Phosphorus NMR Chemical Shifts with Self-Interaction Free, Gradient-Corrected DFT

Abstract

We examine the origin of the orbital localization requirement, commonly imposed on effective potential implementations of the Perdew−Zunger (PZ) self-interaction correction (SIC). We demonstrate that the condition arises because of the presence of irreducible off-diagonal Lagrangian multipliers in the coupled PZ eigenequations. Thus, this condition is essential for obtaining an energy-minimizing solution. Further, we report on an implementation of PZ SIC for the generalized gradient approximation (GGA) to the exchange-correlation energy within density functional theory (DFT). The implementation relies on the Krieger−Li−Iafrate (KLI) approximation to the optimized effective potential (OEP), simplifying the evaluation of molecular properties, such as the NMR chemical shifts. We examine several approaches toward incorporating the frozen core orbitals within the SIC−KLI−OEP scheme. To achieve an accurate description of both the energetic and magnetic properties, core orbitals must be included in the KLI potential on an equal footing with the valence orbitals. Implementation of the frozen core orbitals enables incorporation of relativistic effects via the quasirelativistic Pauli Hamiltonian. As the first application of the SIC−GGA approach, we examine 31P NMR chemical shifts of 18 representative small molecules, as well as the previously reported C, H, N, O, and F SIC-LDA (local density approximation) test set. For C, N, O, and F NMR chemical shifts, SIC−GGA performs similarly to SIC−LDA, whereas a significant improvement is observed for hydrogen. Almost identical results, both for the chemical shifts and absolute shieldings, are obtained with different parent GGAs. For the 31P test set, SIC−revPBE (revised Perdew−Burke−Ernzerhof functional of Zhang and Yang) leads to an root-mean-square (RMS) residual error of 23 ppm, compared to 54 ppm for its parent GGA, and 40 ppm for the SIC−VWN (Vosko−Wilk−Nusair) LDA functional. In particular, SIC−revPBE correctly reproduces the experimental trends in the PF3−PCl3−PBr3−PI3 series, which is described qualitatively incorrectly by VWN, revPBE, and SIC−VWN calculations. A similar improvement is observed for the 31P shielding tensor components. The spurious self-interaction, in modern approximate DFT, appears to be a major, and so far largely overlooked, source of errors in calculations of the NMR shielding tensors.