Forces and Phonons within WIEN2k

2. Forces and Phononswithin WIEN2k Claudia Ambrosch-DraxlInstitute for Theoretical PhysicsUniversity Grazclaudia.ambrosch@uni-graz.at

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

4. The Frozen Phonon Approach fit: harmonic case:

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

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

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

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

9. Forces in the LAPW Method

10. G point mode in Si Example

11. Computational Effort

12. Example: YBa2Cu3O7 yttrium barium copper oxygen Lattice Vibrations

13. Raman Active Phonons YBa2Cu3O7

14. Example: YBa2Cu3O7 Ba / Cu modes Lattice Vibrations oxygen modes