Toward a Comprehensive Treatment of Temperature in Electronic Structure Calculations: Non-zero-Temperature Hartree-Fock and Exact-Exchange Kohn-Sham Methods

Abstract

Non-zero-temperature exact-exchange Kohn-Sham and Hartree-Fock methods are discussed and compared to standard Kohn-Sham methods using exchange-correlation functionals within the generalized gradient approximation (GGA). It is shown that the PBE exchange functional, chosen as representative example of a GGA functional, cannot capture the strong temperature dependence of the exchange energy found in the exact-exchange Kohn-Sham and Hartree-Fock methods. This has the drastic consequences of leading to a qualitatively wrong behavior of the PBE free energy with temperature as compared to the exact-exchange Kohn-Sham and Hartree-Fock methods. This indicates that conventional GGA functionals are unsuitable for the treatment of the electronic structure of matter at high temperatures. The exact evaluation of the exchange energy, the construction of the local multiplicative KS exchange potential as well as the handling of the nonlocal Hartree-Fock exchange potential require integrations in k space for integrands with integrable singularities. It is shown how these singularities can be treated numerically in the case of non-zero temperatures.