Transcript Hier Titel eingeben

Forces and Phonons
within WIEN2k
Claudia Ambrosch-Draxl
Institute for Theoretical Physics
University Graz
[email protected]
Outline
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
Forces in WIEN2k
Atomic Forces
total energy
The Frozen Phonon Approach
fit: Etot (x)E0 E1x E2x 2  .....
F(x)  E1  2E2 x  .....
displacement x
harmonic case:
M 
2
 2Etot
x
2
2 E2
Frozen Phonon Approach
general case:
N atoms: i=1,....,N
3N degrees of freedom
Etot  E0   Ei x i  21   Ei,j xi xj  ....
i,
force constants:
dynamical martrix:
atomic forces:
i, j,
2

  Etot
Fij  Fji    Ei,j
xi x j
  Fij 
diagonalization


D
 MiM j 


 Fi 
E
ui
     phonon
 Ei   Ei,juj  ...
j
Many particle Schrödinger equation
electronic coordinates
ionic coordinates
groundstate wavefunction with respect to fixed ions
Hellmann-Feynman theorem
Many particle system
The Hellmann-Feynman Force
component of the electric field caused by the nuclear charge
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
The Hellmann-Feynman Force
Forces in DFT
Total energy:
Atomic force:
Pulay corrections
Forces in the LAPW Method
Example
G point mode in Si
Computational Effort
yttrium
barium
copper
oxygen
Lattice Vibrations
Example: YBa2Cu3O7
Raman Active Phonons
phonon frequencies [cm-1]
A1g
B2g
B3g
Theory
Experiment
pressure coefficients
[cm-1/GPa]
Ref. [1-2]
optimized
Ref. [3-7]
Ba
105 / 103
123
116-119
Cu(2)
127 / 130
147
145-150
Mode
Theory
Experiment [8]
O(2)-O(3)
312 / 327
338
335-336
O(2)-O(3)
3.6
3.2
O(2)+O(3)
361 / 387
422
435-440
O(2)+O(3)
4.3
4.4
O(4)
513 / 452
487
493-500
O(4)
5.2
5.5
Ba
57
65
70
Cu(2)
133
142
142
O(4)
185
222
210
O(3)
365
389
370
[1] R. E. Cohen et al., Phys. Rev. Lett. 64, 2575 (1990).
O(2)
568
593
579
Ba
72
79
83
[2] R. Kouba et al., Phys. Rev. B 56, 14766 (1997).
[3] T. Strach et al., Phys. Rev. B 51, 16460 (1995).
[4] G. Burns et al., Solid State Commun. 66, 217 (1988).
Cu(2)
133
141
140
O(4)
257
293
303
O(2)
335
372
-
O(3)
524
546
526
[5] K. F. McCarty et al., Phys. Rev. B 41, 8792 (1990).
[6] V. G. Hadjiev et al., Physica C 166, 1107 (1990).
[7] B. Friedl et al., Solid State Commun. 76, 217 (1990).
[8] K. Syassen et al., Physica C 153-155, 264 (1988).
YBa2Cu3O7
Mode
Example: YBa2Cu3O7
oxygen modes
Lattice Vibrations
Ba / Cu modes