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Magnetic susceptibility and paramagnetism-based NMR
Accepted Manuscript Magnetic susceptibility and paramagnetism-based NMR Giacomo Parigi, Enrico Ravera, Claudio Luchinat PII: DOI: Reference:
S0079-6565(19)30019-6 https://doi.org/10.1016/j.pnmrs.2019.06.003 JPNMRS 1481
To appear in:
Progress in Nuclear Magnetic Resonance Spectroscopy
Received Date: Accepted Date:
8 April 2019 17 June 2019
Please cite this article as: G. Parigi, E. Ravera, C. Luchinat, Magnetic susceptibility and paramagnetism-based NMR, Progress in Nuclear Magnetic Resonance Spectroscopy (2019), doi: https://doi.org/10.1016/j.pnmrs.2019.06.003
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Magnetic susceptibility and paramagnetism-based NMR
Giacomo Parigi, Enrico Ravera, Claudio Luchinat*
Magnetic Resonance Center (CERM) and Interuniversity Consortium for Magnetic Resonance of Metallo Proteins (CIRMMP), Via L. Sacconi 6, 50019 Sesto Fiorentino, Italy and Department of Chemistry “Ugo Schiff”, University of Florence, Via della Lastruccia 3, 50019 Sesto Fiorentino, Italy
Edited by Geoffrey Bodenhausen and Gareth A. Morris
* email: [email protected] Tel. +39 055 4574296
Keywords: hyperfine shift, pseudocontact shift, residual dipolar coupling, dipolar relaxation, Curie relaxation
1
Abstract The magnetic interactions between the nuclear magnetic moment and the magnetic moment of unpaired electron(s) depend on the structure and dynamics of the molecules where the paramagnetic center is located and of their partners. The long-range nature of the magnetic interactions is thus a reporter of invaluable information for structural biology studies, when other techniques often do not provide enough data for the atomic-level characterization of the system. This precious information explains the flourishing of paramagnetism-assisted NMR studies in recent years. Many paramagnetic effects are related to the magnetic susceptibility of the paramagnetic metal. Although these effects have been known for more than half a century, different theoretical models and new approaches have been proposed in the last decade. In this review, we have summarized the consequences for NMR spectroscopy of magnetic interactions between nuclear and electron magnetic moments, and thus of the presence of a magnetic susceptibility due to metals, and we do so using a unified notation.
Table of Contents 1.
Introduction
2.
Why bother with magnetic susceptibility?
3.
Angular momenta and magnetic moments
3.1 The Curie spin 4.
The paramagnetic susceptibility
4.1 Saturation at high field 5. The hyperfine shift 5.1 The semi-empirical paramagnetic NMR approach 5.2 Perturbative spin Hamiltonian EPR approach 5.3 Hyperfine shift from first-principles quantum chemistry 2
5.4 The limiting case of D<
Paramagnetism-induced self-orientation
6.1 NMR shifts in self-orienting paramagnetic molecules 6.2 Paramagnetic residual dipolar couplings 7. Paramagnetic relaxation enhancements 7.1 Dipolar relaxation 7.2 Curie spin relaxation 7.3 Fermi contact relaxation 7.4 Anisotropic effects 8. Paramagnetic cross-correlation effects 9. Conclusions Acknowledgements References
3
Glossary CSA
chemical shift anisotropy
NMR nuclear magnetic resonance RDC residual dipolar coupling SBM Solomon-Bloembergen-Morgan TIP
temperature independent paramagnetism
ZFS
zero field splitting
4
1. Introduction The number of research reports dealing with NMR of paramagnetic systems has dramatically increased in the last two decades, mostly thanks to the impressive developments in experimental NMR that have made it easier and more profitable to take account of paramagnetic effects in biological [1–7] and biomedical investigations [8–12], and in material sciences [13–16]. The NMR data (shifts and relaxation rates) collected for systems that contain a paramagnetic center contain valuable information on their structure and dynamics, which becomes particularly relevant for the study of large complexes and/or in the presence of conformational heterogeneity [17–19]. It is also important to recall that technological advances in fast magic angle spinning, robust radiofrequency pulse schemes, and reliable sample preparation techniques have also allowed the study of paramagnetic effects in solids [20–26]. These recently appreciated advantages of paramagnetism have stimulated the development of the theory describing hyperfine shifts and paramagnetic relaxation enhancements [27], both in the framework of semi-empirical quantum chemistry approaches [28–30] and of a first-principles quantum chemistry approach [31–39]. The results obtained in these two frameworks are, however, not always in agreement with one another [40], and experimental validation is needed. A step in this direction has been recently made [41], and the experimental results are in agreement with the semiempirical approach and not with the quantum chemistry approach. Besides providing a critical assessment of recent theoretical work, here we recapitulate in a unified frame the concepts and equations used for the analysis of NMR data of paramagnetic systems using the semi-empirical quantum chemistry approach, keeping the language and the formalism as simple as possible.
2. Why bother with magnetic susceptibility? As outlined in a previous review from our group [42], paramagnetic NMR data are in large part related to the molecular paramagnetic susceptibility. This is easily understood in the following case: 5
a nucleus immersed in a static, constant magnetic field B0 in vacuo experiences a Zeeman splitting (vide infra) which is proportional to B0. If another object is introduced into the same field, in close proximity to the probe nucleus, the object will perturb the magnetic field experienced by our probe nucleus. The object could be diamagnetic, in which case it will act to decrease the static magnetic field along the field axis (and reinforce the field in the plane), or it could be paramagnetic, in which case it will strengthen the static magnetic field along the axis (and reduce it in the plane). In either case, the probe nucleus will sense a different Zeeman splitting. The alteration of the field is usually described by means of the magnetic susceptibility. Moving towards a more practical description, let us imagine that the object is a single paramagnetic center. At a reasonable distance, the field distortion it causes is described by the molecular paramagnetic susceptibility and approximated by a dipolar field. In general, the magnetic susceptibility will be anisotropic, depending on the symmetry around the paramagnetic center. This leads us to an expression for the dipolar shift of the form 1
𝛿dip = 4𝜋𝑟3𝛋 ⋅ 𝛘 ⋅
(
3𝐫𝐫𝑇 𝑟2
)
‒ 𝟏 ⋅ 𝛊,
(1)
where r is the vector between the paramagnetic center and the observed nucleus, the symbol
T
denotes transposition, and and are unit vectors in the direction of the magnetic field and of the quantization axis of the spin I of the observed nucleus, respectively. The rotational average of the dipolar shift is known as the pseudocontact shift. Assuming isotropic reorientation of the molecule with respect to the magnetic field (see Section 6.1), the pseudocontact shift 𝛿pc can be calculated to be
[ (
1
𝛿pc = 12𝜋𝑟3Tr 𝛘 ⋅
3𝐫𝐫𝑇 𝑟2
‒𝟏
)]
(2)
where Tr denotes the trace of a matrix. Other forms of this equation are given in Section 4.1. Nuclear paramagnetic relaxation enhancements (and thus NMR linewidths) can also depend on the paramagnetic susceptibility, through the contribution arising from the dipole-dipole interaction 6
between the averaged electron magnetic moment and the nuclear magnetic moment (Curie spin relaxation). In the absence of anisotropy of the 𝛘 tensor, the Curie contribution to relaxation is (see Section 7.2) 2
R1Curie M
2 3τ cCurie 2 1 ω I2 iso 5 4π r 6 1 ω I2 ( τ cCurie ) 2
2
R
Curie 2M
2 1 1 ωI2 iso 5 4π r6
3τ cCurie 4 τ c 1 ωI2 ( τ cCurie ) 2
(3)
(4)
where τ cCurie is the correlation time for Curie spin relaxation (usually determined by molecular reorientation) and 𝜒iso is the rotational average of the 𝛘 tensor (𝜒iso = Tr(𝛘)/3). More appropriate equations, accounting for the anisotropy of the 𝛘 tensor, are discussed in Section 7.4. The magnetic susceptibility is also important for the description of cross-correlation effects like the one between dipole-dipole relaxation and Curie spin relaxation (see Section 8). Magnetic susceptibility anisotropy also causes different orientations of a molecule in a magnetic field to have different probabilities, thus resulting in partial self-alignment. This induces residual dipolar couplings (RDCs) as a consequence of the non-zero rotational average of the square of the nucleus-nucleus dipole-dipole interaction energy, analogous to the RDCs arising when partial alignment is induced by external orienting effects such as liquid crystal media. Paramagnetic RDCs have the additional advantage that intermolecular interactions with external orienting media that may perturb the structure and dynamics of the investigated molecule are excluded. Paramagnetic RDCs depend on the paramagnetic susceptibility anisotropy Δ𝛘 according to Eq. (5) 12prdc
1 B02 3 I 1 I 2 T r12 χ r12 4 15kT 2r125
(5)
where r12 is the vector between the coupled nuclei 1 and 2. Other forms of this equation are provided in Section 6.2. There is yet another paramagnetic contribution to the NMR shift, the Fermi contact shift, that appears when unpaired electron spin density is present on the observed nucleus. In this case the shift is due to the interaction between the “bare” electron spin magnetic moment and the nuclear magnetic 7
moment, and therefore is not directly related to the magnetic susceptibility. However, it can still be expressed in terms of the paramagnetic susceptibility by dividing the latter by the electron g tensor, according to Eq. (6) 𝐴Fc
𝛿Fc = 3𝜇0ℏ𝛾𝐼𝜇BTr[𝛘 ∙ [𝐠𝑇]
‒1
]
(6)
where 𝐴Fc is the constant for the Fermi contact interaction (see Section 5.1). More comments on this intrinsic asymmetry between the physics behind the contact and the pseudocontact interactions will be given in section 5.3. In most cases, it is very difficult to obtain an experimental measure of the molecular paramagnetic susceptibility, and particularly of its anisotropy. Therefore, all paramagnetic NMR observables have been derived in terms of parameters that are either experimentally observable or that can be directly calculated, either from first-principles or on semi-empirical grounds. Since the magnetic susceptibility of a paramagnetic center is linked to the total effective magnetic moment of the center itself, in the following sections we will show how it is possible within the semi-empirical framework to establish a link between the total effective magnetic moment and paramagnetic NMR observables. First-principles quantum chemistry studies have recently focused on the derivation of the hyperfine shifts, i.e., of the Fermi contact and pseudocontact shifts. Using this theoretical approach, developed with the spin-Hamiltonian formalism, the same equation of Eq. (6) is obtained for the Fermi contact shift, whereas the equation for the pseudocontact shift contains an additional factor 𝑔e
[𝐠𝑇] ‒ 1 (which multiplies the 𝛘 tensor, see Section 5.3) that is missing in Eq. (2). The presence of this factor in the equation derived within the quantum chemistry approach apparently re-establishes a symmetry between the Fermi contact and the pseudocontact terms, but introduces a fundamentally different way of looking at the effect of the presence of a paramagnetic center on distant nuclei. However, we will show that there need not be any symmetry between contact and pseudocontact shifts, and that this asymmetry originates physically from the presence of a contribution from the interaction between the orbital magnetic moment and the nuclear magnetic moment when calculating 8
the psedocontact shift, and the absence of any contributions of this kind when calculating the Fermi contact shift (see Section 5).
3. Angular momenta and magnetic moments The (effective) spin angular momentum 𝐒 of the unpaired electron(s) present in a paramagnetic metal ion has an associated magnetic moment 𝛍𝑆 =‒ 𝜇B𝑔e𝐒.
(7)
The motion of the unpaired electron(s) in their orbitals also introduces an electron orbital angular momentum 𝐋, so that the orbital magnetic moment is 𝛍𝐿 =‒ 𝜇B𝐋.
(8)
We here note that the total orbital angular momentum operator 𝐋 is only Hermitian (corresponding to an observable) in a spherical potential, which is strictly valid only for bare atoms. The following sections assume the validity of the Russell-Saunders (LS) coupling regime, which requires that the spin-spin coupling and orbit-orbit coupling between electrons are much stronger than the spin-orbit coupling. This approximation is considered good for elements up to the first row transition series; for heavier elements, the j-j coupling scheme should be used, because the individual coupling between electrons, via the spin-orbit interaction, is stronger than the electrostatic interaction between them. In the semi-empirical quantum chemistry approach, the interaction between the orbital and spin angular momenta of the electron(s) is represented by a Hamiltonian term of the form 𝐻𝐿𝑆 = 𝜆so𝐋 ∙ 𝐒
(9)
In the first-principles quantum chemistry approach, the orbital and spin angular momenta of the individual electrons are considered, so that the spin-orbit coupling is written as [36] 𝐻𝑆𝑂𝐶 = ∑𝑖𝐳(𝐫𝑖)𝐬𝑖 9
(10)
where 𝐬𝑖 is the spin of the ith electron and 𝐳(𝐫𝑖) contains the three spatial coordinates on which the orbital angular momenta of the electrons depend [35]. For the sake of simplicity, in the following Eq. (9) will be used. We indicate with |𝜓(0) 𝑖 > the ground (i = 0) and excited electronic configurations of the system composed of the paramagnetic metal ion in its coordination scaffold, and with 𝐸(0) 𝑖 the corresponding energy, as determined from crystal field theory. Since for transition metal ions the spin-orbit coupling is usually much smaller than the crystal field energy, it can be considered to cause only a perturbation in the electronic configuration established by the crystal field. Therefore, the eigenstates of the (1) system from first-order perturbation theory are |𝜓𝑖 > = |𝜓(0) 𝑖 > +|𝜓 𝑖 > with
|𝜓(1) 𝑖 > = ∑𝑗 ≠ 𝑖
(0) < 𝜓(0) 𝑗 |𝜆so𝐋 ∙ 𝐒|𝜓 𝑖 > (0) 𝐸(0) 𝑖 ‒𝐸 𝑗
|𝜓(0) 𝑗 >
(11)
This implies that the unperturbed excited states are mixed with the unperturbed ground state by spin(1) (2) orbit coupling. The energy corrections to the second order (𝐸𝑖 = 𝐸(0) 𝑖 +𝐸 𝑖 +𝐸 𝑖 ) are (0) (0) 𝐸(1) 𝑖 = < 𝜓 𝑖 |𝜆so𝐋 ∙ 𝐒|𝜓 𝑖 >
𝐸(2) 𝑖 =
(1) < 𝜓(0) 𝑖 |𝜆so𝐋 ∙ 𝐒|𝜓 𝑖 >
(12)
| < 𝜓(0)𝑖 |𝜆so𝐋 ∙ 𝐒|𝜓(0)𝑗 > |2
= ∑𝑗 ≠ 𝑖
(0) 𝐸(0) 𝑖 ‒𝐸 𝑗
(13)
(0) If the orbital angular momentum of the unperturbed ground state is quenched, < 𝜓(0) 0 |𝐋|𝜓 0 > = (1) 0, because the ground state |𝜓(0) 0 > is orbitally nondegenerate, 𝐸 0 = 0, and 2 (0) | < 𝜓(0) 0 |𝐋|𝜓 𝑗 > | (2) = (0) 2∑ + 𝐸0 = 𝐸(0) +< 𝑚 |𝐒 ∙ ∙ 𝐒|𝑚𝑆' > 𝐸 𝐸 𝜆 𝑆 0 0 0 so 𝑗 ≠ 0 𝐸(0) ‒ 𝐸(0) 0
(14)
𝑗
where 𝑚𝑆 and 𝑚𝑆' indicate the total magnetic (secondary) spin quantum numbers of the unpaired electron(s). We can define a tensor 𝚲, which is sometimes referred to as the dequenching tensor [43], and a tensor 𝐃, called the zero field splitting (ZFS), as 2 (0) | < 𝜓(0) 0 |𝐋|𝜓 𝑗 > |
𝚲 = ∑𝑗 ≠ 0 2 𝐃 = 𝜆so 𝚲
10
(0) 𝐸(0) 0 ‒𝐸 𝑗
(15) (16)
so that 𝐸0 can be rewritten as '
ZFS (0) (0) (0) 𝐸0 = 𝐸(0) 0 +𝐸 0 = 𝐸 0 +< 𝜓 0 ,𝑚𝑆|𝐒 ∙ 𝐃 ∙ 𝐒|𝜓 0 ,𝑚𝑆 >
(17)
The energy 𝐸ZFS 0 can thus be obtained from the application of the Hamiltonian (see Section 5.2 for a more complete description) 𝐻ZFS = 𝐒 ∙ 𝐃 ∙ 𝐒
(18)
to the unperturbed eigenstates. Assuming the validity of the LS coupling regime, a total magnetic moment 𝛍el, the sum of the spin and orbital magnetic moments, can be defined 𝛍el =‒ 𝜇B(𝐋 + 𝑔e𝐒)
(19)
and evaluated from the expectation value of the corresponding quantum mechanical operators over the perturbed orbital ground state,
(
'
(0) ' (0) (0) < 𝜓0,𝑚𝑆 |𝛍el|𝜓0,𝑚'𝑆 > =‒ 𝜇B 𝑔e < 𝜓(0) 0 ,𝑚𝑆|𝐒|𝜓 0 ,𝑚 𝑆 >+ 2𝜆so𝚲 < 𝜓 0 ,𝑚𝑆|𝐒|𝜓 0 ,𝑚 𝑆 >
)
(20) as shown in Panel 1.
----------------------------------------------------------------------------------------------------------------------Panel 1. To evaluate < 𝜓0,𝑚𝑆 |𝛍el|𝜓0,𝑚'𝑆 > we must calculate (1) (1) ' | | (0) 0 ,𝑚𝑆 𝛍el 𝜓 𝑖 + 𝜓 𝑖 ,𝑚 𝑆 >
< 𝜓(0) 0 +𝜓
= ‒ 𝜇 B( (0) ' (0) (1) ' (1) (0) ' (1) (1) ' < 𝜓(0) 0 ,𝑚𝑆|𝐋 + 𝑔e𝐒|𝜓 0 ,𝑚 𝑆 >+ < 𝜓 0 ,𝑚𝑆|𝐋 + 𝑔e𝐒|𝜓 0 ,𝑚 𝑆 >+< 𝜓 0 ,𝑚𝑆|𝐋 + 𝑔e𝐒|𝜓 0 ,𝑚 𝑆 >+ < 𝜓 0 ,𝑚𝑆|𝐋 + 𝑔e𝐒|𝜓 0 ,𝑚 𝑆 >
)
(P1.1) (0) Assuming < 𝜓(0) 0 |𝐋|𝜓 0 > = 0, the first term is (0) ' (0) (0) ' < 𝜓(0) 0 ,𝑚𝑆|𝐋 + 𝑔e𝐒|𝜓 0 ,𝑚 𝑆 > = 𝑔e < 𝜓 0 ,𝑚𝑆|𝐒|𝜓 0 ,𝑚 𝑆 >
(P1.2)
the second and third terms are (using Eq. (11)) (1) < 𝜓(0) 0 ,𝑚𝑆|𝐋 + 𝑔e𝐒|𝜓 0 ,𝑚'𝑆 > = ∑𝑗 ≠ 0
∑
' (0) < 𝜓(0) 𝑗 ,𝑚 𝑆|𝜆so𝐋 ∙ 𝐒|𝜓 0 ,𝑚𝑆 > (0) 𝐸(0) 0 ‒𝐸 𝑗 2 | < 𝜓(0)𝑗 |𝐋|𝜓(0) 0 >|
𝜆 𝑗 ≠ 0 so
11
(0) 𝐸(0) 0 ‒𝐸 𝑗
(0) ' < 𝜓(0) 0 ,𝑚𝑆|𝐋 + 𝑔e𝐒|𝜓 𝑗 ,𝑚 𝑆 > =
< 𝑚𝑆|𝐒|𝑚'𝑆 >
(P1.3)
(0) ' (1) (1) because 𝑔e < 𝜓(0) 0 ,𝑚𝑆|𝐒|𝜓 𝑗 ,𝑚 𝑆 > = 0, and the fourth term in Eq. (P1.1) < 𝜓 0 ,𝑚𝑆|𝐋 + 𝑔e𝐒|𝜓 0 ,𝑚'𝑆 > is (0) proportional to (𝐸(0) 0 ‒𝐸 𝑗 )
‒2
(0) . Therefore, Eq. (P1.1) to the order (𝐸(0) 0 ‒𝐸 𝑗 )
‒1
is reduced to Eq. (20).
-----------------------------------------------------------------------------------------------------------------------
Therefore, if a tensor 𝐠 is defined as 𝐠 = 𝑔e𝟏 + 2𝜆so𝚲
(21)
in the spin Hamiltonian formalism, valid when only the ground state multiplet is thermally populated, the total electron magnetic moment can be written as (22)
𝛍el = ‒ 𝜇B𝐠 ∙ 𝐒
This condition, implying that orbital excited states are far in energy above the ground state with respect to the thermal energy, is commonly met in many transition metal complexes. Eq. (22) is also used as a definition of the electron magnetic moment in first-principles quantum chemistry approaches developed within the spin Hamiltonain EPR formalism, where the 𝐠 tensor accounts for the effect of spin-orbit coupling [32,33,35,36].
3.1 The Curie spin In the presence of a magnetic field, a further Hamiltonian term must be considered, for the Zeeman interaction between the total electron magnetic moment and the magnetic field 𝐁0 (more correctly, this should be called the magnetic flux density, see Section 4): 𝐻Zeeman =‒ 𝐁0 ∙ 𝛍el
(23)
The Zeeman term thus introduces a field dependence, which is proportional to the total electron magnetic moment, into the energy of the system. The total electron magnetic moment can be defined in terms of the derivative of the energy of the electron with respect to the magnetic field d𝐸
d𝐸
(𝜇𝑖 = ‒ d𝐵 , with 𝑖 = {𝑥,𝑦,𝑧}) 0,𝑖
𝛍el = ‒ d𝐁0
12
(24)
B0
S
el
el
Figure 1. An ensemble of electron spin magnetic moments is oriented by a magnetic field, with a thermal average 〈𝛍𝑆〉 parallel to the field. Due to spin-orbit coupling, the total average electron magnetic moment 〈𝛍el〉, comprising contributions from spin and orbital magnetic moments, is not parallel to the field if the 𝐠 tensor is anisotropic. An averaged electron magnetic moment is defined from the population distribution over the electronic energy levels at a given temperature (Fig. 1). The average spin-only electron magnetic moment is thus
〈𝛍𝑆〉 =‒ 𝜇B𝑔e〈𝐒〉 =‒ 𝜇B𝑔e𝐒C
(25)
where 𝐒C is the thermal average of 𝐒, equal to 𝐒C =
[
]
∑ ⟨𝜓𝑖│𝐒│𝜓𝑖⟩exp ‒ 𝐸𝑖 (𝑘𝑇) 𝑖
[
(26)
]
∑ exp ‒ 𝐸𝑖 (𝑘𝑇) 𝑖
and 𝐸𝑖 is the energy of the ith state, resulting from the sum of the crystal field, spin-orbit coupling (and thus ZFS) and Zeeman contributions, k is the Boltzmann constant and T is the temperature. If a single orbitally non-degenerate ground state is populated (because orbital excited states have a high energy), in the absence of ZFS this reduces to 𝐒C =
[
]
∑ ⟨𝜓0,𝑚𝑆,𝑖│𝐒│𝜓0,𝑚𝑆,𝑖⟩exp ‒ 𝐸𝑖Zeeman (𝑘𝑇) 𝑖
[
]
∑ exp ‒ 𝐸𝑖Zeeman (𝑘𝑇) 𝑖
(27)
where (see Eqs. (22) and (23)) 𝐸𝑖Zeeman = ‒ 𝜇B⟨𝜓0,𝑚𝑆,𝑖│𝐁0 ∙ 𝐠 ∙ 𝐒│𝜓0,𝑚𝑆,𝑖⟩ (28) 13
and thus, to the first order in EZeeman/(kT), for the case of isotropic molecular rotation, Eq (27) reduces to the Curie spin
[
∑ ⟨𝜓0,𝑚𝑆,𝑖│𝐒│𝜓0,𝑚𝑆,𝑖⟩ 1 ‒ 𝑖
𝐒C =
𝜇B𝐁 ∙ 𝐠 ∙ ⟨𝜓0,𝑚𝑆,𝑖│𝐒│𝜓0,𝑚𝑆,𝑖⟩ 0 𝑘𝑇
∑1
]
=‒
𝜇B𝑔iso𝑆(𝑆 + 1)𝐵0 3𝑘𝑇
𝑖
𝛋 (29)
where 𝛋 indicates the direction (z axis) of the magnetic field and 𝑔iso is the isotropic average of the 𝐠 tensor (∑𝑖𝑚𝑆,𝑖2 =
𝑆(𝑆 + 1)(2𝑆 + 1) ). 3
Analogously, from Eq. (22), in the spin Hamiltonian formalism the average total electron magnetic moment is
〈𝛍el〉 =‒ 𝜇B𝐠 ∙ 𝐒C
(30)
and, from Eq. (24), performing the same thermal average as in Eq. (26), d𝐸𝑖
〈𝛍el〉 =‒
[
]
∑ d𝐁 exp ‒ 𝐸𝑖 (𝑘𝑇) 𝑖 0
[
(31)
]
∑ exp ‒ 𝐸𝑖 (𝑘𝑇) 𝑖
The Zeeman energy is much smaller than the energy at zero magnetic field given by the crystal field (0) ZFS and ZFS (𝐸(0) 𝑖 = 𝐸 𝑖 + 𝐸 𝑖 ). Therefore, it can be introduced through perturbation theory to the
second-order, providing: Zeeman│ ⟩ 𝜓𝑖 𝐸𝑖 = 𝐸(0) 𝑖 + ⟨𝜓𝑖│𝐻
+
∑ 𝑗≠𝑖
|⟨𝜓𝑖│𝐻Zeeman│𝜓𝑗⟩|2 (0) 𝐸(0) 𝑖 ‒𝐸 𝑗
= 𝐸(0) 𝑖 ‒ ⟨𝜓𝑖│𝐁0 ∙ 𝛍el│𝜓𝑖⟩ +
∑ 𝑗≠𝑖
|⟨𝜓𝑖│𝐁0 ∙ 𝛍el│𝜓𝑗⟩|2 (0) 𝐸(0) 𝑖 ‒𝐸 𝑗
(32) and thus
(
|⟨𝜓𝑖│𝐁0 ∙ 𝛍el│𝜓𝑗⟩|
d ⟨𝜓𝑖│𝐁0 ∙ 𝛍el│𝜓𝑖⟩ ‒ ∑𝑗 ≠ 𝑖
∑
〈𝛍el〉 =
𝑖
(0) (0) 𝐸 𝑖 ‒𝐸 𝑗
d𝐁0
[
]
∑ exp ‒ 𝐸𝑖 (𝑘𝑇) 𝑖
2
)
[
exp ‒ 𝐸𝑖 (𝑘𝑇)
]
.
(33)
If the summation in Eq. (33) is extended only to the manifold of the orbital ground state, |𝜓𝑖 > indicates |𝜓0,𝑚𝑆𝑖 > . Then, to first order in 𝐸𝑖 (𝑘𝑇), for systems with isolated, orbitally nondegenerate ground states (see Panel 2) [40] 14
𝜇B2
〈𝛍el〉 = 𝑘𝑇𝐠 ∙ 〈𝐒𝐒𝑇〉 ∙ 𝐠𝑇 ∙ 𝐁0
(34)
where 〈𝐒𝐒𝑇〉 is the effective electron spin dyadic equal to
⟨𝑆𝛼𝑆𝛽⟩ =
𝑄𝑗𝑖 =
∑ 𝑄𝑗𝑖⟨𝜓𝑖│𝑆𝛼│𝜓𝑗⟩⟨𝜓𝑗│𝑆𝛽│𝜓𝑖⟩ 𝑖𝑗
[
∑ exp ‒ 𝐸(0) (𝑘𝑇) 𝑖 𝑖
]
,
exp [ ‒ 𝐸(0) 𝑖 (𝑘𝑇)]
{
‒
𝑘𝑇
(35)
𝛼,𝛽 = {𝑥,𝑦,𝑧} (0) for 𝐸(0) 𝑖 =𝐸 𝑗
{exp [ ‒ 𝐸(0)𝑗 (𝑘𝑇)] ‒ exp [ ‒ 𝐸(0)𝑖 (𝑘𝑇)]}
(0) for 𝐸(0) 𝑖 ≠𝐸 𝑗
(0) 𝐸(0) 𝑗 ‒𝐸 𝑖
and 𝐸(0) 𝑖 is the energy of the state |𝜓𝑖⟩ at zero magnetic field. Recall that we have assumed here that there are no low-lying orbitally excited states, and thus that only the orbitally non-degenerate ground spin multiplet is populated, and the thermally accessible states are limited to the manifold of the eigenstates of the ZFS Hamiltonian.
----------------------------------------------------------------------------------------------------------------------Panel 2 From Eq. (33), to first order in 𝐸𝑖 (𝑘𝑇), (0) 𝐸 𝑖 ⟨𝜓𝑖│𝑩0 ∙ 𝝁𝑒𝑙│𝜓𝑗⟩⟨𝜓𝑗│𝛍𝐞𝐥│𝜓𝑖⟩ ⟨𝜓𝑖│𝐁0 ∙ 𝛍𝐞𝐥│𝜓𝑖⟩ ∑ ⟨𝜓𝑖│𝛍el│𝜓𝑖⟩ ‒ 2∑ exp ‒ 1+ (0) (0) 𝑘𝑇 𝑘𝑇 𝑖 𝑗≠𝑖
(
〈𝛍el〉 = and, since ∑𝑖⟨𝜓𝑖│𝛍el│𝜓𝑖⟩exp[ ‒ 𝐸
(0) 𝑖
) [ ](
𝐸 𝑖 ‒𝐸 𝑗
[
∑ exp ‒ 𝐸(0) (𝑘𝑇) 𝑖 𝑖
) (P2.1)
]
(𝑘𝑇)] = 0, retaining only the terms linear in 𝐵0 (see Section 4.1), ∑
〈𝛍el〉 =
(
⟨𝜓𝑖│𝛍𝐞𝐥│𝜓𝑖⟩⟨𝜓𝑖│𝛍𝐞𝐥│𝜓𝑖⟩
𝑖
𝑘𝑇
)
⟨𝜓𝑖│𝛍𝐞𝐥│𝜓𝑗⟩⟨𝜓𝑗│𝛍𝐞𝐥│𝜓𝑖⟩ (0) ‒ 2∑𝑗 ≠ 𝑖 exp ‒ 𝐸 𝑖 (𝑘𝑇) (0) (0) 𝐸 𝑖 ‒𝐸 𝑗
[
[
] (P2.2)
∙ 𝐁0
]
∑ exp ‒ 𝐸(0) (𝑘𝑇) 𝑖 𝑖
which is the Van Vleck equation [44]. This can be written as ∑
〈𝛍el〉 =
(
⟨𝝍𝒊│𝛍𝐞𝐥│𝝍𝒊⟩⟨𝜓𝑖│𝛍𝐞𝐥│𝜓𝑖⟩
𝑖
𝑘𝑇
]} )
⟨𝜓𝑖│𝛍𝐞𝐥│𝜓𝑗⟩⟨𝜓𝑗│𝛍𝐞𝐥│𝜓𝑖⟩ (0) (0) (0) exp ‒ 𝐸 𝑖 (𝑘𝑇) ‒ ∑𝑗 ≠ 𝑖 exp ‒ 𝐸 𝑖 (𝑘𝑇) ‒ exp ‒ 𝐸 𝑗 (𝑘𝑇) (0) (0) 𝐸 ‒𝐸
[
]
𝑖
[
{ [
𝑗
]
∑ exp ‒ 𝐸(0) (𝑘𝑇) 𝑖 𝑖
]
[
∙ 𝐁0 .
(P2.3) Eq. (P2.3) corresponds to Eqs. (34) and (35) when the spin Hamoltonian formalism (see also Eq. (22)) is used, i.e., for systems with isolated, orbitally non-degenerate ground states [40]. -----------------------------------------------------------------------------------------------------------------------
15
From Eqs. (30) and (34) 𝜇B
𝐒C =‒ 𝑘𝑇〈𝐒𝐒𝑇〉 ∙ 𝐠𝑇 ∙ 𝐁0
(36)
Eq. (35) provides the following expression for 〈𝐒𝐒𝑇〉 in the approximation to first order in D/(kT) [45]
〈𝐒𝐒𝑇〉 =
𝑆(𝑆 + 1) 𝑆(𝑆 + 1)(2𝑆 ‒ 1)(2𝑆 + 3) 𝟏 ‒ 𝐃 3 30𝑘𝑇
(37)
The traceless ZFS tensor in the frame where it is diagonal is usually written in the form
𝐃=
(
1
‒ 3𝐷 + 𝐸 0
0
0
1
‒ 3𝐷 ‒ 𝐸
0
0 2 3𝐷
0
)
(38)
where D and E are the axial and rhombic ZFS parameters (𝐷 = 𝐷𝑧𝑧 ‒ (𝐷𝑥𝑥 + 𝐷𝑦𝑦)/2, 𝐸 = (𝐷𝑥𝑥 ‒ 𝐷𝑦𝑦 )/2). Of course, Eq. (36) reduces to Eq. (29) when only the ground state is populated in the absence of ZFS. In this case, 〈𝐒𝐒𝑇〉 =
𝑆(𝑆 + 1) 𝟏 3
(and thus Tr(〈𝐒𝐒𝑇〉) = S(S+1)).
Extending the summation of the second order term in Eq. (33) to the orbital excited states, Eq. (34) should be replaced with (see Panel 3),
〈𝛍el〉 =
(
𝜇B2
𝑘𝑇𝐠 ∙
)
〈𝐒𝐒𝑇〉 ∙ 𝐠𝑇 ‒ 2𝜇B2𝚲 ∙ 𝐁0
(39)
and using Eq. (37)
〈𝛍el〉 = 𝜇B2
(
𝑆(𝑆 + 1) 𝑇 𝑆(𝑆 + 1)(2𝑆 ‒ 1)(2𝑆 + 3) 𝐠 ∙ 𝐃 ∙ 𝐠𝑇 ‒ 2𝚲 3𝑘𝑇 𝐠 ∙ 𝐠 ‒ 30(𝑘𝑇)2
)∙𝐁
(40)
0
----------------------------------------------------------------------------------------------------------------------Panel 3 Eq. (P2.2) can be written as ∑
〈𝝁𝑒𝑙〉 =
𝐖𝑰𝒊
(
𝑖 𝑘𝑇
∈ ground 𝐖𝑰𝑰 𝒊𝒋 state manifold
‒ 2∑𝑗 ≠ 𝑖,
∈ excited 𝐖𝑰𝑰 𝒊𝒋 states
‒ 2∑𝑗 ≠ 𝑖,
[
]
∑ exp ‒ 𝐸(0) (𝑘𝑇) 𝑖 𝑖
16
)
[
(0) exp ‒ 𝐸 𝑖 (𝑘𝑇)
] ∙ 𝐁0
(P3.1)
with 𝐖𝑰𝒊 = ⟨𝜓𝑖│𝛍𝑒𝑙│𝜓𝑖⟩⟨𝜓𝑖│𝝁𝑒𝑙│𝜓𝑖⟩ , 𝐖𝑰𝑰 𝒊𝒋 = The terms 𝐖𝑰𝒊 and ∑𝑗 ≠ 𝑖,
∈ ground 𝐖𝑰𝑰 𝒊𝒋 state manifold
⟨𝜓𝑖│𝝁𝑒𝑙│𝜓𝑗⟩⟨𝜓𝑗│𝝁𝑒𝑙│𝜓𝑖⟩
(P3.2)
(0) 𝐸(0) 𝑖 ‒𝐸 𝑗
∈ excited 𝐖𝑰𝑰 𝒊𝒋 states
are those reported in Panel 2. To evaluate the term ∑𝑗 ≠ 𝑖,
the following contributions should be calculated (1) (1) (1) (1) (0) (0) (0) ⟨𝜓𝑖│𝝁𝑒𝑙│𝜓𝑗⟩ = ‒ 𝜇B⟨𝜓(0) 𝑖 + 𝜓 𝑖 │𝐋│𝜓 𝑗 + 𝜓 𝑗 ⟩ ‒ 𝜇B𝑔e⟨𝜓 𝑖 + 𝜓 𝑖 │𝐒│𝜓 𝑗 + 𝜓 𝑗 ⟩
(P3.3)
In this case, Eq. (19), rather than Eq. (22), must be used because the latter is a correct approximation of the 0 ‒1
former only if the orbital ground state alone is considered. If we neglect the terms of the order (𝐸0 ‒ 𝐸 𝑗 ) 0
0 ‒1
or higher (which provides terms of order higher than (𝐸0 ‒ 𝐸 𝑗 ) 0
when inserted in Eq. (P3.2)), we are limited
to the terms connecting the ground to the excited states and we get [46] (0) ⟨𝜓0│𝝁𝑒𝑙│𝜓𝑗⟩ = ‒ 𝜇B⟨𝜓(0) 0 │𝐋│𝜓 𝑗 ⟩ 0 ‒1
Therefore, to the (𝐸0 ‒ 𝐸 𝑗 ) 0
(P3.4)
order (see Eq. (15)),
∈ excited 𝐖𝑰𝑰 𝒊𝒋 states
∑𝑗 ≠ 𝑖,
= 𝜇B2∑𝑗 ≠ 𝑖,
∈ excited states
(0) (0) (0) ⟨𝜓(0) 0 │𝐋│𝜓 𝑗 ⟩⟨𝜓 𝑗 │𝐋│𝜓 0 ⟩ (0) 𝐸(0) 0 ‒𝐸 𝑗
= 𝜇B2𝚲.
(P3.5)
In summary, ∑
〈𝛍el〉 =
𝐖𝑰𝒊
(
𝑖 𝑘𝑇
∈ ground 𝐖𝑰𝑰 𝒊 state manifold
‒ 2∑𝑗 ≠ 𝑖,
)
[
]
(0) exp ‒ 𝐸 𝑖 (𝑘𝑇)
[
∑ exp ‒ 𝐸(0) (𝑘𝑇) 𝑖 𝑖
∙ 𝐁0 ‒ 2𝜇B2𝚲 ∙ 𝐁0 =
]
(
𝜇B2
)
𝐠 ∙ 〈𝐒𝐒𝑇〉 ∙ 𝐠𝑇 ‒ 2𝜇B2𝚲 ∙ 𝐁0
𝑘𝑇
(P3.6) -----------------------------------------------------------------------------------------------------------------------
The contribution ‒ 2𝚲, arising from the presence of excited states, is called temperature-independent paramagnetism (TIP). Although frequently neglected, it can be important in determining the anisotropy of 〈𝛍el〉 even at room temperature, especially for S = 1/2 systems, if there is no ZFS.
4. The paramagnetic susceptibility The induced magnetic moment per molecule 〈𝛍tot〉 established by the anisotropic alignment of the magnetic moments of the electrons of a molecule in the presence of a magnetic field results in a magnetization per unit volume 𝐌 given by 𝐌=
〈𝛍𝐭𝐨𝐭〉𝑁A 𝑉𝑀
,
(41) 17
where 𝑁A is the Avogadro constant and 𝑉𝑀 the molar volume. From the general definition of magnetic susceptibility per unit volume d𝐌
d𝐌
(42)
𝛘𝑽 = d𝐇0 𝜇0d𝐁0
where 𝐇0 is the external magnetic field and 𝐁0 the magnetic flux density (𝐁0 = 𝜇0(𝐇0 + 𝐌), with 𝐌 ≪ 𝐇0), 𝑁A d〈𝛍tot〉
𝛘𝑽 = 𝜇 0𝑉𝑀
(43)
d𝐁0
and thus the magnetic susceptibility per molecule d〈𝛍tot〉
𝑉𝑀
𝛘𝐦𝐨𝐥 = 𝑁A𝛘𝑽 = 𝜇0
(44)
d𝐁0
According to Eq. (44), the paramagnetic contribution to the molecular magnetic susceptibility 𝛘 is d〈𝛍el〉
𝛘 = 𝜇0
d〈𝜇𝑖〉
(𝜒𝑖𝑗 = 𝜇0d𝐵 , with 𝑖,𝑗 = {𝑥,𝑦,𝑧})
d𝐁0
0,𝑗
(45)
Since 〈𝛍el〉 increases linearly with 𝐁0 (Eqs. (34) and (40)), 𝛘 is independent of the magnetic field strength, unless the magnetic field is large enough to reach saturation conditions (see Section 4.1). This implies that
〈𝛍el〉 =
𝛘 ∙ 𝐁0
(46)
𝜇0
In the high temperature approximation (BB0<
𝜇0𝜇B2 𝑘𝑇
𝐠 ∙ 〈𝐒𝐒𝑇〉 ∙ 𝐠𝑇
(47)
or, for D<
(
𝑆(𝑆 + 1) 𝑇 𝑆(𝑆 + 1)(2𝑆 ‒ 1)(2𝑆 + 3) 𝐠 ∙ 𝐃 ∙ 𝐠𝑇 ‒ 2𝚲 3𝑘𝑇 𝐠 ∙ 𝐠 ‒ 30(𝑘𝑇)2
)
(48)
If it is assumed that the 𝐠 and 𝐃 tensor are diagonal in the same frame, the components of the 𝛘 tensor are thus 18
𝜒𝑧𝑧 = 𝜒𝑥𝑥,𝑦𝑦 =
𝑆(𝑆 + 1) 2 3𝑘𝑇 𝑔𝑧𝑧
1)(2𝑆 + 3) 𝐷) ‒ 2𝜇0𝜇B2𝜒𝛬𝑧𝑧 , (1 ‒ (2𝑆 ‒ 15𝑘𝑇
1)(2𝑆 + 3) (𝐷 ∓ 3𝐸)) ‒ 2𝜇0𝜇B2𝜒𝛬𝑥𝑥,𝑦𝑦 (1 + (2𝑆 ‒ 30𝑘𝑇
𝑆(𝑆 + 1) 2 3𝑘𝑇 𝑔𝑥𝑥,𝑦𝑦
(49)
The same expressions were originally obtained by Kurland and McGarvey [28], with the difference that the TIP contribution was neglected. From Eq. (48), in the absence of ZFS and neglecting the TIP contribution, the rotational average of 𝛘 is 𝜒iso =
2 𝜇0𝜇B2𝑔iso 𝑆(𝑆 + 1)
(50)
3𝑘𝑇
which is the Curie law. When orbital excited states should also be considered because they are thermally populated, Eq. (P2.3) should be used instead of Eq. (34), with 𝛍el given by Eq. (19), so that Eq. (45) provides 𝜒𝒌𝒌 =
𝜇0𝜇B2∑𝑖𝑗𝑄𝑗𝑖⟨𝜓𝑖│𝐿𝑘 + 𝑔e𝑆𝑘│𝜓𝑗⟩⟨𝜓𝑗│𝐿𝑘 + 𝑔e𝑆𝑘│𝜓𝑖⟩
[
𝑘𝑇
∑ exp ‒ 𝐸(0) (𝑘𝑇) 𝑖 𝑖
]
(51)
and outside the LS coupling regime 𝜇0 ∑𝑖𝑗𝑄𝑗𝑖⟨𝜓𝑖│𝛍el│𝜓𝑗⟩⟨𝜓𝑗│𝛍el
𝛘 = 𝑘𝑇
[
𝑇
]
∑ exp ‒ 𝐸(0) (𝑘𝑇) 𝑖 𝑖
│𝜓𝑖⟩
.
(52)
which is the general van Vleck expression [44] for the susceptibility tensor. From Eqs. (24) and (45), 𝛘 can also be expressed as d < d𝐸 > d𝐁0d𝐁0
𝛘 =‒ 𝜇0
|
(53) 𝐵0 = 0
4.1 Saturation at high field At very high fields (>20 T) and room temperature, the first-order approximation in 𝐁0 ∙ 𝛍el (𝑘𝑇), invoked on passing from Eq. (33) to Eqs. (P2.1), (34) and (39), may be inaccurate. In this case, Eq. (45) can be rewritten using Eq. (31), so that [47]
19
d𝐸
𝛘=
𝑖 ∑ exp ‒ 𝐸𝑖 (𝑘𝑇) d 𝑖d𝐁0 ‒ 𝜇0d𝐁0 ∑ exp ‒ 𝐸𝑖 (𝑘𝑇) 𝑖
[
]
[
]
d
d
= 𝜇0𝑘𝑇d𝐁0d𝐁0ln∑𝑖exp[ ‒ 𝐸𝑖 (𝑘𝑇)]
(54)
For isolated, orbitally non-degenerate ground states, in the absence of ZFS and g-anisotropy, 𝐸𝑖 =
⟨𝜓𝑖│𝜇B𝑔e𝐁0𝑇 ∙ 𝐒│𝜓𝑗⟩ (Eqs. (22) and (23)) and thus Eq. (54) becomes [48] sinh d d 𝑆 𝜒 = 𝜇0𝑘𝑇d𝐵0d𝐵0ln∑𝑚 =‒ 𝑆exp 𝑆
(‒
𝜇B𝑔e𝑚𝑆𝐵0
)
𝑘𝑇
d d = 𝜇0𝑘𝑇d𝐵0d𝐵0ln
(𝑆 + 12)𝜇B𝑔e𝐵0
(
sinh
[(2𝑆 + 1)coth(
=
1 𝜇 𝑔 𝐵 2 B e 0 𝑘𝑇
(
)
𝜇0𝜇B𝑔e d 2 d𝐵0
)]
(2𝑆 + 1)𝜇B𝑔e𝐵0
𝜇B𝑔e𝐵0
(2𝑘𝑇)
(2𝑘𝑇)
) ‒ coth(
)
𝑘𝑇
(55)
so that 𝜒=
[(2𝑆 + 1) (1 ‒ coth (
𝜇0𝜇B2𝑔2e 4𝑘𝑇
2
2
)) ‒ (1 ‒ coth (
(2𝑆 + 1)𝜇B𝑔e𝐵0 2𝑘𝑇
2
))] .
𝜇B𝑔e𝐵0 2𝑘𝑇
1
(56) 1
Fig. 2 shows the field dependence of 𝜒 as a function of the magnetic field. Since coth (𝑥) = 𝑥 + 3𝑥 ‒ 1 3 45𝑥 + …, to
first order in x,
[
𝜇0𝜇B2𝑔2e
𝜒
4𝑘𝑇
(
(2𝑘𝑇)2
) (
2
(2𝑘𝑇)2
)]
2
= (2𝑆 + 1)2 1 ‒ ( ‒ ‒ 1‒( ‒ (2𝑆 + 1)𝜇B𝑔e𝐵0)2 3 𝜇B𝑔e𝐵0)2 3
in agreement with Eq. (50) under the same conditions.
20
𝜇0𝜇B2𝑔2e𝑆(𝑆 + 1) 3𝑘𝑇
(57)
Magnetic susceptibility / 10-32 m3
2.65 2.64 2.63 2.62 2.61 2.60
0
200
400
600
800
1000
1200
Proton Larmor Frequency / MHz
Figure 2. Field dependence of the magnetic susceptibility (calculated with Eq. (56), S = 1/2, T = 298 K). The dotted line corresponds to the Curie law (Eq. (57)). 5. The hyperfine shift 5.1 The semi-empirical paramagnetic NMR approach In a system composed of a paramagnetic metal ion and a nucleus, we should introduce the spin 𝐈 of the nucleus with its associated magnetic moment 𝛍𝐼 = ℏ𝛾𝐼𝐈
(58)
The resonance frequency of a nuclear transition between states differing by Δ𝑚𝐼 =± 1 leads to the NMR signal. This frequency changes depending on the magnetic field experienced by the nucleus; in turn, such a field is determined not only by the external magnetic field but also by the field generated by the electrons moving in the surrounding atoms. When all electrons in the molecular orbital are paired (a diamagnetic system), this additional field is small, and so is the shift in NMR frequency 𝛿dia (although it may still be measurable). If unpaired electron(s) are present at a distance r from the nucleus, the electron magnetic moment creates a dipolar magnetic field which can cause a significant shift 𝛿para of the nuclear signal. To a good approximation, what is called the hyperfine shift 𝛿hf is the additional shift (𝛿hf = 𝛿para ‒ 𝛿dia), which is experimentally measured as the 21
difference in the NMR shift between a paramagnetic system and its diamagnetic analogue. The latter should ideally be composed of the same atoms, in the same positions, as the paramagnetic system, but with the unpaired electron(s) removed. In practice, diamagnetic analogues are obtained either by reducing the paramagnetic metal to a diamagnetic state, by substituting the paramagnetic metal with a diamagnetic one, or by removing the paramagnetic center. Such systems represent good diamagnetic references if there are no structural rearrangements in the molecule due to removal or substitution of the paramagnetic ion, or to a different electric charge. The hyperfine shift 𝛿hf is thus related to the interaction between the nucleus and the unpaired electron(s) contained in the paramagnetic moiety, and in particular to the energy 𝐸hfc of the magnetic interaction between the nuclear magnetic moment and the spin and orbital magnetic moments of the unpaired electron(s). Since NMR shifts are usually expressed in ppm with respect to the nuclear Larmor frequency (𝛾𝐼𝐵0 2𝜋), the component of the NMR shift corresponding to the hyperfine shift is [28,42] 𝐸hfc
Δ𝐸hfc
𝛿hf = 𝛿para ‒ 𝛿dia =‒ ℏ𝛾𝐼𝐵0𝑚𝐼 =‒ ℏ𝛾𝐼𝐵0 .
(59)
where Δ𝐸hfc corresponds to the difference in the hyperfine energy between the two 𝑚𝐼 states. In writing Eq. (59) we have neglected the residual anisotropic chemical shift 𝛿racs, arising from molecular partial alignment (see Eq. (118)), which is also present in the NMR shifts of paramagnetic molecules (but not in those of diamagnetic molecules when partial orientation is induced by the presence of the paramagnetic moiety (see Section 6)). Unpaired electrons relax much more rapidly than nuclei (see Section 7), so they change their spin state among the possible mS levels much more rapidly than nuclei change their spin states among the possible mI levels. As a result, the hyperfine shift results from the magnetic interaction of the nuclear magnetic moment with the average electron magnetic moment. The hyperfine interaction experienced by the nuclear magnetic moment is conveniently separated into the interaction with the electron magnetic moment delocalized onto the nucleus itself, 22
and the interaction with the average electron magnetic moment outside the nucleus. The former component is called the Fermi contact interaction, the second the dipole-dipole interaction. Therefore, the hyperfine coupling energy is split into the two terms: 𝐸hfc = 𝐸Fc + 𝐸dip
(60)
The energy associated with the dipole-dipole interaction between the nuclear magnetic moment and average total electron magnetic moment can be calculated, in the long-range limit, by applying the point-dipole approximation [49] 𝜇0
𝐸dip =‒ 4𝜋𝑟3〈𝛍el〉 ⋅
(
3𝐫𝐫𝑇 𝑟2
)
‒ 𝟏 ⋅ 𝛍𝐼 ,
(61)
where we have assumed that the interaction between the orbital magnetic moment 𝛍𝐿 and 𝛍𝐼 has the same dipolar form as the interaction between 𝛍𝑆 and 𝛍𝐼 [50,51] (see Section 5.2). The point-dipole approximation introduces small inaccuracies for nuclei at distances of several Å from the paramagnetic center, but can lead to large errors for nuclei at short distances. From Eqs. (46), (59)(61), the equation for the dipolar shift anticipated in Eq. (1) of Section 2 is obtained. From Eq. (1), the dipolar shielding tensor is defined as 𝛘
𝛔dip =‒ 4𝜋𝑟3 ⋅
(
3𝐫𝐫𝑇 𝑟2
)
‒𝟏 .
(62)
The pseudocontact shift is the rotational average of the dipolar shift. Assuming isotropic reorientation of the molecule with respect to the magnetic field (see Section 6.1), in the high-field case, when the nuclear spin is oriented along the magnetic field, the pseudocontact shift 𝛿pc is calculated as already reported in Eq. (2), being 𝛿pc =‒
Tr(𝛔dip) 3
1
[ (
= 12𝜋𝑟3Tr 𝛘 ⋅
3𝐫𝐫𝑇 𝑟2
‒𝟏
)]
(63)
Since the term 3𝐫𝐫𝑇 𝑟2 ‒ 𝟏 is traceless, 𝛿pc is determined simply by the paramagnetic susceptibility anisotropy 𝚫𝛘 = 𝛘 ‒ 𝜒iso𝟏, where 𝜒iso = Tr(𝛘) 3. Other forms of Eq. (2) are reported in Panel 4. In the reference frame where the 𝚫𝛘 tensor is diagonal, Eq. (63) becomes
23
1
[
]
3
𝛿pc = 12𝜋𝑟3 Δ𝜒𝑎𝑥(3cos2𝜃 ‒ 1) + 2Δ𝜒𝑟ℎsin2𝜃cos2𝜑
(64) where Δ𝜒𝑎𝑥 = 𝜒𝑧𝑧 ‒
𝜒𝑥𝑥 + 𝜒𝑦𝑦 2
3
= 2(𝜒𝑧𝑧 ‒ 𝜒iso), (65)
Δ𝜒𝑟ℎ = 𝜒𝑥𝑥 ‒ 𝜒𝑦𝑦
and r, 𝜃 and 𝜑 are the spherical coordinates of the nucleus in the principal frame of the 𝛘 tensor, with the origin in the position of the unpaired electron(s).
----------------------------------------------------------------------------------------------------------------------Panel 4 Equations for the pseudocontact shifts, valid in any frame centered on the metal ion: - in Cartesian coordinates: 𝛿pc =
[
1
4𝜋𝑟3
2𝑧2 ‒ 𝑥2 ‒ 𝑦2
(𝜒𝑧𝑧 ‒ Tr(𝜒)/3)
2𝑟2
𝑥2 ‒ 𝑦2
+ (𝜒𝑥𝑥 ‒ 𝜒𝑦𝑦)
2𝑟2
2𝑥𝑦
+ 𝜒𝑥𝑦
𝑟2
+ 𝜒𝑥𝑧
2𝑥𝑧 𝑟2
2𝑦𝑧
+ 𝜒𝑦𝑧
𝑟2
]
(P4.1)
- in spherical coordinates: 𝛿pc =
[(𝜒
𝑧𝑧 ‒ Tr(𝜒)/3)
1 4𝜋𝑟3
3cos2𝜃 ‒ 1 sin2𝜃cos2𝜑 ( ) + 𝜒 ‒ 𝜒 + 𝜒𝑥𝑦sin2𝜃sin2𝜑 + 𝜒𝑥𝑧sin2𝜃cos𝜑 + 𝜒𝑦𝑧sin2𝜃sin𝜑 𝑥𝑥 𝑦𝑦 2 2
]
(P4.2) - using director cosines (l, m and n, equal to the cosine of the angles between the metal-nucleus vector and the principal directions of the 𝛘 tensor): 𝛿pc =
1
[(𝜒
3
4𝜋𝑟
3𝑛2 ‒ 1 𝑙2 ‒ 𝑚2 ( ) + 𝜒 ‒ 𝜒 + 2𝜒𝑥𝑦𝑙𝑚 + 2𝜒𝑥𝑧𝑙𝑛 + 2𝜒𝑦𝑧𝑚𝑛 𝑥𝑥 𝑦𝑦 2 2
𝑧𝑧 ‒ Tr(𝜒)/3)
]
(P4.3)
Equations for the pseudocontact shifts, valid only in the frame where the 𝛘 tensor is diagonal: - in Cartesian coordinates: 𝛿pc =
1
[
8𝜋𝑟3
2𝑧2 ‒ 𝑥2 ‒ 𝑦2
(𝜒𝑧𝑧 ‒ Tr(𝜒)/3)
𝑟2
]
𝑥2 ‒ 𝑦2
+ Δ𝜒𝑟ℎ
𝑟2
=
1
[
12𝜋𝑟3
Δ𝜒𝑎𝑥
2𝑧2 ‒ 𝑥2 ‒ 𝑦2 𝑟2
3
]
𝑥2 ‒ 𝑦2
+ 2Δ𝜒𝑟ℎ
𝑟2
(P4.4) - in spherical coordinates: 𝛿pc =
1
1
[(𝜒𝑧𝑧 ‒ Tr(𝜒)/3)(3cos2𝜃 ‒ 1) + Δ𝜒𝑟ℎsin2𝜃cos2𝜑] = 12𝜋𝑟3 8𝜋𝑟 3
[Δ𝜒𝑎𝑥(3cos2𝜃 ‒ 1) + 32Δ𝜒𝑟ℎsin2𝜃cos2𝜑] - using director cosines:
24
(P4.5)
𝛿pc =
1
[𝜒𝑥𝑥𝑙2 + 𝜒𝑦𝑦𝑚2 + 𝜒𝑧𝑧𝑛2 ‒ Tr(𝜒)/3] = 12𝜋𝑟 [Δ𝜒𝑎𝑥(3𝑛2 ‒ 1) + 2Δ𝜒𝑟ℎ(𝑙2 ‒ 𝑚2)] (P4.6) 1
4𝜋𝑟3
3
3
-----------------------------------------------------------------------------------------------------------------------
As mentioned above, the hyperfine shift of the nucleus (Eq. (59)) is not only due to the pseudocontact shift, but also, for nuclei with some unpaired electron spin density delocalized onto the nucleus itself, by the Fermi contact shift: 𝛿hf = 𝛿pc + 𝛿Fc
(66)
The Fermi contact energy results from the magnetic interaction between the nuclear magnetic moment and the fraction of the electron magnetic moment corresponding to the electron spin density at the site of the nucleus. This contribution is thus non-zero only for nuclei with non-negligible unpaired spin density in their s orbitals, i.e., those that are a few chemical bonds away from the unpaired electron(s) or anyway close to the metal orbitals. By its nature, the Fermi contact interaction involves only the electron spin magnetic moment and not the orbital magnetic moment, and thus [52,53] 𝐸Fc =‒
𝜇0𝜌𝐼 3𝑆
〈 𝛍𝑆 〉 ⋅ 𝛍𝐼
(67)
where 𝜌𝐼 is the contact electron spin density at the NMR nucleus. Therefore, from Eqs. (25), (36), (59) and (60), implying the presence of an isolated, orbitally non-degenerate ground state, and again assuming isotropic reorientation of the molecule with respect to the magnetic field, 1
[
𝛿Fc = 3Tr
𝜇0𝜇B2𝜌𝐼
〈𝐒𝐒𝑇〉 ∙ 𝐠𝑇
3𝑘𝑇 𝑆 𝑔e
]
(68)
which, using Eq. (47), can be written as [27,28]. 1
[
𝛿Fc = 3Tr
𝜌𝐼
3𝑆𝑔e𝛘 ∙
]
[𝐠𝑇] ‒ 1
(69)
which corresponds to Eq. (6) reported in Section 2. In the case of negligible spin-orbit coupling, Eq. (68) becomes 𝛿Fc =
𝐴Fc𝑔e𝜇B𝑆(𝑆 + 1) ℏ 3𝛾𝐼𝑘𝑇
with the Fermi contact coupling constant 25
(70)
𝜇0
𝜌𝐼
𝐴Fc = 3 ℏ𝛾𝐼𝜇B𝑔e 𝑆 .
(71)
From Eq. (68) it is apparent that anisotropy of the contact shifts can arise due to anisotropy of the 𝐠 and 〈𝐒𝐒𝑇〉 tensors. In the presence of orbital excited states close in energy to the ground state, excited states should also be taken into account in the calculation of 〈𝛍𝑆〉, each of them with its own electron spin density at the nuclear position. Therefore, in the LS coupling regime [28] 𝜇B ∑𝑖𝑗𝑄𝑗𝑖(⟨𝜓𝑖│𝐿𝑘 + 𝑔e𝑆𝑘│𝜓𝑗⟩⟨𝜓𝑗│𝐴𝑖𝑗𝑆𝑘│𝜓𝑖⟩)
1
𝛿Fc = 3∑𝑘ℏ𝛾𝐼𝑘𝑇
[
∑ exp ‒ 𝐸(0) (𝑘𝑇) 𝑖 𝑖
]
.
(72)
The constants 𝐴𝑖𝑗 may depend on the orbitals [54], and represent the proportionality constants which relate the orbital spin density contributions to the spin density at the nucleus for the ground and excited states. The above treatment, developed by McConnell and Robertson [49] and Kurland and McGarvey [28], implies that in the hyperfine Hamiltonian 𝐻hfc = 𝐒 ⋅ 𝐀 ⋅ 𝐈 .
(73)
the hyperfine coupling constant A should be defined as 𝜇0 ℏ𝛾𝐼𝜇B
𝐀 = 4𝜋
3
𝑟
𝐠𝑇 ⋅
(
3𝐫𝐫𝑇 2
𝑟
)
𝜇0
𝜌𝐼
‒ 𝟏 + 3 ℏ𝛾𝐼𝜇B𝑔e 𝑆 𝟏
(74)
where the first and second terms correspond to the point-dipole/point-dipole tensor and to the Fermi contact coupling constant, respectively. In the following section, the derivation of Eq. (74) is presented using the LS coupling approximation and the spin Hamiltonian formalism. This derivation permits one to recover the semi-empirical equation for the hyperfine coupling without involving the magnetic susceptibility anisotropy, using Eq. (87) developed within the first-principles quantum chemistry approach presented in Section 5.3.
5.2 Perturbative EPR spin Hamiltonian approach
26
If all interactions perturbing the crystal field states are considered altogether (Fig. 3), in the spin Hamiltonian formalism and LS coupling regime the following perturbative Hamiltonian terms should at least be considered [55,56]: 𝐻' = 𝐻𝐿𝑆 + 𝐻Zeeman + 𝐻hfc + 𝐻IL + 𝐻ss = 𝜆so𝐋 ∙ 𝐒 + 𝜇B𝐁0 ∙ (𝐋 + 𝑔e𝐒) + 𝐒 ⋅ 𝐀hfc ⋅ 𝐈 + 𝑎IL𝐋 ⋅ 𝐈 + 𝐒 ⋅ ∆𝐬𝐬 ⋅ 𝐒
(75)
where 𝐻IL = 𝑎IL𝐋 ⋅ 𝐈 refers to the interaction of the nuclear magnetic moment with the orbital magnetic moment of the unpaired electron(s) calculated with respect to the nuclear position, with 𝑎IL 𝜇0 ℏ𝛾𝐼𝜇B
= 24𝜋
𝜌3
, where 𝜌 is the nucleus-electron distance (see Panel 5), and 𝐋 = 𝝆 × 𝐩, while 𝐻ss = 𝐒 ⋅ ∆𝐬𝐬
⋅ 𝐒 refers to the dipole-dipole interaction between unpaired electrons (usually negligible with respect to the other terms) present whenever S>1/2. The energy of the interaction between the nuclear magnetic moment and the orbital magnetic moment of the unpaired electron(s) can be calculated by considering that an electric charge moving in a loop generates a dipolar magnetic field at the position of the nucleus that can be expressed in the form 𝐻IL =
𝜇0ℏ𝛾𝐼𝜇B 4𝜋
(3
𝐋·𝐫 𝑇 𝐋⋅𝐈 5𝐫 ⋅ 𝐈 ‒ 𝑟 𝑟3
)
(76)
where 𝐋 is the orbital angular momentum calculated with respect to the metal nucleus (see Panel 5). ----------------------------------------------------------------------------------------------------------------------Panel 5 The dipole-dipole interaction between the nuclear spin I and electron orbital angular momentum 𝐋 is described by the Hamiltonian 𝜇0 ℏ𝛾𝐼𝜇B
𝐻IL = 24𝜋
𝜌3
(P5.1)
𝐋⋅𝐈
where 𝐋 is calculated with respect to the nuclear position (i.e., 𝐋 = ×p, whereas L = R×p is calculated with respect to the metal nucleus) [55].
27
zk I r Me
R
e
=Rr
p
yj
xi Therefore, 𝜇0ℏ𝛾𝐼𝜇B𝛒 × 𝐩
𝐻IL = 2 ∂𝑓
∂𝑓
4𝜋
𝜌3
𝜇0ℏ𝛾𝐼𝜇B 1 ×𝐩⋅𝐈 4𝜋 𝜌
(P5.2)
⋅𝐈=2
∂𝑓
1
(𝑓 = ∂𝑥𝐢 + ∂𝑦𝐣 + ∂𝑧𝐤) where the bar indicates the integral over all electron positions. To first order in R/r, 𝜌 1
≈𝑟+
𝐫·𝐑 𝑟3
and thus 𝐋 𝜌
1
3
𝐩
1
= 𝜌 × 𝐩 = × 𝜌 ‒ 𝜌 × 𝐩 ≈ ×
(
1 𝐫·𝐑 + 3 𝑟 𝑟
)𝐩 = ×
𝐫·𝐑 𝑟3
𝐩
(P5.3)
1
(because 𝐩 × 𝑟 = 0 for a localized, divergenceless electron current distribution, and × 𝐩 = 0). Since 1
𝐫 × 𝐑 × 𝐩 = (𝐫·𝐩)𝐑 ‒ (𝐫·𝐑)𝐩, we get (𝐫·𝐑)𝐩 =‒ 2𝐫 × 𝐑 × 𝐩 [57] and thus 𝐋
1
𝜌3
=‒ 2 ×
𝐫×𝐑×𝐩 𝑟3
1
=‒ 2 ×
𝐫×𝐋 𝑟3
1
= 2 ×
𝐋×𝐫 𝑟3
(P5.4)
Since ×
𝐋×𝐫 3
𝑟
=‒
𝐋 3
𝑟
𝐋·𝐫
(P5.5)
+ 3 5𝐫 𝑟
(because L is independent of x, y, z) Eq. (76) is obtained from Eq. (P5.2). -----------------------------------------------------------------------------------------------------------------------
According to the standard quantum chemistry definition, d2𝐸
𝐀hfc = ∑𝑖d𝐬𝑖d𝐈
|
,
(77)
𝑠𝑖 = 𝐼 = 0
where the total hyperfine coupling tensor is obtained from the electron energy E as a sum of contributions from each electron. The electron-nucleus dipole-dipole interaction is defined to include the electron spin only, thus not comprising the interaction between the electron orbital magnetic moment and the nuclear magnetic moment described by Eq. (76), so that 𝜇0 ℏ𝛾𝐼𝜇B
𝐀hfc = 4𝜋
3
𝑟
(
𝑔e
3𝐫𝐫𝑇 2
𝑟
)
𝜇0
𝜌𝐼
‒ 𝟏 + 3 ℏ𝛾𝐼𝜇B𝑔e 𝑆 𝟏
As a result, to second order the energy corrections to the ligand-field energy are 28
(78)
Zeeman (0) + 𝐻hfc + 𝐻IL + 𝐻ss|𝜓(0) 𝐸(1) 𝑖 = < 𝜓 𝑖 |𝐻𝐿𝑆 + 𝐻 𝑖 >
(79)
Zeeman (0) + 𝐻hfc + 𝐻IL + 𝐻ss|𝜓(1) 𝐸(2) 𝑖 = < 𝜓 𝑖 |𝐻𝐿𝑆 + 𝐻 𝑖 >
(80)
and
where |𝜓(1) 𝑖 >
=∑
Zeeman < 𝜓(0) + 𝐻hfc + 𝐻IL + 𝐻ss|𝜓(0) 𝑗 |𝐻𝐿𝑆 + 𝐻 𝑖 >
𝑗≠𝑖
(0) 𝐸(0) 𝑖 ‒𝐸 𝑗
|𝜓(0) 𝑗 >
(81)
For orbitally non-degenerate ground states, an effective Hamiltonian can be written (see Panel 6) comprising the terms 𝐻' = 𝜇B𝐁0 ∙ 𝐠 ∙ 𝐒 + 𝐒 ⋅ 𝐀 ⋅ 𝐈 + 𝐒 ⋅ 𝐃 ⋅ 𝐒
(82)
with 𝐀 defined as in Eq. (74). Note that the tensor 𝐃 contains contributions in addition to those reported in Eq. (18). Remarkably, it should also be noted that the 𝐠 tensor is not strictly symmetric, due to the contributions from further cross-terms and higher-order perturbation terms.
----------------------------------------------------------------------------------------------------------------------Panel 6 (0) From Eqs. (79)-(81), for orbitally non-degenerate ground states ( < 𝜓(0) 0 |𝐋|𝜓 0 > = 0), (0) hfc ⋅ 𝐈 + 𝐒 ⋅ ∆𝐬𝐬 ⋅ 𝐒|𝜓(0) 𝐸(1) 0 = < 𝜓 0 |𝜇B𝑔e𝐁0 ∙ 𝐒 + 𝐒 ⋅ 𝐀 0 >
𝐸(2) 0 = ∑𝑗 ≠ 0
Zeeman < 𝜓(0) + 𝐻hfc + 𝐻IL + 𝐻ss|𝜓(0) 𝑗 |𝐻𝐿𝑆 + 𝐻 0 > (0) 𝐸(0) 0 ‒𝐸 𝑗
(P6.1)
Zeeman < 𝜓(0) + 𝐻hfc + 𝐻IL + 𝐻ss|𝜓(0) 0 |𝐻𝐿𝑆 + 𝐻 𝑗 > (P6.2)
Since the spin-orbit coupling energy is usually orders of magnitude larger than the other terms, Eq. (P6.2) can be approximated by neglecting all terms non involving 𝐻𝐿𝑆
∑
2 (0) | < 𝜓(0) 0 |𝜆so𝐋 ∙ 𝐒|𝜓 𝑗 > |
𝐸(2) 0 𝑗≠0
∑
+2 (0) 𝐸(0) 0 ‒𝐸 𝑗 𝑗≠0 (0) (0) (0) Zeeman < 𝜓 𝑗 |𝐻𝐿𝑆|𝜓 0 > ( < 𝜓 0 |𝐻 + 𝐻hfc + 𝐻IL + 𝐻ss|𝜓(0) 𝑗 >) (0) 𝐸(0) 0 ‒𝐸 𝑗
(P6.3) where the first term corresponds to the term on the right of Eq. (13), the first product in the second term is (see Eqs. (19) and (23))
29
∑
(0) (0) (0) < 𝜓(0) 𝑗 |𝜆so𝐋 ∙ 𝐒|𝜓 0 >< 𝜓 0 |𝜇B𝐁0 ∙ (𝐋 + 𝑔e𝐒)|𝜓 𝑗 >
𝑗≠0
∙
∑
(0) 𝐸(0) 0 ‒𝐸 𝑗 (0) (0) (0) < 𝜓(0) 𝑗 |𝐋|𝜓 0 >< 𝜓 0 |𝐋|𝜓 𝑗 > (0) 𝐸(0) 0 ‒𝐸 𝑗
𝑗≠0
= 𝜆so𝜇B𝐁0 (P6.4)
∙
< 𝑚𝑆|𝐒|𝑚'𝑆 >
and the third product in the second term is (using the definition of g and 𝚲 in Eqs. (21) and (15), respectively)
|
(0) (0) < 𝜓(0) 𝑗 |𝜆so𝐋 ∙ 𝐒|𝜓 0 >< 𝜓 0
∑
[
∙ ∑𝑗 ≠ 0 =
1𝜇0ℏ𝛾𝐼𝜇B 4𝜋𝑟3
4𝜋
(3
𝐋·𝐫 𝑇 𝐋⋅𝐈 𝐫 ⋅𝐈‒ 3 𝑟5 𝑟
)|𝜓
(0) 𝑗 >
(0) 𝐸(0) 0 ‒𝐸 𝑗
𝑗≠0
𝜇0ℏ𝛾𝐼𝜇B = 𝜆so 4𝜋 < 𝑚𝑆|𝐒|𝑚'𝑆 >
=2
𝜇0ℏ𝛾𝐼𝜇B
(0) (0) (0) < 𝜓(0) 𝑗 |𝐋|𝜓 0 >< 𝜓 0 |𝐋|𝜓 𝑗 > (0) 𝐸(0) 0 ‒𝐸 𝑗
𝜇0ℏ𝛾𝐼𝜇B 4𝜋𝑟3
·𝐫
3𝐫𝑇 𝑟5
[
< 𝑚𝑆|𝐒|𝑚'𝑆 > ∙ 𝜆so𝚲·
[
< 𝑚𝑆|𝐒|𝑚'𝑆 > ∙ (𝐠 ‒ 𝑔e𝟏)·
3𝐫𝐫𝑇 𝑟2
1
(0) (0) (0) < 𝜓(0) 𝑗 |𝐋|𝜓 0 >< 𝜓 0 |𝐋|𝜓 𝑗 >
𝑟
(0) 𝐸(0) 0 ‒𝐸 𝑗
‒ 3∑𝑗 ≠ 0
3𝐫𝐫𝑇 𝑟2
]⋅𝐈
]
‒ 𝜆so𝚲 ⋅ 𝐈
]
(P6.5)
‒ (𝐠 ‒ 𝑔e𝟏) ⋅ 𝐈
Therefore, from Eqs. (P6.1)-(P6.5) 𝜇0ℏ𝛾𝐼𝜇B
2 𝐻' = 𝜇B𝑔e𝐁0 ∙ 𝐒 + 𝐒 ⋅ 𝐀hfc ⋅ 𝐈 + 𝐒 ⋅ ∆𝐬𝐬 ⋅ 𝐒 + 𝐒 ⋅ 𝜆so 𝚲 ⋅ 𝐒 + 2𝜆so𝜇B𝐁0 ⋅ 𝚲 ⋅ 𝐒+2𝜆so𝐒 ∙ 𝐀dip,so ⋅ 𝐈+
𝑔e𝟏)·
[
3𝐫𝐫𝑇 𝑟2
]
4𝜋𝑟3
𝐒 ∙ (𝐠 ‒ (P6.6)
‒ 𝟏 ⋅ 𝐈 + 𝜆so𝐒 ⋅ (𝚲' + 𝚲'') ⋅ 𝐒
where 𝐀dip,so, 𝚲' and 𝚲'' can be calculated from the evaluation of L over the different eigenstates [27,55]. Eq. (P6.6) can be rearranged in the form (see Eq. (78)) 𝐻' = 𝜇B𝐁0 ∙ (𝑔e𝟏 + 2𝜆so𝚲) ∙ 𝐒 + 𝐒 𝜇0 ℏ𝛾𝐼𝜇B 3𝐫𝐫𝑇 𝜇0 𝜌𝐼 𝜇0 ℏ𝛾𝐼𝜇B 3𝐫𝑇 ( ) 𝑔 ‒ 𝟏 ℏ𝛾 𝟏 ‒ 𝟏 + 2𝜆so𝐀dip,so ⋅ 𝐈 ⋅ + 𝜇 𝑔 + 𝐠 ‒ 𝑔 𝟏 · 𝐫 e 4𝜋 𝑟3 e 𝑟2 3 𝐼 B e𝑆 4𝜋 𝑟3 𝑟5
{[
(
]
)
[
}
]
2 + 𝐒 ⋅ (𝜆so 𝚲 + ∆𝐬𝐬 + 𝜆so(𝚲' + 𝚲'')) ⋅ 𝐒
(P6.7) which can be rewritten as in Eq. (82), being the contribution from 𝐀dip,so much smaller than the other terms in A. On passing from Eq. (P6.2) to Eq. (P6.3) we have neglected the term
∑ 𝑗≠0
Zeeman| (0) < 𝜓(0) 𝜓0 > 𝑗 |𝐻 (0) 𝐸(0) 0 ‒𝐸 𝑗
Zeeman| (0) < 𝜓(0) 𝜓 𝑗 >= 𝜇B2 0 |𝐻
∑
(0) < 𝜓(0) 𝑗 |𝐁0 ∙ (𝐋 + 𝑔e𝐒)|𝜓 0 > (0) 𝐸(0) 0 ‒𝐸 𝑗
𝑗≠0
(0) < 𝜓(0) 0 |𝐁0 ∙ (𝐋 + 𝑔e𝐒)|𝜓 𝑗 >
(P6.8) which, using Eq. (P3.4), becomes 𝜇B2𝐁0 ∙ ∑𝑗 ≠ 0
(0) (0) (0) < 𝜓(0) 𝑗 |𝐋|𝜓 0 >< 𝜓 0 |𝐋|𝜓 𝑗 > (0) 𝐸(0) 0 ‒𝐸 𝑗
∙ 𝐁0 = 𝜇B2𝐁0 ∙ 𝚲 ∙ 𝐁0
(P6.9)
This term provides the TIP contribution to the magnetic susceptibility tensor (Eq. (48)), as can be checked using Eq. (53). This is also called the Van Vleck orbital paramagnetism, and can provide non-negligible contributions when the energy of the excited states is not very high. -----------------------------------------------------------------------------------------------------------------------
30
2E-19
1000
2E-20
100
2E-21
10
2E-22
1
2E-23
0.1
2E-24
0.01
2E-25
0.001
2E-26
1E-4
2E-27
1E-5
Energy / J
Energy / cm1
10000
2E-28
Crystal-field Spin-orbit coupling
Zeeman Proton-electron dipole-dipole
Figure 3. Typical orders of magnitude of the energy values of crystal-field, spin-orbit coupling, electron Zeeman (at 20 T) and proton-electron point-dipole point-dipole (r = 3 Å) interactions. 5.3 Hyperfine shift from first-principles quantum chemistry Hyperfine shift expressions analoguous to Eqs. (63) and (68) can also be obtained from the quantum chemistry definition of the isotropic hyperfine shift [33,58] 𝛿hf =‒ 𝛔hf =
d2〈𝐸para〉 d𝐁0d𝛍𝐼
|
Tr(𝛔hf)
(83)
3
(84) 𝐵0 = 𝜇 𝐼 = 0
so that d2
[
𝛔hf = d𝐁0d𝛍𝐼
[
]
∑ 𝐸𝑖exp ‒ 𝐸el,𝑖 (𝑘𝑇) 𝑖
[
]
∑ exp ‒ 𝐸el,𝑖 (𝑘𝑇) 𝑖
]|
(85) 𝐵0 = 𝜇 𝐼 = 0
and, to first order in 𝐸𝑖 (𝑘𝑇) (see Panel 7) in the spin Hamiltonian regime, with 𝐻(1) =‒ 𝐁0𝑇 ∙ 𝛍el + 𝑇 𝐻hfc, 𝐻(1) el =‒ 𝐁0 ∙ 𝛍el,
31
⟨ │ │𝜓 ⟩|
∑ 𝑄𝑗𝑖 𝜓𝑖 𝑖𝑗 1 𝛔hf =‒ 𝑘𝑇
∂𝐻(1) ∂𝐁0
⟨𝜓 │ │𝜓 ⟩|
𝑗
𝑗
𝐵0 = 𝜇 𝐼 = 0
∂𝐻(1) ∂𝛍𝐼
(
𝑖
𝐵0 = 𝜇 𝐼 = 0
(86)
)
∑ exp ‒ 𝐸(0) 𝑘𝑇 𝑖 𝑖
which provides [37,59] 𝜇B
𝛔hf =‒ ℏ𝛾𝐼𝑘𝑇𝐠 ⋅ ⟨𝐒𝐒𝑇⟩ ⋅ 𝐀
(87)
using the definition of g as d2𝐸
1
𝐠 = 𝜇B∑𝑖d𝐬𝑖d𝐁𝟎
|
,
(88)
𝑠𝑖 = 𝐵0 = 0
and of 𝐀 as reported in Eq. (77). If the contribution from spin-orbit coupling is neglected, the tensor 𝐀 is equal to 𝐀hfc. When spin-orbit coupling and the interaction between the nuclear magnetic moment and the orbital magnetic moment are considered, the tensor 𝐀 can be derived in the semi-empirical quantum mechanical framework as shown in Section 5.2 (Eq. 74)). The same expression for the dipolar shielding tensor is actually obtained from Eq. (84) using Eqs. (61) and (34) 𝜇0 𝜇B2
𝛔dip =‒ 4𝜋𝑟3𝑘𝑇𝐠 ∙ 〈𝐒𝐒𝑇〉 ∙ 𝐠𝑇 ⋅
(
3𝐫𝐫𝑇 𝑟2
‒𝟏
)
(89)
----------------------------------------------------------------------------------------------------------------------Panel 7 From Eq. (85)
[
(
|⟨𝜓𝑖│𝐻(1)│𝜓𝑗⟩|
∑ ⟨𝜓𝑖│𝐻(1)│𝜓𝑖⟩ + ∑ 𝑖 𝑗≠𝑖
d2 𝛔hf = d𝐁 d𝛍 0 𝐼
2
‒ ⟨𝜓𝑖│𝐻(1)│𝜓𝑖⟩
(0) (0) 𝐸 𝑖 ‒𝐸 𝑗
[
⟨𝜓𝑖│𝐻(1) el │𝜓𝑖⟩ 𝑘𝑇
)
[
(0) exp ‒ 𝐸 𝑖 (𝑘𝑇)
]
∑ exp ‒ 𝐸(0) (𝑘𝑇) 𝑖 𝑖
]
]|
.
𝐵0 = 𝜇 𝐼 = 0
(P7.1) 2 (1)
∂ 𝐻 0∂𝛍𝐼
The term ∂𝐁
does not provide “paramagnetic” contributions (it provides a contribution approximated with
𝛔dia, if also the nuclear Zeeman term ‒ 𝛾𝐼𝐁0 ∙ (𝟏 ‒ 𝛔orb) ∙ 𝐈 is included in 𝐻(1)). Since ∑ 1 𝛔hf =‒ 𝑘𝑇
(⟨ │ │ ⟩⟨ │
𝑖
𝜓𝑖
∂𝐻(1) el ∂𝐁0
𝜓𝑖 𝜓𝑖
│𝜓 ⟩ ‒ 2∑
∂𝐻(1) ∂𝛍𝐼
𝑖
𝑘𝑇
∂𝐻(1) el ∂𝛍𝐼
⟨𝜓 │ │𝜓 ⟩⟨𝜓 │ │𝜓 ⟩)exp[ ‒ 𝐸(0)𝑖 (𝑘𝑇)]
(0) 𝑗 ≠ 𝑖 (0) 𝐸 𝑗 ‒𝐸 𝑖
𝑖
[
∂𝐻(1) ∂𝐁0
]
∑ exp ‒ 𝐸(0) (𝑘𝑇) 𝑖 𝑖
32
𝑗
𝑗
∂𝐻(1) ∂𝛍𝐼
= 0,
𝑖
(P7.2)
∂𝐻(1) el
and Eq. (86) is obtained ( ∂𝐁 = 0
∂𝐻(1) ∂𝐁0
).
-----------------------------------------------------------------------------------------------------------------------
In the first-principles quantum chemistry approach, 𝐀 in Eq. (87) is 𝐀 = 𝐀hfc + 𝐀so
(90)
where 𝐀hfc is provided by Eq. (78) and 𝐀so is the cross-product between the spin-coupling operator of Eq. (10) and 𝐻pso (corresponding to the term of Eq. (P6.5) in the semi-empirical approach):
⟨𝜓 │ │𝜓 ⟩⟨𝜓 │∑ 𝑗
𝐀so = ∑𝑗0
∂𝐻pso ∂𝐈
0
0
│𝜓 ⟩ + ⟨𝜓 │∑
∂𝐻𝑆𝑂𝐶 𝑖 ∂𝐬𝑖
𝑗
𝑗
(0) 𝐸(0) 0 ‒𝐸 𝑗
where 𝐻pso has the form 𝜇0
│𝜓 ⟩⟨𝜓 │ │𝜓 ⟩
∂𝐻𝑆𝑂𝐶 𝑖 ∂𝐬𝑖
𝐥𝑖𝐼
𝐻pso = 24𝜋ℏ𝛾𝐼𝜇B∑𝑖𝑟 3 ⋅ 𝐈 ,
0
0
∂𝐻pso ∂𝐈
𝑗
(91)
(92)
𝑖𝐼
and 𝐥𝑖𝐼 is the orbital angular momentum of the ith electron relative to the nucleus. Eqs. (87) and (90) should thus allow for the calculation of pseudocontact shifts in systems where only the ground state multiplet is populated. Surprisingly, calculations performed for cobalt(II) systems provided much smaller values than those obtained from Eqs. (63) and (47), which are also valid when only the ground state multiplet is populated [40]. The (much) smaller contribution generally expected for 𝐀so than for 𝐀hfc in Eq. (90) leads to the assumption that the spin-orbital contribution (𝐀so) to the pseudocontact shift can be neglected altogether [31,60,37,61–64]. In this way, the pseudocontact shifts should be calculated to a good approximation using Eqs. (83), (87) and (78). However, the values so calculated are significantly different from those calculated in the semi-empirical framework, where Eqs. (87) 𝑇
and (74) are used. The two models differ by a factor 𝐠 𝑔e, which can be even larger than a factor 2. The first-principles framework for the analysis of the pseudocontact shifts has been recently questioned by a careful comparison of the experimental 𝐠 values of some copper(II) proteins with those derived from the 𝛘 anisotropy tensor obtained from the PCSs measured on the same systems [41]. 33
The different equations for the hyperfine shifts developed within the semi-empirical and the firstprinciples quantum chemistry frameworks are summarized in Table 1. It can be noted that the Fermi contact shift coincides in the two approaches, whereas the pseudocontact shift in the semi-empirical approach, being determined by the 𝛘 tensor, contains contributions from the orbital magnetic moment which are not accounted for by the first-principles quantum chemistry treatment that is limited to the leading-order terms. This results in an “asymmetry” between the contact and pseudocontact terms which is inherent to the different types of interactions which are accounted for by the two contributions (Fig. 4). In the following sections, the semi-empirical model is thus used to describe the pseudocontact shift, as it allows for all contributions to the magnetic susceptibility to be considered.
Paramagnetic susceptibility: 𝛘 𝛘' 𝛘'' 𝛘𝐓𝐈𝐏 Pseudocontact shift 𝛘' 𝛘'' 𝛘𝐓𝐈𝐏
Contact shift
(semi-empirical approach)
𝛘' 𝛘''
(Kurland-McGarvey)
𝛘'
(first principles quantum chemistry)
𝛘'
(semi-empirical approach, Kurland-McGarvey)
𝛘'
(first principles quantum chemistry)
Figure 4. In the semi-empirical approach the pseudocontact shift is proportional to the whole paramagnetic susceptibility (𝛘), whereas in the first-principles quantum chemistry approach it is proportional only to the part of the magnetic susceptibility which is related to the nuclear spin-electron spin hyperfine coupling (𝛘'), thus neglecting contributions from the orbital hyperfine (paramagnetic nuclear spin-electron orbit) coupling (𝛘''). In the Kurland-McGarvey equation [28], the TIP contribution was not considered.
Table 1. Summary of the equations for the hyperfine shift in the semi-empirical and first-principles quantum chemistry approaches Semi-empirical approach 34
hf
𝛿 =
[ (
1
3𝐫𝐫𝑇
Tr 𝛘 ⋅ 12𝜋𝑟3
𝑟2
)]
𝐴Fc ‒1 ‒𝟏 + Tr[𝛘 ∙ [𝐠𝑇] ] 3𝜇0ℏ𝛾𝐼𝜇B
For singly-populated orbitally non-degenerate ground state spin multiplets 𝛘=
𝜇0𝜇B2 𝑘𝑇
𝐠 ∙ 〈𝐒𝐒𝑇〉 ∙ 𝐠𝑇
and thus 𝛿 =
𝜇0𝜇B2
1
hf
12𝜋𝑟3 𝑘𝑇
[
𝑇〉
Tr 𝐠 ∙ 〈𝐒𝐒
𝑇
∙𝐠 ⋅
(
3𝐫𝐫𝑇 𝑟2
)]
‒𝟏 +
𝜇B𝐴Fc
Tr[𝐠 ∙ 〈𝐒𝐒𝑇〉] 3ℏ𝛾𝐼𝑘𝑇
First-principles quantum chemistry approach 𝛿hf =‒
𝜇B
Tr[𝐠 ⋅ ⟨𝐒𝐒𝑇⟩ ⋅ (𝐀hfc + 𝐀so)]
3ℏ𝛾𝐼𝑘𝑇
For singly-populated orbitally non-degenerate ground state spin multiplets, with 𝐀so ≪ 𝐀hfc hfc
𝐀
=
𝜇0 ℏ𝛾𝐼𝜇B 4𝜋 𝑟3
(
𝑔e
3𝐫𝐫𝑇 𝑟2
)
‒ 𝟏 + 𝐴Fc
and thus hf
𝛿 =
1
𝜇0𝜇B2
12𝜋𝑟3 𝑘𝑇
[
𝑇〉
Tr 𝐠 ∙ 〈𝐒𝐒
(
∙ 𝑔e
3𝐫𝐫𝑇 𝑟2
)]
‒𝟏 +
𝜇B𝐴Fc
Tr[𝐠 ∙ 〈𝐒𝐒𝑇〉] 3ℏ𝛾𝐼𝑘𝑇
5.4 The limiting case of D<
𝛔hf =‒ ℏ𝛾𝐼𝑘𝑇𝐠 ⋅
[𝑆(𝑆3+ 1)𝟏 ‒ 𝑆(𝑆 + 1)(2𝑆30𝑘𝑇‒ 1)(2𝑆 + 3)𝐃] ⋅ [4𝜋
𝜇0 ℏ𝛾𝐼𝜇B 𝑟3
𝐠𝑇 ⋅
(
3𝐫𝐫𝑇 𝑟2
)
𝜇0
𝜌𝐼
‒ 𝟏 + 3 ℏ𝛾𝐼𝜇B𝑔e 𝑆 𝟏
]
(93) so that (assuming that the 𝐠 and 𝐃 tensors are parallel) 𝛿Fc =
𝑆(𝑆 + 1)𝐴Fc 𝜇B ∑ 9 ℏ 𝛾𝐼𝑘𝑇 𝒊𝒊
1)(2𝑆 + 3) 𝑔𝑖𝑖𝐷𝑖𝑖] [𝑔𝑖𝑖 ‒ (2𝑆 ‒ 10𝑘𝑇
and using Eq. (38)
35
(ii = xx, yy, zz)
(94)
𝛿Fc 𝑆(𝑆 + 1)𝐴Fc 𝜇B = 9 ℏ 𝛾𝐼𝑘𝑇
[
]
(2𝑆 ‒ 1)(2𝑆 + 3) 1 1 2 (𝑔𝑥𝑥( ‒ 𝐷 + 𝐸) + 𝑔𝑦𝑦( ‒ 𝐷 ‒ 𝐸) + 𝑔𝑧𝑧( 𝐷)) 10𝑘𝑇 3 3 3 Fc 𝜇 𝑔 + 𝑔 + 𝑔 ( ) 𝑆(𝑆 + 1)𝐴 (2𝑆 ‒ 1)(2𝑆 + 3)𝐷 2𝑔𝑧𝑧 ‒ 𝑔𝑥𝑥 ‒ 𝑔𝑦𝑦 + 3𝐸(𝑔𝑥𝑥 ‒ 𝑔𝑦𝑦) B 𝑥𝑥 𝑦𝑦 𝑧𝑧 = 1‒ 3 ℏ 𝛾𝐼𝑘𝑇 3 30𝑘𝑇 𝑔𝑥𝑥 + 𝑔𝑦𝑦 + 𝑔𝑧𝑧 (𝑔𝑥𝑥 + 𝑔𝑦𝑦 + 𝑔𝑧𝑧) ‒
[
]
(95) and (see Panel 8) 𝛿pc
{
[
2 2 𝑔𝑥𝑥 1𝜇0𝑆(𝑆 + 1) 𝜇B (2𝑆 ‒ 1)(2𝑆 + 3)2 2 2 (3cos 𝜃 ‒ 1) 𝑔𝑧𝑧 1 ‒ = 𝐷 ‒ 9 4𝜋 10𝑘𝑇 2 3 𝑘𝑇𝑟3
(
) 𝑔 1)(2𝑆 + 3) 1 1)(2𝑆 + 3) 1 ( ‒ 3𝐷 + 𝐸)) ‒ 2 (1 ‒ (2𝑆 ‒ 10𝑘𝑇 ( ‒ 3𝐷 ‒ 𝐸))] + 32sin (1 ‒ (2𝑆 ‒ 10𝑘𝑇 (2𝑆 ‒ 1)(2𝑆 + 3) 1 (2𝑆 ‒ 1)(2𝑆 + 3) 1 𝜃cos2𝜙[𝑔 (1 ‒ ‒ 𝐷 + 𝐸)) ‒ 𝑔 (1 ‒ ( ( ‒ 3𝐷 ‒ 𝐸 10𝑘𝑇 3 10𝑘𝑇 ))]} 2 𝑦𝑦
2 𝑥𝑥
2
2 𝑦𝑦
(96)
----------------------------------------------------------------------------------------------------------------------Panel 8 From Eq. (93) 𝛿pc
[(
2 𝑔𝑥𝑥 1𝜇0𝑆(𝑆 + 1) 𝜇B 0 = Tr 9 4𝜋 𝑘𝑇𝑟5 0
0 0 (3𝑥2 ‒ 𝑟2)𝑔𝑥𝑥 𝑔𝑦𝑦 0 3𝑥𝑦𝑔𝑦𝑦 0 𝑔𝑧𝑧 3𝑥𝑧𝑔𝑧𝑧
(
(3𝑥2 ‒ 𝑟2)𝑔𝑥𝑥 3𝑥𝑦𝑔𝑦𝑦 3𝑥𝑧𝑔𝑧𝑧
1 ‒ 𝐷+𝐸 3 0 0
0
0
1 ‒ 𝐷‒𝐸 3
0
0
2 𝐷 3
)(
)(
3𝑥𝑦𝑔𝑥𝑥
(3𝑦2 ‒ 𝑟2)𝑔𝑦𝑦 3𝑦𝑧𝑔𝑧𝑧
)
3𝑥𝑧𝑔𝑥𝑥 (2𝑆 ‒ 1)(2𝑆 + 3) 𝑔𝑥𝑥 0 3𝑦𝑧𝑔𝑦𝑦 ‒ 10𝑘𝑇 0 (3𝑧2 ‒ 𝑟2)𝑔𝑧𝑧
3𝑥𝑦𝑔𝑥𝑥 3𝑥𝑧𝑔𝑥𝑥 (3𝑦2 ‒ 𝑟2)𝑔𝑦𝑦 3𝑦𝑧𝑔𝑦𝑦 3𝑦𝑧𝑔𝑧𝑧 (3𝑧2 ‒ 𝑟2)𝑔𝑧𝑧
(
]
)
(P8.1) and thus
36
0 0 𝑔𝑦𝑦 0 0 𝑔𝑧𝑧
)
𝛿pc
[
(
)
2 (3𝑥2 ‒ 𝑟2)𝑔𝑥𝑥 2 1𝜇0𝑆(𝑆 + 1) 𝜇B 2 3𝑥𝑦𝑔𝑦𝑦 = Tr 9 4𝜋 𝑘𝑇𝑟5 2 3𝑥𝑧𝑔𝑧𝑧
(
(
)
1 2 ‒ 𝐷 + 𝐸 (3𝑥2 ‒ 𝑟2)𝑔𝑥𝑥 3 1 2 ‒ 𝐷 ‒ 𝐸 3𝑥𝑦𝑔𝑦𝑦 3
(
)
2 2 3𝑥𝑦𝑔𝑥𝑥 3𝑥𝑧𝑔𝑥𝑥 (2𝑆 ‒ 1)(2𝑆 + 3) 2 2 2 2 (3𝑦 ‒ 𝑟 )𝑔𝑦𝑦 3𝑦𝑧𝑔𝑦𝑦 ‒ 10𝑘𝑇 2 2 3𝑦𝑧𝑔𝑧𝑧 (3𝑧2 ‒ 𝑟2)𝑔𝑧𝑧
(
(
)
1 2 ‒ 𝐷 + 𝐸 3𝑥𝑦𝑔𝑥𝑥 3 1 2 ‒ 𝐷 ‒ 𝐸 (3𝑦2 ‒ 𝑟2)𝑔𝑦𝑦 3
)
2 2𝐷𝑥𝑧𝑔𝑧𝑧
( (
2 2𝐷𝑦𝑧𝑔𝑧𝑧
) )
1 2 ‒ 𝐷 + 𝐸 3𝑥𝑧𝑔𝑥𝑥 3 1 2 ‒ 𝐷 ‒ 𝐸 3𝑦𝑧𝑔𝑦𝑦 3 2 2 𝐷(3𝑧2 ‒ 𝑟2)𝑔𝑧𝑧 3
)] (P8.2)
so that 𝛿pc =
((
2 1𝜇0𝑆(𝑆 + 1) 𝜇B (2𝑆 ‒ 1)(2𝑆 + 3) 2 2 2 (3𝑥2 ‒ 𝑟2)𝑔𝑥𝑥 + (3𝑦2 ‒ 𝑟2)𝑔𝑦𝑦 + (3𝑧2 ‒ 𝑟2)𝑔𝑧𝑧 ‒ 5 9 4𝜋 10𝑘𝑇 𝑘𝑇𝑟 1 1 2 2 2 2 ‒ 𝐷 + 𝐸 (3𝑥2 ‒ 𝑟2)𝑔𝑥𝑥 + ‒ 𝐷 ‒ 𝐸 (3𝑦2 ‒ 𝑟2)𝑔𝑦𝑦 + 𝐷(3𝑧2 ‒ 𝑟2)𝑔𝑧𝑧 3 3 3
[
)
(
)]
)
(P8.3)
and thus, in spherical coordinates, 𝛿pc 2 1𝜇0𝑆(𝑆 + 1) 𝜇B (2𝑆 ‒ 1)(2𝑆 + 3) 1 2 (3sin2𝜃cos2𝜙 ‒ 1)𝑔𝑥𝑥 = 1‒ ‒ 𝐷+𝐸 3 9 4𝜋 10𝑘𝑇 3 𝑘𝑇𝑟
2 + (3sin2𝜃sin2𝜙 ‒ 1)𝑔𝑦𝑦
(
)]
[
1‒
(
(
))
(2𝑆 ‒ 1)(2𝑆 + 3) 1 ‒ 𝐷‒𝐸 10𝑘𝑇 3
(
(
2 + (3cos2𝜃 ‒ 1)𝑔𝑧𝑧 1‒
))
(2𝑆 ‒ 1)(2𝑆 + 3)2 𝐷 10𝑘𝑇 3
(P8.4) Eq. (96) is then obtained using the relationships 3
1
3sin2𝜃cos2𝜙 ‒ 1 = 2sin2𝜃cos2𝜙 ‒ 2(3cos2𝜃 ‒ 1);
(P8.5)
3 1 3sin2𝜃sin2𝜙 ‒ 1 =‒ sin2𝜃cos2𝜙 ‒ (3cos2𝜃 ‒ 1) 2 2 -----------------------------------------------------------------------------------------------------------------------
Note that even the measurement of a very large number of pseudocontact shifts would not allow the determination of the 𝐠 and 𝐃 tensors separately, because all data are related to the nuclear coordinates by a single anisotropic tensor, which is determined by the two anisotropy values Δ𝜒𝑎𝑥 and Δ𝜒𝑟ℎ and the three Euler angles that specify the orientation of this tensor with respect to the frame used to define the nuclear coordinates. Therefore, from any number (larger than 5) of pseudocontact shifts 37
values, Δ𝜒𝑎𝑥 and Δ𝜒𝑟ℎ can be determined, which are linked to the 5 parameters defining the 𝐠 and 𝐃 tensors (assuming that the TIP is negligible) by the following relationships: Δ𝜒𝑎𝑥 =
[
(
(2𝑆 ‒ 1)(2𝑆 + 3)2 2 𝑔𝑧𝑧 1‒ 3𝐷 10𝑘𝑇
2 𝑔𝑥𝑥
)‒ (
(
(2𝑆 ‒ 1)(2𝑆 + 3) 2 1‒ 10𝑘𝑇
2 𝑔𝑦𝑦
)) ‒ 2 (
1 ‒ 3𝐷 + 𝐸
(
(2𝑆 ‒ 1)(2𝑆 + 3) 1‒ 10𝑘𝑇
𝜇0𝜇B2𝑆(𝑆 + 1)
3𝑘𝑇 1 ‒ 3𝐷 ‒ 𝐸
))]
(97) Δ𝜒𝑟ℎ =
𝜇0𝜇B2𝑆(𝑆 + 1) 3𝑘𝑇
[𝑔 (1 ‒ 2 𝑥𝑥
(2𝑆 ‒ 1)(2𝑆 + 3) 10𝑘𝑇
1)(2𝑆 + 3) ( ‒ 13𝐷 + 𝐸)) ‒ 𝑔𝑦𝑦2 (1 ‒ (2𝑆 ‒ 10𝑘𝑇 ( ‒ 13𝐷 ‒ 𝐸))]
(98) in agreement with Eq. (48).
5.5 The case of paramagnetic lanthanoid(III) ions Also for paramagnetic lanthanoid ions, the pseudocontact shifts are usually analysed in the semiempirical framework using Eq. (63), i.e., according to their dependence on the paramagnetic susceptibility anisotropy [45]. For these ions, except for gadolinium(III), the relationship between pseudocontact shifts and EPR quantities (𝐠 and 𝐃) is even more complicated because the spin-orbit interaction couples 𝐋 and 𝐒 strongly, so that 𝐉 = 𝐋 + 𝐒 must be used to describe the total angular momentum, and the electron gfactor is provided by the Landé g-factor (see Table 2) 𝑔𝐽 = 1 +
𝐽(𝐽 + 1) ‒ 𝐿(𝐿 + 1) + 𝑆(𝑆 + 1) 2𝐽(𝐽 + 1)
(99)
and the Zeeman energy becomes (see Eqs. (22) and (23)) 𝐸Zeeman = 𝜇B𝑔𝐽𝐵0𝐽𝑧
Table 2. J quantum numbers and Landé g-factors for lanthanoid ions Ion
J
gJ 38
(100)
Ce3+
5/2
6/7
Pr3+
4
4/5
Nd3+
9/2
8/11
Pm3+
4
3/5
Sm3+
5/2
2/7
Eu3+(Sm2+)
0
–
Gd3+(Eu2+)
7/2
2
Tb3+
6
3/2
Dy3+
15/2
4/3
Ho3+
8
5/4
Er3+
15/2
6/5
Tm3+
6
7/6
Yb3+
7/2
8/7
For these ions, the crystal-field effects are smaller than the spin-orbit interactions. They can be accounted for by a tensor 𝐃 (with the same form as the ZFS tensor in paramagnetic transition metal ions), which removes the degeneracy of the electronic levels at zero magnetic field. As a consequence, Eq. (96) becomes (see Eq. (38)), 2 2 1 𝜇0𝐽(𝐽 + 1)(2𝐽 ‒ 1)(2𝐽 + 3) 𝜇B𝑔 𝐽 3 4𝜋 (𝑘𝑇)2𝑟
{(3cos 𝜃 ‒ 1)[𝐷
𝛿pc =‒ 90
2
𝑧𝑧 ‒
𝐷𝑥𝑥 + 𝐷𝑦𝑦 2
]+
3 2 [ ] 2sin 𝜃cos2𝜙 𝐷𝑥𝑥 ‒ 𝐷𝑦𝑦
}
(101) or, since 𝐷𝑧𝑧 ‒
𝐷𝑥𝑥 + 𝐷𝑦𝑦 2
3
= 2𝐷 (because 𝐷𝑥𝑥 + 𝐷𝑦𝑦 + 𝐷𝑧𝑧 = 0) [45], 𝑧𝑧
2 2 1 𝜇0𝐽(𝐽 + 1)(2𝐽 ‒ 1)(2𝐽 + 3) 𝜇B𝑔 𝐽 3 4𝜋 (𝑘𝑇)2𝑟
𝛿pc =‒ 60
{(3cos2𝜃 ‒ 1)𝐷𝑧𝑧 + sin2𝜃cos2𝜙[𝐷𝑥𝑥 ‒ 𝐷𝑦𝑦]} (102)
As for Eq. (96), this equation is valid in the limit that D, i.e., the splitting of the electronic levels at zero field, is smaller than kT. This condition is often not fulfilled. 39
On the other hand, from Eqs. (25), (59) and (67), the Fermi contact shifts depend only on the spin-dependent part of the electron magnetic moment, 𝐒C = < 𝑆𝑧 > 𝛋; since [44] (103)
𝑆𝑧 = (𝑔𝐽 ‒ 1)𝑚𝐽 for lanthanoid ions Eq. (70) becomes 𝐽(𝐽 + 1)𝐴Fc 𝜇B 3 ℏ 𝛾𝐼𝑘𝑇
𝛿Fc = 𝑔𝐽(𝑔𝐽 ‒ 1)
(104)
For a given ligand nucleus, 𝐴Fc is nearly constant, independently of the lanthanoid ion, whereas the magnitude of the 𝐃 tensor, that determines the pseudocontact shift, depends essentially only on the lanthanoid. This latter condition holds when the crystal field splitting is smaller than kT and much smaller than the spin-orbit coupling; if not, the magnitude and orientation of 𝐃 also depend on the nature of the ligands coordinated to the lanthanoid ion [65–69].
5.6 Beyond the point-dipole approximation Eqs. (1) and (63) for the dipolar and pseudocontact shifts were derived on the assumption that the unpaired electron(s) are localized on the paramagnetic atom, so that the point-dipole approximation can be used. This assumption may be acceptable when analysing the hyperfine shifts of nuclei far from the paramagnetic center: for instance, in a high-spin cobalt(II) system, considering an unpaired electron spin that is positioned at a single point provides good predictions of pseudocontact shifts for nuclei at distances farther than 8 Å from the metal [37]. However, the unpaired electron spin is in reality delocalised over the ligand atoms through molecular orbital overlap, so that dipole-dipole interactions occur between the nuclear magnetic moment and the distributed electron magnetic moment. This corresponds to integrating Eq. (62) over space after multiplication with the electron density function 𝜌(𝐫e) [70,71]: 1
(
𝛔dip =‒ 4𝜋∫ 𝜌(𝐫e)𝛘(𝐫e) ⋅
40
(
3(𝐫e ‒ 𝐫)(𝐫e ‒ 𝐫)𝑇
| 𝐫e ‒ 𝐫|
5
‒|
𝟏
))d 𝐫
𝐫e ‒ 𝐫| 3
3
e
(105)
where 𝐫e is the integration variable and 𝐫 indicates the position of the nucleus detected by NMR in a frame centered on the nucleus of the paramagnetic metal. It has been shown [70] that this integral provides the following expression for the tensor elements: σdip 𝑖𝑗 (𝐫) =
1
∑
2
2
δ𝑖𝑗 ( ) + 𝜒 (𝐫)𝜌(𝐫) 𝑘𝑗 3 𝜒𝑖𝑗(𝐫)𝜌(𝐫) 𝑘𝑖𝑘
(106)
If it is assumed that the 𝛘 tensor is the same for all positions, this can be simplified to provide the following expression for the pseudocontact shift [70]:
(
1 𝑇·𝛘·
𝛿pc(𝐫) =‒ 3
𝑇
·
1
)
‒ 3Tr(𝛘) 𝜌(𝐫)
(107)
These relationships allow one to determine the electron density distribution from the experimental pseudocontact shifts and the atomic coordinates of the molecule [70,71], provided one fulfills some regularization conditions that are needed to take into account the ill-posed nature of the problem.
6. Paramagnetism-induced self-orientation As anticipated in Section 2, the anisotropy of the magnetic susceptibility causes partial selforientation of a paramagnetic molecule. From Eqs. (24) and (46), the field dependence of the energy of the thermally averaged total electron magnetic moment can be expressed as 𝐸𝑎𝑛 =‒ ∫〈𝛍el〉·d𝐁𝟎 =‒
𝐁0 ∙ 𝛘 ∙ 𝐁0 2𝜇0
(108)
In the presence of non-negligible diamagnetic contributions (𝛘𝐝𝐢𝐚) to the overall susceptibility of the molecule, the diamagnetic contribution adds to the paramagnetic one so that 𝐸𝑎𝑛 = ‒
𝐁0 ∙ 𝛘𝐦𝐨𝐥 ∙ 𝐁0 2𝜇0
(109)
with 𝛘𝐦𝐨𝐥 = 𝛘 + 𝛘𝐝𝐢𝐚 (see Section 4). This implies that in the presence of an anisotropic 𝛘𝐦𝐨𝐥 the energy of the molecule changes as a function of its orientation with respect to the magnetic field. Of course, orientations with lower energy will be favored with respect to orientations with higher energy, and thus a partial alignment of the molecular ensemble occurs. To evaluate the extent of this partial alignment, and the effects on the NMR shifts, we can introduce an orientational tensor 𝐏 41
describing the probability that the magnetic field is oriented along the principal components of the 𝛘𝐦𝐨𝐥 tensor E an cos 2 i exp i sin i d i d i kT Pii Eian exp sin d d i i i kT
cos
2
(110)
B2
2 2 mol 2 2 0 ( iimol cos 2 i mol i exp jj sin i cos i kk sin i sin i ) d cos i d i 2 kT 0
B02
exp 2 kT ( 0
mol ii
2 2 mol 2 2 cos 2 i mol jj sin i cos i kk sin i sin i ) d cos i d i
where i and i are the spherical angles describing the orientation of the magnetic field with respect to the main axes i of the 𝛘 tensor (with diagonal components iimol , mol and kkmol ). The result of jj an integration to first order in Ei / kT is mol mol 1 2 B02 mol jj kk Pii 1 ii 3 15 0 kT 2
(111)
so that [72,73], Pii
1 B02 mol 1 iimol iso 3 50 kT
(112)
As expected, Pxx Pyy Pzz 1 . In summary, molecular self-orientation may occur in the presence of an anisotropic susceptibility produced either by paired electrons, especially when there is an anisotropic distribution of aromatic ring planes, or by unpaired electrons, or by a combination of both.
6.1 NMR shifts in self-orienting paramagnetic molecules In the presence of partial molecular alignment, the hyperfine shifts average to values resulting from the projection of the components of the shielding tensor 𝛔hf onto the main directions of the orientational probability tensor 𝐏 (Eq. (112)): 𝛿hf =‒ 𝛔hf ∙ 𝐏
(113)
When self-orientation is caused by the anisotropy of the paramagnetic susceptibility tensor, the 𝛘 and 𝐏 tensor are diagonal in the same frame, and thus the pseudocontact shifts are (see Eq. (62)) 42
xx 0 3xz Pxx 0 0 3xy 0 3x 2 r 2 1 2 2 yz r y xy 3 3 3 0 0 Tr pc 0 Pyy 0 yy 5 4r 2 2 3xz 0 yz z r 3 3 0 zz 0 0 Pzz
(114)
or, in spherical coordinates, using Eq. (112) with 𝛘𝐦𝐨𝐥 = 𝛘, [74]
pc
xx yy 3 rh2 B02 1 2 1 2 1 cos 3 ax zz 2 12r 3 150 kT 4 ax
(115)
B02 3 2 xx 2 yy zz rh sin 2 cos 2 1 2 15 0 kT where θ is the angle between the metal–nucleus vector 𝐫 and the z axis of the diagonal 𝛘 tensor, and
is an angle related to the projection of 𝐫 onto the xy plane of the tensor. The corrections to the B02 ax ax 1 15 0 kT anisotropy values are thus given by
yy 2 zz xx 2
3 rh2 4 ax
,
B02 2 xx 2 yy zz rh rh 1 15 0 kT . This correction to the pseudocontact shift is in most cases minor, and can thus be neglected (see Figure 5), so that Eq. (64) can be safely used in fields of at least up to 20 T.
43
Magnetic susceptibility anisotropy / 10-32 m3
41.0 ax'
40.5 40.0
ax
39.5 -19.5
rh
-20.0 rh'
-20.5 -21.0
0
200 400 600 800 1000 1200 Proton Larmor frequency / MHz
Figure 5. Field dependence of the apparent axial and rhombic magnetic susceptibility anisotropy calculated for the tensor components xx, yy and xx of 5, 7 and 10 ×1031 m3, respectively. Analogously, from Eqs. (68) and (113), in the presence of a partial alignment, the contact shifts are given by [54] 𝜇B𝐴Fc
𝛿 = 𝑘𝑇ℏ𝛾𝐼[(〈𝐒𝐒𝑇〉 ∙ 𝐠𝑇)𝑥𝑥𝑃𝑥𝑥 + (〈𝐒𝐒𝑇〉 ∙ 𝐠𝑇)𝑦𝑦𝑃𝑦𝑦 + (〈𝐒𝐒𝑇〉 ∙ 𝐠𝑇)𝑧𝑧𝑃𝑧𝑧] Fc
(116)
In the presence of a partial alignment, including the self-alignment caused by the anisotropy of the 𝛘 tensor, the NMR shifts are also affected by a contribution resulting from the anisotropy of the chemical shielding tensor, causing residual anisotropic chemical shifts (𝛿racs). Therefore, the NMR shifts in paramagnetic molecules are equal to 𝛿para = 𝛿dia + 𝛿hf + 𝛿racs
(117)
with [75,76]
racs
B02 iiCSA cos 2 ij jj 15 0 kT i , j 44
(118)
where i and j indicate the axes x, y and z, and
ij are the angles between the principal axes of the
magnetic susceptibility anisotropy tensor 𝚫𝛘 and the principal axes of the chemical shift anisotropy tensor CSA. The tensor CSA is defined as the difference between the chemical shielding tensor and its isotropic value. Residual anisotropic chemical shifts can be significant for nuclei with large CSA tensors at high magnetic fields, whereas they are usually negligible for 1H nuclei.
6.2 Paramagnetic residual dipolar couplings In the presence of multiple nuclei, the dipole-dipole interactions occurring between any pair of nuclei have the following energy (see Eq. (61))
(
𝜇0 3𝐫𝑖𝑗𝐫𝒊𝑗𝑇 dip 𝐸 𝑖𝑗 =‒ 4𝜋𝑟 3 𝛍𝐼𝑖 ⋅ 𝑟 2 ‒ 𝟏 𝑖𝑗 𝑖𝑗
)
𝜇0 ℏ2𝛾𝐼𝑖𝛾𝐼𝑗
⋅ 𝛍𝐼𝑗 = ‒ 4𝜋
𝑟𝑖𝑗3
𝛋⋅
(
3𝐫𝑖𝑗𝐫𝒊𝑗𝑇 𝑟𝑖𝑗2
)
‒𝟏 ⋅𝛋
(119)
where 𝛍𝐼𝑖 and 𝛍𝐼𝑗 are the magnetic moments of the ith and jth nuclei, at a distance 𝐫𝑖𝑗 and oriented along the direction 𝛋 with respect to 𝐁0. The averages of 𝐸dip 𝑖𝑗 over isotropic rotation vanish; however, in the presence of a partial alignment, < 𝐸dip 𝑖𝑗 >≠ 0 and residual dipolar couplings (RDCs) arise. The RDC between a pair of nuclei, 1 and 2, is thus [77–79] 12rdc
0 2 I 15 I 2 8 r12
3 x 2 r 2 Tr 3 xy 3 xz
3 xy 3y 2 r 2 3 yz
E12dip h
3 xz Pxx 3 yz 0 3 z 2 r 2 0
0 Pyy 0
0 0 Pzz
(120)
If the partial alignment is induced by anisotropies of both diamagnetic and paramagnetic susceptibility tensors (see Eq. (112)) [80], the RDC can be split into two components 12rdc (χ mol ) 12rdc(dia) (χ dia ) 12prdc (χ )
where the paramagnetic RDC is
45
(121)
3x 2 r 2
prdc 12
0 2 I 15 I 2 Tr 3xy 8 r12
1 B02 1 xx iso 3 5 0 kT 0 0
3xz
3xy 3y 2 r 2 3 yz
3xz 3 yz 3z 2 r 2
0
0
B02 1 1 yy iso 3 5 0 kT
0
0 B02 1 1 zz iso 3 5 0 kT
(122)
Therefore, in the frame where the 𝛘 tensor is diagonal, 12prdc
0 I 1 I 2 B02 xx (3x 2 r 2 ) yy (3 y 2 r 2 ) zz (3z 2 r 2 ) 2 5 8 r12 150 kT
(123)
and, in spherical coordinates [77,80,81], 12prdc
1 B02 I 1 I 2 3 ax (3 cos 2 1) rh sin 2 cos 2 3 4 15kT 2r12 2
(124)
where is the angle between 𝐫12 and the z axis of the diagonal 𝛘 tensor and is the angle which describes the position of the projection of the 𝐫12 vector onto the xy plane of the 𝛘 tensor, relative to the x axis. As already shown for pseudocontact shifts, the equation for the paramagnetic RDC can also be written in a generic reference frame: 12prdc
2 1 B0 3 I 1 I 2 4 15kT 4r123
zz iso (3 cos 2 1) ( xx yy ) sin 2 cos 2 2 xy sin 2 sin 2 2 xz sin 2 cos 2 yz sin 2 sin
(125)
or, using direction cosines, 12prdc
1 B02 3 I 1 I 2 4 15kT 2r123
ij cos i cos j i, j
1 B02 3 I 1 I 2 T r12 χ r12 4 15kT 2r125
(126)
where χ χ iso 1 , the indices i and j run over the three axes x, y and z, and i are the angles between 𝐫12 and each of the three axes. When written in the form of a matrix, Eq. (126) agrees with Eq. (5) reported in Section 2. Although the functional form of the equation for the RDCs is equivalent to that for the 46
pseudocontact shifts (see Panel 4), the information content of these two types of NMR observables is very different because the distances and angles in the equation for the paramagnetic RDCs describe the distance and the orientation of the vector connecting a nuclear pair in the frame defined by the 𝛘 tensor, whereas, in the equation for the pseudocontact shifts, they describe the position of the nucleus in the same reference frame, the origin being at the position of the paramagnetic metal ion. In Eqs. (122)-(126) a generalized Lipari-Szabo order parameter [82], SLS, is often included as a further coefficient, to account for the effect of internal motions with correlation times shorter than the molecular reorientation time. These motions can reduce the paramagnetic RDCs, due to the averaging over multiple orientations of the 𝐫12 vector with respect to the 𝛘 tensor [17]. In total, the splitting of the NMR peaks of covalently bound nuclei in a paramagnetic molecule is given by 12para 1J12 12rdc(dia) 12prdc 12dfs(dia) 12pdfs
(127)
where 1 J12 is the field-independent scalar coupling between the two nuclei, and 12dfs(dia) and 12pdfs are the diamagnetic and paramagnetic dynamic frequency shifts (see Section 8). Therefore, the difference in 12para at two magnetic fields is
para 12
( B0(1) )
para 12
2 2 1 B0(1) B0( 2) I 1 I 2 ( B0 ( 2 ) ) 4 15kT 2r123
3 axmol (3 cos 2 1) rhmol sin 2 cos 2 2
(128)
because, at high magnetic fields, the dynamic frequency shift is almost independent of the field (see Section 8). On the other hand, the difference in the NMR splittings observed at the same field in a paramagnetic sample and in a diamagnetic analogue is 12para 12dia 12prdc 12pdfs
(129)
In theory, Eq. (129) may be used to determine the paramagnetic dipolar frequency shifts from the observed 12para 12dia values and the paramagnetic RDCs, calculated using Eq. (124) and the 𝛘anisotropy tensor obtained from the analysis of the pseudocontact shifts; in practice, the paramagnetic 47
dipolar frequency shifts are so small for nuclei a few Å away from the paramagnetic center that they can be considered to be negligible, so that the differences between the NMR splittings in a paramagnetic sample and in its diamagnetic analogue are usually ascribed to the paramagnetic RDCs alone. Therefore, they must depend on the same 𝛘-anisotropy tensor responsible for the pseudocontact shifts, and the two types of experimental data can be analysed simultaneously and fitted against a unique tensor [83].
7. Paramagnetic relaxation enhancements The hyperfine coupling between the total electron magnetic moment and the nuclear magnetic moment also causes a contribution to nuclear relaxation, known as paramagnetic relaxation enhancement, which adds to the diamagnetic relaxation. The energy of this interaction (described by Eqs. (73) and (74) in the point-dipole approximation), can fluctuate randomly in time through three possible mechanisms: i)
molecular reorientation, causing changes in the orientation of the 𝐫 vector;
ii)
electron relaxation, causing changes in the orientation of the 𝛍(𝐒) vector;
iii)
chemical exchange, causing changes in the length of the 𝐫 vector.
The time scales for these three mechanisms are described by three correlation times: the reorientation correlation time r, the longitudinal and transverse electron relaxation times e1 and e2, and the residence time M. These fluctuations allow for a coupling of the electron-nucleus spin system with the external world (the lattice), thus allowing for energy exchange. The paramagnetic relaxation enhancement can be calculated from the transition probabilities wij between different states of the electron-nucleus spin system. These transition probabilities can be calculated from time-dependent perturbation theory for stochastic perturbations [30,84]: t
wij
2 Re i | H hfc (t ) | j j | H hfc (t ) | i exp(iij )d 2 0
48
(130)
where |𝜓𝑖 >
and |𝜓𝑗 >
indicate the eigenstates of the unperturbed Zeeman Hamiltonian,
characterized by the quantum numbers mS and mI of the electron and the nucleus respectively, ij being the difference between the corresponding eigenvalues divided by , and the bar indicating an ensemble average. If the perturbation is stationary, i.e., independent of time, and the ensemble average decays exponentially with time, which is usually a good approximation, the transition probabilities depend on the autocorrelation functions [52]
Gij ( ) i | H hfc (0) | j j | H hfc ( ) | i i | H hfc (0) | j 2 exp( / c ) (131) where the time constant c depends on the time constants mentioned above (r, e1,2 or M, see Eqs. (146), (151) and (162)). Finally, if the autocorrelation function decays to zero rapidly within the interval of integration, the latter can be safely extended to infinity (Redfield limit) [85–87] wij
2 i | H hfc (0) | j 2 c 2 hfc 2 H Re i | ( 0 ) | exp( / ) d i j ij c 2 2 1 ij2 c2 0
(132) The last term in Eq. (132) is known as a Lorentzian spectral density function J ( , )
1 2 2
(133)
The paramagnetic longitudinal relaxation enhancement, R1M, can be calculated from the sum of the transition probabilities causing a change in the nuclear spin states (the electron spin is assumed to remain in thermal equilibrium on the time scale of nuclear spin relaxation, due to the much more efficient electron relaxation mechanisms) R1M w0 2 w1I w2
(134)
where w0 indicates the probability of transitions causing a decrease in the electron spin quantum number and an increase in the nuclear spin quantum number (or vice versa), w1I indicates the 49
probability of transitions between states with the same electron spin quantum number and different nuclear spin quantum numbers, and w2 indicates the probability of transitions causing a decrease, or an increase, in both the electron and the nuclear spin quantum numbers (see Fig. 6). Therefore, if the energy of the electron-nuclear spin states is determined only by the Zeeman Hamiltonian (i.e., in the absence of ZFS and/or hyperfine coupling between unpaired electron(s) and the metal nucleus), the transition frequencies that determine the transition probabilities w0, w1I and w2 are S I ,
I and
S I , respectively, where 𝜔𝑆 =
< 𝑚𝑆|𝐻Zeeman|𝑚𝑆 >‒< 𝑚𝑆 ‒ 1|𝐻Zeeman|𝑚𝑆 ‒ 1 > ℏ
,
𝜔𝐼 =
|𝑚𝐼 >‒< 𝑚𝐼 ‒ 1|𝐻Zeeman |𝑚𝐼 ‒ 1 > < 𝑚𝐼|𝐻Zeeman 𝐼 𝐼 ℏ
,
(135) = ‒ ℏ𝛾𝐼𝐁0𝑇 ∙ 𝐈 𝐻Zeeman = 𝜇B𝐁0𝑇 ∙ 𝐠 ∙ 𝐒 , 𝐻Zeeman 𝐼 and thus 𝜔𝑆 > 0 and 𝜔𝐼 < 0, see Figure 6. For I = 1/2, the transition probabilities are thus w0
c 2p m S 1,1 / 2 | H hfc (0) | m S ,1 / 2 2 2 mS 1 ( S I ) 2 c2
(136)
w1I
c 2p 1 mS ,1 / 2 | H hfc (0) | mS ,1 / 2 2 2 2 mS 1 I2 c2
(137)
w2
c 2p m S 1,1 / 2 | H hfc (0) | m S ,1 / 2 2 2 mS 1 ( S I ) 2 c2
(138)
where |i,j> indicates |mS,mI>, and p is a normalization factor for the population of the spin states (p = 2/(2S+1), because 2S/(2S+1) is the ratio between the number of unpaired electrons and the number of states - p = 1 in the case of a single unpaired electron, i.e., for S = 1/2).
50
mS 1/2
w1I w0
w1S
w2
mS1, 1/2
w1I
mS 1/2 S
S
w1
( S | I |) ( S | I |)
| I |
mS1, +1/2
Figure 6. Energy levels, transition frequencies and transition probabilities in a dipole-dipole coupled S-I system with four levels mS, mS1, mI = +1/2, mI = 1/2 of the entire manifold of electron and nuclear spin states. The paramagnetic spin-spin relaxation enhancement, R2M, does not depend only on the transition probabilities between states with mI = 1, because spin transitions with equal nuclear energy also contribute to transverse relaxation. R2M can thus be calculated to be [52,88,89] R2 M
R1M p 2 2 mS ,1 / 2 | H hfc (0) | mS ,1 / 2 2 c 2 mS
c p 2 mS ,1 / 2 | H hfc (0) | mS ,1 / 2 2 2 1 S2 c2 mS ,mS
(139)
Due to the proportionality between NMR signal linewidths and transverse relaxation rates, a paramagnetic broadening of the NMR signals occurs:
para R2 M /
(140)
In the following sections, R1M and R2M are calculated by evaluating the terms
i | H hfc (0) | j 2 in Eq. (132). This is done by splitting the hyperfine coupling Hamiltonian into three terms, describing the dipole-dipole interaction between the nuclear magnetic moment and the zero average component of the electron magnetic moment (the dipolar term), the dipole-dipole interaction between the nuclear magnetic moment and the non-zero average component of the electron magnetic moment (the Curie term, Eq. (25)), and the interaction between the nuclear 51
magnetic moment and the electron magnetic moment localized on the nucleus (the Fermi contact term): RiM = RiMdip+RiMCurie+RiMFc.
(141)
Contact contributions can only be present for nuclei with some unpaired electron spin density delocalized in their s orbitals. For nuclei at distances larger than a few Å, only dipolar and Curie contributions can thus remain; they can be safely calculated within the point-dipole approximation, resulting in an r6 distance dependence (see later).
7.1 Dipolar relaxation The paramagnetic relaxation due to the point dipole-point dipole interaction between the nuclear magnetic moment and the electron magnetic moment is described by the Solomon equations, assuming that the static Hamiltonian of the spin system is properly described by considering only terms corresponding to the Zeeman interactions of the nuclear and electron magnetic moments with the external magnetic field [30]. The time-dependent perturbative Hamiltonian is described by Eq. (73) and the first term of Eq. (74), and with isotropic 𝐠 = 𝑔iso𝟏 (for the case of an anisotropic 𝐠 tensor, see Section 7.4) 𝜇0 ℏ𝛾𝐼𝜇B
𝐻dip(𝑡) = 𝐒 ⋅ 4𝜋
3
𝑟
(
𝑔iso
3𝐫𝐫𝑇 𝑟2
)
(142)
‒𝟏 ⋅𝐈
which can be rewritten as
(
𝐻dip(𝑡) = 𝐼𝑧𝑆𝑧 ‒
𝐼+𝑆‒ + 𝐼‒𝑆+ 4
)𝐹
∗ 0 + (𝐼 + 𝑆𝑧 + 𝐼𝑧𝑆 + )𝐹1 + (𝐼 ‒ 𝑆𝑧 + 𝐼𝑧𝑆 ‒ )𝐹 1
+ 𝐼 + 𝑆 + 𝐹2 + 𝐼 ‒ 𝑆 ‒ 𝐹 2∗
3
3
𝜇0 ℏ𝛾𝐼𝜇B𝑔iso
with 𝐹0 = 𝑘(3cos2𝜃 ‒ 1), 𝐹1 = 2𝑘sin𝜃cos𝜃𝑒 ‒ 𝑖𝜑, 𝐹2 = 4𝑘sin2𝜃𝑒 ‒ 2𝑖𝜑, 𝑘 =‒ 4𝜋
𝑟3
(143)
and 𝜃 and 𝜑
are the spherical angles indicating the orientation of the vector connecting the nucleus and the paramagnetic metal ion positions with respect to a molecular frame where the magnetic field is directed along the z axis. 52
----------------------------------------------------------------------------------------------------------------------Panel 9 From Eqs. (136)-(138) and (143), since < 𝑚𝑆 + 1|𝑆 + |𝑚𝑆 > = 𝑆(𝑆 + 1) ‒ 𝑚𝑆(𝑚𝑆 + 1), < 𝑚𝑆 ‒ 1|𝑆 ‒ |𝑚𝑆 > = 𝑆(𝑆 + 1) ‒ 𝑚𝑆(𝑚𝑆 ‒ 1), 2 S ( S 1) mS (mS 1) 2 2S ( S 1)(2S 1) 4S ( S 1) 2 S 1 mS 2S 1 3 3
and
2 2 S ( S 1)(2 S 1) 2 S ( S 1) , mS2 2 S 1 mS 2S 1 3 3 2
w0
2 F0 4 S ( S 1) τ cdip 2 2 3 16 2 1 ( S I ) 2 ( τ cdip 2 )
w1I
τ cdip 2 F1 1 2 S ( S 1) 1 2 2 2 3 1 I2 ( τ cdip 1 )
2
2
τ cdip 2 F2 4 S ( S 1) 2 w2 2 3 1 ( S I ) 2 ( τ cdip 2 2 )
(P9.1)
(P9.2)
(P9.3)
and after integration of Fi2 over the angular coordinates ( F0 2 4 k 2 , F12 F2 2 3 k 2 ), using Eq. (134), 10 5 2
R1dip M
2 μ B2 S ( S 1) 2 0 γ I2 g iso r6 15 4
3τ cdip 6 τ cdip τ cdip 2 1 2 (P9.4) dip 2 2 dip 2 2 dip 2 2 1 I ( τ c1 ) 1 ( S I ) ( τ c 2 ) 1 ( S I ) ( τ c 2 ) Analogously, the paramagnetic enhancement to the spin-spin (transverse) nuclear relaxation rate (Eq. (139)) is 2
R2dip M
2 μ B2 S ( S 1) 1 0 γ I2 g iso 15 4 r6
τ cdip 3τ cdip 6 τ cdip 2 1 2 2 dip 2 2 dip 2 2 dip 2 1 I ( τ c1 ) 1 ( S I ) ( τ c 2 ) 1 ( S I ) ( τ c 2 ) 2
2 2 2 τ cdip2 γ g μ 2 2 S ( S 1) 4 dip 4 4 S ( S 1) 3 0 I iso6 B τ c1 2 dip 2 3 5 4 3 10 1 S ( τ c 2 ) r 4 4
(P9.5)
and thus 2
R2dip M
2 μ B2 S ( S 1) 1 0 γ I2 g iso 15 4 r6
τ cdip 3τ cdip 6 τ cdip 6 τ cdip 2 1 2 2 4 τ cdip 1 2 dip 2 2 dip 2 2 dip 2 2 dip 2 1 ( ) ( τ ) 1 ( τ ) 1 ( ) ( τ ) 1 ( τ ) S I c2 I c1 S I c2 S c2
(P9.6)
-----------------------------------------------------------------------------------------------------------------------
53
Since the electron Larmor frequency S is 658.2 times larger than the proton Larmor frequency I (so that S I S ), the paramagnetic enhancement of the spin-lattice (longitudinal) nuclear relaxation rate is (see Panel 9) [30] 2
dip 1M
R
2 7 τ cdip 3τ cdip μ B2 S ( S 1) 2 μ 0 γ I2 g iso 2 1 1 ω 2 ( τ dip ) 2 1 ω 2 ( τ dip ) 2 15 4π r6 2 c1 c I S
(144)
and the paramagnetic enhancement of the spin-spin (transverse) nuclear relaxation rate is 2
R
dip 2M
2 13τ cdip 3τ cdip μ B2 S ( S 1) dip 1 μ 0 γ I2 g iso 1 2 4 τ 1 c dip 2 2 dip 2 2 15 4π 1 ( ) 1 ( r6 ω τ ω S c2 I τ c1 )
(145)
where the correlation times are
τ
dip -1 c1
τe11 τ r1 τ M1 ,
τ
dip -1 c2
τe21 τ r1 τ M1
(146)
because the dipole-dipole interaction can be modulated by the longitudinal/transverse electron relaxation time, the reorientation time and/or the residence time. It is assumed here that reorientational motions, chemical exchange and electron relaxation processes are not coupled. In the case of lanthanoids and actinoids, the J quantum number replaces the S quantum number and gJ replaces giso. If e = e for nuclear Larmor frequencies smaller than (2 658.2 τ cdip2 ) 1 (corresponding to the condition, which is usually met, that ωS τ cdip2 1 ), or both e and e are much longer than r or M, it follows that Eqs. (144) and (145) can be written as a function of a single correlation time c c1 . dip
dip
As a result, the field dependence of the paramagnetic enhancement of the proton relaxation rates is characterized by two Lorentzian dispersions centered at frequencies
S ( cdip ) 1 and I ( cdip ) 1 ,
and thus separated by a factor 658.2. The electron relaxation time can also depend on the applied magnetic field, and this could lead to a possible field dependence of c . For instance, a field dependence of electron relaxation is dip
expected for systems experiencing ZFS (provided that S>1/2). Collisions due to solvent molecules can cause deformations of the metal coordination polyhedron, which can in turn cause a transient ZFS 54
even in the absence of a static ZFS. This provides a relaxation mechanism for the electron(s), that in the pseudorotational model formulated by Bloembergen and Morgan [90] results in e11
22t 4S ( S 1) 3 τ ν 2 2 4τ ν 2 2 50 1 ωS τ ν 1 4ωS τ ν
(147)
e21
2t 4S ( S 1) 3 3τ ν 5τ ν2 2 2τ ν 2 2 1 ωS τ ν 1 4ωS τ ν 50
(148)
where t is the mean squared fluctuation of the ZFS and τv is the correlation time for the 2
instantaneous distortions of the metal coordination polyhedron. This field dependence of electron relaxation causes a peak (typically at about 1 T for the gadolinium(III) and manganese(II) complexes and higher for e.g. iron(III) and nickel(II) [91]) in the field dependence of the nuclear longitudinal relaxation rate. Eqs. (144)-(148) are usually referred to as the Solomon-Bloembergen-Morgan (SBM) model (together with the following Eqs. (160) and (161)). Notably, different electron transitions may have different relaxation rates, as have been calculated for S = 3/2 and 5/2 systems [92]. Eqs. (147) and (148) thus describe, to a good approximation, the average electron relaxation rates of all allowed transitions. Finally, intramolecular (local) motions can also, at least partially, modulate the dipole-dipole interaction. These fast motions can thus affect the nuclear correlation time, and are usually treated using the Lipari-Szabo model-free approach [82,93].
7.2 Curie spin relaxation As anticipated in Section 2, Curie spin relaxation describes the paramagnetic contribution to nuclear relaxation due to the dipole-dipole interaction between the nuclear magnetic moment and the thermal average of the total electron magnetic moment, i.e., between 𝛍𝐼 and 〈𝛍el〉 (see Eq. (30)). Therefore, using Eq. (73) and the first term of Eq. (74), 𝜇0 ℏ𝛾𝐼𝜇B
𝐻Curie(𝑡) = 𝐒C ⋅ 4𝜋
𝑟3
𝐠𝑇 ⋅
(
3𝐫𝐫𝑇 𝑟2
)
‒𝟏 ⋅𝐈 55
(149)
In the absence of g-anisotropy, the time-dependent perturbative Hamiltonian is given by (see Eq. (142)) 𝜇0 ℏ𝛾𝐼𝜇B
𝐻Curie(𝑡) = 𝐒C4𝜋
𝑟3
𝑔iso ⋅
(
3𝐫𝐫𝑇 𝑟2
)
(150)
‒𝟏 ⋅𝐈
Since the dipole-dipole interaction is in this case only coupled with the average over all electron spin states, it can only be modulated by reorientation and chemical exchange
τ
Curie 1 c
τr1 τ M1
(151)
2 w1I R1Curie M
(152)
and
which provides the I dispersion term of Eq. (144), with S(S+1)/3 replaced by 𝑆C2. Therefore, from Eqs. (144) and (145), [29,94] 2
R1Curie M
2 3τ cCurie 2 μ0 γI2 giso μB2 SC2 1 ωI2 ( τ cCurie ) 2 5 4π r6
2
R2Curie M
2 1 μ0 γI2 giso μB2 SC2 5 4π r6
(153)
Curie 3τ cCurie 4 τ c 2 Curie 2 1 ωI ( τ c )
(154)
and using Eq. (29) 2
Curie 1M
4 μ B4 B02 S 2 ( S 1) 2 τ cCurie 2 μ 0 γ I2 g iso 15 4π k 2T 2 r 6 1 ω I2 ( τ cCurie ) 2
Curie 2M
4 μ B4 B02 S 2 ( S 1) 2 1 μ 0 γ I2 g iso 45 4π k 2T 2 r 6
R
2
R
Curie 3τ cCurie 4 τ c 1 ωI2 ( τ cCurie ) 2
(155)
(156)
Eqs. (155) and (156) can be written as Eqs. (3) and (4), reported in Section 2, using the Curie law, Eq. (50). In the case of lanthanoids and actinoids, the J quantum number replaces the S quantum number and gJ replaces giso. Since 𝑆C2 is much smaller than S(S + 1)/3, Curie spin relaxation can only provide nonCurie negligible contributions when τ cdip , i.e., when the electron relaxation time is much smaller 1 << τ c1
than both the reorientation time and the residence time. 56
7.3 Fermi contact relaxation Fermi contact relaxation describes the paramagnetic contribution to nuclear relaxation due to the dipole-dipole interaction between the nuclear magnetic moment and the electron magnetic moment localized on the nucleus itself. The time-dependent perturbative Hamiltonian is thus described by Eq. (73) and the second term of Eq. (74), i.e., 𝜇0
𝜌𝐼
𝐻Fc(𝑡) = 3 ℏ𝛾𝐼𝜇B𝑔e 𝑆 𝐒 ⋅ 𝐈
(157)
which, using Eq. (71), can be written as
(
𝐻Fc(𝑡) = 𝐴Fc 𝑆𝑧𝐼𝑧 +
𝑆+𝐼‒ + 𝑆‒𝐼+ 2
).
(158)
In this case, only the zero-quantum transition provides a contribution to relaxation (because the spin operators 𝑆𝑧𝐼𝑧, 𝑆 + 𝐼 ‒ and 𝑆 ‒ 𝐼 + contribute only to 𝑤0), so that (see Eq. (136)),
cFc2 2( A Fc ) 2 4 S ( S 1) 3 4 2 1 ( S I ) 2 ( τ cFc2 ) 2
R1FcM w0
(159)
and thus, considering that S>>I [95,96], 2
Fc 1M
AFc 2 τ cFc2 S ( S 1) 2 Fc 2 3 1 ωS ( τ c 2 )
Fc 2M
A Fc 1 S ( S 1) 3
R
R
2
Fc τ cFc2 τ c1 2 Fc 2 1 ωS ( τ c 2 )
(160)
(161)
Since reorientation cannot modulate the contact coupling, the correlation time in this case is
τ
Fc 1 ci
τei1 τ M1
(i = 1,2).
(162)
Fermi contact relaxation has been also shown to occur in the presence of a contact interaction modulated by collisions between solvent molecules and the molecule bearing the paramagnetic center [97]. In this case a distribution of correlation times corresponding to the lengths of the contact times between molecules must be considered (the ‘pulse’ model [98]).
57
Fermi contact coupling can also provide Curie spin relaxation contributions when the residence time is much larger than the electron relaxation time, so that, from Eqs. (161) and (29) [94], 2
2
Fc,Curie 2M
R
g iso B B0 S ( S 1) AFc AFc τM S τ M 3kT 2 C
(163)
where the term in parenthesis is proportional to the contact shift (in hertz, see Eq. (94)).
7.4 Anisotropic effects The previous equations for the paramagnetic enhancements of the nuclear relaxation rates were derived by making a number of approximations: 1) the electron is supposed to reside at a single point and to behave as a magnetic point dipole, thus neglecting electron spin density delocalization (the point-dipole approximation); 2) the perturbation Hamiltonian changes stochastically and is stationary; 3) molecular reorientation is isotropic; 4) molecular reorientation and electron spin relaxation (i.e., the dynamics of the electron magnetic moment) are uncorrelated (the decomposition approximation); 5) electron spin relaxation is a single exponential process (as if all electron spin transitions had the same relaxation rates); 6) the correlation time for nuclear relaxation is much shorter than the nuclear relaxation time, and the correlation time for electron relaxation is much shorter than the electron relaxation time (the Redfield limit); 7) the electron 𝐠 tensor is isotropic, and the static Hamiltonian is dominated by the electron Zeeman interaction. Assumptions 1-5 seem to be acceptable, in the sense that deviations from these approximations either do not cause large changes in the calculated relaxation rates, or corrections can be applied (for 58
instance, Lipari-Szabo model free approaches [82] can be used to treat the case of anisotropic molecular reorientation, or multiexponential electron relaxation can be considered and plugged into the nuclear relaxation equations [92,99]). The effect of the anisotropy of the electron 𝐠 tensor on the relaxation equations was analyzed both in the case of fast rotation [100] and of slow rotation (with respect to the electron relaxation time) [101,102] and it was found to be small. In contrast, beyond the Redfield limit, dramatic changes in the calculated relaxation rates can occur [85,103]. The presence of a static ZFS and/or hyperfine coupling between the electron magnetic moment and the magnetic moment of the metal nucleus (i.e., of the nucleus of the paramagnetic ion itself) can also cause dramatic changes in relaxation rates. When present, these terms should thus be included in the static Hamiltonian of the spin system together with the Zeeman terms, because they can significantly affect the energy of the electron spin states, especially at low fields. As a result, the energies of most electronic spin transitions can be much larger than calculated from the Zeeman interaction only, thus accounting for changes in the relaxation rates. The effects of these anisotropic terms on the nuclear relaxation rates can be calculated by writing the hyperfine coupling Hamiltonian (Eq. 143) in the form 𝐻dip(𝑡) = 𝜇0 ℏ𝛾𝐼𝜇B
where 𝜅 =‒ 4𝜋
𝑟3
24𝜋 𝑚 ∑2 5 𝜅 𝑚 =‒ 2( ‒ 1) 𝑌2,𝑚(𝜃,𝜑)𝑇2, ‒ 𝑚
(164)
, 𝑌2,𝑚(𝜃,𝜑) are normalized spherical harmonics and 𝑇2, ‒ 𝑚 are time independent
tensor operators acting on the spin variables [84]: 𝑇2,0 =
[2𝑔𝑧𝑧𝐼𝑧𝑆𝑧 ‒ 12𝑔 ⊥ (𝐼 + 𝑆 ‒ + 𝐼 ‒ 𝑆 + )], 𝑇2, ± 1 =∓ 12(𝑔𝑧𝑧𝐼 ± 𝑆𝑧 + 𝑔 ⊥ 𝐼𝑧𝑆 ± ), 𝑇2, ± 2 = 12𝑔 ⊥ 𝐼 ±
1 6
𝑆± 𝑌2,0 =
5 2 16𝜋(3cos 𝜃 ‒ 1),
15 8𝜋cos 𝜃sin 𝜃exp ( ± 𝑖𝜑),
𝑌2, ± 1 =∓
𝑌2, ± 2 =
(165)
15 2 32𝜋sin 𝜃exp ( ± 2𝑖𝜑)
(166) and 𝑔𝑧𝑧 and 𝑔 ⊥ are the principal values of the 𝐠 tensor in the case of axial symmetry (for the rhombic case, 𝑔 ⊥ 𝑆 ± must be replaced by 𝑔𝑥𝑥
𝑆+ + 𝑆‒ 2
𝑆+ ‒ 𝑆‒
± 𝑔𝑦𝑦 59
2
[89]). The spherical harmonics depend on
the spherical angles between the vector connecting the positions of the nucleus and the paramagnetic metal ion, and the direction of the magnetic field. If 𝜃 and 𝜑 are the spherical angles reporting on the metal-nucleus direction in the molecular frame, the spherical harmonics can be written in the laboratory frame and then transformed into the molecular frame using Wigner rotation matrices, which contain the Euler angles , and , indicating the magnetic field direction in the molecular frame: 𝑌2,𝑚(𝜃,𝜑) = 𝑅(𝛼,𝛽,𝛾)𝑌2,𝑚(𝜃,𝜑)
(167)
From Eqs. (132)-(134) R1dip M
1 c complex conjugate Re i | H dip | j 2 2 2 2 1 (ij I ) c ij
(168)
where |𝜓𝑖 > and |𝜓𝑗 > are the electron spin eigenstates, and 𝜔𝑖𝑗 =
< 𝜓𝑖|𝐻0|𝜓𝑗 >‒< 𝜓𝑗|𝐻0|𝜓𝑖 > ℏ
,
(169)
where H0 is the unperturbed Hamiltonian. In the presence of ZFS (Eq. (18)), 𝐻 0 = 𝜇 B 𝐁0 ∙ 𝐠 ∙ 𝐒 + 𝐒 ∙ 𝐃 ∙ 𝐒
(170)
and assuming that the 𝐠 and 𝐃 tensors are diagonal in the same molecular frame
(
𝐻0 = 𝜇B𝐵0 𝑔𝑧𝑧𝑆𝑧cos𝛽 ‒ 𝑔𝑥𝑥
𝑆+ + 𝑆‒ 2
𝑆+ ‒ 𝑆‒
sin𝛽cos𝛾 ‒ 𝑖𝑔𝑦𝑦
2
) (
sin𝛽sin𝛾 + 𝐷 𝑆2𝑧 ‒
𝑆 +2 ‒ 𝑆 2‒
)+𝐸
𝑆(𝑆 + 1) 3
2
(171) where , and are the Euler angles that define the orientation of the magnetic field with respect to the molecular frame. The eigenstates in Eq. (168) can be determined from diagonalization of the Hamiltonian of Eq. (171). In solution, isotropic molecular tumbling implies that averaging should be performed over all orientations, as indicated by the bar in Eq. (168). The presence of a static ZFS may significantly affect the nuclear relaxation rates, mainly depending on the magnitude and rhombicity of the ZFS tensor with respect to the Zeeman energy, as well as on the position of the nucleus with respect to the ZFS tensor (Figure 7). Therefore, the paramagnetic enhancement of the nuclear relaxation rate depends not only on the metal-nuclear 60
distance (as predicted by the SBM model, Eqs. (144) and (145), or by the Curie relaxation Eqs. (155) and (156)), but also on the spherical angles defining the metal-nucleus direction in the molecular frame. An additional dispersion in the field dependence of the relaxation rate may appear as a result of the transition from a dominant ZFS to a dominant Zeeman energy as the field increases [104].
10 Solomon profile (no ZFS)
Rdip 1M (a.u.)
8
D=1 cm-1, E=0
=0°
6 4
D=1 cm-1, E/D=1/3
=90° =0°
2 0 0.01
=90°
0.1
1
10
100
1000
Proton Larmor Frequency / MHz Figure 7. Dipole-dipole relaxation (S = 3/2, c
dip
= 0.1 ns) in the absence (Solomon equation) and in
the presence of axial (D = 1 cm1) or rhombic (E/D = 1/3) ZFS for a proton along the z axis ( = 0°) or in the x,y plane ( = 0°). In the presence of a hyperfine coupling between the metal nucleus and unpaired electron(s), the unperturbed Hamiltonian should contain the term [89,105] 𝐴𝑥𝑥
𝐴𝑦𝑦
𝐻0(hyp) = 𝐴𝑧𝑧𝐼𝑧𝑆𝑧 + 4 (𝐼 + 𝑆 + + 𝐼 + 𝑆 ‒ + 𝐼 ‒ 𝑆 + + 𝐼 ‒ 𝑆 ‒ ) + 4 ( ‒ 𝐼+𝑆+ + 𝐼+𝑆‒ + 𝐼‒𝑆+ ‒ 𝐼‒𝑆‒) (172)
where 𝐴𝑥𝑥, 𝐴𝑦𝑦 and 𝐴𝑧𝑧 are the diagonal components of the metal nucleus-electron hyperfine coupling tensor. This effect can be important, for instance, for copper(II) (I = 3/2), oxovanadium(IV) (I = 7/2), manganese(II) (I = 5/2), cobalt(II) (I = 7/2) metal ions. 61
Analogous expressions can be calculated for Fermi contact relaxation using Eq. (157) instead of Eq. (164) [89]. The approach described above assumes that the Euler angles providing the orientation of the electron-nucleus vector (and thus the Wigner rotation matrix) are constant during electron-spin relaxation, i.e., for times of the order of the correlation time c. This is the case of the so-called slow rotation limit (r >> e). Analytical expressions for the nuclear relaxation rates in slowly rotating systems were first obtained to evaluate the effects of 𝐠 anisotropy [101], ZFS (in the limit where the ZFS is much larger than the Zeeman energy) [101,106] and hyperfine coupling between the metal nucleus and the unpaired electron(s) for the case of S = 1/2 and I = 3/2 spin systems (Cu2+) [102], and then generalized to any value of the nuclear spin I [107]. Incidentally, the effect of ZFS was also introduced in the analysis of the nuclear relaxation rates of lanthanoid(III) aqua ions, which is the first case where a field dependence of Curie spin relaxation has been observed [108]. This allowed for a better estimate of their electron relaxation times, which turned out to be longer than would be predicted without considering this effect. To treat the case of slow rotation within the Redfield limit (implying that the relaxation time arising from an interaction can never be as short as the correlation time that describes the fluctuations of that interaction), the “Florence NMRD” program was developed, allowing evaluation of the field dependence of nuclear relaxation rates for any size and rhombicity of ZFS and/or hyperfine coupling between the metal nucleus and unpaired electrons, and for any I and S quantum numbers [89]. A modified version of the Florence NMRD program [109,110] takes into account the effect of the ZFS not only on the nuclear relaxation rates, but also on the electron relaxation rates. Outside the slow rotation and Redfield limits, more complicated approaches should be used [84,85,111–114]. Recently, experimental evidence of the angular dependence of the paramagnetic enhancement of nuclear relaxation was obtained for a paramagnetic lanthanoid(III) complex, as a result of its large static ZFS [115]. These data were analysed using the parametric equation 62
2
dip 1M
R
2 μ 0 γ I2 6 Tr (3rˆrˆ T 1) 2 G (ωI ) 3 4π r
(173)
where rˆ is the unit vector pointing in the direction of r, and the six independent components of the symmetric spectral density tensor G (ωI ) are treated as fitting parameters. Curie spin relaxation may also change in the presence of 𝐠 anisotropy, ZFS and/or interactions with excited states. If Eq. (36) is introduced into Eq. (149), using Eq. (47) we obtain (in agreement with Eq. (61)) ℏ𝛾𝐼
𝐻Curie(𝑡) = ‒ 4𝜋𝑟3𝐁 ∙ 𝛘 ⋅ 0
(
3𝐫𝐫𝑇 𝑟2
)
(174)
‒𝟏 ⋅𝐈
Vega and Fiat [29] derived the longitudinal and transverse paramagnetic relaxation rates resulting from this perturbative Hamiltonian, and thus described the effects on Curie spin relaxation of the anisotropy of the magnetic susceptibility (and in turn of ZFS/g-anisotropy), noting the link between the latter and the pseudocontact shift. Recently, these equations were rewritten as a function of the tensor
1 3rr T 2 1 χ , σ σ0 3 4r r
(175)
where σ 0 is the chemical shielding tensor, so that the equations for Curie and CSA relaxation result in [115] CSA R1Curie M
τr τr 1 2 2 2 I 2 I2 2 2 2 1 9ω I τ r 15 1 ω I2 τ r2
CSA R2Curie M
τr 3τ r 1 2 2 1 2 2 I I 4 τ r 2 2 4 45 1 9ω I τ r 1 ω I2 τ r2
(176)
(177)
where
2 xy yx
2
xz
zx
2
2 2 xx2 yy zz2 xx yy xx zz yy zz
yz
zy
2
3 xy yx 4
(178)
2
xz
zx
2
yz
zy
. 2
These anisotropic effects of Curie spin relaxation were calculated to be rather small and have never 63
been measured. Note that, in contrast to the pseudocontact shift, which arises from the anisotropy of the susceptibility, Curie-spin relaxation is due to the full susceptibility.
8. Paramagnetic cross-correlation effects In the equations previously reported to describe paramagnetic enhancements of nuclear relaxation rates (Eq. (141)), paramagnetic cross-correlation terms were neglected. Cross-correlation effects arise in the presence of multiple correlated interactions, the energies of which are modulated randomly in time due to the same mechanism, and thus with the same correlation time. In fact, the transition probabilities contain the square of the expectation value of all time-dependent Hamiltonian terms (see Eq. (132)) wij
2 i | H1 (0) H1 (0) ... H n (0) | j 2
2
c 1 ij2 c2
(179)
which differs from the sum of the squares of the single terms by the appearance of cross-term products [116]. In paramagnetic systems, the dipole-dipole interaction 𝐻Curie (Eq. (150)) between the magnetic moment of a nucleus and the thermal average of the electron magnetic moment (responsible for Curie spin relaxation) can be modulated by the reorientation of the molecule, which also modulates the (diamagnetic) dipole-dipole interaction between the magnetic moments of two nuclei (Eq. (119)). The resulting cross-terms can be significant, and decrease or increase the nuclear relaxation rates, depending on whether S is in the 1/2 or –1/2 spin state, respectively. The different cross-correlation effects occurring for the two components of a nuclear doublet lead to a difference in their nuclear linewidths [117]. This difference for, e.g., the two components of the amide proton NMR signals in 15N enriched paramagnetic proteins, calculated on the assumption of an isotropic 𝛘 tensor, is [116–119]
ccr 1/2
2 15
2 2 2 2 3 c 0 B0 H N B g iso S ( S 1) 3 cos MHN 1 4 3 c 1 ω 2 2 r 3 rHN kT 2 4 I c 2
64
(180)
3 c 0 B0 H2 N iso 3 cos 2 MHN 1 4 c 2 3 3 2 4 10 r rHN 1 I2 c2
.
where the angle MHN is between the metal-nucleus vector (r) and the vector (rHN) between the 1H nucleus and the 15N nucleus, and c is the correlation time modulating both relaxation processes. The prefactor in Eq. (180) originates from the cross product between the square root of the prefactors of Curie spin relaxation (Eq. (156)) and of the Solomon equation (Eq. (145)) where S = 1/2 (the 1H spin, in this case) and g iso μ B is replaced with N . Equations to evaluate the effects of an anisotropic magnetic susceptibility have also been derived [120]. The dipole-dipole interaction between the nuclear magnetic moment and the thermal average of the electron magnetic moment (responsible for Curie spin relaxation) can also cause a nonnegligible cross-correlation term with nuclear CSA relaxation (see also Eq. 118) [121], which results in decreased or increased nuclear relaxation rates depending on the relative orientation of the principal axes of CSA and of the electron-nucleus dipolar shielding (Eq. (62)) tensors. The magnitude of this cross-correlation effect is usually very small, unless the paramagnetic relaxation enhancement is dominated by Curie spin relaxation rather than by dipolar relaxation. The imaginary part of the spectral density function (see Eq. (132)) related to a crosscorrelation effect is responsible of the dynamic frequency shift [52], introduced in Eq. (127). The dynamic frequency shift associated with the cross-correlation between i) the dipole-dipole interaction between the magnetic moment of a nucleus and the thermal average of the electron magnetic moment and ii) the dipole-dipole interaction between the magnetic moments of two nuclei, for the two components of the amide protons, is [119] 2 S ( S 1) 3 cos 2 MHN 1 I c2 2 0 B0 H2 N B2 g iso 3 5 4 2 r 3 rHN kT 1 I2 c2 2
dfs
65
(181)
B0 H2 N I c2 H N , so that the dynamic For long correlation times and/or at high magnetic fields, 1 I2 c2 frequency shift is to a good approximation independent of the correlation time, of the nucleus observed, and of the magnetic field. dfs is usually negligible with respect to paramagnetic RDC.
9. Conclusions More and more perspectives are opening up in the fields of structural biology and systems biology, as well as in material sciences, thanks to the full exploitation of paramagnetic effects. NMR data on paramagnetic systems offer a rich source of information on structures and dynamics, both on a short and long distance scales. In parallel with the development of experimental applications, there is currently a large effort going on to extend the NMR theory of paramagnetic systems. The new theoretical approaches are questioning the validity of the semi-empirical equations proposed more than half a century ago. In this review we have tried to clarify that a full understanding of the physical meaning of the (spin and orbital) electron magnetic moment and of the magnetic susceptibility is needed to understand NMR shifts and relaxation rates in paramagnetic systems. In our view, the first-principles quantum chemistry framework is still not fully mature, because it does not provide a fully satisfactory treatment of the contributions from the orbital magnetic moment to the observables. Therefore, in this review the semi-empirical framework has been further elaborated upon and fully endorsed, despite its limitations. In summary, all paramagnetic effects observed by NMR (dipolar shifts, pseudocontact shifts, paramagnetic residual dipolar couplings, electron-nucleus dipole-dipole relaxation, Curie relaxation, cross-correlation between nucleus-nucleus dipole-dipole and nucleus-electron Curie relaxation, dynamic frequency shifts) depend on the total electron magnetic moment. Notable exceptions are the Fermi contact shift and relaxation, which depend only on the spin contributions to the electron magnetic moment (Table 3).
66
Table 3. Dependence of paramagnetic NMR observables on the electron magnetic moment and on the g or 𝛘 tensors in the semi-empirical framework. 𝛿dip
∝
< 𝛍el >
∝
𝛘
𝛿pc
∝
< 𝛍el >
∝
𝛘 ‒ 𝛘iso
𝛿Fc
∝
< 𝛍𝑆 >
Δ𝜈prdc 12 ∝
< 𝛍el >
dip 𝑅1,2𝑀
𝛍el2
∝
∝ ∝
𝑔e𝛘 ∙ [𝐠𝑇] 𝛘 ‒ 𝛘iso 𝐠 ∙ 𝐠𝑇
∝
𝑅Curie 1,2𝑀 ∝
< 𝛍el > 2 ∝
Fc 𝑅1,2𝑀
∝
𝛍2𝑆
ccr Δ𝜈1/2
∝
< 𝛍el >
∝
𝛘
∝
< 𝛍el >
∝
𝛘
Δ𝜈dfs
‒1
𝛘 ∙ 𝛘𝑇 𝑔2e
∝
Acknowledgements We thank Ladislav Benda and Maurizio Romanelli for fruitful discussions. The support from Fondazione Cassa di Risparmio di Firenze, MIUR PRIN 2012SK7ASN, and Instruct-ERIC, an ESFRI Landmark, supported by national member subscriptions is also acknowledged. Specifically, we thank the Instruct-ERIC Core Centre CERM, Italy.
67
References
[1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14]
[15]
[16]
M.J. Knight, I.C. Felli, R. Pierattelli, L. Emsley, G. Pintacuda, Magic angle spinning NMR of paramagnetic proteins, Acc. Chem. Res. 46 (2013) 2108–2116. doi:10.1021/ar300349y. E. Ravera, G. Parigi, C. Luchinat, Perspectives on paramagnetic NMR from a life sciences infrastructure, Journal of Magnetic Resonance. 282 (2017) 154–169. doi:10.1016/j.jmr.2017.07.013. C. Nitsche, G. Otting, Pseudocontact shifts in biomolecular NMR using paramagnetic metal tags, Prog Nucl Magn Reson Spectrosc. 98–99 (2017) 20–49. doi:10.1016/j.pnmrs.2016.11.001. P.H.J. Keizers, M. Ubbink, Paramagnetic tagging for protein structure and dynamics analysis, Prog Nucl Magn Reson Spectrosc. 58 (2011) 88–96. C. Luchinat, G. Parigi, E. Ravera, Paramagnetism in experimental biomolecular NMR, 2018. E. Ravera, A. Carlon, M. Fragai, G. Parigi, C. Luchinat, Paramagnetic NMR as a new tool in structural biology, Emerging Topics in Life Sciences. 2 (2018) 19–28. doi:10.1042/ETLS20170084. A. Carlon, E. Ravera, W. Andrałojć, G. Parigi, G.N. Murshudov, C. Luchinat, How to tackle protein structural data from solution and solid state: An integrated approach, Progress in Nuclear Magnetic Resonance Spectroscopy. 92–93 (2016) 54–70. J.H. Ardenkjær-Larsen, B. Fridlund, A. Gram, G. Hansson, L. Hansson, M.H. Lerche, R. Servin, M. Thaning, K. Golman, Increase in signal-to-noise ratio of > 10,000 times in liquidstate NMR, PNAS. 100 (2003) 10158–10163. doi:10.1073/pnas.1733835100. L. Lumata, M.E. Merritt, C.R. Malloy, A.D. Sherry, Z. Kovacs, Impact of Gd3+ on DNP of [1-13C]pyruvate doped with trityl OX063, BDPA, or 4-oxo-TEMPO, J Phys Chem A. 116 (2012) 5129–5138. doi:10.1021/jp302399f. E. Terreno, D.D. Castelli, A. Viale, S. Aime, Challenges for Molecular Magnetic Resonance Imaging, Chemical Reviews. 110 (2010) 3019–3042. doi:10.1021/cr100025t. E. Ravera, D. Shimon, A. Feintuch, D. Goldfarb, S. Vega, A. Flori, C. Luchinat, L. Menichetti, G. Parigi, The effect of Gd on trityl-based dynamic nuclear polarisation in solids, Phys. Chem. Chem. Phys. (2015). doi:10.1039/C5CP04138D. S.A. Walker, D.T. Edwards, T.A. Siaw, B.D. Armstrong, S. Han, Temperature dependence of high field 13C dynamic nuclear polarization processes with trityl radicals below 35 Kelvin, Physical Chemistry Chemical Physics. 15 (2013) 15106. doi:10.1039/c3cp51628h. C.P. Grey, N. Dupré, NMR Studies of Cathode Materials for Lithium-Ion Rechargeable Batteries, Chemical Reviews. 104 (2004) 4493–4512. doi:10.1021/cr020734p. F.C. Strobridge, D.S. Middlemiss, A.J. Pell, M. Leskes, R.J. Clément, F. Pourpoint, Z. Lu, J.V. Hanna, G. Pintacuda, L. Emsley, A. Samoson, C.P. Grey, Characterising local environments in high energy density Li-ion battery cathodes: a combined NMR and first principles study of LiFe x Co 1−x PO 4, J. Mater. Chem. A. 2 (2014) 11948–11957. doi:10.1039/C4TA00934G. I.D. Seymour, D.S. Middlemiss, D.M. Halat, N.M. Trease, A.J. Pell, C.P. Grey, Characterizing Oxygen Local Environments in Paramagnetic Battery Materials via 17 O NMR and DFT Calculations, Journal of the American Chemical Society. 138 (2016) 9405–9408. doi:10.1021/jacs.6b05747. A. Karmakar, M.S. Dodd, S. Agnihotri, E. Ravera, V.K. Michaelis, Cu(II)-Doped Cs 2 SbAgCl 6 Double Perovskite: A Lead-Free, Low-Bandgap Material, Chemistry of Materials. 30 (2018) 8280–8290. doi:10.1021/acs.chemmater.8b03755.
68
[17] M. Fragai, C. Luchinat, G. Parigi, E. Ravera, Conformational freedom of metalloproteins revealed by paramagnetism-assisted NMR, Coordination Chemistry Reviews. 257 (2013) 2652–2667. doi:10.1016/j.ccr.2013.02.009. [18] W. Andrałojć, Y. Hiruma, W.-M. Liu, E. Ravera, M. Nojiri, G. Parigi, C. Luchinat, M. Ubbink, Identification of productive and futile encounters in an electron transfer protein complex, Proceedings of the National Academy of Sciences. 114 (2017) E1840–E1847. doi:10.1073/pnas.1616813114. [19] L. Gigli, W. Andrałojć, A. Dalaloyan, G. Parigi, E. Ravera, D. Goldfarb, C. Luchinat, Assessing protein conformational landscapes: integration of DEER data in Maximum Occurrence analysis, Physical Chemistry Chemical Physics. 20 (2018) 27429–27438. doi:10.1039/C8CP06195E. [20] L.B. Andreas, T. Le Marchand, K. Jaudzems, G. Pintacuda, High-resolution proton-detected NMR of proteins at very fast MAS, Journal of Magnetic Resonance. 253 (2015) 36–49. doi:10.1016/j.jmr.2015.01.003. [21] A. Bertarello, T. Schubeis, C. Fuccio, E. Ravera, M. Fragai, G. Parigi, L. Emsley, G. Pintacuda, C. Luchinat, Paramagnetic Properties of a Crystalline Iron–Sulfur Protein by Magic-Angle Spinning NMR Spectroscopy, Inorganic Chemistry. 56 (2017) 6624–6629. doi:10.1021/acs.inorgchem.7b00674. [22] S. Balayssac, I. Bertini, M. Lelli, C. Luchinat, M. Maletta, K.J. Yeo, Paramagnetic ions provide precious structural restraints in solid-state NMR of proteins, Journal of the American Chemical Society. 129 (2007) 2218–2219. [23] I. Sengupta, P.S. Nadaud, C.P. Jaroniec, Protein Structure Determination with Paramagnetic Solid-State NMR Spectroscopy, Acc. Chem. Res. 46 (2013) 2117–2126. doi:10.1021/ar300360q. [24] I. Sengupta, P.S. Nadaud, J.J. Helmus, C.D. Schwieters, C.P. Jaroniec, Protein fold determined by paramagnetic magic-angle spinning solid-state NMR spectroscopy, Nature Chemistry. 4 (2012) 410–417. doi:10.1038/nchem.1299. [25] A. Bhaumik, C. Luchinat, G. Parigi, E. Ravera, M. Rinaldelli, NMR crystallography on paramagnetic systems: solved and open issues, CrystEngComm. 15 (2013) 8639–8656. doi:10.1039/C3CE41485J. [26] J. Malanho Silva, L. Cerofolini, S. Giuntini, V. Calderone, C.F.G.C. Geraldes, A.L. Macedo, G. Parigi, M. Fragai, E. Ravera, C. Luchinat, Metal centers in biomolecular solid-state NMR, Journal of Structural Biology. (2018). doi:10.1016/j.jsb.2018.11.013. [27] A.J. Pell, G. Pintacuda, C.P. Grey, Paramagnetic NMR in solution and the solid state, Progress in Nuclear Magnetic Resonance Spectroscopy. 111 (2019) 1–271. doi:10.1016/j.pnmrs.2018.05.001. [28] R.J. Kurland, B.R. McGarvey, Isotropic NMR shifts in transition metal complexes: calculation of the Fermi contact and pseudocontact terms, J.Magn.Reson. 2 (1970) 286–301. [29] A.J. Vega, D. Fiat, Nuclear relaxation processes of paramagnetic complexes The slowmotion case, Molecular Physics. 31 (1976) 347–355. doi:10.1080/00268977600100261. [30] I. Solomon, Relaxation Processes in a System of Two Spins, Phys. Rev. 99 (1955) 559–565. doi:10.1103/PhysRev.99.559. [31] A.V. Arbuznikov, J. Vaara, M. Kaupp, Relativistic spin-orbit effects on hyperfine coupling tensors by density-functional theory, J Chem Phys. 120 (2004) 2127–2139. doi:10.1063/1.1636720. [32] T.O. Pennanen, J. Vaara, Nuclear magnetic resonance chemical shift in an arbitrary electronic spin state, Phys. Rev. Lett. 100 (2008) 133002. doi:10.1103/PhysRevLett.100.133002. [33] W. Van den Heuvel, A. Soncini, NMR chemical shift as analytical derivative of the Helmholtz free energy, The Journal of Chemical Physics. 138 (2013) 054113. doi:10.1063/1.4789398. 69
[34] B. Martin, J. Autschbach, Temperature dependence of contact and dipolar NMR chemical shifts in paramagnetic molecules, The Journal of Chemical Physics. 142 (2015) 054108. doi:10.1063/1.4906318. [35] M. Kaupp, M. Bühl, V.G. Malkin, eds., Calculation of NMR and EPR parameters: theory and applications, Wiley-VCH, Weinheim, 2004. [36] F. Neese, Quantum Chemistry and EPR Parameters, in: R.K. Harris, R.L. Wasylishen (Eds.), EMagRes, John Wiley & Sons, Ltd, Chichester, UK, 2017: pp. 1–22. doi:10.1002/9780470034590.emrstm1505. [37] L. Benda, J. Mareš, E. Ravera, G. Parigi, C. Luchinat, M. Kaupp, J. Vaara, Pseudo-Contact NMR Shifts over the Paramagnetic Metalloprotein CoMMP-12 from First Principles, Angew. Chem. Int. Ed. Engl. 55 (2016) 14713–14717. doi:10.1002/anie.201608829. [38] F. Gendron, K. Sharkas, J. Autschbach, Calculating NMR Chemical Shifts for Paramagnetic Metal Complexes from First-Principles, The Journal of Physical Chemistry Letters. 6 (2015) 2183–2188. doi:10.1021/acs.jpclett.5b00932. [39] J. Autschbach, Double perturbation theory: a powerful tool in computational coordination chemistry, Coordination Chemistry Reviews. 238–239 (2003) 83–126. doi:10.1016/S00108545(02)00287-4. [40] G. Parigi, L. Benda, E. Ravera, M. Romanelli, C. Luchinat, Pseudocontact shifts and paramagnetic susceptibility in semiempirical and quantum chemistry theories, J. Chem. Phys. 150 (2019) 144101. doi:10.1063/1.5037428. [41] L. Cerofolini, J.M. Silva, E. Ravera, M. Romanelli, C.F.G.C. Geraldes, A.L. Macedo, M. Fragai, G. Parigi, C. Luchinat, How Do Nuclei Couple to the Magnetic Moment of a Paramagnetic Center? A New Theory at the Gauntlet of the Experiments, J. Phys. Chem. Lett. (2019) 3610–3614. doi:10.1021/acs.jpclett.9b01128. [42] I. Bertini, C. Luchinat, G. Parigi, Magnetic susceptibility in paramagnetic NMR, Progress in Nuclear Magnetic Resonance Spectroscopy. 40 (2002) 249–273. [43] B. Corzilius, Paramagnetic Metal Ions for Dynamic Nuclear Polarization, in: EMagRes, American Cancer Society, 2018: pp. 179–194. doi:10.1002/9780470034590.emrstm1593. [44] J.H. Van Vleck, The theory of electric and magnetic susceptibilities., Oxford, Clarendon Press, 1932. [45] B. Bleaney, Nuclear magnetic resonance shifts in solution due to lanthanide ions, J.Magn.Reson. 8 (1972) 91–100. [46] B.J. Walder, A.M. Patterson, J.H. Baltisberger, P.J. Grandinetti, Hydrogen motional disorder in crystalline iron group chloride dihydrates, The Journal of Chemical Physics. 149 (2018) 084503. doi:10.1063/1.5037151. [47] R. Boča, Mean and differential magnetic susceptibilities in metal complexes, Coordination Chemistry Reviews. 173 (1998) 167–283. doi:10.1016/S0010-8545(98)00139-8. [48] O. Kahn, Molecular magnetism, VCH Publishers Inc., New York, 1993. [49] H.M. McConnell, R.E. Robertson, Isotropic Nuclear Resonance Shifts, The Journal of Chemical Physics. 29 (1958) 1361–1365. doi:10.1063/1.1744723. [50] N.M. Atherton, A.J. Horsewill, Contributions of orbital angular momentum to ligand hyperfine interactions in paramagnetic metal complexes, Journal of the Chemical Society, Faraday Transactions 2. 76 (1980) 660. doi:10.1039/f29807600660. [51] W. Marshall, Hyperfine Interactions with Ligand Nuclei in Transition Ion Complexes, in: W. Low (Ed.), Paramagnetic Resonance, Academic Press, New York, 1963. [52] A. Abragam, The Principles of Nuclear Magnetism, Oxford University Press, Oxford, 1961. [53] H.M. McConnell, D.B. Chesnut, Theory of Isotropic Hyperfine Interactions in π‐Electron Radicals, The Journal of Chemical Physics. 28 (1958) 107–117. doi:10.1063/1.1744052. [54] I. Bertini, C. Luchinat, G. Parigi, The hyperfine shifts in low spin iron(III) hemes: a ligand field analysis, European Journal of Inorganic Chemistry. (2000) 2473–2480. 70
[55] A. Abragam, B. Bleaney, Electron Paramagnetic Resonance of Transition Metal Ions, Clarendon Press, Oxford, 1970. [56] R. Boča, Zero-field splitting in metal complexes, Coordination Chemistry Reviews. 248 (2004) 757–815. doi:10.1016/j.ccr.2004.03.001. [57] J.D. Jackson, Classical electrodynamics, Wiley, New York, 1962. [58] W. Van den Heuvel, A. Soncini, NMR chemical shift in an electronic state with arbitrary degeneracy, Phys. Rev. Lett. 109 (2012) 073001. doi:10.1103/PhysRevLett.109.073001. [59] J. Vaara, S.A. Rouf, J. Mareš, Magnetic Couplings in the Chemical Shift of Paramagnetic NMR, Journal of Chemical Theory and Computation. 11 (2015) 4840–4849. doi:10.1021/acs.jctc.5b00656. [60] S.A. Rouf, J. Mareš, J. Vaara, 1H Chemical Shifts in Paramagnetic Co(II) Pyrazolylborate Complexes: A First-Principles Study, J Chem Theory Comput. 11 (2015) 1683–1691. doi:10.1021/acs.jctc.5b00193. [61] A. Mondal, M.W. Gaultois, A.J. Pell, M. Iannuzzi, C.P. Grey, J. Hutter, M. Kaupp, LargeScale Computation of Nuclear Magnetic Resonance Shifts for Paramagnetic Solids Using CP2K, Journal of Chemical Theory and Computation. 14 (2018) 377–394. doi:10.1021/acs.jctc.7b00991. [62] J. Mareš, J. Vaara, Ab initio paramagnetic NMR shifts via point-dipole approximation in a large magnetic-anisotropy Co( II ) complex, Physical Chemistry Chemical Physics. 20 (2018) 22547–22555. doi:10.1039/C8CP04123G. [63] J. Chyba, M. Novák, P. Munzarová, J. Novotný, R. Marek, Through-Space Paramagnetic NMR Effects in Host–Guest Complexes: Potential Ruthenium(III) Metallodrugs with Macrocyclic Carriers, Inorganic Chemistry. 57 (2018) 8735–8747. doi:10.1021/acs.inorgchem.7b03233. [64] P. Srb, M. Svoboda, L. Benda, M. Lepšík, J. Tarábek, V. Šícha, B. Grüner, K. GrantzŠašková, J. Brynda, P. Řezáčová, J. Konvalinka, V. Veverka, Capturing a dynamically interacting inhibitor by paramagnetic NMR spectroscopy, Physical Chemistry Chemical Physics. 21 (2019) 5661–5673. doi:10.1039/C9CP00416E. [65] I. Bertini, M.B.L. Janik, Y.-M. Lee, C. Luchinat, A. Rosato, Magnetic Susceptibility Tensor Anisotropies for a Lanthanide Ion Series in a Fixed Protein Matrix, Journal of the American Chemical Society. 123 (2001) 4181–4188. [66] I. Bertini, P. Kursula, C. Luchinat, G. Parigi, J. Vahokoski, M. Willmans, J. Yuan, Accurate solution structures of proteins from X-ray data and minimal set of NMR data: calmodulin peptide complexes as examples., Journal of the American Chemical Society. 131 (2009) 5134–5144. [67] A.M. Funk, K.-L.N.A. Finney, P. Harvey, A.M. Kenwright, E.R. Neil, N.J. Rogers, P. Kanthi Senanayake, D. Parker, Critical analysis of the limitations of Bleaney’s theory of magnetic anisotropy in paramagnetic lanthanide coordination complexes, Chemical Science. 6 (2015) 1655–1662. doi:10.1039/C4SC03429E. [68] E.A. Suturina, K. Mason, C.F.G.C. Geraldes, I. Kuprov, D. Parker, Beyond Bleaney’s Theory: Experimental and Theoretical Analysis of Periodic Trends in Lanthanide-Induced Chemical Shift, Angew. Chem. Int. Ed. Engl. 56 (2017) 12215–12218. doi:10.1002/anie.201706931. [69] M. Vonci, K. Mason, E.A. Suturina, A.T. Frawley, S.G. Worswick, I. Kuprov, D. Parker, E.J.L. McInnes, N.F. Chilton, Rationalization of Anomalous Pseudocontact Shifts and Their Solvent Dependence in a Series of C3-Symmetric Lanthanide Complexes, J. Am. Chem. Soc. 139 (2017) 14166–14172. doi:10.1021/jacs.7b07094. [70] E.A. Suturina, I. Kuprov, Pseudocontact shifts from mobile spin labels, Phys. Chem. Chem. Phys. 18 (2016) 26412–26422. doi:10.1039/C6CP05437D. [71] G.T.P. Charnock, I. Kuprov, A partial differential equation for pseudocontact shift, Phys Chem Chem Phys. 16 (2014) 20184–20189. doi:10.1039/c4cp03106g. 71
[72] J.A.B. Lohman, C. Maclean, Alignment effects on high resolution NMR spectra induced by the magnetic field, Chemical Physics. 35 (1978) 269–274. [73] C. Gayathri, A.A. Bothner-By, P.C.M. Van Zijl, C. Maclean, Dipolar magnetic field effects in NMR spectra of liquids, Chem.Phys.Lett. 87 (1982) 192–196. [74] I. Bertini, I.C. Felli, C. Luchinat, High magnetic field consequences on the NMR hyperfine shifts in solution, J.Magn.Reson. 134 (1998) 360–364. [75] M. John, A.Y. Park, G. Pintacuda, N.E. Dixon, G. Otting, Weak Alignment of Paramagnetic Proteins Warrants Correction for Residual CSA Effects in Measurements of Pseudocontact Shifts, Journal of the American Chemical Society. 127 (2005) 17190–17191. doi:10.1021/ja0564259. [76] C. Schmitz, M.J. Stanton-Cook, X.-C. Su, G. Otting, T. Huber, Numbat: an interactive software tool for fitting Δχ-tensors to molecular coordinates using pseudocontact shifts, J Biomol NMR. 41 (2008) 179–189. doi:10.1007/s10858-008-9249-z. [77] J.R. Tolman, J.M. Flanagan, M.A. Kennedy, J.H. Prestegard, Nuclear magnetic dipole interactions in field-oriented proteins: information for structure determination in solution, Proc. Natl. Acad. Sci. U.S.A. 92 (1995) 9279–9283. [78] N. Tjandra, A. Bax, Direct Measurement of Distances and Angles in Biomolecules by NMR in a Dilute Liquid Crystalline Medium, Science. 278 (1997) 1111–1114. doi:10.1126/science.278.5340.1111. [79] F. Tian, H. Valafar, J.H. Prestegard, A dipolar coupling based strategy for simultaneous resonance assignment and structure determination of protein backbones, J. Am. Chem. Soc. 123 (2001) 11791–11796. [80] L. Banci, I. Bertini, J.G. Huber, C. Luchinat, A. Rosato, Partial orientation of oxidized and reduced cytochrome b5 at high magnetic fields: Magnetic susceptibility anisotropy contributions and consequences for protein solution structure determination, Journal of the American Chemical Society. 120 (1998) 12903–12909. [81] N. Tjandra, S. Grzesiek, A. Bax, Magnetic Field Dependence of Nitrogen−Proton J Splittings in 15N-Enriched Human Ubiquitin Resulting from Relaxation Interference and Residual Dipolar Coupling, Journal of the American Chemical Society. 118 (1996) 6264–6272. doi:10.1021/ja960106n. [82] G. Lipari, A. Szabo, Model-Free approach to the interpretation of nuclear magnetic resonance relaxation in macromolecules. 1. Theory and range of validity., Journal of the American Chemical Society. 104 (1982) 4546–4559. [83] M. Rinaldelli, A. Carlon, E. Ravera, G. Parigi, C. Luchinat, FANTEN: a new web-based interface for the analysis of magnetic anisotropy-induced NMR data, Journal of Biomolecular NMR. 61 (2015) 21–34. doi:10.1007/s10858-014-9877-4. [84] J. Kowalewski, L. Maler, Nuclear Spin Relaxation in Liquids: Theory, Experiments, and Applications, Taylor & Francis, 2006. [85] J. Kowalewski, D. Kruk, G. Parigi, NMR relaxation in solution of paramagnetic complexes: recent theoretical progress for S>1, Adv.Inorg.Chem. 57 (2005) 41–104. [86] M. Goldman, Formal Theory of Spin-Lattice Relaxation, J.Magn.Reson. 149 (2001) 160– 187. [87] C.P. Slichter, Principles of magnetic resonance., Springer, Berlin, 1992. [88] R. Kubo, K. Tomita, A General Theory of Magnetic Resonance Absorption, J.Phys.Soc.Japan. 9 (1954) 888–919. [89] I. Bertini, O. Galas, C. Luchinat, G. Parigi, A computer program for the calculation of paramagnetic enhancements of nuclear relaxation rates in slowly rotating systems, Journal of Magnetic Resonance Series A. 113 (1995) 151–158. [90] N. Bloembergen, L.O. Morgan, Proton relaxation times in paramagnetic solutions. Effects of electron spin relaxation, J.Chem.Phys. 34 (1961) 842–850. 72
[91] I. Bertini, C. Luchinat, G. Parigi, E. Ravera, NMR of paramagnetic molecules: applications to metallobiomolecules and models, 2017. [92] M. Rubinstein, A. Baram, Z. Luz, Electronic and nuclear relaxation in solutions of transition metal ions with spin S =3/2 and 5/2, Molecular Physics. 20 (1971) 67–80. doi:10.1080/00268977100100081. [93] P. Caravan, N.J. Cloutier, S.A. McDermid, J.J. Ellison, J.M. Chasse, R.B. Lauffer, C. Luchinat, T.J. McMurry, G. Parigi, M. Spiller, Albumin binding, relaxivity and water exchange kinetics of the diastereoisomers of MS-325 a gadolinium(III) based magnetic resonance angiography contrast agent, Inorganic Chemistry. 46 (2007) 6632–6639. [94] M. Gueron, Nuclear relaxation in macromolecules by paramagnetic ions: a novel mechanism, J.Magn.Reson. 19 (1975) 58–66. [95] N. Bloembergen, Proton relaxation times in paramagnetic solutions, J.Chem.Phys. 27 (1957) 572–573. [96] I. Solomon, N. Bloembergen, Nuclear Magnetic Interactions in the HF Molecule, The Journal of Chemical Physics. 25 (1956) 261–266. doi:10.1063/1.1742867. [97] G. Liu, M. Levien, N. Karschin, G. Parigi, C. Luchinat, M. Bennati, One-thousand-fold enhancement of high field liquid nuclear magnetic resonance signals at room temperature, Nature Chemistry. (2017). doi:10.1038/nchem.2723. [98] W. Müller-Warmuth, R. Vilhjalmsson, P.A.M. Gerlof, J. Smidt, J. Trommel, Intermolecular interactions of benzene and carbon tetrachloride with selected free radicals in solution as studied by 13 C and 1 H dynamic nuclear polarization, Molecular Physics. 31 (1976) 1055– 1067. doi:10.1080/00268977600100811. [99] P.-O. Westlund, A generalized Solomon-Bloembergen-Morgan theory for arbitrary electron spin quantum number S: The dipole-dipole coupling between a nuclear spin I = 1/2 and an electron spin system S = 5/2, Molecular Physics. 85 (1995) 1165–1178. doi:10.1080/00268979500101741. [100] H. Sternlicht, Nuclear Relaxation Induced by Paramagnetic Ions Having Anisotropic g Factors, The Journal of Chemical Physics. 42 (1965) 2250–2251. doi:10.1063/1.1696280. [101] I. Bertini, C. Luchinat, K.V. Vasavada, The effect of magnetic anisotropy on the longitudinal nuclear relaxation time in paramagnetic systems, J.Magn.Reson. 89 (1990) 243–254. [102] I. Bertini, F. Briganti, C. Luchinat, M. Mancini, G. Spina, The electron nucleus dipolar coupling in slow rotating systems (2): the effect of g anisotropy and hyperfine coupling when S=1/2 and I=3/2, J.Magn.Reson. 63 (1985) 41–55. [103] J. Kowalewski, C. Luchinat, T. Nilsson, G. Parigi, Nuclear Spin Relaxation in Paramagnetic Systems: Electron Spin Relaxation Effects under Near-Redfield Limit Conditions and Beyond, J Phys Chem A. 106 (2002) 7376–7382. [104] I. Bertini, O. Galas, C. Luchinat, L. Messori, G. Parigi, A theoretical analysis of the 1H nuclear magnetic relaxation dispersion profiles of diferric transferrin, J.Phys.Chem. 99 (1995) 14217–14222. [105] S.J. Kroes, J. Salgado, G. Parigi, C. Luchinat, G.W. Canters, Electron relaxation and solvent accessibility of the metal site in wild-type and mutated azurins as determined from nuclear magnetic relaxation dispersion experiments, Journal of Biological Inorganic Chemistry. 1 (1996) 551–559. [106] I. Bertini, C. Luchinat, M. Mancini, G. Spina, The electron-nucleus dipolar coupling. The effect of zero-field splitting of an S=3/2 manifold, J.Magn.Reson. 59 (1984) 213–222. [107] I. Bertini, C. Luchinat, Z. Xia, Solvent water 1H NMRD study of oxovanadium(IV) aquaion, J.Magn.Reson. 99 (1992) 235–246. [108] I. Bertini, F. Capozzi, C. Luchinat, G. Nicastro, Z. Xia, Water proton relaxation for some lanthanide aqua ions in solution, J.Phys.Chem. 101 (1993) 198–201.
73
[109] I. Bertini, J. Kowalewski, C. Luchinat, T. Nilsson, G. Parigi, Nuclear spin relaxation in paramagnetic complexes of S=1: Electron spin relaxation effects, J.Chem.Phys. 111 (1999) 5795–5807. [110] D. Kruk, T. Nilsson, J. Kowalewski, Nuclear spin relaxation in paramagnetic systems with zero-field splitting and arbitrary electron spin, Phys.Chem.Chem.Phys. 3 (2001) 4907–4917. [111] T. Larsson, P.O. Westlund, J. Kowalewski, S.H. Koenig, Nuclear-spin relaxation in paramagnetic complexes in the slow-motion regime for the electron spin: the anisotropic pseudorotation model for S=1 and the interpretation of nuclear magnetic relaxation dispersion results for a low-symmetry Ni(II) complex, J.Chem.Phys. 101 (1994) 1116–1128. [112] P.H. Fries, E. Belorizky, Relaxation theory of the electronic spin of a complexed paramagnetic metal ion in solution beyond the Redfield limit, The Journal of Chemical Physics. 126 (2007) 204503. doi:10.1063/1.2730831. [113] S. Rast, P.H. Fries, E. Belorizky, A. Borel, L. Helm, A.E. Merbach, A general approach to the electronic spin relaxation of Gd(III) complexes in solutions. Monte Carlo simulations beyond the Redfield limit, The Journal of Chemical Physics. 115 (2001) 7554–7563. doi:10.1063/1.1392364. [114] E. Belorizky, P.H. Fries, L. Helm, J. Kowalewski, D. Kruk, R.R. Sharp, P.-O. Westlund, Comparison of different methods for calculating the paramagnetic relaxation enhancement of nuclear spins as a function of the magnetic field, The Journal of Chemical Physics. 128 (2008) 052315. doi:10.1063/1.2833957. [115] E.A. Suturina, K. Mason, C.F.G.C. Geraldes, N.F. Chilton, D. Parker, I. Kuprov, Lanthanideinduced relaxation anisotropy, Physical Chemistry Chemical Physics. 20 (2018) 17676– 17686. doi:10.1039/C8CP01332B. [116] M. Goldman, Interference Effects in the Relaxation of a Pair of Unlike Spin-1/2 Nuclei, J.Magn.Reson. 60 (1984) 437–452. [117] I. Bertini, C. Luchinat, M. Piccioli, D. Tarchi, COSY spectra of paramagnetic macromolecules, observability, scalar effects, cross correlation effects, relaxation allowed coherence transfer, Concepts in Magnetic Resonance. 6 (1994) 307–335. [118] I. Bertini, C. Luchinat, D. Tarchi, Are true scalar proton-proton connectivities ever measured in COSY spectra of paramagnetic macromolecules?, Chem.Phys.Lett. 203 (1993) 445–449. [119] R. Ghose, J.H. Prestegard, Electron Spin-Nuclear Spin Cross-Correlation Effects on Multiplet Splittings in Paramagnetic Proteins, J.Magn.Reson. 128 (1997) 138–143. [120] I. Bertini, J. Kowalewski, C. Luchinat, G. Parigi, Cross-correlation between the dipole-dipole interaction and the Curie spin relaxation. The effect of anisotropic magnetic susceptibility, J.Magn.Reson. 152 (2001) 103–108. [121] G. Pintacuda, A. Kaikkonen, G. Otting, Modulation of the distance dependence of paramagnetic relaxation enhancements by CSA×DSA cross-correlation, Journal of Magnetic Resonance. 171 (2004) 233–243. doi:10.1016/j.jmr.2004.08.019.
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Graphical abstract
μI
μ el
B0 0
𝜒
Hyperfine shifts Residual dipolar couplings NMR relaxation
75
Highlights Spin-orbit coupling causes ZFS, g-tensor and magnetic susceptibility anisotropy The average electron magnetic moment depends on the magnetic susceptibility The dipolar interaction with the average electron magnetic moment causes NMR shifts Hyperfine shifts can be expressed as a function of g-tensor and ZFS Hyperfine shifts, RDCs and relaxation depend on the magnetic susceptibility
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S0079-6565(19)30019-6 https://doi.org/10.1016/j.pnmrs.2019.06.003 JPNMRS 1481
To appear in:
Progress in Nuclear Magnetic Resonance Spectroscopy
Received Date: Accepted Date:
8 April 2019 17 June 2019
Please cite this article as: G. Parigi, E. Ravera, C. Luchinat, Magnetic susceptibility and paramagnetism-based NMR, Progress in Nuclear Magnetic Resonance Spectroscopy (2019), doi: https://doi.org/10.1016/j.pnmrs.2019.06.003
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Magnetic susceptibility and paramagnetism-based NMR
Giacomo Parigi, Enrico Ravera, Claudio Luchinat*
Magnetic Resonance Center (CERM) and Interuniversity Consortium for Magnetic Resonance of Metallo Proteins (CIRMMP), Via L. Sacconi 6, 50019 Sesto Fiorentino, Italy and Department of Chemistry “Ugo Schiff”, University of Florence, Via della Lastruccia 3, 50019 Sesto Fiorentino, Italy
Edited by Geoffrey Bodenhausen and Gareth A. Morris
* email: [email protected] Tel. +39 055 4574296
Keywords: hyperfine shift, pseudocontact shift, residual dipolar coupling, dipolar relaxation, Curie relaxation
1
Abstract The magnetic interactions between the nuclear magnetic moment and the magnetic moment of unpaired electron(s) depend on the structure and dynamics of the molecules where the paramagnetic center is located and of their partners. The long-range nature of the magnetic interactions is thus a reporter of invaluable information for structural biology studies, when other techniques often do not provide enough data for the atomic-level characterization of the system. This precious information explains the flourishing of paramagnetism-assisted NMR studies in recent years. Many paramagnetic effects are related to the magnetic susceptibility of the paramagnetic metal. Although these effects have been known for more than half a century, different theoretical models and new approaches have been proposed in the last decade. In this review, we have summarized the consequences for NMR spectroscopy of magnetic interactions between nuclear and electron magnetic moments, and thus of the presence of a magnetic susceptibility due to metals, and we do so using a unified notation.
Table of Contents 1.
Introduction
2.
Why bother with magnetic susceptibility?
3.
Angular momenta and magnetic moments
3.1 The Curie spin 4.
The paramagnetic susceptibility
4.1 Saturation at high field 5. The hyperfine shift 5.1 The semi-empirical paramagnetic NMR approach 5.2 Perturbative spin Hamiltonian EPR approach 5.3 Hyperfine shift from first-principles quantum chemistry 2
5.4 The limiting case of D<
Paramagnetism-induced self-orientation
6.1 NMR shifts in self-orienting paramagnetic molecules 6.2 Paramagnetic residual dipolar couplings 7. Paramagnetic relaxation enhancements 7.1 Dipolar relaxation 7.2 Curie spin relaxation 7.3 Fermi contact relaxation 7.4 Anisotropic effects 8. Paramagnetic cross-correlation effects 9. Conclusions Acknowledgements References
3
Glossary CSA
chemical shift anisotropy
NMR nuclear magnetic resonance RDC residual dipolar coupling SBM Solomon-Bloembergen-Morgan TIP
temperature independent paramagnetism
ZFS
zero field splitting
4
1. Introduction The number of research reports dealing with NMR of paramagnetic systems has dramatically increased in the last two decades, mostly thanks to the impressive developments in experimental NMR that have made it easier and more profitable to take account of paramagnetic effects in biological [1–7] and biomedical investigations [8–12], and in material sciences [13–16]. The NMR data (shifts and relaxation rates) collected for systems that contain a paramagnetic center contain valuable information on their structure and dynamics, which becomes particularly relevant for the study of large complexes and/or in the presence of conformational heterogeneity [17–19]. It is also important to recall that technological advances in fast magic angle spinning, robust radiofrequency pulse schemes, and reliable sample preparation techniques have also allowed the study of paramagnetic effects in solids [20–26]. These recently appreciated advantages of paramagnetism have stimulated the development of the theory describing hyperfine shifts and paramagnetic relaxation enhancements [27], both in the framework of semi-empirical quantum chemistry approaches [28–30] and of a first-principles quantum chemistry approach [31–39]. The results obtained in these two frameworks are, however, not always in agreement with one another [40], and experimental validation is needed. A step in this direction has been recently made [41], and the experimental results are in agreement with the semiempirical approach and not with the quantum chemistry approach. Besides providing a critical assessment of recent theoretical work, here we recapitulate in a unified frame the concepts and equations used for the analysis of NMR data of paramagnetic systems using the semi-empirical quantum chemistry approach, keeping the language and the formalism as simple as possible.
2. Why bother with magnetic susceptibility? As outlined in a previous review from our group [42], paramagnetic NMR data are in large part related to the molecular paramagnetic susceptibility. This is easily understood in the following case: 5
a nucleus immersed in a static, constant magnetic field B0 in vacuo experiences a Zeeman splitting (vide infra) which is proportional to B0. If another object is introduced into the same field, in close proximity to the probe nucleus, the object will perturb the magnetic field experienced by our probe nucleus. The object could be diamagnetic, in which case it will act to decrease the static magnetic field along the field axis (and reinforce the field in the plane), or it could be paramagnetic, in which case it will strengthen the static magnetic field along the axis (and reduce it in the plane). In either case, the probe nucleus will sense a different Zeeman splitting. The alteration of the field is usually described by means of the magnetic susceptibility. Moving towards a more practical description, let us imagine that the object is a single paramagnetic center. At a reasonable distance, the field distortion it causes is described by the molecular paramagnetic susceptibility and approximated by a dipolar field. In general, the magnetic susceptibility will be anisotropic, depending on the symmetry around the paramagnetic center. This leads us to an expression for the dipolar shift of the form 1
𝛿dip = 4𝜋𝑟3𝛋 ⋅ 𝛘 ⋅
(
3𝐫𝐫𝑇 𝑟2
)
‒ 𝟏 ⋅ 𝛊,
(1)
where r is the vector between the paramagnetic center and the observed nucleus, the symbol
T
denotes transposition, and and are unit vectors in the direction of the magnetic field and of the quantization axis of the spin I of the observed nucleus, respectively. The rotational average of the dipolar shift is known as the pseudocontact shift. Assuming isotropic reorientation of the molecule with respect to the magnetic field (see Section 6.1), the pseudocontact shift 𝛿pc can be calculated to be
[ (
1
𝛿pc = 12𝜋𝑟3Tr 𝛘 ⋅
3𝐫𝐫𝑇 𝑟2
‒𝟏
)]
(2)
where Tr denotes the trace of a matrix. Other forms of this equation are given in Section 4.1. Nuclear paramagnetic relaxation enhancements (and thus NMR linewidths) can also depend on the paramagnetic susceptibility, through the contribution arising from the dipole-dipole interaction 6
between the averaged electron magnetic moment and the nuclear magnetic moment (Curie spin relaxation). In the absence of anisotropy of the 𝛘 tensor, the Curie contribution to relaxation is (see Section 7.2) 2
R1Curie M
2 3τ cCurie 2 1 ω I2 iso 5 4π r 6 1 ω I2 ( τ cCurie ) 2
2
R
Curie 2M
2 1 1 ωI2 iso 5 4π r6
3τ cCurie 4 τ c 1 ωI2 ( τ cCurie ) 2
(3)
(4)
where τ cCurie is the correlation time for Curie spin relaxation (usually determined by molecular reorientation) and 𝜒iso is the rotational average of the 𝛘 tensor (𝜒iso = Tr(𝛘)/3). More appropriate equations, accounting for the anisotropy of the 𝛘 tensor, are discussed in Section 7.4. The magnetic susceptibility is also important for the description of cross-correlation effects like the one between dipole-dipole relaxation and Curie spin relaxation (see Section 8). Magnetic susceptibility anisotropy also causes different orientations of a molecule in a magnetic field to have different probabilities, thus resulting in partial self-alignment. This induces residual dipolar couplings (RDCs) as a consequence of the non-zero rotational average of the square of the nucleus-nucleus dipole-dipole interaction energy, analogous to the RDCs arising when partial alignment is induced by external orienting effects such as liquid crystal media. Paramagnetic RDCs have the additional advantage that intermolecular interactions with external orienting media that may perturb the structure and dynamics of the investigated molecule are excluded. Paramagnetic RDCs depend on the paramagnetic susceptibility anisotropy Δ𝛘 according to Eq. (5) 12prdc
1 B02 3 I 1 I 2 T r12 χ r12 4 15kT 2r125
(5)
where r12 is the vector between the coupled nuclei 1 and 2. Other forms of this equation are provided in Section 6.2. There is yet another paramagnetic contribution to the NMR shift, the Fermi contact shift, that appears when unpaired electron spin density is present on the observed nucleus. In this case the shift is due to the interaction between the “bare” electron spin magnetic moment and the nuclear magnetic 7
moment, and therefore is not directly related to the magnetic susceptibility. However, it can still be expressed in terms of the paramagnetic susceptibility by dividing the latter by the electron g tensor, according to Eq. (6) 𝐴Fc
𝛿Fc = 3𝜇0ℏ𝛾𝐼𝜇BTr[𝛘 ∙ [𝐠𝑇]
‒1
]
(6)
where 𝐴Fc is the constant for the Fermi contact interaction (see Section 5.1). More comments on this intrinsic asymmetry between the physics behind the contact and the pseudocontact interactions will be given in section 5.3. In most cases, it is very difficult to obtain an experimental measure of the molecular paramagnetic susceptibility, and particularly of its anisotropy. Therefore, all paramagnetic NMR observables have been derived in terms of parameters that are either experimentally observable or that can be directly calculated, either from first-principles or on semi-empirical grounds. Since the magnetic susceptibility of a paramagnetic center is linked to the total effective magnetic moment of the center itself, in the following sections we will show how it is possible within the semi-empirical framework to establish a link between the total effective magnetic moment and paramagnetic NMR observables. First-principles quantum chemistry studies have recently focused on the derivation of the hyperfine shifts, i.e., of the Fermi contact and pseudocontact shifts. Using this theoretical approach, developed with the spin-Hamiltonian formalism, the same equation of Eq. (6) is obtained for the Fermi contact shift, whereas the equation for the pseudocontact shift contains an additional factor 𝑔e
[𝐠𝑇] ‒ 1 (which multiplies the 𝛘 tensor, see Section 5.3) that is missing in Eq. (2). The presence of this factor in the equation derived within the quantum chemistry approach apparently re-establishes a symmetry between the Fermi contact and the pseudocontact terms, but introduces a fundamentally different way of looking at the effect of the presence of a paramagnetic center on distant nuclei. However, we will show that there need not be any symmetry between contact and pseudocontact shifts, and that this asymmetry originates physically from the presence of a contribution from the interaction between the orbital magnetic moment and the nuclear magnetic moment when calculating 8
the psedocontact shift, and the absence of any contributions of this kind when calculating the Fermi contact shift (see Section 5).
3. Angular momenta and magnetic moments The (effective) spin angular momentum 𝐒 of the unpaired electron(s) present in a paramagnetic metal ion has an associated magnetic moment 𝛍𝑆 =‒ 𝜇B𝑔e𝐒.
(7)
The motion of the unpaired electron(s) in their orbitals also introduces an electron orbital angular momentum 𝐋, so that the orbital magnetic moment is 𝛍𝐿 =‒ 𝜇B𝐋.
(8)
We here note that the total orbital angular momentum operator 𝐋 is only Hermitian (corresponding to an observable) in a spherical potential, which is strictly valid only for bare atoms. The following sections assume the validity of the Russell-Saunders (LS) coupling regime, which requires that the spin-spin coupling and orbit-orbit coupling between electrons are much stronger than the spin-orbit coupling. This approximation is considered good for elements up to the first row transition series; for heavier elements, the j-j coupling scheme should be used, because the individual coupling between electrons, via the spin-orbit interaction, is stronger than the electrostatic interaction between them. In the semi-empirical quantum chemistry approach, the interaction between the orbital and spin angular momenta of the electron(s) is represented by a Hamiltonian term of the form 𝐻𝐿𝑆 = 𝜆so𝐋 ∙ 𝐒
(9)
In the first-principles quantum chemistry approach, the orbital and spin angular momenta of the individual electrons are considered, so that the spin-orbit coupling is written as [36] 𝐻𝑆𝑂𝐶 = ∑𝑖𝐳(𝐫𝑖)𝐬𝑖 9
(10)
where 𝐬𝑖 is the spin of the ith electron and 𝐳(𝐫𝑖) contains the three spatial coordinates on which the orbital angular momenta of the electrons depend [35]. For the sake of simplicity, in the following Eq. (9) will be used. We indicate with |𝜓(0) 𝑖 > the ground (i = 0) and excited electronic configurations of the system composed of the paramagnetic metal ion in its coordination scaffold, and with 𝐸(0) 𝑖 the corresponding energy, as determined from crystal field theory. Since for transition metal ions the spin-orbit coupling is usually much smaller than the crystal field energy, it can be considered to cause only a perturbation in the electronic configuration established by the crystal field. Therefore, the eigenstates of the (1) system from first-order perturbation theory are |𝜓𝑖 > = |𝜓(0) 𝑖 > +|𝜓 𝑖 > with
|𝜓(1) 𝑖 > = ∑𝑗 ≠ 𝑖
(0) < 𝜓(0) 𝑗 |𝜆so𝐋 ∙ 𝐒|𝜓 𝑖 > (0) 𝐸(0) 𝑖 ‒𝐸 𝑗
|𝜓(0) 𝑗 >
(11)
This implies that the unperturbed excited states are mixed with the unperturbed ground state by spin(1) (2) orbit coupling. The energy corrections to the second order (𝐸𝑖 = 𝐸(0) 𝑖 +𝐸 𝑖 +𝐸 𝑖 ) are (0) (0) 𝐸(1) 𝑖 = < 𝜓 𝑖 |𝜆so𝐋 ∙ 𝐒|𝜓 𝑖 >
𝐸(2) 𝑖 =
(1) < 𝜓(0) 𝑖 |𝜆so𝐋 ∙ 𝐒|𝜓 𝑖 >
(12)
| < 𝜓(0)𝑖 |𝜆so𝐋 ∙ 𝐒|𝜓(0)𝑗 > |2
= ∑𝑗 ≠ 𝑖
(0) 𝐸(0) 𝑖 ‒𝐸 𝑗
(13)
(0) If the orbital angular momentum of the unperturbed ground state is quenched, < 𝜓(0) 0 |𝐋|𝜓 0 > = (1) 0, because the ground state |𝜓(0) 0 > is orbitally nondegenerate, 𝐸 0 = 0, and 2 (0) | < 𝜓(0) 0 |𝐋|𝜓 𝑗 > | (2) = (0) 2∑ + 𝐸0 = 𝐸(0) +< 𝑚 |𝐒 ∙ ∙ 𝐒|𝑚𝑆' > 𝐸 𝐸 𝜆 𝑆 0 0 0 so 𝑗 ≠ 0 𝐸(0) ‒ 𝐸(0) 0
(14)
𝑗
where 𝑚𝑆 and 𝑚𝑆' indicate the total magnetic (secondary) spin quantum numbers of the unpaired electron(s). We can define a tensor 𝚲, which is sometimes referred to as the dequenching tensor [43], and a tensor 𝐃, called the zero field splitting (ZFS), as 2 (0) | < 𝜓(0) 0 |𝐋|𝜓 𝑗 > |
𝚲 = ∑𝑗 ≠ 0 2 𝐃 = 𝜆so 𝚲
10
(0) 𝐸(0) 0 ‒𝐸 𝑗
(15) (16)
so that 𝐸0 can be rewritten as '
ZFS (0) (0) (0) 𝐸0 = 𝐸(0) 0 +𝐸 0 = 𝐸 0 +< 𝜓 0 ,𝑚𝑆|𝐒 ∙ 𝐃 ∙ 𝐒|𝜓 0 ,𝑚𝑆 >
(17)
The energy 𝐸ZFS 0 can thus be obtained from the application of the Hamiltonian (see Section 5.2 for a more complete description) 𝐻ZFS = 𝐒 ∙ 𝐃 ∙ 𝐒
(18)
to the unperturbed eigenstates. Assuming the validity of the LS coupling regime, a total magnetic moment 𝛍el, the sum of the spin and orbital magnetic moments, can be defined 𝛍el =‒ 𝜇B(𝐋 + 𝑔e𝐒)
(19)
and evaluated from the expectation value of the corresponding quantum mechanical operators over the perturbed orbital ground state,
(
'
(0) ' (0) (0) < 𝜓0,𝑚𝑆 |𝛍el|𝜓0,𝑚'𝑆 > =‒ 𝜇B 𝑔e < 𝜓(0) 0 ,𝑚𝑆|𝐒|𝜓 0 ,𝑚 𝑆 >+ 2𝜆so𝚲 < 𝜓 0 ,𝑚𝑆|𝐒|𝜓 0 ,𝑚 𝑆 >
)
(20) as shown in Panel 1.
----------------------------------------------------------------------------------------------------------------------Panel 1. To evaluate < 𝜓0,𝑚𝑆 |𝛍el|𝜓0,𝑚'𝑆 > we must calculate (1) (1) ' | | (0) 0 ,𝑚𝑆 𝛍el 𝜓 𝑖 + 𝜓 𝑖 ,𝑚 𝑆 >
< 𝜓(0) 0 +𝜓
= ‒ 𝜇 B( (0) ' (0) (1) ' (1) (0) ' (1) (1) ' < 𝜓(0) 0 ,𝑚𝑆|𝐋 + 𝑔e𝐒|𝜓 0 ,𝑚 𝑆 >+ < 𝜓 0 ,𝑚𝑆|𝐋 + 𝑔e𝐒|𝜓 0 ,𝑚 𝑆 >+< 𝜓 0 ,𝑚𝑆|𝐋 + 𝑔e𝐒|𝜓 0 ,𝑚 𝑆 >+ < 𝜓 0 ,𝑚𝑆|𝐋 + 𝑔e𝐒|𝜓 0 ,𝑚 𝑆 >
)
(P1.1) (0) Assuming < 𝜓(0) 0 |𝐋|𝜓 0 > = 0, the first term is (0) ' (0) (0) ' < 𝜓(0) 0 ,𝑚𝑆|𝐋 + 𝑔e𝐒|𝜓 0 ,𝑚 𝑆 > = 𝑔e < 𝜓 0 ,𝑚𝑆|𝐒|𝜓 0 ,𝑚 𝑆 >
(P1.2)
the second and third terms are (using Eq. (11)) (1) < 𝜓(0) 0 ,𝑚𝑆|𝐋 + 𝑔e𝐒|𝜓 0 ,𝑚'𝑆 > = ∑𝑗 ≠ 0
∑
' (0) < 𝜓(0) 𝑗 ,𝑚 𝑆|𝜆so𝐋 ∙ 𝐒|𝜓 0 ,𝑚𝑆 > (0) 𝐸(0) 0 ‒𝐸 𝑗 2 | < 𝜓(0)𝑗 |𝐋|𝜓(0) 0 >|
𝜆 𝑗 ≠ 0 so
11
(0) 𝐸(0) 0 ‒𝐸 𝑗
(0) ' < 𝜓(0) 0 ,𝑚𝑆|𝐋 + 𝑔e𝐒|𝜓 𝑗 ,𝑚 𝑆 > =
< 𝑚𝑆|𝐒|𝑚'𝑆 >
(P1.3)
(0) ' (1) (1) because 𝑔e < 𝜓(0) 0 ,𝑚𝑆|𝐒|𝜓 𝑗 ,𝑚 𝑆 > = 0, and the fourth term in Eq. (P1.1) < 𝜓 0 ,𝑚𝑆|𝐋 + 𝑔e𝐒|𝜓 0 ,𝑚'𝑆 > is (0) proportional to (𝐸(0) 0 ‒𝐸 𝑗 )
‒2
(0) . Therefore, Eq. (P1.1) to the order (𝐸(0) 0 ‒𝐸 𝑗 )
‒1
is reduced to Eq. (20).
-----------------------------------------------------------------------------------------------------------------------
Therefore, if a tensor 𝐠 is defined as 𝐠 = 𝑔e𝟏 + 2𝜆so𝚲
(21)
in the spin Hamiltonian formalism, valid when only the ground state multiplet is thermally populated, the total electron magnetic moment can be written as (22)
𝛍el = ‒ 𝜇B𝐠 ∙ 𝐒
This condition, implying that orbital excited states are far in energy above the ground state with respect to the thermal energy, is commonly met in many transition metal complexes. Eq. (22) is also used as a definition of the electron magnetic moment in first-principles quantum chemistry approaches developed within the spin Hamiltonain EPR formalism, where the 𝐠 tensor accounts for the effect of spin-orbit coupling [32,33,35,36].
3.1 The Curie spin In the presence of a magnetic field, a further Hamiltonian term must be considered, for the Zeeman interaction between the total electron magnetic moment and the magnetic field 𝐁0 (more correctly, this should be called the magnetic flux density, see Section 4): 𝐻Zeeman =‒ 𝐁0 ∙ 𝛍el
(23)
The Zeeman term thus introduces a field dependence, which is proportional to the total electron magnetic moment, into the energy of the system. The total electron magnetic moment can be defined in terms of the derivative of the energy of the electron with respect to the magnetic field d𝐸
d𝐸
(𝜇𝑖 = ‒ d𝐵 , with 𝑖 = {𝑥,𝑦,𝑧}) 0,𝑖
𝛍el = ‒ d𝐁0
12
(24)
B0
S
el
el
Figure 1. An ensemble of electron spin magnetic moments is oriented by a magnetic field, with a thermal average 〈𝛍𝑆〉 parallel to the field. Due to spin-orbit coupling, the total average electron magnetic moment 〈𝛍el〉, comprising contributions from spin and orbital magnetic moments, is not parallel to the field if the 𝐠 tensor is anisotropic. An averaged electron magnetic moment is defined from the population distribution over the electronic energy levels at a given temperature (Fig. 1). The average spin-only electron magnetic moment is thus
〈𝛍𝑆〉 =‒ 𝜇B𝑔e〈𝐒〉 =‒ 𝜇B𝑔e𝐒C
(25)
where 𝐒C is the thermal average of 𝐒, equal to 𝐒C =
[
]
∑ ⟨𝜓𝑖│𝐒│𝜓𝑖⟩exp ‒ 𝐸𝑖 (𝑘𝑇) 𝑖
[
(26)
]
∑ exp ‒ 𝐸𝑖 (𝑘𝑇) 𝑖
and 𝐸𝑖 is the energy of the ith state, resulting from the sum of the crystal field, spin-orbit coupling (and thus ZFS) and Zeeman contributions, k is the Boltzmann constant and T is the temperature. If a single orbitally non-degenerate ground state is populated (because orbital excited states have a high energy), in the absence of ZFS this reduces to 𝐒C =
[
]
∑ ⟨𝜓0,𝑚𝑆,𝑖│𝐒│𝜓0,𝑚𝑆,𝑖⟩exp ‒ 𝐸𝑖Zeeman (𝑘𝑇) 𝑖
[
]
∑ exp ‒ 𝐸𝑖Zeeman (𝑘𝑇) 𝑖
(27)
where (see Eqs. (22) and (23)) 𝐸𝑖Zeeman = ‒ 𝜇B⟨𝜓0,𝑚𝑆,𝑖│𝐁0 ∙ 𝐠 ∙ 𝐒│𝜓0,𝑚𝑆,𝑖⟩ (28) 13
and thus, to the first order in EZeeman/(kT), for the case of isotropic molecular rotation, Eq (27) reduces to the Curie spin
[
∑ ⟨𝜓0,𝑚𝑆,𝑖│𝐒│𝜓0,𝑚𝑆,𝑖⟩ 1 ‒ 𝑖
𝐒C =
𝜇B𝐁 ∙ 𝐠 ∙ ⟨𝜓0,𝑚𝑆,𝑖│𝐒│𝜓0,𝑚𝑆,𝑖⟩ 0 𝑘𝑇
∑1
]
=‒
𝜇B𝑔iso𝑆(𝑆 + 1)𝐵0 3𝑘𝑇
𝑖
𝛋 (29)
where 𝛋 indicates the direction (z axis) of the magnetic field and 𝑔iso is the isotropic average of the 𝐠 tensor (∑𝑖𝑚𝑆,𝑖2 =
𝑆(𝑆 + 1)(2𝑆 + 1) ). 3
Analogously, from Eq. (22), in the spin Hamiltonian formalism the average total electron magnetic moment is
〈𝛍el〉 =‒ 𝜇B𝐠 ∙ 𝐒C
(30)
and, from Eq. (24), performing the same thermal average as in Eq. (26), d𝐸𝑖
〈𝛍el〉 =‒
[
]
∑ d𝐁 exp ‒ 𝐸𝑖 (𝑘𝑇) 𝑖 0
[
(31)
]
∑ exp ‒ 𝐸𝑖 (𝑘𝑇) 𝑖
The Zeeman energy is much smaller than the energy at zero magnetic field given by the crystal field (0) ZFS and ZFS (𝐸(0) 𝑖 = 𝐸 𝑖 + 𝐸 𝑖 ). Therefore, it can be introduced through perturbation theory to the
second-order, providing: Zeeman│ ⟩ 𝜓𝑖 𝐸𝑖 = 𝐸(0) 𝑖 + ⟨𝜓𝑖│𝐻
+
∑ 𝑗≠𝑖
|⟨𝜓𝑖│𝐻Zeeman│𝜓𝑗⟩|2 (0) 𝐸(0) 𝑖 ‒𝐸 𝑗
= 𝐸(0) 𝑖 ‒ ⟨𝜓𝑖│𝐁0 ∙ 𝛍el│𝜓𝑖⟩ +
∑ 𝑗≠𝑖
|⟨𝜓𝑖│𝐁0 ∙ 𝛍el│𝜓𝑗⟩|2 (0) 𝐸(0) 𝑖 ‒𝐸 𝑗
(32) and thus
(
|⟨𝜓𝑖│𝐁0 ∙ 𝛍el│𝜓𝑗⟩|
d ⟨𝜓𝑖│𝐁0 ∙ 𝛍el│𝜓𝑖⟩ ‒ ∑𝑗 ≠ 𝑖
∑
〈𝛍el〉 =
𝑖
(0) (0) 𝐸 𝑖 ‒𝐸 𝑗
d𝐁0
[
]
∑ exp ‒ 𝐸𝑖 (𝑘𝑇) 𝑖
2
)
[
exp ‒ 𝐸𝑖 (𝑘𝑇)
]
.
(33)
If the summation in Eq. (33) is extended only to the manifold of the orbital ground state, |𝜓𝑖 > indicates |𝜓0,𝑚𝑆𝑖 > . Then, to first order in 𝐸𝑖 (𝑘𝑇), for systems with isolated, orbitally nondegenerate ground states (see Panel 2) [40] 14
𝜇B2
〈𝛍el〉 = 𝑘𝑇𝐠 ∙ 〈𝐒𝐒𝑇〉 ∙ 𝐠𝑇 ∙ 𝐁0
(34)
where 〈𝐒𝐒𝑇〉 is the effective electron spin dyadic equal to
⟨𝑆𝛼𝑆𝛽⟩ =
𝑄𝑗𝑖 =
∑ 𝑄𝑗𝑖⟨𝜓𝑖│𝑆𝛼│𝜓𝑗⟩⟨𝜓𝑗│𝑆𝛽│𝜓𝑖⟩ 𝑖𝑗
[
∑ exp ‒ 𝐸(0) (𝑘𝑇) 𝑖 𝑖
]
,
exp [ ‒ 𝐸(0) 𝑖 (𝑘𝑇)]
{
‒
𝑘𝑇
(35)
𝛼,𝛽 = {𝑥,𝑦,𝑧} (0) for 𝐸(0) 𝑖 =𝐸 𝑗
{exp [ ‒ 𝐸(0)𝑗 (𝑘𝑇)] ‒ exp [ ‒ 𝐸(0)𝑖 (𝑘𝑇)]}
(0) for 𝐸(0) 𝑖 ≠𝐸 𝑗
(0) 𝐸(0) 𝑗 ‒𝐸 𝑖
and 𝐸(0) 𝑖 is the energy of the state |𝜓𝑖⟩ at zero magnetic field. Recall that we have assumed here that there are no low-lying orbitally excited states, and thus that only the orbitally non-degenerate ground spin multiplet is populated, and the thermally accessible states are limited to the manifold of the eigenstates of the ZFS Hamiltonian.
----------------------------------------------------------------------------------------------------------------------Panel 2 From Eq. (33), to first order in 𝐸𝑖 (𝑘𝑇), (0) 𝐸 𝑖 ⟨𝜓𝑖│𝑩0 ∙ 𝝁𝑒𝑙│𝜓𝑗⟩⟨𝜓𝑗│𝛍𝐞𝐥│𝜓𝑖⟩ ⟨𝜓𝑖│𝐁0 ∙ 𝛍𝐞𝐥│𝜓𝑖⟩ ∑ ⟨𝜓𝑖│𝛍el│𝜓𝑖⟩ ‒ 2∑ exp ‒ 1+ (0) (0) 𝑘𝑇 𝑘𝑇 𝑖 𝑗≠𝑖
(
〈𝛍el〉 = and, since ∑𝑖⟨𝜓𝑖│𝛍el│𝜓𝑖⟩exp[ ‒ 𝐸
(0) 𝑖
) [ ](
𝐸 𝑖 ‒𝐸 𝑗
[
∑ exp ‒ 𝐸(0) (𝑘𝑇) 𝑖 𝑖
) (P2.1)
]
(𝑘𝑇)] = 0, retaining only the terms linear in 𝐵0 (see Section 4.1), ∑
〈𝛍el〉 =
(
⟨𝜓𝑖│𝛍𝐞𝐥│𝜓𝑖⟩⟨𝜓𝑖│𝛍𝐞𝐥│𝜓𝑖⟩
𝑖
𝑘𝑇
)
⟨𝜓𝑖│𝛍𝐞𝐥│𝜓𝑗⟩⟨𝜓𝑗│𝛍𝐞𝐥│𝜓𝑖⟩ (0) ‒ 2∑𝑗 ≠ 𝑖 exp ‒ 𝐸 𝑖 (𝑘𝑇) (0) (0) 𝐸 𝑖 ‒𝐸 𝑗
[
[
] (P2.2)
∙ 𝐁0
]
∑ exp ‒ 𝐸(0) (𝑘𝑇) 𝑖 𝑖
which is the Van Vleck equation [44]. This can be written as ∑
〈𝛍el〉 =
(
⟨𝝍𝒊│𝛍𝐞𝐥│𝝍𝒊⟩⟨𝜓𝑖│𝛍𝐞𝐥│𝜓𝑖⟩
𝑖
𝑘𝑇
]} )
⟨𝜓𝑖│𝛍𝐞𝐥│𝜓𝑗⟩⟨𝜓𝑗│𝛍𝐞𝐥│𝜓𝑖⟩ (0) (0) (0) exp ‒ 𝐸 𝑖 (𝑘𝑇) ‒ ∑𝑗 ≠ 𝑖 exp ‒ 𝐸 𝑖 (𝑘𝑇) ‒ exp ‒ 𝐸 𝑗 (𝑘𝑇) (0) (0) 𝐸 ‒𝐸
[
]
𝑖
[
{ [
𝑗
]
∑ exp ‒ 𝐸(0) (𝑘𝑇) 𝑖 𝑖
]
[
∙ 𝐁0 .
(P2.3) Eq. (P2.3) corresponds to Eqs. (34) and (35) when the spin Hamoltonian formalism (see also Eq. (22)) is used, i.e., for systems with isolated, orbitally non-degenerate ground states [40]. -----------------------------------------------------------------------------------------------------------------------
15
From Eqs. (30) and (34) 𝜇B
𝐒C =‒ 𝑘𝑇〈𝐒𝐒𝑇〉 ∙ 𝐠𝑇 ∙ 𝐁0
(36)
Eq. (35) provides the following expression for 〈𝐒𝐒𝑇〉 in the approximation to first order in D/(kT) [45]
〈𝐒𝐒𝑇〉 =
𝑆(𝑆 + 1) 𝑆(𝑆 + 1)(2𝑆 ‒ 1)(2𝑆 + 3) 𝟏 ‒ 𝐃 3 30𝑘𝑇
(37)
The traceless ZFS tensor in the frame where it is diagonal is usually written in the form
𝐃=
(
1
‒ 3𝐷 + 𝐸 0
0
0
1
‒ 3𝐷 ‒ 𝐸
0
0 2 3𝐷
0
)
(38)
where D and E are the axial and rhombic ZFS parameters (𝐷 = 𝐷𝑧𝑧 ‒ (𝐷𝑥𝑥 + 𝐷𝑦𝑦)/2, 𝐸 = (𝐷𝑥𝑥 ‒ 𝐷𝑦𝑦 )/2). Of course, Eq. (36) reduces to Eq. (29) when only the ground state is populated in the absence of ZFS. In this case, 〈𝐒𝐒𝑇〉 =
𝑆(𝑆 + 1) 𝟏 3
(and thus Tr(〈𝐒𝐒𝑇〉) = S(S+1)).
Extending the summation of the second order term in Eq. (33) to the orbital excited states, Eq. (34) should be replaced with (see Panel 3),
〈𝛍el〉 =
(
𝜇B2
𝑘𝑇𝐠 ∙
)
〈𝐒𝐒𝑇〉 ∙ 𝐠𝑇 ‒ 2𝜇B2𝚲 ∙ 𝐁0
(39)
and using Eq. (37)
〈𝛍el〉 = 𝜇B2
(
𝑆(𝑆 + 1) 𝑇 𝑆(𝑆 + 1)(2𝑆 ‒ 1)(2𝑆 + 3) 𝐠 ∙ 𝐃 ∙ 𝐠𝑇 ‒ 2𝚲 3𝑘𝑇 𝐠 ∙ 𝐠 ‒ 30(𝑘𝑇)2
)∙𝐁
(40)
0
----------------------------------------------------------------------------------------------------------------------Panel 3 Eq. (P2.2) can be written as ∑
〈𝝁𝑒𝑙〉 =
𝐖𝑰𝒊
(
𝑖 𝑘𝑇
∈ ground 𝐖𝑰𝑰 𝒊𝒋 state manifold
‒ 2∑𝑗 ≠ 𝑖,
∈ excited 𝐖𝑰𝑰 𝒊𝒋 states
‒ 2∑𝑗 ≠ 𝑖,
[
]
∑ exp ‒ 𝐸(0) (𝑘𝑇) 𝑖 𝑖
16
)
[
(0) exp ‒ 𝐸 𝑖 (𝑘𝑇)
] ∙ 𝐁0
(P3.1)
with 𝐖𝑰𝒊 = ⟨𝜓𝑖│𝛍𝑒𝑙│𝜓𝑖⟩⟨𝜓𝑖│𝝁𝑒𝑙│𝜓𝑖⟩ , 𝐖𝑰𝑰 𝒊𝒋 = The terms 𝐖𝑰𝒊 and ∑𝑗 ≠ 𝑖,
∈ ground 𝐖𝑰𝑰 𝒊𝒋 state manifold
⟨𝜓𝑖│𝝁𝑒𝑙│𝜓𝑗⟩⟨𝜓𝑗│𝝁𝑒𝑙│𝜓𝑖⟩
(P3.2)
(0) 𝐸(0) 𝑖 ‒𝐸 𝑗
∈ excited 𝐖𝑰𝑰 𝒊𝒋 states
are those reported in Panel 2. To evaluate the term ∑𝑗 ≠ 𝑖,
the following contributions should be calculated (1) (1) (1) (1) (0) (0) (0) ⟨𝜓𝑖│𝝁𝑒𝑙│𝜓𝑗⟩ = ‒ 𝜇B⟨𝜓(0) 𝑖 + 𝜓 𝑖 │𝐋│𝜓 𝑗 + 𝜓 𝑗 ⟩ ‒ 𝜇B𝑔e⟨𝜓 𝑖 + 𝜓 𝑖 │𝐒│𝜓 𝑗 + 𝜓 𝑗 ⟩
(P3.3)
In this case, Eq. (19), rather than Eq. (22), must be used because the latter is a correct approximation of the 0 ‒1
former only if the orbital ground state alone is considered. If we neglect the terms of the order (𝐸0 ‒ 𝐸 𝑗 ) 0
0 ‒1
or higher (which provides terms of order higher than (𝐸0 ‒ 𝐸 𝑗 ) 0
when inserted in Eq. (P3.2)), we are limited
to the terms connecting the ground to the excited states and we get [46] (0) ⟨𝜓0│𝝁𝑒𝑙│𝜓𝑗⟩ = ‒ 𝜇B⟨𝜓(0) 0 │𝐋│𝜓 𝑗 ⟩ 0 ‒1
Therefore, to the (𝐸0 ‒ 𝐸 𝑗 ) 0
(P3.4)
order (see Eq. (15)),
∈ excited 𝐖𝑰𝑰 𝒊𝒋 states
∑𝑗 ≠ 𝑖,
= 𝜇B2∑𝑗 ≠ 𝑖,
∈ excited states
(0) (0) (0) ⟨𝜓(0) 0 │𝐋│𝜓 𝑗 ⟩⟨𝜓 𝑗 │𝐋│𝜓 0 ⟩ (0) 𝐸(0) 0 ‒𝐸 𝑗
= 𝜇B2𝚲.
(P3.5)
In summary, ∑
〈𝛍el〉 =
𝐖𝑰𝒊
(
𝑖 𝑘𝑇
∈ ground 𝐖𝑰𝑰 𝒊 state manifold
‒ 2∑𝑗 ≠ 𝑖,
)
[
]
(0) exp ‒ 𝐸 𝑖 (𝑘𝑇)
[
∑ exp ‒ 𝐸(0) (𝑘𝑇) 𝑖 𝑖
∙ 𝐁0 ‒ 2𝜇B2𝚲 ∙ 𝐁0 =
]
(
𝜇B2
)
𝐠 ∙ 〈𝐒𝐒𝑇〉 ∙ 𝐠𝑇 ‒ 2𝜇B2𝚲 ∙ 𝐁0
𝑘𝑇
(P3.6) -----------------------------------------------------------------------------------------------------------------------
The contribution ‒ 2𝚲, arising from the presence of excited states, is called temperature-independent paramagnetism (TIP). Although frequently neglected, it can be important in determining the anisotropy of 〈𝛍el〉 even at room temperature, especially for S = 1/2 systems, if there is no ZFS.
4. The paramagnetic susceptibility The induced magnetic moment per molecule 〈𝛍tot〉 established by the anisotropic alignment of the magnetic moments of the electrons of a molecule in the presence of a magnetic field results in a magnetization per unit volume 𝐌 given by 𝐌=
〈𝛍𝐭𝐨𝐭〉𝑁A 𝑉𝑀
,
(41) 17
where 𝑁A is the Avogadro constant and 𝑉𝑀 the molar volume. From the general definition of magnetic susceptibility per unit volume d𝐌
d𝐌
(42)
𝛘𝑽 = d𝐇0 𝜇0d𝐁0
where 𝐇0 is the external magnetic field and 𝐁0 the magnetic flux density (𝐁0 = 𝜇0(𝐇0 + 𝐌), with 𝐌 ≪ 𝐇0), 𝑁A d〈𝛍tot〉
𝛘𝑽 = 𝜇 0𝑉𝑀
(43)
d𝐁0
and thus the magnetic susceptibility per molecule d〈𝛍tot〉
𝑉𝑀
𝛘𝐦𝐨𝐥 = 𝑁A𝛘𝑽 = 𝜇0
(44)
d𝐁0
According to Eq. (44), the paramagnetic contribution to the molecular magnetic susceptibility 𝛘 is d〈𝛍el〉
𝛘 = 𝜇0
d〈𝜇𝑖〉
(𝜒𝑖𝑗 = 𝜇0d𝐵 , with 𝑖,𝑗 = {𝑥,𝑦,𝑧})
d𝐁0
0,𝑗
(45)
Since 〈𝛍el〉 increases linearly with 𝐁0 (Eqs. (34) and (40)), 𝛘 is independent of the magnetic field strength, unless the magnetic field is large enough to reach saturation conditions (see Section 4.1). This implies that
〈𝛍el〉 =
𝛘 ∙ 𝐁0
(46)
𝜇0
In the high temperature approximation (BB0<
𝜇0𝜇B2 𝑘𝑇
𝐠 ∙ 〈𝐒𝐒𝑇〉 ∙ 𝐠𝑇
(47)
or, for D<
(
𝑆(𝑆 + 1) 𝑇 𝑆(𝑆 + 1)(2𝑆 ‒ 1)(2𝑆 + 3) 𝐠 ∙ 𝐃 ∙ 𝐠𝑇 ‒ 2𝚲 3𝑘𝑇 𝐠 ∙ 𝐠 ‒ 30(𝑘𝑇)2
)
(48)
If it is assumed that the 𝐠 and 𝐃 tensor are diagonal in the same frame, the components of the 𝛘 tensor are thus 18
𝜒𝑧𝑧 = 𝜒𝑥𝑥,𝑦𝑦 =
𝑆(𝑆 + 1) 2 3𝑘𝑇 𝑔𝑧𝑧
1)(2𝑆 + 3) 𝐷) ‒ 2𝜇0𝜇B2𝜒𝛬𝑧𝑧 , (1 ‒ (2𝑆 ‒ 15𝑘𝑇
1)(2𝑆 + 3) (𝐷 ∓ 3𝐸)) ‒ 2𝜇0𝜇B2𝜒𝛬𝑥𝑥,𝑦𝑦 (1 + (2𝑆 ‒ 30𝑘𝑇
𝑆(𝑆 + 1) 2 3𝑘𝑇 𝑔𝑥𝑥,𝑦𝑦
(49)
The same expressions were originally obtained by Kurland and McGarvey [28], with the difference that the TIP contribution was neglected. From Eq. (48), in the absence of ZFS and neglecting the TIP contribution, the rotational average of 𝛘 is 𝜒iso =
2 𝜇0𝜇B2𝑔iso 𝑆(𝑆 + 1)
(50)
3𝑘𝑇
which is the Curie law. When orbital excited states should also be considered because they are thermally populated, Eq. (P2.3) should be used instead of Eq. (34), with 𝛍el given by Eq. (19), so that Eq. (45) provides 𝜒𝒌𝒌 =
𝜇0𝜇B2∑𝑖𝑗𝑄𝑗𝑖⟨𝜓𝑖│𝐿𝑘 + 𝑔e𝑆𝑘│𝜓𝑗⟩⟨𝜓𝑗│𝐿𝑘 + 𝑔e𝑆𝑘│𝜓𝑖⟩
[
𝑘𝑇
∑ exp ‒ 𝐸(0) (𝑘𝑇) 𝑖 𝑖
]
(51)
and outside the LS coupling regime 𝜇0 ∑𝑖𝑗𝑄𝑗𝑖⟨𝜓𝑖│𝛍el│𝜓𝑗⟩⟨𝜓𝑗│𝛍el
𝛘 = 𝑘𝑇
[
𝑇
]
∑ exp ‒ 𝐸(0) (𝑘𝑇) 𝑖 𝑖
│𝜓𝑖⟩
.
(52)
which is the general van Vleck expression [44] for the susceptibility tensor. From Eqs. (24) and (45), 𝛘 can also be expressed as d < d𝐸 > d𝐁0d𝐁0
𝛘 =‒ 𝜇0
|
(53) 𝐵0 = 0
4.1 Saturation at high field At very high fields (>20 T) and room temperature, the first-order approximation in 𝐁0 ∙ 𝛍el (𝑘𝑇), invoked on passing from Eq. (33) to Eqs. (P2.1), (34) and (39), may be inaccurate. In this case, Eq. (45) can be rewritten using Eq. (31), so that [47]
19
d𝐸
𝛘=
𝑖 ∑ exp ‒ 𝐸𝑖 (𝑘𝑇) d 𝑖d𝐁0 ‒ 𝜇0d𝐁0 ∑ exp ‒ 𝐸𝑖 (𝑘𝑇) 𝑖
[
]
[
]
d
d
= 𝜇0𝑘𝑇d𝐁0d𝐁0ln∑𝑖exp[ ‒ 𝐸𝑖 (𝑘𝑇)]
(54)
For isolated, orbitally non-degenerate ground states, in the absence of ZFS and g-anisotropy, 𝐸𝑖 =
⟨𝜓𝑖│𝜇B𝑔e𝐁0𝑇 ∙ 𝐒│𝜓𝑗⟩ (Eqs. (22) and (23)) and thus Eq. (54) becomes [48] sinh d d 𝑆 𝜒 = 𝜇0𝑘𝑇d𝐵0d𝐵0ln∑𝑚 =‒ 𝑆exp 𝑆
(‒
𝜇B𝑔e𝑚𝑆𝐵0
)
𝑘𝑇
d d = 𝜇0𝑘𝑇d𝐵0d𝐵0ln
(𝑆 + 12)𝜇B𝑔e𝐵0
(
sinh
[(2𝑆 + 1)coth(
=
1 𝜇 𝑔 𝐵 2 B e 0 𝑘𝑇
(
)
𝜇0𝜇B𝑔e d 2 d𝐵0
)]
(2𝑆 + 1)𝜇B𝑔e𝐵0
𝜇B𝑔e𝐵0
(2𝑘𝑇)
(2𝑘𝑇)
) ‒ coth(
)
𝑘𝑇
(55)
so that 𝜒=
[(2𝑆 + 1) (1 ‒ coth (
𝜇0𝜇B2𝑔2e 4𝑘𝑇
2
2
)) ‒ (1 ‒ coth (
(2𝑆 + 1)𝜇B𝑔e𝐵0 2𝑘𝑇
2
))] .
𝜇B𝑔e𝐵0 2𝑘𝑇
1
(56) 1
Fig. 2 shows the field dependence of 𝜒 as a function of the magnetic field. Since coth (𝑥) = 𝑥 + 3𝑥 ‒ 1 3 45𝑥 + …, to
first order in x,
[
𝜇0𝜇B2𝑔2e
𝜒
4𝑘𝑇
(
(2𝑘𝑇)2
) (
2
(2𝑘𝑇)2
)]
2
= (2𝑆 + 1)2 1 ‒ ( ‒ ‒ 1‒( ‒ (2𝑆 + 1)𝜇B𝑔e𝐵0)2 3 𝜇B𝑔e𝐵0)2 3
in agreement with Eq. (50) under the same conditions.
20
𝜇0𝜇B2𝑔2e𝑆(𝑆 + 1) 3𝑘𝑇
(57)
Magnetic susceptibility / 10-32 m3
2.65 2.64 2.63 2.62 2.61 2.60
0
200
400
600
800
1000
1200
Proton Larmor Frequency / MHz
Figure 2. Field dependence of the magnetic susceptibility (calculated with Eq. (56), S = 1/2, T = 298 K). The dotted line corresponds to the Curie law (Eq. (57)). 5. The hyperfine shift 5.1 The semi-empirical paramagnetic NMR approach In a system composed of a paramagnetic metal ion and a nucleus, we should introduce the spin 𝐈 of the nucleus with its associated magnetic moment 𝛍𝐼 = ℏ𝛾𝐼𝐈
(58)
The resonance frequency of a nuclear transition between states differing by Δ𝑚𝐼 =± 1 leads to the NMR signal. This frequency changes depending on the magnetic field experienced by the nucleus; in turn, such a field is determined not only by the external magnetic field but also by the field generated by the electrons moving in the surrounding atoms. When all electrons in the molecular orbital are paired (a diamagnetic system), this additional field is small, and so is the shift in NMR frequency 𝛿dia (although it may still be measurable). If unpaired electron(s) are present at a distance r from the nucleus, the electron magnetic moment creates a dipolar magnetic field which can cause a significant shift 𝛿para of the nuclear signal. To a good approximation, what is called the hyperfine shift 𝛿hf is the additional shift (𝛿hf = 𝛿para ‒ 𝛿dia), which is experimentally measured as the 21
difference in the NMR shift between a paramagnetic system and its diamagnetic analogue. The latter should ideally be composed of the same atoms, in the same positions, as the paramagnetic system, but with the unpaired electron(s) removed. In practice, diamagnetic analogues are obtained either by reducing the paramagnetic metal to a diamagnetic state, by substituting the paramagnetic metal with a diamagnetic one, or by removing the paramagnetic center. Such systems represent good diamagnetic references if there are no structural rearrangements in the molecule due to removal or substitution of the paramagnetic ion, or to a different electric charge. The hyperfine shift 𝛿hf is thus related to the interaction between the nucleus and the unpaired electron(s) contained in the paramagnetic moiety, and in particular to the energy 𝐸hfc of the magnetic interaction between the nuclear magnetic moment and the spin and orbital magnetic moments of the unpaired electron(s). Since NMR shifts are usually expressed in ppm with respect to the nuclear Larmor frequency (𝛾𝐼𝐵0 2𝜋), the component of the NMR shift corresponding to the hyperfine shift is [28,42] 𝐸hfc
Δ𝐸hfc
𝛿hf = 𝛿para ‒ 𝛿dia =‒ ℏ𝛾𝐼𝐵0𝑚𝐼 =‒ ℏ𝛾𝐼𝐵0 .
(59)
where Δ𝐸hfc corresponds to the difference in the hyperfine energy between the two 𝑚𝐼 states. In writing Eq. (59) we have neglected the residual anisotropic chemical shift 𝛿racs, arising from molecular partial alignment (see Eq. (118)), which is also present in the NMR shifts of paramagnetic molecules (but not in those of diamagnetic molecules when partial orientation is induced by the presence of the paramagnetic moiety (see Section 6)). Unpaired electrons relax much more rapidly than nuclei (see Section 7), so they change their spin state among the possible mS levels much more rapidly than nuclei change their spin states among the possible mI levels. As a result, the hyperfine shift results from the magnetic interaction of the nuclear magnetic moment with the average electron magnetic moment. The hyperfine interaction experienced by the nuclear magnetic moment is conveniently separated into the interaction with the electron magnetic moment delocalized onto the nucleus itself, 22
and the interaction with the average electron magnetic moment outside the nucleus. The former component is called the Fermi contact interaction, the second the dipole-dipole interaction. Therefore, the hyperfine coupling energy is split into the two terms: 𝐸hfc = 𝐸Fc + 𝐸dip
(60)
The energy associated with the dipole-dipole interaction between the nuclear magnetic moment and average total electron magnetic moment can be calculated, in the long-range limit, by applying the point-dipole approximation [49] 𝜇0
𝐸dip =‒ 4𝜋𝑟3〈𝛍el〉 ⋅
(
3𝐫𝐫𝑇 𝑟2
)
‒ 𝟏 ⋅ 𝛍𝐼 ,
(61)
where we have assumed that the interaction between the orbital magnetic moment 𝛍𝐿 and 𝛍𝐼 has the same dipolar form as the interaction between 𝛍𝑆 and 𝛍𝐼 [50,51] (see Section 5.2). The point-dipole approximation introduces small inaccuracies for nuclei at distances of several Å from the paramagnetic center, but can lead to large errors for nuclei at short distances. From Eqs. (46), (59)(61), the equation for the dipolar shift anticipated in Eq. (1) of Section 2 is obtained. From Eq. (1), the dipolar shielding tensor is defined as 𝛘
𝛔dip =‒ 4𝜋𝑟3 ⋅
(
3𝐫𝐫𝑇 𝑟2
)
‒𝟏 .
(62)
The pseudocontact shift is the rotational average of the dipolar shift. Assuming isotropic reorientation of the molecule with respect to the magnetic field (see Section 6.1), in the high-field case, when the nuclear spin is oriented along the magnetic field, the pseudocontact shift 𝛿pc is calculated as already reported in Eq. (2), being 𝛿pc =‒
Tr(𝛔dip) 3
1
[ (
= 12𝜋𝑟3Tr 𝛘 ⋅
3𝐫𝐫𝑇 𝑟2
‒𝟏
)]
(63)
Since the term 3𝐫𝐫𝑇 𝑟2 ‒ 𝟏 is traceless, 𝛿pc is determined simply by the paramagnetic susceptibility anisotropy 𝚫𝛘 = 𝛘 ‒ 𝜒iso𝟏, where 𝜒iso = Tr(𝛘) 3. Other forms of Eq. (2) are reported in Panel 4. In the reference frame where the 𝚫𝛘 tensor is diagonal, Eq. (63) becomes
23
1
[
]
3
𝛿pc = 12𝜋𝑟3 Δ𝜒𝑎𝑥(3cos2𝜃 ‒ 1) + 2Δ𝜒𝑟ℎsin2𝜃cos2𝜑
(64) where Δ𝜒𝑎𝑥 = 𝜒𝑧𝑧 ‒
𝜒𝑥𝑥 + 𝜒𝑦𝑦 2
3
= 2(𝜒𝑧𝑧 ‒ 𝜒iso), (65)
Δ𝜒𝑟ℎ = 𝜒𝑥𝑥 ‒ 𝜒𝑦𝑦
and r, 𝜃 and 𝜑 are the spherical coordinates of the nucleus in the principal frame of the 𝛘 tensor, with the origin in the position of the unpaired electron(s).
----------------------------------------------------------------------------------------------------------------------Panel 4 Equations for the pseudocontact shifts, valid in any frame centered on the metal ion: - in Cartesian coordinates: 𝛿pc =
[
1
4𝜋𝑟3
2𝑧2 ‒ 𝑥2 ‒ 𝑦2
(𝜒𝑧𝑧 ‒ Tr(𝜒)/3)
2𝑟2
𝑥2 ‒ 𝑦2
+ (𝜒𝑥𝑥 ‒ 𝜒𝑦𝑦)
2𝑟2
2𝑥𝑦
+ 𝜒𝑥𝑦
𝑟2
+ 𝜒𝑥𝑧
2𝑥𝑧 𝑟2
2𝑦𝑧
+ 𝜒𝑦𝑧
𝑟2
]
(P4.1)
- in spherical coordinates: 𝛿pc =
[(𝜒
𝑧𝑧 ‒ Tr(𝜒)/3)
1 4𝜋𝑟3
3cos2𝜃 ‒ 1 sin2𝜃cos2𝜑 ( ) + 𝜒 ‒ 𝜒 + 𝜒𝑥𝑦sin2𝜃sin2𝜑 + 𝜒𝑥𝑧sin2𝜃cos𝜑 + 𝜒𝑦𝑧sin2𝜃sin𝜑 𝑥𝑥 𝑦𝑦 2 2
]
(P4.2) - using director cosines (l, m and n, equal to the cosine of the angles between the metal-nucleus vector and the principal directions of the 𝛘 tensor): 𝛿pc =
1
[(𝜒
3
4𝜋𝑟
3𝑛2 ‒ 1 𝑙2 ‒ 𝑚2 ( ) + 𝜒 ‒ 𝜒 + 2𝜒𝑥𝑦𝑙𝑚 + 2𝜒𝑥𝑧𝑙𝑛 + 2𝜒𝑦𝑧𝑚𝑛 𝑥𝑥 𝑦𝑦 2 2
𝑧𝑧 ‒ Tr(𝜒)/3)
]
(P4.3)
Equations for the pseudocontact shifts, valid only in the frame where the 𝛘 tensor is diagonal: - in Cartesian coordinates: 𝛿pc =
1
[
8𝜋𝑟3
2𝑧2 ‒ 𝑥2 ‒ 𝑦2
(𝜒𝑧𝑧 ‒ Tr(𝜒)/3)
𝑟2
]
𝑥2 ‒ 𝑦2
+ Δ𝜒𝑟ℎ
𝑟2
=
1
[
12𝜋𝑟3
Δ𝜒𝑎𝑥
2𝑧2 ‒ 𝑥2 ‒ 𝑦2 𝑟2
3
]
𝑥2 ‒ 𝑦2
+ 2Δ𝜒𝑟ℎ
𝑟2
(P4.4) - in spherical coordinates: 𝛿pc =
1
1
[(𝜒𝑧𝑧 ‒ Tr(𝜒)/3)(3cos2𝜃 ‒ 1) + Δ𝜒𝑟ℎsin2𝜃cos2𝜑] = 12𝜋𝑟3 8𝜋𝑟 3
[Δ𝜒𝑎𝑥(3cos2𝜃 ‒ 1) + 32Δ𝜒𝑟ℎsin2𝜃cos2𝜑] - using director cosines:
24
(P4.5)
𝛿pc =
1
[𝜒𝑥𝑥𝑙2 + 𝜒𝑦𝑦𝑚2 + 𝜒𝑧𝑧𝑛2 ‒ Tr(𝜒)/3] = 12𝜋𝑟 [Δ𝜒𝑎𝑥(3𝑛2 ‒ 1) + 2Δ𝜒𝑟ℎ(𝑙2 ‒ 𝑚2)] (P4.6) 1
4𝜋𝑟3
3
3
-----------------------------------------------------------------------------------------------------------------------
As mentioned above, the hyperfine shift of the nucleus (Eq. (59)) is not only due to the pseudocontact shift, but also, for nuclei with some unpaired electron spin density delocalized onto the nucleus itself, by the Fermi contact shift: 𝛿hf = 𝛿pc + 𝛿Fc
(66)
The Fermi contact energy results from the magnetic interaction between the nuclear magnetic moment and the fraction of the electron magnetic moment corresponding to the electron spin density at the site of the nucleus. This contribution is thus non-zero only for nuclei with non-negligible unpaired spin density in their s orbitals, i.e., those that are a few chemical bonds away from the unpaired electron(s) or anyway close to the metal orbitals. By its nature, the Fermi contact interaction involves only the electron spin magnetic moment and not the orbital magnetic moment, and thus [52,53] 𝐸Fc =‒
𝜇0𝜌𝐼 3𝑆
〈 𝛍𝑆 〉 ⋅ 𝛍𝐼
(67)
where 𝜌𝐼 is the contact electron spin density at the NMR nucleus. Therefore, from Eqs. (25), (36), (59) and (60), implying the presence of an isolated, orbitally non-degenerate ground state, and again assuming isotropic reorientation of the molecule with respect to the magnetic field, 1
[
𝛿Fc = 3Tr
𝜇0𝜇B2𝜌𝐼
〈𝐒𝐒𝑇〉 ∙ 𝐠𝑇
3𝑘𝑇 𝑆 𝑔e
]
(68)
which, using Eq. (47), can be written as [27,28]. 1
[
𝛿Fc = 3Tr
𝜌𝐼
3𝑆𝑔e𝛘 ∙
]
[𝐠𝑇] ‒ 1
(69)
which corresponds to Eq. (6) reported in Section 2. In the case of negligible spin-orbit coupling, Eq. (68) becomes 𝛿Fc =
𝐴Fc𝑔e𝜇B𝑆(𝑆 + 1) ℏ 3𝛾𝐼𝑘𝑇
with the Fermi contact coupling constant 25
(70)
𝜇0
𝜌𝐼
𝐴Fc = 3 ℏ𝛾𝐼𝜇B𝑔e 𝑆 .
(71)
From Eq. (68) it is apparent that anisotropy of the contact shifts can arise due to anisotropy of the 𝐠 and 〈𝐒𝐒𝑇〉 tensors. In the presence of orbital excited states close in energy to the ground state, excited states should also be taken into account in the calculation of 〈𝛍𝑆〉, each of them with its own electron spin density at the nuclear position. Therefore, in the LS coupling regime [28] 𝜇B ∑𝑖𝑗𝑄𝑗𝑖(⟨𝜓𝑖│𝐿𝑘 + 𝑔e𝑆𝑘│𝜓𝑗⟩⟨𝜓𝑗│𝐴𝑖𝑗𝑆𝑘│𝜓𝑖⟩)
1
𝛿Fc = 3∑𝑘ℏ𝛾𝐼𝑘𝑇
[
∑ exp ‒ 𝐸(0) (𝑘𝑇) 𝑖 𝑖
]
.
(72)
The constants 𝐴𝑖𝑗 may depend on the orbitals [54], and represent the proportionality constants which relate the orbital spin density contributions to the spin density at the nucleus for the ground and excited states. The above treatment, developed by McConnell and Robertson [49] and Kurland and McGarvey [28], implies that in the hyperfine Hamiltonian 𝐻hfc = 𝐒 ⋅ 𝐀 ⋅ 𝐈 .
(73)
the hyperfine coupling constant A should be defined as 𝜇0 ℏ𝛾𝐼𝜇B
𝐀 = 4𝜋
3
𝑟
𝐠𝑇 ⋅
(
3𝐫𝐫𝑇 2
𝑟
)
𝜇0
𝜌𝐼
‒ 𝟏 + 3 ℏ𝛾𝐼𝜇B𝑔e 𝑆 𝟏
(74)
where the first and second terms correspond to the point-dipole/point-dipole tensor and to the Fermi contact coupling constant, respectively. In the following section, the derivation of Eq. (74) is presented using the LS coupling approximation and the spin Hamiltonian formalism. This derivation permits one to recover the semi-empirical equation for the hyperfine coupling without involving the magnetic susceptibility anisotropy, using Eq. (87) developed within the first-principles quantum chemistry approach presented in Section 5.3.
5.2 Perturbative EPR spin Hamiltonian approach
26
If all interactions perturbing the crystal field states are considered altogether (Fig. 3), in the spin Hamiltonian formalism and LS coupling regime the following perturbative Hamiltonian terms should at least be considered [55,56]: 𝐻' = 𝐻𝐿𝑆 + 𝐻Zeeman + 𝐻hfc + 𝐻IL + 𝐻ss = 𝜆so𝐋 ∙ 𝐒 + 𝜇B𝐁0 ∙ (𝐋 + 𝑔e𝐒) + 𝐒 ⋅ 𝐀hfc ⋅ 𝐈 + 𝑎IL𝐋 ⋅ 𝐈 + 𝐒 ⋅ ∆𝐬𝐬 ⋅ 𝐒
(75)
where 𝐻IL = 𝑎IL𝐋 ⋅ 𝐈 refers to the interaction of the nuclear magnetic moment with the orbital magnetic moment of the unpaired electron(s) calculated with respect to the nuclear position, with 𝑎IL 𝜇0 ℏ𝛾𝐼𝜇B
= 24𝜋
𝜌3
, where 𝜌 is the nucleus-electron distance (see Panel 5), and 𝐋 = 𝝆 × 𝐩, while 𝐻ss = 𝐒 ⋅ ∆𝐬𝐬
⋅ 𝐒 refers to the dipole-dipole interaction between unpaired electrons (usually negligible with respect to the other terms) present whenever S>1/2. The energy of the interaction between the nuclear magnetic moment and the orbital magnetic moment of the unpaired electron(s) can be calculated by considering that an electric charge moving in a loop generates a dipolar magnetic field at the position of the nucleus that can be expressed in the form 𝐻IL =
𝜇0ℏ𝛾𝐼𝜇B 4𝜋
(3
𝐋·𝐫 𝑇 𝐋⋅𝐈 5𝐫 ⋅ 𝐈 ‒ 𝑟 𝑟3
)
(76)
where 𝐋 is the orbital angular momentum calculated with respect to the metal nucleus (see Panel 5). ----------------------------------------------------------------------------------------------------------------------Panel 5 The dipole-dipole interaction between the nuclear spin I and electron orbital angular momentum 𝐋 is described by the Hamiltonian 𝜇0 ℏ𝛾𝐼𝜇B
𝐻IL = 24𝜋
𝜌3
(P5.1)
𝐋⋅𝐈
where 𝐋 is calculated with respect to the nuclear position (i.e., 𝐋 = ×p, whereas L = R×p is calculated with respect to the metal nucleus) [55].
27
zk I r Me
R
e
=Rr
p
yj
xi Therefore, 𝜇0ℏ𝛾𝐼𝜇B𝛒 × 𝐩
𝐻IL = 2 ∂𝑓
∂𝑓
4𝜋
𝜌3
𝜇0ℏ𝛾𝐼𝜇B 1 ×𝐩⋅𝐈 4𝜋 𝜌
(P5.2)
⋅𝐈=2
∂𝑓
1
(𝑓 = ∂𝑥𝐢 + ∂𝑦𝐣 + ∂𝑧𝐤) where the bar indicates the integral over all electron positions. To first order in R/r, 𝜌 1
≈𝑟+
𝐫·𝐑 𝑟3
and thus 𝐋 𝜌
1
3
𝐩
1
= 𝜌 × 𝐩 = × 𝜌 ‒ 𝜌 × 𝐩 ≈ ×
(
1 𝐫·𝐑 + 3 𝑟 𝑟
)𝐩 = ×
𝐫·𝐑 𝑟3
𝐩
(P5.3)
1
(because 𝐩 × 𝑟 = 0 for a localized, divergenceless electron current distribution, and × 𝐩 = 0). Since 1
𝐫 × 𝐑 × 𝐩 = (𝐫·𝐩)𝐑 ‒ (𝐫·𝐑)𝐩, we get (𝐫·𝐑)𝐩 =‒ 2𝐫 × 𝐑 × 𝐩 [57] and thus 𝐋
1
𝜌3
=‒ 2 ×
𝐫×𝐑×𝐩 𝑟3
1
=‒ 2 ×
𝐫×𝐋 𝑟3
1
= 2 ×
𝐋×𝐫 𝑟3
(P5.4)
Since ×
𝐋×𝐫 3
𝑟
=‒
𝐋 3
𝑟
𝐋·𝐫
(P5.5)
+ 3 5𝐫 𝑟
(because L is independent of x, y, z) Eq. (76) is obtained from Eq. (P5.2). -----------------------------------------------------------------------------------------------------------------------
According to the standard quantum chemistry definition, d2𝐸
𝐀hfc = ∑𝑖d𝐬𝑖d𝐈
|
,
(77)
𝑠𝑖 = 𝐼 = 0
where the total hyperfine coupling tensor is obtained from the electron energy E as a sum of contributions from each electron. The electron-nucleus dipole-dipole interaction is defined to include the electron spin only, thus not comprising the interaction between the electron orbital magnetic moment and the nuclear magnetic moment described by Eq. (76), so that 𝜇0 ℏ𝛾𝐼𝜇B
𝐀hfc = 4𝜋
3
𝑟
(
𝑔e
3𝐫𝐫𝑇 2
𝑟
)
𝜇0
𝜌𝐼
‒ 𝟏 + 3 ℏ𝛾𝐼𝜇B𝑔e 𝑆 𝟏
As a result, to second order the energy corrections to the ligand-field energy are 28
(78)
Zeeman (0) + 𝐻hfc + 𝐻IL + 𝐻ss|𝜓(0) 𝐸(1) 𝑖 = < 𝜓 𝑖 |𝐻𝐿𝑆 + 𝐻 𝑖 >
(79)
Zeeman (0) + 𝐻hfc + 𝐻IL + 𝐻ss|𝜓(1) 𝐸(2) 𝑖 = < 𝜓 𝑖 |𝐻𝐿𝑆 + 𝐻 𝑖 >
(80)
and
where |𝜓(1) 𝑖 >
=∑
Zeeman < 𝜓(0) + 𝐻hfc + 𝐻IL + 𝐻ss|𝜓(0) 𝑗 |𝐻𝐿𝑆 + 𝐻 𝑖 >
𝑗≠𝑖
(0) 𝐸(0) 𝑖 ‒𝐸 𝑗
|𝜓(0) 𝑗 >
(81)
For orbitally non-degenerate ground states, an effective Hamiltonian can be written (see Panel 6) comprising the terms 𝐻' = 𝜇B𝐁0 ∙ 𝐠 ∙ 𝐒 + 𝐒 ⋅ 𝐀 ⋅ 𝐈 + 𝐒 ⋅ 𝐃 ⋅ 𝐒
(82)
with 𝐀 defined as in Eq. (74). Note that the tensor 𝐃 contains contributions in addition to those reported in Eq. (18). Remarkably, it should also be noted that the 𝐠 tensor is not strictly symmetric, due to the contributions from further cross-terms and higher-order perturbation terms.
----------------------------------------------------------------------------------------------------------------------Panel 6 (0) From Eqs. (79)-(81), for orbitally non-degenerate ground states ( < 𝜓(0) 0 |𝐋|𝜓 0 > = 0), (0) hfc ⋅ 𝐈 + 𝐒 ⋅ ∆𝐬𝐬 ⋅ 𝐒|𝜓(0) 𝐸(1) 0 = < 𝜓 0 |𝜇B𝑔e𝐁0 ∙ 𝐒 + 𝐒 ⋅ 𝐀 0 >
𝐸(2) 0 = ∑𝑗 ≠ 0
Zeeman < 𝜓(0) + 𝐻hfc + 𝐻IL + 𝐻ss|𝜓(0) 𝑗 |𝐻𝐿𝑆 + 𝐻 0 > (0) 𝐸(0) 0 ‒𝐸 𝑗
(P6.1)
Zeeman < 𝜓(0) + 𝐻hfc + 𝐻IL + 𝐻ss|𝜓(0) 0 |𝐻𝐿𝑆 + 𝐻 𝑗 > (P6.2)
Since the spin-orbit coupling energy is usually orders of magnitude larger than the other terms, Eq. (P6.2) can be approximated by neglecting all terms non involving 𝐻𝐿𝑆
∑
2 (0) | < 𝜓(0) 0 |𝜆so𝐋 ∙ 𝐒|𝜓 𝑗 > |
𝐸(2) 0 𝑗≠0
∑
+2 (0) 𝐸(0) 0 ‒𝐸 𝑗 𝑗≠0 (0) (0) (0) Zeeman < 𝜓 𝑗 |𝐻𝐿𝑆|𝜓 0 > ( < 𝜓 0 |𝐻 + 𝐻hfc + 𝐻IL + 𝐻ss|𝜓(0) 𝑗 >) (0) 𝐸(0) 0 ‒𝐸 𝑗
(P6.3) where the first term corresponds to the term on the right of Eq. (13), the first product in the second term is (see Eqs. (19) and (23))
29
∑
(0) (0) (0) < 𝜓(0) 𝑗 |𝜆so𝐋 ∙ 𝐒|𝜓 0 >< 𝜓 0 |𝜇B𝐁0 ∙ (𝐋 + 𝑔e𝐒)|𝜓 𝑗 >
𝑗≠0
∙
∑
(0) 𝐸(0) 0 ‒𝐸 𝑗 (0) (0) (0) < 𝜓(0) 𝑗 |𝐋|𝜓 0 >< 𝜓 0 |𝐋|𝜓 𝑗 > (0) 𝐸(0) 0 ‒𝐸 𝑗
𝑗≠0
= 𝜆so𝜇B𝐁0 (P6.4)
∙
< 𝑚𝑆|𝐒|𝑚'𝑆 >
and the third product in the second term is (using the definition of g and 𝚲 in Eqs. (21) and (15), respectively)
|
(0) (0) < 𝜓(0) 𝑗 |𝜆so𝐋 ∙ 𝐒|𝜓 0 >< 𝜓 0
∑
[
∙ ∑𝑗 ≠ 0 =
1𝜇0ℏ𝛾𝐼𝜇B 4𝜋𝑟3
4𝜋
(3
𝐋·𝐫 𝑇 𝐋⋅𝐈 𝐫 ⋅𝐈‒ 3 𝑟5 𝑟
)|𝜓
(0) 𝑗 >
(0) 𝐸(0) 0 ‒𝐸 𝑗
𝑗≠0
𝜇0ℏ𝛾𝐼𝜇B = 𝜆so 4𝜋 < 𝑚𝑆|𝐒|𝑚'𝑆 >
=2
𝜇0ℏ𝛾𝐼𝜇B
(0) (0) (0) < 𝜓(0) 𝑗 |𝐋|𝜓 0 >< 𝜓 0 |𝐋|𝜓 𝑗 > (0) 𝐸(0) 0 ‒𝐸 𝑗
𝜇0ℏ𝛾𝐼𝜇B 4𝜋𝑟3
·𝐫
3𝐫𝑇 𝑟5
[
< 𝑚𝑆|𝐒|𝑚'𝑆 > ∙ 𝜆so𝚲·
[
< 𝑚𝑆|𝐒|𝑚'𝑆 > ∙ (𝐠 ‒ 𝑔e𝟏)·
3𝐫𝐫𝑇 𝑟2
1
(0) (0) (0) < 𝜓(0) 𝑗 |𝐋|𝜓 0 >< 𝜓 0 |𝐋|𝜓 𝑗 >
𝑟
(0) 𝐸(0) 0 ‒𝐸 𝑗
‒ 3∑𝑗 ≠ 0
3𝐫𝐫𝑇 𝑟2
]⋅𝐈
]
‒ 𝜆so𝚲 ⋅ 𝐈
]
(P6.5)
‒ (𝐠 ‒ 𝑔e𝟏) ⋅ 𝐈
Therefore, from Eqs. (P6.1)-(P6.5) 𝜇0ℏ𝛾𝐼𝜇B
2 𝐻' = 𝜇B𝑔e𝐁0 ∙ 𝐒 + 𝐒 ⋅ 𝐀hfc ⋅ 𝐈 + 𝐒 ⋅ ∆𝐬𝐬 ⋅ 𝐒 + 𝐒 ⋅ 𝜆so 𝚲 ⋅ 𝐒 + 2𝜆so𝜇B𝐁0 ⋅ 𝚲 ⋅ 𝐒+2𝜆so𝐒 ∙ 𝐀dip,so ⋅ 𝐈+
𝑔e𝟏)·
[
3𝐫𝐫𝑇 𝑟2
]
4𝜋𝑟3
𝐒 ∙ (𝐠 ‒ (P6.6)
‒ 𝟏 ⋅ 𝐈 + 𝜆so𝐒 ⋅ (𝚲' + 𝚲'') ⋅ 𝐒
where 𝐀dip,so, 𝚲' and 𝚲'' can be calculated from the evaluation of L over the different eigenstates [27,55]. Eq. (P6.6) can be rearranged in the form (see Eq. (78)) 𝐻' = 𝜇B𝐁0 ∙ (𝑔e𝟏 + 2𝜆so𝚲) ∙ 𝐒 + 𝐒 𝜇0 ℏ𝛾𝐼𝜇B 3𝐫𝐫𝑇 𝜇0 𝜌𝐼 𝜇0 ℏ𝛾𝐼𝜇B 3𝐫𝑇 ( ) 𝑔 ‒ 𝟏 ℏ𝛾 𝟏 ‒ 𝟏 + 2𝜆so𝐀dip,so ⋅ 𝐈 ⋅ + 𝜇 𝑔 + 𝐠 ‒ 𝑔 𝟏 · 𝐫 e 4𝜋 𝑟3 e 𝑟2 3 𝐼 B e𝑆 4𝜋 𝑟3 𝑟5
{[
(
]
)
[
}
]
2 + 𝐒 ⋅ (𝜆so 𝚲 + ∆𝐬𝐬 + 𝜆so(𝚲' + 𝚲'')) ⋅ 𝐒
(P6.7) which can be rewritten as in Eq. (82), being the contribution from 𝐀dip,so much smaller than the other terms in A. On passing from Eq. (P6.2) to Eq. (P6.3) we have neglected the term
∑ 𝑗≠0
Zeeman| (0) < 𝜓(0) 𝜓0 > 𝑗 |𝐻 (0) 𝐸(0) 0 ‒𝐸 𝑗
Zeeman| (0) < 𝜓(0) 𝜓 𝑗 >= 𝜇B2 0 |𝐻
∑
(0) < 𝜓(0) 𝑗 |𝐁0 ∙ (𝐋 + 𝑔e𝐒)|𝜓 0 > (0) 𝐸(0) 0 ‒𝐸 𝑗
𝑗≠0
(0) < 𝜓(0) 0 |𝐁0 ∙ (𝐋 + 𝑔e𝐒)|𝜓 𝑗 >
(P6.8) which, using Eq. (P3.4), becomes 𝜇B2𝐁0 ∙ ∑𝑗 ≠ 0
(0) (0) (0) < 𝜓(0) 𝑗 |𝐋|𝜓 0 >< 𝜓 0 |𝐋|𝜓 𝑗 > (0) 𝐸(0) 0 ‒𝐸 𝑗
∙ 𝐁0 = 𝜇B2𝐁0 ∙ 𝚲 ∙ 𝐁0
(P6.9)
This term provides the TIP contribution to the magnetic susceptibility tensor (Eq. (48)), as can be checked using Eq. (53). This is also called the Van Vleck orbital paramagnetism, and can provide non-negligible contributions when the energy of the excited states is not very high. -----------------------------------------------------------------------------------------------------------------------
30
2E-19
1000
2E-20
100
2E-21
10
2E-22
1
2E-23
0.1
2E-24
0.01
2E-25
0.001
2E-26
1E-4
2E-27
1E-5
Energy / J
Energy / cm1
10000
2E-28
Crystal-field Spin-orbit coupling
Zeeman Proton-electron dipole-dipole
Figure 3. Typical orders of magnitude of the energy values of crystal-field, spin-orbit coupling, electron Zeeman (at 20 T) and proton-electron point-dipole point-dipole (r = 3 Å) interactions. 5.3 Hyperfine shift from first-principles quantum chemistry Hyperfine shift expressions analoguous to Eqs. (63) and (68) can also be obtained from the quantum chemistry definition of the isotropic hyperfine shift [33,58] 𝛿hf =‒ 𝛔hf =
d2〈𝐸para〉 d𝐁0d𝛍𝐼
|
Tr(𝛔hf)
(83)
3
(84) 𝐵0 = 𝜇 𝐼 = 0
so that d2
[
𝛔hf = d𝐁0d𝛍𝐼
[
]
∑ 𝐸𝑖exp ‒ 𝐸el,𝑖 (𝑘𝑇) 𝑖
[
]
∑ exp ‒ 𝐸el,𝑖 (𝑘𝑇) 𝑖
]|
(85) 𝐵0 = 𝜇 𝐼 = 0
and, to first order in 𝐸𝑖 (𝑘𝑇) (see Panel 7) in the spin Hamiltonian regime, with 𝐻(1) =‒ 𝐁0𝑇 ∙ 𝛍el + 𝑇 𝐻hfc, 𝐻(1) el =‒ 𝐁0 ∙ 𝛍el,
31
⟨ │ │𝜓 ⟩|
∑ 𝑄𝑗𝑖 𝜓𝑖 𝑖𝑗 1 𝛔hf =‒ 𝑘𝑇
∂𝐻(1) ∂𝐁0
⟨𝜓 │ │𝜓 ⟩|
𝑗
𝑗
𝐵0 = 𝜇 𝐼 = 0
∂𝐻(1) ∂𝛍𝐼
(
𝑖
𝐵0 = 𝜇 𝐼 = 0
(86)
)
∑ exp ‒ 𝐸(0) 𝑘𝑇 𝑖 𝑖
which provides [37,59] 𝜇B
𝛔hf =‒ ℏ𝛾𝐼𝑘𝑇𝐠 ⋅ ⟨𝐒𝐒𝑇⟩ ⋅ 𝐀
(87)
using the definition of g as d2𝐸
1
𝐠 = 𝜇B∑𝑖d𝐬𝑖d𝐁𝟎
|
,
(88)
𝑠𝑖 = 𝐵0 = 0
and of 𝐀 as reported in Eq. (77). If the contribution from spin-orbit coupling is neglected, the tensor 𝐀 is equal to 𝐀hfc. When spin-orbit coupling and the interaction between the nuclear magnetic moment and the orbital magnetic moment are considered, the tensor 𝐀 can be derived in the semi-empirical quantum mechanical framework as shown in Section 5.2 (Eq. 74)). The same expression for the dipolar shielding tensor is actually obtained from Eq. (84) using Eqs. (61) and (34) 𝜇0 𝜇B2
𝛔dip =‒ 4𝜋𝑟3𝑘𝑇𝐠 ∙ 〈𝐒𝐒𝑇〉 ∙ 𝐠𝑇 ⋅
(
3𝐫𝐫𝑇 𝑟2
‒𝟏
)
(89)
----------------------------------------------------------------------------------------------------------------------Panel 7 From Eq. (85)
[
(
|⟨𝜓𝑖│𝐻(1)│𝜓𝑗⟩|
∑ ⟨𝜓𝑖│𝐻(1)│𝜓𝑖⟩ + ∑ 𝑖 𝑗≠𝑖
d2 𝛔hf = d𝐁 d𝛍 0 𝐼
2
‒ ⟨𝜓𝑖│𝐻(1)│𝜓𝑖⟩
(0) (0) 𝐸 𝑖 ‒𝐸 𝑗
[
⟨𝜓𝑖│𝐻(1) el │𝜓𝑖⟩ 𝑘𝑇
)
[
(0) exp ‒ 𝐸 𝑖 (𝑘𝑇)
]
∑ exp ‒ 𝐸(0) (𝑘𝑇) 𝑖 𝑖
]
]|
.
𝐵0 = 𝜇 𝐼 = 0
(P7.1) 2 (1)
∂ 𝐻 0∂𝛍𝐼
The term ∂𝐁
does not provide “paramagnetic” contributions (it provides a contribution approximated with
𝛔dia, if also the nuclear Zeeman term ‒ 𝛾𝐼𝐁0 ∙ (𝟏 ‒ 𝛔orb) ∙ 𝐈 is included in 𝐻(1)). Since ∑ 1 𝛔hf =‒ 𝑘𝑇
(⟨ │ │ ⟩⟨ │
𝑖
𝜓𝑖
∂𝐻(1) el ∂𝐁0
𝜓𝑖 𝜓𝑖
│𝜓 ⟩ ‒ 2∑
∂𝐻(1) ∂𝛍𝐼
𝑖
𝑘𝑇
∂𝐻(1) el ∂𝛍𝐼
⟨𝜓 │ │𝜓 ⟩⟨𝜓 │ │𝜓 ⟩)exp[ ‒ 𝐸(0)𝑖 (𝑘𝑇)]
(0) 𝑗 ≠ 𝑖 (0) 𝐸 𝑗 ‒𝐸 𝑖
𝑖
[
∂𝐻(1) ∂𝐁0
]
∑ exp ‒ 𝐸(0) (𝑘𝑇) 𝑖 𝑖
32
𝑗
𝑗
∂𝐻(1) ∂𝛍𝐼
= 0,
𝑖
(P7.2)
∂𝐻(1) el
and Eq. (86) is obtained ( ∂𝐁 = 0
∂𝐻(1) ∂𝐁0
).
-----------------------------------------------------------------------------------------------------------------------
In the first-principles quantum chemistry approach, 𝐀 in Eq. (87) is 𝐀 = 𝐀hfc + 𝐀so
(90)
where 𝐀hfc is provided by Eq. (78) and 𝐀so is the cross-product between the spin-coupling operator of Eq. (10) and 𝐻pso (corresponding to the term of Eq. (P6.5) in the semi-empirical approach):
⟨𝜓 │ │𝜓 ⟩⟨𝜓 │∑ 𝑗
𝐀so = ∑𝑗0
∂𝐻pso ∂𝐈
0
0
│𝜓 ⟩ + ⟨𝜓 │∑
∂𝐻𝑆𝑂𝐶 𝑖 ∂𝐬𝑖
𝑗
𝑗
(0) 𝐸(0) 0 ‒𝐸 𝑗
where 𝐻pso has the form 𝜇0
│𝜓 ⟩⟨𝜓 │ │𝜓 ⟩
∂𝐻𝑆𝑂𝐶 𝑖 ∂𝐬𝑖
𝐥𝑖𝐼
𝐻pso = 24𝜋ℏ𝛾𝐼𝜇B∑𝑖𝑟 3 ⋅ 𝐈 ,
0
0
∂𝐻pso ∂𝐈
𝑗
(91)
(92)
𝑖𝐼
and 𝐥𝑖𝐼 is the orbital angular momentum of the ith electron relative to the nucleus. Eqs. (87) and (90) should thus allow for the calculation of pseudocontact shifts in systems where only the ground state multiplet is populated. Surprisingly, calculations performed for cobalt(II) systems provided much smaller values than those obtained from Eqs. (63) and (47), which are also valid when only the ground state multiplet is populated [40]. The (much) smaller contribution generally expected for 𝐀so than for 𝐀hfc in Eq. (90) leads to the assumption that the spin-orbital contribution (𝐀so) to the pseudocontact shift can be neglected altogether [31,60,37,61–64]. In this way, the pseudocontact shifts should be calculated to a good approximation using Eqs. (83), (87) and (78). However, the values so calculated are significantly different from those calculated in the semi-empirical framework, where Eqs. (87) 𝑇
and (74) are used. The two models differ by a factor 𝐠 𝑔e, which can be even larger than a factor 2. The first-principles framework for the analysis of the pseudocontact shifts has been recently questioned by a careful comparison of the experimental 𝐠 values of some copper(II) proteins with those derived from the 𝛘 anisotropy tensor obtained from the PCSs measured on the same systems [41]. 33
The different equations for the hyperfine shifts developed within the semi-empirical and the firstprinciples quantum chemistry frameworks are summarized in Table 1. It can be noted that the Fermi contact shift coincides in the two approaches, whereas the pseudocontact shift in the semi-empirical approach, being determined by the 𝛘 tensor, contains contributions from the orbital magnetic moment which are not accounted for by the first-principles quantum chemistry treatment that is limited to the leading-order terms. This results in an “asymmetry” between the contact and pseudocontact terms which is inherent to the different types of interactions which are accounted for by the two contributions (Fig. 4). In the following sections, the semi-empirical model is thus used to describe the pseudocontact shift, as it allows for all contributions to the magnetic susceptibility to be considered.
Paramagnetic susceptibility: 𝛘 𝛘' 𝛘'' 𝛘𝐓𝐈𝐏 Pseudocontact shift 𝛘' 𝛘'' 𝛘𝐓𝐈𝐏
Contact shift
(semi-empirical approach)
𝛘' 𝛘''
(Kurland-McGarvey)
𝛘'
(first principles quantum chemistry)
𝛘'
(semi-empirical approach, Kurland-McGarvey)
𝛘'
(first principles quantum chemistry)
Figure 4. In the semi-empirical approach the pseudocontact shift is proportional to the whole paramagnetic susceptibility (𝛘), whereas in the first-principles quantum chemistry approach it is proportional only to the part of the magnetic susceptibility which is related to the nuclear spin-electron spin hyperfine coupling (𝛘'), thus neglecting contributions from the orbital hyperfine (paramagnetic nuclear spin-electron orbit) coupling (𝛘''). In the Kurland-McGarvey equation [28], the TIP contribution was not considered.
Table 1. Summary of the equations for the hyperfine shift in the semi-empirical and first-principles quantum chemistry approaches Semi-empirical approach 34
hf
𝛿 =
[ (
1
3𝐫𝐫𝑇
Tr 𝛘 ⋅ 12𝜋𝑟3
𝑟2
)]
𝐴Fc ‒1 ‒𝟏 + Tr[𝛘 ∙ [𝐠𝑇] ] 3𝜇0ℏ𝛾𝐼𝜇B
For singly-populated orbitally non-degenerate ground state spin multiplets 𝛘=
𝜇0𝜇B2 𝑘𝑇
𝐠 ∙ 〈𝐒𝐒𝑇〉 ∙ 𝐠𝑇
and thus 𝛿 =
𝜇0𝜇B2
1
hf
12𝜋𝑟3 𝑘𝑇
[
𝑇〉
Tr 𝐠 ∙ 〈𝐒𝐒
𝑇
∙𝐠 ⋅
(
3𝐫𝐫𝑇 𝑟2
)]
‒𝟏 +
𝜇B𝐴Fc
Tr[𝐠 ∙ 〈𝐒𝐒𝑇〉] 3ℏ𝛾𝐼𝑘𝑇
First-principles quantum chemistry approach 𝛿hf =‒
𝜇B
Tr[𝐠 ⋅ ⟨𝐒𝐒𝑇⟩ ⋅ (𝐀hfc + 𝐀so)]
3ℏ𝛾𝐼𝑘𝑇
For singly-populated orbitally non-degenerate ground state spin multiplets, with 𝐀so ≪ 𝐀hfc hfc
𝐀
=
𝜇0 ℏ𝛾𝐼𝜇B 4𝜋 𝑟3
(
𝑔e
3𝐫𝐫𝑇 𝑟2
)
‒ 𝟏 + 𝐴Fc
and thus hf
𝛿 =
1
𝜇0𝜇B2
12𝜋𝑟3 𝑘𝑇
[
𝑇〉
Tr 𝐠 ∙ 〈𝐒𝐒
(
∙ 𝑔e
3𝐫𝐫𝑇 𝑟2
)]
‒𝟏 +
𝜇B𝐴Fc
Tr[𝐠 ∙ 〈𝐒𝐒𝑇〉] 3ℏ𝛾𝐼𝑘𝑇
5.4 The limiting case of D<
𝛔hf =‒ ℏ𝛾𝐼𝑘𝑇𝐠 ⋅
[𝑆(𝑆3+ 1)𝟏 ‒ 𝑆(𝑆 + 1)(2𝑆30𝑘𝑇‒ 1)(2𝑆 + 3)𝐃] ⋅ [4𝜋
𝜇0 ℏ𝛾𝐼𝜇B 𝑟3
𝐠𝑇 ⋅
(
3𝐫𝐫𝑇 𝑟2
)
𝜇0
𝜌𝐼
‒ 𝟏 + 3 ℏ𝛾𝐼𝜇B𝑔e 𝑆 𝟏
]
(93) so that (assuming that the 𝐠 and 𝐃 tensors are parallel) 𝛿Fc =
𝑆(𝑆 + 1)𝐴Fc 𝜇B ∑ 9 ℏ 𝛾𝐼𝑘𝑇 𝒊𝒊
1)(2𝑆 + 3) 𝑔𝑖𝑖𝐷𝑖𝑖] [𝑔𝑖𝑖 ‒ (2𝑆 ‒ 10𝑘𝑇
and using Eq. (38)
35
(ii = xx, yy, zz)
(94)
𝛿Fc 𝑆(𝑆 + 1)𝐴Fc 𝜇B = 9 ℏ 𝛾𝐼𝑘𝑇
[
]
(2𝑆 ‒ 1)(2𝑆 + 3) 1 1 2 (𝑔𝑥𝑥( ‒ 𝐷 + 𝐸) + 𝑔𝑦𝑦( ‒ 𝐷 ‒ 𝐸) + 𝑔𝑧𝑧( 𝐷)) 10𝑘𝑇 3 3 3 Fc 𝜇 𝑔 + 𝑔 + 𝑔 ( ) 𝑆(𝑆 + 1)𝐴 (2𝑆 ‒ 1)(2𝑆 + 3)𝐷 2𝑔𝑧𝑧 ‒ 𝑔𝑥𝑥 ‒ 𝑔𝑦𝑦 + 3𝐸(𝑔𝑥𝑥 ‒ 𝑔𝑦𝑦) B 𝑥𝑥 𝑦𝑦 𝑧𝑧 = 1‒ 3 ℏ 𝛾𝐼𝑘𝑇 3 30𝑘𝑇 𝑔𝑥𝑥 + 𝑔𝑦𝑦 + 𝑔𝑧𝑧 (𝑔𝑥𝑥 + 𝑔𝑦𝑦 + 𝑔𝑧𝑧) ‒
[
]
(95) and (see Panel 8) 𝛿pc
{
[
2 2 𝑔𝑥𝑥 1𝜇0𝑆(𝑆 + 1) 𝜇B (2𝑆 ‒ 1)(2𝑆 + 3)2 2 2 (3cos 𝜃 ‒ 1) 𝑔𝑧𝑧 1 ‒ = 𝐷 ‒ 9 4𝜋 10𝑘𝑇 2 3 𝑘𝑇𝑟3
(
) 𝑔 1)(2𝑆 + 3) 1 1)(2𝑆 + 3) 1 ( ‒ 3𝐷 + 𝐸)) ‒ 2 (1 ‒ (2𝑆 ‒ 10𝑘𝑇 ( ‒ 3𝐷 ‒ 𝐸))] + 32sin (1 ‒ (2𝑆 ‒ 10𝑘𝑇 (2𝑆 ‒ 1)(2𝑆 + 3) 1 (2𝑆 ‒ 1)(2𝑆 + 3) 1 𝜃cos2𝜙[𝑔 (1 ‒ ‒ 𝐷 + 𝐸)) ‒ 𝑔 (1 ‒ ( ( ‒ 3𝐷 ‒ 𝐸 10𝑘𝑇 3 10𝑘𝑇 ))]} 2 𝑦𝑦
2 𝑥𝑥
2
2 𝑦𝑦
(96)
----------------------------------------------------------------------------------------------------------------------Panel 8 From Eq. (93) 𝛿pc
[(
2 𝑔𝑥𝑥 1𝜇0𝑆(𝑆 + 1) 𝜇B 0 = Tr 9 4𝜋 𝑘𝑇𝑟5 0
0 0 (3𝑥2 ‒ 𝑟2)𝑔𝑥𝑥 𝑔𝑦𝑦 0 3𝑥𝑦𝑔𝑦𝑦 0 𝑔𝑧𝑧 3𝑥𝑧𝑔𝑧𝑧
(
(3𝑥2 ‒ 𝑟2)𝑔𝑥𝑥 3𝑥𝑦𝑔𝑦𝑦 3𝑥𝑧𝑔𝑧𝑧
1 ‒ 𝐷+𝐸 3 0 0
0
0
1 ‒ 𝐷‒𝐸 3
0
0
2 𝐷 3
)(
)(
3𝑥𝑦𝑔𝑥𝑥
(3𝑦2 ‒ 𝑟2)𝑔𝑦𝑦 3𝑦𝑧𝑔𝑧𝑧
)
3𝑥𝑧𝑔𝑥𝑥 (2𝑆 ‒ 1)(2𝑆 + 3) 𝑔𝑥𝑥 0 3𝑦𝑧𝑔𝑦𝑦 ‒ 10𝑘𝑇 0 (3𝑧2 ‒ 𝑟2)𝑔𝑧𝑧
3𝑥𝑦𝑔𝑥𝑥 3𝑥𝑧𝑔𝑥𝑥 (3𝑦2 ‒ 𝑟2)𝑔𝑦𝑦 3𝑦𝑧𝑔𝑦𝑦 3𝑦𝑧𝑔𝑧𝑧 (3𝑧2 ‒ 𝑟2)𝑔𝑧𝑧
(
]
)
(P8.1) and thus
36
0 0 𝑔𝑦𝑦 0 0 𝑔𝑧𝑧
)
𝛿pc
[
(
)
2 (3𝑥2 ‒ 𝑟2)𝑔𝑥𝑥 2 1𝜇0𝑆(𝑆 + 1) 𝜇B 2 3𝑥𝑦𝑔𝑦𝑦 = Tr 9 4𝜋 𝑘𝑇𝑟5 2 3𝑥𝑧𝑔𝑧𝑧
(
(
)
1 2 ‒ 𝐷 + 𝐸 (3𝑥2 ‒ 𝑟2)𝑔𝑥𝑥 3 1 2 ‒ 𝐷 ‒ 𝐸 3𝑥𝑦𝑔𝑦𝑦 3
(
)
2 2 3𝑥𝑦𝑔𝑥𝑥 3𝑥𝑧𝑔𝑥𝑥 (2𝑆 ‒ 1)(2𝑆 + 3) 2 2 2 2 (3𝑦 ‒ 𝑟 )𝑔𝑦𝑦 3𝑦𝑧𝑔𝑦𝑦 ‒ 10𝑘𝑇 2 2 3𝑦𝑧𝑔𝑧𝑧 (3𝑧2 ‒ 𝑟2)𝑔𝑧𝑧
(
(
)
1 2 ‒ 𝐷 + 𝐸 3𝑥𝑦𝑔𝑥𝑥 3 1 2 ‒ 𝐷 ‒ 𝐸 (3𝑦2 ‒ 𝑟2)𝑔𝑦𝑦 3
)
2 2𝐷𝑥𝑧𝑔𝑧𝑧
( (
2 2𝐷𝑦𝑧𝑔𝑧𝑧
) )
1 2 ‒ 𝐷 + 𝐸 3𝑥𝑧𝑔𝑥𝑥 3 1 2 ‒ 𝐷 ‒ 𝐸 3𝑦𝑧𝑔𝑦𝑦 3 2 2 𝐷(3𝑧2 ‒ 𝑟2)𝑔𝑧𝑧 3
)] (P8.2)
so that 𝛿pc =
((
2 1𝜇0𝑆(𝑆 + 1) 𝜇B (2𝑆 ‒ 1)(2𝑆 + 3) 2 2 2 (3𝑥2 ‒ 𝑟2)𝑔𝑥𝑥 + (3𝑦2 ‒ 𝑟2)𝑔𝑦𝑦 + (3𝑧2 ‒ 𝑟2)𝑔𝑧𝑧 ‒ 5 9 4𝜋 10𝑘𝑇 𝑘𝑇𝑟 1 1 2 2 2 2 ‒ 𝐷 + 𝐸 (3𝑥2 ‒ 𝑟2)𝑔𝑥𝑥 + ‒ 𝐷 ‒ 𝐸 (3𝑦2 ‒ 𝑟2)𝑔𝑦𝑦 + 𝐷(3𝑧2 ‒ 𝑟2)𝑔𝑧𝑧 3 3 3
[
)
(
)]
)
(P8.3)
and thus, in spherical coordinates, 𝛿pc 2 1𝜇0𝑆(𝑆 + 1) 𝜇B (2𝑆 ‒ 1)(2𝑆 + 3) 1 2 (3sin2𝜃cos2𝜙 ‒ 1)𝑔𝑥𝑥 = 1‒ ‒ 𝐷+𝐸 3 9 4𝜋 10𝑘𝑇 3 𝑘𝑇𝑟
2 + (3sin2𝜃sin2𝜙 ‒ 1)𝑔𝑦𝑦
(
)]
[
1‒
(
(
))
(2𝑆 ‒ 1)(2𝑆 + 3) 1 ‒ 𝐷‒𝐸 10𝑘𝑇 3
(
(
2 + (3cos2𝜃 ‒ 1)𝑔𝑧𝑧 1‒
))
(2𝑆 ‒ 1)(2𝑆 + 3)2 𝐷 10𝑘𝑇 3
(P8.4) Eq. (96) is then obtained using the relationships 3
1
3sin2𝜃cos2𝜙 ‒ 1 = 2sin2𝜃cos2𝜙 ‒ 2(3cos2𝜃 ‒ 1);
(P8.5)
3 1 3sin2𝜃sin2𝜙 ‒ 1 =‒ sin2𝜃cos2𝜙 ‒ (3cos2𝜃 ‒ 1) 2 2 -----------------------------------------------------------------------------------------------------------------------
Note that even the measurement of a very large number of pseudocontact shifts would not allow the determination of the 𝐠 and 𝐃 tensors separately, because all data are related to the nuclear coordinates by a single anisotropic tensor, which is determined by the two anisotropy values Δ𝜒𝑎𝑥 and Δ𝜒𝑟ℎ and the three Euler angles that specify the orientation of this tensor with respect to the frame used to define the nuclear coordinates. Therefore, from any number (larger than 5) of pseudocontact shifts 37
values, Δ𝜒𝑎𝑥 and Δ𝜒𝑟ℎ can be determined, which are linked to the 5 parameters defining the 𝐠 and 𝐃 tensors (assuming that the TIP is negligible) by the following relationships: Δ𝜒𝑎𝑥 =
[
(
(2𝑆 ‒ 1)(2𝑆 + 3)2 2 𝑔𝑧𝑧 1‒ 3𝐷 10𝑘𝑇
2 𝑔𝑥𝑥
)‒ (
(
(2𝑆 ‒ 1)(2𝑆 + 3) 2 1‒ 10𝑘𝑇
2 𝑔𝑦𝑦
)) ‒ 2 (
1 ‒ 3𝐷 + 𝐸
(
(2𝑆 ‒ 1)(2𝑆 + 3) 1‒ 10𝑘𝑇
𝜇0𝜇B2𝑆(𝑆 + 1)
3𝑘𝑇 1 ‒ 3𝐷 ‒ 𝐸
))]
(97) Δ𝜒𝑟ℎ =
𝜇0𝜇B2𝑆(𝑆 + 1) 3𝑘𝑇
[𝑔 (1 ‒ 2 𝑥𝑥
(2𝑆 ‒ 1)(2𝑆 + 3) 10𝑘𝑇
1)(2𝑆 + 3) ( ‒ 13𝐷 + 𝐸)) ‒ 𝑔𝑦𝑦2 (1 ‒ (2𝑆 ‒ 10𝑘𝑇 ( ‒ 13𝐷 ‒ 𝐸))]
(98) in agreement with Eq. (48).
5.5 The case of paramagnetic lanthanoid(III) ions Also for paramagnetic lanthanoid ions, the pseudocontact shifts are usually analysed in the semiempirical framework using Eq. (63), i.e., according to their dependence on the paramagnetic susceptibility anisotropy [45]. For these ions, except for gadolinium(III), the relationship between pseudocontact shifts and EPR quantities (𝐠 and 𝐃) is even more complicated because the spin-orbit interaction couples 𝐋 and 𝐒 strongly, so that 𝐉 = 𝐋 + 𝐒 must be used to describe the total angular momentum, and the electron gfactor is provided by the Landé g-factor (see Table 2) 𝑔𝐽 = 1 +
𝐽(𝐽 + 1) ‒ 𝐿(𝐿 + 1) + 𝑆(𝑆 + 1) 2𝐽(𝐽 + 1)
(99)
and the Zeeman energy becomes (see Eqs. (22) and (23)) 𝐸Zeeman = 𝜇B𝑔𝐽𝐵0𝐽𝑧
Table 2. J quantum numbers and Landé g-factors for lanthanoid ions Ion
J
gJ 38
(100)
Ce3+
5/2
6/7
Pr3+
4
4/5
Nd3+
9/2
8/11
Pm3+
4
3/5
Sm3+
5/2
2/7
Eu3+(Sm2+)
0
–
Gd3+(Eu2+)
7/2
2
Tb3+
6
3/2
Dy3+
15/2
4/3
Ho3+
8
5/4
Er3+
15/2
6/5
Tm3+
6
7/6
Yb3+
7/2
8/7
For these ions, the crystal-field effects are smaller than the spin-orbit interactions. They can be accounted for by a tensor 𝐃 (with the same form as the ZFS tensor in paramagnetic transition metal ions), which removes the degeneracy of the electronic levels at zero magnetic field. As a consequence, Eq. (96) becomes (see Eq. (38)), 2 2 1 𝜇0𝐽(𝐽 + 1)(2𝐽 ‒ 1)(2𝐽 + 3) 𝜇B𝑔 𝐽 3 4𝜋 (𝑘𝑇)2𝑟
{(3cos 𝜃 ‒ 1)[𝐷
𝛿pc =‒ 90
2
𝑧𝑧 ‒
𝐷𝑥𝑥 + 𝐷𝑦𝑦 2
]+
3 2 [ ] 2sin 𝜃cos2𝜙 𝐷𝑥𝑥 ‒ 𝐷𝑦𝑦
}
(101) or, since 𝐷𝑧𝑧 ‒
𝐷𝑥𝑥 + 𝐷𝑦𝑦 2
3
= 2𝐷 (because 𝐷𝑥𝑥 + 𝐷𝑦𝑦 + 𝐷𝑧𝑧 = 0) [45], 𝑧𝑧
2 2 1 𝜇0𝐽(𝐽 + 1)(2𝐽 ‒ 1)(2𝐽 + 3) 𝜇B𝑔 𝐽 3 4𝜋 (𝑘𝑇)2𝑟
𝛿pc =‒ 60
{(3cos2𝜃 ‒ 1)𝐷𝑧𝑧 + sin2𝜃cos2𝜙[𝐷𝑥𝑥 ‒ 𝐷𝑦𝑦]} (102)
As for Eq. (96), this equation is valid in the limit that D, i.e., the splitting of the electronic levels at zero field, is smaller than kT. This condition is often not fulfilled. 39
On the other hand, from Eqs. (25), (59) and (67), the Fermi contact shifts depend only on the spin-dependent part of the electron magnetic moment, 𝐒C = < 𝑆𝑧 > 𝛋; since [44] (103)
𝑆𝑧 = (𝑔𝐽 ‒ 1)𝑚𝐽 for lanthanoid ions Eq. (70) becomes 𝐽(𝐽 + 1)𝐴Fc 𝜇B 3 ℏ 𝛾𝐼𝑘𝑇
𝛿Fc = 𝑔𝐽(𝑔𝐽 ‒ 1)
(104)
For a given ligand nucleus, 𝐴Fc is nearly constant, independently of the lanthanoid ion, whereas the magnitude of the 𝐃 tensor, that determines the pseudocontact shift, depends essentially only on the lanthanoid. This latter condition holds when the crystal field splitting is smaller than kT and much smaller than the spin-orbit coupling; if not, the magnitude and orientation of 𝐃 also depend on the nature of the ligands coordinated to the lanthanoid ion [65–69].
5.6 Beyond the point-dipole approximation Eqs. (1) and (63) for the dipolar and pseudocontact shifts were derived on the assumption that the unpaired electron(s) are localized on the paramagnetic atom, so that the point-dipole approximation can be used. This assumption may be acceptable when analysing the hyperfine shifts of nuclei far from the paramagnetic center: for instance, in a high-spin cobalt(II) system, considering an unpaired electron spin that is positioned at a single point provides good predictions of pseudocontact shifts for nuclei at distances farther than 8 Å from the metal [37]. However, the unpaired electron spin is in reality delocalised over the ligand atoms through molecular orbital overlap, so that dipole-dipole interactions occur between the nuclear magnetic moment and the distributed electron magnetic moment. This corresponds to integrating Eq. (62) over space after multiplication with the electron density function 𝜌(𝐫e) [70,71]: 1
(
𝛔dip =‒ 4𝜋∫ 𝜌(𝐫e)𝛘(𝐫e) ⋅
40
(
3(𝐫e ‒ 𝐫)(𝐫e ‒ 𝐫)𝑇
| 𝐫e ‒ 𝐫|
5
‒|
𝟏
))d 𝐫
𝐫e ‒ 𝐫| 3
3
e
(105)
where 𝐫e is the integration variable and 𝐫 indicates the position of the nucleus detected by NMR in a frame centered on the nucleus of the paramagnetic metal. It has been shown [70] that this integral provides the following expression for the tensor elements: σdip 𝑖𝑗 (𝐫) =
1
∑
2
2
δ𝑖𝑗 ( ) + 𝜒 (𝐫)𝜌(𝐫) 𝑘𝑗 3 𝜒𝑖𝑗(𝐫)𝜌(𝐫) 𝑘𝑖𝑘
(106)
If it is assumed that the 𝛘 tensor is the same for all positions, this can be simplified to provide the following expression for the pseudocontact shift [70]:
(
1 𝑇·𝛘·
𝛿pc(𝐫) =‒ 3
𝑇
·
1
)
‒ 3Tr(𝛘) 𝜌(𝐫)
(107)
These relationships allow one to determine the electron density distribution from the experimental pseudocontact shifts and the atomic coordinates of the molecule [70,71], provided one fulfills some regularization conditions that are needed to take into account the ill-posed nature of the problem.
6. Paramagnetism-induced self-orientation As anticipated in Section 2, the anisotropy of the magnetic susceptibility causes partial selforientation of a paramagnetic molecule. From Eqs. (24) and (46), the field dependence of the energy of the thermally averaged total electron magnetic moment can be expressed as 𝐸𝑎𝑛 =‒ ∫〈𝛍el〉·d𝐁𝟎 =‒
𝐁0 ∙ 𝛘 ∙ 𝐁0 2𝜇0
(108)
In the presence of non-negligible diamagnetic contributions (𝛘𝐝𝐢𝐚) to the overall susceptibility of the molecule, the diamagnetic contribution adds to the paramagnetic one so that 𝐸𝑎𝑛 = ‒
𝐁0 ∙ 𝛘𝐦𝐨𝐥 ∙ 𝐁0 2𝜇0
(109)
with 𝛘𝐦𝐨𝐥 = 𝛘 + 𝛘𝐝𝐢𝐚 (see Section 4). This implies that in the presence of an anisotropic 𝛘𝐦𝐨𝐥 the energy of the molecule changes as a function of its orientation with respect to the magnetic field. Of course, orientations with lower energy will be favored with respect to orientations with higher energy, and thus a partial alignment of the molecular ensemble occurs. To evaluate the extent of this partial alignment, and the effects on the NMR shifts, we can introduce an orientational tensor 𝐏 41
describing the probability that the magnetic field is oriented along the principal components of the 𝛘𝐦𝐨𝐥 tensor E an cos 2 i exp i sin i d i d i kT Pii Eian exp sin d d i i i kT
cos
2
(110)
B2
2 2 mol 2 2 0 ( iimol cos 2 i mol i exp jj sin i cos i kk sin i sin i ) d cos i d i 2 kT 0
B02
exp 2 kT ( 0
mol ii
2 2 mol 2 2 cos 2 i mol jj sin i cos i kk sin i sin i ) d cos i d i
where i and i are the spherical angles describing the orientation of the magnetic field with respect to the main axes i of the 𝛘 tensor (with diagonal components iimol , mol and kkmol ). The result of jj an integration to first order in Ei / kT is mol mol 1 2 B02 mol jj kk Pii 1 ii 3 15 0 kT 2
(111)
so that [72,73], Pii
1 B02 mol 1 iimol iso 3 50 kT
(112)
As expected, Pxx Pyy Pzz 1 . In summary, molecular self-orientation may occur in the presence of an anisotropic susceptibility produced either by paired electrons, especially when there is an anisotropic distribution of aromatic ring planes, or by unpaired electrons, or by a combination of both.
6.1 NMR shifts in self-orienting paramagnetic molecules In the presence of partial molecular alignment, the hyperfine shifts average to values resulting from the projection of the components of the shielding tensor 𝛔hf onto the main directions of the orientational probability tensor 𝐏 (Eq. (112)): 𝛿hf =‒ 𝛔hf ∙ 𝐏
(113)
When self-orientation is caused by the anisotropy of the paramagnetic susceptibility tensor, the 𝛘 and 𝐏 tensor are diagonal in the same frame, and thus the pseudocontact shifts are (see Eq. (62)) 42
xx 0 3xz Pxx 0 0 3xy 0 3x 2 r 2 1 2 2 yz r y xy 3 3 3 0 0 Tr pc 0 Pyy 0 yy 5 4r 2 2 3xz 0 yz z r 3 3 0 zz 0 0 Pzz
(114)
or, in spherical coordinates, using Eq. (112) with 𝛘𝐦𝐨𝐥 = 𝛘, [74]
pc
xx yy 3 rh2 B02 1 2 1 2 1 cos 3 ax zz 2 12r 3 150 kT 4 ax
(115)
B02 3 2 xx 2 yy zz rh sin 2 cos 2 1 2 15 0 kT where θ is the angle between the metal–nucleus vector 𝐫 and the z axis of the diagonal 𝛘 tensor, and
is an angle related to the projection of 𝐫 onto the xy plane of the tensor. The corrections to the B02 ax ax 1 15 0 kT anisotropy values are thus given by
yy 2 zz xx 2
3 rh2 4 ax
,
B02 2 xx 2 yy zz rh rh 1 15 0 kT . This correction to the pseudocontact shift is in most cases minor, and can thus be neglected (see Figure 5), so that Eq. (64) can be safely used in fields of at least up to 20 T.
43
Magnetic susceptibility anisotropy / 10-32 m3
41.0 ax'
40.5 40.0
ax
39.5 -19.5
rh
-20.0 rh'
-20.5 -21.0
0
200 400 600 800 1000 1200 Proton Larmor frequency / MHz
Figure 5. Field dependence of the apparent axial and rhombic magnetic susceptibility anisotropy calculated for the tensor components xx, yy and xx of 5, 7 and 10 ×1031 m3, respectively. Analogously, from Eqs. (68) and (113), in the presence of a partial alignment, the contact shifts are given by [54] 𝜇B𝐴Fc
𝛿 = 𝑘𝑇ℏ𝛾𝐼[(〈𝐒𝐒𝑇〉 ∙ 𝐠𝑇)𝑥𝑥𝑃𝑥𝑥 + (〈𝐒𝐒𝑇〉 ∙ 𝐠𝑇)𝑦𝑦𝑃𝑦𝑦 + (〈𝐒𝐒𝑇〉 ∙ 𝐠𝑇)𝑧𝑧𝑃𝑧𝑧] Fc
(116)
In the presence of a partial alignment, including the self-alignment caused by the anisotropy of the 𝛘 tensor, the NMR shifts are also affected by a contribution resulting from the anisotropy of the chemical shielding tensor, causing residual anisotropic chemical shifts (𝛿racs). Therefore, the NMR shifts in paramagnetic molecules are equal to 𝛿para = 𝛿dia + 𝛿hf + 𝛿racs
(117)
with [75,76]
racs
B02 iiCSA cos 2 ij jj 15 0 kT i , j 44
(118)
where i and j indicate the axes x, y and z, and
ij are the angles between the principal axes of the
magnetic susceptibility anisotropy tensor 𝚫𝛘 and the principal axes of the chemical shift anisotropy tensor CSA. The tensor CSA is defined as the difference between the chemical shielding tensor and its isotropic value. Residual anisotropic chemical shifts can be significant for nuclei with large CSA tensors at high magnetic fields, whereas they are usually negligible for 1H nuclei.
6.2 Paramagnetic residual dipolar couplings In the presence of multiple nuclei, the dipole-dipole interactions occurring between any pair of nuclei have the following energy (see Eq. (61))
(
𝜇0 3𝐫𝑖𝑗𝐫𝒊𝑗𝑇 dip 𝐸 𝑖𝑗 =‒ 4𝜋𝑟 3 𝛍𝐼𝑖 ⋅ 𝑟 2 ‒ 𝟏 𝑖𝑗 𝑖𝑗
)
𝜇0 ℏ2𝛾𝐼𝑖𝛾𝐼𝑗
⋅ 𝛍𝐼𝑗 = ‒ 4𝜋
𝑟𝑖𝑗3
𝛋⋅
(
3𝐫𝑖𝑗𝐫𝒊𝑗𝑇 𝑟𝑖𝑗2
)
‒𝟏 ⋅𝛋
(119)
where 𝛍𝐼𝑖 and 𝛍𝐼𝑗 are the magnetic moments of the ith and jth nuclei, at a distance 𝐫𝑖𝑗 and oriented along the direction 𝛋 with respect to 𝐁0. The averages of 𝐸dip 𝑖𝑗 over isotropic rotation vanish; however, in the presence of a partial alignment, < 𝐸dip 𝑖𝑗 >≠ 0 and residual dipolar couplings (RDCs) arise. The RDC between a pair of nuclei, 1 and 2, is thus [77–79] 12rdc
0 2 I 15 I 2 8 r12
3 x 2 r 2 Tr 3 xy 3 xz
3 xy 3y 2 r 2 3 yz
E12dip h
3 xz Pxx 3 yz 0 3 z 2 r 2 0
0 Pyy 0
0 0 Pzz
(120)
If the partial alignment is induced by anisotropies of both diamagnetic and paramagnetic susceptibility tensors (see Eq. (112)) [80], the RDC can be split into two components 12rdc (χ mol ) 12rdc(dia) (χ dia ) 12prdc (χ )
where the paramagnetic RDC is
45
(121)
3x 2 r 2
prdc 12
0 2 I 15 I 2 Tr 3xy 8 r12
1 B02 1 xx iso 3 5 0 kT 0 0
3xz
3xy 3y 2 r 2 3 yz
3xz 3 yz 3z 2 r 2
0
0
B02 1 1 yy iso 3 5 0 kT
0
0 B02 1 1 zz iso 3 5 0 kT
(122)
Therefore, in the frame where the 𝛘 tensor is diagonal, 12prdc
0 I 1 I 2 B02 xx (3x 2 r 2 ) yy (3 y 2 r 2 ) zz (3z 2 r 2 ) 2 5 8 r12 150 kT
(123)
and, in spherical coordinates [77,80,81], 12prdc
1 B02 I 1 I 2 3 ax (3 cos 2 1) rh sin 2 cos 2 3 4 15kT 2r12 2
(124)
where is the angle between 𝐫12 and the z axis of the diagonal 𝛘 tensor and is the angle which describes the position of the projection of the 𝐫12 vector onto the xy plane of the 𝛘 tensor, relative to the x axis. As already shown for pseudocontact shifts, the equation for the paramagnetic RDC can also be written in a generic reference frame: 12prdc
2 1 B0 3 I 1 I 2 4 15kT 4r123
zz iso (3 cos 2 1) ( xx yy ) sin 2 cos 2 2 xy sin 2 sin 2 2 xz sin 2 cos 2 yz sin 2 sin
(125)
or, using direction cosines, 12prdc
1 B02 3 I 1 I 2 4 15kT 2r123
ij cos i cos j i, j
1 B02 3 I 1 I 2 T r12 χ r12 4 15kT 2r125
(126)
where χ χ iso 1 , the indices i and j run over the three axes x, y and z, and i are the angles between 𝐫12 and each of the three axes. When written in the form of a matrix, Eq. (126) agrees with Eq. (5) reported in Section 2. Although the functional form of the equation for the RDCs is equivalent to that for the 46
pseudocontact shifts (see Panel 4), the information content of these two types of NMR observables is very different because the distances and angles in the equation for the paramagnetic RDCs describe the distance and the orientation of the vector connecting a nuclear pair in the frame defined by the 𝛘 tensor, whereas, in the equation for the pseudocontact shifts, they describe the position of the nucleus in the same reference frame, the origin being at the position of the paramagnetic metal ion. In Eqs. (122)-(126) a generalized Lipari-Szabo order parameter [82], SLS, is often included as a further coefficient, to account for the effect of internal motions with correlation times shorter than the molecular reorientation time. These motions can reduce the paramagnetic RDCs, due to the averaging over multiple orientations of the 𝐫12 vector with respect to the 𝛘 tensor [17]. In total, the splitting of the NMR peaks of covalently bound nuclei in a paramagnetic molecule is given by 12para 1J12 12rdc(dia) 12prdc 12dfs(dia) 12pdfs
(127)
where 1 J12 is the field-independent scalar coupling between the two nuclei, and 12dfs(dia) and 12pdfs are the diamagnetic and paramagnetic dynamic frequency shifts (see Section 8). Therefore, the difference in 12para at two magnetic fields is
para 12
( B0(1) )
para 12
2 2 1 B0(1) B0( 2) I 1 I 2 ( B0 ( 2 ) ) 4 15kT 2r123
3 axmol (3 cos 2 1) rhmol sin 2 cos 2 2
(128)
because, at high magnetic fields, the dynamic frequency shift is almost independent of the field (see Section 8). On the other hand, the difference in the NMR splittings observed at the same field in a paramagnetic sample and in a diamagnetic analogue is 12para 12dia 12prdc 12pdfs
(129)
In theory, Eq. (129) may be used to determine the paramagnetic dipolar frequency shifts from the observed 12para 12dia values and the paramagnetic RDCs, calculated using Eq. (124) and the 𝛘anisotropy tensor obtained from the analysis of the pseudocontact shifts; in practice, the paramagnetic 47
dipolar frequency shifts are so small for nuclei a few Å away from the paramagnetic center that they can be considered to be negligible, so that the differences between the NMR splittings in a paramagnetic sample and in its diamagnetic analogue are usually ascribed to the paramagnetic RDCs alone. Therefore, they must depend on the same 𝛘-anisotropy tensor responsible for the pseudocontact shifts, and the two types of experimental data can be analysed simultaneously and fitted against a unique tensor [83].
7. Paramagnetic relaxation enhancements The hyperfine coupling between the total electron magnetic moment and the nuclear magnetic moment also causes a contribution to nuclear relaxation, known as paramagnetic relaxation enhancement, which adds to the diamagnetic relaxation. The energy of this interaction (described by Eqs. (73) and (74) in the point-dipole approximation), can fluctuate randomly in time through three possible mechanisms: i)
molecular reorientation, causing changes in the orientation of the 𝐫 vector;
ii)
electron relaxation, causing changes in the orientation of the 𝛍(𝐒) vector;
iii)
chemical exchange, causing changes in the length of the 𝐫 vector.
The time scales for these three mechanisms are described by three correlation times: the reorientation correlation time r, the longitudinal and transverse electron relaxation times e1 and e2, and the residence time M. These fluctuations allow for a coupling of the electron-nucleus spin system with the external world (the lattice), thus allowing for energy exchange. The paramagnetic relaxation enhancement can be calculated from the transition probabilities wij between different states of the electron-nucleus spin system. These transition probabilities can be calculated from time-dependent perturbation theory for stochastic perturbations [30,84]: t
wij
2 Re i | H hfc (t ) | j j | H hfc (t ) | i exp(iij )d 2 0
48
(130)
where |𝜓𝑖 >
and |𝜓𝑗 >
indicate the eigenstates of the unperturbed Zeeman Hamiltonian,
characterized by the quantum numbers mS and mI of the electron and the nucleus respectively, ij being the difference between the corresponding eigenvalues divided by , and the bar indicating an ensemble average. If the perturbation is stationary, i.e., independent of time, and the ensemble average decays exponentially with time, which is usually a good approximation, the transition probabilities depend on the autocorrelation functions [52]
Gij ( ) i | H hfc (0) | j j | H hfc ( ) | i i | H hfc (0) | j 2 exp( / c ) (131) where the time constant c depends on the time constants mentioned above (r, e1,2 or M, see Eqs. (146), (151) and (162)). Finally, if the autocorrelation function decays to zero rapidly within the interval of integration, the latter can be safely extended to infinity (Redfield limit) [85–87] wij
2 i | H hfc (0) | j 2 c 2 hfc 2 H Re i | ( 0 ) | exp( / ) d i j ij c 2 2 1 ij2 c2 0
(132) The last term in Eq. (132) is known as a Lorentzian spectral density function J ( , )
1 2 2
(133)
The paramagnetic longitudinal relaxation enhancement, R1M, can be calculated from the sum of the transition probabilities causing a change in the nuclear spin states (the electron spin is assumed to remain in thermal equilibrium on the time scale of nuclear spin relaxation, due to the much more efficient electron relaxation mechanisms) R1M w0 2 w1I w2
(134)
where w0 indicates the probability of transitions causing a decrease in the electron spin quantum number and an increase in the nuclear spin quantum number (or vice versa), w1I indicates the 49
probability of transitions between states with the same electron spin quantum number and different nuclear spin quantum numbers, and w2 indicates the probability of transitions causing a decrease, or an increase, in both the electron and the nuclear spin quantum numbers (see Fig. 6). Therefore, if the energy of the electron-nuclear spin states is determined only by the Zeeman Hamiltonian (i.e., in the absence of ZFS and/or hyperfine coupling between unpaired electron(s) and the metal nucleus), the transition frequencies that determine the transition probabilities w0, w1I and w2 are S I ,
I and
S I , respectively, where 𝜔𝑆 =
< 𝑚𝑆|𝐻Zeeman|𝑚𝑆 >‒< 𝑚𝑆 ‒ 1|𝐻Zeeman|𝑚𝑆 ‒ 1 > ℏ
,
𝜔𝐼 =
|𝑚𝐼 >‒< 𝑚𝐼 ‒ 1|𝐻Zeeman |𝑚𝐼 ‒ 1 > < 𝑚𝐼|𝐻Zeeman 𝐼 𝐼 ℏ
,
(135) = ‒ ℏ𝛾𝐼𝐁0𝑇 ∙ 𝐈 𝐻Zeeman = 𝜇B𝐁0𝑇 ∙ 𝐠 ∙ 𝐒 , 𝐻Zeeman 𝐼 and thus 𝜔𝑆 > 0 and 𝜔𝐼 < 0, see Figure 6. For I = 1/2, the transition probabilities are thus w0
c 2p m S 1,1 / 2 | H hfc (0) | m S ,1 / 2 2 2 mS 1 ( S I ) 2 c2
(136)
w1I
c 2p 1 mS ,1 / 2 | H hfc (0) | mS ,1 / 2 2 2 2 mS 1 I2 c2
(137)
w2
c 2p m S 1,1 / 2 | H hfc (0) | m S ,1 / 2 2 2 mS 1 ( S I ) 2 c2
(138)
where |i,j> indicates |mS,mI>, and p is a normalization factor for the population of the spin states (p = 2/(2S+1), because 2S/(2S+1) is the ratio between the number of unpaired electrons and the number of states - p = 1 in the case of a single unpaired electron, i.e., for S = 1/2).
50
mS 1/2
w1I w0
w1S
w2
mS1, 1/2
w1I
mS 1/2 S
S
w1
( S | I |) ( S | I |)
| I |
mS1, +1/2
Figure 6. Energy levels, transition frequencies and transition probabilities in a dipole-dipole coupled S-I system with four levels mS, mS1, mI = +1/2, mI = 1/2 of the entire manifold of electron and nuclear spin states. The paramagnetic spin-spin relaxation enhancement, R2M, does not depend only on the transition probabilities between states with mI = 1, because spin transitions with equal nuclear energy also contribute to transverse relaxation. R2M can thus be calculated to be [52,88,89] R2 M
R1M p 2 2 mS ,1 / 2 | H hfc (0) | mS ,1 / 2 2 c 2 mS
c p 2 mS ,1 / 2 | H hfc (0) | mS ,1 / 2 2 2 1 S2 c2 mS ,mS
(139)
Due to the proportionality between NMR signal linewidths and transverse relaxation rates, a paramagnetic broadening of the NMR signals occurs:
para R2 M /
(140)
In the following sections, R1M and R2M are calculated by evaluating the terms
i | H hfc (0) | j 2 in Eq. (132). This is done by splitting the hyperfine coupling Hamiltonian into three terms, describing the dipole-dipole interaction between the nuclear magnetic moment and the zero average component of the electron magnetic moment (the dipolar term), the dipole-dipole interaction between the nuclear magnetic moment and the non-zero average component of the electron magnetic moment (the Curie term, Eq. (25)), and the interaction between the nuclear 51
magnetic moment and the electron magnetic moment localized on the nucleus (the Fermi contact term): RiM = RiMdip+RiMCurie+RiMFc.
(141)
Contact contributions can only be present for nuclei with some unpaired electron spin density delocalized in their s orbitals. For nuclei at distances larger than a few Å, only dipolar and Curie contributions can thus remain; they can be safely calculated within the point-dipole approximation, resulting in an r6 distance dependence (see later).
7.1 Dipolar relaxation The paramagnetic relaxation due to the point dipole-point dipole interaction between the nuclear magnetic moment and the electron magnetic moment is described by the Solomon equations, assuming that the static Hamiltonian of the spin system is properly described by considering only terms corresponding to the Zeeman interactions of the nuclear and electron magnetic moments with the external magnetic field [30]. The time-dependent perturbative Hamiltonian is described by Eq. (73) and the first term of Eq. (74), and with isotropic 𝐠 = 𝑔iso𝟏 (for the case of an anisotropic 𝐠 tensor, see Section 7.4) 𝜇0 ℏ𝛾𝐼𝜇B
𝐻dip(𝑡) = 𝐒 ⋅ 4𝜋
3
𝑟
(
𝑔iso
3𝐫𝐫𝑇 𝑟2
)
(142)
‒𝟏 ⋅𝐈
which can be rewritten as
(
𝐻dip(𝑡) = 𝐼𝑧𝑆𝑧 ‒
𝐼+𝑆‒ + 𝐼‒𝑆+ 4
)𝐹
∗ 0 + (𝐼 + 𝑆𝑧 + 𝐼𝑧𝑆 + )𝐹1 + (𝐼 ‒ 𝑆𝑧 + 𝐼𝑧𝑆 ‒ )𝐹 1
+ 𝐼 + 𝑆 + 𝐹2 + 𝐼 ‒ 𝑆 ‒ 𝐹 2∗
3
3
𝜇0 ℏ𝛾𝐼𝜇B𝑔iso
with 𝐹0 = 𝑘(3cos2𝜃 ‒ 1), 𝐹1 = 2𝑘sin𝜃cos𝜃𝑒 ‒ 𝑖𝜑, 𝐹2 = 4𝑘sin2𝜃𝑒 ‒ 2𝑖𝜑, 𝑘 =‒ 4𝜋
𝑟3
(143)
and 𝜃 and 𝜑
are the spherical angles indicating the orientation of the vector connecting the nucleus and the paramagnetic metal ion positions with respect to a molecular frame where the magnetic field is directed along the z axis. 52
----------------------------------------------------------------------------------------------------------------------Panel 9 From Eqs. (136)-(138) and (143), since < 𝑚𝑆 + 1|𝑆 + |𝑚𝑆 > = 𝑆(𝑆 + 1) ‒ 𝑚𝑆(𝑚𝑆 + 1), < 𝑚𝑆 ‒ 1|𝑆 ‒ |𝑚𝑆 > = 𝑆(𝑆 + 1) ‒ 𝑚𝑆(𝑚𝑆 ‒ 1), 2 S ( S 1) mS (mS 1) 2 2S ( S 1)(2S 1) 4S ( S 1) 2 S 1 mS 2S 1 3 3
and
2 2 S ( S 1)(2 S 1) 2 S ( S 1) , mS2 2 S 1 mS 2S 1 3 3 2
w0
2 F0 4 S ( S 1) τ cdip 2 2 3 16 2 1 ( S I ) 2 ( τ cdip 2 )
w1I
τ cdip 2 F1 1 2 S ( S 1) 1 2 2 2 3 1 I2 ( τ cdip 1 )
2
2
τ cdip 2 F2 4 S ( S 1) 2 w2 2 3 1 ( S I ) 2 ( τ cdip 2 2 )
(P9.1)
(P9.2)
(P9.3)
and after integration of Fi2 over the angular coordinates ( F0 2 4 k 2 , F12 F2 2 3 k 2 ), using Eq. (134), 10 5 2
R1dip M
2 μ B2 S ( S 1) 2 0 γ I2 g iso r6 15 4
3τ cdip 6 τ cdip τ cdip 2 1 2 (P9.4) dip 2 2 dip 2 2 dip 2 2 1 I ( τ c1 ) 1 ( S I ) ( τ c 2 ) 1 ( S I ) ( τ c 2 ) Analogously, the paramagnetic enhancement to the spin-spin (transverse) nuclear relaxation rate (Eq. (139)) is 2
R2dip M
2 μ B2 S ( S 1) 1 0 γ I2 g iso 15 4 r6
τ cdip 3τ cdip 6 τ cdip 2 1 2 2 dip 2 2 dip 2 2 dip 2 1 I ( τ c1 ) 1 ( S I ) ( τ c 2 ) 1 ( S I ) ( τ c 2 ) 2
2 2 2 τ cdip2 γ g μ 2 2 S ( S 1) 4 dip 4 4 S ( S 1) 3 0 I iso6 B τ c1 2 dip 2 3 5 4 3 10 1 S ( τ c 2 ) r 4 4
(P9.5)
and thus 2
R2dip M
2 μ B2 S ( S 1) 1 0 γ I2 g iso 15 4 r6
τ cdip 3τ cdip 6 τ cdip 6 τ cdip 2 1 2 2 4 τ cdip 1 2 dip 2 2 dip 2 2 dip 2 2 dip 2 1 ( ) ( τ ) 1 ( τ ) 1 ( ) ( τ ) 1 ( τ ) S I c2 I c1 S I c2 S c2
(P9.6)
-----------------------------------------------------------------------------------------------------------------------
53
Since the electron Larmor frequency S is 658.2 times larger than the proton Larmor frequency I (so that S I S ), the paramagnetic enhancement of the spin-lattice (longitudinal) nuclear relaxation rate is (see Panel 9) [30] 2
dip 1M
R
2 7 τ cdip 3τ cdip μ B2 S ( S 1) 2 μ 0 γ I2 g iso 2 1 1 ω 2 ( τ dip ) 2 1 ω 2 ( τ dip ) 2 15 4π r6 2 c1 c I S
(144)
and the paramagnetic enhancement of the spin-spin (transverse) nuclear relaxation rate is 2
R
dip 2M
2 13τ cdip 3τ cdip μ B2 S ( S 1) dip 1 μ 0 γ I2 g iso 1 2 4 τ 1 c dip 2 2 dip 2 2 15 4π 1 ( ) 1 ( r6 ω τ ω S c2 I τ c1 )
(145)
where the correlation times are
τ
dip -1 c1
τe11 τ r1 τ M1 ,
τ
dip -1 c2
τe21 τ r1 τ M1
(146)
because the dipole-dipole interaction can be modulated by the longitudinal/transverse electron relaxation time, the reorientation time and/or the residence time. It is assumed here that reorientational motions, chemical exchange and electron relaxation processes are not coupled. In the case of lanthanoids and actinoids, the J quantum number replaces the S quantum number and gJ replaces giso. If e = e for nuclear Larmor frequencies smaller than (2 658.2 τ cdip2 ) 1 (corresponding to the condition, which is usually met, that ωS τ cdip2 1 ), or both e and e are much longer than r or M, it follows that Eqs. (144) and (145) can be written as a function of a single correlation time c c1 . dip
dip
As a result, the field dependence of the paramagnetic enhancement of the proton relaxation rates is characterized by two Lorentzian dispersions centered at frequencies
S ( cdip ) 1 and I ( cdip ) 1 ,
and thus separated by a factor 658.2. The electron relaxation time can also depend on the applied magnetic field, and this could lead to a possible field dependence of c . For instance, a field dependence of electron relaxation is dip
expected for systems experiencing ZFS (provided that S>1/2). Collisions due to solvent molecules can cause deformations of the metal coordination polyhedron, which can in turn cause a transient ZFS 54
even in the absence of a static ZFS. This provides a relaxation mechanism for the electron(s), that in the pseudorotational model formulated by Bloembergen and Morgan [90] results in e11
22t 4S ( S 1) 3 τ ν 2 2 4τ ν 2 2 50 1 ωS τ ν 1 4ωS τ ν
(147)
e21
2t 4S ( S 1) 3 3τ ν 5τ ν2 2 2τ ν 2 2 1 ωS τ ν 1 4ωS τ ν 50
(148)
where t is the mean squared fluctuation of the ZFS and τv is the correlation time for the 2
instantaneous distortions of the metal coordination polyhedron. This field dependence of electron relaxation causes a peak (typically at about 1 T for the gadolinium(III) and manganese(II) complexes and higher for e.g. iron(III) and nickel(II) [91]) in the field dependence of the nuclear longitudinal relaxation rate. Eqs. (144)-(148) are usually referred to as the Solomon-Bloembergen-Morgan (SBM) model (together with the following Eqs. (160) and (161)). Notably, different electron transitions may have different relaxation rates, as have been calculated for S = 3/2 and 5/2 systems [92]. Eqs. (147) and (148) thus describe, to a good approximation, the average electron relaxation rates of all allowed transitions. Finally, intramolecular (local) motions can also, at least partially, modulate the dipole-dipole interaction. These fast motions can thus affect the nuclear correlation time, and are usually treated using the Lipari-Szabo model-free approach [82,93].
7.2 Curie spin relaxation As anticipated in Section 2, Curie spin relaxation describes the paramagnetic contribution to nuclear relaxation due to the dipole-dipole interaction between the nuclear magnetic moment and the thermal average of the total electron magnetic moment, i.e., between 𝛍𝐼 and 〈𝛍el〉 (see Eq. (30)). Therefore, using Eq. (73) and the first term of Eq. (74), 𝜇0 ℏ𝛾𝐼𝜇B
𝐻Curie(𝑡) = 𝐒C ⋅ 4𝜋
𝑟3
𝐠𝑇 ⋅
(
3𝐫𝐫𝑇 𝑟2
)
‒𝟏 ⋅𝐈 55
(149)
In the absence of g-anisotropy, the time-dependent perturbative Hamiltonian is given by (see Eq. (142)) 𝜇0 ℏ𝛾𝐼𝜇B
𝐻Curie(𝑡) = 𝐒C4𝜋
𝑟3
𝑔iso ⋅
(
3𝐫𝐫𝑇 𝑟2
)
(150)
‒𝟏 ⋅𝐈
Since the dipole-dipole interaction is in this case only coupled with the average over all electron spin states, it can only be modulated by reorientation and chemical exchange
τ
Curie 1 c
τr1 τ M1
(151)
2 w1I R1Curie M
(152)
and
which provides the I dispersion term of Eq. (144), with S(S+1)/3 replaced by 𝑆C2. Therefore, from Eqs. (144) and (145), [29,94] 2
R1Curie M
2 3τ cCurie 2 μ0 γI2 giso μB2 SC2 1 ωI2 ( τ cCurie ) 2 5 4π r6
2
R2Curie M
2 1 μ0 γI2 giso μB2 SC2 5 4π r6
(153)
Curie 3τ cCurie 4 τ c 2 Curie 2 1 ωI ( τ c )
(154)
and using Eq. (29) 2
Curie 1M
4 μ B4 B02 S 2 ( S 1) 2 τ cCurie 2 μ 0 γ I2 g iso 15 4π k 2T 2 r 6 1 ω I2 ( τ cCurie ) 2
Curie 2M
4 μ B4 B02 S 2 ( S 1) 2 1 μ 0 γ I2 g iso 45 4π k 2T 2 r 6
R
2
R
Curie 3τ cCurie 4 τ c 1 ωI2 ( τ cCurie ) 2
(155)
(156)
Eqs. (155) and (156) can be written as Eqs. (3) and (4), reported in Section 2, using the Curie law, Eq. (50). In the case of lanthanoids and actinoids, the J quantum number replaces the S quantum number and gJ replaces giso. Since 𝑆C2 is much smaller than S(S + 1)/3, Curie spin relaxation can only provide nonCurie negligible contributions when τ cdip , i.e., when the electron relaxation time is much smaller 1 << τ c1
than both the reorientation time and the residence time. 56
7.3 Fermi contact relaxation Fermi contact relaxation describes the paramagnetic contribution to nuclear relaxation due to the dipole-dipole interaction between the nuclear magnetic moment and the electron magnetic moment localized on the nucleus itself. The time-dependent perturbative Hamiltonian is thus described by Eq. (73) and the second term of Eq. (74), i.e., 𝜇0
𝜌𝐼
𝐻Fc(𝑡) = 3 ℏ𝛾𝐼𝜇B𝑔e 𝑆 𝐒 ⋅ 𝐈
(157)
which, using Eq. (71), can be written as
(
𝐻Fc(𝑡) = 𝐴Fc 𝑆𝑧𝐼𝑧 +
𝑆+𝐼‒ + 𝑆‒𝐼+ 2
).
(158)
In this case, only the zero-quantum transition provides a contribution to relaxation (because the spin operators 𝑆𝑧𝐼𝑧, 𝑆 + 𝐼 ‒ and 𝑆 ‒ 𝐼 + contribute only to 𝑤0), so that (see Eq. (136)),
cFc2 2( A Fc ) 2 4 S ( S 1) 3 4 2 1 ( S I ) 2 ( τ cFc2 ) 2
R1FcM w0
(159)
and thus, considering that S>>I [95,96], 2
Fc 1M
AFc 2 τ cFc2 S ( S 1) 2 Fc 2 3 1 ωS ( τ c 2 )
Fc 2M
A Fc 1 S ( S 1) 3
R
R
2
Fc τ cFc2 τ c1 2 Fc 2 1 ωS ( τ c 2 )
(160)
(161)
Since reorientation cannot modulate the contact coupling, the correlation time in this case is
τ
Fc 1 ci
τei1 τ M1
(i = 1,2).
(162)
Fermi contact relaxation has been also shown to occur in the presence of a contact interaction modulated by collisions between solvent molecules and the molecule bearing the paramagnetic center [97]. In this case a distribution of correlation times corresponding to the lengths of the contact times between molecules must be considered (the ‘pulse’ model [98]).
57
Fermi contact coupling can also provide Curie spin relaxation contributions when the residence time is much larger than the electron relaxation time, so that, from Eqs. (161) and (29) [94], 2
2
Fc,Curie 2M
R
g iso B B0 S ( S 1) AFc AFc τM S τ M 3kT 2 C
(163)
where the term in parenthesis is proportional to the contact shift (in hertz, see Eq. (94)).
7.4 Anisotropic effects The previous equations for the paramagnetic enhancements of the nuclear relaxation rates were derived by making a number of approximations: 1) the electron is supposed to reside at a single point and to behave as a magnetic point dipole, thus neglecting electron spin density delocalization (the point-dipole approximation); 2) the perturbation Hamiltonian changes stochastically and is stationary; 3) molecular reorientation is isotropic; 4) molecular reorientation and electron spin relaxation (i.e., the dynamics of the electron magnetic moment) are uncorrelated (the decomposition approximation); 5) electron spin relaxation is a single exponential process (as if all electron spin transitions had the same relaxation rates); 6) the correlation time for nuclear relaxation is much shorter than the nuclear relaxation time, and the correlation time for electron relaxation is much shorter than the electron relaxation time (the Redfield limit); 7) the electron 𝐠 tensor is isotropic, and the static Hamiltonian is dominated by the electron Zeeman interaction. Assumptions 1-5 seem to be acceptable, in the sense that deviations from these approximations either do not cause large changes in the calculated relaxation rates, or corrections can be applied (for 58
instance, Lipari-Szabo model free approaches [82] can be used to treat the case of anisotropic molecular reorientation, or multiexponential electron relaxation can be considered and plugged into the nuclear relaxation equations [92,99]). The effect of the anisotropy of the electron 𝐠 tensor on the relaxation equations was analyzed both in the case of fast rotation [100] and of slow rotation (with respect to the electron relaxation time) [101,102] and it was found to be small. In contrast, beyond the Redfield limit, dramatic changes in the calculated relaxation rates can occur [85,103]. The presence of a static ZFS and/or hyperfine coupling between the electron magnetic moment and the magnetic moment of the metal nucleus (i.e., of the nucleus of the paramagnetic ion itself) can also cause dramatic changes in relaxation rates. When present, these terms should thus be included in the static Hamiltonian of the spin system together with the Zeeman terms, because they can significantly affect the energy of the electron spin states, especially at low fields. As a result, the energies of most electronic spin transitions can be much larger than calculated from the Zeeman interaction only, thus accounting for changes in the relaxation rates. The effects of these anisotropic terms on the nuclear relaxation rates can be calculated by writing the hyperfine coupling Hamiltonian (Eq. 143) in the form 𝐻dip(𝑡) = 𝜇0 ℏ𝛾𝐼𝜇B
where 𝜅 =‒ 4𝜋
𝑟3
24𝜋 𝑚 ∑2 5 𝜅 𝑚 =‒ 2( ‒ 1) 𝑌2,𝑚(𝜃,𝜑)𝑇2, ‒ 𝑚
(164)
, 𝑌2,𝑚(𝜃,𝜑) are normalized spherical harmonics and 𝑇2, ‒ 𝑚 are time independent
tensor operators acting on the spin variables [84]: 𝑇2,0 =
[2𝑔𝑧𝑧𝐼𝑧𝑆𝑧 ‒ 12𝑔 ⊥ (𝐼 + 𝑆 ‒ + 𝐼 ‒ 𝑆 + )], 𝑇2, ± 1 =∓ 12(𝑔𝑧𝑧𝐼 ± 𝑆𝑧 + 𝑔 ⊥ 𝐼𝑧𝑆 ± ), 𝑇2, ± 2 = 12𝑔 ⊥ 𝐼 ±
1 6
𝑆± 𝑌2,0 =
5 2 16𝜋(3cos 𝜃 ‒ 1),
15 8𝜋cos 𝜃sin 𝜃exp ( ± 𝑖𝜑),
𝑌2, ± 1 =∓
𝑌2, ± 2 =
(165)
15 2 32𝜋sin 𝜃exp ( ± 2𝑖𝜑)
(166) and 𝑔𝑧𝑧 and 𝑔 ⊥ are the principal values of the 𝐠 tensor in the case of axial symmetry (for the rhombic case, 𝑔 ⊥ 𝑆 ± must be replaced by 𝑔𝑥𝑥
𝑆+ + 𝑆‒ 2
𝑆+ ‒ 𝑆‒
± 𝑔𝑦𝑦 59
2
[89]). The spherical harmonics depend on
the spherical angles between the vector connecting the positions of the nucleus and the paramagnetic metal ion, and the direction of the magnetic field. If 𝜃 and 𝜑 are the spherical angles reporting on the metal-nucleus direction in the molecular frame, the spherical harmonics can be written in the laboratory frame and then transformed into the molecular frame using Wigner rotation matrices, which contain the Euler angles , and , indicating the magnetic field direction in the molecular frame: 𝑌2,𝑚(𝜃,𝜑) = 𝑅(𝛼,𝛽,𝛾)𝑌2,𝑚(𝜃,𝜑)
(167)
From Eqs. (132)-(134) R1dip M
1 c complex conjugate Re i | H dip | j 2 2 2 2 1 (ij I ) c ij
(168)
where |𝜓𝑖 > and |𝜓𝑗 > are the electron spin eigenstates, and 𝜔𝑖𝑗 =
< 𝜓𝑖|𝐻0|𝜓𝑗 >‒< 𝜓𝑗|𝐻0|𝜓𝑖 > ℏ
,
(169)
where H0 is the unperturbed Hamiltonian. In the presence of ZFS (Eq. (18)), 𝐻 0 = 𝜇 B 𝐁0 ∙ 𝐠 ∙ 𝐒 + 𝐒 ∙ 𝐃 ∙ 𝐒
(170)
and assuming that the 𝐠 and 𝐃 tensors are diagonal in the same molecular frame
(
𝐻0 = 𝜇B𝐵0 𝑔𝑧𝑧𝑆𝑧cos𝛽 ‒ 𝑔𝑥𝑥
𝑆+ + 𝑆‒ 2
𝑆+ ‒ 𝑆‒
sin𝛽cos𝛾 ‒ 𝑖𝑔𝑦𝑦
2
) (
sin𝛽sin𝛾 + 𝐷 𝑆2𝑧 ‒
𝑆 +2 ‒ 𝑆 2‒
)+𝐸
𝑆(𝑆 + 1) 3
2
(171) where , and are the Euler angles that define the orientation of the magnetic field with respect to the molecular frame. The eigenstates in Eq. (168) can be determined from diagonalization of the Hamiltonian of Eq. (171). In solution, isotropic molecular tumbling implies that averaging should be performed over all orientations, as indicated by the bar in Eq. (168). The presence of a static ZFS may significantly affect the nuclear relaxation rates, mainly depending on the magnitude and rhombicity of the ZFS tensor with respect to the Zeeman energy, as well as on the position of the nucleus with respect to the ZFS tensor (Figure 7). Therefore, the paramagnetic enhancement of the nuclear relaxation rate depends not only on the metal-nuclear 60
distance (as predicted by the SBM model, Eqs. (144) and (145), or by the Curie relaxation Eqs. (155) and (156)), but also on the spherical angles defining the metal-nucleus direction in the molecular frame. An additional dispersion in the field dependence of the relaxation rate may appear as a result of the transition from a dominant ZFS to a dominant Zeeman energy as the field increases [104].
10 Solomon profile (no ZFS)
Rdip 1M (a.u.)
8
D=1 cm-1, E=0
=0°
6 4
D=1 cm-1, E/D=1/3
=90° =0°
2 0 0.01
=90°
0.1
1
10
100
1000
Proton Larmor Frequency / MHz Figure 7. Dipole-dipole relaxation (S = 3/2, c
dip
= 0.1 ns) in the absence (Solomon equation) and in
the presence of axial (D = 1 cm1) or rhombic (E/D = 1/3) ZFS for a proton along the z axis ( = 0°) or in the x,y plane ( = 0°). In the presence of a hyperfine coupling between the metal nucleus and unpaired electron(s), the unperturbed Hamiltonian should contain the term [89,105] 𝐴𝑥𝑥
𝐴𝑦𝑦
𝐻0(hyp) = 𝐴𝑧𝑧𝐼𝑧𝑆𝑧 + 4 (𝐼 + 𝑆 + + 𝐼 + 𝑆 ‒ + 𝐼 ‒ 𝑆 + + 𝐼 ‒ 𝑆 ‒ ) + 4 ( ‒ 𝐼+𝑆+ + 𝐼+𝑆‒ + 𝐼‒𝑆+ ‒ 𝐼‒𝑆‒) (172)
where 𝐴𝑥𝑥, 𝐴𝑦𝑦 and 𝐴𝑧𝑧 are the diagonal components of the metal nucleus-electron hyperfine coupling tensor. This effect can be important, for instance, for copper(II) (I = 3/2), oxovanadium(IV) (I = 7/2), manganese(II) (I = 5/2), cobalt(II) (I = 7/2) metal ions. 61
Analogous expressions can be calculated for Fermi contact relaxation using Eq. (157) instead of Eq. (164) [89]. The approach described above assumes that the Euler angles providing the orientation of the electron-nucleus vector (and thus the Wigner rotation matrix) are constant during electron-spin relaxation, i.e., for times of the order of the correlation time c. This is the case of the so-called slow rotation limit (r >> e). Analytical expressions for the nuclear relaxation rates in slowly rotating systems were first obtained to evaluate the effects of 𝐠 anisotropy [101], ZFS (in the limit where the ZFS is much larger than the Zeeman energy) [101,106] and hyperfine coupling between the metal nucleus and the unpaired electron(s) for the case of S = 1/2 and I = 3/2 spin systems (Cu2+) [102], and then generalized to any value of the nuclear spin I [107]. Incidentally, the effect of ZFS was also introduced in the analysis of the nuclear relaxation rates of lanthanoid(III) aqua ions, which is the first case where a field dependence of Curie spin relaxation has been observed [108]. This allowed for a better estimate of their electron relaxation times, which turned out to be longer than would be predicted without considering this effect. To treat the case of slow rotation within the Redfield limit (implying that the relaxation time arising from an interaction can never be as short as the correlation time that describes the fluctuations of that interaction), the “Florence NMRD” program was developed, allowing evaluation of the field dependence of nuclear relaxation rates for any size and rhombicity of ZFS and/or hyperfine coupling between the metal nucleus and unpaired electrons, and for any I and S quantum numbers [89]. A modified version of the Florence NMRD program [109,110] takes into account the effect of the ZFS not only on the nuclear relaxation rates, but also on the electron relaxation rates. Outside the slow rotation and Redfield limits, more complicated approaches should be used [84,85,111–114]. Recently, experimental evidence of the angular dependence of the paramagnetic enhancement of nuclear relaxation was obtained for a paramagnetic lanthanoid(III) complex, as a result of its large static ZFS [115]. These data were analysed using the parametric equation 62
2
dip 1M
R
2 μ 0 γ I2 6 Tr (3rˆrˆ T 1) 2 G (ωI ) 3 4π r
(173)
where rˆ is the unit vector pointing in the direction of r, and the six independent components of the symmetric spectral density tensor G (ωI ) are treated as fitting parameters. Curie spin relaxation may also change in the presence of 𝐠 anisotropy, ZFS and/or interactions with excited states. If Eq. (36) is introduced into Eq. (149), using Eq. (47) we obtain (in agreement with Eq. (61)) ℏ𝛾𝐼
𝐻Curie(𝑡) = ‒ 4𝜋𝑟3𝐁 ∙ 𝛘 ⋅ 0
(
3𝐫𝐫𝑇 𝑟2
)
(174)
‒𝟏 ⋅𝐈
Vega and Fiat [29] derived the longitudinal and transverse paramagnetic relaxation rates resulting from this perturbative Hamiltonian, and thus described the effects on Curie spin relaxation of the anisotropy of the magnetic susceptibility (and in turn of ZFS/g-anisotropy), noting the link between the latter and the pseudocontact shift. Recently, these equations were rewritten as a function of the tensor
1 3rr T 2 1 χ , σ σ0 3 4r r
(175)
where σ 0 is the chemical shielding tensor, so that the equations for Curie and CSA relaxation result in [115] CSA R1Curie M
τr τr 1 2 2 2 I 2 I2 2 2 2 1 9ω I τ r 15 1 ω I2 τ r2
CSA R2Curie M
τr 3τ r 1 2 2 1 2 2 I I 4 τ r 2 2 4 45 1 9ω I τ r 1 ω I2 τ r2
(176)
(177)
where
2 xy yx
2
xz
zx
2
2 2 xx2 yy zz2 xx yy xx zz yy zz
yz
zy
2
3 xy yx 4
(178)
2
xz
zx
2
yz
zy
. 2
These anisotropic effects of Curie spin relaxation were calculated to be rather small and have never 63
been measured. Note that, in contrast to the pseudocontact shift, which arises from the anisotropy of the susceptibility, Curie-spin relaxation is due to the full susceptibility.
8. Paramagnetic cross-correlation effects In the equations previously reported to describe paramagnetic enhancements of nuclear relaxation rates (Eq. (141)), paramagnetic cross-correlation terms were neglected. Cross-correlation effects arise in the presence of multiple correlated interactions, the energies of which are modulated randomly in time due to the same mechanism, and thus with the same correlation time. In fact, the transition probabilities contain the square of the expectation value of all time-dependent Hamiltonian terms (see Eq. (132)) wij
2 i | H1 (0) H1 (0) ... H n (0) | j 2
2
c 1 ij2 c2
(179)
which differs from the sum of the squares of the single terms by the appearance of cross-term products [116]. In paramagnetic systems, the dipole-dipole interaction 𝐻Curie (Eq. (150)) between the magnetic moment of a nucleus and the thermal average of the electron magnetic moment (responsible for Curie spin relaxation) can be modulated by the reorientation of the molecule, which also modulates the (diamagnetic) dipole-dipole interaction between the magnetic moments of two nuclei (Eq. (119)). The resulting cross-terms can be significant, and decrease or increase the nuclear relaxation rates, depending on whether S is in the 1/2 or –1/2 spin state, respectively. The different cross-correlation effects occurring for the two components of a nuclear doublet lead to a difference in their nuclear linewidths [117]. This difference for, e.g., the two components of the amide proton NMR signals in 15N enriched paramagnetic proteins, calculated on the assumption of an isotropic 𝛘 tensor, is [116–119]
ccr 1/2
2 15
2 2 2 2 3 c 0 B0 H N B g iso S ( S 1) 3 cos MHN 1 4 3 c 1 ω 2 2 r 3 rHN kT 2 4 I c 2
64
(180)
3 c 0 B0 H2 N iso 3 cos 2 MHN 1 4 c 2 3 3 2 4 10 r rHN 1 I2 c2
.
where the angle MHN is between the metal-nucleus vector (r) and the vector (rHN) between the 1H nucleus and the 15N nucleus, and c is the correlation time modulating both relaxation processes. The prefactor in Eq. (180) originates from the cross product between the square root of the prefactors of Curie spin relaxation (Eq. (156)) and of the Solomon equation (Eq. (145)) where S = 1/2 (the 1H spin, in this case) and g iso μ B is replaced with N . Equations to evaluate the effects of an anisotropic magnetic susceptibility have also been derived [120]. The dipole-dipole interaction between the nuclear magnetic moment and the thermal average of the electron magnetic moment (responsible for Curie spin relaxation) can also cause a nonnegligible cross-correlation term with nuclear CSA relaxation (see also Eq. 118) [121], which results in decreased or increased nuclear relaxation rates depending on the relative orientation of the principal axes of CSA and of the electron-nucleus dipolar shielding (Eq. (62)) tensors. The magnitude of this cross-correlation effect is usually very small, unless the paramagnetic relaxation enhancement is dominated by Curie spin relaxation rather than by dipolar relaxation. The imaginary part of the spectral density function (see Eq. (132)) related to a crosscorrelation effect is responsible of the dynamic frequency shift [52], introduced in Eq. (127). The dynamic frequency shift associated with the cross-correlation between i) the dipole-dipole interaction between the magnetic moment of a nucleus and the thermal average of the electron magnetic moment and ii) the dipole-dipole interaction between the magnetic moments of two nuclei, for the two components of the amide protons, is [119] 2 S ( S 1) 3 cos 2 MHN 1 I c2 2 0 B0 H2 N B2 g iso 3 5 4 2 r 3 rHN kT 1 I2 c2 2
dfs
65
(181)
B0 H2 N I c2 H N , so that the dynamic For long correlation times and/or at high magnetic fields, 1 I2 c2 frequency shift is to a good approximation independent of the correlation time, of the nucleus observed, and of the magnetic field. dfs is usually negligible with respect to paramagnetic RDC.
9. Conclusions More and more perspectives are opening up in the fields of structural biology and systems biology, as well as in material sciences, thanks to the full exploitation of paramagnetic effects. NMR data on paramagnetic systems offer a rich source of information on structures and dynamics, both on a short and long distance scales. In parallel with the development of experimental applications, there is currently a large effort going on to extend the NMR theory of paramagnetic systems. The new theoretical approaches are questioning the validity of the semi-empirical equations proposed more than half a century ago. In this review we have tried to clarify that a full understanding of the physical meaning of the (spin and orbital) electron magnetic moment and of the magnetic susceptibility is needed to understand NMR shifts and relaxation rates in paramagnetic systems. In our view, the first-principles quantum chemistry framework is still not fully mature, because it does not provide a fully satisfactory treatment of the contributions from the orbital magnetic moment to the observables. Therefore, in this review the semi-empirical framework has been further elaborated upon and fully endorsed, despite its limitations. In summary, all paramagnetic effects observed by NMR (dipolar shifts, pseudocontact shifts, paramagnetic residual dipolar couplings, electron-nucleus dipole-dipole relaxation, Curie relaxation, cross-correlation between nucleus-nucleus dipole-dipole and nucleus-electron Curie relaxation, dynamic frequency shifts) depend on the total electron magnetic moment. Notable exceptions are the Fermi contact shift and relaxation, which depend only on the spin contributions to the electron magnetic moment (Table 3).
66
Table 3. Dependence of paramagnetic NMR observables on the electron magnetic moment and on the g or 𝛘 tensors in the semi-empirical framework. 𝛿dip
∝
< 𝛍el >
∝
𝛘
𝛿pc
∝
< 𝛍el >
∝
𝛘 ‒ 𝛘iso
𝛿Fc
∝
< 𝛍𝑆 >
Δ𝜈prdc 12 ∝
< 𝛍el >
dip 𝑅1,2𝑀
𝛍el2
∝
∝ ∝
𝑔e𝛘 ∙ [𝐠𝑇] 𝛘 ‒ 𝛘iso 𝐠 ∙ 𝐠𝑇
∝
𝑅Curie 1,2𝑀 ∝
< 𝛍el > 2 ∝
Fc 𝑅1,2𝑀
∝
𝛍2𝑆
ccr Δ𝜈1/2
∝
< 𝛍el >
∝
𝛘
∝
< 𝛍el >
∝
𝛘
Δ𝜈dfs
‒1
𝛘 ∙ 𝛘𝑇 𝑔2e
∝
Acknowledgements We thank Ladislav Benda and Maurizio Romanelli for fruitful discussions. The support from Fondazione Cassa di Risparmio di Firenze, MIUR PRIN 2012SK7ASN, and Instruct-ERIC, an ESFRI Landmark, supported by national member subscriptions is also acknowledged. Specifically, we thank the Instruct-ERIC Core Centre CERM, Italy.
67
References
[1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14]
[15]
[16]
M.J. Knight, I.C. Felli, R. Pierattelli, L. Emsley, G. Pintacuda, Magic angle spinning NMR of paramagnetic proteins, Acc. Chem. Res. 46 (2013) 2108–2116. doi:10.1021/ar300349y. E. Ravera, G. Parigi, C. Luchinat, Perspectives on paramagnetic NMR from a life sciences infrastructure, Journal of Magnetic Resonance. 282 (2017) 154–169. doi:10.1016/j.jmr.2017.07.013. C. Nitsche, G. Otting, Pseudocontact shifts in biomolecular NMR using paramagnetic metal tags, Prog Nucl Magn Reson Spectrosc. 98–99 (2017) 20–49. doi:10.1016/j.pnmrs.2016.11.001. P.H.J. Keizers, M. Ubbink, Paramagnetic tagging for protein structure and dynamics analysis, Prog Nucl Magn Reson Spectrosc. 58 (2011) 88–96. C. Luchinat, G. Parigi, E. Ravera, Paramagnetism in experimental biomolecular NMR, 2018. E. Ravera, A. Carlon, M. Fragai, G. Parigi, C. Luchinat, Paramagnetic NMR as a new tool in structural biology, Emerging Topics in Life Sciences. 2 (2018) 19–28. doi:10.1042/ETLS20170084. A. Carlon, E. Ravera, W. Andrałojć, G. Parigi, G.N. Murshudov, C. Luchinat, How to tackle protein structural data from solution and solid state: An integrated approach, Progress in Nuclear Magnetic Resonance Spectroscopy. 92–93 (2016) 54–70. J.H. Ardenkjær-Larsen, B. Fridlund, A. Gram, G. Hansson, L. Hansson, M.H. Lerche, R. Servin, M. Thaning, K. Golman, Increase in signal-to-noise ratio of > 10,000 times in liquidstate NMR, PNAS. 100 (2003) 10158–10163. doi:10.1073/pnas.1733835100. L. Lumata, M.E. Merritt, C.R. Malloy, A.D. Sherry, Z. Kovacs, Impact of Gd3+ on DNP of [1-13C]pyruvate doped with trityl OX063, BDPA, or 4-oxo-TEMPO, J Phys Chem A. 116 (2012) 5129–5138. doi:10.1021/jp302399f. E. Terreno, D.D. Castelli, A. Viale, S. Aime, Challenges for Molecular Magnetic Resonance Imaging, Chemical Reviews. 110 (2010) 3019–3042. doi:10.1021/cr100025t. E. Ravera, D. Shimon, A. Feintuch, D. Goldfarb, S. Vega, A. Flori, C. Luchinat, L. Menichetti, G. Parigi, The effect of Gd on trityl-based dynamic nuclear polarisation in solids, Phys. Chem. Chem. Phys. (2015). doi:10.1039/C5CP04138D. S.A. Walker, D.T. Edwards, T.A. Siaw, B.D. Armstrong, S. Han, Temperature dependence of high field 13C dynamic nuclear polarization processes with trityl radicals below 35 Kelvin, Physical Chemistry Chemical Physics. 15 (2013) 15106. doi:10.1039/c3cp51628h. C.P. Grey, N. Dupré, NMR Studies of Cathode Materials for Lithium-Ion Rechargeable Batteries, Chemical Reviews. 104 (2004) 4493–4512. doi:10.1021/cr020734p. F.C. Strobridge, D.S. Middlemiss, A.J. Pell, M. Leskes, R.J. Clément, F. Pourpoint, Z. Lu, J.V. Hanna, G. Pintacuda, L. Emsley, A. Samoson, C.P. Grey, Characterising local environments in high energy density Li-ion battery cathodes: a combined NMR and first principles study of LiFe x Co 1−x PO 4, J. Mater. Chem. A. 2 (2014) 11948–11957. doi:10.1039/C4TA00934G. I.D. Seymour, D.S. Middlemiss, D.M. Halat, N.M. Trease, A.J. Pell, C.P. Grey, Characterizing Oxygen Local Environments in Paramagnetic Battery Materials via 17 O NMR and DFT Calculations, Journal of the American Chemical Society. 138 (2016) 9405–9408. doi:10.1021/jacs.6b05747. A. Karmakar, M.S. Dodd, S. Agnihotri, E. Ravera, V.K. Michaelis, Cu(II)-Doped Cs 2 SbAgCl 6 Double Perovskite: A Lead-Free, Low-Bandgap Material, Chemistry of Materials. 30 (2018) 8280–8290. doi:10.1021/acs.chemmater.8b03755.
68
[17] M. Fragai, C. Luchinat, G. Parigi, E. Ravera, Conformational freedom of metalloproteins revealed by paramagnetism-assisted NMR, Coordination Chemistry Reviews. 257 (2013) 2652–2667. doi:10.1016/j.ccr.2013.02.009. [18] W. Andrałojć, Y. Hiruma, W.-M. Liu, E. Ravera, M. Nojiri, G. Parigi, C. Luchinat, M. Ubbink, Identification of productive and futile encounters in an electron transfer protein complex, Proceedings of the National Academy of Sciences. 114 (2017) E1840–E1847. doi:10.1073/pnas.1616813114. [19] L. Gigli, W. Andrałojć, A. Dalaloyan, G. Parigi, E. Ravera, D. Goldfarb, C. Luchinat, Assessing protein conformational landscapes: integration of DEER data in Maximum Occurrence analysis, Physical Chemistry Chemical Physics. 20 (2018) 27429–27438. doi:10.1039/C8CP06195E. [20] L.B. Andreas, T. Le Marchand, K. Jaudzems, G. Pintacuda, High-resolution proton-detected NMR of proteins at very fast MAS, Journal of Magnetic Resonance. 253 (2015) 36–49. doi:10.1016/j.jmr.2015.01.003. [21] A. Bertarello, T. Schubeis, C. Fuccio, E. Ravera, M. Fragai, G. Parigi, L. Emsley, G. Pintacuda, C. Luchinat, Paramagnetic Properties of a Crystalline Iron–Sulfur Protein by Magic-Angle Spinning NMR Spectroscopy, Inorganic Chemistry. 56 (2017) 6624–6629. doi:10.1021/acs.inorgchem.7b00674. [22] S. Balayssac, I. Bertini, M. Lelli, C. Luchinat, M. Maletta, K.J. Yeo, Paramagnetic ions provide precious structural restraints in solid-state NMR of proteins, Journal of the American Chemical Society. 129 (2007) 2218–2219. [23] I. Sengupta, P.S. Nadaud, C.P. Jaroniec, Protein Structure Determination with Paramagnetic Solid-State NMR Spectroscopy, Acc. Chem. Res. 46 (2013) 2117–2126. doi:10.1021/ar300360q. [24] I. Sengupta, P.S. Nadaud, J.J. Helmus, C.D. Schwieters, C.P. Jaroniec, Protein fold determined by paramagnetic magic-angle spinning solid-state NMR spectroscopy, Nature Chemistry. 4 (2012) 410–417. doi:10.1038/nchem.1299. [25] A. Bhaumik, C. Luchinat, G. Parigi, E. Ravera, M. Rinaldelli, NMR crystallography on paramagnetic systems: solved and open issues, CrystEngComm. 15 (2013) 8639–8656. doi:10.1039/C3CE41485J. [26] J. Malanho Silva, L. Cerofolini, S. Giuntini, V. Calderone, C.F.G.C. Geraldes, A.L. Macedo, G. Parigi, M. Fragai, E. Ravera, C. Luchinat, Metal centers in biomolecular solid-state NMR, Journal of Structural Biology. (2018). doi:10.1016/j.jsb.2018.11.013. [27] A.J. Pell, G. Pintacuda, C.P. Grey, Paramagnetic NMR in solution and the solid state, Progress in Nuclear Magnetic Resonance Spectroscopy. 111 (2019) 1–271. doi:10.1016/j.pnmrs.2018.05.001. [28] R.J. Kurland, B.R. McGarvey, Isotropic NMR shifts in transition metal complexes: calculation of the Fermi contact and pseudocontact terms, J.Magn.Reson. 2 (1970) 286–301. [29] A.J. Vega, D. Fiat, Nuclear relaxation processes of paramagnetic complexes The slowmotion case, Molecular Physics. 31 (1976) 347–355. doi:10.1080/00268977600100261. [30] I. Solomon, Relaxation Processes in a System of Two Spins, Phys. Rev. 99 (1955) 559–565. doi:10.1103/PhysRev.99.559. [31] A.V. Arbuznikov, J. Vaara, M. Kaupp, Relativistic spin-orbit effects on hyperfine coupling tensors by density-functional theory, J Chem Phys. 120 (2004) 2127–2139. doi:10.1063/1.1636720. [32] T.O. Pennanen, J. Vaara, Nuclear magnetic resonance chemical shift in an arbitrary electronic spin state, Phys. Rev. Lett. 100 (2008) 133002. doi:10.1103/PhysRevLett.100.133002. [33] W. Van den Heuvel, A. Soncini, NMR chemical shift as analytical derivative of the Helmholtz free energy, The Journal of Chemical Physics. 138 (2013) 054113. doi:10.1063/1.4789398. 69
[34] B. Martin, J. Autschbach, Temperature dependence of contact and dipolar NMR chemical shifts in paramagnetic molecules, The Journal of Chemical Physics. 142 (2015) 054108. doi:10.1063/1.4906318. [35] M. Kaupp, M. Bühl, V.G. Malkin, eds., Calculation of NMR and EPR parameters: theory and applications, Wiley-VCH, Weinheim, 2004. [36] F. Neese, Quantum Chemistry and EPR Parameters, in: R.K. Harris, R.L. Wasylishen (Eds.), EMagRes, John Wiley & Sons, Ltd, Chichester, UK, 2017: pp. 1–22. doi:10.1002/9780470034590.emrstm1505. [37] L. Benda, J. Mareš, E. Ravera, G. Parigi, C. Luchinat, M. Kaupp, J. Vaara, Pseudo-Contact NMR Shifts over the Paramagnetic Metalloprotein CoMMP-12 from First Principles, Angew. Chem. Int. Ed. Engl. 55 (2016) 14713–14717. doi:10.1002/anie.201608829. [38] F. Gendron, K. Sharkas, J. Autschbach, Calculating NMR Chemical Shifts for Paramagnetic Metal Complexes from First-Principles, The Journal of Physical Chemistry Letters. 6 (2015) 2183–2188. doi:10.1021/acs.jpclett.5b00932. [39] J. Autschbach, Double perturbation theory: a powerful tool in computational coordination chemistry, Coordination Chemistry Reviews. 238–239 (2003) 83–126. doi:10.1016/S00108545(02)00287-4. [40] G. Parigi, L. Benda, E. Ravera, M. Romanelli, C. Luchinat, Pseudocontact shifts and paramagnetic susceptibility in semiempirical and quantum chemistry theories, J. Chem. Phys. 150 (2019) 144101. doi:10.1063/1.5037428. [41] L. Cerofolini, J.M. Silva, E. Ravera, M. Romanelli, C.F.G.C. Geraldes, A.L. Macedo, M. Fragai, G. Parigi, C. Luchinat, How Do Nuclei Couple to the Magnetic Moment of a Paramagnetic Center? A New Theory at the Gauntlet of the Experiments, J. Phys. Chem. Lett. (2019) 3610–3614. doi:10.1021/acs.jpclett.9b01128. [42] I. Bertini, C. Luchinat, G. Parigi, Magnetic susceptibility in paramagnetic NMR, Progress in Nuclear Magnetic Resonance Spectroscopy. 40 (2002) 249–273. [43] B. Corzilius, Paramagnetic Metal Ions for Dynamic Nuclear Polarization, in: EMagRes, American Cancer Society, 2018: pp. 179–194. doi:10.1002/9780470034590.emrstm1593. [44] J.H. Van Vleck, The theory of electric and magnetic susceptibilities., Oxford, Clarendon Press, 1932. [45] B. Bleaney, Nuclear magnetic resonance shifts in solution due to lanthanide ions, J.Magn.Reson. 8 (1972) 91–100. [46] B.J. Walder, A.M. Patterson, J.H. Baltisberger, P.J. Grandinetti, Hydrogen motional disorder in crystalline iron group chloride dihydrates, The Journal of Chemical Physics. 149 (2018) 084503. doi:10.1063/1.5037151. [47] R. Boča, Mean and differential magnetic susceptibilities in metal complexes, Coordination Chemistry Reviews. 173 (1998) 167–283. doi:10.1016/S0010-8545(98)00139-8. [48] O. Kahn, Molecular magnetism, VCH Publishers Inc., New York, 1993. [49] H.M. McConnell, R.E. Robertson, Isotropic Nuclear Resonance Shifts, The Journal of Chemical Physics. 29 (1958) 1361–1365. doi:10.1063/1.1744723. [50] N.M. Atherton, A.J. Horsewill, Contributions of orbital angular momentum to ligand hyperfine interactions in paramagnetic metal complexes, Journal of the Chemical Society, Faraday Transactions 2. 76 (1980) 660. doi:10.1039/f29807600660. [51] W. Marshall, Hyperfine Interactions with Ligand Nuclei in Transition Ion Complexes, in: W. Low (Ed.), Paramagnetic Resonance, Academic Press, New York, 1963. [52] A. Abragam, The Principles of Nuclear Magnetism, Oxford University Press, Oxford, 1961. [53] H.M. McConnell, D.B. Chesnut, Theory of Isotropic Hyperfine Interactions in π‐Electron Radicals, The Journal of Chemical Physics. 28 (1958) 107–117. doi:10.1063/1.1744052. [54] I. Bertini, C. Luchinat, G. Parigi, The hyperfine shifts in low spin iron(III) hemes: a ligand field analysis, European Journal of Inorganic Chemistry. (2000) 2473–2480. 70
[55] A. Abragam, B. Bleaney, Electron Paramagnetic Resonance of Transition Metal Ions, Clarendon Press, Oxford, 1970. [56] R. Boča, Zero-field splitting in metal complexes, Coordination Chemistry Reviews. 248 (2004) 757–815. doi:10.1016/j.ccr.2004.03.001. [57] J.D. Jackson, Classical electrodynamics, Wiley, New York, 1962. [58] W. Van den Heuvel, A. Soncini, NMR chemical shift in an electronic state with arbitrary degeneracy, Phys. Rev. Lett. 109 (2012) 073001. doi:10.1103/PhysRevLett.109.073001. [59] J. Vaara, S.A. Rouf, J. Mareš, Magnetic Couplings in the Chemical Shift of Paramagnetic NMR, Journal of Chemical Theory and Computation. 11 (2015) 4840–4849. doi:10.1021/acs.jctc.5b00656. [60] S.A. Rouf, J. Mareš, J. Vaara, 1H Chemical Shifts in Paramagnetic Co(II) Pyrazolylborate Complexes: A First-Principles Study, J Chem Theory Comput. 11 (2015) 1683–1691. doi:10.1021/acs.jctc.5b00193. [61] A. Mondal, M.W. Gaultois, A.J. Pell, M. Iannuzzi, C.P. Grey, J. Hutter, M. Kaupp, LargeScale Computation of Nuclear Magnetic Resonance Shifts for Paramagnetic Solids Using CP2K, Journal of Chemical Theory and Computation. 14 (2018) 377–394. doi:10.1021/acs.jctc.7b00991. [62] J. Mareš, J. Vaara, Ab initio paramagnetic NMR shifts via point-dipole approximation in a large magnetic-anisotropy Co( II ) complex, Physical Chemistry Chemical Physics. 20 (2018) 22547–22555. doi:10.1039/C8CP04123G. [63] J. Chyba, M. Novák, P. Munzarová, J. Novotný, R. Marek, Through-Space Paramagnetic NMR Effects in Host–Guest Complexes: Potential Ruthenium(III) Metallodrugs with Macrocyclic Carriers, Inorganic Chemistry. 57 (2018) 8735–8747. doi:10.1021/acs.inorgchem.7b03233. [64] P. Srb, M. Svoboda, L. Benda, M. Lepšík, J. Tarábek, V. Šícha, B. Grüner, K. GrantzŠašková, J. Brynda, P. Řezáčová, J. Konvalinka, V. Veverka, Capturing a dynamically interacting inhibitor by paramagnetic NMR spectroscopy, Physical Chemistry Chemical Physics. 21 (2019) 5661–5673. doi:10.1039/C9CP00416E. [65] I. Bertini, M.B.L. Janik, Y.-M. Lee, C. Luchinat, A. Rosato, Magnetic Susceptibility Tensor Anisotropies for a Lanthanide Ion Series in a Fixed Protein Matrix, Journal of the American Chemical Society. 123 (2001) 4181–4188. [66] I. Bertini, P. Kursula, C. Luchinat, G. Parigi, J. Vahokoski, M. Willmans, J. Yuan, Accurate solution structures of proteins from X-ray data and minimal set of NMR data: calmodulin peptide complexes as examples., Journal of the American Chemical Society. 131 (2009) 5134–5144. [67] A.M. Funk, K.-L.N.A. Finney, P. Harvey, A.M. Kenwright, E.R. Neil, N.J. Rogers, P. Kanthi Senanayake, D. Parker, Critical analysis of the limitations of Bleaney’s theory of magnetic anisotropy in paramagnetic lanthanide coordination complexes, Chemical Science. 6 (2015) 1655–1662. doi:10.1039/C4SC03429E. [68] E.A. Suturina, K. Mason, C.F.G.C. Geraldes, I. Kuprov, D. Parker, Beyond Bleaney’s Theory: Experimental and Theoretical Analysis of Periodic Trends in Lanthanide-Induced Chemical Shift, Angew. Chem. Int. Ed. Engl. 56 (2017) 12215–12218. doi:10.1002/anie.201706931. [69] M. Vonci, K. Mason, E.A. Suturina, A.T. Frawley, S.G. Worswick, I. Kuprov, D. Parker, E.J.L. McInnes, N.F. Chilton, Rationalization of Anomalous Pseudocontact Shifts and Their Solvent Dependence in a Series of C3-Symmetric Lanthanide Complexes, J. Am. Chem. Soc. 139 (2017) 14166–14172. doi:10.1021/jacs.7b07094. [70] E.A. Suturina, I. Kuprov, Pseudocontact shifts from mobile spin labels, Phys. Chem. Chem. Phys. 18 (2016) 26412–26422. doi:10.1039/C6CP05437D. [71] G.T.P. Charnock, I. Kuprov, A partial differential equation for pseudocontact shift, Phys Chem Chem Phys. 16 (2014) 20184–20189. doi:10.1039/c4cp03106g. 71
[72] J.A.B. Lohman, C. Maclean, Alignment effects on high resolution NMR spectra induced by the magnetic field, Chemical Physics. 35 (1978) 269–274. [73] C. Gayathri, A.A. Bothner-By, P.C.M. Van Zijl, C. Maclean, Dipolar magnetic field effects in NMR spectra of liquids, Chem.Phys.Lett. 87 (1982) 192–196. [74] I. Bertini, I.C. Felli, C. Luchinat, High magnetic field consequences on the NMR hyperfine shifts in solution, J.Magn.Reson. 134 (1998) 360–364. [75] M. John, A.Y. Park, G. Pintacuda, N.E. Dixon, G. Otting, Weak Alignment of Paramagnetic Proteins Warrants Correction for Residual CSA Effects in Measurements of Pseudocontact Shifts, Journal of the American Chemical Society. 127 (2005) 17190–17191. doi:10.1021/ja0564259. [76] C. Schmitz, M.J. Stanton-Cook, X.-C. Su, G. Otting, T. Huber, Numbat: an interactive software tool for fitting Δχ-tensors to molecular coordinates using pseudocontact shifts, J Biomol NMR. 41 (2008) 179–189. doi:10.1007/s10858-008-9249-z. [77] J.R. Tolman, J.M. Flanagan, M.A. Kennedy, J.H. Prestegard, Nuclear magnetic dipole interactions in field-oriented proteins: information for structure determination in solution, Proc. Natl. Acad. Sci. U.S.A. 92 (1995) 9279–9283. [78] N. Tjandra, A. Bax, Direct Measurement of Distances and Angles in Biomolecules by NMR in a Dilute Liquid Crystalline Medium, Science. 278 (1997) 1111–1114. doi:10.1126/science.278.5340.1111. [79] F. Tian, H. Valafar, J.H. Prestegard, A dipolar coupling based strategy for simultaneous resonance assignment and structure determination of protein backbones, J. Am. Chem. Soc. 123 (2001) 11791–11796. [80] L. Banci, I. Bertini, J.G. Huber, C. Luchinat, A. Rosato, Partial orientation of oxidized and reduced cytochrome b5 at high magnetic fields: Magnetic susceptibility anisotropy contributions and consequences for protein solution structure determination, Journal of the American Chemical Society. 120 (1998) 12903–12909. [81] N. Tjandra, S. Grzesiek, A. Bax, Magnetic Field Dependence of Nitrogen−Proton J Splittings in 15N-Enriched Human Ubiquitin Resulting from Relaxation Interference and Residual Dipolar Coupling, Journal of the American Chemical Society. 118 (1996) 6264–6272. doi:10.1021/ja960106n. [82] G. Lipari, A. Szabo, Model-Free approach to the interpretation of nuclear magnetic resonance relaxation in macromolecules. 1. Theory and range of validity., Journal of the American Chemical Society. 104 (1982) 4546–4559. [83] M. Rinaldelli, A. Carlon, E. Ravera, G. Parigi, C. Luchinat, FANTEN: a new web-based interface for the analysis of magnetic anisotropy-induced NMR data, Journal of Biomolecular NMR. 61 (2015) 21–34. doi:10.1007/s10858-014-9877-4. [84] J. Kowalewski, L. Maler, Nuclear Spin Relaxation in Liquids: Theory, Experiments, and Applications, Taylor & Francis, 2006. [85] J. Kowalewski, D. Kruk, G. Parigi, NMR relaxation in solution of paramagnetic complexes: recent theoretical progress for S>1, Adv.Inorg.Chem. 57 (2005) 41–104. [86] M. Goldman, Formal Theory of Spin-Lattice Relaxation, J.Magn.Reson. 149 (2001) 160– 187. [87] C.P. Slichter, Principles of magnetic resonance., Springer, Berlin, 1992. [88] R. Kubo, K. Tomita, A General Theory of Magnetic Resonance Absorption, J.Phys.Soc.Japan. 9 (1954) 888–919. [89] I. Bertini, O. Galas, C. Luchinat, G. Parigi, A computer program for the calculation of paramagnetic enhancements of nuclear relaxation rates in slowly rotating systems, Journal of Magnetic Resonance Series A. 113 (1995) 151–158. [90] N. Bloembergen, L.O. Morgan, Proton relaxation times in paramagnetic solutions. Effects of electron spin relaxation, J.Chem.Phys. 34 (1961) 842–850. 72
[91] I. Bertini, C. Luchinat, G. Parigi, E. Ravera, NMR of paramagnetic molecules: applications to metallobiomolecules and models, 2017. [92] M. Rubinstein, A. Baram, Z. Luz, Electronic and nuclear relaxation in solutions of transition metal ions with spin S =3/2 and 5/2, Molecular Physics. 20 (1971) 67–80. doi:10.1080/00268977100100081. [93] P. Caravan, N.J. Cloutier, S.A. McDermid, J.J. Ellison, J.M. Chasse, R.B. Lauffer, C. Luchinat, T.J. McMurry, G. Parigi, M. Spiller, Albumin binding, relaxivity and water exchange kinetics of the diastereoisomers of MS-325 a gadolinium(III) based magnetic resonance angiography contrast agent, Inorganic Chemistry. 46 (2007) 6632–6639. [94] M. Gueron, Nuclear relaxation in macromolecules by paramagnetic ions: a novel mechanism, J.Magn.Reson. 19 (1975) 58–66. [95] N. Bloembergen, Proton relaxation times in paramagnetic solutions, J.Chem.Phys. 27 (1957) 572–573. [96] I. Solomon, N. Bloembergen, Nuclear Magnetic Interactions in the HF Molecule, The Journal of Chemical Physics. 25 (1956) 261–266. doi:10.1063/1.1742867. [97] G. Liu, M. Levien, N. Karschin, G. Parigi, C. Luchinat, M. Bennati, One-thousand-fold enhancement of high field liquid nuclear magnetic resonance signals at room temperature, Nature Chemistry. (2017). doi:10.1038/nchem.2723. [98] W. Müller-Warmuth, R. Vilhjalmsson, P.A.M. Gerlof, J. Smidt, J. Trommel, Intermolecular interactions of benzene and carbon tetrachloride with selected free radicals in solution as studied by 13 C and 1 H dynamic nuclear polarization, Molecular Physics. 31 (1976) 1055– 1067. doi:10.1080/00268977600100811. [99] P.-O. Westlund, A generalized Solomon-Bloembergen-Morgan theory for arbitrary electron spin quantum number S: The dipole-dipole coupling between a nuclear spin I = 1/2 and an electron spin system S = 5/2, Molecular Physics. 85 (1995) 1165–1178. doi:10.1080/00268979500101741. [100] H. Sternlicht, Nuclear Relaxation Induced by Paramagnetic Ions Having Anisotropic g Factors, The Journal of Chemical Physics. 42 (1965) 2250–2251. doi:10.1063/1.1696280. [101] I. Bertini, C. Luchinat, K.V. Vasavada, The effect of magnetic anisotropy on the longitudinal nuclear relaxation time in paramagnetic systems, J.Magn.Reson. 89 (1990) 243–254. [102] I. Bertini, F. Briganti, C. Luchinat, M. Mancini, G. Spina, The electron nucleus dipolar coupling in slow rotating systems (2): the effect of g anisotropy and hyperfine coupling when S=1/2 and I=3/2, J.Magn.Reson. 63 (1985) 41–55. [103] J. Kowalewski, C. Luchinat, T. Nilsson, G. Parigi, Nuclear Spin Relaxation in Paramagnetic Systems: Electron Spin Relaxation Effects under Near-Redfield Limit Conditions and Beyond, J Phys Chem A. 106 (2002) 7376–7382. [104] I. Bertini, O. Galas, C. Luchinat, L. Messori, G. Parigi, A theoretical analysis of the 1H nuclear magnetic relaxation dispersion profiles of diferric transferrin, J.Phys.Chem. 99 (1995) 14217–14222. [105] S.J. Kroes, J. Salgado, G. Parigi, C. Luchinat, G.W. Canters, Electron relaxation and solvent accessibility of the metal site in wild-type and mutated azurins as determined from nuclear magnetic relaxation dispersion experiments, Journal of Biological Inorganic Chemistry. 1 (1996) 551–559. [106] I. Bertini, C. Luchinat, M. Mancini, G. Spina, The electron-nucleus dipolar coupling. The effect of zero-field splitting of an S=3/2 manifold, J.Magn.Reson. 59 (1984) 213–222. [107] I. Bertini, C. Luchinat, Z. Xia, Solvent water 1H NMRD study of oxovanadium(IV) aquaion, J.Magn.Reson. 99 (1992) 235–246. [108] I. Bertini, F. Capozzi, C. Luchinat, G. Nicastro, Z. Xia, Water proton relaxation for some lanthanide aqua ions in solution, J.Phys.Chem. 101 (1993) 198–201.
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[109] I. Bertini, J. Kowalewski, C. Luchinat, T. Nilsson, G. Parigi, Nuclear spin relaxation in paramagnetic complexes of S=1: Electron spin relaxation effects, J.Chem.Phys. 111 (1999) 5795–5807. [110] D. Kruk, T. Nilsson, J. Kowalewski, Nuclear spin relaxation in paramagnetic systems with zero-field splitting and arbitrary electron spin, Phys.Chem.Chem.Phys. 3 (2001) 4907–4917. [111] T. Larsson, P.O. Westlund, J. Kowalewski, S.H. Koenig, Nuclear-spin relaxation in paramagnetic complexes in the slow-motion regime for the electron spin: the anisotropic pseudorotation model for S=1 and the interpretation of nuclear magnetic relaxation dispersion results for a low-symmetry Ni(II) complex, J.Chem.Phys. 101 (1994) 1116–1128. [112] P.H. Fries, E. Belorizky, Relaxation theory of the electronic spin of a complexed paramagnetic metal ion in solution beyond the Redfield limit, The Journal of Chemical Physics. 126 (2007) 204503. doi:10.1063/1.2730831. [113] S. Rast, P.H. Fries, E. Belorizky, A. Borel, L. Helm, A.E. Merbach, A general approach to the electronic spin relaxation of Gd(III) complexes in solutions. Monte Carlo simulations beyond the Redfield limit, The Journal of Chemical Physics. 115 (2001) 7554–7563. doi:10.1063/1.1392364. [114] E. Belorizky, P.H. Fries, L. Helm, J. Kowalewski, D. Kruk, R.R. Sharp, P.-O. Westlund, Comparison of different methods for calculating the paramagnetic relaxation enhancement of nuclear spins as a function of the magnetic field, The Journal of Chemical Physics. 128 (2008) 052315. doi:10.1063/1.2833957. [115] E.A. Suturina, K. Mason, C.F.G.C. Geraldes, N.F. Chilton, D. Parker, I. Kuprov, Lanthanideinduced relaxation anisotropy, Physical Chemistry Chemical Physics. 20 (2018) 17676– 17686. doi:10.1039/C8CP01332B. [116] M. Goldman, Interference Effects in the Relaxation of a Pair of Unlike Spin-1/2 Nuclei, J.Magn.Reson. 60 (1984) 437–452. [117] I. Bertini, C. Luchinat, M. Piccioli, D. Tarchi, COSY spectra of paramagnetic macromolecules, observability, scalar effects, cross correlation effects, relaxation allowed coherence transfer, Concepts in Magnetic Resonance. 6 (1994) 307–335. [118] I. Bertini, C. Luchinat, D. Tarchi, Are true scalar proton-proton connectivities ever measured in COSY spectra of paramagnetic macromolecules?, Chem.Phys.Lett. 203 (1993) 445–449. [119] R. Ghose, J.H. Prestegard, Electron Spin-Nuclear Spin Cross-Correlation Effects on Multiplet Splittings in Paramagnetic Proteins, J.Magn.Reson. 128 (1997) 138–143. [120] I. Bertini, J. Kowalewski, C. Luchinat, G. Parigi, Cross-correlation between the dipole-dipole interaction and the Curie spin relaxation. The effect of anisotropic magnetic susceptibility, J.Magn.Reson. 152 (2001) 103–108. [121] G. Pintacuda, A. Kaikkonen, G. Otting, Modulation of the distance dependence of paramagnetic relaxation enhancements by CSA×DSA cross-correlation, Journal of Magnetic Resonance. 171 (2004) 233–243. doi:10.1016/j.jmr.2004.08.019.
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Graphical abstract
μI
μ el
B0 0
𝜒
Hyperfine shifts Residual dipolar couplings NMR relaxation
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Highlights Spin-orbit coupling causes ZFS, g-tensor and magnetic susceptibility anisotropy The average electron magnetic moment depends on the magnetic susceptibility The dipolar interaction with the average electron magnetic moment causes NMR shifts Hyperfine shifts can be expressed as a function of g-tensor and ZFS Hyperfine shifts, RDCs and relaxation depend on the magnetic susceptibility
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