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

Averaging of vectors and tensors

The HFI Interaction has two contributions:

If the molecules are arbitrary oriented (e.g. due to rotation) then a nucleus "feels" only the averaged interaction with the spin. In this case the averaged HFI tensor should be used.

The dipole-dipole HFI tensor can be easy derived from the Hdd Hamiltonian

 The g-tensor and nuclear-electron distances are defined in the molecular coordinate system. Below they are marked with a bar. We are interested in the average values in the laboratory coordinate system.

Let us define the transformation of vector components from the molecular to the laboratory system 

Obviously the transformation for the products is

The two-index tensors obey the same transformation as the products above 

For the averaging the two approaches can be used

1. Direct averaging

One of the possible choices for the transformation matrix can be the following

The averaging is the integration over all orientation of the molecule

Exercise. Perform averaging using the direct approach.

2. Symmetry use

Some obvious preliminary remarks:

i) The length of a vector does not depend on orientation and coordinate system

ii) If all orientations are equivalent then all projections are equivalent

iii) different projections are uncorrelated

This is enough to derive the following expressions


Now it easy to get

The second term in the dipole-dipole tensor can be rewritten as

The vectors and are fixed in the molecular coordinate system. In the laboratory system they keep the same relative orientation (and, obviously, the scalar or dot product). Similar to the above products of r-coordinates, the following formula takes place 

Substituting the    with its definition we get

If in addition, the molecular coordinate axes are chosen along the principal axes of the g-tensor then the tensor is diagonal and the above expression is further simplified and we get

In the  the polar coordinates 

it reduces to the well known expression

Here the axial and equatorial g-values are introduced. The first term of the Add is called an axial term and the second one is called a rhombic term.

Back to Chemical Shifts.

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