by Nikolai V. Shokhirev
Prof. F. Ann Walker Research Group, Department of Chemistry, University of Arizona, Tucson, Arizona 85721, USA
The Hamiltonian of a nucleus in the external magnetic field (nuclear Zeeman interaction) can be written in the following form
Here the nuclear g-factor gN is considered as a constant and the shielding factor (tensor) is introduced to take into account chemical surrounding of the nucleus (chemical shift).
Let the molecule under consideration can be in two states
The probabilities to find the molecule in the states 1 and 2 are
However, the manifestation of the chemical shift averaging in NMR spectrum depends on the kinetic properties of the system.
1. Slow chemical exchange. The transition rates between the states are slow than the distance between lines in the frequency scale:
The molecule lives long enough in each state that NMR "sees" the two kind of molecules but the intensity of lines are proportional to the probabilities.
2. Fast chemical exchange. The inequality is opposite. The spectrum consists of one line at the average position is determined by the Boltzmann probabilities
The following picture illustrate these cases
Slow chemical exchange
Fast chemical exchange
The latter case is typical for paramagnetic complexes due to fast electron relaxation
©Nikolai V. Shokhirev, F. Ann Walker, 2002