Phonons from 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

$H(\mathbf{k}) | \psi_{n\mathbf{k}} \rangle= e_{n\mathbf{k}}S(\mathbf{k}) | \psi_{n\mathbf{k}} \rangle,$

where ...

Taking the derivative with respect to the ionic positions $R_i^a$, we obtain the Sternheimer equations

$\left[ H(\mathbf{k}) - e_{n\mathbf{k}}S(\mathbf{k}) \right] | \partial_{u_i^a}\psi_{n\mathbf{k}} \rangle = -\partial_{u_i^a} \left[ H(\mathbf{k}) - e_{n\mathbf{k}}S(\mathbf{k})\right] | \psi_{n\mathbf{k}} \rangle$

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

$| \psi^{u^a_i}_\lambda \rangle = | \psi \rangle + \lambda | \partial_{u^a_i}\psi \rangle.$

The second-order response to ionic displacement, i.e., the force constants or Hessian matrix, are then computed using

$\Phi^{ab}_{ij}= \frac{\partial^2E}{\partial u^a_i \partial u^b_j}= -\frac{\partial F^a_i}{\partial u^b_j} \approx -\frac{ \left( \mathbf{F}[\{\psi^{u^b_j}_{\lambda/2}\}]- \mathbf{F}[\{\psi^{u^b_j}_{-\lambda/2}\}] \right)^a_i}{\lambda},$

where $\mathbf{F}$ yields the forces for a given set of KS orbitals.

The internal strain tensor is computed using

$\Xi^a_{il}=\frac{\partial^2 E}{\partial \eta^a_i \partial u^b_j}= \frac{\partial \sigma_l}{\partial u^a_i} \approx \frac{ \left( \sigma[\{\psi^{u^a_i}_{\lambda/2}\}]- \sigma[\{\psi^{u^a_i}_{-\lambda/2}\}] \right)_l }{\lambda}.$