Solid State NMR Studies of Biopolymers

The content of this volume has been added to eMagRes (formerly Encyclopedia of Magnetic Resonance) - the ultimate online resource for NMR and MRI.

The field of solid state NMR of biological samples [ssNMR] has blossomed in the past 5-10 years, and a cohesive overview of the technology is needed for new practitioners in industry and academia. This title provides an overview of Solid State NMR methods for studying structure dynamics and ligand-binding in biopolymers, and offers an overview of RF pulse sequences for various applications, including not only a systematic catalog but also a discussion of theoretical tools for analysis of pulse sequences. Practical examples of biochemical applications are included, along with a detailed discussion of the many aspects of sample preparation and handling that make spectroscopy on solid proteins successful.

About EMR Handbooks / eMagRes Handbooks 

The Encyclopedia of Magnetic Resonance (up to 2012) and eMagRes (from 2013 onward) publish a wide range of online articles on all aspects of magnetic resonance in physics, chemistry, biology and medicine. The existence of this large number of articles, written by experts in various fields, is enabling the publication of a series of EMR Handbooks / eMagRes Handbooks on specific areas of NMR and MRI. The chapters of each of these handbooks will comprise a carefully chosen selection of articles from eMagRes. In consultation with the eMagRes Editorial Board, the EMR Handbooks / eMagRes Handbooks  are coherently planned in advance by specially-selected Editors, and new articles are written (together with updates of some already existing articles) to give appropriate complete coverage. The handbooks are intended to be of value and interest to research students, postdoctoral fellows and other researchers learning about the scientific area in question and undertaking relevant experiments, whether in academia or industry.

Have the content of this Handbook and the complete content of eMagRes at your fingertips!
Visit: www.wileyonlinelibrary.com/ref/eMagRes

View other eMagRes publications here

• Language: English
• Publisher: Wiley
• Release date: Dec 19, 2012
• ISBN: 9781118588888

Book preview

PART A

Fundamentals of Solid-State NMR Spectroscopy

Chapter 1

Internal Spin Interactions and Rotations in Solids

Michael Mehring

2 Physikalisches Institut, Universitäat Stuttgart, Pfaffenwaldring 57, Stuttgart, 70569, Germany

1.1 INTRODUCTION

NMR spectroscopy in the solid state has evolved from CW spectroscopy¹,² via the pulsed and spin echo experiments³ of the early days to a highly sophisticated spectroscopic technique which allows one to distinguish and determine very subtle nuclear spin interactions either with other nuclear spins or with electrons in any type of solid material.⁴–⁶ Because of the local property of these interactions, information not only on crystalline solids but also on disordered materials (amorphous solids, glasses, polymers, etc.) can be obtained. This makes NMR spectroscopy in the solid state complementary to many other spectroscopic techniques, such as optical spectroscopy. With the advent of special high-resolution techniques for solids,⁷–¹⁰ different nuclear spin interactions could be resolved, and detailed information on the spatial and electronic structure of solids has become available. The sevenfold ways¹⁰ that nuclear spins can interact with their environment in solid materials are summarized schematically in Figure 1.1.

External magnetic fields B0 and B1 are usually applied to the sample and also to the nuclear spins (path 1). These fields basically serve the purpose of creating and manipulating the quantum states of the nuclear spins. The most elementary spin–spin interaction is the direct dipole–dipole interaction between nuclear spins (path 2). The dominant contribution to internal interactions of nuclear spins in solids is, however, due to the electron–nucleus interaction (path 3). It is responsible for such interactions as the chemical shift,¹¹ the Knight shift,¹² quadrupolar interactions,¹³ and indirect spin–spin interactions.¹⁴–¹⁶ The coupling to phonons (path 4) occurs only in an indirect way and is rather weak, and in general leads only to relaxation phenomena.

Relaxation will not be discussed in this chapter. I shall restrict myself to the discussion of spectral features and give an introduction to how internal spin interactions influence NMR spectra. In addition, I shall give a brief account of how these quantities might be calculated. Spatial and spin space rotations play an important role in solid-state NMR. I shall address the question of how rotations affect the spectra and what kind of mathematical techniques are required to deal with these. The discussion will be restricted to basic NMR spectra that are typically obtained by Fourier transformation of the FID after pulsed excitation. In many cases, however, special multiple pulse techniques are a prerequisite in order to obtain basic NMR spectra of the kind discussed here, because unwanted interactions have to be eliminated first. Nevertheless, no multiple pulse techniques will be discussed in this chapter. The interested reader is referred to the appropriate articles in the Encyclopedia of Magnetic Resonance as well as to special reviews.⁹,¹⁰,¹⁷,¹⁸

c01f001

Figure 1.1. The sevenfold ways a nuclear spin can interact with its environment. The different pathways (1–7) are discussed in the text. (Reproduced by permission of Springer-Verlag from M. Mehring, Principles of High Resolution NMR in Solids, Springer-Verlag: Berlin, 1983, p. 8.)

1.2 NUCLEAR SPIN HAMILTONIAN OPERATORS

NMR spectra are determined by the initial density matrix¹⁹ c01p004_1 and the spin Hamiltonian operators Ĥ, which we will refer to in the following simply as the Hamiltonian. As in usual spectroscopic theory the eigenvalues Ej and eigenstates |j〉 of the Hamiltonian can be calculated, leading to dipole-allowed transitions at frequency ωij = (Ei − Ej)/ħ with strength |〈i|Î+j〉|², which results in the NMR line intensity

(1.1) 1.1

where pi and pj are the populations of the eigenstates or, equivalently, the diagonal elements of the initial density matrix c01p004_1 . Because of the flexibility of NMR in manipulating the initial density matrix and the effective Hamiltonian by well-defined pulse sequences the calculation of NMR spectra proceeds, however, in a more advanced way via the calculation of the response function in the time domain (FID) followed by a Fourier transformation. The FID is, of course, governed by the effective Hamiltonian of the internal interaction. The most common Hamiltonian, that for internal interactions in solids, will thus be described in this section.

Guided by Figure 1.1 we distinguish spin interactions caused by external and internal fields,

(1.2) 1.2

where the external fields are the static magnetic field B0, which is usually applied parallel to the z axis of the laboratory frame, and Brf, which is a time-dependent radiofrequency field applied in the(x,y) plane of the laboratory frame, leading to the external Hamiltonian:

(1.3) 1.3

which can be summarized as

(1.4) 1.4

where ω0 = −γ B0 and ωrf = −γ Brf. The internal interactions can be represented as a sum of different contributions:

(1.5) 1.5

where the first term is due to shielding contributions, the second term represents the quadrupolar interaction caused by electric field gradients, and the last three terms summarize direct and indirect spin–spin interactions, where two different types of nuclear spins, namely I and S spins, are considered. In order to compare different Hamiltonians, we define

(1.6) 1.6

as the magnitude of a Hamiltonian, where Tr refers to the trace operation. In solids, most of the interactions can be represented by second-rank tensors, leading to the following general formulation of internal interactions in Cartesian coordinates of the laboratory frame (xyz):

(1.7) 1.7

where Î is a nuclear spin operator, A is the interaction tensor, and c01p005_1 may be a magnetic field or another nuclear spin or angular momentum operator. In general, nine different components of the interaction Hamiltonian exist, depending on the local symmetry of the nucleus and the type of interaction. Since rotations either in real space or in spin space play an important role, it is convenient to express the interaction Hamiltonian in addition in the form of spherical tensors A(l,m) and spherical tensor operators c01p023_1 (for details see the literature⁹,¹⁰,²⁰,²¹),

(1.8) 1.8

which, when expanded explicitly, is represented by nine different terms (for lmax = 2):

(1.9) 1.9

where the A(0,0) component represents the isotropic and the A(1,m) components the antisymmetric parts of the interaction, respectively. Although the antisymmetric part may not be zero (as in chemical shift and indirect spin–spin interactions) it does not contribute to the NMR spectrum. In spin–lattice relaxation, however, these parts do contribute. Since we are concentrating on NMR spectra here, the A(1,q) terms will be neglected in the following. In 1.5.4 the spherical tensor representation of the internal interactions discussed in this chapter is summarized. The basic advantage of spherical tensor notation over Cartesian tensors is their transformation properties under rotations. These will be utilized in 1.2.5.

1.2.1 Chemical Shift and Knight Shift

Soon after the discovery of NMR, it was realized that the resonance frequency of the nuclear spins deviates from the bare nuclear Larmor frequency.¹¹,²²,²³ The chemical shift¹¹ and the Knight shift¹² belong to the class of spin interactions characterized by spin Hamiltonians that are linear in spin operators and can be expressed in Cartesian form by the generalized shift Hamiltonian responsible for line shifts of the spectra,

(1.10a) 1.10a

with

(1.10b) 1.10b

which contains both the chemical shift and the Knight shift. This is justified because of their similar spin structure in the Hamiltonian. After all, both result from magnetic electron–nuclear coupling; the chemical shift from closed shell electrons, and the paramagnetic or Knight shift from open shell or conduction electron spin interactions with the nuclear spin. This type of paramagnetic shift should not be confused with the paramagnetic contribution to the chemical shift, which is a closed shell property (see 1.5.1). The different sign convention is historical, and stems from the early observation of mostly diamagnetic chemical shifts, which are negative (shielding), in contrast to paramagnetic and Knight shifts, which are, in most cases, positive (depending, however, on the sign of the hyperfine interaction). Both the chemical and Knight shifts can be expressed phenomenologically as ratios of field or frequency shifts with respect to the static magnetic field or Larmor frequency:

(1.11) 1.11

With the conventional orientation of the magnetic field B0 = (0,0,B0), we obtain

(1.12) 1.12

Note that this notation for the shift Hamiltonian is exact in the sense that no high field approximation has been invoked. The influence of the shift Hamiltonian on NMR spectra and their analysis is briefly discussed in 1.3.1. The physical origin of the different parts of the shift Hamiltonian will be outlined briefly in 1.5.1 and 1.5.2.

1.2.2 Quadrupolar Interactions

The quadrupolar interaction arises from the coupling of an electric field gradient V = {Vαβ} (where α,β = x,y,z) with the nuclear quadrupole moment Q. The corresponding spin Hamiltonian is bilinear in the spin operator, and can be formally expressed in Cartesian form as

(1.13) 1.13

where

(1.14) 1.14

with KQ = eQ/[2I(2I − 1)ħ] and V = {Vαβ}, is the quadrupolar interaction tensor. In the principal axis system (X,Y,Z) of the traceless electric field gradient tensor, the quadrupolar Hamiltonian can be written as¹³,²⁴,²⁵

(1.15) 1.15

with η = (VXX − VYY)/VZZ, where the quadrupolar frequency ωQ is defined as

(1.16) 1.16

The definition of the quadrupolar frequency used here is consistent with the one employed by Cohen and Reif,²⁴ Abragam,⁴ and others.²⁶ There exist other definitions in the literature, however, where ωQ differs by a factor of two compared with the definition used here.

The formulation in terms of spherical tensor operators is given in 1.5.4. The high field approximation will be discussed in 1.2.4. NMR spectra resulting from quadrupolar interactions are shown in 1.3.2. The physical origin of electric field gradients is outlined briefly in 1.5.3.

1.2.3 Spin–Spin Couplings

Direct as well as indirect spin–spin interactions¹⁴–¹⁶ both lead to a bilinear spin Hamiltonian. In a solid, a large number of nuclei are usually coupled to each other. The spin–spin coupling Hamiltonian can therefore be expressed in Cartesian form as

(1.17) 1.17

where the coupling occurs pairwise either between two homonuclear (Ij, Ik) or heteronuclear spins (Ij, Sk). The distinction between heteronuclear (|ω0I – ω0S| ≫ ωIS) and homonuclear (|ω0I – ω0S| ≪ ωIS) spins is based on their difference in resonance frequencies, where we define the spin–spin interaction frequency as ωIS = ‖ ĤIS ‖ ħ. This distinction becomes important only when a magnetic field is applied, a case which is discussed in 1.2.4 and 1.3.3. We shall use in the following sections the form ĤII for homonuclear spins exclusively and the form ĤIS for heteronuclear spins as well as for homonuclear spins if the corresponding expression applies to both. It should be kept in mind also that even between spins of the same type (e.g., two ¹³C spins) the interaction can be of heteronuclear type if the difference frequency between the two ¹³C spins is larger than their interaction frequency.

Let us first discuss the (direct) dipole–dipole interaction tensor between two nuclear spins Ij and Sk which can be expressed as

(1.18) 1.18

with c01p006_1 and c01p006 This interaction depends only on the internuclear distance rjk and the product of direction cosines eαeβ(α,β = x,y,z) of the distance vector with the (x,y,z) axes. The complete Hamiltonian can be represented in the form of the dipolar alphabet⁴ as

(1.19) 1.19

with

(1.19a) 16.19a

(1.19b) 16.19b

(1.19c) 16.19c

(1.19d) 16.19d

(1.19e) 16.19e

(1.19f) 16.19f

which, for a large number of homonuclear spins, can be summarized as

(1.20) 1.20

The same expression applies to heteronuclear spins Ij and Sk where the sum is over j, k. Note that the dipole–dipole interaction tensor is both traceless and axially symmetric, i.e. it can be represented by a single quantity, namely the dipolar interaction strength djk.

If the spin–spin coupling is mediated by the surrounding electrons (indirect interaction) the representation in equation (1.18) applies, but with an interaction tensor {Dαβ} that is neither traceless nor symmetric in the general case. All nine components of the interaction tensor must be considered. Although NMR spectra are only affected by the symmetric part of the interaction tensor, relaxation processes also involve the antisymmetric part. Relaxation will not be discussed here, however. In those cases where the trace of the indirect spin–spin interaction tensor is dominant (isotropic spin–spin coupling), we can express the corresponding Hamiltonian as

(1.21) 1.21

where Jjk = c01p007_6 Tr {Dαβ}. Often local orbital symmetry allows the indirect interaction to be axially symmetric. In those cases the anisotropic spin Hamiltonian has the same spin structure as the dipole–dipole Hamiltonian (equation 1.20), with the only difference that the coupling constant djk is no longer purely geometric, but is a molecular orbital (MO) property. In the general case, the complete symmetric part of the indiret spin interaction tensor must be diagonalized, resulting in three principal values (D11, D22, D33) and three principal axes. The spherical tensor form of spin–spin interactions is summarized in 1.5.4.

1.2.4 Spin Interactions in Large Magnetic Fields

NMR in solids is usually performed for two extreme cases, namely zero or large magnetic fields. In a zero magnetic field, those spin Hamiltonians that are independent of the magnetic field, such as the quadrupolar and spin–spin interaction Hamiltonians, lead directly to resolved spectral lines at the difference frequency of their principal elements even in a powder sample. This is what makes zero field NMR/NQR so attractive.²⁷ However, information on the orientation of the principal elements is lost. In this section, we preferentially cover NMR in large magnetic fields where ‖Ĥ0‖ ≫ Ĥint‖, leading to a separation of the interaction Hamiltonian into secular c01p007 and nonsecular terms c01p007

(1.22) 1.22

where c01p007 and c01p007_3 The physical reason for this separation is the rapid rotation of the spins by the introduction of the large magnetic field. Those terms of the interaction Hamiltonian that are time-independent under this rotation are secular, whereas all time dependent terms are called nonsecular. This suggests transforming the density matrix c01p007 l from the laboratory frame into the rotating frame, leading to the following equation of motion for the density matrix c01p007 :

(1.23) 1.23

where c01p007 =exp[(i/ħ) Ĥ0(t)] c01p007 l(t) exp[–(i/ħ) and c01p007 The secular part of the interaction Hamiltonian can now be extracted by the procedure

(1.24) 1.24

where ω0tc = 2π. In spherical tensor operator notation, terms of the type T(l,m) = exp(±imω0t) appear in the time dependent Hamiltonian. When performing the time averaging (equation 1.24), all terms vanish except those with m = 0, resulting in the general secular interaction Hamiltonian:

(1.25) 1.25

where, however, the antisymmetric term A(1,0) c01p005 (1,0) is irrelevant for the spectrum, as mentioned earlier, and will be dropped. In order to be more specific, we shall now discuss secular parts of different interaction Hamiltonians in turn.

1.2.4.1 Heteronuclear

We begin with heteronuclear spin–spin interactions, where ω0I ≠ ω0S, which leads to

(1.26) 1.26

In the case of dipole–dipole interaction, the z component Dzz can be expressed as

(1.27) 1.27

1.2.4.2 Homonuclear

The corresponding expression for homonuclear (ω0I = ω0S) spin–spin interaction is slightly more complicated and can be written out in general as separating the isotropic part from the anisotropic part:

(1.28) 1.28

with J = c01p007 Tr {Dαβ}. In the case of dipole–dipole interaction, the secular spin–spin interaction Hamil-tonian reduces to

(1.29) 1.29

with c01p013_1 . = –djk(3 cos² ϑjk – 1) The secular quadrupolar Hamiltonian assumes a similar form:

(1.30) geg01

with ωQz[3eQ/2I(2I – 1)ħ]Vzz. Vzz = Vzz[ c01p008 (3 cos² β – 1)+ c01p008 η sin² β cos 2α]. Note that ωQz as defined here still contains the asymmetry parameter η and the full orientational dependence.

In the case of the shift Hamiltonian, the situation is rather simple, because only the z component survives the rotating-frame transformation

(1.31) 1.31

leading to a relative shift in resonance frequency of δzz = Kzz − σzz. In order to keep the treatment general, both shift contributions, namely the Knight shift and the chemical shift, are included here. For closed shell materials, only the chemical shift has to be considered. Note that the tensor character (anisotropy) of the shift interactions is still contained in the z component of the tensor. We shall see in the next section how this leads to an orientational dependence of the resonance line.

1.2.5 Transformation Properties of Spin Interactions

Transformations in NMR are usually rotations either in real space or spin space. The corresponding transformations are represented by unitary matrices. Rotations can be visualized either as rotations of an object like a vector or a tensor (matrix) or alternatively as a counter-rotation of the coordinate system. Both views lead to the same physical result. Here we express the rotation of a vector by r' = Rr, whereas a tensor (matrix) rotation is expressed as

(1.32) 1.32

where the rotation matrix R may be represented (in Cartesian notation) in terms of the direction cosines between the vectors r' and r as

(1.33) 1.33

with {rij} = {rji}−1. The matrix multiplication

(1.34) 1.34

results, where i,j represent the coordinates of the new (i,j) frame, whereas (m, l) labels the coordinates of the old frame. In the high-field approximation, the z component of the tensor governs the NMR spectrum. If the tensor A is given in the principal axis frame (X,Y,Z) by the principal elements (AXX, AYY, AZZ) the z component of the corresponding tensor in the laboratory (rotating) frame (x,y,z) is given by

(1.35) 1.35

which is readily expressed in terms of Euler angles (α,β) by

(1.36) 1.36

This is the typical form all anisotropic secular spin interactions assume, where the tensor elements Aαα (α = X, Y, Z) in equation (1.36) may be replaced by those of the shift, quadrupolar, or spin–spin interaction tensor. The orientational dependence of the spin interaction as represented in equation (1.36) determines the orientational dependence of the lineshift, as well as the quadrupolar and dipolar broadening and splitting.

When discussing rotations, we use the sign convention and the Euler angle definition of Rose,²⁸ where a counterclockwise rotation is defined as positive. Such a rotation by Euler angles (αβγ ) can be expressed as a rotation about the z axis by the angle α followed by a rotation about the new y′ axis by the angle β, and the final rotation about the new z″ axis by the angle γ , which can be summarized as

(1.37) 1.37

In terms of the original axes, the same rotation can be written as

(1.38) 1.38

In order to provide an example in spin space, we consider the rotation operator about the x axis by an angle α:

(1.39) 1.39

The application of these rotation operators to spin operators follows simple transformation rules. The example given here can readily be extended to the operator set (Îx,Îy,Îz). The following example follows directly from the vector picture of spin operators:

(1.40) 1.40

It can readily be extended to other rotations by applying classical vector rotation rules. Note, however, that these transformations are quantum mechanically exact and not a classical analogue. Moreover, they apply to any spin operator independent of the spin quantum number.

Since irreducible spherical tensor operators are specially designed to have convenient rotational properties, we can express rotations of these in the compact form⁹,¹⁰,²⁰,²¹,²⁸,²⁹

(1.41) 1.41

by using the Wigner rotation matrix

(1.42) 1.42

The reduced rotation matrices c01p009 are real and are listed for different values of l in 1.5.5. These transformation properties will be applied in 1.4 in order to discuss rotational properties of NMR spectra in solids.

1.3 NMR SPECTRA

NMR spectra can be recorded in a number of different ways. Several articles in the Encyclopedia of Magnetic Resonance and in this Handbook deal with special techniques, and there are textbooks which discuss, for example, special high-resolution techniques in solid-state NMR spectroscopy. In this chapter I restrict myself to basic spectra which are obtained by, for example, Fourier transformation of the free induction decay after c01p008 π pulse excitation. The time evolution of this FID is governed by the internal spin interactions discussed in 1.2, and can be expressed as

(1.43) 1.43

where ρ(0) represents the density matrix at t = 0 directly after the c01p008 π pulse and Ĥ is the time-independent effective Hamiltonian governing the spin evolution, and Î+ = Îx + iÎy is the detection operator, which includes the two orthogonal (x,y) detection channels. The Hamiltonian used in equation (1.45) must be considered an effective Hamiltonian depending on the type of pulse (or multiple pulse) experiment performed. In the simple case where the Hamiltonian is diagonal in the Zeeman representation, the trace operation is conveniently expressed by

(1.44) 1.44

where Hm ρm,m+1 = m〈|Ĥ/ħ\|〉 and 〈m\| c01p007 (0)|m+1〉 This case is realized in the high-field approximation for all interactions discussed here besides the homonuclear spin–spin interaction. After Fourier transformation, the spectrum

(1.45) 1.45

results. This corresponds to line spectra consisting of delta functions at the frequencies

(1.46a) 1.46a

with intensities

(1.46b) 1.46b

In the general case the Hamiltonian must be diagonalized and the eigenstates must be used in the evaluation of the trace. In solids, the eigenvalues Hm and the matrix elements ρm,m+1 usually depend on the orientation of the magnetic field with respect to the principal axes of the interaction tensor (see 1.4).

1.3.1 Lineshift Spectra

Shift spectra are usually recorded in large magnetic fields, i.e., the high-field approximation applies (see 1.2.4). In a single crystal, the NMR spectrum consists of a single line for every nuclear site. Owing to symmetry, several nuclear sites in a molecule or in the unit cell can be magnetically equivalent, i.e., they give rise to a single line. The line is shifted with respect to the nuclear Larmor frequency and appears at frequency ω=ω0 + δ. In practice, a reference sample is used, to which the observed shift is related by δobs = δ − δref. In a solid with low nuclear site symmetry, this shift depends on the orientation of the magnetic field with respect to the principal axis system of the shift tensor. By using a goniometer as discussed in detail in 1.4.1, so-called rotation plots of the orientation-dependent shifts are obtained, from which the shift tensor can be determined. An example of a shift tensor determination from single crystal rotation plots, in which multiple pulse line-narrowing techniques were used in order to obtain resolved spectral lines, was presented by Griffin et al.³⁰

In Figure 1.2 different NMR line spectra of ¹³C, obtained from a single crystal of the organic conductor (fluoranthenyl)2PF6, are plotted for different orientations of the magnetic field.³¹ Note the variation of the lineshifts with the orientation. The functional form of these rotation plots will be discussed in 1.4.1. Besides the pronounced orientational dependence shown in Figure 1.2, it is of importance that the lineshifts are caused by both the chemical shift and the Knight shift tensors, which are additive.

In a powder sample the orientation of the principal axis system with respect to the magnetic field occurs with probability PΩ, where Ω is the solid angle (αβ). The powder lineshape

(1.47) 1.47

results where dΩ = sin β dβ dα and I(ω,Ω) is the NMR spectrum for orientation Ω. For an isotropic powder P(Ω) = c01p010 . Typical examples of powder spectra of axially and nonaxially symmetric shift tensors are shown in Figure 1.3 (top and middle). Note the edges and the center singularities, which are characteristic features of these spectra and from which the principal elements of the shift tensor can be determined.

In practice it is, however, more appropriate to make a computer fit of the experimental spectra to the calculated lineshape, which can be expressed as

(1.48) 1.48

with m = (ω22 − ω11)(ω33 − ω)/(ω33 − ω22)(ω − ω11),

(1.49) 1.49

with m = (ω − ω11)(ω33 − ω22)/(ω33 − ω)(ω22 − ω11), and I(ω) = 0 for ω > ω33 and ω < ω11. K(m) is the complete elliptic integral of the first kind:

(1.50) 1.50

c01f002

Figure 1.2. Carbon-13 NMR single crystal spectra of the 1D organic conductor (fluoranthenyl)2PF6 for different orientations of the magnetic field perpendicular to the stacking axis. (Reproduced by permission of The American Physical Society from D. Köngeter and M. Mehring, Phys. Rev. B, 1989, 39, 6361.)

c01f003

Figure 1.3. Calculated powder spectra of anisotropic spin interactions, where the dotted lines correspond to unbroadened (ideal) spectra, and the spectra indicated by thick lines are additionally broadened in order to mimic experimental spectra. Top: axially symmetric shift tensor with ν11 = ν22 = −50 a.u. and ν33 = 100 a.u. Middle: nonaxially symmetric shift tensor with ν11 = −70 a.u., ν22 = −30 a.u. and ν33 = 100 a.u. Bottom: Pake pattern spectra³⁷ either due to a spin I = 1 axially symmetric quadrupole interaction or dipole–dipole coupling of two spin-½ nuclei. The maximum coupling strength (ν33, B0‖ unique axis) is 100 a.u.

In order to compare this lineshape with the experimentally observed lineshape, it must be convoluted with a line-broadening function. In practice, Lorentzian or Gaussian functions are used as line-broadening functions. In Figure 1.3, a Lorentzian broadening function was applied, leading to a rounding off of the edges and singularities (thick lines). In order to speed up the numerical calculation of powder spectra, Aldermann et al.³² have proposed a fast algorithm.

The type of powder pattern discussed here was first observed by Bloembergen and Rowland³³ in metallic thallium, where the shift anisotropy is larger than any other interaction. In ordinary solids, this is usually not the case because of dominant dipolar broadenings. With the advent of high-resolution NMR techniques, chemical shift powder patterns became observable in many materials where without these methods only structureless broad lines appeared in the NMR spectra.⁸ An early example of such a resolution enhanced powder spectrum is shown in Figure 1.4.

Resolution was enhanced in Figure 1.4 by applying a multiple pulse experiment of the WAHUHA type, which eliminates the dipole–dipole interaction among the ¹⁹F spins on the average.³⁴ A summary of experimentally determined chemical shift tensors of different nuclei can be found in the literature,¹⁰ in particular in a review on ¹³C chemical shift tensors by Duncan.³⁵

c01f004

Figure 1.4. Experimental ¹⁹F powder spectrum of fluo-ranil (C6F4O2 at 300 K). The nonaxially symmetric shift tensor elements σ11, σ22, and σ33 are indicated. The powder spectrum corresponds to the theoretical spectrum shown in Figure 1.3 (middle). (Reproduced by permission of The American Physical Society from M. Mehring, R. G. Griffin, and J. S. Waugh, J. Chem. Phys., 1971, 59, 746.)

1.3.2 Quadrupole Spectra

In cubic solids, the electric field gradient vanishes because of symmetry. The quadrupolar interaction has nevertheless been observed in cubic solids, caused by point defects and dislocations.³⁶ Surprisingly, it was found that for point defects a Lorentzian line-shape (exponential FID) results, which is usually taken as evidence for homogeneous broadening.³⁶ Here we restrict ourselves to solids where the local symmetry is low enough to observe quadrupolar splittings directly. In zero magnetic field, the transition frequencies between the different eigenstates of the quadrupole Hamiltonian do not depend on the orientation of the electric field gradient tensor. Therefore, narrow lines are observed even in powder samples. This has been developed into a special kind of zero field spectroscopy, covered separately in the Encyclopedia of Magnetic Resonance.

In large magnetic fields only the secular part of the Hamiltonian contributes to the spectrum (see 1.2.4). In the case of the quadrupolar interaction, this results in first-order quadrupolar spectra with transition frequencies

(1.51) geg01_051

Because of the spin symmetry, the spectra correspond to pairs of lines symmetrical about ω0 where the additional line in the center (at ω0) appears for half-integer spins. The corresponding intensities follow from equation (1.47). The simplest situation is encountered for a nuclear spin I = 1 (e.g., ²D), which results in a doublet of lines separated by the quadrupolar frequency. The corresponding powder spectrum depends on the symmetry of the electric field gradient. If the electric field gradient is axially symmetric, the typical Pake pattern³⁷ (Figure 1.3 (bottom)) results, which consists of the sum of the axially symmetric shift pattern (Figure 1.3 (top)) and its mirror image symmetric to zero frequency. For a nonaxially symmetric field gradient, the original and the mirror image of the nonaxially symmetric shift pattern (Figure 1.3 (middle)) must be added. For half-integer nuclear spins, the procedure is similar, but more complicated spectra result. Some examples are shown in Figure 1.5.

If the first-order approximation is no longer valid because of an increased quadrupole interaction, it is best to perform a numerical diagonalization of the Hamiltonian and to calculate the corresponding spectrum. Although this procedure is numerically rather trivial, second-order quadrupolar transition frequencies have been calculated in the past in order to cope with this situation. For completeness, we include here the transition frequency for the central transition ( – c01p008 → c01p008 ) of half-integer spins

(1.52) geg01_052

where

(1.52a) geg01

(1.52b) geg01_052b

(1.52c) geg01

Equations (1.52)–(1.52c) were first derived by Narita et al.³⁸ and applied to the calculation of the second-order powder pattern of the central transition for arbitrary half-integer spin- c01p008 nuclei. Powder spectra of the central transition in half-integer spin systems are shown in Figure 1.6 for different values of η. Further details about quadrupolar spectra can be found in the recent review by Freude and Haase.²⁶

c01f005

Figure 1.5. Powder spectra caused by first-order quadrupole interaction (central transition omitted) for nuclear spin I = 3/2 (top) and I = 5/2 (bottom) for two different values of the asymmetry parameter η = 0.1 and 0.6. (Reproduced by permission of Springer-Verlag from D. Freude and J. Haase, in NMR Basic Principles and Progress, eds P. Diehl, E. Fluch, and R. Kosfeld, Springer-Verlag: Berlin, 1993, Vol. 29, p. 25.)

c01f006

Figure 1.6. Powder spectra of the central transition (−½ → ½) caused by the second-order quadrupole interaction of arbitrary half-integer spin nuclei for different values of the asymmetry parameter η. (Reproduced by permission of Springer-Verlag from D. Freude and J. Haase in NMR Basic Principles and Progress, eds P. Diehl, E. Fluch, and R. Kosfeld, Springer-Verlag: Berlin, 1993, Vol. 29, p. 27.)

1.3.3 Spin–Spin Splittings

Spin–spin couplings in the high-field approximation produce line splittings, as is evident from the corresponding Hamiltonian (see 1.2.4). We must distinguish two different cases, namely

1. heteronuclear interactions: |ω0I − ω0S| ≫ |ωIS|;

2. homonuclear interactions: |ω0I − ω0S| ≪ |ωIS|.

ωIS is the corresponding interaction frequency between spin I and spin S. These conditions do not require, however, that I and S be different types of nuclear spins, merely that their resonance frequency difference with respect to the interaction frequency determines whether they must be considered as homo- or heteronuclear.

1.3.3.1 Heteronuclear Interaction

The symmetric part of the secular interaction Hamiltonian can be expressed in this case as

(1.53) geg01

as was shown in 1.2.4. Let us assume that we are interested in the spectrum of the I spins. Under the assumption that the S–S spin coupling is much weaker than the I–S coupling, an exact expression can be obtained for the FID of the Ij spin coupled to Ns S spins,¹⁰

(1.54) geg01

which takes the following special forms for some representative values of S:

(1.54a) geg01

(1.54b) geg01

(1.54c) geg01

The corresponding spectra are readily obtained after Fourier transformation. Note that c01p013 does include the isotropic interaction J. In the case of a single spin S = c01p008 coupled to a spin I, a pair of lines at frequencies

(1.55) geg01

with c01p013 results. In the case of a powder sample, an average over all angles ϑIS must be performed, leading to a Pake pattern as displayed in Figure 1.3 (bottom), where the outermost edges appear at ω0 ± dIS.

1.3.3.2 Homonuclear Interaction

In this case, it is convenient to separate the isotropic from the anisotropic part, leading to

(1.56) geg01

with c01p014 for the spin–spin coupling between two nuclear spins Ij and Ik. Unfortunately, this Hamiltonian is no longer diagonal, and thus a diagonalization procedure has to be applied for the calculation of the FID and the corresponding spectrum. In the case of the dipole–dipole interaction the isotropic part vanishes, and for two coupled spin- c01p008 nuclei a pair of lines results at frequencies

(1.57) geg01

with c01p014 . For a powder sample, the typical Pake pattern again results (see Figure 1.3 (bottom)). In contrast to the heteronuclear case, here the outermost edges appear at ω0 ± 0.75dII.

1.3.4 The Multispin Lineshape Problem

The multispin lineshape problem for heteronuclear interactions has been discussed in the previous section. Here we consider N homonuclear spins of type I. The truncated (high field) interaction Hamiltonian can be expressed as

(1.58) geg01

For a limited number of spins, a numerical diagonalization is feasible. However, already for 10 spin- c01p008 nuclei, a 1024 × 1024 Hamiltonian matrix must be diagonalized. For much larger spin clusters, supercomputer power is needed, and reaches the current computational limit very quickly. In practical cases, one therefore has to resort to approximation schemes. The most obvious procedure is Van Vleck’s moment expansion,³⁹ which is in fact exact in the case where an infinite number of moments is taken into account. It follows from the symmetry of the interaction Hamiltonian that only even order moments must be considered. The FID can in this case be expressed as

(1.59) geg01

where higher order terms become exceedingly difficult to calculate.⁴⁰ In more or less evenly distributed nuclear spins in a solid, the observed lineshape is often Gaussian, as is the FID. In this case, a very good approximation is

(1.60) geg01

with the corresponding lineshape

(1.61) geg01

The full width at half-height (FWHH) of this line-shape is given by c01p014 The calculation of the second and fourth moments can be performed rigorously, and results in⁴

(1.62) geg01

(1.63) geg01

where ∑j,k,m≠ means that all indices must be unequal. Note that a population factor p was introduced in equations (1.62) and (1.63), which takes the isotopic abundance p of nuclear spins I into account, where p = 1 for 100% abundance. The summation is performed over all possible sites of spin I. Instead of calculating higher order terms a practical approach is to assume special functional forms of the FID, whose second and fourth moments are set to the values calculated according to equations (1.62) and (1.63). Prompted by the idea of Abragam, who used a truncated Lorentzian in order to obtain a linewidth expression based only on the second and fourth moments, an attempt was made to derive a simple linewidth expression by applying a memory function approach, which resulted in the expression¹⁰

(1.64) geg01

where c01p015_1 is a dimensionless parameter, which equals 3 in the case of a Gaussian function, and which assumes large values for µ ≫ 3 if the character of the lineshape is more Lorentzian. Equation (1.64) is valid to a good approximation also in the intermediate regime. Note that in a diluted spin system (p ≪ 1), the moment ratio µ becomes large according to equations (1.62) and (1.63), resulting in a narrow line of more Lorentzian character as for p = 1.

1.4 ROTATIONS

Rotation operations are performed by unitary transformations applied to real space tensors or to spin space tensor operators as introduced in 1.2.5. We shall apply the rules quoted there in order to work out the corresponding frequency shifts or splittings observed in NMR spectra in solids. The most general scheme of transformations necessary for obtaining the spatial tensor (operator) A1 in the laboratory frame (x y z) is

(1.65) geg01

We start from the principal axis system, i.e., the frame in which the symmetric part of the tensor is diagonal, and distinguish different reference frames such as the crystal frame, the goniometer frame, and the laboratory frame. The goniometer frame is a synonym for any frame with a unique axis about which rotations (flippings) can occur. This includes a real goniometer as well as a spinner and a molecular rotation frame. Each type of rotation will be discussed in turn in the following sections.

There may not always be a crystal frame (as in glasses and polymers). In this case, the two transformations c01p015 p and c01p015 c can be combined into a single one. The total transformation required when transforming from the principal axis system to the laboratory frame can be written as the product of single step transformations:

(1.66) geg01

In terms of Wigner rotation matrices, the total transformation leads to

(1.67) geg01

Very often, rotation operations can be contracted into a single operation, and only a subset of the total transformation is required. This will be applied partially in the next sections.

1.4.1 Sample Rotation

Single crystal investigations were performed in the early days of NMR.⁴¹ They can reveal the complete symmetric part of the shift tensor if performed properly. Consider a sample that is rotated about a specific axis which might be tilted by an angle ϑ with respect to the static magnetic field. A set of Euler angles (ϕ ϑ ψ) is appropriate to describe these rotations. Spectra taken in the laboratory frame are proportional to c01p015_3 and c01p015 , where c01p015 represents the isotropic part and is therefore invariant under rotation, whereas the anisotropic part c01p015 is transformed according to

(1.68) geg01

which leads to

(1.69) geg01

where Ag represents the tensor in the goniometer frame. Note that the angle ψ becomes irrelevant in this case, and for a fixed tilt angle ϑ only the rotation angle Ω is varied from one goniometer setting to the next. In the most general case, c01p015 can be expressed as

(1.70) geg01

with c01p015 , which leads to the following Ω dependence of the z component of the chemical shift tensor:

(1.71) geg01

(1.71a) geg01

(1.71b) geg01

Related expressions can be obtained for the dipolar and quadrupolar splittings. Since the am contain a combination of the six tensor elements of the symmetric part of the shift tensor all symmetric tensor elements can be determined in a single rotation plot if σiso is known. This requires, however, that the tilt angle ϑ is chosen appropriately in order to avoid degeneracies. It is more common to chose ϑ = c01p008 π, in which case equation (1.71) reduces to

(1.72) geg01

with c01p016 , and c01p016 , where the relation between spherical tensor components c01p016 and the Cartesian tensor elements of the shift tensor in the goniometer frame are given in Table 1.1. In order to determine the complete shift tensor in the crystal frame, usually three different orthogonal settings of the crystal with respect to the goniometer are required. Let us label the corresponding Euler angles (αβγ ) as outlined in the transformation scheme, equation (1.65), at the beginning of this section. Table 1.1 summarizes the parameters A, S, and C in terms of the Carte-sian chemical shift tensor elements, where only the symmetric part of the shift tensor (σjk = σkj) is considered. In the case of quadrupolar interactions the corresponding field gradient tensor elements, and in the case of spin–spin interactions the corresponding spin–spin coupling tensor elements, must be inserted.

An experimental example of rotation plots according to equation (1.71) has been shown in Figure 1.2. Fitting the Ω dependence of the observed line shifts to equation (1.71) allows one to determine the parameter set (A, S, C). If this procedure is followed for the three orthogonal axes (x, y, z) of the crystal setting with respect to the goniometer z axis, the column vector A = (Ax, Sx, Cx, Ay, Sy, Cy, Az, Sz, Cz) results, which can be related according to Table 1.1 to the column vector s = (σxx, σxy, σxz, σyy, σyz, σzz) or to the corresponding tensor elements of other types of interactions. Although the chemical shift tensor elements can be directly determined from the different Ak, Sk, Ck with k = x,y,z according to Table 1.1, it is more practical to determine the elements of the vector s from the following set of linear equations:

(1.73) geg01

where the matrix

(1.74) geg01

follows directly from Table 1.1. If other crystal settings are used, the matrix M is changed. The overdetermination of the linear equations can be utilized in order to correct for misalignment of the crystal and other experimental errors. As a result of the whole procedure, the symmetric part of the chemical shift tensor is obtained in the crystal axis frame. As a final step, a diagonalization of the symmetric part of the chemical shift tensor is performed, which leads to the principal elements and the principal axes of the chemical shift tensor in the crystal axis frame.

Table 1.1. Relation between the parameters A, C , S in equation 1.72 and the Cartesian chemical shift tensor elements for three different orthogonal settings of the goniometer axis with respect to the crystal axes

c01t001

1.4.2 Sample Spinning

Whereas sample rotation as discussed in the preceding section is performed by rotating the goniometer axis in steps while taking spectra after each step, sample spinning is performed continuously where the spinner axis, which may be tilted by an arbitrary angle ϑ with respect to the magnetic field, is rotated with angular frequency ωR. Andrew et al.⁴² and Lowe⁴³ proposed that considerable narrowing of NMR lines could be achieved by sufficiently fast rotation of a sample, where the spinner axis is tilted with respect to the magnetic field axis. Schaefer et al.⁴⁴ later applied this technique to polymers. At this point, the reader is referred to Chapter 5. Here we restrict ourselves to the tensor character of spin interactions under rotation. In order to analyze the spectra, we assume that the angle Ω in equations (1.69)–(1.71) is replaced by Ω = ωRt, which results in a time-dependent frequency

(1.75) geg01

which generates oscillatory behavior of the FID and leads to sidebands in the spectrum. Let us first consider the average frequency ϖ, which corresponds to neglecting the sidebands:

(1.76) geg01

We note that this average frequency corresponds to the sum of the isotropic shift and a scaled anisotropy, where the scaling factor is determined by the tilt angle. At the magic angle, i.e. when cos² ϑ = c01p007 (θ = 54° 44′), only the isotropic term survives.

The chemical shift anisotropy is contained in σzz, where the z axis refers to the spinner axis. If we label the principal values of the chemical shift tensor (σ11 σ22 σ33) and perform the transformation from the principal axis system to the spinner frame by the Euler angles (α β 0), we can express σzz by

(1.77) geg01

In other words, a scaled anisotropic powder pattern is obtained where the shape and width depends on the angle θ. This procedure has led to the method of variable angle spinning (VAS), which is discussed in a separate article in the Encyclopedia of Magnetic Resonance. A typical example for the type of sideband spectrum which is observed under magic angle spinning is shown in Figure 1.7.

If the full time dependence is considered, the FID can be expressed by the product of three phase factors,¹⁰,⁴⁵,⁴⁶

(1.78) geg01

where

(1.78a) geg01

(1.78b) geg01

with the parameters S1, S2, C1, and C2 as defined in the previous section (see equation (1.71)).

The first term in equation (1.78) sets the initial phase, the second one represents the isotropic part of the shift tensor, and the third one the anisotropic part. All these parameters depend on the tilt angle ϑ of the spinner and the Euler angles (α β γ ). In the case of a powder sample, the FID must be averaged over the angles (α β γ ), which are conveniently expressed with respect to the principal axis system. The anisotropic part Φa(t) = Φa(t − n2π/ωR) with n = 0, 1, 2, … shows cyclic behavior, which leads to the creation of sidebands in the spectrum,⁴⁵,⁴⁶

(1.79) geg01

where the sideband intensity c01p017_1 can be calculated from¹⁰

(1.80) geg01

Herzfeld and Berger⁴⁶ have calculated graphs from which the shift tensor elements can be determined from sideband intensities.

1.4.3 Rapid Molecular Rotation

The spectra corresponding to rapid molecular rotation follow directly from the treatment presented in 1.4.2. Instead of the spinner axis, a molecular rotation axis must be chosen. Because the rotation is considered to be rapid, i.e., ωR ≫ Δω, where Δω represents the frequency spread or anisotropy of the interaction, only the average frequency according to equation (1.76) contributes to the spectrum. In the case of a powder sample, the appropriate powder average must be calculated.

c01f007

Figure 1.7. Phosphorus-31 powder spectra of solid dipalmitoyl-phosphatidylcholine under magic angle spinning conditions for different spinning frequencies νRot including νRot = 0 (top). (Reproduced by permission of The American Physical Society from J. Herzfeld and A.Berger, J. Chem. Phys., 1980, 73, 6021. Courtesy of R. G. Griffin.)

However, there is no free rotation in solids. Molecular reorientation always proceeds through jumps between potential wells that correspond to different orientations of the molecule. The corresponding lineshift or splitting for a particular orientation of the molecule can be readily calculated by using the transformation properties of tensor interactions, as outlined in 1.4.1. Under rapid hopping conditions, an average tensor is observed that results from the average over all possible orientations. In the special case where a unique axis can be defined for the reorientation of the molecule and where the symmetry of this axis is at least threefold, this unique axis plays the role of a rotation axis, and the same equations as in 1.4.2 can be applied. As an example of rapid flipping, one can consider the limiting spectra in Figure 1.8.

The procedure for the general case of molecular reorientation is discussed in the following section.

1.4.4 Molecular Rotational Exchange

Molecular jumps between different orientations numbered j result in different eigenfrequencies ωj. For simplicity, we discuss only single lines as observed in chemical shift spectra. However, the extension to line splitting as observed in quadrupolar or spin–spin coupling spectra is straightforward if only single quantum transitions are considered. In this case, the corresponding spectra can be treated as single lines. In fact, soon after the discovery of NMR, line narrowing due to molecular rotation in the solid state was observed by Andrew and Eades⁴⁷ in solid benzene. Since then, it was realized that characteristic changes in the NMR/ESR spectra occur under molecular reorientation, and lineshape calculations became mandatory in order to extract the molecular dynamics from the spectra.⁴⁸–⁵³ Surprisingly, in inorganic solids, molecular rotations were also observed in temperature-dependent NMR shift spectra.⁵⁴

c01f008

Figure 1.8. Calculated powder spectra of an axially symmetric shift tensor under 180° jumps about an axis which is tilted by 54° 44' (magic angle) with respect to the symmetry axis. The relative jump rate κ with respect to the total anisotropy (ω' − ω⊥) is indicated. Note that in the fast motion limit (κ = 100) a nonaxially symmetric average tensor results. (Reproduced by permission of Springer-Verlag from M. Mehring, Principles of High Resolution NMR in Solids, Springer-Verlag: Berlin, 1983, p. 58.)

As an example of a simple molecular two-site jump, we consider an axially symmetric shift or electric field gradient tensor. The corresponding powder spectrum for the shift tensor is shown in Figure 1.3 (top), while the spectrum for the electric field gradient tensor (spin I = 1) would resemble the Pake pattern shown in Figure 1.3 (bottom). Under rapid 180° molecular jumps about an axis tilted at the magic angle (54° 44' ) with respect to the unique axis of the tensor, a complete nonaxially symmetric (η = 1) tensor results. This could be observed for the shift tensor of ¹H in H2O and for the electric field gradient tensor of ²D in D2O, for example. The variation of the lineshape for different jump rates κ between the extreme situations (κ = 0 and κ = ∞) is shown in Figure 1.8.

In order to calculate spectra like those displayed in Figure 1.8, we need to write down and solve the corresponding equation of motion. For every orientational site j, a site-specific FID Mj(t) contributes to the total FID, with a probability pj corresponding to the a priori probability that site j is occupied. Microscopic reversibility requires that κjkpk = κkjpj where c01p019 is the jump rate from site k to site j. The equation of motion for Mj(t) can now be written as

(1.81) geg01

where additional transversal spin relaxation processes are collected in T2j. By defining the column vector M(t) consisting of N elements Mj(t) with j = 1, 2, … , N, equation (1.88) can be expressed in matrix form as

(1.82) geg01

where Ω represents a diagonal matrix with diagonal elements ωj and Γ corresponds to the jump matrix with elements c01p019 , where the diagonal elements are replaced by κjj − 1/T2j. Since the total FID consists of the sum of Mj(t) over all j, the solution of equation (1.89) can be formally expressed as

(1.83) geg01

where 1 = (1, 1, …1) is an N-dimensional row one-vector and p = (p1, p2, … pN) represents the a priori probability vector. The NMR spectrum is obtained after Fourier transformation of equation (1.83), which leads to the lineshape expression

(1.84) geg01

where A(ω) = i(ω E − Ω) + Γ and E is the N × N unit matrix.

Although the matrix A(ω) is a non-Hermitian complex matrix, there are numerical procedures to diagonalize it. Let us assume a transformation matrix S exists:

(1.85) geg01

where λij = λjδij, with eigenvalues λj. FID g(t) and lineshape I(ω) can be expressed as a single sum over N elements as⁵⁵

(1.86a) geg01

(1.86b) geg01

In the case of powder samples, the appropriate powder average has to be performed. An example of such a calculation is shown in Figure 1.9. Subtle details of the molecular motion, including the jump angles, can be determined from such an analysis. This type of molecular exchange spectroscopy turned out to be very useful in the investigation of molecular motions in polymers.⁵⁶

c01f009

Figure 1.9. (a) Calculated and (b) measured powder spectra for a random jumping tetrahedron with jump rates τ−1.The measured spectra (b) obtained from ³¹P in solid white phosphorus in the β phase compare very well with the calculated spectra (a). The values for the jump rates τ−1 of the experimental spectra were obtained from T1 data. (Reproduced by permission of Elsevier Publishing Co. from H. W. Spiess, R. Grosescu, and U. Haeberlen, Chem. Phys., 1974, 6, 226.)

1.5 ORIGIN OF INTERNAL INTERACTIONS

There are articles in the Encyclopedia of Magnetic Resonance and in this Handbook on the theoretical calculation of spin interactions. Therefore, I shall only outline some essential features of these interactions here.

1.5.1 Chemical Shift

Shielding of the nuclear spins in condensed matter was noticed and treated theoretically soon after the discovery of NMR.¹¹,⁵⁷,⁵⁸ The local magnetic field Blocal at the nuclear site deviates from the externally applied field B0 due to the induced magnetic field Bind

(1.87) geg01

(with 1 the unit tensor), where

(1.88) geg01

Which can be viewed as being caused by induced orbital currents:

(1.89) geg01

It is convenient to separate the chemical shift tensor and the corresponding currents into diamagnetic (σd, jd) or shielding and paramagnetic (σp, jp) or deshielding contributions. The diamagnetic part is caused by diamagnetic orbital currents resulting from Lenz’s rule (shielding), which leads to positive values (by definition) of σd, whereas the paramagnetic part is caused by paramagnetic orbital currents due to partial dequenching of the orbital angular momentum (second-order effect), which leads to an enhanced local field and therefore negative values of σp. From the definition of these orbital currents,

(1.90) geg01

(1.91) geg01

with vector potential A, and

(1.92) geg01

it follows that knowledge of the electronic wave function in the presence of the static magnetic field is required in order to calculate the chemical shift tensor. Under the condition div j = 0 and A = c01p008 B × (r − R), the induced field vector can be calculated from the expression¹¹,⁵⁹–⁶³

(1.93) geg01

where a summation over the electronic ground state |0〉 as well as excited states |n〉 is required. In deriving equation (1.93), it is assumed that the magnetic field is applied parallel to the z axis and only terms linear in B0 are retained. c01p021 is the angular momentum operator of electron j. Equation (1.93) was derived by Ramsey.¹¹ It consists of the (negative) diamagnetic part, which contains only a summation over electrons in the ground state, as well as the (positive) paramagnetic part, which involves summation over excited states.

The calculation of chemical shift tensors is by no means a trivial task. For reliable calculations on molecules and other delocalized electronic systems gauge independent atomic orbitals (GIAO) must be applied in the LCAO–MO scheme. One of the very successful and more recent methods applies an individual gauge for localized orbitals (IGLO). For a review of this technique consult Kutzelnigg et al.⁶⁴,⁶⁵ From their work, a plot of the diamagnetic, the paramagnetic, and the total orbital current in the benzene molecule is displayed in Figure 1.10. Note that a simple ring current picture, as is sometimes invoked, is not applicable.

1.5.2 Knight Shift

The Knight shift is caused by the hyperfine interaction of nuclear spins with electron spins as discussed in 1.2.1. The complete electron–nucleus interaction is summarized in the Hamiltonian

(1.94) geg01

where the electron spins Sj couple to the nuclear spins Ii via the Fermi contact term (first part) and the dipolar term (second and third parts). An NMR lineshift results only if the electronic spin motion is much faster than the hyperfine interaction. This is the situation in metals, but also in some rapidly exchanging or diffusing electron spin systems. Although the corresponding NMR lineshift was first reported in metals by W. Knight, it also applies to magnetic systems if the requirement of rapid spin diffusion is met. The classical Knight shift refers to the isotropic Knight shift, which is caused by the Fermi contact term,

(1.95) geg01

where the electronic wavefunction ψk = uk(r)eik·t for wavevector k is a Bloch function consisting of a lattice periodic function uk(r) and a planewave function eikr. Since an average over the Fermi surface is performed, the Knight shift in its classical form is a measure of the s-wave character of the conduction electrons at the nuclear site and the conduction electron susceptibility χs. ne is the number density of the conduction electrons, which must be replaced by Avogadro’s number when the molar susceptibility is used. However, in general conductors, including organic conductors, a number of non-s orbitals contribute to the hyperfine interaction, and the Fermi contact term makes only a minor contribution to the Knight shift. For the isotropic Knight shift, core polarization effects usually give larger contributions than the Fermi contact term. Often the dipolar part is significant, and leads to Knight shift anisotropy. Nevertheless, one can rigorously summarize all these contributions in the simple relation

(1.96) geg01

where, besides known parameters, the product of the z component of the local hyperfine tensor A(j)zz (in rad s−1) at nuclear site j and the reduced electron spin susceptibility χ(q = 0) (in J−1) results in the z component of the Knight shift tensor. Note that the full symmetric part of the Knight shift tensor is contained in the orientational dependence of Kzz. The reduced electron spin susceptibility χ(q = 0) depends on the density of states at the Fermi level D(EF), and is in case of uncorrelated conduction electrons given by χ(q = 0) = ¼ D(EF). It is related to the ordinary electron spin susceptibility χs by χs = µ0neγ²eħ²χ(q = 0), where ne is the electron number density. Once the local hyperfine interaction is known, the Knight shift allows a determination of the electronic spin susceptibility, even in the superconducting state, where the bulk susceptibility is dominated by diamagnetic shielding. Note that all quantities used here are given in SI units and χs is a dimensionless parameter.

c01f010

Figure 1.10. (a) Diamagnetic, (b) paramagnetic, and (c) total orbital currents in benzene, derived by the IGLO method, in an external magnetic field B0 perpendicular to the benzene ring. Chemical shift tensors for protons and¹³C can be calculated according to the Biot–Savart law (equation 1.88). (Reproduced by permission of Kluwer from W. Kutzelnigg, C.v. Wüllen, U. Fleischer, R. Franke, and T-v. Mourik, in Nuclear Magnetic Shielding and Molecular Structure, ed. J. Tossell, Kluwer: Dordrecht, 1993, p. 141.)

Since the susceptibility is a global parameter, the site selective Knight shift reflects the hyperfine interaction and therefore the electron spin density at the particular nuclear site. This has been utilized in the determination of the site selective electron spin density, i.e. the amplitude of the conduction electron wave function, as in the organic conductor (fluranthenyl)2PF6 displayed in Figure 1.11. For more details, the reader is referred to the specialized literature.⁴,⁵,⁶⁶,⁶⁷

c01f011

Figure 1.11. Spin density distribution of the conduction electrons at different carbon sites (numbered 1–9) of the fluoranthene molecule in the one-dimensional organic conductor (fluoranthenyl)2PF6. (Reproduced by permission of The American Physical Society from D. Köngeter and M. Mehring, Phys. Rev. B, 1989, 39, 6361.)

1.5.3 Electric Field Gradients

Quadrupolar interaction parameters such as ωQ and η determined from NMR/NQR spectra directly lead to the electric field gradient at the nuclear site.⁶⁸ In order to calculate this electric field gradient, the charge distribution ρ(r) must be known. The calculation of the electric field gradient Vzz is conveniently separated into an ionic contribution where summation takes place over all ions with charge qj at position j and an electronic contribution where integration over the electronic charge density is considered:⁶⁹–⁷¹

(1.97) geg01

The parameters γ ∞ and γ (r) are so-called Sternheimer factors which take the enhancement of the electric field gradient at the nuclear site due to core polarization into account. The electronic charge density ρ(r) = eψ*ψ is derived from an MO-type calculation which must provide accurate wave functions ψ. Valuable information has been obtained from quadrupole interactions in different types of solids²⁵ and, in particular, in metals.⁶⁶,⁶⁷,⁷²

In recent years, local density functional-type calculations have provided accurate quadrupole coupling parameters in molecules as well as in solids.a As an example, Figure 1.12 shows the charge density difference at different nuclear sites in the unit cell of the high temperature superconductor YBa2Cu3O7, which was calculated within the local density approximation (LDA) without any adjustable parameters and which resulted in quadrupolar coupling constants in agreement with experimentally determined values from NMR/NQR experiments.⁷³,⁷⁴

c01f012

Figure 1.12. Difference electron density between the crystalline (calculated by the LAPW method) and the superposed ionic charge density in YBa2Cu3O7, together with a drawing of the unit cell. The charge density contour plot corresponds to a plane parallel to the b and c axes through the Cu(1)–O(1) chains. Maxima and minima are labelled in units of e Å−3. Dashed lines correspond to negative charge density, and solid lines to positive charge density. (Reproduced by permission of the American Physical Society from K. Schwarz, C. Ambrosch-Draxl, and P.Blaha, Phys. Rev. B, 1990, 42, 2052. Courtesy of K. Schwarz.)

Table 1.2. General relation between spherical tensors A(l,m) and Cartesian tensors Aαβ(αβ = x,y,z) of internal spin interactions

c01t002

Table 1.3. Relation between spherical tensor operators c01p023 and the Cartesian operators Îα and