Spin correlations in a magnetic electron gas. II

The third moments of some correlation functions based on our recent work (henceforth referred to as I) are compared with the corresponding exact expressions derived by Goodman and Sjölander (GS). A moment-conserving (MC) scheme proposed by GS must be modified for the magnetic system in order for it to obey an exact relationship derived in I; the GS scheme is generalized by using a MC scheme presented in I but employing the exact results for the third moments. In the paramagnetic limit, our scheme gives the same results as those of GS. The second and third moments of transverse spin fluctuations, ${\chi }_{+-}(q,\omega )$, in a magnetic electron gas are derived and these are believed to be new. (GS did not consider ${\chi }_{+-}$ at all.) The exact evaluations of moments serve as precise tests of any calculation based on approximation schemes. We also derive and discuss an exact expression for the second moment of the spectral weight function of one-particle Green's function of this system.