NMR spectroscopy

NMR spectroscopy, MR spectroscopy, ( often abbreviated as NMR) measures the response of nuclear spin magnetization to radio frequency electromagnetic radiation, in the presence of a magnetic field, to obtain static (structure and composition) as well as dynamic information. These methods are used extensively for structural elucidation of proteins, DNA and many other natural products.

NMR spectroscopy has also been used to obtain information regarding interactions between nuclear spins within the same molecule as well as to obtain information regarding intermolecular interactions.

NMR spectroscopy has been used for the study of matter in disordered, ordered and partially ordered systems such as gases, liquids, quantum fluids , superconductors, solutions, amorphous solids, crystalline solids, liquid crystals, membranes and living organisms. NMR spectroscopic methods have also found use in quantum computing.

Principles of Nuclear Magnetic resonance

Nuclear magnetic resonance is a consequence of a property possessed by the nucleus known as nuclear spin angular momentum. Some properties associated with nuclear spin angular momentum are similar to those of a spinning macroscopic body, however, nuclear spin angular momentum is a fundamental property that cannot be explained in terms of any other fundamental property such as mass, charge, etc.

Nuclear spin angular momentum—like the angular momentum of any other fundamental particle —is a quantized vectorial quantity. Its magnitude is restricted to certain fixed values and its direction is also restricted to certain directions in the presence of a magnetic field. In the absence of a magnetic field, it is not possible to obtain any information regarding its direction.

Nuclei that have an even mass number and an even atomic number do not exhibit nuclear magnetic resonance, e.g., O-16, C-12. Some common nuclei that do exhibit nuclear magnetic resonance are: H-1, C-13, N-15, F-19. For a detailed list Catalog of magnetic nuclei.

The nuclear spin angular momentum is characterized by a quantum number I known as the nuclear spin angular momentum quantum number (often briefly referred to as "nuclear spin"). For example, the proton has a nuclear spin angular momentum quantum number of 1/2 and is known as a spin-1/2 particle. Similarly, the N-14 nucleus has a nuclear spin angular momentum quantum number of 1 and is known as a spin-1 nucleus. The magnitude of the nuclear spin angular momentum with quantum number I is,

$\scriptstyle {\sqrt {I(I+1)}}\,\hbar ,$

where $\scriptstyle \hbar =h/(2\pi )$ is Planck's reduced constant.

In the presence of an external homogeneous magnetic field (i.e., a magnetic field that has same magnitude everywhere in the space of interest; the magnetic field has no transverse components since the z-direction is chosen to point along the direction of the magnetic field), the z-component of the nuclear spin angular momentum vector is restricted to certain values mh/(2π), where m is the spin magnetic quantum number, and can be any one of the values from +I to −I, that differ from each other in integral steps.

For example,

If I=1, then m = +1, 0 or −1

If I=1/2, then m = +1/2 or −1/2

(Note: Difference between different values of m should be integral; however, actual values of m may be integers or half-integers)

As a consequence of the restrictions on the magnitude of the z-component, the nuclear spin angular momentum vector can only point in certain (allowed) directions with reference to the external magnetic field. In the absence of other fields, there are no restrictions on the allowed directions in the x-y plane. The net result is that the spin angular momentum vectors of different nuclei point along the surface of cones that have a fixed angle with respect to the external magnetic field. The nuclear spin magnetic moment is proportional to the nuclear spin angular momentum and the constant of proportionality is known as the magnetogyric ratio. The magnetogyric ratio may be either positive or negative. Therefore, the nuclear spin magnetic moment vector is either parallel or antiparallel to the nuclear spin angular momentum vector.

The different allowed values of m define the allowed orientations of the nuclear spin angular momentum and each of these spin states is associated with a different energy. This is due to the fact that the energy of the spin states is proportional to the scalar product of the nuclear spin magnetic moment and the external magnetic field vectors. Electromagnetic radiation can efficiently cause transitions between the nuclear spin states if the frequency of the electromagnetic radiation, ν, is equal to the energy difference ΔE between the nuclear spin states divided by Planck's constant h.

NMR spectroscopy: Observable parameters

The Chemical shift:

The nuclei in molecules are surrounded by electrons. The applied magnetic field induces a circulation of electrons, which in turn produces an additional magnetic field. This effect is called 'shielding', because the magnetic field induced by the circulation of charge is in opposite direction to the field (external) that was responsible for the circulation of charge. Therefore, the nuclei in a given molecule are subjected to a net magnetic field which is the sum of the applied magnetic field and the induced magnetic field.

$B_{induced}=-\sigma B_{applied}\quad \quad \quad B_{net}=B_{applied}+B_{induced}$

where $\scriptstyle \sigma$ is a tensor; however for molecules rotating rapidly in all directions, such as molecules in a gas, liquid or solution, $\sigma$ can be approximated by a scalar and is known as the shielding constant.

$\scriptstyle \sigma$ depends upon the electron density as well as polarizability which vary substantially depending upon the nature of the molecule and the location within the molecule. As a consequence, for a given value of the applied field, nuclei in different molecules and even nuclei in different parts of the same molecule may encounter different net values of the magnetic field - this results in a variation in the resonance frequency.

$\nu _{i}={\frac {\gamma B_{net}}{2\pi }}$

where $\scriptstyle \nu _{i}$ is the resonance frequency for the nucleus labeled 'i' and $\scriptstyle \gamma$ is the magnetogyric ratio.

The relative change in the resonance frequency is called the chemical shift.

$\delta _{i}(ppm)={\frac {(\nu _{i}-\nu _{ref})*10^{6}}{\nu _{ref}}}$

where $\scriptstyle \nu _{ref}$ is the resonance frequency of a standard. The chemical shift $\delta$ is generally quoted in ppm. In the NMR spectroscopy of organic compounds, Tetramethylsilane was chosen as the standard.

The chemical shifts of functional groups in many molecules have been measured and provide a basis for identification of functional groups in novel molecules.

Signal area:

The area of a signal is proportional to the total number of contributing nuclei. e.g. In the 1H NMR spectrum of a pure compound containing a single methyl group and a single methylene group, the relative areas of the corresponding signals will be in the ratio 3:2, because there are 3 protons in the methyl group and 2 in the methylene group.

If the pure compound can exist in multiple conformations, and if the transitions between the different conformational states occur on a time scale that is slow with reference to the relevant NMR time scale (typically msec) then it may be possible to observe separate signals corresponding to the different conformations in the NMR spectrum. In such a case, the relative areas of the signals corresponding to different conformations will be proportional to the fractions of the molecules that exist in each conformational state during the time that the NMR spectrum was acquired.

Many small molecules can exist in multiple conformations due to rotation about single bonds, however, such rotation is generally rapid at room temperature and hence separate signals are usually not observed for the different rotational isomers that exist at equilibrium. For large molecules, e.g. proteins the close packing of the constituent amino acids results results in restricted rotation about many single bonds and complete loss of conformational freedom for many other single bonds. Such restricted rotation about single bonds results in separate NMR signal for each conformer and the areas of the NMR signals are again proportional to the populations of the conformers.

In the NMR spectra of mixtures, the relative areas of signals can be used to deduce information regarding the relative concentration of each compound. The very high resolution of NMR spectra enables us to resolve a very large number of compounds, and hence determine the relative concentrations of a large number of species. This is of particular use in vivo spectroscopy. The ability to resolve the signals of a large number of chemicals/biochemicals can be enhanced significantly by using 2D NMR.

Systems undergoing physical and chemical changes can also be monitored by NMR spectroscopy as the interconverting species usually have different chemical shifts. The changes in the relative areas of the reactants and products can be used to obtain information regarding the time course of the reaction. This is possible if the time scale of the conversion is slow with respect to the NMR time scale.

J-coupling and multiplicity

J-coupling, also known as scalar coupling, is due to the interaction between different nuclei in the same molecule that is mediated through electrons in chemical bonds. Usually, the J-coupling interaction is observable between nuclei that are separated from each other by three or fewer bonds. The effect of this interaction on the observable spectrum is that the signals of a given nuclei are 'split' if the nucleus has a J-coupling interaction with neighboring nuclei. If this coupling is weak, the resulting pattern can be used to deduce information regarding the number of neighboring nuclei. This information plays a critical role in structural elucidation of small organic molecules. In addition, the magnitude of the coupling constant provides information regarding the conformation of a molecule.

When weak coupling is applicable, the observed multiplicity of a signal provides useful structural information. The multiplicity of a signal is expected to be 2nI+1 where n is the number of nuclei with spin angular momentum quantum number (spin) I that have J coupling with the nucleus responsible for the signal being observed. If all magnetically active nuclei are protons, then the multiplicity is equal to n+1, because I=1/2 for protons. In addition, the relative intensities of the peaks within a multiplet is given by the coefficients of the binomial expansion. This is known as the n+1 rule. In using the n+1 rule, couplings between magnetically equivalent nuclei are not considered because the effects of J-coupling are not observable in a normal 1D NMR spectrum if the nuclei involved in the interaction are magnetically equivalent. e.g. in the fragment CH3-CH2-O- , the signal corresponding to the methyl (CH3) protons will have a multiplicity of three, because each of the methyl hydrogen has a J-coupling to each of the two methylene (CH2) protons, hence n=2, I=1/2 and 2nI+1 = 3. The signal corresponding to the methylene group of protons will be a quartet, as each methylene proton is expected to have J coupling to each of the three methyl (CH3) protons, hence n=3, I=1/2 and 2nI+1 = 4. In this case the pairs of nuclei that have observable J-coupling are separated by three bonds. Note that interactions between magnetically equivalent nuclei, such as interactions between the different protons of the same methyl group (separated by only two bonds), need not be considered when predicting the multiplicity. In this example the three protons in the methyl group are treated as magnetically equivalent and the two protons in the methylene group are treated as magnetically equivalent by assuming that free rotation occurs about the single bond separating the methyl and methylene groups. If this is not the case, e.g. if restricted rotation occurs about the single bond and if the fragment is placed in an asymmetric environment (such as part of a rigid macromolecule) then the methylene protons may no longer be assumed to be magnetically equivalent.

The 2nI+1 rule assumes that all the neighboring nuclei (i.e. those with observable J-coupling to nucleus being observed) are equivalent. If this is not the case, then the multiplet structure can still be predicted by considering first the coupling to one set of equivalent J-coupled nuclei and predicting that the effect of the remaining nuclei will cause additional splitting of the multiplet generated by interaction with the first set.

The magnitude of the J-coupling depends on the number of bonds separating the interacting nuclei as well as on the geometry. For ³J coupling, i.e., coupling between nuclei separated by three bonds, the magnitude of the coupling between the interacting nuclei is given by the Karplus equation

$^{3}J=A+B\cos {\theta }+C\cos ^{2}{\theta }\qquad or\qquad ^{3}J=A+B\cos {\theta }+C\cos {2\theta }$

where θ is the dihedral angle involving the nuclei of interest. The values of A,B and C depend on the chemical nature of the fragment. The Karplus equation is useful for conformational analysis of molecules, particurly biopolymers such as polypeptides and nucleic acids. The parameters A,B and C of the Karplus equation have been determined for most combinations of nuclei that occur in organic compounds, by comparing experimental NMR and X-ray crystallographic data. If relevant experiment data is not available the magnitude of the J coupling may be calculated by using quantum chemical calculations; however the prediction accuracy is often lower than that obtained from empirical parametrization.

The net Magnetization vector

In the presence of an external magnetic field B, the number of the nuclei in the different allowed nuclear spin states follows the Boltzmann distribution at equilibrium. i.e. for a two state system

${\frac {n_{\beta }}{n_{\alpha }}}=e^{\frac {E_{\alpha }-E_{\beta }}{kT}}$

where

$n_{\beta }=$ number of nuclei in state $\beta$; $n_{\alpha }=$ number of nuclei in state $\alpha$

$E_{\beta }=$ Energy of state $\beta$; $E_{\alpha }=$ Energy of state $\alpha$

T = Absolute temperature in Kelvin; $\quad$ k=Boltzmann constant

If the energy of state $\beta$ is greater than that of state $\alpha$, then fewer nuclei will be present in the higher energy state $\beta$. However, since the energy difference between these two states is small the population difference is quite small and the macroscopically observable net magnetization vector for the entire population will be the vectorial sum of the magnetic moments of all the nuclei present in the sample. At equilibrium, in a homogeneous magnetic field, although the orientation of the spin magnetic moments of individual nuclei is restricted to certain directions with respect to the applied magnetic field, there is no preference for any direction for the projection of the magnetic moment onto the x-y plane. The X- and Y-components of the net magnetization vector of the sample should be zero at equilibrium because the X- and Y-components of the spin magnetic moments of individual nuclei are randomly oriented and the net magnetization vector is the vectorial sum of the spin magnetic moments of individual nuclei. Therefore, the equilibrium magnetization will be

$M_{equil}=M_{z,equil}=N<\mu _{z}>={\frac {N\sum {g_{i}\mu _{zi}e^{-{\frac {E_{i}}{kT}}}}}{\sum {e^{-{\frac {E_{i}}{kT}}}}}}$

where

N = total number of nuclei in sample; $M_{equil}=$ Magnetization vector (net) at equilibrium

$E_{i}=$ energy of state i; $\quad \quad \quad \mu _{zi}=$ z-component of magnetic moment of state i

$g_{i}=$ degeneracy of state i

For a two state system, the $\mu _{zi}$ are equal in magnitude but opposite in direction. However, there are more nuclei in the lower energy state at the temperature of the experiment T, hence the numerator in the sum above is nonzero. Therefore, at equilibrium, the net magnetization vector has only one non-zero component (the $M_{z,equil}$-component) when the system is placed in a static homogeneous magnetic field.

In the presence of a time dependent homogeneous magnetic field, represented by the vector B, the magnetization vector experiences a torque and the time dependence of the net magnetization vector M (neglecting relaxation effects) is

${\frac {dM}{dt}}=M\times \gamma B$

The Bloch equations describe the time dependence of the different components of the net magnetization vector subject to relaxation, in the presence of a time dependent magnetic field.

Relaxation time

After a collection of nuclei in a magnetic field that is at equilibrium with its surroundings is perturbed in some manner (usually by a pulse of electromagnetic radiation) the system requires a certain amount of the time to return to equilibrium. If this process is exponential, the rate constant is called relaxation rate. The relaxation rate is inversely proportional to relaxation time. T1 relaxation time characterizes the return to equilibrium of the longitudinal component of the magnetization of the collection of nuclei. Similarly, T2 relaxation time characterizes the return to equilibrium of the transverse component of the magnetization of the collection of nuclei that are being studied. In a static homogeneous magnetic field, the transverse component of the net magnetization vector for a sufficiently large collection of nuclei is always zero, at equilibrium.

The dominant mechanism of relaxation is usually the dipolar interaction with the closest neighbor. For a pair of identical spins in a homogeneous magnetic field, undergoing isotropic rotational motion in solution, the relaxation rates are:

${\frac {1}{T_{1}}}={\frac {\gamma ^{4}\hbar ^{2}}{10r^{6}}}(3J(w_{o})+12J(2w_{o}))$

${\frac {1}{T_{2}}}={\frac {\gamma ^{4}\hbar ^{2}}{20r^{6}}}(9J(0)+15J(w_{o})+6J(2w_{o}))$

where $J(w)={\frac {\tau _{c}}{1+w^{2}\tau _{c}^{2}}}$ is the spectral density function and $\tau _{c}$ is the mean time required for rotational motion of the pair of nuclei by one radian.

For a pair of nonidentical spins I and S, :

${\frac {d<I_{z}>}{dt}}=-\rho _{I}(<I_{z}>-<I_{z,equil}>)-\sigma (<S_{z}>-<S_{z,equil}>)$

${\frac {d<S_{z}>}{dt}}=-\rho _{S}(<S_{z}>-<S_{z,equil}>)-\sigma (<I_{z}>-<I_{z,equil}>)$

$\sigma$ is called the cross relaxation rate and is responsible for the Nuclear overhauser effect.

$\rho _{I}={\frac {\gamma _{I}^{2}\gamma _{S}^{2}\hbar ^{2}}{10r^{6}}}(J(w_{I}-w_{S})+3J(w_{I})+6J(w_{I}+w_{S}))$

$\sigma ={\frac {\gamma _{I}^{2}\gamma _{S}^{2}\hbar ^{2}}{10r^{6}}}(-J(w_{I}-w_{S})+6J(w_{I}+w_{S})))$

${\frac {1}{T_{2}}}={\frac {\gamma ^{2}\gamma _{S}^{2}\hbar ^{2}}{20r^{6}}}(4J(0)+J(w_{I}-w_{S})+3J(w_{I})+6J(w_{I}+w_{S})+6J(w_{S}))$

Nuclear Overhauser effect

Irradiation at the resonance frequency of one nucleus in the molecule may cause changes in the intensity of a signal at a different frequency corresponding to another nucleus - this is called the Nuclear overhauser effect (Noe). Nuclear overhauser effect is due to dipole-dipole interactions between the magnetic moments of the pair of nuclei. Unlike J-coupling, this interaction is not mediated through bonds and hence it may be possible to observe the Nuclear Overhauser effect between pairs of nuclei separated by many bonds provided that they are in spatial proximity. The strength of the observable Nuclear overhauser effect for molecules in solution is proportional to the inverse of the sixth power of the distance between the two nuclei due to averaging caused by rotational motion. Both the magnitude as well as the sign of the nuclear overhauser effect depend on the rotational frequencies of the pair of nuclei with respect to the applied magnetic field. The Noe enhancement, $\eta$, in the steady state, when the resonance frequency of spin I is irradiated and the intensity of spin S is monitored is $\eta$ = ($\sigma$ $\gamma$_I )/ ($\rho$_S $\gamma$_S). This indicates that considerable enhancement in the intensity of the S signal can be obtained by irradiation at the frequency of the I spin, provided that $\gamma$_I/$\gamma$_S is greater than 1, because $\sigma$/$\rho$_S is approximately 1/2 when w$\tau _{c}$ is much smaller than 1 .

However, when w$\tau _{c}$ is much greater than 1, $\sigma$/$\rho$_S approaches -1 and negative Noe enhancements are obtained.

The sign of $\eta$ changes from positive to negative when w$\tau _{c}$ is close to one and under such conditions the Noe effect may not be observable. This happens for rigid molecules with relative molecular mass about 500 at room temperature e.g. many hexapeptides.

Pulsed (r.f.) NMR spectroscopy

Spin echoes

A pulse of r.f. radiation is applied for sufficient duration to tip the net magnetization vector into the x-y plane, this is followed by a short delay $\tau$, followed by another r.f.pulse that inverts the magnetization around the x-axis, followed by another short delay $\tau$. The magnetization at the end of this pulse-sequence is

$M(2\tau )=M_{o}e^{-{\frac {2\tau }{T_{2}}}}$

The transverse relaxation time can be measured by repeating the experiment with different value of $\tau$, and plotting $log_{e}M(2\tau )$ vs $2\tau$. The resulting plot is expected to be a straight line with slope $-1/T_{2}$.

If the above experiment is carried out with a magnetic field gradient during the two delays $\tau$ then

$M(2\tau )=M_{o}e^{-{\frac {2\tau }{T_{2}}}}e^{-{\frac {2DG^{2}\tau ^{3}}{3}}}$

where D is the diffusion coefficient and G is the magnetic field gradient. Repeating this experiment with different values of G enables us to measure the Diffusion coefficient D.

FTNMR

Fourier transform NMR spectroscopy (Ernst ) : A pulse (or a pulse train) of electromagnetic radiation (usually radiofrequency electromagnetic radiation is abbreviated as RF) is used to cause a perturbation in the sytem. The time dependent response of the system is recorded (this time dependent response is often referred to as the free induction decay, abbreviated as fid). A fourier transform of the response gives information regarding the frequency response. In the case of a single pulse perturbation, the fourier transform of the time dependent response is equivalent to a 1D NMR spectrum.

fid		NMR spectrum
free induction decay	Fourier transformation of the time response (fid) gives the frequency response (NMR spectrum). →	NMR Spectrum

nD NMR

Multidimensional NMR spectroscopy (Jeener ; Aue et al.) The nuclei in a magnetic field are subjected to a series of pulses of electromagnetic radiation separated by delays. The time dependent response of the system is recorded. The delays between the pulses may be fixed or incremented systematically between different repetitions of the experiment. The number of variable delays determines the dimensionality of the experiment. A multidimensional Fourier Transform of the entire data set characterizes the frequency responses of the system and enables a correlation between different NMR parameters. Different combinations of the pulses and delays known as 'pulse sequences' enable us to correlate and measure different types of NMR parameters. Some common examples of multidimensional NMR spectroscopy experiments (2D, 3D and 4D): COSY, NOESY, TOCSY, EXSY, HSQC, HNHA, HSQC-NOESY, HNCA, HNCO, HNCACO, HSQC-NOESY-HSQC. For a longer list of NMR experiments see http://en.citizendium.org/wiki/List_of_Nuclear_Magnetic_Resonance_experiments

This 2-D COSY spectrum shows two diagonal peaks and two cross peaks. The observed cross peaks are used to identify (correlate) the chemical shifts of nuclei that are J-coupled.

Two dimensional correlation spectroscopy (2D-COSY): Correlates chemical shifts of J-coupled nuclei.

Two dimensional nuclear overhauser effect spectroscopy (2D-NOESY): correlates chemical shifts of nuclei that exhibit significant Nuclear Overhauser effect. For molecules that experience free rotation along all three dimensions, the Nuclear Overhauser effect is generally observable between nuclei that are less than 5 angstroms apart.

Biomolecular NMR spectroscopy

NMR spectroscopy can be used to determine the structure of macromolecules and to obtain information regarding their dynamics. However, the NMR spectra of macromolecules are much more complicated than those of small molecules and it is usually necessary to use multidimensional NMR spectroscopy in order to obtain data that can be used for structural analysis. The Sequential resonance assignment method was developed in order to associate specific nuclei in a protein with the observed resonance frequencies. Subsequently, the information obtained from quantitative and qualitative analysis of Nuclear Overhauser effects, J-coupling and chemical shifts is converted into geometric restraints. These geometric restraints are then subsequently used to build a model of the molecule.

Applications of Magnetic resonance in Pharmacology, Physiology and Medicine

NMR spectroscopy is a useful tool in drug design and development. It is used to determine the structure of ligands and receptors; to study the interactions of ligands with receptors , even when the binding is weak and transient ; and to build structure activity relationships..

MR spectroscopy is of particular use in physiological studies, because the system of interest can be studied in a non-invasive manner, and also because it is possible to quantify the changes in a large set of metabolites simultaneously. Metabonomics is an attempt to characterize the physiological changes by quantifying the entire set of metabolites in an organism or its components.

Information regarding spatial distribution of NMR parameters may be obtained by using magnetic field gradients. This is the basis of Magnetic Resonance Imaging (MRI) a tool that has found extensive applications in medical diagnostics.

Applications of Magnetic Resonance in Food technology

A few examples of application of magnetic resonance techniques in food science/technology are:

• NMR spectroscopy has been used to assess the quality of wines.
• Magnetic resonance techniques have been used to monitor the ripening of cheese. In addition, spectroscopic information has been used to determine the time dependence of the degradation of packed cheese.
• Magnetic resonance techniques have been applied to obtain process information in the baking of bread.

In such applications, the ability of NMR spectroscopy to resolve complex mixtures and its applicability to matter in a wide variety of states is particularly useful.

Summary

NMR spectroscopy is a versatile analytical technique for the study of structure and dynamics of disordered, partially ordered and highly ordered systems in pure substances, mixtures as well as in live organisms.

Further reading

Fundamentals of Physics. Halliday, Resnick and Walker.

Biophysical Chemistry. Cantor and Schimmel.

Principles of Nuclear Magnetic Resonance spectroscopy in one and two dimensions. (1987). R.R.Ernst, G. Bodenhausen and A.Wokaun. Clarendon Press. Oxford.

NMR of proteins and nucleic acids. K.Wuthrich. Wiley.

Quantum description of high resolution NMR in liquids. M.Goldman. Oxford.

Principles of Magnetic Resonance (1996) Charles P. Slichter. Springer Series in Solid-State Sciences.

Protein NMR spectroscopy: Principles and Practice. (2006) J.Cavanagh, W.J. Fairbrother, A6G. Palmer, N.J. Skelton, M.Rance. Academic Press.

Understanding NMR spectroscopy. (2005) J.Keeler. Wiley

Spectrometric Identification of Organic Compounds. (2005) Robert M. Silverstein, Francis X. Webster, and David Kiemle.

NMR in Drug Design. David J. Craik. (1995). Crc Series in Analytical Biotechnology.

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