The influence of all effective two- and three-particle interactions appearing in second order of perturbation theory was investigated for the 5 d-shell atoms Hf, Ta, W, Re, and Os. The radial parts of the effective operators were treated as free parameters which were determined by a least squares fit of calculated energy levels to the observed ones. An appreciable reduction of the mean deviation was achieved in all spectra considered. A test of the eigenvectors of the effective Hamiltonian was performed by a comparison of calculated gJ values and gJ values, which were measured with high precision by radiofrequency spectroscopy. The agreement found is typically better than a few parts in 104. © 1986 Springer-Verlag.