Implementation of Kohn-Sham LCAO Method in Crystals Calculations

1 Implementation of Kohn—Sham LCAO Method in Crystals Calculations 

7 Kohn-Sham LCAO Method for Periodic Systems 

The HF LCAO method for periodic systems was considered in Sections 4.1.5 and 4.1.6. We discuss here the KS LCAO method for crystals in comparison with the HF LCAO approach. The electronic energy of the crystttl (per primitive cell) as calculated within the HF approximation (Ehf) and DFT E pt) can be expressed in terms of the one-electron density matrix (DM) of the crystal dehned as P k) in terms of Bloch sums of AOs, (4.125) or as p R,R ) in coordinate space, (4.126). These expressions are  

H k) is a one-electron operator that describes the motion of an electron in the crystal and is equal to the sum of the kinetic-energy operator and the Coulomb interaction operator between the electron and fixed atomic nuclei and J k) and X k) are the Coulomb and exchange operators, respectively, which describe the interaction of the given electron with the other electrons of the crystal. 

In the LCAO basis, both the Hartree-Fock and Kohn-Sham equations are the same (see (4.57))  

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