Challenging example integer spin

How do we know or decide what terms to put in the spin Hamiltonian This is a question of rather far-reaching importance because, since we look at our biomolecular systems through the framework of the spin Hamiltonian, our initial choice very much determines the quality limits of our final results. In other branches of spectroscopy this is sometimes referred to as a sporting activity. We are guided (one would hope) by a fine balance of intellectual inspection, (bio)chemical intuition, and practical considerations. In a more hypochondriacal vein, one could also call this the Achilles heel of the spectroscopy a wrong choice of the model (the spin Hamiltonian) will not lead to an accurate description of nature represented by the paramagnetic biomolecule. 

For the sake of completeness we note that spin-Hamiltonian theory (Pake and Esde 1973 95) states that any interaction of the form Bb Ss P is allowed to occur, i.e., 

To keep the analysis tractable by eliminating insignificantly small interactions, the following practical restrictions are usually added 

We now single out one of these interactions for our discussion of the integer-spin 5 = 2 system, and we defer an explanation of this deliberate choice to the end of this section. We write the spin Hamiltonian for an isolated system (i.e., no interactions between paramagnets) with 5 = 2 and 7=0 (i.e., no hyperfine interactions) as 

We use these relations to write out the energy matrix initially in zero field (i.e., ignoring the Zeeman interaction)  

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