by Nikolai V. Shokhirev

Prof. F. Ann Walker Research Group, Department of Chemistry, University of Arizona, Tucson, Arizona 85721, USA

Chemical shifts

Diamagnetic molecules (with zero spin)

The chemical shift of diamagnetic molecules is purely caused by the magnetic field from the electric current induced by the external magnetic field. In theoretical papers the shifts of diamagnetic molecules is divided into diamagnetic and paramagnetic contributions. The latter is not associated with molecular spin. Avoiding this ambiguity, we will refer to the chemical shift of diamagnetic molecules as "diamagnetic" shift: .

Paramagnetic molecules (non-zero spin)

The chemical shift of paramagnetic molecules (radicals, complexes) has additional contribution due to the hyperfine interaction of a nucleus with unpaired electron spin. The Hamiltonian is

Here is the tensor of the hyperfine interaction. The magnitude of the HFI interaction is larger than the nuclear Zeeman interaction. It could cause huge broadening of the NMR spectra. Fortunately, the electron relaxation is very fast (in NMR time scale) and a nucleus "feels" only the averaged spin 

(how to average electronic spin using the Boltzmann distribution see Curie dependence). 

Comparing with the above Zeeman Hamiltonian, we can derive that the spin contribution the chemical shift of paramagnetic molecules ("real" paramagnetic shift) is described by the following expression

This type of temperature dependence (linear in the inverse temperature) is called the Curie law or Curie dependence. 

The averaging of the electron spin effectively replace 2S+1 sub-levels with one level but with temperature-dependent magnetic moment.

For the description of liquid-phase NMR experiments, the paramagnetic shifts should be further averaged over orientation of molecules (see here how to average vectors and tensors over orientations). 

©Nikolai V. Shokhirev, F. Ann Walker, 2002

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