Electric field response from density-functional-perturbation theory

Density-functional-perturbation theory (DFPT) provides a way to compute the second-order linear response to ionic displacement, strain, and electric fields.

The starting point is density-functional theory (DFT), where one has to 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 deals with 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 the derivative with respect to an external, finite electric field

[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 the polarization vector, [math]\displaystyle{ \vec\beta_{n\mathbf k} }[/math], being given by

[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]

where [math]\displaystyle{ Q_{ij} }[/math] and [math]\displaystyle{ \vec\tau_{ij} }[/math] are the norm and the dipole moments of the one-center charge densities, each respectively expressed as

[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]

and

[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]

The perturbed orbitals, [math]\displaystyle{ \psi^{\mathcal{E}_\alpha}_\lambda }[/math], under a small, finite electric field of magnitude [math]\displaystyle{ \lambda }[/math] are given by

[math]\displaystyle{ |\psi^{\mathcal{E}_\alpha}_\lambda\rangle = |\psi\rangle + \lambda |\partial_{\mathcal{E}_\alpha}\psi\rangle }[/math]

and can be used to compute the 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_j}{\partial \mathcal{E}_i} \approx\frac{1}{e}\frac{ \left( \mathbf{F}[\{\psi^{\mathcal{E}_i}_{\lambda/2}\}]- \mathbf{F}[\{\psi^{\mathcal{E}_i}_{-\lambda/2}\}] \right)^a_j}{\lambda} }[/math]

where [math]\displaystyle{ \mathbf{F}[\psi] }[/math] are the forces on each atom, [math]\displaystyle{ a }[/math], for a given set of perturbed KS orbitals.

Finally, the piezoelectric tensor [math]\displaystyle{ \overline{e}_{ij} }[/math] can be computed using

[math]\displaystyle{ \overline{e}_{ij} =-\frac{\partial \sigma_j}{\partial \mathcal{E}_i} \approx -\frac{ \left( \sigma[\{\psi^{\mathcal{E}_i}_{\lambda/2}\}]- \sigma[\{\psi^{\mathcal{E}_i}_{-\lambda/2}\}] \right)_j }{\lambda} }[/math]

with [math]\displaystyle{ \mathbf{\sigma}[\psi_{n\mathbf{k}}] }[/math] being the strain tensor.

Related tags and sections

LEPSILON, LRPA, LPEAD, Born effective charges