 1.1 The Effect of Paramagnetism on NMR Spectra
 1.1.1 The Hyperfine Coupling
 1.1.2 The Curie Spin
 1.1.3 The Magnetic Susceptibility
 1.2 The Hyperfine Shift
 1.2.1 The Fermicontact Shift
 1.2.2 The Pseudocontact Shift
 1.2.3 Simplified Expressions in Limiting Cases
 1.2.4 The Effect of Partial Selforientation
 1.3 The Paramagnetic Residual Dipolar Couplings
 1.4 The Paramagnetic Relaxation Enhancements
 1.4.1 Dipolar Relaxation
 1.4.2 Curie Spin Relaxation
 1.4.3 Fermicontact Relaxation
 1.5 Paramagnetic Cross Correlation Effects
 1.6 First Principles Calculation of Hyperfine Shifts
 1.7 Metal Ion Dependence of the Paramagnetic Effects
 1.8 The Overhauser Effect in Paramagnetic Systems
Chapter 1: NMR Consequences of the Nucleus–Electron Spin Interactions

Published:03 Aug 2018

Special Collection: 2018 ebook collectionSeries: New Developments in NMR
G. Parigi and C. Luchinat, in Paramagnetism in Experimental Biomolecular NMR, ed. C. Luchinat, G. Parigi, and E. Ravera, The Royal Society of Chemistry, 2018, ch. 1, pp. 141.
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This chapter introduces readers to the principles of paramagnetism and to the effects of paramagnetism on NMR spectra. It provides an indepth review of the equations describing the hyperfine shifts, composed of the contact and pseudocontact contributions, the paramagnetisminduced partial selfalignment and the subsequent paramagnetic residual dipolar couplings, and the paramagnetic relaxation enhancements to the nuclear relaxation rates. The dependence of these effects on the selected paramagnetic metal ion is also discussed.
1.1 The Effect of Paramagnetism on NMR Spectra
The presence of unpaired electrons in molecules makes them paramagnetic and largely affects their NMR spectra. The main effects related to the presence of a paramagnetic center, i.e., of atoms or ions with unpaired electrons, are (see Figure 1.1):^{1 }
 the NMR shifts are perturbed, so that the shifts measured for the paramagnetic molecule (δ) and for a diamagnetic analogue (δ^{dia}) (i.e., for the same molecule without the paramagnetic center or with the paramagnetic metal ion substituted by a diamagnetic one) differ. These differences are called hyperfine shifts (δ^{hs}):δ=δ^{dia}+δ^{hs}Equation 1.1
 the nuclear relaxation rates of the paramagnetic molecule (R_{1}, longitudinal; R_{2}, transverse) are increased with respect to those of a diamagnetic analogue (R$1dia$ and R$2dia$). This difference is called paramagnetic relaxation enhancement (R_{1M} and R_{2M}):R_{1}=R$1dia$+R_{1M}, R_{2}=R$2dia$+R_{2M}Equation 1.2
 the probabilities for the different orientations of the paramagnetic molecule in a magnetic field are not the same, so a partial selforientation occurs. This partial selforientation is responsible for the occurrence of residual dipolar couplings (Δν^{rdc}), analogously to what happens when partial molecular orientation is driven by external devices. The presence of paramagnetic residual dipolar couplings affects the Jcoupling between nuclei:^{1}J_{12} (B_{0})=^{1}J_{12}+Δv^{dia}+Δν^{rdc}Equation 1.3
This chapter deals with the description of these effects, and of their physical origin, which is related to the socalled hyperfine coupling. In eqn (1.1)–(1.3) other minor effects, namely residual anisotropic chemical shifts, cross relaxation terms, and dynamic frequency shifts, respectively, have been neglected (see later).
1.1.1 The Hyperfine Coupling
The interaction between the magnetic moment associated to a nuclear spin I and the magnetic moment associated to an unpaired electron spin S is called hyperfine coupling and is traditionally described by the Hamiltonian
The energy corresponding to this interaction fluctuates with time (and thus this interaction represents a source of relaxation for the nuclear spin, see Section 1.4) as a result of the molecular motions and of changes in the electron spin state. The other terms describing the contributions to the energy of the system are:
(i) the Zeeman interaction between the applied magnetic field B_{0} and the nuclear magnetic moment μ_{I}=ħγ_{I}I, where ħ is Planck's constant divided by 2π and γ_{I} is the nuclear magnetogyric ratio,
(ii) the Zeeman interaction between B_{0} and an ‘effective’ electron magnetic moment taking into account both the electron spin and the orbital magnetic moments. This is parameterised using the g tensor, with the effective electron magnetic moment given by μ_{S}=−μ_{B}g · S, where μ_{B} is the electron Bohr magneton. Therefore, the g tensor couples the ‘effective’ electron spin to B_{0}, thus accounting for the presence of the anisotropic orbital contributions to the electron magnetic moment. As a result, the latter depends on the orientation of the molecule with respect to the magnetic field direction,
(iii) the zerofield splitting (ZFS) interaction, due to the spin–orbit coupling, for systems with more than one unpaired electron (S>1/2), and
(iv) all other diamagnetic interactions present in the molecule.
The whole Hamiltonian describing a paramagnetic system is thus composed of the terms
where D is the ZFS tensor.
The A tensor in H_{hc} is composed of two terms: the isotropic Fermicontact term and the anisotropic electron–nucleus dipole–dipole interaction term
where A^{FC} is a scalar and A^{DIP} is a traceless symmetric tensor. The scalar term, called the contact coupling constant, is given by
where μ_{0} is the permeability of the vacuum, g_{e} is the freeelectron g factor and ρ is the electron spin density at the nucleus (normalized to one electron). Anisotropic orbital contributions are excluded from A^{FC} because the coupling responsible for the contact interaction is only with the unpaired electron spin density residing on the nucleus (and thus in s orbitals).
In the pointdipole approximation (see Section 1.2.2), the dipolar term S · A^{DIP} · I should describe the energy related to the dipolar magnetic field generated by the magnetic moments, which, in the classical approach, is^{2 }
where the effective electron magnetic moment is considered to take into account the presence of the orbital contributions. Therefore,
where the superscript T indicates the matrix transpose, and r is the vector connecting the positions of the nuclear and electron spins. Of note, the components of the g tensor are not the experimental EPR values obtained for the ground state, but the molecular g values along the main directions of the tensor.
In quantum chemistry computations, a different formulation for the A^{DIP} tensor is used (see Section 1.6).
1.1.2 The Curie Spin
The tiny difference in population between the nuclear spin states in the presence of a magnetic field, at room temperature, is responsible for the NMR signal originated by that nucleus. In fact, this difference causes an induced average magnetic moment, which is proportional to the observed NMR signal (see Section 8.1.1). The larger the magnetic field (and the smaller the temperature), the larger the difference in spin population, and thus the induced average magnetic moment.
The difference in population between the different spin states is much larger for electrons than for protons, due to their larger magnetogyric ratio (γ_{S}/γ_{H}=658.2). This difference is also of paramount importance for the understanding of the NMR spectra of paramagnetic systems. The electron spin S is thus conveniently separated into two terms, corresponding to the null average part s (〈s〉=0) and to the thermal average of S, S_{C}:
where κ is the versor of S_{C}, which in general differs from the magnetic field direction (see later). Therefore, the electron (thermal average) induced magnetic moment is given by
In quantum mechanics formalism the magnetic moment is defined as the partial derivative of the energy with respect to the magnetic field
so that
where E_{i} is the energy of the electron ith state, k is the Boltzmann constant and T is the temperature. Following van Vleck, let us suppose that E_{i} can be written to a good approximation through perturbation theory limited to secondorder correction to the ground state energy
In our case H^{(1)} is the Zeeman energy (H^{(1)}=−B_{0} · μ_{S}), so that E^{(1)} and E^{(2)} for relatively small B_{0} are much smaller than the eigenvalues of H^{(0)}, E^{0}, independent of the magnetic field. Therefore,
Then, to the first order in E_{i}/(kT)
Since , and retaining only the linear terms in B_{0}
In the second term of the numerator, the average of the values calculated over the i and k states is
so thatThis equation is derived on the assumption that all states at zero field are not degenerate. In the presence of states (indicated by the indexes λ and η) with degeneracy (indicated by n and m),
Eqn (1.18) and (1.19) can be written as
where 〈SS〉 is the effective electron spin dyadic equal to
Using eqn (1.11) and (1.20),
Therefore, 〈μ_{S}〉 and S_{C} are anisotropic quantities, i.e., they have different values for the different directions of the magnetic field with respect to the molecular frame.
As an example, we can consider the case of an S=1 system with the states +1〉 and −1〉 higher in energy than the state 0〉 of a quantity D (axial ZFS). Let us suppose that excited states have energy much larger than kT. Recalling that S_{x}=(S_{+}+S_{−})/2, S_{y}=(S_{+}−S_{−})/2i, S_{z}m_{s}〉=m_{s}m_{s}〉 and , the nonnull elements of 〈SS〉 are:
and thus1.1.3 The Magnetic Susceptibility
As we have seen, a magnetic field induces magnetic moments with nonnull thermal average (eqn (1.20)). Usually, the larger the field, the larger the magnetization (until saturation conditions are met). The magnetic susceptibility is defined as the derivative of the magnetization of a substance with respect to the magnetic field strength; if the magnetic field is weak enough not to reach the saturation conditions and thus the magnetization increases linearly with B_{0}, the magnetic susceptibility is independent of the magnetic field strength. In paramagnetic systems, the magnetization is determined by the induced electron magnetic moments; therefore,
and thus (using eqn (1.20))
In the linear regime, the proportionality constant relating the induced electron magnetic moment and the magnetic field is thus dictated by the magnetic susceptibility per molecule χ according to the following equation:
The tensor χ takes into account both the anisotropy of g and the anisotropy of S_{C} due to ZFS or coupling of the ground state with excited states, and it is thus an anisotropic quantity. The principal values of χ (χ_{xx}, χ_{yy}, χ_{zz}) are labelled in order to have the χ_{zz} component as the most separated of the three components, so that the anisotropy parameters
obey the relationship Δχ_{rh}≤2Δχ_{ax}/3.
The energy related to the (thermal average) induced electron magnetic moment (see eqn (1.12)) is thus
and therefore it depends on the orientation of B_{0} with respect to the molecule.
In the case of S=1 and axial ZFS (see eqn (1.23)),
Analogous equations can be calculated for other spin multiplicities.^{3 }
1.2 The Hyperfine Shift
The hyperfine shift is the difference in the resonance frequency of a nuclear transition between states differing by Δm_{I}=±1 in the absence of the paramagnetic center and in its presence, scaled by the nuclear Larmor frequency (γ_{I}B_{0}/2π). It is thus related to the energy of the hyperfine coupling (eqn (1.4)) between nuclear and electron magnetic moments.
Electrons change their spin state much faster than protons, due to their more efficient relaxation pathways. As a consequence, the nucleus sees the unpaired electron(s) changing its state rapidly among the possible m_{S} levels before being able to change its own m_{I} energy level. This implies that the nuclear magnetic moment actually interacts with the electron induced magnetic moment, calculated from the average of their population distribution at the temperature of the system (see previous section). Therefore, using eqn (1.4) and (1.22), the hyperfine shift is
where indicates the magnetic field direction. It is convenient to define a hyperfine shielding tensor σ^{hs}:^{4 }
from which the shifts can be easily calculated for the different directions of the magnetic field. In the case of isotropic molecular reorientation, i.e. of isotropically oriented magnetic fields, the hyperfine shift (called in this case the isotropic hyperfine shift) becomes
where Tr indicates the trace, i.e. the sum of the diagonal terms, of the σ^{hs} matrix.^{5 } In writing eqn (1.32) it was also assumed that the magnetic field direction changes on a time scale slower than the electron relaxation time, i.e. that the molecular reorientation time is longer than the electron relaxation time.
In the quantum chemistry formalism, the shielding tensor is defined as the second partial derivative of the thermally averaged energy with respect to the magnetic field and to the nuclear magnetic moment, calculated at zero magnetic field and zero magnetic moment:^{6,7 }
From eqn (1.5), it is found that^{8–10 }
As already anticipated in Section 1.1, the operator A is split in the Fermicontact (A^{FC}) and dipolar (A^{DIP}) components, which in solution provide the contact (δ^{cs}) and pseudocontact (δ^{pcs}) shifts, respectively.
1.2.1 The Fermicontact Shift
The contact shift arises from the delocalization of the unpaired electron spin(s), in particular at the site of the detected nucleus. It is thus proportional to the unpaired electron spin density, ρ, present at the nucleus (i) through direct overlap between the molecular orbital(s) containing the unpaired electron(s) and the s orbitals (allowing for a nonzero electron density at the nucleus) of the observed nucleus, and (ii) due to spin polarization of paired electrons. When an unpaired electron in a d orbital of a metal is partially delocalized to an overlapping p or d orbital of a covalently bound atom, in fact, the spin polarization mechanism induces unpaired electron spin density at the nucleus of this atom because, among the paired electrons in an s orbital, the electron with spin parallel to the unpaired electron spin has a slight preference to occupy the region of space closer to that of the unpaired electron (Hund's rule). In this way the electron with spin antiparallel to the unpaired electron has a larger spin density on the nucleus, and the difference in the spin densities corresponding to the two states provides the unpaired spin density. In the presence of direct molecular overlap with an s orbital, the spin density at the nucleus is positive, whereas in the case of spin polarization, it is negative.
As is clear from eqn (1.6), (1.7), (1.31), (1.32) and (1.34), in isotropic systems the contact shift is given by^{3,11 }
where
The last expression in eqn (1.36) clarifies that the contact shifts do not depend on the nuclear parameters; they depend on the electronic parameters and on the unpaired spin(s) density at the nucleus.
When g is orientation dependent, in isotropic systems
In S=1 systems with axial ZFS (see eqn (1.23)),
which, to the first order in D/(kT), is equal to^{3 }
This results in a temperature dependence which is not linear in T^{−1}, but has also a T^{−2} contribution. The presence of such a (usually minor) contribution can be detected in a temperature dependence plot by noting that, assuming a linear dependence, the contact shift does not extrapolate to zero for T^{−1}=0. Analogous equations can be calculated for other spin multiplicities.^{3 }
Only nuclei experiencing a nonzero unpaired electron spin density experience contact shifts. This implies that contact shifts are present only for atoms within a few bonds from the paramagnetic center. The magnitude of the shift depends, of course, on the A^{FC} constant, which in turn depends on the geometrical arrangement of the overlapping orbitals. Equations of the Karplus type have been introduced to describe the contact shifts as a function of dihedral angles θ defined according to the binding directions,
or
(depending on the relative orientation of the involved orbitals), where a, b, and c are constants.^{12–14 }
Euristic Karplustype equations were also obtained for hyperfine shifts (comprising both the contact and the pseudocontact contributions) of methyl protons in low spin iron(iii) heme proteins with one or two histidines as axial ligands,^{15 } and in high spin iron(ii) heme proteins.^{16 } In these proteins, the methyl proton hyperfine shifts depend on the angle(s) formed by the axial histidine plane(s) and the planar ligands.
1.2.2 The Pseudocontact Shift
The pseudocontact shift (PCS) results from the dipole–dipole interaction between the nucleus and the electron magnetic moment occurring through space, and thus not requiring molecular binding. For an accurate evaluation of this NMR observable, the distribution of the electron spin over the space should be considered. However, to simplify the problem, the electron spin is commonly considered located at a single point (i.e., the metal nucleus in a paramagnetic metal ion) in the socalled metal centered pointdipole approximation (Section 1.2.2.1). We will then see how the problem can be faced outside this approximation (Section 1.2.2.2).
1.2.2.1 The Pointdipole Approximation
As is clear from eqn (1.6), (1.9), (1.31), (1.32) and (1.34), in isotropic systems the PCS is given by
where
and thus, using eqn (1.25),
Therefore, in the absence of preferential orientations of the molecule with respect to the magnetic field
which can be rewritten as
or, using the spherical coordinates r, θ and ϕ (see Figure 1.2(a)) to identify the position of a nucleus, with the electron assumed in the origin of the reference frame,
where
Eqn (1.47) can also be written as^{17 }
In the case where the axes of the reference frame are chosen to coincide with the main directions of the χ tensor (i.e., χ_{xy}, χ_{xz} and χ_{yz} in eqn (1.46) are all equal to zero), eqn (1.47) simplifies to
where Δχ_{ax} and Δχ_{rh} are defined as in eqn (1.27). Of note, the PCSs do not depend on the magnitude of the magnetic susceptibility tensor but only on its anisotropy, i.e., for an isotropic χ, . They are also independent of the nuclear parameters, being the same for, e.g., ^{1}H, ^{13}C or ^{15}N nuclei, if placed in the same position. Figure 1.2(b) shows the surfaces with constant PCS value for different choices of Δχ_{rh}/Δχ_{ax}. The dependence of PCSs on the nuclear coordinates expressed in a common frame makes these NMR observables useful for molecular structural characterization (see Sections 3.3 and 4.2).^{18–32 }
If α, β and γ are the Euler angles describing the rotation from any molecular frame to the frame with axes coinciding with the main directions of the χ tensor, the anisotropic part of the χ tensor in the molecular frame is given by
where R is the rotation matrix allowing for the change of the reference frame.^{33 } In summary, the PCSs can be calculated from the nuclear coordinates of the paramagnetic molecule and (i) the five components of the anisotropic part of the symmetric χ tensor (eqn (1.47)) or (ii) the two anisotropic parameters Δχ_{ax} and Δχ_{rh} and the three Euler angles describing the orientation of the main axes of the χ tensor (eqn (1.50) and (1.51)).
For an S=1 system with axial ZFS (from eqn (1.50) and (1.29)), to the first order in D/(kT),
Different to contact shifts (eqn (1.39)), for PCSs the term in T^{−2} is usually larger than the term in T^{−1}. Analogous equations can be calculated for other spin multiplicities.^{3 }
1.2.2.2 Breaking the Pointdipole Approximation
Unpaired electrons are certainly not localized on a single atom but spread over the ligand atoms through molecular overlap. The resulting effect is that the dipole–dipole interaction between the nucleus and all fractions of the distributed electron magnetic moment should be considered. In a semiclassical approach, this has been done by calculating the sum of pointdipole contributions arising from the fractional magnetic moments located in corresponding positions with high electron density.^{34 } Of course, the further the nucleus is from the paramagnetic center, the smaller the error in neglecting such an electron distribution: the dipolar interaction for nuclei far from the paramagnetic center is well approximated by considering the unpaired electron at a single point. It was shown, for instance, that in a high spin cobalt(ii) system, the pointdipole approximation provides well predicted shifts for nuclei at distances further than 8 Å from the metal.^{35 }
Recently, it has been shown that the hyperfine shifts outside the pointdipole approximation can be calculated by integration over space of eqn (1.44) after multiplication of the electron density function ρ(r_{e}):^{17 }
This corresponds to calculating the convolution
which can be done using the Fourier transform and the differential operators (corresponding to vectors in the reciprocal space). Numerical solutions of the inverse problem can provide the electron density function (under some regularization conditions, due to the illposed nature of the problem) from the pseudoconctact shifts and the atomic coordinates.^{17,36 }
1.2.3 Simplified Expressions in Limiting Cases
In the absence of lowlying electronically excited states, the thermally accessible electronic states are limited to the eigenstates of the ZFS, and when ZFS is absent (and thus Tr(〈SS〉)=S(S+1)). Therefore, in the limit of no ZFS and no interactions of the ground S multiplet with the excited states (second order Zeeman mixing), eqn (1.20) indicates that the three values of 〈μ_{S}〉for the three main directions of the gframe are
and, if there are also no orbital contributions (g=g_{e}1), from eqn (1.10) and (1.22),
In this simple case, S_{C} is directed along B_{0}.
Only in the case of no ZFS, of a singly populated S manifold, and of the validity of eqn (1.55) (electron Zeeman energy is much smaller than thermal energy, excited states are not coupled to the ground state, ), the principal axes of χ and g coincide, and eqn (1.25) provides
and if g is also isotropic (g=g_{e}1), the magnetic susceptibility becomes isotropic and equal to
which is known as Curie law. In these conditions, σ^{FC} is also isotropic and the contact shift (eqn (1.35) and (1.36)) is given by^{37 }
whereas the PCS is zero. On the other hand, in the case of the validity of eqn (1.57), from eqn (1.50),
1.2.4 The Effect of Partial Selforientation
The presence of anisotropy in the magnetic susceptibility tensor makes the energy of the different orientations of a molecule in a magnetic field also anisotropic. In fact, the energy related to the induced electron magnetic moment depends on the orientation of 〈μ_{S}〉 with respect to B_{0}, and thus, it is different depending on the magnetic field direction, as described by eqn (1.28). Therefore, the molecule does not have the same probability to sample each orientation, and a partial alignment occurs. The degree of alignment can be described by a probability tensor P. The latter indicates the probabilities that the magnetic field forms the angles α_{i} with the main directions i of the χ tensor. Therefore, using eqn (1.28), along the principal components of the χ tensor
where α_{x} and β_{x}, α_{y} and β_{y}, and α_{z} and β_{z} are the spherical angles describing the orientation of the magnetic field with respect to the x, y and z main axes of the χ tensor, respectively. The results of the integrals at the first order in E_{i}/kT are
i.e.,
As expected, P_{xx}+P_{yy}+P_{zz}=1. In eqn (1.61)–(1.63) the magnetic susceptibility to be considered is the overall magnetic susceptibility tensor, composed of the paramagnetic and the diamagnetic contributions. In fact, the interaction of the magnetic field with the motions of paired electrons in their orbitals may produce an anisotropic magnetic moment, due to the possible anisotropy of the electron current, which causes an anisotropic χ^{dia}. Therefore, the χtensor components contained in the σ tensor are not equal to those contained in the P tensor, although for simplicity they are indicated here with the same symbols. The diamagnetic susceptibility is roughly proportional to the molecular weight, and thus χ^{dia} might not be negligible in macromolecules or when multiple aromatic planes are stacked together, as in nucleic acids or bicelles.
The hyperfine shifts average to values which depend on the components of the shielding tensor σ along the main directions of the probability tensor P. Therefore, the PCSs are (see eqn (1.44))
and thus, in spherical coordinates,
where θ is the angle between the metal–nucleus vector r and the z axis of the diagonal χ tensor, and φ is an angle related to the projection of r on the xy plane of the tensor.
This correction to the PCS value is in most cases minor, and thus usually neglected. At very high fields (>20 T), when it may become important, the paramagnetic susceptibility becomes field dependent, as described by the Brillouin equation, and this affects the PCSs as much as the selforientation effect.^{38 } An analogous correction should also be applied to the contact shift, but again it is usually not taken into account because it is considered smaller than the error of the experimental data.
NMR shifts in paramagnetic molecules can also be affected by the residual anisotropic chemical shifts (RACS), which arise in the presence of partial alignment, and thus also of the selfalignment caused by the anisotropy of the χ tensor. These contributions to the shift originate from the anisotropy of the chemical shift (CSA) tensor, and should thus be evaluated for an accurate calculation of the PCSs, when the latter are obtained from the difference of the NMR shifts between the paramagnetic molecule and a diamagnetic analogue.^{39,40 }
1.3 The Paramagnetic Residual Dipolar Couplings
Partial alignment of a molecule causes the occurrence of residual dipolar couplings (RDCs). RDCs in fact result from the nonzero average of the dipole–dipole interaction energy between nuclear magnetic moments, occurring when not all molecular orientations have the same probability. Since the dipolar energy is equal to
where r_{12} is the vector connecting the coupled nuclei I_{1} and I_{2}, with magnetic moments oriented along the direction of B_{0}, the RDCs are given by
Therefore, paramagnetic RDCs (pRDCs) are described, in the main frame of the χ tensor, by
which, since =0 if , can be written as
and thus, in spherical coordinates (see previous section),^{41,42 }
where α is the angle between r_{12} and the z axis of the diagonal χ tensor and β is the angle which describes the position of the projection of the r_{12} vector on the xy plane of the χ tensor, relative to the x axis (Figure 1.3).
Eqn (1.70) has clearly the same functional form of the equation for the PCSs (eqn (1.50)); however the angles in the equation for the PCSs describe the position of the nucleus in the frame defined by the χ tensor, whereas the angles in the equation for the RDCs describe the orientation of the vector connecting the coupled nuclei in the same reference frame. Actually, the partial selforientation of a molecule depends on the overall χ tensor, composed of the paramagnetic and the diamagnetic parts. However, the RDCs arising from the diamagnetic anisotropy of χ are present also in the J measured for the diamagnetic sample, and thus, if the pRDCs are obtained from the difference in the Jcouplings measured for the paramagnetic and the diamagnetic samples, the diamagnetic contribution is subtracted. In this way, only the paramagnetic contribution to the anisotropy of χ, i.e. the same as is responsible for the PCSs, determines the obtained pRDCs. A further contribution to the Jcouplings should actually be considered, related to the presence of the dynamics frequency shift (see Section 1.5). Of note, the pRDCs, different to the PCSs, do not depend on the distance of the metal ion from the observed nuclei.
As shown for the PCSs, also the pRDCs can be calculated from the nuclear coordinates of the paramagnetic molecule and the magnetic susceptibility anisotropy tensor. The latter can be defined by (i) the five components of the anisotropic part of the symmetric χ tensor or (ii) the two anisotropic parameters Δχ_{ax} and Δχ_{rh} and the three Euler angles describing the orientation of the main axes of the χ tensor. In fact, the equation for the pRDCs can also be written in a generic reference frame, analogously to that of the PCSs (eqn (1.47)):
or, using the direction cosines,
where the indices i and j run over the three axes x, y and z, and φ_{i} are the angles between the r_{12} and each of the three axes. Eqn (1.72) can also be written as
In eqn (1.68)–(1.73) a further S_{LS} coefficient is often considered to indicate a generalized Lipari–Szabo order parameter.^{43 } The S_{LS} coefficient accounts for the effect of internal motions with correlation times shorter than the molecular reorientation time, and is thus able to reduce the observed pRDCs as a result of averaging due to the multiple orientations of the r_{12} vector with respect to the χ tensor.^{44 } Eqn (1.70), for instance, thus becomes
The pRDCs can be used together with the PCSs as restraints in molecular structure determination^{21,41,42,45 } (see Figure 1.3, and Sections 3.4 and 4.2).
1.4 The Paramagnetic Relaxation Enhancements
The magnetic interaction between electron and nucleus is also responsible for a mechanism for nuclear relaxation which adds to those present in diamagnetic molecules. When the interaction is with the electron spin density localized on the nucleus, the paramagnetic enhancement is indicated as nuclear contact (or scalar) relaxation, whereas when it is with the rest of the electron spin density, it is indicated as nuclear dipolar relaxation, and it is usually treated in the pointdipole approximation.
Paramagnetic relaxation occurs because the hyperfine coupling energy (eqn (1.4) and (1.8)) fluctuates randomly with time possibly due to three mechanisms:
(i) molecular reorientation, so that the orientation of the r vector changes with time;
(ii) electron relaxation, so that the orientation of the S vector changes with time (and thus the orientation of the dipolar field generated by the electron magnetic moment to the nuclear position);
(iii) chemical exchange, so that the magnitude of the r vector changes with time.
The time scales for these three mechanisms are described by three correlation times, i.e., the reorientation correlation time (τ_{r}), the electron relaxation time (τ_{e}) and the lifetime (τ_{M}), respectively.
The paramagnetic enhancements to the relaxation rates depend on the transition probabilities w_{ij} between the different states of an electron–nucleus spin system. These transition probabilities can be calculated using timedependent perturbation theory for stochastic perturbations:^{46,47 }
where i and j indicate the eigenstates of the unperturbed Zeeman Hamiltonian, ω_{ij} is the difference between the corresponding eigenvalues divided by ħ, and the bar denotes the ensemble average. In eqn (1.75) we suppose that the ensemble average is stationary, i.e. independent of the initial time, and that it decays to zero rapidly with respect to the integration time, so that the latter can be extended to infinity (Redfield limit).^{48–50 } The transition probabilities thus depend on the autocorrelation functions defined as
where it was assumed that the timecorrelation function decays exponentially,^{51 } with the abovementioned correlation times as the possible time constant τ_{c}. Finally,
where the last term is called the Lorentzian spectral density function
The longitudinal and transverse relaxation rate enhancements for a nucleus in a fixed position within a paramagnetic molecule, called paramagnetic relaxation enhancements (PREs), are indicated with R_{1M} and R_{2M} (R_{iM}=R$iMdip$+R$iMCurie$+R$iMFC$, see later). In the standard Solomon–Bloembergen–Morgan (SBM) model, largely used in the analysis of the relaxation rates, R_{1M} and R_{2M} are calculated from the dipolar and Fermicontact relaxation equations as provided later in eqn (1.84), (1.85), (1.87), (1.98) and (1.99). Due to the proportionality between NMR signal linewidth and transverse relaxation rate, a paramagnetic broadening of the NMR signal occurs, equal to
In solutions of paramagnetic molecules, we should consider the case that the observed nucleus (for instance, the water proton) is in a ligand which may reside in two types of environment: in the coordination sphere of the paramagnetic center and in the bulk. In these cases, the longitudinal and transverse relaxation rates and the NMR shifts measured for the ligand nuclei in solution differ from the values measured in the absence of the paramagnetic center. If R_{1p}, R_{2p} and Δω_{p} indicate the differences in relaxation rates and NMR shifts of solvent ligand nuclei between a paramagnetic solution and a diamagnetic reference solution:^{52–55 }
where Δω_{M} is the difference in NMR shift of the nucleus in the coordination sphere of the paramagnetic center and in a diamagnetic analogue molecule, and f_{M} is the mole fraction of ligand nuclei in bound positions. The chemical exchange rate, 1/τ_{M}, of the ligand bearing the observed nucleus between bulk and coordination sphere of the paramagnetic center thus acts as a limiting factor for the propagation of the effect from the nuclei at the paramagnetic site to the bulk nuclei.
For relatively small concentrations of the paramagnetic center, the paramagnetic relaxation enhancements of solvent nuclei (sPREs) are linear with respect to the concentration of the paramagnetic center. The relaxivity r_{1} is defined as the paramagnetic enhancement of the solvent nuclear relaxation rates in the presence of 0.001 mol dm^{−3} of paramagnetic centers in solution.^{56 } Therefore, the measured relaxation rate R_{1} of the solvent nuclei can be written as
where R$1dia$ is the diamagnetic relaxation and c is the concentration of the paramagnetic centers. Note that R_{1} in eqn (1.2) indicates the relaxation rate of a specific nucleus in the molecule, whereas in eqn (1.83) it indicates the relaxation rate of the solvent nuclei. The relationship between R_{1p} and R_{1M} is given in eqn (1.80).
1.4.1 Dipolar Relaxation
The paramagnetic relaxation due to the point dipole–point dipole interaction between the nuclear magnetic moment and the electron magnetic moment is traditionally split into two components, resulting from the separation of the electron spin S into a part s with zero average and the remaining part with thermal average S_{C} (see eqn (1.10)). The first term provides the Solomon equation, the second the Curie spin relaxation.
1.4.1.1 Solomon Equation
Solomon calculated the contribution to relaxation due to the dipole–dipole interaction between an electronic spin s with isotropic g=g_{e}1 and a nuclear spin, in the hypothesis that the static Hamiltonian contains only the Zeeman interactions of the nuclear and electron magnetic moments with the external magnetic field.^{46 } The difference in energy in the zero quantum (w_{0}), single quantum (w_{1}) and double quantum (w_{2}) transitions are ω_{S} – ω_{I}, ω_{I} and ω_{S}+ω_{I}, respectively (ω_{S}>0, ω_{I}<0, see Figure 1.4 and eqn (1.77)) and since the electron Larmor frequency ω_{S} is 658.2 times larger than the nuclear Larmor frequency ω_{I} (so that ω_{S}±ω_{I}≈ω_{S}), it was found
The dipole–dipole interaction can be modulated by τ_{r}, τ_{1e}/τ_{2e} (the longitudinal/transverse electron relaxation time) and/or τ_{M} because each of them can contribute to the decay of the correlation function. Therefore, the correlation times are given by
If τ_{1e}=τ_{2e} (= τ_{e}), or both τ_{1e} and τ_{2e} are much larger than τ_{r} or τ_{M}, τ$cdip$=τ$c1dip$=τ$c2dip$. The field dependences of the relaxation rates are shown in Figure 1.5, and are characterized by the presence of two Lorentzian dispersions centered at frequencies ω_{S}=(τ$cdip$)^{−1} and ω_{I}=(τ$cdip$)^{−1}, and thus separated by a factor 658.2. In the case of lanthanoids and actinoids, the J quantum number substitutes the S quantum number and g_{J} substitutes g_{e}.
When electron relaxation is important in the definition of the correlation time, its field dependence should also be taken into account. A field dependent τ_{e} is expected, for instance, for all systems with S>1/2 and thus possibly experiencing ZFS. Deformations of the coordination polyhedron by collision with solvent molecules in fact cause a transient ZFS even in high symmetry systems without static ZFS. This transient ZFS allows for the coupling of distortional motions with spin transitions and provides a relaxation mechanism for the electron(s), so that^{57,58 }
where Δ$t2$ is the mean squared fluctuation of the ZFS and τ_{v} is the correlation time for the instantaneous distortions of the metal coordination polyhedron. The field dependence of the nuclear longitudinal relaxation rate in the presence of field dependent electron relaxation shows a typical relaxation peak at about 1 T, due to the increase in the electron relaxation time with the increase of the field, described by eqn (1.87), and the subsequent decrease due to the ω_{I}τ$cdip$=1 dispersion present in eqn (1.84) (see Figure 1.6).
Internal (intramolecular) motions can also change the r vector and thus be able to (at least partially) modulate the dipole–dipole interaction. In the simple Lipari–Szabo modelfree treatment^{43,59 } (see Section 1.3), eqn (1.84) should thus be replaced by
where (τ$fdip$)^{−1}=(τ$cdip$)^{−1}+τ$l\u22121$ and τ_{l} is the correlation time for the local (fast) reorientation of the nucleusunpaired electron(s) vector. An analogous equation can be written for transverse relaxation.
1.4.1.2 Anisotropic Effects
Eqn (1.84) and (1.85) were derived by assuming (i) an isotropic electron g tensor and (ii) that the electron spin system is dominated by the electronic Zeeman interaction (besides the fact that reorientation, chemical exchange and electron relaxation are uncoupled and that the electron spin behaves as a pointdipole). The deviations with respect to the Solomon behaviour in the presence of an anisotropic electron g tensor were analyzed both in the case of fast rotation^{60 } and of slow rotation (with respect to the electron relaxation time)^{61 } and it was found that they are usually small.
In the presence of static ZFS and/or hyperfine coupling between the electron(s) and the metal nuclear magnetic moment, the energy of the nuclear and electron spin states can be very different from what is predicted using only the Zeeman term. Therefore, the energy of most electronic spin transitions can be much larger than calculated in the absence of these terms, and thus ω_{S}τ_{c} can be much larger than one even at relatively large frequency. Furthermore, an additional dispersion in the field dependence of the relaxation rate may appear corresponding to the transition from a dominant ZFS to a dominant Zeeman energy.^{62 } The extent of these effects largely depends on the magnitude and rhombicity of static ZFS and/or hyperfine coupling between electron(s) and the metal nucleus, as well as on the position of the nucleus with respect to these tensors (Figure 1.7). In the socalled slow rotation limit (τ_{r}≫τ_{e}) and within the Redfield limit, the (modified) Florence NMRD program can be used to evaluate the field dependent relaxation rates as a function of these parameters.^{63–65 } Outside the slow rotation limit and/or the Redfield limit, more complicated approaches should be used.^{47,48,66–68 }
1.4.1.3 Outersphere Relaxation
In the absence of ligand molecules coordinated to the paramagnetic center, but diffusing around it, there is still a paramagnetic contribution to the relaxation of the solvent ligand nuclei. This contribution, called the outersphere relaxation, depends on the diffusional correlation time τ_{D} defined as
where d is the distance of closest approach between the ligand and the paramagnetic center, and D_{M} and D_{L} are the diffusion coefficients of the paramagnetic molecule and of the ligand molecule, respectively. Assuming that unpaired electron(s) and the ligand nucleus are in the center of hard sphere spherical molecules, according to one of the most commonly used models for diffusion^{69,70 }
where N_{A} is the Avogadro's constant, [M] is the concentration of the paramagnetic centers, expressed in mol dm^{−3}, and the spectral density functions are given by
with
or, in the case of τ_{D} and τ_{e} having comparable values,^{71 }
1.4.2 Curie Spin Relaxation
The paramagnetic relaxation due to the point dipole–point dipole interaction between the nuclear magnetic moment and thermal averaged electron spin S_{C} is described by the following equations (see eqn (1.56)),
where only the contribution from the single quantum (w_{1}) transition is present. These equations were derived assuming an isotropic g tensor (g=g_{e}1) and in the absence of ZFS and interactions of the ground S multiplet with the excited states.^{72 } Equations were also derived in order to consider the presence of g anisotropy, ZFS and/or interactions with excited states, and in these cases they contain the elements of the χ tensor (A · S_{C}∞g · 〈SS〉 · g^{T} · B_{0}∞χ · B_{0}).^{73 } Since in Curie spin relaxation the dipolar interaction is only with the average over all electron spin states, it can be modulated by reorientation and chemical exchange only:
In the case of lanthanoids and actinoids, the J quantum number substitutes the S quantum number and g_{J} substitutes g_{e}. Even better, the term g$J4$μ$B4$J^{2}(J+1)^{2} can be replaced with the experimental μ$eff2$ value, when available.^{74 }
Although the value of 〈S_{z}〉^{2} is much smaller than the value of S(S+1)/3, which appears in the Solomon equation, Curie spin relaxation may be of the same order or larger than dipolar relaxation (eqn (1.84) and (1.85)), especially for the transverse relaxation time, when the correlation time in the Solomon equation is determined by the electron relaxation time because it is much smaller than the reorientation time. Furthermore, Curie spin relaxation increases with the square of the magnetic field, i.e. of the nuclear Larmor frequency ω_{I}=γ_{I}B_{0}.
1.4.3 Fermicontact Relaxation
In the case of relaxation by contact coupling, only the contribution from the zero quantum (w_{0}) transition is present and the relaxation rates, when g=g_{e}1 and in the absence of ZFS and coupling of the ground state with excited states, are equal to^{75 }
The field dependence of the contact relaxation rates is characterized by a single Lorentzian dispersion centered at frequency ω_{S}=(τ$cFC$)^{−1} (Figure 1.8). In this case, reorientation cannot modulate the contact coupling, and thus only chemical exchange and electron relaxation can determine the correlation time, so that
In the presence of possible internal (intramolecular) reorientation with correlation time τ_{l} able to change the electron spin density on the nucleus, contributions from spectral density functions containing the correlation time
may also be present and accounted for with the Lipari–Szabo modelfree approach^{43 } (see Section 1.4.1.1).
In theory, Curie spin relaxation may also occur through contact coupling. Since Curie spin relaxation requires a correlation time much larger than the electron relaxation time τ$cFC(Curie)$=τ_{M} and thus (for ω_{S}τ_{M}≫1), using eqn (1.56),
where the term in parenthesis is related to the contact shift. Again, possible contributions arising with (τ$M\u22121$+ τ$l\u22121$)^{−1} instead of τ_{M} should be considered in the presence of intramolecular motions.
1.5 Paramagnetic Cross Correlation Effects
Cross correlation effects arise in the presence of two (or more) different types of interactions modulated by the same motion. This results in an interference between the two interactions, related to the fact that the transition probabilities contain the square of the expectation value of all timedependent Hamiltonian terms (in eqn (1.77) only H_{hc} is indicated), which differs from the sum of the squares of the single terms. In paramagnetic systems, one of the most important cross correlation effects is that arising in the presence of dipole–dipole interaction between the magnetic moments of two nuclei and the dipole–dipole interaction between the magnetic moment of one nucleus and the average electron magnetic moment (responsible for Curie spin relaxation). Both of these interactions can in fact be modulated by the reorientation correlation time. In contrast, when electron relaxation is faster than molecular reorientation, there is no cross correlation arising from the dipole–dipole interaction between the magnetic moments of two nuclei and the dipole–dipole interaction between the magnetic moment of one nucleus and the electron magnetic moment (responsible for Solomon relaxation).
The cross term either decreases or increases the relaxation rates, depending on S being in the 1/2 or −1/2 spin state, and this causes a difference in nuclear linewidths.^{76 } This difference for, e.g., the two proton dimension components of the HN doublet in HSQC spectra of ^{15}N enriched paramagnetic proteins, calculated in the assumption of isotropic χ tensor, is given by^{76–79 }
The angle θ_{MHN} is that between the HN axis and the Hmetal axis, and τ_{c} is the correlation time modulating both relaxation processes. Eqn (1.103) can also be generalized to treat the case of anisotropic magnetic susceptibility.^{80 }
The imaginary part of the cross correlation term causes a dynamic frequency shift,^{51 } which results in a change of the position of the individual lines of a multiplet. This effect sums to the effect arising in the presence of partial selforientation (called RDCs, see Section 1.3, which in general can have both a diamagnetic and a paramagnetic origin), so that
and^{79 }
For large correlation times and/or at high magnetic fields, the dynamic frequency shift is independent of the value of the correlation time, of the nucleus observed, and of the magnetic field, as . The magnitude of Δv^{dfs} is usually very small with respect to the paramagnetic line broadening and pRDCs.
Also the interference between Curie and CSA relaxation causes a cross correlation effect,^{81 } which results in decreased or increased nuclear relaxation rates depending on the relative orientation of the principal axes of CSA and shielding tensors. The magnitude of this cross correlation effect is also usually very small, unless the Curie spin is so large that paramagnetic relaxation is predominantly due to the Curie rather than the dipolar contribution.
1.6 First Principles Calculation of Hyperfine Shifts
In quantum chemistry computations, the A^{DIP} tensor, in the nonrelativistic approximation, refers to the interaction between the electron spin magnetic moment, without considering the orbital contributions, and the nuclear magnetic moment, so that^{6 }
If g is separated into (g_{e}+Δg_{iso})1+Δg, the contact shift (see eqn (1.35) and (1.36)) is split into two components (g_{e}A^{FC}〈SS〉 and Δg_{iso}A^{FC}〈SS〉) with nonnull isotropic average and one traceless component (A^{FC}Δg · 〈SS〉).^{4,8,82 } The electron spin–nuclear spin dipole–dipole shift (see eqn (1.42) and (1.43)) is provided by the term Δg · 〈SS〉 · A^{DIP}, the trace of which can be different from zero, thus providing a nonnull isotropic average. Further traceless anisotropic components are provided by the terms g_{e}〈SS〉 · A^{DIP} and Δg_{iso}〈SS〉 · A^{DIP}, as well as by spin–orbit correction terms.^{4,8,82 } The spin–orbit correction provides three further components of the hyperfine coupling tensor, namely the spin–orbit contribution to the isotropic hyperfine coupling constant and to the hyperfine anisotropy, and an antisymmetric spin–orbit contribution.^{83 } All together these terms, however, do not recover the interaction between the orbital magnetic moment and the nuclear magnetic moment, as described in Kurland and McGarvey.^{3 }
First principles calculations of the hyperfine shifts require the evaluation of the g, D (ZFS, for S>1/2) and A tensors by taking into account all thermally accessible energy levels of the electron spin states. In the pointdipole approximation, the electron spin–nuclear spin dipole–dipole shifts can be predicted from the molecular structure after calculation of the g and D tensors (D is needed to calculate 〈SS〉). Contact shifts require the determination of g, 〈SS〉 and A^{FC} (see eqn (1.36)). Of course, an accurate treatment of spin–orbit coupling is necessary for g tensor calculations. Whereas the hyperfine coupling tensor can be calculated using densityfunctional theory (DFT), in several cases DFT is inaccurate for the evaluation of the D tensor and the g tensor.^{84–86 } These tensors can be better estimated using multireference wavefunction theories (WFT) at the complete active space selfconsistent field (CASSCF) and nelectron valence state perturbation theory 2 (NEVPT2) levels,^{4 } and a molecular model comprising the nuclei in closest proximity to the paramagnetic center. Using this approach, the electron spin–nuclear spin dipole–dipole shifts were calculated for the nuclei of a highspin cobalt(ii) protein from its crystal structure, and were found in agreement with experimental PCSs.^{35 } The contribution to the PCSs from the dipole–dipole interaction between electron orbital magnetic moment and nuclear magnetic moment is likely masked by inaccuracies in the calculation of the D tensor, also arising from possible slight inaccuracies in the protein structural model.
In the absence of ZFS, DFT calculations of hyperfine coupling constants, g tensor and orbital shieldings gave accurate enough results to provide contact shifts in good agreement with the experimental data, for instance, for nitroxide radicals,^{82,87 } chromium(iii), manganese(iii), iron(iii), and copper(ii) acetylacetonates, with the metal ions in the high spin states (Cr: S=3/2, Mn: S=2, Fe: S=5/2, Cu: S=1/2),^{87 } after geometry optimization of a structural model at the B3LYP/ccpVTZ level, and for paramagnetic iron–sulfur proteins, as in the case of iron(iii) rubredoxin.^{88 }
1.7 Metal Ion Dependence of the Paramagnetic Effects
The key parameters determining the magnitude of PCSs, pRDCs and PREs are the magnetic susceptibility anisotropy and the electron relaxation time.^{89 } Metal ions with very low paramagnetic susceptibility anisotropy are gadolinium(iii), manganese(ii), and type2 copper(ii). Therefore, they produce very small (usually undetectable) PCSs and pRDCs. On the other hand, these metal ions have a long electron relaxation time, which induces relatively large PREs (see Table 1.1 and Figure 1.9). In fact, in macromolecules with large reorientation time, the correlation time τ_{c} is often determined by the electron relaxation time. As is clear from eqn (1.85), the larger τ_{c}, the larger R_{2M}, and thus the paramagnetic broadening of the observed NMR signals. The PREs may make the signals of nuclei relatively close to the metal so broad as to be undetectable.
.  g_{e}/g_{J} .  S/J .  τ_{e} [ps] .  χ [10^{−32} m^{3}] .  Δχ_{ax} [10^{−32} m^{3}] . 

Ti(iii)  2.0  1/2  40  2.6  0.4 
VO^{2+}  2.0  1/2  500–2000  2.6  0.1 
Cr(iii)  2.0  3/2  1000  13  <0.1 
Mn(ii)  2.0  5/2  10 000  31  <0.1 
Mn(iii), high spin  2.0  2  50  21  0.5 
Fe(iii), high spin  2.0  5/2  10–1000  31  0.6–3 
Fe(iii), low spin  2.0  1/2  1  2.6  2.4 
Fe(ii), high spin  2.0  2  1  21  2.1 
Co(ii), high spin, hexacoord.  2.0  3/2  1  13  7 
Co(ii), high spin, tetracoord.  2.0  3/2  10  13  3 
Co(ii), low spin  2.0  1/2  100  2.6  1 
Ni(ii), hexacoord.  2.0  1  100  7  2 
Ni(ii), tetracoord.  2.0  1  1  7  1.5 
Type1 Cu(ii)  2.0  1/2  500  2.6  0.6 
Type2 Cu(ii)  2.0  1/2  3000  2.6  0.6 
Ce(iii)  6/7  5/2  0.1  5.6  2 
Pr(iii)  4/5  4  0.1  11  3.5 
Nd(iii)  8/11  9/2  0.2  11  1.7 
Sm(iii)  2/7  5/2  0.1  0.6  0.2 
Eu(iii)  —  0a  0.01  0a  2.4 
Gd(iii)  2.0  7/2  10 000  55  0.2 
Tb(iii)  3/2  6  0.3  83  40 
Dy(iii)  4/3  15/2  0.5  99  35 
Ho(iii)  5/4  8  0.3  99  20 
Er(iii)  6/5  15/2  0.3  80  12 
Tm(iii)  7/6  6  0.5  50  25 
Yb(iii)  8/7  7/2  0.3  18  8 
.  g_{e}/g_{J} .  S/J .  τ_{e} [ps] .  χ [10^{−32} m^{3}] .  Δχ_{ax} [10^{−32} m^{3}] . 

Ti(iii)  2.0  1/2  40  2.6  0.4 
VO^{2+}  2.0  1/2  500–2000  2.6  0.1 
Cr(iii)  2.0  3/2  1000  13  <0.1 
Mn(ii)  2.0  5/2  10 000  31  <0.1 
Mn(iii), high spin  2.0  2  50  21  0.5 
Fe(iii), high spin  2.0  5/2  10–1000  31  0.6–3 
Fe(iii), low spin  2.0  1/2  1  2.6  2.4 
Fe(ii), high spin  2.0  2  1  21  2.1 
Co(ii), high spin, hexacoord.  2.0  3/2  1  13  7 
Co(ii), high spin, tetracoord.  2.0  3/2  10  13  3 
Co(ii), low spin  2.0  1/2  100  2.6  1 
Ni(ii), hexacoord.  2.0  1  100  7  2 
Ni(ii), tetracoord.  2.0  1  1  7  1.5 
Type1 Cu(ii)  2.0  1/2  500  2.6  0.6 
Type2 Cu(ii)  2.0  1/2  3000  2.6  0.6 
Ce(iii)  6/7  5/2  0.1  5.6  2 
Pr(iii)  4/5  4  0.1  11  3.5 
Nd(iii)  8/11  9/2  0.2  11  1.7 
Sm(iii)  2/7  5/2  0.1  0.6  0.2 
Eu(iii)  —  0a  0.01  0a  2.4 
Gd(iii)  2.0  7/2  10 000  55  0.2 
Tb(iii)  3/2  6  0.3  83  40 
Dy(iii)  4/3  15/2  0.5  99  35 
Ho(iii)  5/4  8  0.3  99  20 
Er(iii)  6/5  15/2  0.3  80  12 
Tm(iii)  7/6  6  0.5  50  25 
Yb(iii)  8/7  7/2  0.3  18  8 
Eu(iii) is paramagnetic due to the excited states.
Metal ions with short electron relaxation times, in contrast, determine small PREs (if Curie spin relaxation can be neglected) and thus a small broadening of the NMR lines, so that the quality of the NMR spectra of the diamagnetic analogues is preserved. This is the case of lowspin iron(iii), highspin iron(ii), titanium(iii), nickel(ii), cobalt(ii) and type1 copper(ii). Most of these also have a relatively small, but still sizable, paramagnetic susceptibility anisotropy, so that PCSs can be measured. High spin iron(iii) has a paramagnetic susceptibility anisotropy of similar magnitude, but PCSs and pRDCs can hardly be observed in large macromolecules due to the occurrence of Curie spin relaxation.
Paramagnetic lanthanoid(iii) ions except gadolinium(iii) may have paramagnetic susceptibility anisotropies from small to very large (see Section 2.2.2).^{90–92 } They also have electron relaxation times smaller than 10^{−12} s. However, the R_{2M} in lanthanoid(iii) macromolecules, and thus the paramagnetic line broadening, can be very large due to Curie spin relaxation (which also depends on J and g_{J}). Paramagnetic lanthanoid(iii) ions with large paramagnetic susceptibility anisotropy are terbium(iii), dysprosium(iii), thulium(iii) and holmium(iii). They thus produce large PCSs and pRDCs outside the large blind region of these metal ions, where no signal can be detected due to the very large line broadening. Cerium(iii), neodymium(iii) and europium(iii) have a much smaller paramagnetic susceptibility anisotropy, and also a smaller Curie spin relaxation, and erbium(iii) and ytterbium(iii) are characterized by intermediate values.
1.8 The Overhauser Effect in Paramagnetic Systems
The nuclear Overhauser effect (NOE) is the result of the transfer of spin population (called polarization transfer) between two nuclei which are dipole–dipole coupled, upon irradiation of one of the nuclei. When the equilibrium population of the two energy levels (with different spin state) of one nucleus is altered by radiofrequency irradiation at the frequency corresponding to the transition between the two spin states, also the population of the second, dipole–dipole coupled nucleus is perturbed. Therefore, selective irradiation of one signal produces a change in the intensity in the other signal.
If the nuclear spin I is observed, and the dipole–dipole coupled nuclear spin J is saturated for a time long enough that the system reaches a new steady state equilibrium, the steady state NOE η_{I( J)} is given by
where ρ_{I} represents the longitudinal relaxation rate of spin I and σ_{I(J)} is called the crossrelaxation rate
The nucleus I is coupled to the nucleus J and to many other nuclei. In paramagnetic systems, the nucleus I is also coupled to the paramagnetic center, so that ρ_{I} is the sum of the contributions arising from all these interactions:
where ρ_{I(J)} represents the contribution to the longitudinal relaxation rate of spin I due to the interaction with spin J upon reciprocal reorientation (see Section 1.4.1.1)
and R$1MI$ is its paramagnetic relaxation rate. Therefore, the homonuclear NOE is
Of course, the larger the denominator in eqn (1.111), the smaller the NOE. Therefore, when the R_{1M} term is larger than the other terms, relaxation can be very effective and the NOE very small.
The support from Fondazione Cassa di Risparmio di Firenze, MIUR PRIN 2012SK7ASN, European Commission project pNMR No. 317127, and InstructERIC, a Landmark ESFRI project, is acknowledged. Discussions with Ladislav Benda, Martin Kaupp, Jozef Kowalewski, Jiři Mareš, Guido Pintacuda and Juha Vaara are acknowledged.