Density Functional Theory (DFT) is a theory of interacting many-body systems of electrons. It
has gained widespread use because of the promise of exact solutions for certain properties of the
many-body system using tractable independent-particle methods, together with important
practical realizations based upon approximate functionals. The organization of this tutorial is:
1) The Hohenberg -Kohn theorems and Levy-Lieb construction - proofs that all properties of
interacting electrons are functionals of the ground state density.
2) The ansatz of Kohn and Sham - replacing interacting electrons with a corresponding problem
involving independent fermions, which is the basis of essentially all DFT calculations. The
correspondence is designed to reproduce only the ground state density and energy.
3) Representative classes of approximate functionals for exchange and correlation - LDA, GGA
and orbital dependent functionals.
4) Locality principles that can be used to construct linear scaling algorithms in DFT (and other
many-body approaches) for insulators and scaling principles that should make possible linear
scaling calculations in metals.
5) An example that brings out deep questions and reveals a case where the density is not
sufficient in the Kohn-Sham approach - the electric polarization of a crystal, e.g., a
ferroelectric. Is this a violation of the Hohenberg-Kohn theorem? Of the Kohn-Sham ansatz?
The result of our analysis is that (in the absence of a macroscopic electric field) the
polarization is a bulk quantity, the Hohenberg-Kohn theorem is satisfied, but the Kohn-Sham
expression for the polarization is not correct because it cannot be determined from the bulk
density alone. There is extreme non-locality in the Kohn-Sham density functional, and a
physically meaningful functional can be constructed only by defining a functional of the
polarization density. (See R. M. Martin and G. Ortiz, Phys. Rev. B56, 1124-1140 (1997)
which gives references to earlier work.)
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