1. Brief outline of classical molecular dynamics
   • Verlet algorithm for numerical integration of coupled F=ma equations
   • Simulated annealing to lowest energy state
2. Forces on atoms from the electrons - the Hellmann-Feynman theorem
   • The electron density alone is sufficient to determine forces
3. The Car-Parrinello Methods
   • References
     R. Car and M. Parrinello, Phys. Rev. Lett. 55, 2471 (1985)
     (the original - very readable)
     J. Garner, Computers in Physics 4, 395 (1990). Nice discussion; simple exampl es.
     M. Payne, et. al., Rev Mod. Phys. 64, 1045, (1992). Most complete recent large review
     
   • Solution of quantum equations by classical molecular dynamics
     • Solving quantum equations by using a "fictitious mass" and a fictitious set of Newtonian equations
     • Annealing by reducing the temperature leads system to ground state - the same solution as found by matrix diagonalization
     • Examples of simple quantum systems
       • A simple 2 x 2 Hamiltonian - solution by Car-Parrinello molecular dynamics and "simulated annealing"
       • Particle in a box
   • FFT's instead of dense matrix multiplication
   • Examples of use in large scale calculations
     • "Ab Initio" Molecular dynamics of solids and liquids
     • The real Newtonian equations for atoms with forces from electrons from the Helmann-Feynman theorem
     • Example: melting of Carbon

Next time: Continue Car-Parrinello discussion as needed
Conclusion of first part of course on independent particle problem