Abstract

We present a derivation and computations of the paramagnetic enhancement of the nuclear magnetic resonance (NMR) spin–spin coupling, which may be expressed in terms of the hyperfine coupling (HFC) and (for systems with multiple unpaired electrons) zero-field splitting (ZFS) tensors. This enhancement is formally analogous to the hyperfine contributions to the NMR shielding tensor as formulated by Kurland and McGarvey. The significance of the spin–spin coupling enhancement is demonstrated by using a combination of density-functional theory and correlated ab initio calculations, to determine the HFC and ZFS tensors, respectively, for two paramagnetic 3d metallocenes, a Cr^II(acac)₂ complex, a Co(II) pyrazolylborate complex, and a lanthanide system, Gd–DOTA. Particular attention is paid to relativistic effects in HFC tensors, which are calculated using two methods: a nonrelativistic method supplemented by perturbational spin–orbit coupling corrections, and a fully relativistic, four-component matrix–Dirac–Kohn–Sham approach. The paramagnetic enhancement lacks a direct dependence on the distance between the coupled nuclei, and represents more the strength and orientation of the individual hyperfine couplings of the two nuclei to the spin density distribution. Therefore, the enhancement gains relative importance as compared to conventional coupling as the distance between the nuclei increases, or generally in the cases where the conventional coupling mechanisms result in a small value. With the development of the experimental techniques of paramagnetic NMR, the more significant enhancements, e.g., of the ¹³C¹³C couplings in the Gd–DOTA complex (as large as 9.4 Hz), may eventually become important.