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

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In density functional theory we solve the Hamiltonian
The phonon calculations using [[Phonons:_Theory#Density_functional_perturbation_theory|density-functional-perturbation theory (DFPT)]] are carried out by setting [[IBRION#ibrion78|'''IBRION'''=7 or 8]] in the {{FILE|INCAR}} file.
{{NB|mind|Only zone-center (Γ-point) frequencies are calculated.}}


:<math>
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.
H(\mathbf{k}) | \psi_{n\mathbf{k}} \rangle=
Therefore, this approach offers few advantages over computing [[Phonons_from_finite_differences|phonons from finite differences]].
e_{n\mathbf{k}}S(\mathbf{k}) | \psi_{n\mathbf{k}} \rangle
In particular, the DFPT routines are limited to LDA and GGA functionals, and they do not determine the elastic tensors, since the perturbation with respect to the strain tensor is not implemented.
</math>
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 within DFPT and then switch to the [[Phonons from finite differences|finite differences approach]] in order to determine the largest displacement that will produce results compatible with the linear response routines.


Taking derivatives with respect to a perturbation <math>\lambda</math> we obtain the Sternheimer equation
A few technical comments: 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.
# Second, after VASP has determined the first-order change of the orbitals, it computes all [[Phonons:_Theory#ForceConstantsDFPT|second derivatives using finite displacements]].


:<math>
To this end, 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.
\left[ H(\mathbf{k}) - e_{n\mathbf{k}}S(\mathbf{k}) \right]
It can be shown that this yields precisely the [[Phonons:_Theory#InternalStrainDFPT|second-order force constants]] and the [[Phonons:_Theory#InternalStrainDFPT|internal strain tensor]], respectively.
| \partial_\lambda\psi_{n\mathbf{k}} \rangle
=
-\partial_\lambda \left[ H(\mathbf{k}) - e_{n\mathbf{k}}S(\mathbf{k})\right]
| \psi_{n\mathbf{k}} \rangle
</math>


Once the derivative of the orbitals is computed from the Sternheimer equation we can write
== Input ==
:<math>
| \psi^{R^a_i}_\lambda \rangle =  
| \psi \rangle +
\lambda | \partial_{R^a_i}\psi \rangle
</math>


The force constants are then computed using
To use DFPT, set the tag [[IBRION#ibrion78|'''IBRION'''=7 or 8]] in the {{TAG|INCAR}} file.
:<math>
There are two options for using the DFPT routines to compute the second-order force-constants
\Phi^{ab}_{ij}=
* {{TAG|IBRION}}=7, all the atoms are displaced in all three Cartesian directions,
\frac{\partial^2E}{\partial R^a_i \partial R^b_j}=
* {{TAG|IBRION}}=8, uses symmetry to reduce the number of displacements.
-\frac{\partial F^a_i}{\partial R^b_j}
\approx
-\frac{
\left(
  \mathbf{F}[\{\psi^{R^b_j}_{\lambda/2}\}]-
  \mathbf{F}[\{\psi^{R^b_j}_{-\lambda/2}\}]
\right)^a_i}{\lambda}
</math>
where <math>\mathbf{F}</math> yields the forces for a given set of orbitals.


At the end of the calculation if {{TAG|LEPSILON}}=.TRUE., the Born effective charges are computed using Eq. (42) of Ref. {{cite|gonze:prb:1997}}.
If {{TAG|LEPSILON}}=.TRUE. or {{TAG|LCALCEPS}}=.TRUE., additional dielectric properties are computed.  


:<math>
== Output ==
Z^{a*}_{ij} =  
2 \frac{\Omega_0}{(2\pi)^3}
\int_\text{BZ}
\sum_n
\langle
  \partial_{R^a_i}\psi_{n\mathbf{k}} | \nabla_{\mathbf{k}} \tilde{u}_{n\mathbf{k}}
\rangle d\mathbf{k}
</math>


where <math>a</math> is the atom index, <math>i</math> the direction of the displacement of atom and <math>j</math> the polarization direction.
The second derivates of the total energy with respect to ionic displacements (interatomic force constants) are computed,
The results should be equivalent to the ones obtained using {{TAG|LCALCEPS}} and {{TAG|LEPSILON}}.
the [[Phonons:_Theory#DynamicalMatrix|dynamical matrix]] is constructed, diagonalized and the phonon modes and frequencies of the system are reported after the following lines:


When {{TAG|IBRION}}=7 or 8 VASP solves the Sternheimer equation above with an ionic displacement perturbation.
  Eigenvectors and eigenvalues of the dynamical matrix
If {{TAG|IBRION}}=7 no symmetry is used and the displacement of all the ions is computed.
  ----------------------------------------------------
When {{TAG|IBRION}}=8 only the irreducible displacements are computed with the physical quantities being reconstructed by symmetry.
 
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.
 
It is possible to [[Computing_the_phonon_dispersion_and_DOS|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 use the results of a density-functional-perturbation theory calculation done with VASP.{{cite|phonopy_dfpt}}
{{NB|mind|{{TAG|IBRION}}{{=}}7 or 8 are supported by VASP.5.1 and later versions.}}
 
== Related tags and sections ==
{{TAG|IBRION}},
{{TAG|LEPSILON}},
[[Phonons: Theory]],
[[Phonons from finite differences]]


== References ==
== References ==
<references/>
<references/>


[[Category:Phonons]]
[[Category:Phonons]][[Category:Howto]]

Latest revision as of 10:14, 18 October 2024

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, this approach offers few advantages over computing phonons from finite differences. In particular, the DFPT routines are limited to LDA and GGA functionals, and they do 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 within DFPT and then switch to the finite differences approach in order to determine the largest displacement that will produce results compatible with the linear response routines.

A few technical comments: 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 this end, 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 precisely the second-order force constants and the internal strain tensor, respectively.

Input

To use DFPT, set the tag IBRION=7 or 8 in the INCAR file. There are two options for using 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. or LCALCEPS=.TRUE., additional dielectric properties are computed.

Output

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 after the following lines: 

 Eigenvectors and eigenvalues of the dynamical matrix
 ----------------------------------------------------

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 or 8 are supported by VASP.5.1 and later versions.

Related tags and sections

IBRION, LEPSILON, Phonons: Theory, Phonons from finite differences

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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