Revisiting the $r_s$ expansion of the uniform electron gas in $d$ dimensions: $\log r_s$ terms from coincident Mellin poles

Abstract

The high-density expansion of the correlation energy of the uniform electron gas contains $\log r_s$ terms, where $r_s$ is the Wigner--Seitz radius. Their form depends on the dimension $d$. Adapting the Mellin-transform analysis of Feynman integrals in relativistic field theory, we write the energy of the ring series and of diagrams linear in one screened interaction line in $d$ dimensions as Mellin convolutions of a radial kernel, fixed by the diagram, with a scaling function. The scaling function depends on the momentum transfer only through its ratio to a density scale. In the Mellin variable $s$, screening places poles at $s=N(d-1)$ with $N=1,2,\dots$, and the diagram places a pole at $s=d+1$ for the rings and at $s=2$ for a polarization insertion. Once the factors $\log (1/q)$, with $q$ the momentum transfer, cancel between the vertex and self-energy corrections, as the Ward identity requires, a $\log r_s$ term appears where two poles coincide. This rule reproduces the known $\log r_s$ terms in two and three dimensions. In two dimensions it gives the coefficient of the next $\log r_s$ term, $r_s^2\log r_s$, in closed form, $7/\pi +4\log 2/\pi -160/(9{\pi }^2)-16G/{\pi }^2=-0.17546$~Ry with $G$ Catalan's constant. In three dimensions the screened second-order exchange diagram alone has an $r_s{\log }^2{r}_s$ term at high density and an $r_s^{-1/4}$ term at low density, and the Fock self-energy corrections cancel both.

arXiv commentv4: substantially rewritten and extended, with a new title. Supersedes v1-v3, which treated only the screened second-order exchange (kite) diagram in three dimensions with a single-pole or static screened line. The spin-polarized kite is now in a companion paper (ChemRxiv, doi:10.26434/chemrxiv.15008489/v1). 96 pages including the Supporting Information, 6 figures