Tensorial sets for coupled pairs of spin-1/2 nuclei

A new basis set for describing multipulse NMR experiments is presented and discussed. It is shown that unit spherical tensors Ûk q (I 1 I 2) can be used to simplify the description of AA and AB coupled spin-1/2 systems evolving between radio-frequency pulses, in the presence of chemical shift and scalar coupling. For comparison the new results are compared with those obtained using coupled irreducible tensor operators [Tcirc]k q (k 1 k 2). It is shown that 86 (respectively 94) terms are required to describe the evolution of the general density matrix for the AB system if unit spherical tensors (respectively irreducible tensor operators) are used. For the AA system the gains are more substantial in that only 26 (respectively 54) terms are required. Such reductions have clear advantages for handling systems of three or four coupled spins. It is also argued that in dealing with coupled spin systems of three or more spins, the use of the unit spherical tensors Ûk q (Ii j ) offer significant mathematical advantages over concatenated tensor operators [Tcirc]k q (k 1…kn ). In the case of the Ûk q (IiIj ) tensors only one Clebsch-Gordan coefficient is required, regardless of the number of coupled spins. However, for the [Tcirc]k q (k 1…kn ) tensors, it is necessary to evaluate ‘nested’ Clebsch-Gordan coefficients repeatedly.