Electric field response 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
- [math]\displaystyle{ H(\mathbf{k}) | \psi_{n\mathbf{k}} \rangle= e_{n\mathbf{k}}S(\mathbf{k}) | \psi_{n\mathbf{k}} \rangle, }[/math]
where [math]\displaystyle{ H(\mathbf{k}) }[/math] is the DFT Hamiltonian, [math]\displaystyle{ S(\mathbf{k}) }[/math] is the overlap operator and, [math]\displaystyle{ | \psi_{n\mathbf{k}} \rangle }[/math] and [math]\displaystyle{ e_{n\mathbf{k}} }[/math] are the KS eigenstates.
To compute the response with respect to an external electric field [math]\displaystyle{ \mathcal{E_\alpha} }[/math] one has to solve two Sternheimer equations. The first one to obtain the derivative of the orbitals with respect to [math]\displaystyle{ \mathbf{k} }[/math]
- [math]\displaystyle{ \left[ H(\mathbf{k}) - e_{n\mathbf{k}}S(\mathbf{k}) \right] |\partial_\mathbf{k}\tilde{u}_{n\mathbf{k}}\rangle = -\partial_\mathbf{k} \left[ H(\mathbf{k}) - e_{n\mathbf{k}}S(\mathbf{k})\right] |\tilde{u}_{n\mathbf{k}}\rangle }[/math]
and a second one with respect to an external electric field [1]
- [math]\displaystyle{ \left[ H(\mathbf{k}) - e_{n\mathbf{k}}S(\mathbf{k}) \right] |\partial_\mathcal{E_\alpha}\tilde{u}_{n\mathbf{k}} \rangle = -\Delta H_{\text{SCF}}(\mathbf{k}) |\tilde{u}_{n\mathbf{k}}\rangle -\mathbf{\hat{q}_\alpha}\cdot |\vec{\beta}_{n\mathbf{k}}\rangle }[/math]
with
- [math]\displaystyle{ |\vec{\beta}_{n\mathbf{k}}\rangle= \left( 1+\sum_{ij} |\tilde{p}_{i\mathbf{k}}\rangle Q_{ij} \langle\tilde{p}_{j\mathbf{k}}| \right) |\partial_\mathbf{k} \tilde{u}_{n\mathbf{k}}\rangle+ i\left( \sum_{ij} |\tilde{p}_{i\mathbf{k}}\rangle Q_{ij} (\mathbf{r}-\mathbf{R}_i)-\vec{\tau}_{ij} \langle\tilde{p}_{j\mathbf{k}}| \right)|\tilde{u}_{n\mathbf{k}}\rangle }[/math]
- [math]\displaystyle{ Q_{ij} = \int_{\Omega_\text{PAW}} \left[ \phi_i(\mathbf{r}) \phi_j(\mathbf{r}) - \tilde{\phi}_i(\mathbf{r}) \tilde{\phi}_j(\mathbf{r}) \right] d^3 \mathbf{r} }[/math]
- [math]\displaystyle{ \vec{\tau}_{ij} = \int_{\Omega_\text{PAW}} (\mathbf{r}-\mathbf{R}_i) \left[ \phi_i(\mathbf{r}) \phi_j(\mathbf{r}) - \tilde{\phi}_i(\mathbf{r}) \tilde{\phi}_j(\mathbf{r}) \right] d^3 \mathbf{r} }[/math]
After obtaining the derivatives of the Kohn-Sham orbitals with respect to the electric field [math]\displaystyle{ |\partial_\mathcal{E_\alpha}\tilde{u}_{n\mathbf{k}} \rangle }[/math] the dielectric tensor can then be computed
- [math]\displaystyle{ \epsilon^\infty(\hat{\mathbf{q}},0) = 1-\frac{8\pi e^2}{\Omega} \sum_{v\mathbf{k}} 2 w_\mathbf{k} \langle \mathbf{\hat{q}}\vec{\beta}_{n\mathbf{k}} | \partial_{\mathcal{E}_\alpha} \tilde{u}_{n\mathbf{k}} \rangle. }[/math]
One can compute the orbital with a finite electric field with a small magnitude [math]\displaystyle{ \lambda }[/math]
- [math]\displaystyle{ |\psi^{\mathcal{E}_\alpha}_\lambda\rangle = |\psi\rangle + \lambda |\partial_{\mathcal{E}_\alpha}\psi\rangle }[/math]
and compute born effective charges using
- [math]\displaystyle{ Z^a_{ij}=\frac{\Omega}{e}\frac{\partial P_i}{\partial u^a_j} =\frac{1}{e}\frac{\partial F^a_i}{\partial \mathcal{E}_j} \approx\frac{1}{e}\frac{ \left( \mathbf{F}[\{\psi^{\mathcal{E}_j}_{\lambda/2}\}]- \mathbf{F}[\{\psi^{\mathcal{E}_j}_{-\lambda/2}\}] \right)^a_i}{\lambda} }[/math]
with [math]\displaystyle{ \mathbf{F}[\psi] }[/math] retrieves the set of forces from the [math]\displaystyle{ \psi_{n\mathbf{k}} }[/math] Kohn-Sham orbitals.
The piezoelectric tensor can be computed using
- [math]\displaystyle{ \overline{e}_{ij} =-\frac{\partial \sigma_i}{\partial \mathcal{E}_j} \approx -\frac{ \left( \sigma[\{\psi^{\mathcal{E}_j}_{\lambda/2}\}]- \sigma[\{\psi^{\mathcal{E}_j}_{-\lambda/2}\}] \right)_i }{\lambda} }[/math]
with [math]\displaystyle{ \mathbf{\sigma}[\psi_{n\mathbf{k}}] }[/math] returning the strain tensor.