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Smooth, highly accurate analytical representations of Fermi-Dirac (FD)integral combinations important in free-energy density functional calculationsare presented. Specific forms include those that occur in the local densityapproximation (LDA), generalized gradient approximation (GGA), and fourth-ordergradient expansion of the non-interacting free energy as well as in the LDA andsecond-order gradient expansion for exchange. By construction, all therepresentations and their derivatives of any order are continuous on the fulldomains of their independent variables. The same type of technique provides ananalytical representation of the function inverse to the FD integral of order$1/2$. It plays an important role in physical problems related to the electrongas at finite temperature. From direct evaluation, the quality of theseimproved representations is shown to be substantially superior to existingones, many of which were developed before the era of large-scale computation orearly in the era.

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