Phonons from density-functional-perturbation theory

In density functional theory we solve the Hamiltonian

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

Taking derivatives with respect to the ionic positions $R_i^a$ we obtain the Sternheimer equation

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

Once the derivative of the orbitals is computed from the Sternheimer equation we can write

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

The second-order force constants 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 orbitals.

The internal strain tensor is computed using

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

At the end of the calculation if LEPSILON=.TRUE., the Born effective charges are computed using Eq. (42) of Ref. .

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

where $a$ is the atom index, $i$ the direction of the displacement of atom and $j$ the polarization direction. The results should be equivalent to the ones obtained using LCALCEPS and LEPSILON.

When IBRION=7 or 8 VASP solves the Sternheimer equation above with an ionic displacement perturbation. If IBRION=7 no symmetry is used and the displacement of all the ions is computed. When IBRION=8 only the irreducible displacements are computed with the physical quantities being reconstructed by symmetry.