Relativistic Theory of EPR and (p)NMR

Paramagnetic Effects in NMR Spectroscopy of Transition-Metal Complexes: Principles and Chemical Concepts

“…14,15 This mapping of pNMR to EPR tensors provides us with a more natural language for chemical analysis, as it allows the use of DFT methodologies and it rationalizes the separation of the total pNMR shift into orbital and hyperfine terms; see eq 1. We refer the reader to a recent review for a discussion of the latest theoretical understanding of pNMR theory 12 and to examples of application of the pNMR theory to systems with higher than doublet multiplicity or low-lying excited states. 16−21 In the case of vanishing or negligible zero-field splitting (ZFS), 22 and when the degenerate ground state with 2S + 1 multiplicity is well separated from any excited states, the hyperfine contribution to the isotropic NMR shift (δ L HF ) introduced in eq 1 can be related directly to the EPR quantities electronic g-tensor g and hyperfine coupling A-tensor A as follows (in Hartree atomic units):…”

“…The quantum mechanical theory of pNMR has been known for more than 50 years, but only recently has it been realized that the proper starting point for any temperature-dependent molecular property (including pNMR) is the Helmholtz free energy and its derivatives. , Paramagnetic NMR theory was originally developed within an exact-state formalism, however it is possible to express the pNMR theory in terms of parameters of the EPR effective spin Hamiltonian. , This mapping of pNMR to EPR tensors provides us with a more natural language for chemical analysis, as it allows the use of DFT methodologies and it rationalizes the separation of the total pNMR shift into orbital and hyperfine terms; see eq . We refer the reader to a recent review for a discussion of the latest theoretical understanding of pNMR theory and to examples of application of the pNMR theory to systems with higher than doublet multiplicity or low-lying excited states. − …”

“…This is analogous to perturbed electronic populations in closed-shell (diamagnetic) systems, for example, by additional nuclei with nonzero nuclear spin (FC mechanism of J coupling) or by spin–orbit coupling and magnetic field (SO-HALA NMR shift) . One can write the FC contribution to δ L HF through A FC , see eq , as bold-italicA u v FC ( L ) = γ normalL S 1 2 c 8 π 3 true∫ ρ u false( J v false) δ D false( r⃗ − R⃗ normalL false) dV = γ normalL S 4 π 3 c ρ u ( J v , R⃗ L ) Here, δ D is a Dirac delta function, R⃗ normalL is the position of nucleus L, J⃗ is the orientation of the magnetization (spin) of the electronic system, and ρ u is the component of the Gordon spin density described in ref . Equation follows from the Gordon decomposition of the current density , and is a valid expression in four-component (4c) relativistic theory .…”

“…8−10 Paramagnetic NMR (pNMR) shifts are excellent indicators of the underlying (particularly long-range) electron−nucleus hyperfine interactions governed by the molecular structure. Within the approximations discussed in Section 2, the total pNMR shift of atom L can be expressed as the sum of the orbital and hyperfine (also known as Curie) contributions in eq 1: 11,12 = +…”

“…The situation is fundamentally different for paramagnetic (open-shell) systems, where the NMR resonances are greatly shifted due to the imbalanced magnetic contributions of α/β electron spin at finite temperature and broadened by paramagnetic relaxation. − Paramagnetic NMR (pNMR) shifts are excellent indicators of the underlying (particularly long-range) electron–nucleus hyperfine interactions governed by the molecular structure. Within the approximations discussed in Section , the total pNMR shift of atom L can be expressed as the sum of the orbital and hyperfine (also known as Curie) contributions in eq : , δ normalL tot = δ normalL orb + δ normalL HF The orbital contribution to the NMR shift (δ L orb , approximately temperature-independent) is related to the NMR shift of its diamagnetic counterpart. In contrast, the hyperfine contribution (δ L HF , temperature-dependent) originates from the presence of unpaired electrons in the molecule and results in NMR resonances in large spectral regions.…”