Analytic energy gradients for the orbital-optimized third-order Moller-Plesset perturbation theory (OMP3) [U. Bozkaya, J. Chem. Phys. 135, 224103 (2011)] are presented. The OMP3 method is applied to problematic chemical systems with challenging electronic structures. The performance of the OMP3 method is compared with those of canonical second-order Moller-Plesset perturbation theory (MP2), third-order Moller-Plesset perturbation theory (MP3), coupled-cluster singles and doubles (CCSD), and coupled-cluster singles and doubles with perturbative triples [CCSD(T)] for investigating equilibrium geometries, vibrational frequencies, and open-shell reaction energies. For bond lengths, the performance of OMP3 is in between those of MP3 and CCSD. For harmonic vibrational frequencies, the OMP3 method significantly eliminates the singularities arising from the abnormal response contributions observed for MP3 in case of symmetry-breaking problems, and provides noticeably improved vibrational frequencies for open-shell molecules. For open-shell reaction energies, OMP3 exhibits a better performance than MP3 and CCSD as in case of barrier heights and radical stabilization energies. As discussed in previous studies, the OMP3 method is several times faster than CCSD in energy computations. Further, in analytic gradient computations for the CCSD method one needs to solve.-amplitude equations, however for OMP3 one does not since lambda(ij(1))(ab) = t(ij)(ab(1)) and lambda(ij(2))(ab) = t(ij)(ab(2)). Additionally,one needs to solve orbital Z-vector equations for CCSD, but for OMP3 orbital response contributions are zero owing to the stationary property of OMP3. Overall, for analytic gradient computations the OMP3 method is several times less expensive than CCSD (roughly similar to 4-6 times). Considering the balance of computational cost and accuracy we conclude that the OMP3 method emerges as a very useful tool for the study of electronically challenging chemical systems. (C) 2013 AIP Publishing LLC.