Valentin V. Karasiev ∗, Debajit Chakraborty, S.B. Trickey. *Improved analytical representation of combinations of Fermi–Dirac integrals for finite-temperature density functional calculations*

**Computer Physics Communications 192 (2015) 114–123**, 2015-03-09

Abstract: Smooth, highly accurate analytical representations of Fermi–Dirac (FD) integral combinations important

in free-energy density functional calculations are presented. Specific forms include those that occur in the

local density approximation (LDA), generalized gradient approximation (GGA), and fourth-order gradient

expansion of the non-interacting free energy as well as in the LDA and second-order gradient expansion

for exchange. By construction, all the representations and their derivatives of any order are continuous

on the full domains of their independent variables. The same type of technique provides an analytical

representation of the function inverse to the FD integral of order 1/2. It plays an important role in physical

problems related to the electron gas at finite temperature. From direct evaluation, the quality of these

improved representations is shown to be substantially superior to existing ones, many of which were

developed before the era of large-scale computation or early in the era.