The lecture follows the Classnotes on Periodic Crystals discussed in the previous lecture, Classnotes on Mathieu Equation and the F90 program given in the Fortran Directory for the class.

Outline

1. Matrix equations in a plane wave basis
   • Reciprocal Lattice G(n1,n2.n3)
   • Crystal momentum k in one cell of Reciprocal Lattice
   • The Bloch theorem
   • Hamiltonian matrix in G,G' for each k
   • Hermitian complex or real symmetric matrices
   • Kinetic energy - diagonal |k+G|²
   • Potential - form factor, structure factor
     V(G-G') = S(G-G') V_atom (|G-G'|)
2. Needed information to specify problem
   • translation vectors, a(i,j)
   • positions, types of atoms in unit cell
   • form of potential
   • number of electrons
3. Output Information
   • Bands - eigenvalues as function of k in Brillouin Zone
   • eigenvectors, .....
4. Fortran 90 program
   • Modules for realted operations required
   • Allocation/deallocation of memory
   • Subroutines for operations
   • Keyword input - read by "Taghandler"
5. Subroutines needed for all crystals
   • Reading information
   • Generation of the reciprocal lattice basis vectors
6. Subroutines specialized to plane waves
   • Generating the reciprocal space basis within the spherical cutoff
   • Setting up hamiltonian matrix for each k
7. Packages for diagonalization - the same for all bases
   • Methods discussed in Numerical Recipies
   • Packages available for general symmetric matrices
   • Diagonalization routine from Lapack - on EWS workstations
   • Information on Lapack routines on network
8. What to expect
   • One dimensional examples - Mathieu Equation
   • Nearly free electron metals
   • Insulators like He
   • Semiconductors like silicon, diamond
   • Examples of bands - empirical fitting by Chelikowsky & Cohen
   • General features of metals, insulators
9. Issues involved in plane calculations
   • How many plane waves are needed?
   • Huge numbers for core electrons!
   • Sucessful only because of pseudopotentials
   • Theory of pseudopotentials - not done here - PROJECT FOR SOMEONE?