Lecture 3: Density Functional Theory I
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Link to pdf file for lecture

Reading: Text: Chapters 6,7.

What is so important about DFT that it led to the 1998 Nobel Prize in Chemistry for Walter Kohn?  Why did it happen in 1998? Why in Chemistry? 

DFT is by far the main method used for realistic calculations in solids.  Under what circumstances does it appear to be successful?  When are present approximations not successful?  This lecture and the next are devoted to the theory and description of some typical results.

  1. The Hohenberg-Kohn Theorems
    1. These are exact theorems that ALL properties of electron systems are functionals of the density
    2. Holds the promise of solving many-electron problems WITHOUT dealing with the full many-body wave-function
    3. But no hint of how to accomplish this! Must still solve the many-body problem to find the functional
    4. Analogy with P-V relation of a classical liquid - Legendre transformation between V and P as independent variables
  2. What are reasons that transforming to in terms of density instead` of potential?
    1. The hard parts to calculate are effects of correlation among the electrons. This is intrinsically a function of density (how far apart the the electrons are). It is only an indirect functional of the potential.
  3. Kohn-Sham Auxiliary system
    1. The H-K theorems expressions in terms of many-body expectation values, but is this useful?
    2. Exact expressions in terms of many-body expectation values, but is this useful?
    3. Replace interacting-electron problem with non-interacting particle problem
    4. Require the density be the same as interacting-electron system (not proven in general that this is possible)
    5. Leads to density and ground state energy of the interacting-electron system by solving appropriate non-interacting particle equations called the Kohn-Sham equations
    6. NO other properties are guaranteed to be given correctly by solution of Kohn-Sham equations
  4. Force`and Stress are also given exactly in principle. They are ground state properties given by the "force theorem" ("Helmann-Feynman theorem", even though a small displacement of an atom can be expressed in terms of perturbation theory involving excitations. Similarly for the "stress theorem"
    1. This is valid in an interacting particle system. It is independent of the HK theorem, but the analogy can be instructive.
  5. Conclusions