Zero- and Double-Quantum Operators

The MQC intermediate state in coherence ( INEPT ) transfer can also be used to clean up the spectrum. In this case, we can apply a double-quantum filter (using either gradients or a phase cycle) to kill all coherences at the intermediate step that are not DQC. We will see the usefulness of this technique in the DQF (double-quantum filtered) COSY experiment (Chapter 10). As with the spoiler gradient applied to the 2IZSZ intermediate state, a doublequantum filter destroys any unwanted magnetization, leaving only DQC that can then be carried on to observable antiphase magnetization in the second step of INEPT transfer. 

In either case, whether we do the 90° pulse first or the 13C 90° pulse first, we are simply choosing the order of the two processes (Fig. 7.27) the lH operator (Ix) in the product moves from the x-y plane to the z axis (lH 90° pulse) and the 13C operator (Sz) in the product moves from the z axis to the x-y plane (13C 90° pulse). We can bump up the operator to the z axis first, resulting in both operators on the z axis, and then knock down the 13C operator to the x-y plane. Alternatively, we can first knock down the 13C operator from the z axis to the x-y plane, resulting in both operators in the product in the x-y plane (MQC), and then bump up the operator from the x-y plane to the 

Product operators in which both components are in the x -y plane represent zero-quantum and double-quantum coherences (collectively called multiple-quantum coherences). DQC is a superposition of the spin states aid s and /3i/3s which involves promotion of both nuclei I and S simultaneously from the a state to the P state or vice versa. ZQC is a superposition of the spin states a Ps and /fias, which involves nucleus I flipping from a to p while nucleus S flips from the p state to the a state, or the reverse process. Neither of these coherences can be directly observed, but we can convert them into observable (single-quantum) coherence and see the effect of evolution during the time spent as zero- and double-quantum coherences. In product operator notation they look like this 

DQC precesses under the influence of chemical shifts at a rate determined by the sum of the two chemical shifts, whereas ZQC precesses at a rate determined by the difference (Fig. 7.28). This can be demonstrated by plugging in [Ix cos( ir) + Iy sin( ir)] for lx and [Sy cos( sr) — Sx sin( sr)] for Sy, etc., and multiplying the expressions together. The math is rather messy (although very satisfying) so we will not go through it here. 

Note that the four single-quantum transitions have energy differences corresponding to their exact frequencies in the 1H spectrum  

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