Phonons: Theory: Difference between revisions

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[[Category:Phonons]]
[[Category:Phonons]][[Category:Theory]]

Revision as of 09:08, 10 August 2022

Phonons are the collective excitation of nuclei in an extended periodic system.

To compute them we start by Taylor expanding the total energy () around the set of equilibrium positions of the nuclei ()

where the positions of the nuclei. The first derivative of the total energy with respect to the nuclei corresponds to the forces

,

and the second derivative to the second-order force-constants


Changing variables in the Taylor expansion of the total energy with that corresponds to the displacement of the atoms with respect to their equilibrium position leads to

If we take the system to be in equilibrium, the forces on the atoms are zero and then the Hamiltonian of the system is

with the mass of the -th nucleus. The equation of motion is then given by

We then look for solutions of the form of plane waves traveling parallel to the wave vector , i.e.

where are the phonon mode eigenvectors and the amplitudes. Replacing it in the equation of motion we obtain the following eigenvalue problem

with

the dynamical matrix in the harmonic approximation. Now by solving the eigenvalue problem above we can obtain the phonon modes and frequencies at any arbitrary q point.

Finite differences

The second-order force constants are computed using finite differences of the forces when each ion is displaced in each independent direction. This is done by creating systems with finite ionic displacement of atom in direction with magnitude , computing the orbitals and the forces for these systems. The second-order force constants are then computed using

where corresponds to the displacement of atom in the cartesian direction and retrieves the set of forces acting on all the ions given the KS orbitals.

Similarly, the internal strain tensor is

where computes the strain tensor given the KS orbitals.

Density functional perturbation theory

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

where is the DFT Hamiltonian, is the overlap operator and, and are the KS eigenstates.

Taking the derivative with respect to the ionic displacements , we obtain the Sternheimer equations

Once the derivative of the KS orbitals is computed, we can write

where is a small numeric value to use in the finite differences formulas below.

The second-order response to ionic displacement, i.e., the force constants or Hessian matrix can be computed using the same equation used in the case of the finite differences approach

where again yields the forces for a given set of KS orbitals.

Similarly, the internal strain tensor is computed using

where computes the strain tensor given the KS orbitals. The Born effective charges are then computed using Eq. (42) of Ref. [1].

where is the atom index, the direction of the displacement of the atom, the polarization direction, and is the polarization vector defined in Eq. (30) in Ref. [2]. The results should be equivalent to the ones obtained using LCALCEPS and LEPSILON.

References