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For the spin‐unpolarized uniform electron gas, rigorous theorems are used (Migdal, Galitskii‐Migdal, Hellmann‐Feynman) which allow the calculation of the pair density, g(r), or equivalently its Fourier transform, the static structure factor, S(q), from the dynamical 1‐body self‐energy Σ (k, ω), supposing the self‐energy is (approximately) known as a functional, depending on the kinetic energy of a single electron, t( k ), and on the bare Coulomb repulsion between two electrons, v( q ). With the momentum distribution, n(k), and with the kinetic (t) and potential (v) components of the total energy e = t + v, the respective steps are: (i) Σ (k, ω) → n(k) → t, (ii) Σ (k, ω) → v, (iii) t + v = e→ S(q). How this general scheme works  in detail  is shown explicitly for the high‐density limit (as an illustration). For this case the ring‐diagram partial summation or random‐phase approximation applies. In this way, the results of Macke (1950), Gell‐Mann/Brueckner (1957), Daniel/Vosko (1960), Kulik (1961), and Kimball (1976) are summarized in a coherent manner. Besides, several identities were brought to the light, e.g. the Kimball function for S(q) proves to be identical with Macke's momentum transfer function I(q) for e.

annalen der physik

The high‐density electron gas: How momentum distribution  n  ( k ) and static structure factor S(q) are mutually related through the off‐shell self‐energy Σ (k, ω)