UNESCO - EOLSS SAMPLE CHAPTER

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1 NMR SPECTROSCOPY Juan Carlos Paniagua and Miquel Pons University of Barcelona, Spain Keywords: chemical shift, correlation spectroscopy, diffusion coefficient, Fourier transform spectroscopy, imaging, liquid crystals, magic angle spinning, magnetic properties, medical diagnosis, molecular dynamics, molecular structure, MRI, multidimensional NMR, NMR, nuclear Overhauser effect, nuclear spin, quality control, spectroscopy, structure determination. Contents 1. Introduction 2. Classical Description 3. Quantum Description 4. Multidimensional NMR 5. Dynamic Aspects of NMR 6. Spatial Information from NMR 7. Solid, Liquid, and Partially Oriented Samples 8. The Impact of NMR Acknowledgments Glossary Bibliography Biographical Sketches To cite this chapter Summary Nuclear magnetic resonance (NMR) is a powerful and versatile spectroscopic technique for investigating molecular structure and dynamics. It involves reorientations of nuclear spins with respect to an applied static magnetic field. The overall process can be understood by using classical arguments, but a quantum treatment is needed in order to comprehend the details of the technique, especially those related to spin spin interactions. While the proton is the most frequently studied nucleus, almost every element has an isotope active in NMR. Multipulse and multidimensional Fourier transform NMR techniques provide an unlimited diversity of possibilities for simplifying and interpreting complicated spectra, thus allowing resolution of the three-dimensional structure of macromolecules, such as proteins and nucleic acids. As a non-destructive analytical technique, it finds many applications not only in chemistry but also in agriculture, food science, and the characterization of materials. Dynamical aspects ranging from the nanosecond to the reciprocal second can be characterized. Transient magnetic-field gradients applied across a sample allow the acquisition of images of great value for medical diagnosis.

2 1. Introduction Nuclear magnetic resonance (NMR) is a spectroscopic technique that provides information about the structure and some dynamic properties of a sample subjected to a static magnetic field, from the analysis of the interaction between the magnetic moments of sample nuclei and an applied electromagnetic wave. Nuclear magnetic moments will be introduced before proceeding to analyze their interaction with electromagnetic radiation. A nucleus is built up of protons and neutrons moving in a very small volume (nucleus radii range from to m). Each of these particles can have orbital angular momentum associated with its motion and spin angular momentum intrinsic to the particle. The modulus of the spin angular momentum takes a fixed value for each kind s s+ 1, of particle (like the mass or the charge) and is usually expressed in the form ( ) where is the Plank constant divided by 2π and s is a spin quantum number. This number, usually referred to as the particle spin, takes the value 1/2 for both protons and neutrons. The sum of the orbital and spin angular momenta of the particles within a nucleus is called nuclear spin vector (I) and its modulus is I( I + 1) where I takes one of the values 0, 1/2, 1, 3/2, and so on. (I values greater than 4 are rare). Although I depends on the nuclear state, nuclei are normally in their ground state and the value of I for this state will be considered an intrinsic nuclear property. Nuclei with even numbers of protons and neutrons (like 12 C and 16 O) have a null spin, but most nuclei with odd numbers of protons and/or neutrons have I 0. These nuclei have a magnetic (dipole) moment proportional to the spin vector: μ = γ I (1) where the gyromagnetic ratio γ is a constant characteristic of each nuclide. γ can be positive or negative, and takes its largest values for 3 H ( T 1 s 1 ) and 1 H ( T 1 s 1 ). Nuclear magnetic moments interact with an applied magnetic field as a magnet does: its energy depends on the moduli and the relative orientation of the magnetic moment and magnetic field vectors. Some effects of this interaction can be understood by means of simple classical arguments, but many important aspects must be accounted for in a quantum framework. The classical description (usually referred to as the vector model ) will be presented first, so that readers not acquainted with quantum mechanics can grasp the physical basis of the technique. This picture is useful even in the context of the quantum treatment, which will be considered afterwards. 2. Classical Description In a sample with no external fields, the nuclear magnetic moments are oriented at random giving a null resultant. If a static magnetic field B 0 is applied to the sample, the energy of interaction between the magnetic moment µ of each nucleus and BB0 is E= -μ B 0 (2)

3 Since this energy is minimum when both vectors are parallel, the nuclei will tend to lose energy by orienting their magnetic moments in the direction of the field. Thermal agitation opposes this tendency so that, at room temperature, only a very small net orientation is achieved. Nevertheless, a non-null resultant parallel to BB0 will now remain, and one says that the sample has acquired some spin-polarization. The polarized sample acts as a magnet, and its magnetization (M) is defined as the total magnetic moment per unit volume. The magnetic field exerts a torque on each nucleus given by the vector product τ = μ B (3) 0 If the vector I is treated as a classical (orbital) angular momentum, one can use Newton s law to relate its time derivative with the applied torque: di τ = τ = dt From Eqs. (1), (3), and (4) one obtains dμ = μ B 0 (5) dt Thus, the torque exerted by the magnetic field on the nucleus produces a change in its magnetic moment perpendicular to the plane formed by the vectors µ and BB0, that is, a rotation or precession of µ round B 0B (Figure 1). (4) Figure 1. Precession of µ in the laboratory system under a static magnetic field BB0 for a nucleus with positive γ.

4 As frequently happens in NMR, this problem is conveniently viewed from a rotating reference system. Let us consider one that rotates with angular velocity ω. The relationship between the time derivative of µ in the rotating system (δµ/δt) and that in the laboratory frame (dµ/dt) is the same as that relating the position derivatives (that is, the linear velocities): δμ d δ t = μ dt + μ ω (6) Using Eq. (5) one obtains δ μ ω = μ γ B 0 + (7) δt γ As a consequence, in the rotating system everything happens as in an inertial system (Eq. (5)) except for a substitution of the actual magnetic field BB0 by an effective one: Beff = B 0 + ω (8) γ By choosing the angular velocity ω = γb (9) one gets δ μ 0 δ t = 0 That is, the magnetic moment remains static in a frame rotating at the angular velocity γbb0; thus it will precess in the laboratory frame with that speed. The precession phases (ϕ) of the sample nuclei are distributed at random ( incoherently ), so that the transversal (that is, perpendicular to B 0B ) components of their magnetic moments give a null resultant and the magnetization is parallel to the rotation axis. (10) Let us now apply to the sample an electromagnetic wave whose magnetic field BB1 rotates clockwise with angular velocity ω rad (for radiation ) in a plane perpendicular to the vector B 0B, which will be taken to define the z axis (Figure 2): ( 1 1 rad rad ) B = B icosω t jsinω t (11)

5 Figure 2. Magnetic field of a circularly polarized electromagnetic wave of angular frequency ω rad. In practice, a linearly-polarized wave is used, the magnetic field of which has a constant direction, but this kind of wave is equivalent to a superposition of two circularly-polarized waves whose magnetic fields rotate in opposite directions, and only one of these (represented with a solid line in the figure) has the right angular momentum to affect nuclear magnetic moments; that is, angular momentum conservation rules out its interaction with the other (the dashed arrow). It will be assumed that γ is positive (for negative γ values the sign of ω rad should be reversed in every equation). The effective field in a frame that rotates with angular velocity ω is now obtained by substituting the total field BB0+B 1B for BB0 in Eq. (8). (See Figure 3.) Beff = B0 + B 1 + ω (12) γ By choosing the modulus of ω equal to the angular velocity of the radiation field ( ω = ω rad ) this field will remain static in the rotating frame, so the argument following Eq. (10) can be applied to conclude that the magnetic moment µ of each nucleus will precess around BBeff in that frame with angular velocity γb eff B. If all the nuclei with spin are of the same type (same nuclide ), they will have the same γ value and they will precess at the same angular velocity; hence, their sum M will also precess at the angular velocity γbbeff in the rotating frame.

6 Figure 3. Precession of µ and M in a frame rotating at the angular velocity (ω rad ) of the radiation field (BB1) during a pulse of angle θ = γb eff B t p, where t p is the pulse duration (γ is assumed positive). Since the magnetization is initially directed along B 0, the radiation field causes a maximum effect on it when B eff is perpendicular to B 0 (and equal to B 1 ). This implies (see Eq. (12)): ω 0 γ = B (13) and leads to the resonance condition ωrad = γ B0 (14) As ω rad moves away from γbb0, B eff B acquires a component along B 0 and the pulse effect on M tends to vanish. One normally uses the frequency rather than the angular velocity to characterize the radiation, so the resonance or Larmor frequency for the nuclide being considered will be (dropping the subscript rad ): γ B0 ν = 2π (15)

7 If the radiation acts during a time interval t p the rotation angle of M in radians will be (see Figure 3) θ = γ B t (16) eff p and it is said that a pulse of angle θ has been applied to the sample. Most common pulses are those of 90 and 180. As the direction of M separates from that of B 0 the energy of interaction with the static field ( M BB0) becomes more positive, which means that the sample is absorbing energy from the radiation. After the pulse, BB0 is anew the only applied field but, in contrast with the initial equilibrium situation, M is no longer directed along the z axis. By adding up equations like Eq. (5) for every nucleus of a (unit volume) sample one obtains a similar equation for M: dm = M γ B 0 (17) dt so the conclusions above drawn for µ (see the discussion following Eq. (10)) can now be extended to M: the magnetization vector will precess about BB0 with angular velocity γb 0B in the laboratory frame (Figure 4). Figure 4. Precession of M in the laboratory frame after a pulse of angle θ for nuclei with positive γ.

8 After the pulse, the precession phases of the individual nuclear spin vectors are not distributed at random, as in the initial equilibrium state, but are grouped around the phase of M. It is thus said that the nuclei precess coherently, or that the pulse has generated some phase-coherence in the sample (there are other types of coherences relevant to NMR that cannot be visualized with the present classical model). The precession of M can be detected by measuring the current induced by the transversal magnetization in a coil surrounding the sample. The resulting time-domain signal is digitalized and converted into a frequency-domain plot by carrying out a mathematical Fourier transformation on a digital computer (Figure 5). Figure 5. The magnetization component on an axis perpendicular to BB0 is registered as a function of time (a) and Fourier transformed to give a frequency function (b). The faster the time-domain signal decays (shorter T 2 ), the wider the frequency-domain signal results.

9 Relaxation tends to restore thermodynamic equilibrium, so that the magnetization will tend to recover the direction of BB0 and the time-domain signal will eventually disappear, blurred by the spectrometer noise (Figure 5a). The time-domain signal is thus referred to as the free induction decay (FID). Two different relaxation mechanisms bring the longitudinal (that is, parallel to BB0) and transversal components of M towards their equilibrium values. Both are approximately first-order processes with characteristic evolution times: the longitudinal or spinlattice relaxation time (T 1 ) and the transversal or spin spin relaxation time (T 2 ). By including the corresponding terms in the equation for the evolution of M in the rotating frame (which results from adding up equations like Eq. (7) for the nuclei), one obtains the so-called Bloch equation : δ δ M M M i + M M eq z x y = M γ B eff + k (18) t T1 T2 Spin-lattice relaxation restores the equilibrium value (M eq ) of the z component of M by means of a non-radiative energy transfer between nuclear spins and the surroundings: the lattice vibrations in the case of solid samples, and molecular motions (mainly rotations) for liquids and gases. Typical T 1 values range from 10 2 to 10 2 s. Of course, the transversal magnetization must have disappeared by the time M z has reached its equilibrium value, but the interactions between nuclear spins introduce random variations in their precession frequencies, which result in a coherence loss and speed up the transversal-magnetization decay. Transversal relaxation times (T 2 ) determine the line-width of the frequency-domain signals: the faster the FID decays the wider the signal results after Fourier transformation (Figure 5). The magnetic field inhomogeneities over the sample also contribute to fade the transversal magnetization (and to increase line-width), since variations in BB0 from one region of the sample to another produce some spread of resonance frequencies (see Eq. (15)). This leads to an effective transversal relaxation time (T * 2 ) shorter than T 2. Field inhomogeneities are partially averaged out by rapidly spinning the sample. Nuclei with I > 1/2 have an electric quadrupole moment whose interaction with the electric field gradient at the nuclear site further accelerates the transversal relaxation. If the molecules of the sample have different nuclides with I 0 (e. g. 1 H and 13 C) the above reasoning can be applied to every nuclide, and a specific signal could be detected for each one by conveniently adjusting the radiation frequency: j ν H γ HB0 γ CB 0 =, ν C =,etc. 2π 2π (19) Magnetic field intensities of present day commercial spectrometers range from 1.4 to T, which leads to resonance frequencies of 60 to 900 MHz for 1 H and 15.1 to MHz for 13 C (whose gyromagnetic ratio is almost four times smaller). These fall

10 in the radiofrequency region of the electromagnetic spectrum. Spectrometers are usually classified according to their proton resonance frequency. Each nuclide would certainly give a unique signal in a low-resolution spectrum but, under the usual high-resolution conditions, nuclei of the same type in different chemical environments (an OH bond, a methyl group, and so on) can be distinguished thanks to small differences in their resonance frequencies. These are due to the small magnetic field produced on each nucleus by the rest of the particles of the molecule. Let us first consider the effect of the electrons. These have orbital and spin angular momenta, whose associated magnetic moments generate magnetic fields. However, many molecules have a closed shell electronic structure with no net angular momentum and, in the absence of external magnetic fields, no significant magnetic interaction with the nuclei. Nevertheless, the external static field distorts the electron distribution in such a way that a net electronic magnetic moment remains. The magnetic field produced by this moment depends on the nuclear position, which gives different total fields and resonance frequencies for identical nuclei in different environments. Thus, if the electrons produce a field BBel,i at the position of a nucleus i whose gyromagnetic ratio is γ i, its resonance frequency will be: γ i B + B ν i = ν i = 2π 0 el, i The field produced by the electrons at the position of a nucleus is proportional to the external field. In general, it is not parallel to BB0, so that the proportionality relationship must be expressed in terms of a tensorial magnitude, the shielding or screening tensor at nucleus i (σ i ): B = σ B (21) el, i i 0 For protons, BBel,i is mainly due to an induced electronic current that opposes the external field, whence the term shielding and the negative sign in Eq. (21). Molecular rotation in gas or (not very viscous) liquid samples produces an averaging of this tensor that leads to a scalar magnitude: the shielding constant σ i : B = σ B (22) el, i i 0 (20) The resonance frequency of the nucleus i is then: γ i(1 σi) B νi = 2π 0 (23) The values of σ i are much smaller than 1, so that the resonance frequencies of the same nuclide in different molecular environments are very close to each other compared with typical frequency differences of distinct nuclides. Consequently, NMR spectra of different nuclides rarely overlap.

11 The magnetization of a sample with identical nuclei in different environments is a sum of components that precess at slightly different frequencies. This leads to a complicated FID that, after Fourier transformation, shows up the component frequencies. Before 1970, NMR spectra were recorded either by sweeping the radiation frequency at a fixed BB0 or by varying this field intensity at a fixed radiofrequency. Nowadays, those continuous-wave operation modes have been superseded by pulse techniques, in which nuclei with different shielding constants are simultaneously irradiated. In fact, the radiation used is essentially monochromatic, but one can show that a short pulse is equivalent (by a Fourier transform) to a mixture of infinite monochromatic waves whose frequencies spread over an interval centered on the pulse frequency. This makes it possible to obtain the whole spectrum of a nuclide (or a substantial part of it) with a single pulse, and to obtain many spectra in a short time by repeating the sequence pulse- FID registration. These are then digitally added to improve the signal to noise ratio. It is not easy to measure accurate absolute values of shielding constants and one usually gives values relative to that of a reference substance added to the sample (for 1 H and 13 C tetramethylsilane, commonly referred to as TMS, is normally used). These are obtained from the adimensional chemical shift, which is independent of the spectrometer staticfield intensity BB0: νi νi, ref 6 σi, ref σi δ i = δi = 10 = 10 ν 1 σ i,ref i, ref 6 Here the subscript ref refers to the reference substance and the factor 10 6 is included to obtain convenient values (the unit ppm is usually added to δ i values). By taking into account that σ i,ref << 1, chemical shifts can be identified with relative shielding constants: 6 (,ref ) 10 δ = σ σ (25) i i i Chemical shifts of protons in typical organic compounds range from 0 to 16 ppm; for 13 C the range is about 300 ppm. Conventionally, NMR spectra are plotted with δ i increasing from right to left. Let us look at the evolution of nuclear spins after a pulse of, say, 90 from a frame that rotates at a reference frequency (which may be that of the radiation source). A nucleus with exactly that Larmor frequency will remain static, but one with chemical shift δ i will precess at an offset angular frequency (see Eq. (23)): (24) 6 ( 1 ) B ( 1 ) B B Ω = ω ω = γ σ γ σ = γ δ 10 (26) i i i,ref i i 0 i i,ref 0 i i 0 Offset frequencies, which lie in the audio frequency region of the spectrum, are what NMR spectrometers actually measure.

12 Nuclei that can be interchanged by a molecular symmetry operation or a rapid conformational change have the same chemical shift, and are said to be chemically equivalent or isochronous. Chemical shift values can be related to molecular structure, either by theoretical calculations or, most conveniently for routine applications, by using tabulated empirical correlations. Thus, chemical shift information is a valuable aid for structure determination. Another important source of information, present in most NMR spectra, is related to the interaction between nuclear spins. While the electronic environment produces a shift in the resonance frequency with respect to that of the bare nucleus, the spins of other nuclei can split that frequency into two or more signals. Quantum arguments are needed to understand this effect. Actually, the reason why classical mechanics works well for predicting NMR spectra of independent nuclei is that the quantum evolution equation (the time-dependent Schrödinger equation) leads, in that case, to the classical result shown in Eq. (5); however, important divergences appear for coupled nuclei. 3. Quantum Description Some quantum mechanical results, such as the quantization of angular momentum components and the selection rules, are essential to fully understand NMR. Moreover, some aspects of the technique that can be explained by classical arguments (as the origin of the resonance conditions Eq. (14) or Eq. (23)) have a simpler explanation within quantum theory. Quantum mechanics restricts the values that can result when measuring any component of the nuclear spin angular momentum to m, where the quantum number m can take any of the following 2I+1 values m= I, I 1, I 2,... I (27) This leads to a quantization of the energy of interaction of a nucleus i with an static magnetic field BBi = kb ib : ( 1 ) ( 1 ) Ei = μi Bi = μz, ibi = γiiz, i σi B0 = m γi σi B0 = mhν i (28) where, for the time being, ν i should be considered as a parameter defined by Eq. (23). Hence, the nucleus has 2I+1 equally spaced energy levels, the difference between two consecutive ones being hν i (Figure 6).

13 Figure 6. The four spin states of a spin 3/2 nucleus split into four energy levels under a static magnetic field BB0. The three allowed transitions between those energy levels appear at the same frequency (positive γ is assumed). When the sample is exposed to radiofrequency radiation, transitions can be induced between those energy levels. Quantum mechanics imposes a selection rule that restricts such transitions to those involving a unit change in m: Δ m =± 1 (29) All the possible transitions will then have the same energy: Δ Ei = hν i (30) and will give a unique signal in the spectrum at a frequency (see Figure 6): γ i(1 σi) B νi = 2π 0 (31) The resonance condition (Eq. (23)) thus appears as a consequence of energy conservation: the absorbed-photon energy (hν i ) must match the molecular energy change ( E i ). The sharpness of NMR signals is related to the fact that the energy levels involved in the transition have a well-defined energy. According to the time energy

14 uncertainty relationship, the low uncertainty of these energy levels is, in turn, related to their relatively long average lifetimes. Let us now consider spin spin interactions. Two nuclear magnetic moments interact though a direct or dipolar coupling mechanism in the same way that a pair of magnets do: the magnetic field produced by one of them interacts with the magnetic moment of the other. The classical interaction energy thus depends on the magnitude of the nuclear magnetic moments, their relative orientation, and the distance between the nuclei, falling off as the cube of this distance. In the quantum treatment, the quantization of I z leads to a splitting of the NMR signals produced by each nucleus. This can be understood by a qualitative argument based on two spin-1/2 nuclei: each spin can adopt two orientations relative to the z axis (I z = ±1/2 ), so there will be two possible values (with opposite signs) of the magnetic field produced by one of the nuclei on the other. In one case this field will add to the external plus electronic field, thus increasing the resonance frequency of the latter, and in the other this frequency will decrease. As a consequence, the signal of each nucleus splits into a doublet. By putting this argument into a quantitative basis one finds that the splitting of the signal of nucleus A produced by the dipolar coupling to nucleus X is proportional to Aγ X 2 3 ( AX γ 3cos θ 1) (32) r where r AX is the distance between the nuclei and θ is the angle formed by the vector connecting them and B 0. The splitting disappears for 3cos 2 θ = 1, that is θ = which is known as the magic angle. When a crystal is considered, the successive shells of atoms surrounding each nucleus produce splittings of decreasing magnitude, which results in a strong signal broadening that usually erases chemical shift information. This broadening can be minimized by fast sample spinning around an axis forming the magic angle with BB0. Special pulse techniques can also be used to recover chemical shift information. In gaseous and (not very viscous) liquid samples the magnetic field caused at the position of one nucleus by other nuclear moments is averaged over the rapid molecular rotations. This produces a null average dipolar coupling, leading to narrow NMR lines that allow chemical shifts to be observed. Nevertheless, the dipolar coupling plays an important role in nuclear spin relaxation. (33) A weaker spin spin coupling can still be observed in fluid samples whose origin is subtler than the direct one. The bonding electrons transmit this indirect coupling. Consider two nuclei (A and B) connected by a bond formed by two paired electrons. The interaction of the electronic magnetic moments with the spin of nucleus A stabilizes one electron with respect to the other. This produces a slight breakdown of the electron pairing: the electron stabilized by nucleus A tends to approach this nucleus, while the other will preferably move near nucleus B. This stabilizes one orientation of this

15 nucleus spin with respect to the other. Being an electronically transmitted effect, the intensity of the indirect coupling between two nuclei decreases with the number of bonds connecting them, and can usually be neglected for more than three bonds. Obviously, this is an intramolecular interaction whose effect can be studied by applying quantum mechanics to a single molecule. The indirect spin spin interaction is also termed scalar coupling because its quantum operator is independent of molecular orientation, which is the reason why it is not averaged out by molecular rotations. For two coupled nuclei, A and B, this operator takes the form: 2 ( ) J h `I `I (34) AB A B where J AB is a coupling constant (with units of frequency) that can be positive or negative and is independent of the magnetic field acting on the molecule. The spin Hamiltonian operator of the two nuclei system (see Schrödinger Equation and Quantum Chemistry) will include the terms for the interaction of their magnetic moments with the static field ( free precession or chemical shift terms) plus that for the scalar coupling: `H = `μ B (1 σ ) `μ B (1 σ ) + ( hj / ) I I (35) A 0 A B 0 B AB A or, dividing by h and using Eq. (31), H h I I `I `I B ν ν ` za zb A = A B + JAB 2 The corresponding time-independent Schrödinger equation can be solved exactly. For the spin-1/2 case four energy levels are obtained, and the selection rules allow four transitions between them. The resulting NMR spectrum consists of two doublets at both sides of the mean resonance frequency of the non-coupled nuclei, (ν A +ν B )/2 (Figure 7). The doublet splitting (J AB ) is constant, while the separation between both doublets ( ) 2 2 ν ν J A B AB B (36) + (37) increases with BB0 (see Eq. (31)). If the nuclei are chemically equivalent (ν A = ν B ) the internal lines of either doublet of Figure 7 collapse into a single line of double intensity. Likewise, the intensity of the outer lines vanishes, so that the spectrum consists of a unique line at the non-coupled frequency ν A. Thus, the coupling between equivalent nuclei does not show up in the spectrum. On the other hand, the doublets collapse into two single lines at the noncoupled frequencies ν A and ν B for J AB = 0, as was to be expected.

16 Figure 7. NMR spectrum of two spin-1/2 nuclei with coupling constant J and Larmor frequencies (in the absence of coupling) ν A and ν B. For the strong fields of modern NMR spectrometers one usually has ν A ν B >> J AB, in which case it is said that the nuclei are weakly coupled and that the spectrum is of firstorder type. Letters well separated in the alphabet are used for weakly coupled nuclei, so the spectrum of two weakly coupled nuclei is designated AX and the term AB refers to a strongly coupled pair. In first-order spin systems the scalar coupling can be accounted for by using first-order perturbation theory (instead of solving exactly the Schrödinger equation). Energy levels can then be associated to particular combinations of I z values, and the NMR spectrum consists of multiplets centered very close to the non-coupled frequencies, which greatly eases the assignment of the multiplets. Thus, for an AX spin- 1/2 system, first order perturbation theory leads to: mamx A A X X A X AX E = m hν m hν + m m hj (38) and the NMR spectrum consists of a pair of doublets centered at ν A and ν X (Figure 8). If J AX is positive, the resonance frequency of a nucleus A coupled to a nucleus X with positive/negative m X value (ν A /ν A ) will be somewhat smaller/larger than the noncoupled frequency ν A : ν J 2 AX A = ν A (39)

17 ν J 2 AX A = ν A + (40) and the same holds for nuclei X. Figure 8. Energy levels and allowed NMR transitions for two weakly coupled spin-1/2 nuclei, A and X, with positive coupling constant J. This first-order coupling case also admits a simple picture within the vector model: the nuclei A and A give two independent contributions to the magnetization that precess with frequencies differing by ν A ν A = J AX (see Figure 9). Some of the preceding conclusions can be extended to more complex coupling cases: 1. A group of n chemically equivalent independent nuclei (A n ) gives a unique line whose intensity is proportional to n. 2. A system formed by a group of n chemically equivalent, spin-1/2 nuclei (A n ) weakly coupled to another group of m chemically equivalent, spin-1/2 nuclei (X m ) with the same coupling constant (J AX ) between any nucleus A and any nucleus X (in which case it is said that nuclei A are magnetically equivalent or isogamous, and the same holds for nuclei X) gives a set of m+1 lines centered about ν A and a set of n+1 lines centered about ν B. The spacing between lines of either group is J AX. The total intensities of the A n and X m multiplets are

18 proportional to n and m respectively. The relative intensities of the A n lines are m proportional to the binomial coefficients,, with k = 0,1, 2, m, and those k n of the Xm lines are proportional to, with k = 0,1, 2, n. k 3. The spectrum of a group of n magnetically equivalent nuclei (A n ) weakly coupled to another group of m magnetically equivalent nuclei (M m ) which, in turn, is weakly coupled to a third group of l magnetically equivalent nuclei (X l ), all with spin 1/2, can be obtained by first applying rule (2) to A n and M m and then applying again that rule to obtain the splitting of each component of the M m multiplet due to the coupling with the group X l. 4. Similar rules can be deduced for nuclei with I > 1/2. Figure 9. Evolution of the two components of the magnetization due to a spin-1/2 nucleus A weakly coupled to another spin-1/2 nucleus X with positive coupling constant J (the phase difference has been exaggerated for clarity s sake). Scalar coupling is a valuable source of information about the way nuclei are bonded. However, it is sometimes convenient to remove its effect in order to simplify an NMR spectrum. One way of eliminating the signal splitting of a nucleus A caused by its coupling to a nucleus X (that is, to decouple A from X) is to apply a strong radiofrequency at ν X while the spectrum of A is recorded. This saturates the signal of X and makes the lines of the A multiplet collapse into a single one at ν A.

19 It has been already pointed out that strong static magnetic fields are needed to obtain first-order NMR spectra. Another benefit of such fields is sensitivity, as can be seen by evaluating the population difference between the two levels of a spin-1/2 nucleus. The Boltzmann distribution law gives the fraction of molecules in each level: N ± 1/2 N = e e E / kt ± 1/2 E / kt E / kt + e 1/2 1/2 (41) where N ±1/2 are the numbers of molecules per unit volume in the levels m = ±1/2, N = N 1/2 + N 1/2 and k is Boltzmann constant. According to Eq. (28), E ±1/2 = hν/2 and, even for the largest frequencies used in NMR, hν/2 << kt; thus, for ν = 900 MHz and T = 298 K, hν/2kt = This allows substituting each exponential in Eq. (41) by the two first terms of its Taylor expansion: N N ± 1/2 hν 1± 2kT 1 hν = ± (42) hν hν kT 2kT 2kT The lower energy level (m = 1/2) is only slightly more populated than the upper one (m = 1/2), so that the net absorption (absorption minus emission), which is proportional to N 1/2 N 1/2, will be relatively weak. The same conclusion is reached by considering the equilibrium magnetization, which, once converted into observable transversal magnetization, is what is registered by an NMR spectrometer: ( μ ) ( μ ) ( γ ) ( γ ) M = N + N = N I + N I z 12 z z z z 12 ( 12) γ ( 12) ( )( 2) = N γ + N = N N The sensitivity problem grows worse for nuclides with low natural abundance (that of 13 C is about 1%) and/or low gyromagnetic ratio (for 39 K it is 21.4 times smaller than that of the proton), as follows from Eq. (31) and Eq. (42): N1/2 N 1/2 hν γ (1 σ ) B0 = = (44) N 2kT 2kT (43) Moreover, spontaneous emission is completely negligible at radio frequencies and the non-radiative relaxation mechanisms are not very efficient, so saturation problems can arise. Eq. (44) shows that stronger static magnetic fields lead to higher population differences and better sensitivities. Sensitivity improvement for low-γ nuclei can be derived from magnetization transfer from high-γ nuclei (usually protons) in the same molecule. This can be achieved by cross-relaxation between the two nuclei after the population difference of the two states of the high-γ nucleus is removed from the equilibrium value, for example, by saturation using a strong radiofrequency. The resulting change in the intensity of the low-γ nucleus

20 signal is referred to as the nuclear Overhauser effect (NOE). NOE between nuclei of the same type is also possible and provides useful geometrical information, as the efficiency of this process depends on the inverse sixth power of the internuclear distance. Magnetization transfer can also be effected in a coherent fashion when both nuclei are coupled. This can be achieved by manipulating non-equilibrium states created by an initial radiofrequency pulse with subsequent pulses separated by delays during which the system evolves under the influence of spin spin couplings. Unwanted evolution during these delays (for example, because of chemical shift differences or magnetic field inhomogeneities) can be eliminated by introducing the so-called spin-echoes. Echo formation can be understood by taking as an example the suppression of a chemical shift effect. At a time t after a 90 initial pulse, the components of transverse magnetization corresponding to identical nuclei with different local magnetic fields will have precessed different angles ϕ i = Ω i t in the rotating frame (see Eq. (26) and Figure 10). Application of a 180 pulse with the magnetic field directed along the x-axis causes inversion of the spin y-component of all nuclei. As a consequence, all nuclear moments will join together after a time 2t, irrespectively of their chemical shifts. Figure 10. Evolution in a frame that rotates at the radiation frequency, of the magnetization due to two uncoupled nuclei with different chemical shifts during a spinecho experiment (the duration t p of the pulses is negligible compared to t).

21 4. Multidimensional NMR The simplest NMR experiments entail only the observation of the evolution following a single radiofrequency pulse applied to a system in equilibrium. The long relaxation times typical of NMR allow much more complex experiments, in which additional pulses are applied to non-equilibrium magnetization. In multipulse experiments the spin system evolves through different coherent states, until transverse magnetization is finally measured. The spin-echo experiment mentioned in the previous section is one example of this type of experiment. The NMR signals are obtained in the time-domain by registering the evolution of the spin systems from a non-equilibrium situation. The relevant frequencies (chemical shifts, and coupling constants) and line-shapes are obtained by Fourier transformation. Practical implementations of this scheme require the digitalization of the FID; in other words, the continuous time evolution is sampled at discrete time intervals. The resulting list of points is a faithful representation of the time-domain signal evolving under a range of frequencies determined by the sampling interval. Individual points in this list, however, do not necessarily have to be acquired sequentially: multidimensional experiments require that many multipulse one-dimensional spectra are recorded with one or more time delays systematically changed. The essentials of multidimensional NMR spectroscopy can be illustrated by analyzing one of the simplest bi-dimensional techniques: heteronuclear COrrelation SpectroscopY (heteronuclear COSY). Let us consider a sample with molecules containing two weakly coupled nuclei of 1 H and 13 C (both with spin 1/2) and no other nuclei with spin. Two 90 pulses of frequency ν H are applied along the x-axis with a delay t 1 between them; then, a 90 pulse of frequency ν C is applied and the FID of 13 C is registered. The first pulse converts the longitudinal equilibrium H-magnetization into a transversal one (Figure 11 (a) and (b)). Then its two components M H and M H begin to precess around BB0 at the frequencies given by Eq. (39) and Eq. (40), so they will gradually separate from each other (Figures 9 and 11 (c)). Let us consider an evolution time t 1 such that M H and M H have opposite projections along the y-axis of the rotating frame, as represented in Figure 11(c). The second pulse moves the two components from the xy plane to the xz one, so that the y- projections are converted into z-projections. According to Eq. (43) the component with negative z-projection ( M H ) requires that the populations of the two levels involved in the corresponding transition be inverted with respect to the equilibrium situation (Figure 12), since there must be more spin-down ( ) than spin-up ( ) H nuclei.

22 Figure 11. Evolution of the H-magnetization in a frame rotating at the radiation frequency for the two first pulses of an H,C-COSY experiment (strictly speaking frequencies should be replaced by offset frequencies). Figure 12(b) also reveals a population inversion in the C transition, which implies that the C-magnetization must have a component with negative z-projection M. Hence, although the pulses have so far been applied at the hydrogen Larmor frequency ν H, they have affected the equilibrium magnetization of both nuclei in such a way that some magnetization (or polarization) has been transferred from the easily polarizable H- nuclei to the C-nuclei. The final 90 pulse at frequency ν C transforms the C- magnetization z-components into transversal for detection. Fourier transformation of the corresponding FID (as a function of the time after the third pulse, t 2 ) gives a onedimensional NMR spectrum with two lines at frequencies ν 2 = ν C ± JHC/2. C

23 Figure 12. Population distribution at equilibrium (a) and after the two first pulses (b) of an H,C-COSY experiment. The level populations are roughly proportional to the thickness of the horizontal lines that represent them, and the upward/downward direction of the transition arrows indicates net absorption/emission. Figure 11(d) shows that the z-projections of proportional to cos[2π ( ν H ± JHC/2) t1] ; these projections determine the population changes produced on the spin levels, which in turn determine the z-projections of M and M C M H and M H after the second pulse are. In fact, the mathematical treatment shows that these are modulated by the same factors as those of the H-magnetization: cos[2π ( ν H ± J HC /2) t 1 ]. By repeating the experiment for different values of t 1 one obtains a series of one-dimensional NMR spectra, each one showing two signals (at frequencies ν 2 = ν C ± J HC /2 ) whose amplitudes oscillate with t 1, and a Fourier transform with respect to this time variable gives two signals at frequencies ν 1 = ν H ± J HC /2. The representation of those signals with contour plots in a plane of coordinates ( ν 1, ν 2 ) gives a two-dimensional NMR spectrum (Figure 13). C

24 Figure 13. Schematic H,C-COSY spectrum of two weakly coupled 1 H and 13 C nuclei. A more sophisticated version of this experiment eliminates the splitting, leading to a single peak with coordinates ( ν H, ν C ) for each pair of coupled nuclei; this allows a simple assignation of HC bonds. The magnetization evolution in an homonuclear A,X-COSY experiment is more difficult to understand (a more sophisticated quantum treatment is indeed needed), but the resulting bi-dimensional spectrum is still easy to interpret: if the nuclei are scalar coupled, each one can transfer some polarization to the other and, disregarding multiplicity, four signals are obtained with coordinates: ( ν A, νa), ( νx, νa), ( νa, νx), ( νx, νx) and ( νm, νm). Signals that have the same frequency in both dimensions are called diagonal peaks. Signals with different frequencies in each dimension are called cross peaks. A non-coupled nucleus M gives a single diagonal peak of coordinates ( ν M, ν M) ; hence, a cross peak of coordinates ν A and ν X permits an easy identification of nuclei A and X as coupled partners (see Figure 14).

25 Figure 14. Schematic proton COSY spectrum of methylethylketone (line shapes are different for cross peaks in the simplest COSY experiment). Since protons I are isolated they give only a diagonal peak, but protons II and III are coupled, thus producing cross peaks. Three-dimensional or four-dimensional experiments contain two and three delays that are incremented independently, and thus the final signals are characterized by three and four frequencies, respectively. COSY spectra belong to the most widely used class of multidimensional NMR experiments: the so-called correlation experiments. In these, the two coordinates of each cross peak are the chemical shifts of two groups of interacting nuclear spins. There are many different correlation experiments that differ in the type of interaction existing between the spins. They can be grouped in two main classes: those involving coherent evolution of groups of spins under the influence of coupling constants (for example, the COSY experiment) and those that imply incoherent magnetization transfer. An example of the latter is the Nuclear Overhauser Effect correlation SpectroscopY (NOESY) in

26 which cross peaks identify spins that are close enough in space for them to affect each other s relaxation. The frequencies sampled during each delay of a multidimensional experiment reflect the interactions that are active in it. Different terms of the Hamiltonian (for example, the chemical shifts or some coupling constants) can be effectively suppressed or scaled by the application of radiofrequency pulses during the evolution. Thus, multidimensional experiments offer a way of separating different interactions and simplifying the analysis. For example, in J-resolved spectroscopy, one of the dimensions carries the chemical shift information only, and thus all signals are singlets, while the second dimension contains only the coupling information (multiplicity and coupling constants). 5. Dynamic Aspects of NMR The time evolution of the non-equilibrium magnetization is affected by the dynamics of the molecules at very different time scales. 1. In liquid samples, fast isotropic reorientation averages out all interactions that depend on the specific orientation of a structural unit (bonds, or internuclear vectors). These include strong dipolar interactions between spins that are close in space or the anisotropic parts of the chemical shift or quadrupolar interaction tensors (see Sections 2 and 3). This greatly simplifies the spectra of liquids. The fluctuations of these interactions at frequencies close to those corresponding to energy differences between states of the spin system (sums and differences of the Larmor frequencies of the nuclei involved) do affect, however, the relaxation rates. Measurement of the relaxation rates allows the characterization of motions in the nano- picosecond time scale that include the overall reorientation of the molecule and fast internal motions. 2. Chemical exchange is a generic term that includes conformational fluctuations, isomer interconversion or ligand binding. Exchange at frequencies commensurate with the chemical shift differences between the interconverting chemical forms can provide an additional contribution to transverse relaxation (T 2 ) that can be reflected as line broadening or can be detected by specific experiments aiming at measuring the relaxation rate and separating the exchange contribution from other relaxation mechanisms. The time scale of these processes is of micro- to milliseconds. 3. Chemical exchange that is slower than the chemical shift differences between the interconverting species does not affect the spectra of the individual species. However, if the exchange is faster than the relaxation time of the nuclei, the interconversion of particular sites occupied by the same nucleus in the two states can be mapped (for instance, by cross peaks that correlate the two chemical shifts). This often allows the characterization of the exchange mechanisms with rates of the order of a reciprocal second. 6. Spatial Information from NMR The chemical shift of a nucleus depends on the external magnetic field experienced by the nucleus particular point of space. This is the reason for the extremely high

27 homogeneity requirements for NMR spectroscopy. By applying a well-defined magnetic field gradient the spatial localization of the spins in the sample is encoded in their frequency. As a result, the magnetic field at a particular position will be given by 0 0( ) 0( 0 db B x = B ) + dx x (45) And the frequency of a nucleus situated at this position will be (see Eq. 31) ν γ(1 σ) db x 2π dx (46) 0 ( x) B (0) = 0 + The frequency information is thus transformed to images and the technique is known as magnetic resonance imaging (MRI). The spatial resolution depends on the strength of the gradient. In addition, gradients are applied for short periods during the evolution of the spins (field gradient pulses). The technical limitations that hampered this type of experiment for a long time were the eddy currents caused by fast switching of strong gradients. They are minimized nowadays by the use of the so-called actively shielded gradients. Shielded field gradients have found widespread application also in NMR spectroscopy as a way of selecting a particular flow of magnetization in an NMR experiment (and therefore the information content of that particular experiment). These applications are based on the so-called gradient-echoes. When a gradient is applied for a certain time, nuclei that belong to the same chemical entity evolve at different frequencies, depending on their location in the sample. Destructive interference between them results in the effective disappearance of the signal. If an identical gradient of opposite sign is applied, the evolution of each spin is reversed and the signal is recovered. The effect of the gradient depends obviously on the spin state and the gyromagnetic ratio of the nuclei. Thus, instead of reversing the gradient, the same effect is obtained by inverting the spin state, and if the magnetization has been transferred from one nucleus type to another, echo formation will require applying a gradient with a strength scaled by the relative gyromagnetic ratio of the two nuclei. Gradient echoes can also be used to obtain information on the translational diffusion of a molecule. If molecules move away from their original position in the time between the two gradients, the compensation will not be complete and the intensity of the echo will decrease. By measuring the echo intensity as a function of gradient strength, the translational diffusion coefficient can be determined. In more qualitative applications, gradient echoes can be used to selectively observe only large molecules, with low diffusion coefficients in mixtures of molecules of different sizes (typically ligands that can bind to macromolecules). 7. Solid, Liquid, and Partially Oriented Samples A number of relevant NMR parameters (shielding tensors, coupling constants either through bonds or through space, quadrupolar interactions) are tensor quantities that

28 contain a richness of information concerning details of the electronic and threedimensional structure of the molecule. In liquids these interactions are averaged and the information is lost but the spectra of even complex molecules can be easily analyzed. In solid samples the spectra can become extremely complex and impossible to analyze except for some very simple systems, unless special experimental techniques are used. One of the approaches involves averaging of some of the interactions by rapid sample spinning at the magic angle (Eq. 33). Spinning must be fast compared with the spread of frequencies associated to the interaction that has to be eliminated. High spinning speeds pose a hard engineering problem and are ultimately limited by the speed of sound. Averaging by pulse trains that effectively rotate the Hamiltonian is an alternative approach. Internuclear interactions can also be suppressed by high power decoupling. Solid-state NMR techniques are rapidly extending and offer promises for the NMR study of non-soluble large biological systems, such as membrane proteins. The assignment of a small protein in the solid state has already been achieved. Dipolar couplings contain valuable structural information as they depend on both the distance and the angle formed by the internuclear vector connecting the coupled nuclei. In isotropic liquids, dipolar couplings are averaged to zero. However, if the molecule is not oriented isotropically, the average is not complete. If the degree of alignment is small, this residual dipolar coupling can be easily measured using standard liquid-state NMR techniques. Partial sample alignment can be induced by high magnetic fields in samples with substantial anisotropic susceptibility. This is particularly important in paramagnetic samples. Alternatively, the addition of diluted liquid crystal systems can cause the orientation of the molecule of interest by collisions or electrostatic interactions. Bicelles, formed by mixtures of phospholipids with different lengths of the fatty acid chain, viruses, or purple membrane patches, among other systems, have been shown to provide the desired degree of alignment in proteins. 8. The Impact of NMR NMR is one of the most versatile spectroscopic methods and has had substantial impact in a large range of fields in chemistry, biochemistry, medicine, food science, material science, and solid-state physics or quantum computing. In chemistry, NMR has become the standard spectroscopic technique for the determination of the constitution and the stereochemistry (static and dynamic) of organic and inorganic molecules. While proton NMR, due to its much higher sensitivity and the widespread presence of protons in all types of molecules, is the most widely used, NMR active nuclei (I > 0) cover nearly all the periodic table. However, a clear distinction has to be made between nuclei with spin 1/2 (like the proton) and those with higher spin. The latter have quadrupole moments that cause fast relaxation and broad lines that limit their use as routine. Natural abundance of a particular NMR active isotope introduces an additional limitation. NMR is now an established method for the determination of three-dimensional structures of proteins and nucleic acids. Unlike the alternative technique (X-ray diffraction), NMR does not require crystallization of the molecule and can provide