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)

Abstract

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: (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), GellMann/Brueckner (1957, Daniel/Vosko (1960), Kulik (1961), andKimball (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.

Momentum distribution and structure factors of a high-density homogeneous electron gas from its cumulant two-body reduced density matrix

“…Important short-range correlation parameters are g a (0), the depth of the Coulomb hole, and g p (0), the on-top (zero electron-electron distance) curvature of the Fermi hole. It holds that 0 g a (0) 1 and 0 g p (0) 4 3 t. The corresponding cumulant pair densities h a,p (r) are defined by…”

“…To each direct diagram d 1 with two open particle-hole lines [one running from r 1 to r 1 and another one from r 2 to r 2 as for example in Fig. 1(a)] belongs a matrix χ d 1 (r 1 |r 1 ,r 2 |r 2 ) and an exchange diagram x 1 with χ x 1 (r 1 |r 1 ,r 2 |r 2 ) = χ d 1 (r 1 |r 2 ,r 2 |r 1 ), such that for r 2 = r 2 and d 3 r 2 there remains only one line running from r 1 to r 1 (like the diagrams of the self-energy ; see, for example, [4]). With these building elements χ d = χ d 1 + χ d 2 + · · · and χ x = χ x 1 + χ x 2 + · · · the singlet (triplet) components are…”

“…So contraction finally means r 2 = r 2 , d 3 r 2 , and adding a factor sgn(1 − k). How (5.9) is related to the Migdal formula [4] has to be studied.…”

“…2(c), as shown in Appendix C. The approach therein seems easy and straightforward, but in reality it is a difficult process. For further details, including the calculation of t r , the RPA part of t, see [4,28]. For the momentum distribution n(k) in second order, there is a still missing term, arising from a ladder diagram.…”

Cumulant Structure Factors of Jellium

“…Equation (B.1) in [11]. This leads to Equation (B.5) taking the form ( ) v does not diverge (needs no RPA renormalization).…”