Theory of ESR hyperfine tensor calculations

There are many sources that contribute to an ESR spectrum. Apart from the applied external field, local magnetic fields are also created within the sample by other unpaired electrons in the molecule or the surroundings, or from magnetic nuclei in the molecule/ion. There may also be contributions to the hf structure arising from the orbital angular momentum of the electron, which will influence the electronic g-value. In the present calculations, the g-value of the free electron, g = 2.0023, is assumed throughout. 

The most important interactions between the unpaired electron and the magnetic nuclei can be of two kinds dipole-dipole (anisotropic) interactions, that depend on the molecular orientation with respect to the external field, and Fermi contact or isotropic interactions (spherically symmetric). The latter are purely quantum mechanical, and arise when there is a non-zero probability of finding the electron at a particular magnetic nucleus. Hence, the interaction (coupling) will in principle be larger the more s-character there is in the singly occupied molecular orbital on the particular atom (JV). 

Usually, only diamagnetic contributions to the hyperfine tensor TN associated with the operator H puv in Eqn. (12) are taken into account. The 

Here we have separated the isotropic and magnetic dipolar parts hence Tnuv denotes the part of the hyperfine tensor containing the anisotropic components. The hyperfine tensor TNuv is related to the reduced hyperfine tensor ANuv in Eqn. (1) by 

The Fermi contact term arises from coupling between the magnetic moments of the electron ( ) and the nuclei through a contact interaction. Introducing the normalized spin density, Sx -1 the following 

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