The lecture follows the discussion in Thijssen, Chapter 9 except that we postphone the discussion of molecular dynamics methods (especially 9..2,9.3) until next time. The subject is iterative methods to project out the low energy states of the time-independent Schrodinger Eq., which are also related to methods for evolution of the time-dependent Schrodinger Eq.
Also note the relation to the imaginary time projection used in the Monte Carlo solutions of many-body problems to be discussed later

A set of notes prepared by Prof. Martin is also provided.

Note: Several possible PROJECTS FOR STUDENTS

Outline

1. Key point #1: replace matrix diagonalization by iterative methods for solving the Schrodinger Eq.
   • All iterative methods generate an "iterative space" by a starting vector psi₀ and an operator (such as H): {psi₀; psi₁ = H psi₀; psi₂ = H psi₁; ...}
     • Called a "Krylov Subspace"
     • May use only the last vectors created, several vectors, or the complete subspace generated up to step n.
   • Solution by minimization of the energy
     • Steepest descent and conjugate gradient methods
     • Constraint of orthonormalization
     • Widely used in electronic structure codes today
   • Solution by minimization of a "residual"
     • Converges to states with eigenvalues closest to the specified energy
     • Does not require constraint of orthonormalization
     • Widely used in electronic structure codes today
   • Solution by a "fake" time dependence
     • Equivalent to evolution in "imaginary time" or inverse temperature
     • Similarities differences from real time dynamics
     • Example of use in quantum electronic device simulation programs
     • Will be used later in Monte Carlo solutions
   • Lanczos - very general method - we will postphone to the section on many body methods
   • Comments on relation to the widely used Car-Parrinello methods for finding the ground state by a fictious classical dynamics
2. Key Point #2 - efficient operation of H*psi
   • Sparse matrix operations - needed for large bases
   • H never explicitly stored as 2x2 matrix
   • Can be accomplished in plane wave methods by FFTs
   • H represented by banded matrices in real space methods