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

Conspectus Magnetic resonance techniques represent a fundamental class of spectroscopic methods used in physics, chemistry, biology, and medicine. Electron paramagnetic resonance (EPR) is an extremely powerful technique for characterizing systems with an open-shell electronic nature, whereas nuclear magnetic resonance (NMR) has traditionally been used to investigate diamagnetic (closed-shell) systems. However, these two techniques are tightly connected by the electron–nucleus hyperfine interaction operating in paramagnetic (open-shell) systems. Hyperfine interaction of the nuclear spin with unpaired electron(s) induces large temperature-dependent shifts of nuclear resonance frequencies that are designated as hyperfine NMR shifts (δHF). Three fundamental physical mechanisms shape the total hyperfine interaction: Fermi-contact, paramagnetic spin–orbit, and spin–dipolar. The corresponding hyperfine NMR contributions can be interpreted in terms of through-bond and through-space effects. In this Account, we provide an elemental theory behind the hyperfine interaction and NMR shifts and describe recent progress in understanding the structural and electronic principles underlying individual hyperfine terms. The Fermi-contact (FC) mechanism reflects the propagation of electron-spin density throughout the molecule and is proportional to the spin density at the nuclear position. As the imbalance in spin density can be thought of as originating at the paramagnetic metal center and being propagated to the observed nucleus via chemical bonds, FC is an excellent indicator of the bond character. The paramagnetic spin–orbit (PSO) mechanism originates in the orbital current density generated by the spin–orbit coupling interaction at the metal center. The PSO mechanism of the ligand NMR shift then reflects the transmission of the spin polarization through bonds, similar to the FC mechanism, but it also makes a substantial through-space contribution in long-range situations. In contrast, the spin–dipolar (SD) mechanism is relatively unimportant at short-range with significant spin polarization on the spectator atom. The PSO and SD mechanisms combine at long-range to form the so-called pseudocontact shift, traditionally used as a structural and dynamics probe in paramagnetic NMR (pNMR). Note that the PSO and SD terms both contribute to the isotropic NMR shift only at the relativistic spin–orbit level of theory. We demonstrate the advantages of calculating and analyzing the NMR shifts at relativistic two- and four-component levels of theory and present analytical tools and approaches based on perturbation theory. We show that paramagnetic NMR effects can be interpreted by spin-delocalization and spin-polarization mechanisms related to chemical bond concepts of electron conjugation in π-space and hyperconjugation in σ-space in the framework of the molecular orbital (MO) theory. Further, we discuss the effects of environment (supramolecular interactions, solvent, and crystal packing) and demonstrate applications of hyperfine shifts in determining the structure of paramagnetic Ru(III) compounds and their supramolecular host–guest complexes with macrocycles. In conclusion, we provide a short overview of possible pNMR applications in the analysis of spectra and electronic structure and perspectives in this field for a general chemical audience.


INTRODUCTION
Nuclear magnetic resonance (NMR) spectroscopy has been utilized for decades to solve various problems in the chemistry of transition-metal (TM) compounds.Myriad papers have described its application to diamagnetic (closed-shell) systems, and this topic has been extensively reviewed. 5,6The NMR dependencies for ligand atoms in diamagnetic TM compounds resemble those known for organic molecules except for those situations where relativistic heavy-atom effects on the NMR resonances of neighboring light atoms play a significant role. 7−10 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 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.Three physical mechanisms contribute to δ L HF : Fermi-contact, spin−dipolar, and paramagnetic spin−orbit.We provide an elemental theory behind the hyperfine shifts and describe recent progress in understanding the structural and electronic principles underlying the individual hyperfine mechanisms.

THEORY
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. 12,13Paramagnetic 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. 14,15This 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.−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): Here, kT represents the thermal energy, c is the speed of light, and γ L is the gyromagnetic ratio of nucleus L. In the following, we use the separation of δ L HF that is governed by the physical terms of the A-tensor, 23 i.e., the Fermi-contact (FC), 1,23 paramagnetic spin−orbit (PSO), 3,23 and spin−dipolar (SD) 23,24 terms:

S S ckT
where X in eq 4 is FC, PSO, or SD.The approximate sign in eq 3 expresses the fact that we analyze the terms on the RHS of the equation using perturbation theory (PT), where we include only the dominant terms, and that a small relativistic contribution that appears in four-component expressions is neglected. 25PT is most suited for chemical analysis because it allows for the use of nonrelativistic or scalar-relativistic molecular orbitals, the familiar quantity and language of chemists.

Fermi-Contact (FC) Mechanism.
The Fermicontact contribution to δ L HF in eq 3 originates in the FC interaction between the nuclear and electron spin at the position of the nucleus (Figure 1).
Obviously, this can occur only for the imbalanced α/β ground-state electron population in the paramagnetic systems.This is analogous to perturbed electronic populations in closed-shell (diamagnetic) systems, for example, by additional

Accounts of Chemical Research
nuclei with nonzero nuclear spin (FC mechanism of J coupling) 26 or by spin−orbit coupling and magnetic field (SO-HALA NMR shift). 7One can write the FC contribution to δ L HF through A FC , see eq 4, as Here, δ D is a Dirac delta function, R L is the position of nucleus L, J is the orientation of the magnetization (spin) of the electronic system, 12 and ρ u is the component of the Gordon spin density described in ref 27.Equation 5follows from the Gordon decomposition of the current density 28,29 and is a valid expression in four-component (4c) relativistic theory. 27The expression can thus easily be rewritten into a two-component (2c) or one-component (1c) theory by expressing the spin density in the respective form.Note that in this work the label 1c will encompass both nonrelativistic (NR) and scalar relativistic (SR) theories.Because of the spin−orbit coupling (SOC) interaction, the 4c A FC has both isotropic and anisotropic nonzero components.However, in 1c theory the A FC becomes isotropic, because ρ u (J v ) = (ρ α − ρ β )δ uv K , with δ uv K being Kronecker delta and ρ α,β being the charge densities of the α and β electrons, respectively.In such case, A FC becomes a pure isotropic tensor and thus contributes only to the so-called contact shift; see ref 30.Because the SR contributions usually dominate over the contributions from SOC, A FC is mostly isotropic and thus δ L FC is influenced primarily by the isotropic part of the g-tensor, δ L FC ∝ A FC,iso g iso (see the decomposition of δ L in ref 31).
2.1.2.Paramagnetic Spin−Orbit (PSO) Mechanism.The PSO mechanism originates in the interaction of the nuclear magnetic moment with the electronic orbital current density (Figure 2), which is, however, nonzero only if SOC is included in consideration.
Because the 4c expression for A PSO involves 4c spinors, its use in chemical analysis is nontrivial.To analyze A PSO , one finds the perturbation theory to be a more convenient approach, utilizing the more suitable 1c MOs.The leadingorder contribution to A PSO involves linear SOC interaction, as follows: Here, Z M is the nuclear charge of the single heavy atom M in the system, ) and φ a α,β (ε a α,β ) denote occupied and vacant molecular orbitals (one-electron energies), respectively, l ̂u X is a component of the angular momentum operator centered on nucleus X, and r X is the relative distance between the electron and nucleus X with X = M, L. For a more detailed discussion of eq 6, see ref 3. Counterintuitively, the PSO mechanism contributes significantly to the long-distance limit, 32 although one would expect that SOC would influence the electronic structure only close to the heavy atom M.
2.1.3.Spin−Dipolar (SD) Mechanism.The SD mechanism is a consequence of the magnetic dipole−dipole interaction between the nuclear and electron spins (Figure 3).
Similar to the FC mechanism, the 4c expression in eq 7 can be derived from Gordon decomposition of the current density.In a similar way, one can reduce it to a 1c framework, where it holds ρ u (J v ) = (ρ α − ρ β )δ uv K .In such a case, A SD becomes purely anisotropic, and although with the inclusion of SOC A SD will have a nonzero isotropic part, the anisotropic part usually dominates; therefore, δ L SD is influenced primarily by the anisotropic part of the g-tensor, δ L SD ∝ Tr(A SD,ani g ani ) (see the decomposition of δ L in ref 31).At the long-distance limit, the combined PSO and SD mechanisms lead to the wellknown pseudocontact shift, which is important when investigating interactions between a paramagnetic center and a spectator nucleus L at long distances.

Long-Distance (LD) Limit of the A-Tensor and the Hyperfine NMR Shift.
Recently, it has been shown from first-principles that the long-distance limit of the A-tensor within the perturbation theory (and linear SOC effects) has the following form: 32,33 where R L is the distance between the paramagnetic center M and the nucleus L. Based on refs 34, 35, and 32, one can easily show that eq 8 holds even at the four-component level of theory.As expected, in the nonrelativistic limit, where the gtensor is just the free-electron g-value g e , eq 8 gives a purely   anisotropic A LD (L), and by inclusion of the SOC interaction, A LD (L) will get nonzero isotropic parts as well.
Using the LD limit of the hyperfine coupling tensor, eq 8, in the expression for the hyperfine shift, eq 2, one obtains the well-known pseudocontact shift 36,30,33,32 that can be written as follows where (R L R L T ) uv = (R L ) u (R L ) v and χ Curie refers to the temperature-dependent part of the susceptibility tensor.Note that the LD limit in eqs 9 and 10 is also known in the literature as the point-dipole approximation. 37,36,30The full susceptibility tensor, χ = χ orb + χ Curie , would enter eq 9 when LD is made the limit of δ L tot .However, one obtains δ L pc from experiments as the difference between the total pNMR shift δ L tot and its diamagnetic analogue δ L orb,dia , which approximates the orbital part of the NMR shift (δ L orb ); therefore, one can write . Note that from eq 9, one can conclude that δ L pc arises solely due to relativistic effects, because in the nonrelativistic limit g = g e 1 and the tensor in round brackets is anisotropic, i.e., it has zero trace.Therefore, one must include at least linear SO effects to obtain a reasonable description of the δ L pc , and as discussed in the next section, in many cases, one must include quadratic SO effects as well.

Electronic g-Tensor
The final component in the EPR-based approach, as outlined in eqs 2−4, is the electronic g-tensor.The deviation of the molecular g-tensor from the g-factor of the free electron, the gshift (Δg), originates in SOC and is governed by the orbital-Zeeman (OZ) and spin-Zeeman (SZ) mechanisms.Interestingly, the leading-order contribution is linear for the OZ interaction but quadratic for the SZ interaction.Applying perturbation theory of second and third order and accommodating some reasonable assumptions (see refs 3 and 38) to simplify the expressions, one gets formulas useful for chemical analysis: To further simplify the discussion, we assume that the system is oriented on the principal axes of the g-tensor, i.e., only the diagonal elements of the g-tensor are nonzero.Whereas the quadratic contributions are formally c −2 smaller than the linear contributions, in heavy-element compounds, they may actually be similar to or even larger than the linear ones.Equations 12−16 reveal a useful mechanism of SOC on the g-tensor.Let us assume that there is an efficient occupied−vacant MO coupling on the metal governed by the l ̂x M operator, one can then expect sizable linear SO effects on Δg x SO/OZ and quadratic SO effects on Δg y SOd 2 /SZ and Δg z SOd 2 /SZ (for a schematic representation, see Figure 4).We direct the reader to ref 38 for a detailed discussion and analysis of both linear and quadratic SO effects on the g-tensor, with general conclusions when the interplay of these effects results in a more isotropic or anisotropic g-tensor.
In Section 2 we have shown that relativistic effects (in particular, SOC) are mandatory in a correct description of pNMR shifts at short or long distances.Although relativity adds significant complexity to the problem at hand, one can apply perturbation theory, which uses only standard chemical tools, NR MOs, and angular momentum operators.One can therefore explain SO effects on pNMR shifts by extending usual chemical concepts to the relativistic domain.

CHEMICAL CONCEPTS
The nature of the hyperfine interaction can be used to characterize the electronic structure and geometrical arrangements of molecules and supramolecular assemblies.The electronic g-tensor and metal (M) hyperfine coupling tensor are related to the electronic structure around atom M and are pertinent to EPR spectroscopy.In contrast, weaker hyperfine interaction on distant ligand atoms (L) can be detected by NMR and gives an excellent indication of the molecular geometry and bonding.

"Through-Bond" Hyperfine Interaction
We start our discussion of concepts with the FC mechanism arising from the propagation of spin density from the paramagnetic center M toward the spectator atom L via chemical bonds. 39This includes spin delocalization based on electron conjugation (π-space) and hyperconjugation (σ- space) interactions and is further complemented by neighboratom or resonance spin polarization steps manifesting Hund's rule of maximum multiplicity and Pauli's exclusion principle. 40.1.1.π-Conjugation Delocalization.The spin delocalization can be defined as the distribution of α-spin density over the atoms with nonzero atomic coefficients in singly occupied molecular orbital(s), SOMO(s), calculated at the spinrestricted level. 40This can be clearly seen in the example of the N-methylen-4-methyl-pyridinium radical (NMP) depicted in Figure 5a.
In the spin-restricted calculation, the α-spin density is delocalized to carbon atoms at positions ortho and para relative to the N-methyl group (Figure 5a).The system is polarized in spin-unrestricted calculations resulting in an increased α-spin density at positions ortho and para (Hund's rule), and consequently, the induction of β-spin density at positions meta and ipso.In other words, the overabundance of α-spin density on atoms contributing to the SOMO in the restricted calculation (delocalization) is responsible for a further accumulation of α density on these atoms, maximizing the exchange interaction.This results in an overabundance of β density on the neighboring atoms from which the α density is pulled (polarization).The propagation of spin polarization through delocalized π-space can be explained from the perspective of perturbation theory. 41nalogous behavior is obtained for the Cr(acac) 3 compound, where three unpaired electrons reside in degenerate SOMOs oriented in π-space (d xy , d xz , and d yz ) relative to individual bonds with ligands 2 (Figure 5b).The α-spin density in the spin-restricted calculation is delocalized from M via πspace to ligand atoms O−C with atomic coefficients contributing to the SOMO.In the spin-polarization step enabled in the spin-unrestricted calculation, the metal center M pulls further α density predominantly in σ-space (see Neighbor-Atom and Resonance Polarization below) because of the symmetry of the SOMOs, thus leaving an overabundance of β density on the M−O bond.

σ-Hyperconjugation Delocalization.
The situation is different for systems with different symmetries of the SOMO relative to the pyridine π-space and the absence of its conjugation.This applies to the NMP with a CH 2 group rotated by 90°in the transition state 40 shown in Figure 6a.
Here, the α-spin density in the restricted calculation is delocalized to positions ortho (and meta) of the aromatic ring via σ-hyperconjugation.As a result of the spin polarization in the unrestricted calculations, these atoms accumulate additional α density and leave β density at the neighboring position ipso.
An analogous situation is observed for the square-planar Cu(acac) 2 compound, Figure 6b, where the α-spin density in the SOMO (d x 2 −y 2 ) oriented in the M−L bonds is delocalized to the carbons of the CH 3 groups via σ-hyperconjugation 2 (see also σ delocalization in the high-spin Fe 2+ complex 20 ).The spin polarization in the unrestricted calculation induces β density at the position of the carbonyl carbon.

Neighbor-Atom and Resonance Polarization.
In the absence of π-conjugation or σ-hyperconjugation delocalization discussed above, the only mechanism of spin transmission is polarization. 40The presence of α density at M results in pulling additional α density to M in the unrestricted calculation (Figure 7). 43It leaves β density at the neighboring atoms as a manifestation of Pauli's principle. 44his neighbor-atom polarization can propagate further in a consecutive manner, although the σ-polarization pathway is inefficient because of the "hard" nature of σ bonds compared to the "soft" nature of the π system discussed above. 45lternative mechanisms involve σ or π resonance polarization as described in ref 40.

Solvent Effects.
We demonstrated that the efficiency of σ vs π resonance polarization is greatly influenced by the solvent. 1,23This can even result in switching the sign of the FC mechanism, 1 as shown in the example of the meta  carbon (+19 ppm vs −14 ppm; visualization of the π and σ resonance polarizations in Figure 8) of the pyridine in the Ru(III) complex.Therefore, the calculation of solution-state data should preferably be done using at least an implicit solvent model.

Crystal Packing Effects. Supramolecular interac-
tions have large effects on the hyperfine shifts in the solid state, where neighboring molecules not only influence the electronic structure of the investigated system but also can be sources of additional hyperfine fields (spin centers that create additional through-space hyperfine interactions).This can be exemplified by the 13 C NMR of the methyl group calculated in vacuo, in the crystal environment approximated by the two diamagnetic neighbors (cluster-d), and in the environment of the two paramagnetic neighbors (cluster-p) 2 (Figure 9).
The 13 C NMR shift of +1168 ppm calculated in vacuo is reduced by approximately 80 ppm in the presence of two diamagnetic neighbors.Further, recovering the electronic doublet character of each of the two neighboring molecules results in an additional shielding of more than 100 ppm, thus reaching the value of 988 ppm (the experimental resonance at 929 ppm). 2

"Through-Space" Hyperfine Interaction
In the absence of electron sharing or charge transfer between the atoms or fragments discussed in Section 3.1, the hyperfine interaction propagates exclusively "through-space".This arises from the PSO and SD mechanisms, and at long distances corresponds to the pseudocontact shift (see Section 2.1.4). 46owever, the through-space interaction can dominate even at short distances in compounds with ionic or only weakly covalent bonds lacking notable electron sharing (inefficient FC mechanism; e.g., f-block lanthanide complexes). 47However, in d-block TM complexes, these short-range shifts are dominated by the FC term.Therefore, only through-space effects at longer distances are discussed further for the TM compounds.2. The contributions of the PSO and SD to the hyperfine shift can be calculated at the 2c or 4c level of theory using eq 2 involving both g-and A-tensors.However, the size of the system may prohibit the usage of 4c and even 2c computational methods for full systems.In the past, various methods have been tested to find approximate and computationally more feasible approaches.For example, only the surroundings of M have been used to calculate the g-tensor at the 2c level, and the distant L hyperfine coupling has been obtained at the 1c level for the full system. 24Although such a calculation yields artificially small values due to the missing contributions from the PSO, the trends in δ L HF remain intact in systems with small g-tensor anisotropy (Section 2), 24,48 as shown in Figure 10.3.2.2.Through-Space Shift at Long Distances.At long distances, through-space hyperfine shielding can be calculated from the less expensive magnetic susceptibility tensor obtained via the electronic g-tensor.Note that this approach has been termed the long-distance (LD) limit or point-dipole approximation (PDA). 32,49,30In practice, it applies to situations where the Fermi-contact contribution is vanishingly small and the spatial distribution of spin density around M relative to L can be neglected (for applications, see Section 4.2).The character of bonding along the transmission pathway from M is crucial for the spin delocalization and polarization toward the ligand L. The degree of covalency (electron sharing) can be characterized using regular chemical tools, for example, by the delocalization index (QTAIM) 50 or NOCV channels. 1,23n prototypical Ru(III) compounds 51 with pyridine ligands, concepts of σ-donation and π-backdonation have been used to describe the bond between the ruthenium and nitrogen atoms (Figure 11). 23Note that the spin delocalization is quenched in this arrangement because of the unfavorable symmetry of the SOMO. 40The first σ-channel represents a classical donation of the lone pair of electrons of the nitrogen atom to M. This is slightly more efficient in the α-space (Hund's rule), thus leaving an overabundance of β density in the σ-space of pyridine.The second π-backdonation channel is more efficient in β-space, thus leaving α density at M and transmitting β density toward the ortho and para positions of the π system.

Effect of a trans
Ligand Attached to M. The character of the ligands can influence the distribution of spin density around the magnetic M and the transmission of spin density toward the spectator atom(s).This can be classified as a cis or equatorial-to-axial ligand effect (weak) involving mostly neighbor-atom polarization and a hyperfine trans effect (strong) 40 with two ligands competing for one d orbital (e.g., d z 2 ) of the M. 52 This leads to changes in the M−L bond distance and covalency upon the change of the trans ligand and simultaneously modulates the mechanism of the spin transmission and the ligand hyperfine coupling, as shown by the example of the iridium compounds in Figure 12.Because of the relatively weak bond between the trans chloride and the iridium in compound IrCl, the Ir−N1 bond is strong, with significant delocalization of the spin density into the π-space of N1.This is propagated further with altered signs, as described for conjugated systems in Section 3.1.In contrast, because of the strong triple bond between the trans nitrogen and iridium in compound IrN, the Ir−N1 bond is relatively weak, hindering direct spin delocalization.Therefore, the spectator atom N1 is spin-polarized in the π-space and the resulting spindensity pattern around the N1−C2−C3 fragment in IrN is the inverse of that in IrCl.Strong host−guest (HG) complexes give NMR resonance lines resolved from those of the free host and guest components.Therefore, the interpretation of their pNMR spectra is straightforward and can be used to determine the relative HG distance and orientation.It has been shown 24 that the ruthenium guest Ru1 with the pyridine linker to an adamantyl anchor has a distance from the oxygen portal of cucurbit [7]uril (CB7) of 763 pm, whereas this distance is much shorter (469 pm) for the guest Ru2 with an imidazole linker.This would imply stronger through-space effects and a larger δ L HF in Ru2.Indeed, this has been observed for the theoretically calculated values visualized in Figure 14.However, because of the larger tilting (26°vs 37°) of the guest inside the host, both shielding and deshielding hyperfine effects on the equivalent atoms of CB7 are observed for Ru2.As a result of the HG dynamics in solution (rotational tumbling), the shielding−deshielding effects are almost averaged out, and the experimentally observed complexation δ L HF for hydrogen Ha in Ru2 is negligible (Figure 14).In other words, whereas the portal atoms Ha and Ha′ in Ru1 are shielded upon complexation as expected, Ha in Ru2 is shifted only marginally because the shielding and deshielding effects are averaged.

NMR Regime of Fast Chemical Exchange.
In the fast-exchange regime, both the free and bound forms of the guest and host contribute to the (population-weighted) averaged NMR signals.Therefore, the averaged signals are affected by various factors, such as the HG affinity (binding constant and free−bound ratio), the geometry of the HG assembly, and the magnetic anisotropy of the paramagnetic center.pNMR spectroscopy has been used to characterize the relative position of the cyclodextrin (CD) host in the hyperfine field of the ruthenium guest. 4The NMR spectra for CD at various HG ratios with compound Ru3 and the calculated hyperfine shielding maps are shown in Figure 15.This approach allowed the determination of head versus tail HG binding.

SUMMARY AND OUTLOOK
The NMR spectroscopy of paramagnetic systems has developed as an extremely powerful technique in investigating hyperfine interactions.In this Account, we build on the separation of the hyperfine shift into "through-bond" and "through-space" contributions that are associated with essential physical hyperfine mechanisms.We aim to discuss these mechanisms, Fermi-contact (FC), paramagnetic spin−orbit (PSO), and spin−dipolar (SD), for a general chemical  in their HG assemblies with CB7. 24(c) NMR spectra of free CB7, Ru1@CB7, and Ru2@CB7.Adapted with permission from ref 24.Copyright 2018 American Chemical Society.readership employing standard chemical concepts involving molecular orbitals.We provide several models to demonstrate the principles of spin delocalization and polarization in σand π-space to rationalize the distribution patterns of the spin density in ligands.We discuss the effects of the solvent and environment on the Fermi-contact mechanism of hyperfine shifts.Further, we demonstrate the applicability of long-range hyperfine interactions (PSO and SD) to determining the geometry of supramolecular host−guest complexes.
Even after decades of progress in interpreting NMR spectra for both diamagnetic and paramagnetic substances, the interpretation of pNMR spectra still lags behind that of diamagnetic NMR.Obviously, this is due to the inherent physical complexity underlying the pNMR spectra.In contrast to the NMR shifts of diamagnetic compounds, pNMR shifts are influenced by multiple nonadditive interactions characterized by nonlocal character and strong dependence on relativistic effects.Our Account is a step toward a deeper understanding of the chemical concepts underlying hyperfine interaction and its manifestation in pNMR spectra.

■ ACKNOWLEDGMENTS
This work has received support from the Czech Science Foundation (Grant 21-06991S to R.M.) and the Slovak Grant Agency APVV (Grant APVV-19-0516 to S.K.).

Figure 1 .
Figure 1.Schematic representation of the FC mechanism of the hyperfine interaction.

Figure 2 .
Figure 2. Schematic representation of the PSO mechanism of the hyperfine interaction.

Figure 3 .
Figure 3. Schematic representation of the SD mechanism of the hyperfine interaction.

Figure 4 .
Figure 4. Schematic representation of the linear OZ and quadratic SZ mechanisms of the electronic g-tensor.

Figure 5 .
Figure 5. Visualization of the conjugation delocalization of α-spin density in π-space and total atomic spin populations of (a) the NMP radical and (b) Cr(acac) 3 calculated at the spin-restricted DFT level (left) accompanied by polarization shown by the spin density and atomic spin populations calculated using the spin-unrestricted (right) approach.Adapted with permission from refs 40 and 2. Copyright 2021 and 2024 American Chemical Society.

Figure 6 .
Figure 6.Visualization of the hyperconjugation delocalization of the α-spin density in σ-space and the total atomic spin populations of (a) the transition state of NMP and (b) Cu(acac) 2 calculated at the spinrestricted DFT level (left), with the polarization shown at the spinunrestricted approach (right).Note the β-spin polarization of the outer shell of the Cu atom. 42,43Adapted with permission from refs 40 and 2. Copyright 2021 and 2024 American Chemical Society.

Figure 7 .
Figure 7. Visualization of the spin density and atomic spin populations on the M and neighboring ligand atoms in the model [RuCl 2 (NH 3 ) 2 (NHMe 2 )(DMSO)] + calculated at the restricted (left) and unrestricted (right) DFT levels.Adapted with permission from ref 40.Copyright 2024 American Chemical Society.

Figure 8 .
Figure 8. Visualization of the spin density and hyperfine shifts for carbon atoms of the pyridine in Na + [Ru III Cl 4 (4-CN-pyridine)-(DMSO)] − calculated in vacuo (left) and using an implicit solvent model (right).The in-plane spin polarization of the pyridine is represented as a detailed slice of the spin density at the lower threshold (0.00005 au).Adapted with permission from ref 1.Copyright 2016 American Chemical Society.

Figure 9 .
Figure 9. 13 C NMR shift of the methyl group in Cu(acac) 2 calculated in vacuo (vacuo), in a cluster with two diamagnetic neighbors (clustd), in a cluster with two paramagnetic neighbors (clust-p), and the experimental value (exp).Adapted with permission from ref 2. Copyright 2021 American Chemical Society.

Figure 10 .
Figure 10.Hyperfine 1 H NMR shifts of a CB7 host obtained experimentally (EXP) and calculated by DFT: 2c g and A values (A-2c), 2c g and 1c A (A-1c), and using PDA.Data from ref 24.

4 . 1 . 3 . 4 . 2 .
Effect of Substituents on the Aromatic Ligand.The character (electronic properties) of the substituents influences the distribution of electron density on the ligand and by that also the spin transmission from M toward the ligand atoms.The effect of the substituent on the electron (and spin) density is shown by correlating the δ L HF with the Hammett constant,52 Hirshfeld charge, 1 and atomic spin population1 in Figure 13.Depending on the position of the electron donor (acceptor), the concentration (depletion) of the atomic electron density can enhance or diminish the spin polarization and the FC contribution to δ L HF .Through-Space: Host−Guest Assemblies and the SD and PSO Terms 4.2.1.NMR Regime of Slow Chemical Exchange.

Figure 11 .
Figure 11.Visualization of the two most important NOCV channels (α and β parts) for the formation of the Ru−N bond.The value 0.5 × 10 −3 au was used to plot the isosurface of the electron deformation density.Adapted with permission from ref 23.Copyright 2018 American Chemical Society.

Figure 12 .
Figure 12.Visualization of the spin density in compounds IrCl (left) and IrN (right), highlighting the switch in the transmission mechanism induced by the trans ligand and the oxidation state of the Ir atom.Adapted with permission from ref 3.Copyright 2019 American Chemical Society.

Figure 13 .
Figure 13.Correlation of the (a) Hammett constant, (b) Hirschfeld charge, and (c) atomic spin population with the hyperfine 1 H or 13 C NMR shift.Adapted with permission from refs 52 and 1.Copyright 2016 and 2023 American Chemical Society.

Corresponding Authors Stanislav
Email: stanislav.komorovsky@savba.skRadek Marek − CEITEC − Central European Institute of Technology, Masaryk University, CZ-625 00 Brno, Czechia; Department of Chemistry, Faculty of Science, Masaryk University, CZ-625 00 Brno, Czechia; orcid.org/0000-0002-3668-3523;Email: radek.marek@ceitec.muni.cz& editing; Stanislav Komorovsky conceptualization, funding acquisition, methodology, writing-original draft, writing-review & editing; Radek Marek conceptualization, funding acquisition, methodology, project administration, resources, writingoriginal draft, writing-review & editing.Jan Novotny studied biomolecular chemistry at the Masaryk University in Brno, Czechia and obtained his PhD with Radek Marek in 2015.As a postdoc in the group of Radek Marek, he received the Petr Sedmera Award in NMR spectroscopy in 2018.After a two-year research stay with Stanislav Komorovsky in Bratislava he returned to his alma mater in 2022 as a senior researcher.He is currently teaching NMR spectroscopy and computational quantum chemistry.His main research interests include NMR spectroscopy, paramagnetic effects in NMR, TM complexes, supramolecules, and noncanonical forms of nucleic acids.Stanislav Komorovsky is the head of the Department of Theoretical Chemistry at the Institute of Inorganic Chemistry of the Slovak Academy of Sciences.He received his PhD (2009) in chemical physics under the supervision of Vladimir Malkin.His research focuses on the development of relativistic methods for the calculation of spectroscopic properties with stress on the magnetic properties of open-shell systems.He is one of the main authors of the ReSpect program (respectprogram.org),which is dedicated to calculating the properties of molecules and solids using first-principle relativistic DFT methods.Radek Marek is a Professor of Chemistry and Group Leader at the Masaryk University in Brno, Czechia.He obtained his PhD in 1995 and did his postdoc in NMR spectroscopy with Vladimir Sklenaŕ.He started his professorship at the Masaryk University in 2011, teaching NMR spectroscopy and structural and computational chemistry.The main research areas of his group are paramagnetic and relativistic effects in NMR spectroscopy, aspects of chemical bonding, DFT applied to bonding and molecular properties, TM compounds, metallodrugs, supramolecular host−guest chemistry, and noncanonical nucleic acids.