Homonuclear product operators

The chemical shift evolution (precession of spins as a result of chemical shift) can be represented as a circle with the rotation rate (in radians per second) written in the center (Fig. 7.6). This is the same as the motion of the net magnetization vector, viewed from the axis. Homonuclear product operators (Ha represented by Ia and Hb represented by Ib) undergo chemical shift evolution in the same way ... 

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 ... 

UNDERSTANDING COSY WITH PRODUCT OPERATORS 9.5.1 Analyzing the Homonuclear Front End ... 

The terms in (la) and (lb) both involve sums of single nuclear spin operators Iz. In contrast, the terms in (lc) involve pairwise sums over the products of the nuclear spin operators of two different nuclei, and are thus bilinear in nuclear spin. If the two different nuclei are still of the same isotope and have the same NMR resonant frequency, then the interactions are homonuclear if not, then heteronuclear. The requirements of the former case may not be met if the two nuclei of the same isotope have different frequencies due to different chemical or Knight shifts or different anisotropic interactions, and the resulting frequency difference exceeds the strength of the terms in (lc). In this case, the interactions behave as if they were heteronuclear. The dipolar interaction is proportional to 1/r3, where r is the distance between the two nuclei. Its angular dependence is described below, after discussing the quadrupolar term. 

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