Lecture 11: Atom Centered Orbital Bases: Full DFT calculations
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Link to HTML version of Power Point Slides for Lecture 11

Link to TBPW - simple pedagogical plane wave code.

Link to SIESTA - Kohn-Sham DFT local-orbital code. This code is available only upon request to the developers. Anyone in this class can get the source code since R. Martin is on of the development team.

The solution of the self-consistent Kohn-Sham equations in local orbital bases is a widely-used method, with many codes used in chemistry and a growing use in physics applications.  This approach provides simple interpretations and is often much more efficient than plane waves.  However, more care is needed in choosing the basis. If the orbitals are chosen to be atom centered then this is the self-consistent version of the empirical tight-binding methods discussed before.  We will discuss the general features of the approach and describe specific calculations using the SIESTA code as examples.

See text, Chapters 14 and 15. A good reference for The general formulation and methods used in chemistry is: A. Szabo and N. S. Ostlund, "Modern Quantum Chemistry" (Paperback by Dover, 1996)

  1. Kohn-Sham Equations in a  local orbital basis
    1. Calculate matrix elements of H in real space:  Hij = <i|H|j>
    2. In general orbitals are not orthogonal
      • Thus Sij = <i|j>
      • Must solve generalized eigenvalue equation H-ES = 0
    3. Advantages
      • Can have a small basis
      • Physically interpretable (like empirical tight-binding)
      • Problems with eigenvalues
    4. Disadvantages
      • Generalized Eigenvalue equation
      • Difficult to converge - overcomplete if one attempts to converge with many functions
      • Must express potential in form convenient for calculating matrix elements

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  2. Gaussian Bases
    1. All integrals analytic (if potential is expressed as Coulomb or Gaussians)

      2. (See Szabo and Ostlund)
    2. Poisson equation solved using analytic potentials due to Gaussians
    3. Widely used in chemistry - packages like Gaussian, Dmol
    4. Can describe core and valence electrons
    5. Many Gaussians required on each atom

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  3. Gaussian type orbitals  GTO-nG
    1. Sum of n Gaussians that approximates an atomic-like radial function function
      1. Closer to atomic function than a single Gaussian
      2. Vary only one coefficient per GTO
      3. Integrals analytic
    2. Standard Basis in Chemistry (See Szabo and Ostlund)

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  4. Numerical basis orbitals
    1. Tabulated on a radial grid
    2. Used in Chemistry program package Dmol
    3. New Solid State program SIESTA written by Spanish group (our collaborators)
    4. Must calculate all integrals numerically
    5. Must express potential numerically (And Solve Poisson Equation to get Hartree potential)
    6. Done in SIESTA on a grid exactly like in the plane wave calculations

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  5. Examples of results for molecules and solids
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  6. Conclusions