# Nuclear Overhauser effect

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The nuclear Overhauser effect[1] (nOe or more often NOE) is widely used in the field of NMR spectroscopy, particularly in the field of structural biology in which interatomic distances derived from various forms of homonuclear and heteronuclear-edited NOESY spectra are used to constrain structural calculations for DNA, RNA, proteins, and even small molecules. However, for small molecules ( < 500 Dalton) a rotating frame version of this effect (ROESY) is often used.

The NOE effect causes changes in the intensity of a signal at one frequency when the resonance frequency of a different nucleus is irradiated, due to dipole-dipole interactions between the magnetic moments of the pair of nuclei. Unlike J-coupling, this interaction is not mediated through (electrons) 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 ( < ~ 5-6 Angstrom). 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 is quantitatively defined as

$\eta = \frac{<S_z> - <S_{z,equil}>}{<S_{z,equil}>} \,\!$
where, $<S_z>$ is the magnetization of spin S after irradiation at the frequency of spin I,
$<S_{z,equil}>$ is the magnetization of spin S at equilibrium

In the steady state, when the resonance frequency of spin I is irradiated and the intensity of spin S is monitored[2],

$\eta = \frac{<S_z> - <S_{z,equil}>}{<S_{z,equil}>} = \frac{\sigma}{\rho_S}\frac{\gamma_I}{\gamma_S}$

where, $\sigma$ is the rate of cross-correlation
$\rho_S$ is auto-relaxation rate for spin S
$\gamma_I$ and $\gamma_S$ are the gyromagnetic ratios of spins I and S respectively

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 $\frac{\gamma_I}{\gamma_S} > 1$, because $\frac{\sigma}{\rho_S} \rightarrow 1/2$ when $w\tau_c << 1$. However, when $w\tau_c >> 1$, $\frac{\sigma}{\rho_S} \rightarrow -1$ 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 and other molecules with similar correlation times e.g. many hexapeptides.

## References

1. Overhauser, A. W. (1953). Phys. Rev. 92, 411
2. Quantum description of high resolution NMR in liquids. M.Goldman. Oxford.