 1.1 Spin Interactions: A General Introduction
 1.1.1 Magnetic Moments
 1.1.2 Zeeman Interaction
 1.1.3 Internal NMR Spin Interactions
 1.2 Paramagnetic Interactions in NMR
 1.2.1 Magnetic Interactions in Battery Materials
 1.2.2 Hyperfine Interactions in NMR
 1.3 Calculating Paramagnetic NMR Parameters
 1.3.1 NMR Shifts of Paramagnetic Molecules
 1.3.2 NMR Shifts of Paramagnetic Battery Materials
 References
CHAPTER 1: NMR Principles of Paramagnetic Materials

Published:17 Jun 2021

Series: New Developments in NMR
L. Peng, R. J. Clément, M. Lin, and Y. Yang, in NMR and MRI of Electrochemical Energy Storage Materials and Devices, ed. Y. Yang, R. Fu, and H. Huo, The Royal Society of Chemistry, 2021, pp. 170.
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This chapter gives a general introduction to NMR interactions in solids. More details on paramagnetic interactions of NMR, often dominating the spectra of paramagnetic solids, are then presented. A brief introduction to the firstprinciples calculations for NMR spectra of paramagnetic solids is also provided, which play an important role in spectral assignments.
1.1 Spin Interactions: A General Introduction
In 1945, American physicists Felix Bloch^{1 } and Edward Mills Purcell^{2 } discovered the phenomenon of nuclear magnetic resonance (NMR) independently, which can be observed when nuclear spins are placed in a magnetic field. With the development of pulsed Fourier transform NMR and multidimensional NMR spectroscopy by Richard Robert Ernst,^{3,4 } NMR has gradually become one of the most powerful analytical methods that provides detailed structural information of matter at atomic/molecular scale, as well as physical/chemical/electronic properties. The major reason behind this fact is that there are many different types of interaction involving the nuclear spins in the sample, which can be exploited to provide such information. The NMR interactions can be treated with quantum mechanical approaches and readers are encouraged to find more details in some valuable books and review papers.^{5–13 } In this section, only a brief introduction on these NMR interactions will be presented.
1.1.1 Magnetic Moments
A nucleus possesses a nuclear spin angular momentum I, leading to a magnetic dipole moment µ_{I},
where ℏ is Planck's constant divided by 2π and γ_{I} is the nuclear gyromagnetic ratio, a constant depending on the nuclear species. The value of γ_{I} can be positive or negative, meaning that the nuclear magnetic moment µ_{I} is parallel or antiparallel to the spin I, respectively (see Figure 1.1).
For paramagnetic systems, where there are unpaired electrons, it is also important to describe the electronic magnetic moment, since these electrons will interact with the nuclear species of interest and have an effect on the NMR spectra (shift, linewidth, relaxation, etc.). Similar to the case for the nucleus, the electron magnetic moment µ_{S} is related to the electronic spin S by
where γ_{S} is the electron gyromagnetic ratio, which is negative, indicating the electron magnetic moment µ_{S} is always antiparallel to the electronic spin S (see Figure 1.1b).
In electron paramagnetic resonance (EPR), the relationship between the electronic magnetic moment µ_{S} and electron spin S is also expressed as
where g_{e} and µ_{B} are the gvalue of the free electron (approx. 2.002319) and Bohr magneton, respectively. Therefore,
1.1.2 Zeeman Interaction
A magnetic dipole moment µ interacts with a magnetic field B, and the energy of this interaction, E, is equal to their scalar product, therefore,
This equation clearly shows that the interaction energy is dependent on the relative orientation of the magnetic moment µ with respect to the magnetic field B. The corresponding expression for the Hamiltonian is
where is the magnetic moment operator.
In NMR spectroscopy, the sample is placed in an external magnetic field B_{0}, thus the nuclear magnetic moment interacts with B_{0}, leading to Zeeman interaction. Its Hamiltonian Ĥ_{Z} is
where Î is the nuclear spin operator.
The external magnetic field B_{0} in NMR is usually set along the zdirection of the laboratory frame, i.e., B_{0} = (0, 0, B_{0}). Therefore eqn (1.7) becomes
where Î_{z} is the operator for the component of the spin along the z axis.
A single nuclear spin I has (2I + 1) states according its magnetic quantum number, m_{I}, where m_{I} = −I, (−I + 1),…, (I − 1), I. These states, which can be expressed as m_{I}〉, are the eigenstates of Zeeman Hamiltonian, and the eigenvalues of Ĥ_{Z}, or the possible energy levels, are
In the simplest case, I = 1/2 (e.g., ^{1}H, ^{13}C or ^{19}F), and m_{I} = −1/2 or +1/2. At zero external magnetic field, the two states (e.g., −1/2〉 and +1/2〉) are degenerate. However, with a nonzero magnetic field, the energy levels of the two states split (see Figure 1.2). The separation between the two energy levels is γ_{I}ℏB_{0}, which increases linearly with the external magnetic field. Classically, transitions between −1/2〉 and +1/2〉 (or in general, Δm_{I} = ±1), can occur and an NMR signal can be observed by applying an electromagnetic radiation with a frequency that matches the energy difference. This frequency, ω_{0}, is Larmor or resonance frequency (rad^{−1}), which can be derived from
Apparently, ν_{0}, Larmor or resonance frequency in Hz, is
In a real sample containing an ensemble of nuclear spins, it can be expected that the states with a higher energy is less populated. The populations of the two states follow the Boltzmann distribution at equilibrium, which is given by
where N is the total number of nuclear spins, N_{±1/2} are the number of spins at corresponding states, T is the absolute temperature and k_{B} is the Boltzmann's constant. Under high temperature approximation, eqn (1.12) becomes
The population difference, N_{+1/2} − N_{−1/2} is therefore
The magnetization of the sample, M, which is the net nuclear magnetic moment contributed from each individual spin, and the NMR signal can be further analyzed. The equilibrium magnetization M_{0} and the NMR signal S_{NMR}, are given by^{9 }
where V_{c} is the sample volume. Eqn (1.14)–(1.16) also provide some key insights into the characteristics of NMR spectroscopy. Since is small at common external magnetic field and temperature, the sensitivity of NMR is unfortunately low and it is often required to have a relatively large number of spins (N). In order to obtain stronger NMR signals, the nucleus with a larger gyromagnetic ratio (γ_{I}) can be chosen for investigation and usually a higher external magnetic field (B_{0}) is preferred.
Similarly, there is Zeeman interaction between an electron spin S and an external magnetic field B_{0}.
1.1.3 Internal NMR Spin Interactions
In addition to the external magnetic field, the total energy of a nucleus is also affected by different internal spin interactions between the nucleus and its local environment. Therefore, if the relaxation part is not considered, the total Hamiltonian of nuclear spin, Ĥ_{Total}, can be expressed as
where Ĥ_{Ext} is the Hamiltonian for external interactions including Zeeman interactions between the spin and external magnetic fields (static and vertical field B_{0} and radiofrequency field B_{1}), while Ĥ_{Int} is the Hamiltonian for internal interactions. The latter can be expressed by
where Ĥ_{CS}, Ĥ_{D}, Ĥ_{J}, Ĥ_{Q} and Ĥ_{HFC} represent chemical shielding interactions, direct dipolar interactions, indirect spin–spin interaction, quadrupolar interaction (for quadrupolar nucleus only, I > 1/2) and hyperfine interaction (for paramagnetic samples), respectively.
It has been shown that all the NMR interactions can be described by a Hamiltonian in the following form:
where K_{λ}, R_{λ}and Â_{λ} are the constant, the secondrank tensor related to spatial orientation, and the coupling partner of spin vector Î, which can be another spin vector of the same or different spin, or a magnetic field, for a specific interaction λ, respectively. It clearly shows that internal NMR interactions are anisotropic, meaning that they depend on the relative orientation of the spin vector and its coupling partner. In solution NMR, fast molecular tumbling averages the anisotropic interactions and sharp resonances can be observed. However, in solidstate NMR spectroscopy, these anisotropic interactions lead to broad peaks due to lack of rapid molecular motion. In order to obtain highresolution NMR data and extract detailed structural information of solid materials, such interactions must be carefully treated with rational design and application of a variety of solidstate NMR techniques.
1.1.3.1 Chemical Shielding Interaction
The electrons surrounding the nucleus circulate around due to the presence of an external magnetic field. This movement produces an induced magnetic field, which modifies the effective field experienced by the nucleus. In a diamagnetic material, it is often the case that the induced field is in the opposite direction to the external field, making the nucleus experience a smaller magnetic field. Therefore, a “shielding” effect arises from the interaction between the nuclear spin and the field induced by the surrounding electrons. Since it reflects the local electronic environments, this interaction is called the “chemical shielding interaction” and its Hamiltonian is given by
where σ is the chemical shielding tensor,
a secondrank Cartesian tensor with a 3 × 3 matrix defining its relative orientation with respect to the laboratory frame, while B_{local} is the local magnetic field induced. Therefore, considering the external magnetic field B_{0} used in NMR (along the zdirection) and the shielding effect B_{local}, the actual magnetic B the nucleus experienced is
where 1 is the unit matrix.
In common NMR experiments, strong external magnetic field B_{0} is applied in the zdirection and Zeeman interaction can be considered dominating. Eqn (1.22) shows that σ_{zz} is the only component aligned with B_{0}, therefore only this term gives major contributions and now the Hamiltonian of the chemical shielding interaction becomes
and resonance frequency of the nucleus changes accordingly.
The chemical shielding tensor σ can be transformed to a diagonalized tensor σ^{PAS} in its principal axis system (PAS), in which σ_{11}, σ_{22}, σ_{33} (σ_{33} ≥ σ_{22} ≥ σ_{11}) are the eigenvalues of the tensor, appearing on the diagonal of the matrix and mostly determining the NMR spectra.
The value of σ_{zz} is related to the three eigenvalues by
where θ and ϕ are the polar angles relating the orientation of the laboratory frame with PAS (see Figure 1.3). In an isotropic, liquid sample, σ_{zz} can be replaced by isotropic chemical shielding σ_{iso}, given by
owing to rapid molecular tumbling, which averages out the anisotropic effects of chemical shielding interactions. In solids, however, lack of this type of motion leads to anisotropic effect which is related to the relative orientation of the tensor of the nucleus.
In NMR experiments, it is conventional to use chemical shift to describe the local environment of a nucleus. Chemical shift and chemical shielding are related by
where δ, ν and σ are the chemical shift, resonant frequency and chemical shielding (δ_{reference} = 0). Therefore, a diagonalized chemical shift tensor with eigenvalues of δ_{11}, δ_{22} and δ_{33} (δ_{11} ≥ δ_{22} ≥ δ_{33}) can be obtained according to the chemical shielding tensor. The isotropic average of this tensor is thus
The parameter's anisotropy, span (Ω), and asymmetry parameter, skew (κ), can be defined as
respectively, to describe the chemical shift anisotropy (CSA) of the chemical shielding interactions.
In a solid sample, all the possible orientations of the crystal (or the possible values of θ and ϕ) contribute to the observed NMR signal, leading to the powder pattern (see Figure 1.4a), which can be used to extract the values of δ_{11}, δ_{22} and δ_{33}, as well as δ_{iso}, Ω and κ. δ_{iso} is often used as a “finger print” for the examination of the environment of the nucleus, while CSA parameters Ω and κ can also provide key information on the asymmetry of the local structure the nucleus. However, it is difficult to obtain δ_{iso} when different sites are present and their resonances overlap to produce featureless spectra. Since there is a common term (3cos^{2}θ − 1) describing the dependence of the interactions (chemical shielding, direct dipolar and firstorder quadrupolar) on the polar angle (vide infra), it has been shown that rotating the sample at a “magic angle” of 54.736° with respect to the external magnetic field can significantly reduce the effects of anisotropic interactions on the spectrum.^{14 } This commonly used method to acquire highresolution data in solidstate NMR is called magic angle spinning (MAS), which can be considered as introducing fast motion to the molecules in the solid sample. With a slow MAS rate, the spinning sideband manifolds resemble the powder pattern. With increasing MAS rate, anisotropic interactions are reduced to a larger extent, leading to simpler spectrum (see Figure 1.4b).
1.1.3.2 Direct Dipolar Interaction
The magnetic dipole moment of spin I_{1} and I_{2} interact with each other directly. It can be considered as the interaction between the magnetic moment of a nuclear spin and the local magnetic field generated by a spin nearby. The Hamiltonian of this direct dipolar interaction (direct dipole–dipole interaction) is given by
where D is the dipolar coupling tensor, a symmetric and traceless tensor in Cartesian coordinates, while r̂ is the dipolar vector between spins I_{1} and I_{2}, r is the internuclear distance and b_{12} is the dipolar coupling constant, which is defined as
where γ_{1} and γ_{2} are the gyromagnetic ratio of spins I_{1} and I_{2}, respectively and µ_{0} is the permeability constant (4π × 10^{−7} kg m s^{−2} A^{−2}).
In common NMR experiments, the highfield approximation can be applied and the Hamiltonian can be simplified as
where θ is the polar angle related to the orientation of the dipolar vector r and the external magnetic field (see Figure 1.5a), while I_{1±} = I_{1x} ± iI_{1y} and I_{2±} = I_{2x} ± iI_{2y}.
The term I_{1z}I_{2z} represent the secular dipolar coupling, which can be treated as perturbations to Zeeman interaction, leading to broad resonances in NMR. (I_{1+}I_{2−} + I_{1−}I_{2+}) is the “flipflop” term, which is associated transitions between different states in this spin pair. In order to have a significant probability for the “flipflop” process, it must be energy conserving, which requires that the spins I_{1} and I_{2} are the same nucleus (i.e., homonuclear dipolar coupling). For heteronuclear spin pairs, the “flipflop” term vanishes and eqn (1.33) can be further simplified as
Eqn (1.31)–(1.34) clearly show that the direct dipolar interactions are angular dependent. In solution NMR, fast random motion of molecules removes direct dipolar interactions. In solid state NMR, however, the fixed dipolar vectors in the solid are randomly oriented with respect to the external magnetic field, generating broad resonances (Pake doublet, see Figure 1.5b)^{16 } arising from the same species. Therefore, techniques such as MAS should be used to obtain high resolution spectra.
The direct dipolar interaction, however, depends strongly on the internuclear distance r as shown in eqn (1.31). Therefore, investigations of direct dipolar coupling provide key structural information such as the distance between two atoms or through space connectivity. It is common practice to reintroduce direct dipolar coupling under MAS in order to extract structural information at high spectral resolution. Details of such techniques can be found elsewhere.^{8–10 }
1.1.3.3 Indirect Spin–Spin Interaction: Scalar Coupling
In addition to direct dipolar coupling (throughspace), the magnetic moment of spins can also be coupled via the electrons in the chemical bond connecting nuclei. This type of throughbond interaction is indirect spin–spin interaction, also known as Jcoupling or scalar coupling. The Hamiltonian for this interaction is
where J is the tensor that reflects the orientation dependence of the Jcoupling with respect to the external magnetic field. Unlike the direct dipolar interaction, the trace of the tensor J is not zero. Therefore, the isotropic Jcoupling (scalar coupling constant) is given by
and in the heteronuclear case, the Hamiltonian of Jcoupling is
Because the Jcoupling is not averaged to zero by fast tumbling of molecules, it may have an observable effect in both solution and solidstate NMR spectra. The Jcoupling between two spins leads to the splitting of the peak in the spectrum to generate lines evenly spaced (see Figure 1.6a). This “fine structure” shown in the spectrum can provide important information in chemical bonding. For example, small doublets with a separation of 550 Hz (corresponding to Jcoupling of ^{1}H–^{31}P) can be observed in nondecoupled ^{31}P MAS NMR spectrum of zeolite HY adsorbed with (CH_{3})_{3}P, indicating the formation of chemisorbed (CH_{3})_{3}P–H^{+}, while the doublets disappear in decoupled spectrum (Figure 1.6b).^{17 } In many cases, however, Jcoupling is only tens of Hz and much weaker than other NMR interactions in solid, and is not observed.
1.1.3.4 Quadrupolar Interaction
The nucleus with a spin quantum number I > 1/2 (70% in the periodic table) is quadrupolar, indicating that the electric charge distribution of the nucleus is not spherical, leading to a nonzero nuclear electric quadrupole moment (eQ). This quadrupole moment further interacts with the electric field gradient (EFG) at the nucleus, which arises from the distribution of surrounding nuclei and electrons, resulting in quadrupolar interaction. The Hamiltonian for this interaction is given by
where V is the EFG tensor, a symmetric traceless Cartesian tensor in its PAS, described by only three components in the diagonal, V_{XX}, V_{YY} and V_{ZZ} (Figure 1.1.3a). The magnitude these components are ordered as V_{ZZ} > V_{YY} > V_{XX}, while V_{XX} + V_{YY} + V_{ZZ} = 0. The size of EFG is defined as
while the shape of the tensor can be described by the asymmetry parameter, η, which is given by
A quadrupolar coupling constant, C_{Q} (in Hz), which a measure of the size of quadrupolar interaction, can be defined as
C_{Q} can be used to describe local distortion around the nucleus, while η provides the information of the local symmetry, specifically, the deviation of EFG from axial symmetry of the nucleus of interest.
In NMR spectroscopy, quadrupolar interaction is often treated as a small perturbation of Zeeman interaction (firstorder and secondorder quadrupolar interactions). Details on the treatment of quadrupolar interaction (i.e., the energy levels, powder pattern, etc.) can be found elsewhere in this book. In general, MAS is often used to remove firstorder quadrupolar coupling to achieve high resolution observation. However, secondorder quadrupolar coupling results in characteristic line shape and additional shift. The shift of the center of gravity (δ_{CG}) of the central line is at a lower frequency than the isotropic chemical shift δ_{iso} (fieldindependent) because of secondorder quadrupolar coupling and they are related by
where ν_{0} is the Larmor frequency in Hz.^{18,19 }Eqn (1.42) clearly shows that the experimentally observed shift is more negative with a higher C_{Q} and at a smaller external magnetic field. Since quadrupolar interaction is usually the largest line broadening factor in solidstate NMR spectrum of quadrupolar nuclei and secondorder quadrupolar coupling is inversely proportional to the external field, a high B_{0} is preferred to obtain high resolution NMR spectra. Figure 1.7 shows the ^{27}Al MAS NMR spectra of aluminoborate 9Al_{2}O_{3} + 2B_{2}O_{3} (A_{9}B_{2}) at different external magnetic fields. Since the Al ions in A_{9}B_{2} are associated with large C_{Q}s, the linewidths are greatly reduced at a higher external field, providing spectra with better resolution.
1.1.3.5 Hyperfine Interaction
Many materials in energy applications are paramagnetic, in which the hyperfine interaction between the nucleus and the unpaired electrons plays an important role in determining the features in the NMR spectrum. The two major contributions to the hyperfine interaction are Fermicontact interaction and dipolar coupling, which can be considered as the Jcoupling and direct dipolar interaction, respectively, between a nucleus and the unpaired electrons, instead of another nuclear spin.
Fermicontact interaction is dependent on the unpaired electron spin density transferred to the nuclear spin of interest, ρ(r = 0), where r is the distance to the center of the nucleus. Because the electron spin density can be transferred through chemical bond, Fermicontact interaction provides important information on the chemical bonding of the paramagnetic center. Unlike dipolar coupling between two nuclear spins, the nuclear spins can only interact with the timeaveraged magnetic moment of the electrons or the paramagnetic ions, leading to a line shape similar to CSA rather than a Pake doublet. Again, the size of hyperfine dipolar coupling is inversely proportional to the cube of the distance between the nuclear and electronic spins.
The Hamiltonian of hyperfine coupling can be written as
where A is the hyperfine coupling tensor and the contributions from dipolar coupling and Fermicontact interaction can be more clearly seen in the following expression
where χ is the magnetic susceptibility, and ρ(r) is the total spinunpaired electron density at position r.
Fermicontact interaction and dipolar coupling broaden the peaks and cause additional shifts, making the spectra of paramagnetic materials more difficult to analyze than diamagnetic materials. Eqn (1.44) indicates that the size of hyperfine interaction is proportional to both the gyromagnetic ratio γ_{I} and the external magnetic field B_{0}, and its dependence on B_{0} is different from the direct dipolar and indirect spin–spin interactions between nuclear spins. Therefore, in order to decrease the size of hyperfine interaction and simplify the spectra, it is better to perform NMR of paramagnetic materials with the nucleus associated with a lower gyromagnetic ratio, at lower external field with faster MAS rate.^{21 } Figure 1.8 shows the comparison of ^{6}Li and ^{7}Li MAS NMR spectra of paramagnetic LiMn_{2}O_{4} material. ^{6}Li, which has a smaller gyromagnetic ratio thus less sensitivity, is associated with smaller hyperfine interactions. Therefore, the ^{6}Li NMR spectrum is better resolved than the corresponding ^{7}Li NMR spectrum.
1.2 Paramagnetic Interactions in NMR
As discussed in the previous section, electronnuclear spin interactions strongly impact the NMR spectra of paramagnetic materials, including battery electrodes that contain redoxactive transition metal species. To correctly interpret paramagnetic NMR spectra, an understanding of the spin, orbital and spin–orbit contributions to the electronic magnetic moment, which in turn dictates the magnetic properties of the material, is required. In this section, electron–nuclear spin interactions in the case of magnetically isotropic systems are described before the more complicated case of materials with magnetic anisotropy is examined. A comparison of the spectral signatures of paramagnetic interactions in the case of magnetic isotropy and anisotropy is also provided.
NMR is a noninvasive and quantitative technique which allows sequences of events in rechargeable battery materials to be followed by studying electrochemical cells operando,^{23–29 } or, more commonly, electrode samples stopped at different stages of charge and discharge ex situ.^{30–34 } As a local structure probe, it is well suited for the investigation of amorphous or disordered systems, such as charged/discharged electrode materials. NMR is a sitespecific technique, and crystallographicallyunique sites or local environments in a material can be distinguished on the basis of their resonant frequencies or chemical shifts, provided that these frequency/shift differences are larger than the linewidth of the signals. This is usually not an issue for battery electrodes containing openshell transition metals, as the paramagnetic chemical shift of a species, such as ^{7}Li, ^{23}Na, ^{31}P or ^{19}F, is typically large and highly sensitive to the local geometry and to the number and oxidation state of nearby transition metals (TMs), resulting in well separated signals. Hence, NMR is ideal for the study of local distortions and variations in the electronic structure and in the oxidation states of redoxactive TM species during charge and discharge.^{21 } Paramagnetic NMR can also monitor charge ordering transitions and the migration of electrochemically active or inactive species in electrode materials.
1.2.1 Magnetic Interactions in Battery Materials
We begin this section by presenting the fundamental magnetic properties of TMcontaining compounds, thereby laying the conceptual foundations for understanding paramagnetic interactions in common battery electrodes.
The contribution of nuclear spins to the magnetism of substances is negligible except at very low temperatures. At most temperatures, the macroscopic magnetic properties of atoms, molecules and solids depend on the number and distribution of paired and unpaired electrons.^{35 } In the case of battery electrode materials, these properties are by and large determined by the redoxactive TMs, and more specifically, by microscopic magnetic moments associated with unpaired electrons borne by the TMs. The next few sections describe the various contributions to the magnetic moment of openshell TM ions.
1.2.1.1 Orbital and Spin Magnetic Moments
Openshell TM ions exhibit a magnetic moment originating from the orbiting motion of unpaired electrons with a finite charge around the nucleus. The microscopic magnetic moment associated with each electron can be viewed classically as the magnetic field generated by a circulating current. Namely, a current i flowing around an elementary oriented loop of area dS will create a magnetic moment
with units of A m^{2}.^{36 } The length of the dS vector is equal to the area of the loop, and its direction is normal to the loop and set by the righthand rule with respect to the direction of the current around the loop, as shown in Figure 1.9.^{36 }
For a loop of finite size, the magnetic moment µ is obtained by summing up the magnetic moments associated with a large number of infinitesimal current loops distributed throughout the area of the loop,^{36 } as shown in Figure 1.9b, and
The orbiting motion of electrons around the nucleus not only leads to a charge current, but also to the motion of particles with a finite mass. As such, a magnetic moment is always associated with angular momentum. The orbital angular momentum is described by the quantum number L, which takes integer values. The application of an external magnetic field leads to a Zeeman interaction between the angular momentum and the field, and each level L splits into 2L + 1 nondegenerate states labelled by a unique magnetic quantum number m_{L}, which takes integer values between −L and +L.
In TM ions, the magnetic moment µ_{L} associated with an orbiting electron lies along the direction of the angular momentum L of that electron and is proportional to it:
where γ_{S} is the gyromagnetic ratio of the electron, ℏ is Planck's constant divided by 2π, and µ_{B} is the Bohr magneton. Since the electron has a negative charge, µ_{L} is antiparallel to L and the electron gyromagnetic ratio, γ_{S}, is negative:
where −e and m_{e} are the electron charge and electron mass, respectively. The Bohr magneton,
represents the magnetic moment of an electron orbiting around the nucleus of a hydrogen atom in its ground state, i.e., with an angular momentum of ℏ and in a circular orbit of radius r = a_{0}.^{36 }
Electrons possess a second and intrinsic source of angular momentum due to their spin S. The spin magnetic moment of a free electron is given by:
where g_{e} is the free electron gfactor (g_{e} ≈ 2.002319), and S always takes the value of ½. Upon application of an external magnetic field, the Zeeman interaction splits the S state into 2S + 1 = 2 nondegenerate states with magnetic quantum numbers .
While µ_{S} results in spatially isotropic magnetic properties, µ_{L} introduces anisotropy, whereby the magnetic response of the system to the application of an external magnetic field depends on the orientation of the field with respect to that of the principal axis of the molecule or crystal under consideration. Importantly, since L = S = 0 for closedshell orbitals, the magnetic properties of TM ions arise from the interactions between the magnetic moments of unpaired electrons in openshell valence orbitals.
1.2.1.2 Interactions Between Angular Momenta
In multielectron systems, electronic angular momenta interact via couplings between individual orbital angular momenta (L_{i}–L_{j}), between individual spin angular momenta (S_{i}–S_{j}), and between orbital and spin angular momenta (L_{i}–S_{j}). These socalled spin–orbit coupling (SOC) interactions are a major source of magnetic anisotropy and greatly impact the NMR properties of TMcontaining systems.
While the spin and orbital angular momenta of isolated electrons are conserved within the spherically symmetric field of the nucleus, such that L̂^{2} and Ŝ^{2} commute with the Hamiltonian Ĥ ([L̂^{2}, Ĥ] = 0, [Ŝ^{2}, Ĥ] = 0), interactions between them break this symmetry and individual angular momenta are no longer conserved individually but the angular momentum of the system as a whole is conserved. A total angular momentum J is therefore defined, where J is a good quantum number as Ĵ^{2} commutes with Ĥ. As for the L and S angular momenta, upon application of an external magnetic field, each level J splits into 2J+1 nondegenerate states labelled by a unique magnetic quantum number m_{J} that takes integer values between +J and −J.
The presence of radiallycontracted 3d orbitals leads to strong interelectron interactions. Hence, in battery systems containing 3d TM ions, L_{i}–S_{i} couplings are generally much smaller than L_{i}–L_{j} and S_{i}–S_{j} couplings and can be considered as a perturbation. The Russell–Saunders coupling scheme can therefore be applied, where individual electron spin momenta (S_{i}) couple strongly to give a total spin , and individual orbital momenta (L_{i}) couple strongly to give a total orbital contribution . In this limit, the total angular momentum J is determined through coupling of the S and L vectors: J = L + S, as shown in Figure 1.10a. The quantum number J varies in integer steps between L + S and L − S, resulting in (2L + 1)(2S + 1) possible electronic states labeled with term symbols of the form ^{2S+1}L_{j}.
1.2.1.3 gfactors
Similar to the gyromagnetic ratio, the gfactor relates the magnetic moment to the angular momentum. While the free electron (S = 1/2) gfactor g_{e} ≈ 2.002319, gfactors of individual particles and TM ions can deviate substantially from g_{e}, due to the presence of orbital angular momentum and SOC. For a multielectron TM ion, the total magnetic moment operator , depicted in Figure 1.10b, is given by
where g_{L} = 1 and g_{e} ≈ 2 are the gfactors of the orbital and spin angular momenta, respectively.^{34 }
As shown in Figure 1.10, the vectors µ and J are not parallel because g_{e} ≠ 1. Yet, only the component of µ parallel to J is conserved, and the component of the magnetic moment operator that appears in the interaction Hamiltonian is that which commutes with Ĵ and its matrix elements are proportional to those of Ĵ.^{11 } Applying the projection theorem,^{11 } the total magnetic moment operator can be rewritten as:
where g_{J} is the Landé gfactor given by
g_{J} takes values between 1 and 2, such that g_{J} = 1 for S = 0 and g_{J} = 2 for L = 0. A complete derivation of g_{J} can be found in ref. 11 and 36.
As discussed in Section 1.1, the magnetic spin dipole moment operator of a nucleus, _{I}, is given by:
where Î is the spin angular momentum operator and γ_{I} is the nuclear gyromagnetic ratio. The product
where g_{I} is the dimensionless nuclear gfactor, which depends on the isotope and takes the same sign as γ_{I}. µ_{N} is the nuclear magneton:
where e is the elementary charge, and m_{p} is the rest mass of the proton. Substituting the expression for ℏγ_{I}, an alternative expression for the nuclear spin magnetic moment, in terms of the nuclear gfactor, is obtained:
1.2.1.4 Ligand Field Splitting
The magnetic properties of isolated TM ions, or free ions, can be fully described in terms of the spin, orbital, and SOC contributions to the magnetic moments of unpaired electrons occupying degenerate valence d orbitals. As described by ligand field theory,^{37 } in molecules and solids the presence of nearby ligand atoms leads to strong interactions between TM d orbitals and ligand orbitals. These interactions, in turn, result in a splitting of the degenerate d manifold of the free ion into nondegenerate sets of d orbitals.
For openshell TM species, lifting of the orbital degeneracy results in partial or complete quenching of the orbital angular momentum, L, and strongly influences their magnetic properties. For example, for a high spin d^{3} TM ion octahedrally and tetrahedrallycoordinated to ligand atoms, the ligand field alters the d orbital energies in such a way that full quenching (whereby the effective orbital angular momentum, L_{eff}, is zero) and partial quenching (L_{eff} = 1) of L is observed, respectively, as shown in Figure 1.11.
For systems where the orbital angular momentum is, to a first approximation, quenched (L_{eff} = 0), the resulting magnetic moment depends solely on the spin angular momentum S, which remains unaffected by the ligand field. However, if the spin and orbital magnetic moments are coupled by SOC, the ligand field is unable to effect a perfect separation of electronic states with the same spin multiplicity. As a result, the ground state may gain some orbital angular momentum through mixing in of excited states of the same spin multiplicity, as shown in Figure 1.12c. The extent of state mixing depends on the energy separation of the excited state from the ground state, and on the strength of SOC, quantified by the SOC constant λ.
1.2.1.5 Zerofield Splitting
In multielectron systems, the presence of a noncubic ligand field and strong spin–spin interactions can lead to zerofield splitting (ZFS) effects. ZFS results from interactions between electronic magnetic moments and occurs in the absence of a magnetic field, as its name suggests. An energy level diagram illustrating ZFS effects is presented in Figure 1.12, where, for convenience, we set and L = 0. In an axiallysymmetric ligand field, the fourfold degeneracy of the Zeeman sublevels (Figure 1.12a) is partially lifted, with the states separated by an amount 2D (in units of energy) from the states (Figure 1.12b).^{42 } The population of the energy levels can be described by a Boltzmann distribution and depends on the relative magnitudes of the axial ZFS parameter D and the thermal energy kT. Figure 1.12b and c represent a situation where D > 0, resulting in ±½ states that are lower in energy than the ±3/2 states.^{40 }
While departure from a sphericallysymmetric electronic spin density around the TM ion, as in a noncubic ligand field, is a necessary condition for ZFS, orbital angular momentum and SOC are important contributors to the splitting and make it fairly large. Similarly to the orbital angular momentum and SOC, ZFS results in magnetic anisotropy and has profound effects on magnetic properties, and therefore on paramagnetic NMR properties.
1.2.1.6 Magnetic Susceptibility
In typical systems composed of a large number of nuclear and electronic spins, the application of a magnetic field (denoted H for reasons that will become clear shortly) results in a Boltzmann distribution of the spins across the Zeeman energy levels. The lowest energy state with a magnetic moment aligned with the field becomes more populated than the highest energy state with a magnetic moment antialigned with the field. This Boltzmann distribution leads to a finite induced magnetic moment µ_{ind} along H.
Let us consider materials with isotropic magnetic properties, i.e., containing spinonly TMs in a cubic ligand field. The total magnetic moment per unit volume is the net bulk magnetization M which, for many materials (termed linear materials), is proportional to the applied magnetic field H:^{42 }
where χ_{V} is the dimensionless magnetic susceptibility per unit volume. A magnetic flux density B can be defined as:
where µ_{0} = 4π × 10^{−7} H m^{−1} is the permeability of free space. Apart from ferromagnets, χ_{V} ≪ 1 and
Hereafter, we restrict ourselves to materials for which eqn (1.60) is valid, i.e., linear materials with χ_{V} ≪ 1, and refer to B as the external static magnetic field. The Hamiltonian of a system of nuclear and electronic spins in the presence of the constant B field is^{36 }
where Ĥ_{0} is the unperturbed Hamiltonian (without the magnetic field), the second term is the paramagnetic term, which vanishes when all electron spins are paired (L = S = 0), and the third term is the diamagnetic term present in any material with paired and/or unpaired electrons.
We proceed to evaluate the various terms in the Hamiltonian above. From statistical mechanics, the Helmholtz free energy is defined as dF = d(E − TS) = −SdT − pdV − MdB. F can be expressed in terms of the partition function as F = −Nk_{B}T·lnZ, and . By convention, B is set along the z axis: B=(0, 0, B).
1.2.1.6.1 The Curie–Weiss Paramagnetic Susceptibility χ_{para}
To obtain the paramagnetic part of the susceptibility, we consider the partition function of a system with total angular momentum magnetic quantum number m_{J} and gfactor g_{J}. The magnetic moments associated with individual J states take values of µ_{J} = m_{J}g_{J}µ_{B}, with corresponding energies E_{J} = m_{J}g_{J}µ_{B}B.^{36 } Hence, the partition function becomes: . Setting , the partition function is a geometric progression with initial term a = e^{−Jx} and multiplying term r = e^{x}. Using the formula ,where b is the number of terms in the series (here, b = 2J + 1), and after some maths, the partition function can be written in compact form:^{36 }.
The average magnetic moment µ_{J} is given by:
Using the expression of Z derived above,
The magnetization M is then:
with n the number of individual <µ_{J}> moments in the system. Setting , we find
where M_{sat} is the saturation magnetization when all n paramagnetic centres occupy the lowest energy J state with m_{J} = J, hence M_{sat} = ng_{J}µ_{B}J. B_{J}(y) is the Brillouin function:^{36,40 }
The expressions above are valid for any temperature and magnetic field and can be simplified in two limiting cases. In the saturation regime, i.e., when the field is sufficiently high or the temperature is sufficiently low that m_{J}g_{J}µ_{B}B ≫ kT, the coth function and B_{J}(y) tend to unity and M = M_{sat}. Conversely, in the hightemperature limit typical of NMR experiments, E_{J} = m_{J}g_{J}µ_{B}B ≪ kT, i.e., y ≪ 1, and the Brillouin function can be simplified using a Maclaurin expansion for coth(y):^{36 }
Hence, for low magnetic fields, the susceptibility is given by
where
is the effective magnetic moment.^{36 } The paramagnetic susceptibility, χ_{para}, takes the form of Curie's law, with a 1/T dependence. For materials with interacting magnetic moments µ_{J}, such as ferromagnets and antiferromagnets, the susceptibility takes a Curie form at temperatures above the magnetic ordering temperature, i.e., in the paramagnetic regime. For ferromagnets, this is the Curie temperature T_{C}, and , and for antiferromagnets, this is the Néel temperature T_{N}, and . Thus, a general expression for the magnetic susceptibility is given by the Curie–Weiss law:
where
is the Curie constant, Θ the Weiss temperature and α is a temperatureindependent term. Θ can, in practice, be very different from the magnetic transition temperature, T_{C} or T_{N}. For Curie paramagnets (no internal interactions between magnetic spins), the Weiss temperature Θ = 0, while Θ > 0 for ferromagnets and Θ < 0 for antiferromagnets.
1.2.1.6.2 The Diamagnetic Susceptibility χ_{dia}
For the diamagnetic term, B × r_{i} = B(−y_{i},x_{i},0) and (B × r_{i})^{2} = B^{2}(x$i2$ + y$i2$). Within firstorder perturbation theory, the change in the ground state energy due to the diamagnetic term is
where ϕ_{0}〉 is the ground state wavefunction and Z is the number of electrons per atom.^{36 } The last equality assumes a sphericallysymmetric atom, such that . For a diamagnetic solid composed of N atoms (each with Z electrons of mass m) in volume V, the magnetization at T = 0 K can be derived:^{36 }
The diamagnetic susceptibility, χ_{dia}, can be extracted from eqn (1.58) and (1.60): (assuming that χ_{dia} ≪ 1). We obtain:^{15 }
This expression demonstrates that the diamagnetic susceptibility is largely temperature independent.
1.2.1.6.3 The Van Vleck Paramagnetic Susceptibility χ_{Van Vleck}
Van Vleck paramagnetism is a small, secondorder perturbation to the magnetic susceptibility of a system with total angular momentum quantum number J = 0 in the ground stateϕ_{0}〉. In this case, firstorder paramagnetic effects vanish since 〈ϕ_{0}ϕ_{0}〉 = g_{J}µ_{B}〈ϕ_{0} Ĵϕ_{0}〉 = 0. However, mixing of excited states with J ≠ 0 can affect the ground state energy E_{0} according to secondorder perturbation theory. The change in E_{0} for a TM ion with J = 0 is^{36 }
The sum in the first term is taken over all excited states of the system, and the second term is the diamagnetic term discussed earlier. The Van Vleck susceptibility is obtained from the first term as
and is positive because E_{n} − E_{0} > 0.^{36 } Van Vleck paramagnetism, like diamagnetism, is temperatureindependent.
1.2.1.6.4 General Comments for Compounds Containing Openshell TM Ions
For paramagnetic systems, the diamagnetic susceptibility term is very small compared to paramagnetic terms and is hereafter neglected. Four cases can be distinguished:
Case 1: For paramagnetic compounds containing spinonly TM ions in a perfectly cubic environment (A_{1} ground term symbol), the magnetic susceptibility is purely isotropic and the Van Vleck term vanishes. As a result, the temperatureindependent α term in eqn (1.70) for the Curie–Weiss susceptibility becomes negligible and depends on the spinonly effective magnetic moment,
Case 2: For paramagnetic systems where the orbital angular momentum of the ground electronic state is quenched (L_{eff,gs} = 0), e.g., by a ligand field, but SOC mixes in lowlying excited states of the same spin multiplicity with a nonzero orbital angular momentum (L_{eff,es} ≠ 0), the ground state gains some orbital angular momentum and the magnetic moment deviates from the spinonly formula.
Case 3: The presence of SOC in the ground electronic state (L_{eff,gs} ≠ 0) adds an anisotropic component to the susceptibility. In the Russell–Saunders coupling regime (weak L − S coupling), and in the absence of ZFS, the magnetic moment depends on the J quantum number and the Landé gfactor, g_{J}, such that . As in the spinonly case, the magnetic susceptibility is of the Curie–Weiss form, with, .
Case 4: In the presence of ZFS effects, an anisotropic, temperatureindependent Van Vleck paramagnetic susceptibility term is added to χ_{CW}, which will be discussed in greater detail later.
1.2.1.6.5 The Case of 3d TM Ions
The SOC constant λ, which quantifies the strength of spin–orbit interactions, depends on the oneelectron SOC constant ζ proportional to (Z^{*})^{4}, where Z^{*} is the effective nuclear charge. For 3d TM ions, Z^{*} is relatively low, and ζ_{3d} constants are fairly small, varying from ∼150 cm^{−1} for Ti^{3+} to ∼870 cm^{−1} for Cu^{2+}, as shown in Table 1.1 (for comparison, oneelectron SOC constants for lanthanide ions, ζ_{4f}, are typically in the range of 650–3000 cm^{−1}). SOC constants generally decrease when an ion is placed in a ligand field. Hence, even in the presence of residual orbital angular momentum (L_{eff} = 1), SOC effects are typically small compared to ligand field interactions for systems containing openshell 3d TM species, and experimental magnetic moments measured on TM complexes do not always agree with effective moments predicted from Russell–Saunders coupling of L and S angular momenta.
3d ion .  d^{n} .  ζ_{3d} cm^{−1−1} .  S .  L_{eff} .  J_{eff} .  g_{J} .  µ_{eff,RS}/µ_{B} .  µ_{eff,S+L}/µ_{B} .  µ_{eff,so}/µ_{B} .  µ_{exp}/µ_{B} . 

V^{4+}, Ti^{3+}  d^{1}  253, 158  ½  1  0.58  2.24  1.73  1.6–1.8  
V^{3+}  d^{2}  220  1  1  0  —  0  3.17  2.83  2.7–2.9 
Cr^{3+}, V^{2+}  d^{3}  296, 187  0  2  3.87  3.88  3.88  3.7–3.9  
Mn^{3+}, Cr^{2+}  d^{4}  338, 256  2  0  2  2  4.90  4.90  4.90  4.7–5.0 
Fe^{3+}, Mn^{2+}  d^{5}  499, 343  0  2  5.92  5.92  5.92  5.6–6.1  
Fe^{2+}  d^{6}  441  2  1  3  5.77  5.10  4.90  5.1–5.7  
Co^{2+}  d^{7}  561  1  4.73  4.13  3.88  4.3–5.2  
Ni^{2+}  d^{8}  703  1  0  1  2  2.83  2.83  2.83  2.8–3.5 
Cu^{2+}  d^{9}  870  ½  0  ½  2  1.73  1.73  1.73  1.7–2.2 
Zn^{2+}  d^{10}  —  0  0  0  —  0  0  0  0 
3d ion .  d^{n} .  ζ_{3d} cm^{−1−1} .  S .  L_{eff} .  J_{eff} .  g_{J} .  µ_{eff,RS}/µ_{B} .  µ_{eff,S+L}/µ_{B} .  µ_{eff,so}/µ_{B} .  µ_{exp}/µ_{B} . 

V^{4+}, Ti^{3+}  d^{1}  253, 158  ½  1  0.58  2.24  1.73  1.6–1.8  
V^{3+}  d^{2}  220  1  1  0  —  0  3.17  2.83  2.7–2.9 
Cr^{3+}, V^{2+}  d^{3}  296, 187  0  2  3.87  3.88  3.88  3.7–3.9  
Mn^{3+}, Cr^{2+}  d^{4}  338, 256  2  0  2  2  4.90  4.90  4.90  4.7–5.0 
Fe^{3+}, Mn^{2+}  d^{5}  499, 343  0  2  5.92  5.92  5.92  5.6–6.1  
Fe^{2+}  d^{6}  441  2  1  3  5.77  5.10  4.90  5.1–5.7  
Co^{2+}  d^{7}  561  1  4.73  4.13  3.88  4.3–5.2  
Ni^{2+}  d^{8}  703  1  0  1  2  2.83  2.83  2.83  2.8–3.5 
Cu^{2+}  d^{9}  870  ½  0  ½  2  1.73  1.73  1.73  1.7–2.2 
Zn^{2+}  d^{10}  —  0  0  0  —  0  0  0  0 
To demonstrate this, effective magnetic moments predicted from (1) the Russell–Saunders (RS) coupling scheme (), from (2) the combined, albeit noninteracting, theoretical spin and orbital magnetic moments (), and from (3) the spinonly (SO) model, which assumes complete quenching of the orbital angular momentum (L_{eff} = 0 and ), are compared to experimental moments obtained on high spin O_{h} 3d TM complexes in Table 1.1. Notably, µ_{eff, S+L} corresponds to the situation where the orbital angular momentum makes its full contribution to the total magnetic moment,^{43 } while µ_{eff, RS} also accounts for SOC.
As shown in Table 1.1, for high spin d^{3}, d^{4}, d^{5}, d^{8}, d^{9} and d^{10} ions, the application of an ideal O_{h} ligand field quenches the orbital angular momentum (L_{eff} = 0), and effective moments predicted using the spinonly, S + L, and RS models become equal and broadly consistent with experimental values. The small discrepancies observed between the theoretical and experimental µ_{eff} values may be due to deviations from the ideal O_{h} TM coordination geometry (low symmetry ligand fields and possible ZFS), and/or to mixing in of lowlying excited states with L_{eff} ≠ 0.
The moments tabulated above further indicate that complexes containing octahedral V^{3+/4+} and Ti^{3+}, with a theoretical partiallyquenched angular momentum of L_{eff} = 1, appear to choose a ground state where L_{eff} = 0, such that the spinonly formula provides a better picture for the ground state effective magnetic moment. In reality, for ions with a less than halffilled d shell (d^{1} to d^{4}), SOC leads to spin and orbital angular momenta opposing one another and to a reduction of the net magnetic moment. In cases where the net moment reduction due to SOC effects roughly cancels out the orbital contribution, the resulting moment is fortuitously close to its spinonly value. For d^{6} to d^{9} ions with a more than halffilled d shell, the moment is increased due to spin and orbital contributions reinforcing each other, and in Table 1.1, complexes containing octahedral Fe^{2+} (d^{6}) and Co^{2+} (d^{7}) exhibit magnetic moments closer to the theoretical values predicted using the RS and ‘S + L’ schemes. It is important to note that the experimental moments presented here were obtained on TM complexes in solution, where interactions between paramagnetic TM centres are fairly weak. In the case of battery materials composed of an extended network of coupled TM ions, interactions between magnetic momenta on neighboring ions are often exacerbated and SOC effects can become significant.
Overall, the above analysis indicates that, for the purpose of describing the magnetic properties of battery electrode materials comprising 3d TM ions, a simple “spinonly” approach is appropriate in the case of complete orbital quenching in a ligand field (L_{eff} = 0), with little mixing in of excited states with a finite orbital contribution. The spinonly approach is also a good approximation for early 3d openshell TM ions (3d^{1}3d^{4}) with a partially quenched orbital angular momentum (L_{eff} = 1). The paramagnetic NMR properties of spinonly systems are described in Section 1.2.2.2. On the other hand, for systems containing 3d^{6}3d^{9} ions, SOC must be accounted for in the magnetic response, and the paramagnetic properties of such systems are presented in Section 1.2.2.3. The situation is much less clear cut for the 4d and 5d TM series, because the heavier ions have a larger spin–orbit splitting, and the effects of the ligand field, ZFS and of the spin–orbit interaction can be comparable.^{36 }
1.2.2 Hyperfine Interactions in NMR
In the NMR experiment, a sample is placed in an external magnetic field B_{0}, by convention applied along z (B_{0}=(0, 0, B_{0})). The Zeeman interaction between the nuclear magnetic moments µ_{I} and the B_{0} field leads to a splitting of the I nuclear energy levels into 2I + 1 nondegenerate Zeeman states distinguished by their magnetic spin quantum number m_{I}, which can take any integer value between −I and +I (−I, −I+1, …, I−1, I), as shown in Figure 1.13a. For a paramagnetic TM ion with total spin S obtained by summing over all n unpaired electron spins, , the Zeeman interaction leads to a ladder of electron spin energy levels, as shown in Figure 1.13b.
On closer inspection, each electron Zeeman spectroscopic term can be split further according to the total angular momentum J, due to coupling of the L and S angular momenta, leading to fine structure of the signal lineshape in electron paramagnetic resonance (EPR). In addition, the nuclear spin I and the electron total angular momentum J also couple and result in hyperfine structure. Hence, the expressions hyperfine and paramagnetic interactions are used interchangeably to designate the magnetic interactions between nuclei and electrons.
1.2.2.1 The Total Hamiltonian Relevant to the NMR Experiment
The total Hamiltonian representing NMR interactions can be written as
The external contribution, Ĥ_{Ext}, comprises the Zeeman term due to the interaction of the NMR nucleus and the external static B_{0} field along z, Ĥ_{Z}, as well as a term resulting from the interaction with the radiofrequency (RF) field B_{1}.
where Î_{z} is the operator representing the component of the spin along the z axis and
is the Larmor frequency. ω_{0} is the nuclear precession frequency and corresponds to the energy separation (in units of ℏ) between the m_{I} Zeeman sublevels, as shown in Figure 1.13a. The transient RF field B_{1} causes transitions between m_{I} spin states when the RF frequency ω_{1} = ω_{0}.
As discussed in Section 1.1, the internal Hamiltonian,
has contributions from: chemical shielding interactions, Ĥ_{CS}; dipolar interactions between two nuclei, Ĥ_{D}; scalar couplings between two nuclei, Ĥ_{J}; quadrupolar interactions, Ĥ_{Q}; and hyperfine interactions between an electron and a nucleus, Ĥ_{HFC}. Typical NMR experiments are performed in the high field limit, where the interaction between spins with the external magnetic field dominate all internal interactions. In this limit, interactions between nuclear spins or between electron spins and nuclear spins can be treated as perturbations of the Zeeman levels, greatly simplifying the corresponding Hamiltonians.
In strongly paramagnetic materials, the internal part of the NMR Hamiltonian of nuclei with I = ½ (e.g., ^{1}H, ^{13}C, ^{19}F, ^{31}P), Ĥ_{Int}, is largely dominated by the hyperfine Ĥ_{HFC} term. For quadrupolar nuclei with I ≥ 3/2 (e.g., ^{17}O, ^{23}Na, ^{27}Al), the contribution from the quadrupolar Ĥ_{Q} term can also be significant, although many quadrupolar nuclei present in battery systems (e.g., ^{6/7}Li, and in some cases, ^{23}Na) exhibit a small quadrupole moment and/or occupy cubic environments so that the Ĥ_{Q} term and their NMR properties are dominated by the paramagnetic effects discussed here.
1.2.2.2 Hyperfine Interactions in Magnetically Isotropic Systems
The presence of redoxactive, openshell TM species in battery electrodes leads to hyperfine interactions between nuclear spins and unpaired d electron spins on the TMs. While an overview of the relevant coupling terms is provided below, the reader is referred to reviews by Pell et al.,^{11,45 } Bertini et al.^{41 } and Kaupp and Köhler^{46 } for additional details. This section focuses on magnetically isotropic paramagnets containing spinonly TM ions in a perfectlycubic ligand field environment, with unpaired electrons occupying a nondegenerate (singlet) orbital state (e.g., high spin d^{5} Mn^{2+} and Fe^{3+}), and no added complication from orbital (L = 0), SOC or ZFS effects.
1.2.2.2.1 Derivation of the Hyperfine Interaction Hamiltonian
The hyperfine Hamiltonian can be derived either as the interaction between the nuclear magnetic moment µ_{I} with a magnetic field due to the unpaired electron, or equivalently as the interaction of the electronic magnetic moment µ_{S} with a magnetic field due to the nucleus.^{16 } Abragam and Bleaney^{47 } show that
where ∇ is the vector derivative operator del, . (x, y, z) are the coordinates of a point at distance from the nucleus.^{11 }
The hyperfine Hamiltonian can be divided into two parts
where the first term describes the electronnuclear spin dipolar coupling and the second term describes the Fermi contact interaction. Let us assume that the unpaired electron is spatially delocalized while the nucleus is localized at r = 0, as appropriate for 3d electrons of firstrow TM ions that are delocalized into ligand orbitals.^{11 } The magnetic interaction is calculated by multiplying Ĥ_{HFC} with the unpaired electron density at position r,
where ψ(r) is the electron wavefunction, and then integrating over all space.^{11 }
Electron–Nuclear Spin Dipolar Coupling
To derive an expression for the spin dipolar coupling term, we consider the situation where the unpaired electron and nucleus are well separated (r ≠ 0), i.e., the electron occupies an orbital other than an s orbital, and the second term in eqn (1.83) integrates to zero. The resulting hyperfine coupling term
where r is the vector displacement of the unpaired electron with respect to the nucleus and e the corresponding unit vector, describes the spin dipolar interaction.^{11 }
1.2.2.2.1.2 Fermi Contact Interaction
To derive an expression for the Fermi contact interaction term, we instead consider the situation where the unpaired electron is in the immediate vicinity of the nucleus, i.e., in an s orbital, such that r = 0 and the spin dipolar coupling is equal to zero. Integrating the second term in eqn (1.83) leads to
which is the Fermi contact term. This term is purely isotropic and arises due to delocalization of the unpaired electron onto the nucleus, as indicated by the dependence of the size of the interaction on ψ(0)^{2}, which is the unpaired electron density at the nuclear position (r= 0).^{11 }
The Full Hyperfine Hamiltonian
The full Hamiltonian is obtained by summing over Ĥ_{D} and Ĥ_{FC}:
In paramagnetic battery materials, we generally consider the interaction between a nucleus and a paramagnetic TM ion containing multiple unpaired electrons, with total electronic spin S. The Hamiltonian in eqn (1.87) is modified so that we sum over all electrons:
where _{s,i} and ψ_{i}(r_{i}) are the magnetic moment and wavefunction of electron i at position r_{i}. This expression can be simplified by writing the sum over the electrons i in terms of the average spin density per electron , where ρ^{α−β}(r) is the total unpaired spin electron density at position r, and 2S is the number of unpaired electrons. The Fermi contact term can then be rewritten as
where is the magnetic moment operator for the total electron spin Ŝ.^{11 } Similarly, the spin dipolar term can be rewritten as
The total hyperfine interaction Hamiltonian is thus
Orientationdependence and Dynamics of Hyperfine Interactions
As shown in Figure 1.14 and as evidenced by eqn (1.91), paramagnetic interactions depend on the spatial orientation of µ_{I} and µ_{S}, and therefore on the orientation of I and S. In practice, electron spin relaxation dynamics (i.e., transitions between m_{s} levels) are fast (on the order of 10^{−14} to 10^{−8} s)^{48 } on the timescale of nuclear spin relaxation (in the range 10^{−4} to 10^{1} s),^{49 } and the nuclear spin I (µ_{I}) does not interact with the unpaired electron spin S (µ_{S}), but rather with its average value 〈S〉 (〈µ_{S}〉).^{11 } As a result, hyperfine coupling between I and S does not lead to a splitting of the NMR resonance, as would be expected for an interaction between two different magnetic dipoles (e.g., heteronuclear dipolar coupling). It instead takes the form of a chemical shift between I and 〈S〉 and causes a shift of the resonant frequency, as well as chemical shift anisotropy.^{11 }
1.2.2.2.2 The Hyperfine Hamiltonian in Terms of Î and B_{0}
The Average Electron Spin Vector 〈S
To obtain an expression for the expectation value or time average of the electron spin vector, 〈S〉, we assume that it is equivalent to the average over the entire ensemble of paramagnetic centres.^{11 } In this case, 〈S〉 is equal to the Boltzmann average of the components S_{i} of the spin:
where k is the Boltzmann constant and T is the temperature. In eqn (1.92) we use the fact that the electron Zeeman term is by far the dominant contribution to the total Hamiltonian in eqn (1.78) and approximate the energy of the system to that of the spinonly electron Zeeman level defined by quantum numbers (S,m_{S}), E_{S, mS} = g_{e}µ_{B}m_{S}B_{0}. It then becomes evident that 〈S_{x}〉 = 〈S_{y}〉 = 0, because these components are perpendicular to the applied field and there is no driving force for inducing a magnetic moment in these directions.^{11 } Yet, the magnetic moment along z is nonzero and
which becomes:
By analogy with eqn (1.65), 〈S_{z}〉 can be expressed in terms of the Brillouin function. For spinonly systems, 〈S_{z}〉 = −SB_{S}(y), with .
In the hightemperature limit typical of NMR experiments E_{S, MS} = g_{e}µ_{B}M_{S}B_{0} ≪ k_{B}T and the expression for 〈S_{z}〉 becomes
This value of 〈S_{z}〉 is referred to as the Curie spin, as it exhibits a Curie temperature dependence of 1/T. In the hightemperature limit, both 〈S_{z}〉 and the corresponding hyperfine Hamiltonian, 〈Ŝ〉.A.Î, depend linearly on B_{0}. 〈S_{z}〉 can be related to the expectation value of the electronic magnetic moment 〈µ_{S}〉 along B_{0}. However, instead of considering the spin S of a single free electron (as in eqn (1.54)), we consider the total effective spin S̃ of a spinonly paramagnet containing N unpaired electrons. The energy levels of the electrons are then conveniently described by the single effective spin S̃, which has the same multiplicity 2S̃ + 1 as the true states.^{47 } Here, we consider 3d TMs, for which the pseudospin S̃ is the same as the true spin S, so we drop the tilde.
The components of the expectation value of the electronic magnetic moment, 〈µ_{S,i}〉, are proportional to those of the average spin, 〈S_{i}〉, such that 〈µ_{S,x}〉 = 〈µ_{S,y}〉 = 0 and
where we have related the average induced magnetic moment 〈µ_{S}〉 to the expectation value of µ_{S,z}.^{11 }〈 µ_{S}〉 can also be written as the ensemble average of the induced magnetic moment per paramagnetic metal ion:
where N_{A} is Avogadro's number, V_{m} is the molar volume. Using eqn (1.58) and (1.60), we find that . 〈_{S}〉 can then be related to the magnetic susceptibility by substituting this expression into eqn (1.97):
where χ_{m} = V_{M}χ_{V} is the magnetic susceptibility per mole (m^{3} mol^{−1}).
Combining eqn (1.96) and (1.98), the average electron spin along B_{0} can be expressed in terms of χ_{m}:
The magnetization of the sample, M, can be rewritten as
And from eqn (1.99) and (1.100), the molar Curie susceptibility is derived:
This expression is similar to that in eqn (1.68) and (1.69), albeit for a spinonly system where , with .
The Hyperfine Interaction Hamiltonian
We proceed to derive an expression for the hyperfine interaction Hamiltonian in terms of the spin operator Î and the external magnetic field B_{0}, which in turn allows us to make predictions of NMR data. For this, we start from eqn (1.91) and account for the fact that the nuclear magnetic moment couples to the average electronic magnetic moment operator, 〈_{S}〉:^{11 }
〈_{S}〉 is the operator equivalent of 〈µ_{S}〉 in eqn (1.98). Expressed in terms of the magnetic susceptibility per TM ion, χ (where χ = Vχ_{V} = N_{A}χ_{m} in m^{3}), and for an arbitrary direction of the magnetic field B_{0},
Using this expression and _{I} = ℏγ_{I} Î, we obtain the expression for the hyperfine interaction Hamiltonian presented in Section 1.1, namely
1.2.2.2.3 The Hyperfine Coupling Tensor
The hyperfine Hamiltonian Ĥ_{HFC} can be written in compact form as
where A is the hyperfine coupling tensor and expresses how much the nuclear spin I and the average electron spins 〈S〉 sense each other.
The full Hamiltonian, in the high field limit, is then^{11 }
where the first two terms are the nuclear and electron Zeeman interactions, respectively, and the third term is the hyperfine coupling.
The hyperfine coupling tensor A is composed of an isotropic Fermi contact term (A^{FC}) and a traceless and symmetric spin dipolar coupling tensor (A^{D}):
where 1 is the identity matrix.
If the electrons are delocalized onto ligand orbitals, as appropriate for unpaired electrons in 3d TM orbitals, A^{FC} and the Cartesian components of A^{D} are given by^{11 }
where i and j are equal to x, y or z, and δ_{ij} is the Kronecker delta.
1.2.2.2.4 The Fermi Contact Shift
We now describe the Fermi contact interaction depicted in Figure 1.14a. Starting from the hyperfine coupling tensor A presented in eqn (1.107), the isotropic part of the tensor, A^{FC}, leads to a Fermi contact interaction between S and I. Since A^{FC} is a constant (eqn (1.108)), the corresponding Fermi contact Hamiltonian can be written as Ĥ_{FC} = A^{FC}〈Ŝ〉.Î. For spinonly systems, this interaction is purely isotropic and leads to a shift of the resonant frequency of the nucleus. The Fermi contact shift, δ_{FC}, in ppm, is obtained by dividing the contact coupling energy by the nuclear Zeeman energy:
with
where S is the total spin of the paramagnetic (TM) centre, and γ_{I} and ω_{0} are the gyromagnetic ratio and Larmor frequency, respectively, of the nucleus. A lower γ_{I} results in lower sensitivity, but also in smaller paramagnetic shifts. The size and sign of A^{FC} determine the magnitude and direction, respectively, of the chemical shift observed. Through its dependence on A^{FC}, the Fermi contact shift is proportional to ρ^{α−β}(0). Hence, δ_{FC} necessarily results from a nonzero unpaired electron spin density at the position of the nucleus, via electron delocalization and polarization mechanisms discussed in the next section.
Replacing 〈S_{Z}〉 in eqn (1.110) by its expression in (1.99), we derive an expression for δ_{FC} in terms of the isotropic molar susceptibility, χ_{m}:
The equation above indicates that paramagnetic TM ions that exhibit a stronger tendency for their electronic magnetic moments to align with the external magnetic field (large susceptibility) also give rise to larger Fermi contact shifts.^{11 }
Equivalently, using the expression for 〈S_{Z}〉 in eqn (1.96):
1.2.2.2.5 Unpaired Spin Density Transfer Pathways and the Goodenough–Kanamori Rules
The throughbond Fermi contact interaction leads to a chemical shift proportional to the magnitude of the unpaired electron spin density at the position of the nucleus of interest, ρ^{α−β}(0). For battery materials containing a paramagnetic TM ion, unpaired spin density transfer occurs from the TM d orbitals to the s orbitals of the species A of interest. This transfer occurs through direct TM(d)A(s) orbital overlap or involves additional orbitals on bridging atoms.
An example of a local environment for nucleus A is shown in Figure 1.15. Each paramagnetic TM ion in the vicinity of the A nucleus (generally within a ≈5 Å radius from A, based on recent first principles studies^{34,50,51 }) transfers a finite amount of unpaired electron spin density to the s orbital on A, via a spin density transfer pathway P_{i}. This spin density transfer results in a finite contribution to the overall Fermi contact shift, δ_{Pi}. In Figure 1.15, three distinct P_{i} pathways give rise to three distinct Fermi contact shift contributions.
A number of solidstate NMR studies on paramagnetic materials^{22,50–54 } have shown that individual shift contributions from nearby TM ions are additive, and that the overall isotropic Fermi contact shift observed experimentally can be obtained by summing over all individual TM⋯A bond pathway shift contributions δ_{Pi} (in ppm), where the TM ion is within the first few coordination shells around A:
The δ_{Pi} contributions can be positive or negative. The additivity of Fermi contact shift contributions entails that a careful assignment of the features in paramagnetic NMR spectra can provide a wealth of information on the system, ranging from detailed insight into the local environments experienced by the NMR nucleus (including the number and nature of the TM ions within ≈5 Å from the nucleus) to the extent of spin density transfer.^{45 } In practice, individual TM⋯A shift contributions are first determined experimentally or computationally on simple model compounds with wellknown structures. They are then used to interpret the shifts of more complex materials.
1.2.2.2.4.1 How Do We Predict the Sign and Magnitude of Individual P_{i} Shift Contributions?
We take the example of alkali transition metal oxides, which account for a large fraction of the cathode materials used in commercial Liion battery systems. In these compounds, unpaired electron spin density transfer from the TM d orbitals to the alkali (A) s orbitals involves bridging oxygen (O) p orbitals. The limiting 90° and 180° TM–O–A spin density transfer mechanisms, shown in Figure 1.16, can be rationalized using the Goodenough–Kanamori (GK) rules.^{55–58 } These rules, originally designed to predict the sign and approximate magnitude of the 90° and 180° isotropic super exchange coupling between TM d electrons, indicate that the sign and magnitude of spin density transfers depend on the geometry (bond lengths and angles) and covalency of the relevant TM–O–A bond pathways. The sign and magnitude of spin density transfers, and therefore of the Fermi contact shift contributions, also depend on the magnetic susceptibility of the material and are sensitive to both orbital occupation and the oxidation state of the paramagnetic TM ions. As such, the Fermi contact shift not only provides detailed information on the local structure of electrode materials, but also on the redox processes taking place on charge and discharge.^{41,50,51,59,60 } Importantly, predicting the sign and magnitude of Fermi contact shift contributions using the simple empirical model outlined here rapidly becomes difficult when TM–O–A bond pathways deviate from the limiting 90° or 180° geometries. For this reason, density functional theorybased computational approaches have been developed for the determination of accurate P_{i} values.^{51,52 } These will be discussed in section 1.3.
1.2.2.2.6 The Spin Dipolar Coupling
The anisotropic part of the A tensor in eqn (1.107), A^{D}, leads to an orientationdependent coupling interaction between nuclear and electron spin dipoles, as shown in Figure 1.14b. In spinonly systems, this interaction can be described by a CSA and leads to broadening of the NMR resonance with no shift of the resonant frequency.
The spin dipolar coupling is a throughspace interaction that is proportional to the unpaired electronic spin density at position r from the nucleus, ρ^{α−β}(r), and to the isotropic magnetic susceptibility χ, as shown in eqn (1.104). Unlike the shortrange (∼5 Å) Fermi contact interaction, which requires the nucleus to be within the coordination environment of the paramagnetic centre for electron density to be delocalized onto the s orbital of the nucleus, the spin dipolar coupling is longer range and nuclei exhibit a shift anisotropy even when located several 10s of Å away from the paramagnetic centre.^{11 } Hence, the point dipole approximation, which assumes that unpaired electrons are completely localized in the form of point dipoles situated at the TM nuclear positions,^{50 } is often applied to simplify the evaluation of the spin dipolar interaction, and electron delocalization need only be considered to compute the Fermi contact term.^{11 } In the point dipole approximation, the Cartesian components of A^{D} are given by
where i and j are equal to x, y or, z e_{i} and e_{j} represent the x, y or z components of a unit vector pointing from the nuclear spin to the electron spin in a chosen coordinate system, and δ_{ij} is the Kronecker delta. R is the position of the paramagnetic centre where all electrons are localized (ψ_{i}(r)^{2} = δ(r − R), and δ is the Dirac delta function), hence, R is the distance between the nucleus and the paramagnetic centre. The expression above indicates that spin dipolar couplings have a distance dependence of .^{11 } The errors in the point dipole approximation are not great at large distances from the paramagnetic centre, typically when R > 4 Å, but can lead to inaccuracies for nuclei that are closer.^{11,61 }
Rather than using the dipolar hyperfine coupling tensor, A^{D}, whereby
the electron–nuclear spin dipolar Hamiltonian is often expressed in terms of the dipolar coupling tensor 〈D̃^{en}〉 (in frequency units), such that^{62,63 }
For an isotropic, spinonly TM system, eqn (1.98) and (1.101) can be used to show that . In the point dipole approximation, the dipolar tensor is defined in terms of its matrix elements by^{5,13 }
The geometric part of the dipolar coupling tensor in eqn (1.117) is represented in spherical polar coordinates as:^{62,63 }
where θ and φ denote the polar and azimuthal angles that the vector connecting the electron and nuclear magnetic dipoles makes with the x, y, z, axes of a Cartesian axis system.
Assuming that the nuclear spin is coupled to many electron spins which act as many independent local fields (Curie moments), the total coupling matrix D̃^{en} can be written as the sum over all electron spins k within a chosen cutoff radius of the nucleus:
provided that all tensors are expressed in a common axis system.
The total dipolar coupling Hamiltonian, Ĥ_{D}, is then:^{63 }
In the highfield approximation, and using the convention that B_{0} lies along the z Cartesian axis, this expression simplifies to:
The equation above clearly indicates a 1 − 3cos^{2}θ angular dependence of the electron–nuclear dipolar interaction, in a similar manner to the nuclear–nuclear dipolar interaction discussed in Section 1.1. In solution, fast molecular tumbling completely averages the Ĥ_{D} interaction to zero. In the solid state and under static conditions, the dipolar vectors are randomly oriented with respect to B_{0}, leading to broadening of the NMR lines. Hence, to increase resolution, NMR experiments on paramagnetic solids are generally performed under MAS conditions, whereby the sample is spun at an angle of θ ≈ 54.736° from the direction of the B_{0} field. Since typical electron–nuclear interaction strengths are greater than currentlyaccessible spinning speeds, anisotropic dipolar interactions are only partially averaged by MAS, and broad paramagnetic resonances are broken into a series of spinning sidebands separated by the spinning speed.
The traceless dipolar tensor in eqn (1.120) can be transformed into its principal axis frame (PAF) by diagonalization of the final coupling matrix to obtain the eigenvalues of the shift anisotropy, δ_{zz}, δ_{xx} and δ_{yy}. Here, we use the convention δ_{zz} − δ_{iso} ≥ δ_{xx} − δ_{iso} ≥ δ_{yy} − δ_{iso} to define the principal components of the tensor. The shift anisotropy, Δδ, and the asymmetry, η, are obtained from the eigenvalues as follows:^{63 }
and
The sign and magnitude of Δδ depend on the site symmetry and the strength of the dipolar interaction, respectively, while η takes values between 0 and 1 and quantifies the deviation of the spectral pattern from axial symmetry. Starting from the case of axial symmetry (η = 0), an oblatetype arrangement is represented by a negative value of Δδ, and a prolate arrangement by a positive Δδ value.
1.2.2.3 Hyperfine Interactions in Magnetically Anisotropic Systems
We have so far assumed that the electronic magnetic moment arises from the electron spin angular momentum S. This approach is only strictly correct when the paramagnetic centres behave as spinonly ions, but can provide a satisfactory description of more complex systems, including paramagnetic battery materials with isotropic shifts dominated by the Fermi contact contribution, and shift anisotropies largely dictated by spin dipolar interactions.^{50,51,64,65 } In such systems, the interpretation of paramagnetic NMR data is relatively straightforward, with the isotropic shift and shift anisotropy providing information on the spin transfer pathways from the unpaired electrons to the nucleus and on the geometry of the system, respectively.^{11 }
Systems that contain TM ions exhibiting SOC effects, and in coordination environments of noncubic symmetry, are magnetically anisotropic. This is the case for certain battery electrodes, as has been shown for LiMPO_{4} (M = Fe, Co)^{66,67 } and Li_{x}V_{2}(PO_{4})_{3}.^{68,69 } The theory by which orbital, SOC and ZFS terms affect paramagnetic NMR properties is by no means simple, and we shall give here only a superficial and pragmatic account of the subject. The reader is referred to work by Bertini et al.,^{21 } Vaara and coworkers,^{70,71 } Kaupp and Köhler^{46 } and Pell et al.^{11 } for a more detailed account of magnetically anisotropic systems.
1.2.2.3.1 Describing Magnetic Anisotropy
The Total Magnetic Moment Operator
As mentioned earlier, in the presence of both spin and orbital angular momentum, the total electronic magnetic moment operator of a TM subject to RussellSaunders (L–S) coupling, is given by = −µ_{B}(g_{L}L̂ + g_{e}Ŝ) (eqn (1.55)), where g_{L} = 1 and g_{e} ≈ 2 are the gfactors for the orbital and spin angular momenta, respectively. In a crystalline solid, the orbital contribution to the magnetic moment of a TM ion has its own orientation with respect to the crystal axes. The ligand field experienced by the TM determines the crystal axis along which the orbital magnetic moment is oriented. As a result, the projection of the total magnetic moment along B_{0} depends on the orientation of the crystal and is therefore anisotropic.^{72 }
For the purpose of interpreting paramagnetic NMR data, the Hamiltonian, and therefore the total magnetic moment operator, are expressed in terms of spin operators. To this effect, the magnetic moment anisotropy due to L is represented by replacing the isotropic freeelectron value, g_{e}, by a gtensor, g, acting on Ŝ, such that
The difference between eqn (1.55) and (1.125) lies in the description of the spatial dependence of the magnetic moment operator. Eqn (1.55) contains L̂, giving an intrinsic spatial dependence. In eqn (1.125), the explicit dependence on L̂ is removed and only spin operators are retained, while the spatial dependence is encoded in the gtensor anisotropy.^{11 }
The χ and 〈〉 Tensors
Magnetic anisotropy can be represented by replacing the isotropic magnetic susceptibility (χ) and average magnetic moment (〈µ̂ 〉) values by χ and 〈〉 tensors (in this section, all values are given per TM ion). By analogy with eqn (1.98), 〈〉 depends on the orientation of the χ tensor with respect to the direction of the B_{0} field,
as depicted in Figure 1.17.
Since the Zeeman interaction energy is E = −B_{0}., the infinitesimal change in the average energy 〈E〉 per paramagnetic centre resulting from an infinitesimal change in magnetic field (at constant entropy and constant volume) is . After integration, the average energy per paramagnetic centre is obtained as
from which we see that the χ tensor must be symmetric. Eqn (1.126) provides a straightforward way to evaluate the components of the secondrank 〈 〉 tensor as
where the χ_{ij} are the components of the symmetric, secondrank χ tensor and (B_{0x}, B_{0y}, B_{0z}) are the three components of the B_{0} vector. Eqn (1.126) clearly shows that, while 〈〉 is proportional to B_{0} in the case of an isotropic χ, magnetic anisotropy leads to an orientationallydependent proportionality constant χ_{ij}.
The symmetric magnetic susceptibility tensor can be rewritten as
where χ_{iso} is the isotropic part, and Δχ is the traceless and symmetric susceptibility anisotropy. The anisotropy is parametrized according to one of two conventions.^{11 } The first is in terms of the axial and rhombic anisotropies, Δχ_{ax} and Δχ_{rh}, which are defined in terms of the PAF components _{ii} as
The principal components are defined so that _{zz} ≥ _{yy} ≥ _{xx}. Alternatively, the principal components can be defined to satisfy_{zz}− χ_{iso} ≥ _{xx} − χ_{iso} ≥ _{yy} − χ_{iso}. The anisotropy Δχ and asymmetry η_{χ} parameters are given by
and
If we consider a collection of paramagnetic ions, Δχ is a measure of the degree of spatial anisotropy in the tendency for their electronic magnetic moments to align with the external magnetic field.^{11 }
Simple Case of an Orbitally NonDegenerate System
In systems that are orbitally nondegenerate, magnetic anisotropy can be adequately represented by an anisotropic gtensor, whereby the g̃_{ii} coefficients of the tensor, for any direction ii of the magnetic field, are solutions of the equation
where ϕ stands for the electron Zeeman eigenfunctions.^{41 } The values of 〈〉 along the three main directions of the reference frame are then computed as
where E_{n,ii} = g̃_{ii}µ_{B}M_{S,n}B_{0} is the Zeeman energy of the n_{th} eigenfunction, and the exponential has been approximated to first order, as appropriate in the hightemperature limit (E_{n,ii} ≪ k_{B}T). The principal values of the magnetic susceptibility (χ) tensor are:^{21 }
In cases when the Zeeman energy (E_{n,ii}) is not much smaller than the thermal energy (k_{B}T), the firstorder approximation used in eqn (1.135) is no longer valid, and different equations can be obtained by analogy with the Brillouin function (eqn (1.66)).^{41 }
The equations above hold rigorously for orbitally nondegenerate systems.^{50 } For multielectron systems (), they hold as long as the magnetic spin ground state with spin quantum number S is well separated from higher energy states, i.e., thermal population of excited spin states can be neglected, and in the absence of ZFS discussed below.
Magnetic Anisotropy in the Presence of Zerofield Splitting Effects
In the absence of SOC but in the presence of ZFS effects, which act to split the S states in the absence of an external magnetic field, the total energy of the system equals the Zeeman energy plus an E^{0} ZFS term that does not depend on the applied magnetic field. The E^{0} term typically dominates, and Zeeman effects can be treated using perturbation theory.^{41 } In such cases, the effect of ZFS on the magnetic susceptibility can be computed using the Van Vleck equation:^{73,74 }
The sums are taken over all excited states of the system, with energies E$i0$. The anisotropic, temperatureindependent Van Vleck paramagnetic susceptibility term is added to χ_{CW}, and the overall magnetic susceptibility deviates from that predicted by the Curie–Weiss law in the temperature range where ZFS effects are comparable or greater than the thermal term (k_{B}T). While beyond the scope of this chapter, a more complete derivation of the ZFS interaction Hamiltonian can be found in works by Pell et al.^{11 } and Abragam and Bleaney.^{47 }
Magnetic Anisotropy in the Presence of Zerofield Splitting and Spin–Orbit Coupling
In the presence of both SOC and ZFS, the electronic magnetic moment operator and magnetic susceptibility tensor can be expressed in terms of the g and D tensors, which describe the effects of SOC and ZFS, respectively. For an ensemble of TM ions, the electron magnetic moment operator takes the form
and we can approximate the susceptibility to second order in :^{11 }
This expression describes the susceptibility tensor in the hightemperature limit, where ZFS effects are small compared to the thermal term (k_{B}T).^{11 }
1.2.2.3.2 The Total Hamiltonian in the Presence of Magnetic Anisotropy
A general form of the Hamiltonian describing a paramagnetic system with magnetic anisotropy is^{11 }
where the first two terms represent the nuclear and electron Zeeman interactions, respectively, the third term is the hyperfine contribution discussed earlier, and the fourth term describes ZFS effects, where D is the ZFS coupling tensor. The nuclear Zeeman term contains the chemical shielding tensor, σ_{orb}, which accounts for the fact that the external magnetic field causes electrons to move in such a way that they produce their own induced magnetic field, modifying the actual field experienced by the nucleus.^{11 } In turn, σ_{orb} leads to a deviation of the nuclear resonance frequency from ω_{0}. The electron Zeeman term contains the orbital and spin contributions to the total electron magnetic moment encountered in previous sections.
Ĥ_{aniso} includes electron spin, nuclear spin and electron–nuclear spin–spin interactions to fully describe the magnetic response of the system to the applied B_{0} field. The electron Zeeman and ZFS interactions do not involve nuclear spins and cannot be probed directly by NMR, yet they have a direct impact on the average electron spin angular momentum, 〈Ŝ〉, thus on electron–nuclear hyperfine interactions and must be accounted for. In the case of the ZFS interaction, the Ŝ.D.Ŝ coupling does not affect the resonant frequency ω_{0} of the nucleus under observation, as it shifts all the S spin energy levels by the same amount in the same direction,^{11 } but it can impact nuclear spin relaxation^{75 } and therefore NMR lineshapes.
In previous sections, we have used the highfield approximation to simplify the NMR Hamiltonian and retained only those terms that commute with the unperturbed Zeeman interaction. This approximation is only valid if the Zeeman interaction is several orders of magnitude larger than the other interactions, which is not necessarily the case here as ZFS can dominate.^{11 } Hence, the full Hamiltonian is used in the rest of this section.
While magnetic anisotropy stems from the presence of both spin and orbital angular momenta, the Hamiltonian shown above only contains spin operators and can readily be used to interpret paramagnetic NMR spectra. The process of converting the interaction Hamiltonian Ĥ_{aniso} from a form containing both L̂ and Ŝ operators to one that solely depends on Ŝ is described in detail by Pell et al.^{11 } This process involves the application of first and secondorder perturbation theory to the orbital ground state of the electron, and results in nonrelativistic terms, as well as relativistic terms resulting from SOC. The formalism adopted here is generally used in EPR spectroscopy, although it has been used extensively to describe hyperfine interactions in NMR,^{46,70,71 } and involves g, Ŝ, A and D tensors to describe magnetic anisotropy. Alternatively, the susceptibility formalism makes use of the susceptibility tensor χ to account for anisotropy.^{41,48 } A comparison of these two formalisms can be found in Pell et al.'s review.^{11 }
The g, A and D tensors can be expanded as Taylor series in terms of the fine structure constant α, which takes the value 1/137.036.^{11,76 } The g tensor is composed of a nonrelativistic (NR) and a relativistic SOC term:^{77 }
where g_{NR} is of order O(α^{0}) and g_{SOC} is of order O(α^{2}). The NR term is simply the isotropic freeelectron gfactor,
and the SOC term is the sum of an isotropic term and a traceless anisotropic term that is not necessarily symmetric,^{11 }
Grouping isotropic and anisotropic terms together, the overall expression for the g tensor is:
Similarly, the hyperfine coupling tensor can be expanded as:^{78 }
The NR contribution,
is the sum of the isotropic Fermi contact term (A^{FC}) and of the symmetric and anisotropic electron–nuclear spin dipolar part (A^{D}) previously introduced for magnetically isotropic systems. The SOC contribution,
contains an isotropic part A^{FC,2}, which is referred to as the secondorder Fermi contact coupling constant,^{11 } a symmetric and anisotropic component, A^{D,2}, denoted the secondorder dipolar interaction, and an antisymmetric anisotropic contribution A^{as}.^{11 } Grouping isotropic and anisotropic terms together:
The ZFS tensor takes the form:^{79 }
where the NR part is due to the electron spin–spin interaction, and is usually small compared to the SOC term D_{SOC}. D is symmetric and traceless, and is equal to zero either for electronic spins S < 1, or for TMs in perfectly cubic environments.^{11 }
As summarized in Table 1.2, only nonrelativistic terms contribute in the absence of SOC, while additional terms are needed in the presence of SOC. We note that the expressions for the hyperfine interactions derived earlier for magneticallyisotropic systems did not account for the NR contribution to the ZFS, and are therefore only approximations for S > ½ systems.^{11 }
Magnetic isotropy .  g, A and D tensor components . 

, isotropic  g = g_{e}, A = A^{FC}1+A^{D}, D = 0 
, anisotropic  g = g_{e}, A = A^{FC}1+A^{D}, D = D_{NR} 
Anisotropic with SOC  g = (g_{e}+Δg_{iso})1+Δg, A=(A^{FC} + A^{FC,2})1 + A^{D} + A^{D,2} + A^{as}, D = D_{NR} + D_{SOC} 
Magnetic isotropy .  g, A and D tensor components . 

, isotropic  g = g_{e}, A = A^{FC}1+A^{D}, D = 0 
, anisotropic  g = g_{e}, A = A^{FC}1+A^{D}, D = D_{NR} 
Anisotropic with SOC  g = (g_{e}+Δg_{iso})1+Δg, A=(A^{FC} + A^{FC,2})1 + A^{D} + A^{D,2} + A^{as}, D = D_{NR} + D_{SOC} 
1.2.2.3.3 Hyperfine Interactions for Systems with Magnetic Anisotropy
For magnetically isotropic systems, NR expressions for the isotropic Fermi contact shift and shift anisotropy due to electron–nuclear spin dipolar interactions can be used. While these interactions have been discussed in previous sections, the expressions provided below allow for an easy comparison with the magnetically anisotropic case that follows. In the absence of SOC or ZFS, the isotropic Fermi contact shift, δ_{FC}, in Hz, is given by^{11 }
which is similar to eqn (1.113) encountered earlier, save from the 10^{6} factor which yields shift values in ppm. We can also define the Fermi contact shielding, σ_{FC}, such that^{11 }
noting that the only difference between the chemical shift and chemical shielding conventions is a change of sign. Hence, these two formalisms can be used interchangeably.
The anisotropy of the spin dipolar shielding interaction, σ_{D}, is parameterized in terms of the shielding anisotropy, given by^{11 }
and asymmetry
where ΔA^{D} and η_{D} are the anisotropy and asymmetry of the nonrelativistic spin dipolar interaction. The overall shielding tensor therefore has the same PAF and anisotropic properties as the nonrelativistic spin dipolar coupling tensor.^{11 }
The impact of SOC on electron–nuclear spin couplings are described below for 3d TM systems, where SOC effects are rather small. Expressions for the various hyperfine interactions are obtained using the g, A and D tensors presented earlier. For comparison, we also provide expressions for selected interactions using the susceptibility formalism. All of these terms are discussed in greater detail in the review by Pell et al.,^{11 } which also deals with more complex ions such as lanthanides and actinides. In the presence of magnetic anisotropy, the expressions for the various isotropic and anisotropic interactions contain tensor products corresponding to crossterms between SOC, ZFS and other interactions (hyperfine, spin–spin, electron Zeeman). In general, each tensor has its own PAF, which does not necessarily coincide with the PAF of other tensor properties. Hence, the overall anisotropy and asymmetry of the interaction under consideration are not simple combinations of the tensor contributions, but are instead obtained via matrix diagonalization.^{11 }
The Isotropic Contact Shift
In the hightemperature limit relevant to NMR experiments, the expression for the isotropic contact shift, δ_{con, iso}, takes the form^{11 }
The equation above indicates that both terms involving g_{e}, which includes the SOC contribution g_{e}A^{FC,2}, as well as the SOC contribution to the Fermi contact interaction Δg_{iso}A^{FC}, appear to first order in (1/k_{B}T). Conversely, the ganisotropy and ZFS splitting only appear to second order in .^{11 } In the absence of ZFS, the isotropic contact shift becomes
which is a simple modification of the nonrelativistic expression, with changes to the isotropic gfactor and to the contact coupling interaction due to SOC.^{11 }
In the susceptibility formalism, the isotropic contact shift depends on the isotropic part of the magnetic susceptibility tensor, χ_{iso} and on the unpaired electron spin density at the nuclear position:^{11 }
The Contact Shift Anisotropy
In the hightemperature limit, the anisotropic contact shielding is given by^{11 }
where Δg^{sym} and {Δg.D}^{sym} are the symmetric anisotropic parts of those tensors. These terms are given by^{11 }
The first term in eqn (1.159) depends on the ZFS tensor, with the anisotropy and asymmetry being proportional to those of the D tensor, ΔD and η^{D}, respectively, and the same PAF as the ZFS. The second term depends on the anisotropic gtensor, with an anisotropy proportional to Δg, an asymmetry parameter equal to η^{g}, and the same PAF as the symmetric gtensor. The third term is more complicated, with an anisotropy, asymmetry parameter, and PAF that depend on the product of the anisotropic parts of D and g.^{11 }
The ZFS only contributes to second order in (1/k_{B}T), while the ganisotropy contributes to first order. If there is no ZFS, the following expression is obtained for the contact shielding anisotropy:^{11 }
In this case, σ_{con,aniso} depends on the anisotropic properties of the g tensor and can be parameterized in terms of the anisotropy (Δσ_{con,aniso}) and asymmetry (η_{con}) parameters
and
In the susceptibility formalism, the form of the contact shielding anisotropy is similar to that of the isotropic contact shift, and σ_{con,aniso} depends on the susceptibility anisotropy, Δχ, and on the unpaired spin density at the nuclear position. The anisotropy and asymmetry parameters are given by:
and
The first expression indicates that TM ions with a greater degree of magnetic anisotropy lead to larger contact shift anisotropies.^{11 }
The Isotropic Pseudocontact Shift
The isotropic pseudocontact shift (PCS) arises from the electron–nuclear spin dipolar interaction and has a very different form to that of the contact shift. Mathematically, it results from the matrix product of the spin dipolar tensor (A^{D} or A^{D,2}) with at least one other anisotropic and symmetric tensor.^{11 } In the hightemperature limit, the following expression is obtained for the PCS:
The first term depends on the orientation of the PAF of the nonrelativistic spin dipolar tensor relative to the PAF of the symmetric part of the gtensor. The second and fourth terms depend on the orientation of the PAF of the nonrelativistic spin dipolar tensor relative to the PAF of the ZFS tensor. The third term is similar, but depends on the orientation of the PAF of the SOC spin dipolar tensor relative to the PAF of the ZFS tensor. The fifth term depends on the orientation of the nonrelativistic spin dipolar PAF relative to the PAF of the symmetric part of the product Δg·D. The ganisotropy, Δg, is present in terms with both a first and secondorder temperature dependence (1/k_{B}T), whilst the ZFS only affects the PCS to second order.^{11 }
In the absence of ZFS, the PCS takes a simple form:^{11 }
In the susceptibility formalism, a simple expression for δ_{PCS, iso} can be obtained by using the fact that, at TM ionnucleus distances greater than 4 Å, the point dipole approximation can be employed. The PCS then takes the form^{11 }
where θ and ϕ are the polar and azimuthal angles relating the PAF of the susceptibility tensor relative to the TM ionnucleus vector.^{11 } The expression above indicates that, similarly to the contact shift, the PCS depends on the interaction with the unpaired spin density. Yet, while the contact shift results from the spin density at the nuclear site, the PCS is a longer range effect that depends on the spatial position of the nucleus with respect to the paramagnetic centre. Eqn (1.167) also shows that the PCS depends only on the susceptibility anisotropy, hence it only arises for TM ions subject to SOC.^{11 }
The Spin Dipolar Shift Anisotropy
The anisotropic shielding tensor that arises from the spin dipolar interaction, in the high temperature approximation, takes the form^{11 }
The first four terms depend on the dipolar coupling tensors and ganisotropy, and have a temperature dependence of (1/k_{B}T). The other terms, which also depend on the ZFS tensor, exhibit a second order dependence on (1/k_{B}T).^{11 } In the absence of ZFS, σ_{D,sym} becomes:
The Antisymmetric Hyperfine Shift Anisotropy
The antisymmetric hyperfine shift anisotropy, σ_{as,sym}, does not contribute to the isotropic paramagnetic shift but contributes to the shielding anisotropy. In the hightemperature limit,^{11 }
The temperature dependence is second order in (1/k_{B}T). The anisotropic properties depend on the tensor product D·A^{as}, while the ganisotropy does not play a role. Hence, in the absence of ZFS, this term disappears altogether.^{11 }
1.2.2.3.4 Main Takeaways
The main differences between NMR spectra collected on spinonly and spin–orbit coupled paramagnetic systems are illustrated in Figure 1.18 below. For magnetically isotropic TM systems, the throughbond Fermi contact interaction is purely isotropic and leads to a shift δ_{FC} of the resonant frequency of the nucleus under consideration away from the reference chemical shift, conventionally set at 0 ppm. Concurrently, the throughspace electron–nuclear spin dipolar interaction is purely anisotropic and leads to a shift anisotropy. In the presence of SOC, both the Fermi contact and electron–nuclear spin dipolar interaction have an isotropic and anisotropic component. The chemical shift due to isotropic contact interactions is δ_{con,iso}, and the shift due to the isotropic spin dipolar coupling is the pseudocontact shift δ_{PCS,iso}.
The presence of magnetic anisotropy leads to complex expressions for hyperfine interactions. Hence, it is helpful to summarize the impact of the g, A and D tensor properties on the form of the isotropic shift and shift anisotropy, and to review the temperature dependence and size of the various contributions for different systems.^{11 }
Systems with No Spin–Orbit Coupling
In systems where SOC effects are negligible, the isotropic shift stems from the interplay between purely isotropic g, ZFS and hyperfine terms through the product g_{e}D_{iso}A^{FC}. The shift depends on contact interactions and therefore on throughbond transfer of unpaired electron spin density onto the nucleus under observation. The shift anisotropy is given by g_{e}D_{iso}A^{D}, and is similarly dependent on isotropic g and ZFS terms, but it results from anisotropic throughspace coupling of nuclear and electron spins. These interactions have a (1/k_{B}T) temperature dependence.^{11 }
Systems with Spin–Orbit Coupling
SOC mixes thermallypopulated electronic energy levels, adding contributions to the g, hyperfine and ZFS tensors. In this case, throughspace electron–nuclear spin dipolar terms can contribute to the isotropic hyperfine chemical shift (e.g., through the PCS), while throughbond contact terms contribute to the chemical shift anisotropy. The isotropic paramagnetic shift is the sum of the contact shift and PCS. These two interactions provide complementary information on the throughbond and throughspace interactions with the unpaired electrons, respectively.^{11 } In situations where there is good overlap between TM d orbitals and s orbitals on the nucleus of interest, a significant amount of unpaired spin density may be transferred to the nucleus and the contact shift is likely to dominate the overall isotropic shift. On the other hand, in situations where orbital overlap is less favourable, or if the nucleus is outside of the immediate (∼5 Å) coordination environment of the TM ion, the contact shift becomes very small and the PCS is more important. Finally, in the hightemperature limit, the effect of the ZFS tensor on hyperfine interactions exhibits a 1/(k_{B}T)^{2} temperature dependence. Hence, for TM ions where the ZFS is small, its contribution can be neglected.^{11 }
1.3 Calculating Paramagnetic NMR Parameters
Complex interactions reflected in NMR spectroscopy of paramagnetic ions can provide rich structural information of the solids, while they also increase the difficulty of the spectral analysis, as the two sides of the coin. Calculation of NMR shifts with firstprinciples method is very helpful for spectral assignments. In this section, the paramagnetic NMR (PNMR) theory for paramagnetic molecules is presented briefly, then the corresponding implementation methods in the calculation of chemical shifts for paramagnetic solids are induced briefly though a series of phosphate battery materials. More details on the calculation of chemical shifts in paramagnetic solids can be found in some excellent review papers and books.^{11,46,76 }
1.3.1 NMR Shifts of Paramagnetic Molecules
The PNMR theory is developed and implemented for paramagnetic molecules at the beginning, particularly in transition metal coordination compounds.^{70,73,81 } The most promising rechargeable battery cathode materials usually contain redoxactive transition metal ions, thus it attracts abundant NMR study. Nevertheless, stateoftheart formalism used for paramagnetic solid is more or less a straightforward modification from the theory developed for simple molecules.^{67 } Herein, a brief introduction for the developing of paramagnetic theory for molecules is presented first, although the paramagnetic molecules are not the main interest of NMR study of energy storage materials.
In the analysis of PNMR spectra, PNMR chemical shift is usually decomposed into three parts: the orbital shift δ^{orb}, the Fermi contact shift δ^{FC}, and pseudocontact shift δ^{PC}:
The orbital shift is taken as usual NMR chemical shift in the diamagnetic system, this term is approximately temperatureindependent and equal to the chemical shift in a same diamagnetic environment. The orbital shift arises from the induced current of paired electrons in the external magnetic field. According to the Biot–Savart law, a current density j(r) induces a magnetic field at position s:
where c is the speed of light in a vacuum.
The δ^{FC} comes from interaction between the nuclear magnetic moment and the average spin density at the location of the nucleus. In the simplest case, it is given by:^{82 }
µ_{B} and µ_{N} are the Bohr and nuclear magnetons, respectively. S is the spin quantum number, A$isoI$ is the isotropic hyperfine coupling constant that is determined by spin density calculated though density functional theory (DFT) methods, g is the rotationally averaged electronic gvalue, g$NI$ is the nuclear gvalue, k_{B} is the Boltzmann constant and T is the absolute temperature. The longrange dipolar interaction between the nuclear magnetic moment and the induced magnetic moment at the paramagnetic centre gives the δ_{PC}. In the simplest form, it is given by:^{83 }
Here, if paramagnetic centre is taken as coordinate origin, Ω is the angle between its principal symmetry axis and the direction to the observed nucleus. R is the distance between the induced magnetic moment and the nucleus. F(g) is an algebraic function of the gtensor values, which subsumes the relative magnitudes of various relaxation time. There is an assumption that R is large enough, thus the paramagnetic centre can be treated as a point dipole.
There are also other assumptions^{73 } in the derivation of eqn (1.173) and (1.174) that still restrict their wide application: (1) The paramagnetic molecule has only one thermally populated energy level in the absence of the magnetic field, and the states of this ground level could be assigned a spin quantum number, S; (2) Splitting of the ground level in the absence of the field (zero field splitting, ZFS) could be ignored when S ≥ 1; (3) the orbital contribution to the isotropic NMR shift could be taken into account indirectly through the use of gtensor components in eqn (1.173) and (1.174). This assumption requires that there be at most only a firstorder orbital contribution to the magnetic moment of paramagnetic centre, brought about by spin–orbital mixing of the ground level with nonpopulated excited states. However, abundant transition metal complexes do not satisfy all those assumptions. Therefore, Kurland and McGarvey develop an approximate density matrix treatment to handle these situations.^{12 } It is a milestone of paramagnetic NMR theory, involving both spin–orbit coupling (SOC) and ZFS effect in the formalism, while the formalism is not accessible though firstprinciples computational properties.
Moon and Patchkovskii^{76 } derived the formalism of PNMR shift for molecules having only one unpaired electron (doublet electronic state). The main difference from the simplest case is that the hyperfine shielding is expressed by the matrix product of g and A tensors, rather than a rotationally averaged electronic gvalue. It is given by:
Both the A and g tensor could be expressed via the isotropic and anisotropic parts:
Then the total isotropic part is given by:
The σ$orb,isoI$, g_{e}A$FCI\u2009$1 + Δg_{iso}A$FCI$1 and Δg̃A$dipI$ in brackets correspond to orbital shift, Fermi contact shift and pseudocontact shift, respectively. Moreover, Δg_{iso}A$FCI$1 corresponds to an orbital magnetic moment induced δFC in this way the spin–orbit effect is included. This formalism should also satisfy for the molecules with small ZFS effect.
Vaara^{70 } derived a complete and DFT methods accessible PNMR shift representation for paramagnetic molecules, which not only satisfies the hightemperature region but also the low temperature region by including the ZFS effect correctly. It is given by:
〈SS〉 is a spin dyadic with the components 〈S_{ε}S_{τ}〉 evaluated in the manifold of eigenstates q〉 with eigenenergies E_{q} of the ZFS Hamiltonian, H_{ZFS}. The offdiagonal elements of the symmetric matrix Q_{pq} incorporate magnetic couplings between the eigenstates of the ZFS Hamiltonian, which is necessary for the correct behavior when going to low temperatures.
1.3.2 NMR Shifts of Paramagnetic Battery Materials
The computation of the NMR shift for a paramagnetic solid is a greater challenge than paramagnetic molecules. It is still in an early stage, the main difficulty for the complete calculations comes from both theoretical and empirical implemental aspects. In contrast to the neglectable interaction between intermolecular paramagnetic centres in molecule systems, the solid multiparamagnetic centres are closely correlated, usually leading to a Curie–Wiess magnetic susceptibility in the room temperature region. The interaction between them is incorporated by introducing the Wiess constant in the stateofart formalism, which is a quite straightforward modification of Vaara's formalism. The lack of periodic code and high computational resource demanded for gtensor, zerofieldsplitting tensor and exchange coupling constant between paramagnetic centres, are their limitations. A simple classification of current implementation in paramagnetic battery materials according to shift mechanism is shown in Figure 1.19.
A careful comparison of different implementations for the PNMR shift of a series lithium transition metal phosphate shows the importance of including the spin–orbit and ZFS effect. The first direct δFC calculation for lithiumion batteries (LIBs) cathode materials, which are typical Fe(iii) phosphates, were carried out by Kim et al..^{50 } The formula is given by:
µ_{eff} is the effective spin moment, Θ is the Weiss constant, both are obtained in the magnetic susceptibility measurement. Comparing with eqn (1.173), S(S + 1) and T are replaced by µ_{eff}^{2} and (T − Θ), thus including the effect of SOC and residual exchange couplings respectively. The computational hyperfine constants are implemented in CRYSTAL with B3LYP based hybrid functional, which is based on all electron linear combinations of the atomic orbitals (LCAO) method. It turns out that the δFC is highly dependent on Hartree–Fock (HF) exchange contents, which determine the localized extent of the computational electronic state (Figure 1.20). Moreover, using XRD (SPE) or DFT optimized structures for δFC calculation yields remarkable shift differences, which indicates that the hyperfine constant is highly sensitive to atomic positions. The XRD can usually get cell parameters precisely, whereas the small local structure distortions are sometime missing refinement, thus it is necessary to do the cross check upon different geometry schemes. Nevertheless, the calculated δFC and experimental results are generally consistent, especially for the ^{31}P shift. This shows that the δFC dominates the PNMR shift. A possible reason could be the small SOC and ZFS effect of Fe(iii) 3dorbital, which is a highspin state because of the symmetric half filled 3d orbital in the tetrahedral field, leading to a small δPC. It is a clever choice for the purpose of comparing with the experimental shift at the very beginning.
Boucher et al.^{83 } has calculated the δFC of a series of LiMPO_{4} (M = Mn, Fe, Co, Ni) using the same formalism as Kim et al. The hyperfine field calculation is implemented in fullpotential linearized augmented plane wave (FPLAPW) code, WIEN2k. The δFC are tested with PBE, DFT + U and PBE based onsite hybrid functional. Comparing with experiment shift, the onsite hybrid functional applied on TM and oxygen gives the best agreement for TM = Mn, Fe, Ni cases, however the LiCoPO_{4} shift is a significant overestimate (Figure 1.21).
It suggests that the limitation of neglecting δPC has emerged.
Pigliapochi et al.^{66 } recalculate the PNMR shift of LiMPO_{4} (M = Mn, Fe, Co, Ni). Again, the residual exchange coupling is included using the experimental Weiss constant, as shown in eqn (1.181).The formalism is modified from the doublet electronic state that also satisfies the small ZFS cases. The gtensor calculation is implemented in QUANTUM EXPRESSO with PAW and pseudopotential methods. It is the first time that the SOC effect for cathode materials NMR shifts by firstprinciples EPR parameters rather than the experimental effective magnetic moment has been incorporated. The computational ^{7}Li shift of LiCoPO_{4} is still an overestimate, leading to a ^{7}Li shift sequence: δ_{LiMnPO4} > δ_{LiFePO4} > δ_{LiCoPO4} > δ_{LiNiPO4}, while it disagrees with the experimental ^{7}Li shift results: δ_{LiMnPO4} > δ_{LiFePO4} > δ_{LiNiPO4} > δ_{LiCoPO4}. The existing qualitative error suggests that the ZFS effect plays an important role in the PNMR shift of LiCoPO_{4}.
Kaupp^{15 } included the ZFS effect using the following formula:
It is still a straightforward modification from Vaara's formalism because the residual exchange coupling is incorporated by the experimental Weiss constant. Comparing with molecule systems, another difference is that paramagnetic solids contain multiple paramagnetic centres. Only the hyperfine interaction between the first neighborhood paramagnetic centre and the observed nucleus are involved in the formula, n are the first neighborhood numbers, as shown in Figure 1.22.
Periodic GAPW based code cp2k is used for the hyperfine tensor, while the g and Dtensor are implemented in the nonperiodic ORCA program with the single paramagnetic centre incremental cluster model and high level postHF method, multiconfigurational selfconsistent field (MCSCF), Nelectron valence state perturbation theory (NEVPT) methods.^{68,80 } The reason for this is that the gtensor and Dtensor are associated with electronic excited states, which is still a challenge for DFT methods. When the relativistic effect is neglectable, the paramagnetic part of the NMR shift is composed of six terms with different physical meanings:
The results in Figure 1.23 show that the ^{7}Li shift of LiCoPO_{4} is dominated by negative δPC, which is neglected^{83 } or underestimated^{66 } in the previous work. Finally, qualitatively correct computational results of the series of transition metal phosphate are obtained. However, comparing with the situation for molecules, this approach still has some shortcomings: (1) The previously suggested use of the experimental Weiss constant to incorporate magnetic couplings within the Curie–Weiss temperature regime into the computations seems to work well but obviously introduces a semiempirical aspect and a potential doublecounting of singleion ZFS. This could be done by combining quantumchemical computations of exchangecoupling constants with statistical treatments. (2) The cluster model employed for the g and Dtensor computations clearly still had deficiencies that should be improved using periodic approaches. (3) Spin–orbit effects on hyperfine couplings could not yet be computed properly for solids. (4) Use of beyondDFT approaches or of improved density functionals would be desirable in the computation of hyperfine couplings, but this is also still a problem for the molecules.
In a short conclusion, computational methods provide a big opportunity to simulate and explanations of the experimental NMR spectra, however, we should be clear whether the utilized computational methods are suitable or not in calculating the PNMR shift of the target systems, this requires not only an understanding of the limitation of the methods, but also obtaining the available magnetic and electronic information of the materials.
This author wrote Section 1.1.
This author wrote Section 1.2.
Those authors wrote Section 1.3.