Lecture 7: Atoms and Peudopotentials
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Spherical atoms are the simplest systems for application of DFT, and calculations on atoms provide tests of approximate forms of the exchange-correlation functional.  In addition, atomic calculations provide the framework for generation of "ab initio" pseudopotentials that are essential for important calculations on molecules and solids.

  • Atoms
    1. The Kohn-Sham Equations for atoms
      1. Self-consistent radial equation
      2. Examples of solutions
      3. A difficult test for functionals
        • Obviously not very close the a homogeneous electron gas!
        • Recently (1990's and 2000's) improved functionals have made major advances in accuracy
        • Very important in Chemistry
    2. Why is the local approximation reasonable even in an atom?
  • Pseudopotentials
    1. The big picture: "Partitioning" the core and valence electrons
      1. Separation of core and valence electrons in independent electron methods
      2.  Not so clear in full theory where all electrons are identical
    2. Why eliminate core electrons?
      1. MANY aspects become much easier
      2. Hellmann-Feynman Forces, Car-Parrinello type simulations, Quantum Monte Carlo, Green's function many-body methods
    3. Eliminating the core electrons with pseudopotentials
      1. The OPW method an the Phillips-Kleinman Pseudopotential Operator
        1.  The Cancellation Theorem and the feasibility of the plane wave approach
        2. Empirical Pseudopotentials
      2. Model potentials - historically the first and now the modern form used today
      3. "Ab Initio" Norm-Conserving Pseudopotentials
        1. Relation to logarithmic energy derivatives
        2. Kerker Form of potentials  (Troullier-Martins extension)
        3. See ElectronicStructure.org for pseudopotential software.
          For example, a site for a working Troullier-Martins program:  http://bohr.inesc.pt/~jlm/pseudo.html
          and a site for on-line generation of Troullier-Martins potentials:  http://www.tddft.org/programs/octopus/pseudo.php
    4. Examples of results for solids
    5. Extensions
      1. Extended Norm Conservation
      2. Many-Body quasiparticle pseudopotentials
    6. Conclusions