Nuclear Overhauser effect

The nuclear Overhauser effect is the transfer of nuclear spin polarization from one population of spin-active nuclei to another via cross-relaxation. A phenomenological definition of the NOE in nuclear magnetic resonance spectroscopy is the change in the integrated intensity of one NMR resonance that occurs when another is saturated by irradiation with an RF field. The change in resonance intensity of a nucleus is a consequence of the nucleus being close in space to those directly affected by the RF perturbation.
The NOE is particularly important in the assignment of NMR resonances, and the elucidation and confirmation of the structures or configurations of organic and biological molecules. The ¹H two-dimensional NOE spectroscopy experiment and its extensions are important tools to identify stereochemistry of proteins and other biomolecules in solution, whereas in solid form crystal x-ray diffraction typically used to identify stereochemistry. The heteronuclear NOE is particularly important in ¹³C NMR spectroscopy to identify carbons bonded to protons, to provide polarization enhancements to such carbons to increase signal-to-noise, and to ascertain the extent the relaxation of these carbons is controlled by the dipole-dipole relaxation mechanism.

History

The NOE developed from the theoretical work of American physicist Albert Overhauser who in 1953 proposed that nuclear spin polarization could be enhanced by the microwave irradiation of the conduction electrons in certain metals. The electron-nuclear enhancement predicted by Overhauser was experimentally demonstrated in ⁷Li metal by T. R. Carver and C. P. Slichter also in 1953. A general theoretical basis and experimental observation of an Overhauser effect involving only nuclear spins in the HF molecule was published by Ionel Solomon in 1955. Another early experimental observation of the NOE was used by Kaiser in 1963 to show how the NOE may be used to determine the relative signs of scalar coupling constants, and to assign spectral lines in NMR spectra to transitions between energy levels. In this study, the resonance of one population of protons in an organic molecule was enhanced when a second distinct population of protons in the same organic molecule was saturated by RF irradiation. The application of the NOE was used by Anet and Bourn in 1965 to confirm the assignments of the NMR resonances for β,β-dimethylacrylic acid and dimethyl formamide, thereby showing that conformation and configuration information about organic molecules in solution can be obtained. Bell and Saunders reported direct correlation between NOE enhancements and internuclear distances in 1970 while quantitative measurements of internuclear distances in molecules with three or more spins was reported by Schirmer et al.
Richard R. Ernst was awarded the 1991 Nobel Prize in Chemistry for developing Fourier transform and two-dimensional NMR spectroscopy, which was soon adapted to the measurement of the NOE, particularly in large biological molecules. In 2002, Kurt Wuthrich won the Nobel Prize in Chemistry for the development of nuclear magnetic resonance spectroscopy for determining the three-dimensional structure of biological macromolecules in solution, demonstrating how the 2D NOE method can be used to constrain the three-dimensional structures of large biological macromolecules. Professor Anil Kumar was the first to apply the two-dimensional Nuclear Overhauser Effect experiment to a biomolecule, which opened the field for the determination of three-dimensional structures of biomolecules in solution by NMR spectroscopy.

Relaxation

The NOE and nuclear spin-lattice relaxation are closely related phenomena. For a single spin- nucleus in a magnetic field there are two energy levels that are often labeled α and β, which correspond to the two possible spin quantum states, + and -, respectively. At thermal equilibrium, the population of the two energy levels is determined by the Boltzmann distribution with spin populations given by P_α and P_β. If the spin populations are perturbed by an appropriate RF field at the transition energy frequency, the spin populations return to thermal equilibrium by a process called spin-lattice relaxation. The rate of transitions from α to β is proportional to the population of state α, P_α, and is a first order process with rate constant W. The condition where the spin populations are equalized by continuous RF irradiation is called saturation and the resonance disappears since transition probabilities depend on the population difference between the energy levels.
In the simplest case where the NOE is relevant, the resonances of two spin- nuclei, I and S, are chemically shifted but not J-coupled. The energy diagram for such a system has four energy levels that depend on the spin-states of I and S corresponding to αα, αβ, βα, and ββ, respectively. The W's are the probabilities per unit time that a transition will occur between the four energy levels, or in other terms the rate at which the corresponding spin flips occur. There are two single quantum transitions, W₁^I, corresponding to αα ➞ βα and αβ ➞ ββ; W₁^S, corresponding to αα ➞ αβ and βα ➞ ββ; a zero quantum transition, W₀, corresponding to βα ➞ αβ, and a double quantum transition corresponding to αα ➞ ββ.
While rf irradiation can only induce single-quantum transitions giving rise to observable spectral lines, dipolar relaxation may take place through any of the pathways. The dipolar mechanism is the only common relaxation mechanism that can cause transitions in which more than one spin flips. Specifically, the dipolar relaxation mechanism gives rise to transitions between the αα and ββ states and between the αβ and the βα states.
Expressed in terms of their bulk NMR magnetizations, the experimentally observed steady-state NOE for nucleus I when the resonance of nucleus S is saturated is defined by the expression:
where is the magnetization of nucleus at thermal equilibrium. An analytical expression for the NOE can be obtained by considering all the relaxation pathways and applying the Solomon equations to obtain
where
is the total longitudinal dipolar relaxation rate of spin I due to the presence of spin s, is referred to as the cross-relaxation rate, and and are the magnetogyric ratios characteristic of the and nuclei, respectively.
Saturation of the degenerate W₁^S transitions disturbs the equilibrium populations so that P_αα = P_αβ and P_βα = P_ββ. The system's relaxation pathways, however, remain active and act to re-establish an equilibrium, except that the W₁^S transitions are irrelevant because the population differences across these transitions are fixed by the RF irradiation while the population difference between the W_I transitions does not change from their equilibrium values. This means that if only the single quantum transitions were active as relaxation pathways, saturating the resonance would not affect the intensity of the resonance. Therefore to observe an NOE on the resonance intensity of I, the contribution of and must be important. These pathways, known as cross-relaxation pathways, only make a significant contribution to the spin-lattice relaxation when the relaxation is dominated by dipole-dipole or scalar coupling interactions, but the scalar interaction is rarely important and is assumed to be negligible. In the homonuclear case where, if is the dominant relaxation pathway, then saturating increases the intensity of the resonance and the NOE is positive, whereas if is the dominant relaxation pathway, saturating decreases the intensity of the resonance and the NOE is negative.

Molecular motion

Whether the NOE is positive or negative depends sensitively on the degree of rotational molecular motion. The three dipolar relaxation pathways contribute to differing extents to the spin-lattice relaxation depending a number of factors. A key one is that the balance between ω₂, ω₁ and ω₀ depends crucially on molecular rotational correlation time,, the time it takes a molecule to rotate one radian. NMR theory shows that the transition probabilities are related to and the Larmor precession frequencies,, by the relations:
where is the distance separating two spin- nuclei.
For relaxation to occur, the frequency of molecular tumbling must match the Larmor frequency of the nucleus. In mobile solvents, molecular tumbling motion is much faster than. The so-called extreme-narrowing limit where ). Under these conditions the double-quantum relaxation W₂ is more effective than W₁ or W₀, because τ_c and 2ω₀ match better than τ_c and ω₁. When ω₂ is the dominant relaxation process, a positive NOE results.
This expression shows that for the homonuclear case where I = S, most notably for ¹H NMR, the maximum NOE that can be observed is 1\2 irrespective of the proximity of the nuclei. In the heteronuclear case where I ≠ S, the maximum NOE is given by 1\2, which, when observing heteronuclei under conditions of broadband proton decoupling, can produce major sensitivity improvements. The most important example in organic chemistry is observation of ¹³C while decoupling ¹H, which also saturates the ¹J resonances. The value of γ_S/γ_I is close to 4, which gives a maximum NOE enhancement of 200% yielding resonances 3 times as strong as they would be without NOE. In many cases, carbon atoms have an attached proton, which causes the relaxation to be dominated by dipolar relaxation and the NOE to be near maximum. For non-protonated carbon atoms the NOE enhancement is small while for carbons that relax by relaxation mechanisms by other than dipole-dipole interactions the NOE enhancement can be significantly reduced. This is one motivation for using deuteriated solvents in ¹³C NMR. Since deuterium relaxes by the quadrupolar mechanism, there are no cross-relaxation pathways and NOE is non-existent. Another important case is ¹⁵N, an example where the value of its magnetogyric ratio is negative. Often ¹⁵N resonances are reduced or the NOE may actually null out the resonance when ¹H nuclei are decoupled. It is usually advantageous to take such spectra with pulse techniques that involve polarization transfer from protons to the ¹⁵N to minimize the negative NOE.