The 2-matrix of the spin-polarized electron gas: contraction sum rules and spectral resolutions

The spin-polarized homogeneous electron gas with densities ${\rho }_{\uparrow }$ and ${\rho }_{\downarrow }$ for electrons with spin up' ($\uparrow$) and spin down’ ($\downarrow$), 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 ${\gamma }_{\uparrow \uparrow }$, ${\gamma }_{\downarrow \downarrow }$, ${\gamma }_a$ are 4-point functions for electron pairs with spins $\uparrow \uparrow$, $\downarrow \downarrow$, and antiparallel, respectively. From them, three functions $G_{\uparrow \uparrow }(x,y)$, $G_{\downarrow \downarrow }(x,y)$, $G_a(x,y)$, depending on only two variables, are derived. These functions contain not only the pair densities but also the 1-body reduced density matrices. The contraction properties of the 2-body reduced density matrices lead to three sum rules to be obeyed by the three key functions $G_{ss}$, $G_a$. These contraction sum rules contain corresponding normalization sum rules as special cases. The momentum distributions $n_{\uparrow }(k)$ and $n_{\downarrow }(k)$, following from $f_{\uparrow }(r)$ and $f_{\downarrow }(r)$ by Fourier transform, are correctly normalized through $f_s(0)=1$. In addition to the non-negativity conditions $n_s(k),g_{ss}(r),g_a(r)\ge 0$ [these quantities are probabilities], it holds $n_s(k)\le 1$ and $g_{ss}(0)=0$ due to the Pauli principle and $g_a(0)\le 1$ 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.