Density Functional Theory and the Kohn-Sham Equation

2 Density Functional Theory and the Kohn-Sham Equation 

In the yearly 1970s, a new electronic structure approach found its way from the physics to the chemistry community in the form of density functional theory (DFT), where the total energy of an electronic system is expressed as a functional of the total electronic density. The basic notion in DFT that the energy of an electronic system can be expressed in terms of its electronic density is almost as old as quantum mechanics and dates back to the early work by P.A.M. Dirac [32], E. Fermi [33] and L.H. Thomas [34], 

Slater [35] embrace in part this idea in 1951 when he replaced the Hartree-Fock exchange energy appearing as the last term on the right hand side of (2) with the simple expression 

The Thomas-Fermi method and the Xa scheme were at the time of their inceptions considered as useful models based on the notion that the energy of an electronic system can be expressed in terms of its density. A formal proof of this notion came in 1964 when it was shown by Hohenberg and Kohn [38] that there is a unique relation between density and energy. The year after Kohn and Sham put forward a practical variational DFT approach in which they replaced E of (2) with a combined exchange and correlation term 

N (Kohn-Sham) orbitals can be expanded into atomic orbitals according to (4). Furthermore, the expansion coefficients C, can be determined by requiring that they optimize the total (Kohn-Sham) energy. This results in the (Kohn-Sham) matrix equation similar to (5) 

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In this section, we discuss first the basic formalism of density functional theory, and the Kohn-Sham equations whose solution yields the ground state density and energy of a system. We next describe a two-electron system, called Hooke s atom, which has the highly unusual feature of an analytic ground state, and is therefore of considerable pedagogical value. Lastly we discuss the notation and organization of this article. 

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