The Kohn-Sham Single-particle Equations

Within the framework of Kohn-Sham (KS) DFT, the intractable many-body problem of interacting electrons in a static external potential is reduced to a tractable problem of noninteracting electrons moving in an effective potential. The functional in (7.8) is written as a fictitious density functional of a noninteracting system 

one can solve the so-called Kohn-Sham equations of this auxihary noninteracting system with the effective Hamiltonian 

There are only single-particle operators in (7.11). Therefore, the solution to the Schrodinger equation for a model system of noninteracting electrons can be written exactly as a single Slater determinant S = p, p2,.. , PnV where the single-particle orbitals pi are determined as solutions of the single-particle equation 

Furthermore, p r) = where the summation runs over the N orbitals 

HK theorems and KS equations can be extended to the spin-polarized systems where the electron- density components pa T),Pi r) for spin-up and spin-down orbitals differ i.e. the spin-density p (r) = Pc t) — pp r) is nonzero. 

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DFT is then used to calculate the ground state electronic distribution for interacting electrons in this external potential. This is achieved by solving the Kohn-Sham single-particle equations,... 

The computational method used to study the infinite, periodic polymers has been described in detail elsewhere [4,22,23] and shall therefore only briefly be described here. It is based in the theorem of Hohemberg and Kohn [31], where the Kohn-Sham single-particle equations [32]... 

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