1. Physics 250-06 “Advanced Electronic Structure” Frozen Phonon and Linear Response Calcuations of Lattice Dynamics Contents: 1. Total Energies and Forces 2. Density Functional Linear Response. 3. Applications.

2. Frozen Phonon Method Given vector q generates displacements field If a supercell can be found such that the vector q becomes reciprocal lattice vector in the new structure, one restores periodicity and can study lattice dynamics with the frozen phonon method

3. Evaluating the forces Trying to evaluate first derivative of the energy analytically would mean that i.e numerically more stable

4. Hellmann-Feynman Theorem DFT expression for the total energy would assume Non-self-consistency force which is difficult to evaluate!

5. Pulay forces due to incomplete basis sets Change in the eigenvalues only when (Pulay, 1967)

6. However, if We obtain If the basis does not depend on atomic positions Pulay force disappears! If it is centered on atoms

7. Outcome: Pulay forces need to be kept when working with atom centered basis sets (TB, LAPWs, LMTOs, LCAOs, etc) They disappear for plane wave basis sets, not because Plane waves are complete, but because they don’t depend on atomic positions (even obviously incomplete basis with a single plane wave will not have Pulay force!) Pulay forces are very large! If neglected, the Hellman-Feynman Force alone can be 1-2 order of magnitude larger than the real force. On the other hand, since numerical derivatives of the total energy produce correct numerical force, the HF+Pulay force obtained analytically is very accurate!

8. Linear Response Lattice Dynamics Main advantage: no need for supercells, works for any q! Consider external perturbation made of two traveling waves

9. External perturbation causes change in the electronic density where change in the electronic (KS) wave function is One then obtains (Sham, 1969):

10. One needs to assume self-consistency because which changes and obtains new In other words:

11. Problem with standard perturbation theory Main problem is the perturbation theory using original basis set: Consider rigid shift of the lattice: Only if basis is complete How to repair perturbation theory?

12. If force is given by The dynamical matrix is the next variation (Sham, 1969) We obtained a compact linear response expression for the dynamical matrix valid for any wave vector q. Unfortunately, it is only valid for plane wave basis sets. when using atom centered basis sets, this Hellmann-Feynman based formula has huge errors (~1000%)

13. Perturbation Theory with adjustable basis If we expect Where the important term is given by the change in the basis! This is absent in standard perturbation theory (SS, PRL 1992)

14. We obtain which goes back to standard perturbation theory if basis is complete: second term is equal zero, while first and third terms cancel out!

15. Consider acoustic sum rule Equal zero automatically! We never used completeness property to show that acoustic sum rule is satisfied.

16. Expression for dynamical matrix is more complicated. Basically One needs to start from the energy and gets the force which accounts for the Pulay part Dynamical Matrix Now the dynamical matrix is the second order variation

17. Dynamical matrix can be thought as a functional of first order Change in charge density. It is stationary with respect to variations of this quantity as directly follows from DFT! If that is the case, any change in the dynamical matrix with Respect to external parameter can be computed 2n+1 Theorem (Gonze, et.al PRL 1992) Equal to zero! Thus, here is the 2n+1 theorem: knowledge of ground state Density allows to evaluate energy and force (n=0), Knowledge of first order change in density (n=1) allows to evaluate dynamical matrix and third-order anharmonicity constants (2n+1=3 here), knowledge of second-order density variation will allow to evaluate anharmonicity coefficents of Forth and fifth order (2n+1=5), etc.

18. (after SS, PRB1996)

24. Superconductivity in MgB2 was recently studied using density functional linear response (after Y. Kong et.al. PRB 64, 020501 (R) 2002)

25. Phosphorus under pressure and its implications for spintronics. In collaboration with Ostanin, Trubitsin, Staunton, PRL 91, 087002 (2003)

26. Phonons and Linewidths in MgCNi 3 <- Carbon modes <- Mg based modes <- Ni based modes Harmonic ~ 0.9 from all modes except for soft mode. Soft mode cannot be neglected. (after Ignatov, SS, Tyson, PRB 2004)

27. Nearly Instable Mode at q=(0.5,0.5,0.0) Soft Mode view Octahedral interstitial viewSoft mode as antibreating Antiperovskite Lattice

28. EXAFS in MgCNi 3 Recent EXAFS data (Ignatov et.al, PRB 2003) can be used: MgCNi 3 EXAFS does not show static distortions in MgCNi 3 EXAFS shows a double well with depth =20 K,  0 =70K, and  1 =150K

29. M point Frozen Phonon (after Ignatov, SS, Tyson, PRB 2004)

30. Phonons in  -Pu C 11 (GPa) C 44 (GPa) C 12 (GPa) C'(GPa) Theory 34.56 33.03 26.81 3.88 Experiment 36.28 33.59 26.73 4.78 (after Dai, Savrasov, Kotliar,Ledbetter, Migliori, Abrahams, Science, 9 May 2003) (experiments from Wong et.al, Science, 22 August 2003)

31. Phonons in doped CaCuO2 (after SS, Andersen, PRL 1996)