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Keywords:

  • Spin-polarized electron gas;
  • density matrices;
  • cumulant expansion;
  • sum rules;
  • geminals

Abstract

The spin-polarized homogeneous electron gas with densities ρ[UPWARDS ARROW] and ρ[DOWNWARDS ARROW] for electrons with spin ‘up’ ([UPWARDS ARROW]) and spin ‘down’ ([DOWNWARDS ARROW]), respectively, is systematically analyzed with respect to its lowest-order reduced densities and density matrices and their mutual relations. The three 2-body reduced density matrices γ[UPWARDS ARROW][UPWARDS ARROW], γ[DOWNWARDS ARROW][DOWNWARDS ARROW], γa are 4-point functions for electron pairs with spins [UPWARDS ARROW][UPWARDS ARROW], [DOWNWARDS ARROW][DOWNWARDS ARROW], and antiparallel, respectively. From them, three functions G[UPWARDS ARROW][UPWARDS ARROW](x,y), G[DOWNWARDS ARROW][DOWNWARDS ARROW](x,y), Ga(x,y), depending on only two variables, are derived. These functions contain not only the pair densities according to g[UPWARDS ARROW][UPWARDS ARROW](r) = G[UPWARDS ARROW]uarr;(0,r), g[DOWNWARDS ARROW][DOWNWARDS ARROW](r) = G[DOWNWARDS ARROW][DOWNWARDS ARROW](0,r), ga(r) = Ga(0,r) with r = |r1 - r2|, but also the 1-body reduced density matrices γ[UPWARDS ARROW] and γ[DOWNWARDS ARROW] being 2-point functions according to γs = ρsfs and fs(r) = Gss(r, ∞) with s = [UPWARDS ARROW],[DOWNWARDS ARROW] and r = |r1 - r1|. The contraction properties of the 2-body reduced density matrices lead to three sum rules to be obeyed by the three key functions Gss, Ga. These contraction sum rules contain corresponding normalization sum rules as special cases. The momentum distributions n[UPWARDS ARROW](k) and n[DOWNWARDS ARROW](k), following from f[UPWARDS ARROW](r) and f[DOWNWARDS ARROW](r) by Fourier transform, are correctly normalized through fs(0) = 1. In addition to the non-negativity conditions ns(k),gss(r),ga(r) ≥ 0 [these quantities are probabilities], it holds ns(k) ≤ 1 and gss(0) = 0 due to the Pauli principle and ga(0) ≤ 1 due to the Coulomb repulsion. Recent parametrizations of the pair densities of the spin-unpolarized homogeneous electron gas in terms of 2-body wave functions (geminals) and corresponding occupancies are generalized (i) to the spin-polarized case and (ii) to the 2-body reduced density matrix giving thus its spectral resolutions.