Phonons from density-functional-perturbation theory: Difference between revisions
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e_{n\mathbf{k}}S(\mathbf{k}) | \psi_{n\mathbf{k}} \rangle, | e_{n\mathbf{k}}S(\mathbf{k}) | \psi_{n\mathbf{k}} \rangle, | ||
</math> | </math> | ||
where ... | where ... (MTH: please define all quantities.) | ||
Taking the derivative with respect to the ionic positions <math>R_i^a</math>, we obtain the Sternheimer equations | Taking the derivative with respect to the ionic positions <math>R_i^a</math>, we obtain the Sternheimer equations | ||
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</math> | </math> | ||
where <math>\mathbf{F}</math> yields the [[:Category:Forces|forces]] for a given set of KS orbitals. | where <math>\mathbf{F}</math> yields the [[:Category:Forces|forces]] for a given set of KS orbitals. | ||
MTH: Here, it would be good to explicitly write the eigenvalue equation that is solved to obtain phonon frequencies. | |||
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MTH: The following is not concerning the calculation of phonons: | |||
The internal strain tensor is computed using | The internal strain tensor is computed using | ||
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</math> | </math> | ||
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}}. | 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}}. | ||
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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. | 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 results should be equivalent to the ones obtained using {{TAG|LCALCEPS}} and {{TAG|LEPSILON}}. | The results should be equivalent to the ones obtained using {{TAG|LCALCEPS}} and {{TAG|LEPSILON}}. | ||
MTH: This should be part of a more extensive discussion in a how-to article. The beginning of the article seemed more like a theory article, so I propose to move it in that category. | |||
When {{TAG|IBRION}}=7 or 8 VASP solves the Sternheimer equation above with an ionic displacement perturbation. | When {{TAG|IBRION}}=7 or 8 VASP solves the Sternheimer equation above with an ionic displacement perturbation. | ||
If {{TAG|IBRION}}=7 no symmetry is used and the displacement of all the ions is computed. | 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. | When {{TAG|IBRION}}=8 only the irreducible displacements are computed with the physical quantities being reconstructed by symmetry. | ||
-- | --> | ||
== References == | == References == | ||
<references/> | <references/> | ||
[[Category:Phonons]] | [[Category:Phonons]][[Category:Theory]] | ||
Revision as of 14:59, 20 July 2022
Density-functional-perturbation theory provides a way to compute the second-order linear response to ionic displacement, strain, and electric fields. The equations are derived as follows.
In density-functional theory, we solve the Kohn-Sham (KS) equations
- [math]\displaystyle{ H(\mathbf{k}) | \psi_{n\mathbf{k}} \rangle= e_{n\mathbf{k}}S(\mathbf{k}) | \psi_{n\mathbf{k}} \rangle, }[/math]
where ... (MTH: please define all quantities.)
Taking the derivative with respect to the ionic positions [math]\displaystyle{ R_i^a }[/math], we obtain the Sternheimer equations
- [math]\displaystyle{ \left[ H(\mathbf{k}) - e_{n\mathbf{k}}S(\mathbf{k}) \right] | \partial_{u_i^a}\psi_{n\mathbf{k}} \rangle = -\partial_{u_i^a} \left[ H(\mathbf{k}) - e_{n\mathbf{k}}S(\mathbf{k})\right] | \psi_{n\mathbf{k}} \rangle }[/math]
Once the derivative of the KS orbitals is computed from the Sternheimer equations, we can write
- [math]\displaystyle{ | \psi^{u^a_i}_\lambda \rangle = | \psi \rangle + \lambda | \partial_{u^a_i}\psi \rangle. }[/math]
The second-order response to ionic displacement, i.e., the force constants or Hessian matrix, are then computed using
- [math]\displaystyle{ \Phi^{ab}_{ij}= \frac{\partial^2E}{\partial u^a_i \partial u^b_j}= -\frac{\partial F^a_i}{\partial u^b_j} \approx -\frac{ \left( \mathbf{F}[\{\psi^{u^b_j}_{\lambda/2}\}]- \mathbf{F}[\{\psi^{u^b_j}_{-\lambda/2}\}] \right)^a_i}{\lambda}, }[/math]
where [math]\displaystyle{ \mathbf{F} }[/math] yields the forces for a given set of KS orbitals.
MTH: Here, it would be good to explicitly write the eigenvalue equation that is solved to obtain phonon frequencies.