4.2. Double- and zero-quantum experiments

Double- and zero-quantum coherence (DQ/ZQ) techniques (Braunschweiler et al. 1983; Rexroth et al. 1995; Permi et al. 1999a) require at least a three-spin system A, M, and X. Let us assume that A and M correspond to ¹⁵N and ¹³C’, which are both coupled to a common passive spin X, corresponding to ¹H^N (Figure 2). If it is possible to invoke DQ/ZQ coherence between spins A and M, followed by a free precession period, the DQ coherence evolves with the sum of the chemical shifts of A and M. More importantly, DQ coherence evolves with the sum of the couplings J_AX and J_MX. The corresponding ZQ coherence evolves with the difference in the chemical shifts of A and M, whereas the coupling evolves with the difference of J_AX and J_MX.

It is now obvious that J_AX equals 0.5*(J_DQ+J_ZQ) and J_MX 0.5*(J_DQ-J_ZQ), where J_DQ and J_ZQ are DQ and ZQ splittings, respectively. In this example, J_DQ would correspond to ¹J_H^NN + ²J_H^NC’ at the chemical shift of ω_N+C’, and J_ZQ would be ¹J_H^NN - ²J_H^NC’ at the chemical shift of ω_N-C’. Addition of J_DQ and J_ZQ would give ¹J_H^NN, whereas subtraction yields ²J_H^NC’.

The DQ/ZQ experiments have numerous advantages over the traditional single-quantum coherence-based methods. The most important of these is a good tolerance against the effects of differential relaxation. The differential relaxation of in-phase and antiphase magnetization results in a decrease of the apparent coupling constants. The error in the experimental coupling is inversely proportional to the magnitude of the coupling. Thus, it would be advantageous to measure small coupling from large splitting. As J_DQ ≠ J_ZQ, (if J_AX and J_MX ≠ 0), the effect of differential relaxation will be eliminated as it is inversely proportional to the magnitude of the DQ and ZQ splittings (Rexroth et al. 1995). Additionally, the DQ/ZQ experiments give information on the relative sign of the coupling constants. Since J_DQ evolves with the sum of two couplings and J_ZQ with the difference, it is rather straightforward to extract the relative signs of J_AX and J_MX. This is very useful in the case of residual dipolar couplings where dipolar contributions can be either positive or negative, depending on the relative orientation of internuclear bond vectors. Finally, the random measurement error is smaller than for the J-resolved experiments since the coupling constant of interest is extracted from two splittings, and the sum or difference is divided by two. A minor complication is the assignment of cross-peaks now resonating at shifts differing from the usual single-quantum coherence chemical shift frequencies.