Editing Nuclear Overhauser effect (section)

==Molecular motion==
Whether the NOE is positive or negative depends sensitively on the degree of rotational molecular motion.<ref name="Claridge"/> 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 ω<sub>2</sub>, ω<sub>1</sub> and ω<sub>0</sub> depends crucially on molecular [[rotational correlation time]], <math>\tau_{c}</math>, the time it takes a molecule to rotate one radian. NMR theory shows that the transition probabilities are related to <math>\tau_{c}</math> and the [[Gyromagnetic ratio#Larmor precession|Larmor precession frequencies]], <math>\omega</math>, by the relations:
:::<math>W_{1}^{I} \propto  \frac {3\tau_c }{(1+\omega_{I}^2\tau_{c}^2)} \frac {1}{r^{6}}</math>

:::<math>W_{0} \propto  \frac {2\tau_c }{(1+(\omega_{I}-\omega_S)^2\tau_{c}^2)} \frac {1}{r^{6}}</math>

:::<math>W_{2} \propto  \frac {12\tau_c }{(1+(\omega_{I}+\omega_S)^2\tau_{c}^2)} \frac {1}{r^{6}}</math>

where <math>r</math> is the distance separating two spin-{{frac|1|2}} 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 <math>\omega</math>. The so-called extreme-narrowing limit where <math>\omega\tau_{c} \ll 1 </math>). Under these conditions the double-quantum relaxation W<sub>2</sub> is more effective than W<sub>1</sub> or W<sub>0</sub>, because τ<sub>c</sub> and 2ω<sub>0</sub> match better than τ<sub>c</sub> and ω<sub>1</sub>. When ω<sub>2</sub> is the dominant relaxation process, a positive NOE results.

:::<math>W_{1}^{I} \propto \gamma_{I}^{2}\gamma_{S}^{2}\frac {3\tau_{c}}{r^{6}  }</math>
:::<math>W_{0} \propto \gamma_{I}^{2}\gamma_{S}^{2}\frac {2\tau_{c}}{r^{6}  }</math>
:::<math>W_{2}\propto \gamma_{I}^{2}\gamma_{S}^{2}\frac {12\tau_{c}}{r^{6}  }</math>

:::<math> \eta_{I}^{S}(max) = \frac{\gamma_S }{\gamma_I }\left[\frac { \frac {12\tau_{c}}{r^{6}} - \frac{2\tau_{c}}{r^{6}} }
{ \frac {2\tau_{c}}{r^{6}} + 2\frac{3\tau_{c}}{r^{6}} + \frac {12\tau_{c}}{r^{6}}                                                                                   } \right] =\frac{\gamma_S }{\gamma_I }\left[ \frac {12-2 }{2+6+12 } \right] = \frac{\gamma_S }{\gamma_I }\frac{1}{2}</math>

This expression shows that for the homonuclear case where ''I'' = ''S'', most notably for <sup>1</sup>''H'' 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 (''&gamma;''<sub>S</sub>/''&gamma;''<sub>I</sub>), 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 <sup>13</sup>C while decoupling <sup>1</sup>H, which also saturates the <sup>1</sup>J resonances. The value of ''&gamma;''<sub>S</sub>/''&gamma;''<sub>I</sub> is close to 4, which gives a maximum NOE enhancement of 200% yielding resonances 3 times as strong as they would be without NOE.<ref name="Derome">{{cite book
 |last1=Derome
 |first1=Andrew E.
 |year=1987
 |title=Modern NMR Techniques for Chemistry Research
 |url=https://archive.org/details/modernnmrtechniq00dero
 |url-access=limited
 |publisher=Pergamon
 |page=[https://archive.org/details/modernnmrtechniq00dero/page/n52 106] <!-- or pages= -->
 |isbn = 978-0080325149 }}</ref> 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 (e.g. [[deuterated chloroform|CDCl<sub>3</sub>]]) in <sup>13</sup>C NMR. Since deuterium relaxes by the quadrupolar mechanism, there are no cross-relaxation pathways and NOE is non-existent. Another important case is <sup>15</sup>N, an example where the value of its magnetogyric ratio is negative. Often <sup>15</sup>N resonances are reduced or the NOE may actually null out the resonance when <sup>1</sup>H nuclei are decoupled. It is usually advantageous to take such spectra with pulse techniques that involve polarization transfer from protons to the <sup>15</sup>N to minimize the negative NOE.