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
![{\displaystyle
H(\mathbf{k}) | \psi_{n\mathbf{k}} \rangle=
e_{n\mathbf{k}}S(\mathbf{k}) | \psi_{n\mathbf{k}} \rangle,
}](/wiki/index.php?title=Special:MathShowImage&hash=4b8755bfb2c1c8878595281ee75ca2ba&mode=mathml)
where ...
Taking the derivative with respect to the ionic positions
, we obtain the Sternheimer equations
![{\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
}](/wiki/index.php?title=Special:MathShowImage&hash=bd43691b16fbb321b1620d483149d0d0&mode=mathml)
Once the derivative of the KS orbitals is computed from the Sternheimer equations, we can write
![{\displaystyle
| \psi^{u^a_i}_\lambda \rangle =
| \psi \rangle +
\lambda | \partial_{u^a_i}\psi \rangle.
}](/wiki/index.php?title=Special:MathShowImage&hash=ad039a87b232ca6ac37cd965ef4a105c&mode=mathml)
The second-order response to ionic displacement, i.e., the force constants or Hessian matrix, are then computed using
![{\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},
}](/wiki/index.php?title=Special:MathShowImage&hash=3660be86dfd9e6d112303e5b0a890207&mode=mathml)
where
yields the forces for a given set of KS orbitals.
The internal strain tensor is computed using
![{\displaystyle
\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}.
}](/wiki/index.php?title=Special:MathShowImage&hash=fae823c7104eea6b915b4f4d1d20c25e&mode=mathml)