The Ursell‐function structure of the memory function

The integrodifferential equation which defines a memory function f (t) in terms of the time−correlation function g (t) generates ordered Ursell functions having the cluster property. Thus f (t) has an expansion f⁽²⁾ (t) + f⁽³⁾ (t) + ... in which f⁽ⁿ⁾ (t) is an n−1 fold time integral whose integrand vanishes when the interval between any two successive times is much greater than a certain correlation time of the system. An interaction representation is used in deriving these results and it is found that f⁽ⁿ⁾ (t) depends, in a certain sense, upon the nth power of the interaction term H₁ of the Hamiltonian. The integrand of f⁽ⁿ⁾ (t) is closely related to the ’’cumulant averaged’’ Liouville operators introduced by Kubo and Tomita and developed further by Kubo, Freed, and van Kampen. Thus, in the Markoffian limit f⁽²⁾ (t) is simply related to the Redfield relaxation matrix. However, the peculiar time−ordering problems of the cumulant expansion theory do not appear here. Except in very simple cases, all of these results depend upon identifying g (t) as a correlation matrix (Kubo) and the f (t) is the corresponding memory matrix. As a first application the memory function for the EPR relaxation of aqueous Ni⁺⁺ is calculated in terms of the spin parameters. It is assumed that fluctuating zero−field splitting causes the relaxation, but it is not assumed that the EPR relaxation obeys Bloch’s equations.