The development and implementation of sum-over-states density-functional-perturbation theory (SOS-DFPT) [V.G. Malkin, O.L. Malkina, M.E. Casida, and D.R. Salahub, J. Am. Chem. Soc. 116, 5898 (1994)] has allowed a significant improvement in the accuracy of nuclear magnetic resonance (NMR) chemical shift values over the Hartree–Fock approximation. Furthermore, due to its computational efficiency, SOS-DFPT has opened the way to the study of systems of increased size compared to those that may be approached by more sophisticated but also computationally more intensive methods, such as Møller–Plesset perturbation theory or coupled-cluster theory. The success of SOS-DFPT relies on the introduction of an ad hoc correction to the excitation energy that improves the calculation of the paramagnetic component of the NMR shielding tensor. The lack of a clear physical basis for this approximation has left the SOS-DFPT open to some criticism. We have shown in a previous article [E. Fadda, M.E. Casida, and D.R. Salahub, Int. J. Quantum Chem. 91, 67 (2003)] that the electric field and magnetic field responses are given by equivalent expressions within the Tamm–Dancoff approximation of time-dependent density-functional theory (TD-DFT). This provides an SOS-DFPT expression which, upon restriction to diagonal contributions, yields a new rigorous “Loc.3” approximation. In this article, we more than double our original test set of 10 molecules for 13C, 15N, and 17O chemical shifts to a set of 25 molecules. In addition, we compare the results of “Loc.3” SOS-DFPT with the results of promising recent functionals for DFT calculations of chemical shifts. The results show not only that the “Loc.3” approximation represents the rigorous physical connection between SOS-DFPT and TD-DFT, but also that it has very good potential for the prediction of NMR shielding constants, opening the way to further developments in DFT-based NMR parameter calculations.