Generalized Cumulant Expansions and Spin-Relaxation Theory

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Abstract
The generalized cumulant expansion method of Kubo is applied to an analysis of spin‐relaxation theory appropriate for NMR and ESR studies of molecular systems. It leads to a general, formal solution of the equation of motion of a suitably averaged magnetization operator (and also the spin‐density matrix). This solution permits a convenient perturbation expansion in the region of motional narrowing; i.e., when the random functional spin perturbation H 1 (t) obeys | H 1 (t) |τ c <1 , where τ c is a correlation time for the random process. It is shown how this expansion valid for times t≫τ c generates the time‐independent R or relaxation matrix to all orders in | 1 (t) |τ c , and detailed expressions are given through fourth order. While the R matrix supplies the linewidth and dynamic frequency shift of the main Lorentzian resonance line, it is found that by formulating the cumulant method without the restriction t≫τ c , weak subsidiary Lorentzian lines are predicted. These subsidiary lines appear as second‐order perturbation corrections in | H 1 (t)|τ c . They have widths given by τ c −1 and, in general, large frequency shifts. The problem of (anisotropic) ‐rotational diffusion is discussed in detail, and it is shown that the spin‐Hamiltonian approximation for liquids rests upon a Born–Oppenheimer‐type approximation appropriate for random modulation of the nuclear coordinates, i.e., τ c −1 ≪ω n,o , where ℏω n,o is the energy separation between the ground electronic state and the nth excited state coupled by the angular‐momentum operator L. The cumulant method is then used to obtain higher‐order corrections to the g‐tensor line‐broadening mechanism.

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