Linear Scaling Pseudo Fermi-Operator Expansion for Fractional Occupation

Recursive Fermi-operator expansion methods for the calculation of the idempotent density matrix are valid only at zero electronic temperature with integer occupation numbers. We show how such methods can be modified to include fractional occupation numbers of an approximate or pseudo Fermi-Dirac distribution and how the corresponding entropy term of the free energy is calculated. The proposed methodology is demonstrated and evaluated for different electronic structure methods, including density functional tight-binding theory, Kohn-Sham density functional theory using numerical orbitals, and quantum chemistry Hartree-Fock theory using Gaussian basis functions.