Phonons from density-functional-perturbation theory: Difference between revisions

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The phonon calculations using [[Phonons:_Theory#Density_functional_perturbation_theory|density-functional-perturbation theory (DFPT)]] are carried out by setting {{TAG|IBRION}}=7 or 8 in the {{FILE|INCAR}} file.
{{NB|mind|Only zone-center (Γ-point) frequencies are calculated.}}


This approach determines the Hessian matrix (matrix of second derivatives with respect to the position of the ions) using density-functional-perturbation theory (DFPT). To use DFPT the tag {{TAG|IBRION}} has to be set in the {{TAG|INCAR}} file. Setting {{TAG|IBRION}}=7 does not exploit the symmetry of the crystal, whereas {{TAG|IBRION}}=8 uses symmetry to reduce the number of needed displacements. The output is similar as for [[IBRION#ibrion56|'''IBRION'''=5 and 6]]. Specifically, the second derivates of the total energy with respect to ionic displacements (interatomic force constants) and the mixed second derivative with respect to the strain and the ionic displacement (internal strain tensor) are evaluated. Although the contributions from the ionic relaxations to the elastic tensor are calculated, the ion-clamped elastic tensor (rigid ion) is not determined. Born effective charges, piezoelectric constants, and the ionic contributions to the dielectric tensor are calculated if {{TAG|LEPSILON}}=.TRUE. is specified in the {{TAG|INCAR}} file.
In general, the DFPT routines in VASP are somewhat rudimentary and only support displacements commensurate with the supercell, i.e., so-called q=0 phonons.
{{NB|mind|Only zone-center (Γ-point) frequencies are calculated.}}
Therefore, the code offers few advantages over the [[Phonons_from_finite_differences|finite differences methods]].
In general, the DFPT routines in VASP are somewhat rudimentary and only support displacements commensurate with the supercell, i.e., so-called q=0 phonons. In other words, VASP can only determine phonon frequencies at the Gamma point of the supercell. Therefore, the code offers few advantages over the finite differences methods discussed above. In particular, the linear response is limited to LDA and GGA functionals and it does not determine the elastic tensors, since the linear response with respect to the strain tensor is not implemented. The only advantage of the linear response routines is that they eliminate the need to choose the magnitude of the finite displacement. Therefore, it might be helpful to first calculate phonon frequencies using linear response and then switch to [[Phonons from finite differences|finite differences]] and determine the largest displacement that will produce results compatible with the linear response routines.
In particular, the DFPT routines are limited to LDA and GGA functionals and it does not determine the elastic tensors, since the perturbation with respect to the strain tensor is not implemented.
The only advantage of the linear response routines is that they eliminate the need to choose the magnitude of the finite displacement {{TAG|POTIM}}.
Therefore, it might be helpful to first calculate phonon frequencies using linear response and then switch to [[Phonons from finite differences|finite differences]] and determine the largest displacement that will produce results compatible with the linear response routines.


A few technical comments are in order at this point. VASP solves the [[Phonons:_Theory#IonSternheimer|linear Sternheimer equation]] to determine the linear response of the orbitals. Hence, unoccupied orbitals are not required. Internally, the VASP routines for linear response rely on finite differences in two places:
A few technical comments are in order at this point. VASP solves the [[Phonons:_Theory#IonSternheimer|linear Sternheimer equation]] to determine the linear response of the orbitals.  
Hence, unoccupied orbitals are not required.  
Internally, the VASP routines for linear response rely on finite differences in two places:
#  The first place is the determination of the second derivative of the [[:Category:Exchange-correlation_functionals|exchange-correlation functional]]: Since most functionals do not support an algebraic determination of second derivatives, VASP always resorts to finite differences to determine the second-order change of the exchange correlation-potential and the PAW one-center terms for each atomic displacement.  
#  The first place is the determination of the second derivative of the [[:Category:Exchange-correlation_functionals|exchange-correlation functional]]: Since most functionals do not support an algebraic determination of second derivatives, VASP always resorts to finite differences to determine the second-order change of the exchange correlation-potential and the PAW one-center terms for each atomic displacement.  
# Second, after VASP has determined the first-order change of the orbitals, it computes all second derivatives using finite displacements.  
# Second, after VASP has determined the first-order change of the orbitals, it computes all [[Phonons:_Theory#ForceConstantsDFPT|second derivatives using finite displacements]].
 
To do this, VASP displaces the selected atom in the selected directions [[Phonons:_Theory#FiniteDiffWF|adds the calculated linear response to the orbitals]], and finally determines the differences in the forces and the stress tensor for positive and negative displacements.
It can be shown that this yields exactly the [[Phonons:_Theory#InternalStrainDFPT|second-order force constants]] and the [[Phonons:_Theory#InternalStrainDFPT|internal strain tensor]], respectively.
 
== Input ==
 
To use DFPT the tag {{TAG|IBRION}}=7 or 8 has to be set in the {{TAG|INCAR}} file.
There are two options to use the DFPT routines to compute the second order force-constants
* {{TAG|IBRION}}=7, all the atoms are displaced in all three Cartesian directions,
* {{TAG|IBRION}}=8, uses symmetry to reduce the number of displacements.
 
If {{TAG|LEPSILON}}=.TRUE. is specified in the {{TAG|INCAR}} file then Born effective charges, piezoelectric constants, and the ionic contributions to the dielectric tensor are calculated.
 
== Output ==
 
The output is similar as for [[IBRION#ibrion56|'''IBRION'''=5 and 6]].
 
The second derivates of the total energy with respect to ionic displacements (interatomic force constants) are computed,
the [[Phonons:_Theory#DynamicalMatrix|dynamical matrix]] is constructed, diagonalized and the phonon modes and frequencies of the system are reported.


To do this, VASP displaces the selected atom in the selected directions adds the calculated linear response to the orbitals, and finally determines the differences in the forces and the stress tensor for positive and negative displacements. It can be shown that this yields exactly the [[Phonons:_Theory#InternalStrainDFPT|second-order force constants]] and the [[Phonons:_Theory#InternalStrainDFPT|internal strain tensor]], respectively.
The mixed second derivative with respect to the strain and the ionic displacement (internal strain tensor) are evaluated and reported.
Although the contributions from the ionic relaxations to the elastic tensor are calculated, the ion-clamped elastic tensor (rigid ion) is not determined because the perturbation with respect to the strain tensor is not implemented.


Furthermore, the [[Phonons:_Theory#BornChargeDFPT|Born effective charges]] are determined "analytically" by contracting the linear response of the orbitals over the "polarization" vector Eq. (30) in Ref. {{cite|gajdos:prb:2006}}. These should agree well with the Born effective charges that were previously determined when the linear response with respect to external fields {{TAG|LEPSILON}}=.TRUE. was calculated (there are two different routes to calculate mixed derivatives). The final summary output towards the end of the {{TAG|OUTCAR}} file writes the Born effective charges determined from the linear response with respect to external fields.
Furthermore, the [[Phonons:_Theory#BornChargeDFPT|Born effective charges]] are determined analytically by contracting the linear response of the orbitals over the "polarization" vector Eq. (30) in Ref. {{cite|gajdos:prb:2006}}.
These should agree well with the Born effective charges that were previously determined when the linear response with respect to external fields {{TAG|LEPSILON}}=.TRUE. was calculated (there are two different routes to calculate mixed derivatives).
The final summary output towards the end of the {{TAG|OUTCAR}} file writes the Born effective charges determined from the linear response with respect to external fields.


It is possible to [[Computing_the_phonon_dispersion |obtain the phonon dispersion at different '''q''' points]] by computing the force constants on a sufficiently large supercell and Fourier interpolating the dynamical matrices in the primitive cell.
It is possible to [[Computing_the_phonon_dispersion |obtain the phonon dispersion at different '''q''' points]] by computing the force constants on a sufficiently large supercell and Fourier interpolating the dynamical matrices in the primitive cell.


It is also possible to use phonopy{{cite|phonopy}} to post-process the results of a finite differences calculation done with VASP.{{cite|phonopy_dfpt}}
It is also possible to use phonopy{{cite|phonopy}} to use the results of a density-functional-perturbation theory calculation done with VASP.{{cite|phonopy_dfpt}}
{{NB|mind|{{TAG|IBRION}}{{=}}7 and {{TAG|IBRION}}{{=}}8 are supported from VASP.5.1 and later versions.}}
{{NB|mind|{{TAG|IBRION}}{{=}}7 and {{TAG|IBRION}}{{=}}8 are supported from VASP.5.1 and later versions.}}


Revision as of 07:37, 12 August 2022

The phonon calculations using density-functional-perturbation theory (DFPT) are carried out by setting IBRION=7 or 8 in the INCAR file.

Mind: Only zone-center (Γ-point) frequencies are calculated.

In general, the DFPT routines in VASP are somewhat rudimentary and only support displacements commensurate with the supercell, i.e., so-called q=0 phonons. Therefore, the code offers few advantages over the finite differences methods. In particular, the DFPT routines are limited to LDA and GGA functionals and it does not determine the elastic tensors, since the perturbation with respect to the strain tensor is not implemented. The only advantage of the linear response routines is that they eliminate the need to choose the magnitude of the finite displacement POTIM. Therefore, it might be helpful to first calculate phonon frequencies using linear response and then switch to finite differences and determine the largest displacement that will produce results compatible with the linear response routines.

A few technical comments are in order at this point. VASP solves the linear Sternheimer equation to determine the linear response of the orbitals. Hence, unoccupied orbitals are not required. Internally, the VASP routines for linear response rely on finite differences in two places:

  1. The first place is the determination of the second derivative of the exchange-correlation functional: Since most functionals do not support an algebraic determination of second derivatives, VASP always resorts to finite differences to determine the second-order change of the exchange correlation-potential and the PAW one-center terms for each atomic displacement.
  2. Second, after VASP has determined the first-order change of the orbitals, it computes all second derivatives using finite displacements.

To do this, VASP displaces the selected atom in the selected directions adds the calculated linear response to the orbitals, and finally determines the differences in the forces and the stress tensor for positive and negative displacements. It can be shown that this yields exactly the second-order force constants and the internal strain tensor, respectively.

Input

To use DFPT the tag IBRION=7 or 8 has to be set in the INCAR file. There are two options to use the DFPT routines to compute the second order force-constants

  • IBRION=7, all the atoms are displaced in all three Cartesian directions,
  • IBRION=8, uses symmetry to reduce the number of displacements.

If LEPSILON=.TRUE. is specified in the INCAR file then Born effective charges, piezoelectric constants, and the ionic contributions to the dielectric tensor are calculated.

Output

The output is similar as for IBRION=5 and 6.

The second derivates of the total energy with respect to ionic displacements (interatomic force constants) are computed, the dynamical matrix is constructed, diagonalized and the phonon modes and frequencies of the system are reported.

The mixed second derivative with respect to the strain and the ionic displacement (internal strain tensor) are evaluated and reported. Although the contributions from the ionic relaxations to the elastic tensor are calculated, the ion-clamped elastic tensor (rigid ion) is not determined because the perturbation with respect to the strain tensor is not implemented.

Furthermore, the Born effective charges are determined analytically by contracting the linear response of the orbitals over the "polarization" vector Eq. (30) in Ref. [1]. These should agree well with the Born effective charges that were previously determined when the linear response with respect to external fields LEPSILON=.TRUE. was calculated (there are two different routes to calculate mixed derivatives). The final summary output towards the end of the OUTCAR file writes the Born effective charges determined from the linear response with respect to external fields.

It is possible to obtain the phonon dispersion at different q points by computing the force constants on a sufficiently large supercell and Fourier interpolating the dynamical matrices in the primitive cell.

It is also possible to use phonopy[2] to use the results of a density-functional-perturbation theory calculation done with VASP.[3]

Mind: IBRION=7 and IBRION=8 are supported from VASP.5.1 and later versions.

References

  1. ↑ M. Gajdoš, K. Hummer, G. Kresse, J. Furthmüller, and F. Bechstedt, Phys. Rev. B 73, 045112 (2006).
  2. ↑ http://phonopy.github.io/phonopy/index.html (2022).
  3. ↑ http://phonopy.github.io/phonopy/vasp-dfpt.html (2022).
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