Composite Liouville space, renormalization

Equation (116) has a form which is similar to that of the equation (35) of motion for non-exchanging spin systems. The analogy is even closer, as is shown later, since a judicious renormalization of the vectors in the composite Liouville space can convert equation (116) into one in which all the superoperators become Hermitian. Firstly we wish to draw attention to some of the properties of the exchange superoperator X. ... 

In composite Liouville space, the expectation values of renormalized observables, calculated by using the renormalized a matrix, are expressed by ... 

In the renormalized composite Liouville space the superoperator F , defined by equation (121), commutes with all the (Hermitian) superoperators Hf, R, and X. Therefore, upon the proper rearrangement of the basis set in this space, one can obtain a factorization of the equation (137) of motion into blocks which are connected with individual eigenvalues of the superoperator F°. This resembles the analogous procedure in the case of static NMR spectra, i.e. those for non-exchanging spin systems (Section II.E.2). The equations for the free induction decay M ID and for the lineshape of an unsaturated steady-state spectrum, in terms of quantities from composite Liouville space, are therefore obtained for exchanging spin systems in a way which is analogous to that for non-exchanging systems (Section II.F). 

One should note that in reports on intermolecular spin exchange there are often errors in the renormalization of the vectors in composite Liouville space. (14, 51, 78) Such errors lead to erroneous lineshape equations in cases where the numbers of nuclei in the components of an equilibrium are not equal. 

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