Exact irreducible moments of the Landau collision operator in the random-velocity moment expansion

Exact moments of the Landau collision operator are calculated for the irreducible Hermite polynomials written in terms of the random-velocity variable. We present closed, algebraic formulas that reproduce the results for the total-velocity moment expansion\footnote{J.-Y. Ji and E. D. Held, Phys. Plasmas {\bf 13}, 102103 (2006).} and for the random-velocity moment expansion with the small mass-ratio approximation\footnote{J.-Y. Ji and E. D. Held, Phys. Plasmas {\bf 15}, 102101 (2008).}. The collisional moments can be applied in the derivations of Braginskii and integral closures for arbitrary relative flow velocity between electrons and ions. Modifications to Braginskii closures are discussed.