Summary of Two-Spin Operators

Now look at the energy differences corresponding to the double-quantum and zero-quantum transitions  

The J terms cancel out in both cases because the interaction between Ha and Hb is the same it remains opposed (ZQ transition) or it remains aligned (DQ transition). 

If there is another spin besides I and S, which is coupled to either I or S, this passive coupling will undergo /-coupling evolution. It is only the / coupling between the two nuclei involved in the DQC or ZQC (in this case I and S), which does not lead to any evolution. 

As you can see, the bookkeeping for MQCs gets pretty messy, and we will see later (Chapter 10, Section 10.4) that another kind of operator called spherical operators is neater and easier to visualize for MQCs. 

For a system of two kinds of spins (I and S for heteronuclear, or Ia and Ib for homonuclear systems), there are 16 product operators, formed from the 16 matrix elements of the density matrix. So far we have discussed 14 of them  

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