Improved analytical representation of combinations of Fermi-Dirac integrals for finite-temperature density functional calculations

TL;DR

This paper introduces highly accurate, smooth analytical representations of Fermi-Dirac integral combinations crucial for finite-temperature density functional calculations, improving upon previous models with better accuracy and continuity.

Contribution

The authors develop new analytical forms for Fermi-Dirac integrals and their inverses, ensuring continuity of derivatives and superior accuracy over existing representations.

Findings

Representations are highly accurate and smooth across full domains.

The inverse FD integral function is effectively represented.

Improved models outperform earlier approximations in accuracy.

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.