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

$\left[H(\mathbf {k} )-e_{n\mathbf {k} }S(\mathbf {k} )\right]|\partial _{R_{i}^{a}}\psi _{n\mathbf {k} }\rangle =-\partial _{R_{i}^{a}}\left[H(\mathbf {k} )-e_{n\mathbf {k} }S(\mathbf {k} )\right]|\psi _{n\mathbf {k} }\rangle$

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

$|\psi _{\lambda }^{R_{i}^{a}}\rangle =|\psi \rangle +\lambda |\partial _{R_{i}^{a}}\psi \rangle$

The force constants are then computed using

$\Phi _{ij}^{ab}={\frac {\partial ^{2}E}{\partial R_{i}^{a}\partial R_{j}^{b}}}=-{\frac {\partial F_{i}^{a}}{\partial R_{j}^{b}}}\approx -{\frac {\left(\mathbf {F} [\{\psi _{\lambda /2}^{R_{j}^{b}}\}]-\mathbf {F} [\{\psi _{-\lambda /2}^{R_{j}^{b}}\}]\right)_{i}^{a}}{\lambda }}$

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

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

$Z_{ij}^{a*}=2{\frac {\Omega _{0}}{(2\pi )^{3}}}\int _{\text{BZ}}\sum _{n}\langle \partial _{R_{i}^{a}}\psi _{n\mathbf {k} }|\nabla _{\mathbf {k} }{\tilde {u}}_{n\mathbf {k} }\rangle d\mathbf {k}$

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.