NMR is among the most versatile tools to probe chemical bonding. Paramagnetic NMR (pNMR) refers to the NMR shifts in paramagnetic systems such as open-shell metal complexes, which are very different from their diamagnetic counterparts and poorly understood—despite decades of research [2,3]. The pNMR shift may be split (using pseudo-relativistic terminology) into a contact, a spin-dipolar (or ‘pseudo-contact’), and a paramagnetic spin—orbital (PSO) term. The dipolar term arises from the dipolar magnetic interaction between the electron spin magnetic moment of the paramagnetic center and the magnetic moment of the probe nucleus. The contact term is caused by a non-zero spin density at the probe nucleus. For a ligand atom in a paramagnetic metal complex, the contact term probes the spin density delocalization between the metal center and the ligand, and the spin polarization, and therefore the degree of metal-ligand covalency [1,4]. The PSO mechanism probes the magnetic interaction of the nucleus with the electron orbital angular momentum, if present. For actinide complexes, PSO can be dominant [5]. The main issue is that reliable theoretical approaches are urgently needed in order to advance the broad use of pNMR, for example to quantify experimentally the degree of covalency of the metal-ligand bonds. Without reliable theoretical calculations, the assignment of experimental pNMR spectra is very difficult, or simply not possible.
First principles theoretical calculations of pNMR shifts are extremely challenging, since they confront several of the grand challenges of quantum chemistry : i) open-shell systems, in particular when they are paramagnetic, require a treatment by multi-reference methods. ii) for systems with heavy elements, such as d- and f-block metals that are of very high interest in chemistry, physics, and materials science, relativistic effects must be included in the calculations, both scalar and spin-orbit. iii)The pNMR shift is a response property, and therefore very sensitive to the dynamic electron correlation. Addressing points i), ii), and iii) simultaneously is a major frontier in contemporary quantum chemistry.The calculation of pNMR shifts adds two more difficulties: the coupling with the nuclear spin momenta through the hyperfine coupling and the delocalisation of the spin density on the whole molecule in order to describe correctly the contact term. It necessitates an accurate description of both the anisotropic magnetic properties of the paramagnetic center and of the delocalisation of the spin density on the ligands, since the contact shielding is caused by a non-zero spin density at the nucleus of interest. The benefits of addressing these issues goes well beyond calculating pNMR shifts.
The first quantum-chemical methods to calculate pNMR shifts that went beyond the pure contact shifts were only devised in the early 2000s and were based on density functional theory (DFT) calculations [6,7,8]. They are mostly suited for the description of transition metal complexes, were the ground state manifold is a pure spin state split by spin-orbit coupling. The modeling of pNMR relies typically on approximate expressions for the chemical shift in terms of electron paramagnetic resonance (EPR) pseudo-spin Hamiltonian parameters (g and hyperfine coupling tensors) or the magnetic susceptibility. The former approach misses the coupling with the excited states which may be the leading terms in the case of low lying states (Van Vleck contribution) while the latter is restricted to the description of the dipolar term. Some authors [7] have augmented DFT-based calculations with the aforementioned coupling terms and multi-reference g-tensor calculations, but severe limitations remain due to the underlying DFT calculations of the hyperfine coupling tensors. Soncini et al. recently derived the NMR shielding tensor as a temperature-dependent bilinear derivative of the Helmholtz free energy, providing a very general expression [9]. Multi-reference wavefunction-based pNMR calculations based on Soncini’s expression, without recourse to a spin Hamiltonian mapping, have been reported in the meantime. However, these calculations are far from routine, and presently suffer from a poor description of the spin-polarization (which is very sensitive to the dynamic correlation) [5,10].
DFT based approaches are suitable to describe the spin polarization-delocalization in a paramagnetic system, but they are limited to states that can be described reasonably well with a single determinant, and the magnetic coupling between states requires an elaborate treatment in a response theory setup. Wave Function Theory (WFT) based approaches describe correctly the magnetic properties of a paramagnetic metal center, but with the currently available approaches it is difficult to treat the spin polarization-delocalization. One may foresee range-separation approaches which combine the advantages of DFT and WFT [11] or DMRG methods [12] which permit to handle large active spaces for a proper description of pNMR shifts, or a treatment of the spin polarization via perturbation theory.