Prof. F. Ann Walker Research Group, Department of Chemistry, University of Arizona, Tucson, Arizona 85721, USA
The HFI Interaction has two contributions:
here is the electronic spin density on a nucleus
here is the vector from a nucleus to electron in the molecular coordinate frame.
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.
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.
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©Nikolai V. Shokhirev, F. Ann Walker, 2002