LiO2, C3(+), and NO2 molecules. For harmonic vibrational frequencies, the OMP2 method eliminates the singularities arising from the abnormal response contributions observed for MP2 in case of symmetry-breaking problems, and provides significantly improved vibrational frequencies for the above molecules. We also consider the hydrogen transfer reactions between several free radicals, for which MP2 provides poor reaction energies. The OMP2 method again exhibits a considerably better performance than MP2, providing a mean absolute error of 2.3 kcal mol(-1), which is more than 5 times lower than that of MP2 (13.2 kcal mol(-1)). Overall, the OMP2 method seems quite helpful for electronically challenging chemical systems such as symmetry-breaking molecules, hydrogen transfer reactions, or other cases where standard MP2 proves unreliable. For such systems, we recommend using OMP2 instead of MP2 as a more robust method with the same computational scaling.