Symmetry and Hartmann-Hahn Transfer

Symmetry considerations play a role on several levels in the analysis of Hartmann-Hahn experiments. In the presence of rotational symmetry and permutation symmetry, the effective Hamiltonian often can be simplified by using symmetry-adapted basis functions (Banwell and Primas, 1963 Corio, 1966). For example, any zero-quantum mixing Hamiltonian can be block-diagonalized in a set of basis functions that have well defined magnetic quantum numbers. Block-diagonalization of the effective Hamiltonian simplifies the analysis of Hartmann-Hahn experiments (Muller and 

1979 Chingas et al., 1981 Cavanagh et al., 1990). Similarly, the diagonalization of the mixing Hamiltonian superoperator can be facilitated if a set of basis operators is used that is completely reduced with respect to rotation, permutation, and particle number (Listerud and Drobny, 1989 Listerud et al., 1993). 

Under the idealized zero-quantum coupling topologies (see Section V.B), the transfer of magnetization between two spins 1 /2 that are part of an arbitrary coupling network is identical in both directions (see Section VI). This symmetry with respect to the direction of the transfer is related to the symmetry of homonuclear, two-dimensional Hartmann-Hahn spectra with respect to the diagonal (Griesinger et al., 1987a). In Hartmann-Hahn experiments, the properties of the multiple-pulse sequence can induce additional symmetry constraints (Ernst et al., 1991). 

In this section, we restrict the discussion to constants of motion during Hartmann-Hahn experiments that are induced by the symmetry of the mixing Hamiltonian and to selection rules for cross-peak signals in two-dimensional isotropic-mixing experiments. 

Constants of Motion during Hartmann-Hahn Mixing 

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