G. wider/Progress in Nuclear Magnetic Resonance Spectroscopy 32(1998)193-275 lue to the counter- rotating field the term"non-reso- than their mutual scalar coupling J; this situation nant effect was introduced [34]. Since this additional often referred to as weak spin-spin coupling whereas field may be rather strong and close to the resonance strong spin-spin coupling specifies the case where frequency, this effect can become quite large and is close to or even smaller than J. With these assump- needs to be compensated [35](Section 2.2.1) tions simple rules can be calculated which describe the evolution of spin operators under the action of 2. 1. 2. Operators, coherence, and product operator chemical shift, J coupling and RF pulses. The form formalism alism combines the exact quantum mechanical treat- The description of NMr experiments by the Bloch ment with an illustrative classical interpretation and is equations and by magnetization vectors in the rotating the basis for the development of many NMR experi- frame has significant limitations particularly for the ments. However, some parts of experimental schemes description of multipulse experiments. On the other for example ToCSY sequences(Section 4.2.1), can hand, a full quantum mechanical treatment which only be described with a full quantum mechanical describes the state of the system by calculation of treatment. Calculations with the formalism are not the time evolution of the density operator under the fficult, but many terms may have to be treated an action of the appropriate Hamiltonian can be cumber aplementations of the formalism within computer some. In a quantum mechanical description an R programs are very helpful in such situations [36,37 pulse applied to the equilibrium state creates a coher- Although most of the experiments applied in bio- ent superposition of eigenstates which differ in their molecular NMR correlate three or more spins, the magnetic quantum number by one, often simply majority of interactions can still be understood based referred to as a coherence. In more complex experi- on an analysis of two spins. For two spins I and S the ments the magnetic quantum numbers between states operator basis for the formalism contains 16 elements differ by a value g different from one, leadi Two sets of basis operator, cartesian and shift opera a q quantum coherence with at least q spins involved. tors, have proven very useful for the description of However, only in-phase single quantum coherences experimental schemes and are used in parallel. The (Table 2)are observable and correspond to the classi- two basis sets and the nomenclature used to character- al magnetization detected during the acquisition of ize individual states are summarized in Table 2 [28] an NMR experiment. Multiple quantum coherences The operators /, and S are identical in the two basis cannot be observed directly, but they influence the sets and a simple relationship exists between the two spin state and this information can be transferred to other cartesian and shift operators bservable magnetization In a step towards a full quantum mechanical treat- lx=(7++/)/2 I=l+il (4) ment, the product operator formalism for spin , nuclei as introduced [28]. In this approach it is assumed I-ilv at there is no relaxation and that the difference in the Three operators, which represent the action of the resonance frequency Ar of two nuclei is much larger Hamiltonians for chemical shift, scalar coupling and Table 2 Product operator basis for a two-spin system Cartesian operator basis Nomenclature Shift operator basis Longitudinal magnetization 、l,SS In-phase transverse magnetization 1,,S+,S 2/S:2/S Anti-phase I spin magnetization 2/+S,2S 25/252 Anti-phase S spin magnetization 1, 2S I. 2/Sx2Sx21/S,2/.S 2/*5+,2/S, 2/S,2/"S 21.S ongitudinal two-spin order