Sum rules and atomic correlations in classical liquids. IV. The coherent and incoherent scattering functions

The continued-fraction representation of the generalized Langevin equation developed by Mori is used to calculated the coherent and incoherent scattering functions of simple classical liquids. A three-pole approximation, together with the assumption that the decay of the correlation function of the third random force is a function of $t^2$, is used to derive a microscopic expression for the Maxwell relaxation time for the longitudinal mode. No adjustable parameters are introduced, and the corresponding expression for $S(q, \omega )$ has the correct large wave-vector dependence at zero energy transfer. The method is extended to calculate the incoherent scattering function $S_s(q, \omega )$. Numerical predictions for the relaxation time in liquid argon are compared with other existing estimates, the present results are in very good agreement with the experimental observations. It is also found that our results give a better description of the behavior of $S(q, \omega )$ and $S_s(q, \omega )$ in liquid argon as revealed by the neutron inelastic scattering measurements of Sköld et al.