Momentum distribution of the uniform electron gas: improved parametrization and exact limits of the cumulant expansion

The momentum distribution of the unpolarized uniform electron gas in its Fermi-liquid regime, n(k,r_s), with the momenta k measured in units of the Fermi wave number k_F and with the density parameter r_s, is constructed with the help of the convex Kulik function G(x). It is assumed that $n(0,r_s), n(1^\pm, r_s)$, the on-top pair density g(0,r_s) and the kinetic energy t(r_s) are known (respectively, from effective-potential calculations, from the solution of the Overhauser model, and from Quantum Monte Carlo calculations via the virial theorem). Information from the high- and the low-density limit, corresponding to the random-phase approximation and to the Wigner crystal limit, is used. The result is an accurate parametrization of n(k,r_s), which fulfills most of the known exact constraints. It is in agreement with the effective-potential calculation of Takada and Yasuhara [1991 {\it Phys. Rev.} B {\bf 44} 7879], is compatible with Quantum Monte Carlo data, and is valid in the density range $r_s \lesssim 12$. The corresponding cumulant expansions of the pair density and of the static structure factor are discussed, and some exact limits are derived.