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Forces and Phonons within WIEN2k. Claudia Ambrosch-Draxl Institute for Theoretical Physics University Graz claudia.ambrosch@uni-graz.at. Atomic Forces. Phonons. Molecular Dynamics. Geometry optimization The frozen phonon approach The Hellmann-Feynman theorem
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Forces and Phononswithin WIEN2k Claudia Ambrosch-DraxlInstitute for Theoretical PhysicsUniversity Grazclaudia.ambrosch@uni-graz.at
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
The Frozen Phonon Approach fit: harmonic case:
Frozen Phonon Approach general case: N atoms: i=1,....,N 3N degrees of freedom force constants: dynamical martrix: atomic forces:
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
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
Total energy: Forces in DFT Atomic force: Pulay corrections
G point mode in Si Example
Example: YBa2Cu3O7 yttrium barium copper oxygen Lattice Vibrations
Raman Active Phonons YBa2Cu3O7
Example: YBa2Cu3O7 Ba / Cu modes Lattice Vibrations oxygen modes