Lecture 5: The three basic methods I - Atom Centered Orbital Bases: LCAO and tight-binding
Return to Main Page

Link to pdf file for slides

Link to SIESTA - Kohn-Sham DFT local-orbital code. This code is available free, but you must register with the SIESTA group.

The solution of independent-particle equations - such as the Kohn-Sham equations - can be done in local orbital bases.  This approach provides simple interpretations and is valuable for understanding electronic properties as well as a widely-used method for calculations. This lecture focusses on the basic expressions in terms of localized orbitals, a third derivation of the Bloch theorem, and the empirical tight-binding methods in which the matrix elements are taken as parameters.  Tight-binding provides a useful, intuitive description of electronic states, as shown in important examples.

See text, Chapters 14 and 15. Good references for practical tight-binding calculations are books by Harrison and by Papconstantopoulos (see reference list). 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)
    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
  2. Form of the Hamiltonian
    1. Classic Work of Slater and Koster, 1950's, originally viewed as interpolation
    2. Classification of matrix elements by symmetry
    3. 2-center, 3-center forms
    4. Orthogonal, non-orthogonal. orbitals
  3. Simplfied forms for tight-binding matrix elements
    1. Huckel form
    2. Harrison "Solid State Table"
  4. Accurate Fitting of Matrix elements to data or other calculations
    1. Book by Papconstantopoulos and the NRL web site
  5. Description of Bands
    1. s,p band metals, semiconductors
    2. d band metals, magnetic materials
    3. Transition metal oxides
  6. TBPW - computer program to calculate bands in tight-binding form (as well as plane wave)
    1. Uses same lattice and k-point information and codes as the plane wave code
    2. Find neighbors of each atom
    3. Sum over neighbors to construct tight-binding hamiltonian
    4. Diagonalize tight-binding hamiltonian to find eigenstates
  7. Examples of aplications
    1. Si and C in diamond structure
    2. Graphite
    3. Nanotubes
    4. Ni - d-band transition metal
  8. Total energy in tight-binding form
  9. Conclusions