2. Claudia Ambrosch-Draxl Institute for Theoretical Physics University Graz claudia.ambrosch@uni-graz.at Forces and Phonons within WIEN2k

3. Outline Forces in WIEN2k Atomic Forces Geometry optimization The frozen phonon approach The Hellmann-Feynman theorem Forces within density functional theory Pulay corrections Phonons Computational effort Examples Molecular Dynamics Inputs Limitations

4. The Frozen Phonon Approach fit: harmonic case:

5. Frozen Phonon Approach atomic forces: 3N degrees of freedom N atoms: i=1,....,N force constants: dynamical martrix: general case:

6. The Hellmann-Feynman Force Many particle Schrödinger equation Many particle system electronic coordinatesionic coordinates groundstate wavefunction with respect to fixed ions Hellmann-Feynman theorem

7. The Hellmann-Feynman Force Hellmann-Feynman force: total classical Coulomb force acting on the nucleus  stemming from all other charges of the system = electrostatic force stemming from all other nuclei + electrostatic force stemming from the electronic charges Many particle system component of the electric field caused by the nuclear charge

8. Forces in DFT Atomic force: Pulay corrections Total energy:

9. Forces in the LAPW Method

10.  point mode in Si Example

11. Computational Effort

12. Example: YBa 2 Cu 3 O 7 yttrium barium copper oxygen Lattice Vibrations

13. Raman Active Phonons YBa 2 Cu 3 O 7

14. Example: YBa 2 Cu 3 O 7 Lattice Vibrations oxygen modes Ba / Cu modes