Static linear response: theory
Let’s consider three types of static perturbations
- atomic displacements Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): {\textstyle u_m} with Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): {\displaystyle m=I\alpha} with Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): {\displaystyle I=\{1..N_\text{atoms}\}} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): {\displaystyle \alpha=\{1..3\}}
- homogeneous strains Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): {\textstyle \eta_j} with Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): {\textstyle j=\{1..6\}}
- static electric field Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): {\textstyle \mathcal{E}_\alpha} with Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): {\textstyle \alpha=\{1..3\}}
By performing a Taylor expansion of the total energy $E$ in terms of these perturbations we obtain[1]
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): {\displaystyle \begin{aligned} E(u,\mathcal{E},\eta) = &E_0 + \\ &\frac{\partial E}{\partial u_m} u_m + \frac{\partial E}{\partial \mathcal{E}_\alpha} \mathcal{E}_\alpha+ \frac{\partial E}{\partial \eta_j} \eta_j + \\ &\frac{1}{2} \frac{\partial^2 E} {\partial u_m \partial u_n } u_m u_n + \frac{1}{2} \frac{\partial^2 E} {\partial \mathcal{E}_\alpha \partial \mathcal{E}_\beta } \mathcal{E}_\alpha \mathcal{E}_\beta + \frac{1}{2} \frac{\partial^2 E} {\partial \eta_j \partial \eta_k} \eta_j \eta_k + \\ &\frac{\partial^2 E}{\partial u_m \partial \mathcal{E}_\alpha} u_m \mathcal{E}_\alpha + \frac{\partial^2 E}{\partial u_m \partial \eta_j} u_m \eta_j + \frac{\partial^2 E}{\partial \mathcal{E}_\alpha \partial \eta_j} \mathcal{E}_\alpha \eta_j + \text{terms of higher order} \end{aligned} }
The derivatives of the energy with respect to an electric field are the polarization, with respect to atomic displacements are the forces, with respect to changes in the lattice vectors are the stress tensor.
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): {\displaystyle P_\alpha = -\frac{\partial E}{\partial \mathcal{E}_\alpha} \qquad \text{polarization} }
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): {\displaystyle F_m = -\Omega_0\frac{\partial E}{\partial u_m} \qquad \text{forces} }
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): {\displaystyle \sigma_j = \frac{\partial E}{\partial \eta_j} \qquad \text{stresses} }
This leads to the following ‘clamped-ion’ or ‘frozen-ion’ definitions:
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): {\displaystyle \overline{\chi}_{\alpha\beta} = - \frac{\partial^2 E}{\partial \mathcal{E}_\alpha \partial \mathcal{E}_\beta} |_{u,\eta} \qquad \text{dielectric susceptibility} }
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): {\displaystyle \overline{C}_{jk} = \frac{\partial^2 E}{\partial \eta_j \partial \eta_k} |_{u,\mathcal{E}} \qquad \text{elastic tensor} }
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): {\displaystyle \Phi_{mn}=\Omega_0 \frac{\partial^2 E}{\partial u_m \partial u_n} |_{\mathcal{E},\eta} \qquad \text{force-constants} }
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): {\displaystyle \overline{e}_{\alpha k} = \frac{\partial^2 E}{\partial \mathcal{E}_\alpha \partial \eta_k} |_{u} \qquad \text{piezoelectric tensor} }
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): {\displaystyle Z^*_{m\alpha}=-\Omega_0 \frac{\partial^2 E}{\partial u_m \partial \mathcal{E}_\alpha} |_{\eta} \qquad \text{Born effective charges} }
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): {\displaystyle \Xi_{mj}=-\Omega_0 \frac{\partial^2 E}{\partial u_m \partial \eta_j} |_{\mathcal{E}} \qquad \text{force response internal strain tensor} }
To compare with experimental results, however, the static response properties should take into account the ionic relaxation. This follows from the Taylor expansion above by looking at the ionic positions where the energy is minimal:
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): {\displaystyle \tilde{E}(\mathcal{E},\eta) = \text{min}_u E(u,\mathcal{E},\eta) }
The physical ‘relaxed-ion’ tensors are
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): {\displaystyle \begin{aligned} \chi_{\alpha\beta} &= \overline{\chi}_{\alpha\beta} + \Omega_0^{-1} Z^*_{m\alpha} (\Phi)^{-1}_{mn} Z^*_{n\beta} \qquad \text{dielectric susceptibility}\\ C_{jk} &= \overline{C}_{jk} + \Omega_0^{-1} \Xi_{mj} (\Phi)^{-1}_{mn} \Xi_{nk} \qquad \text{elastic tensor}\\ e_{\alpha j} &= \overline{e}_{\alpha j} + \Omega_0^{-1}Z^*_{m\alpha} (\Phi)^{-1}_{mn} \Xi_{nj} \qquad \text{piezoelectric tensor} \end{aligned} }
The second term on the right-hand side of each of these equations is called the ionic contributions to the dielectric susceptibility, elastic tensor, and piezoelectric tensor.
The ionic contributions to the dielectric tensor are: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): {\displaystyle \epsilon^{\text{ion}}_{ij}=\frac{4\pi}{\Omega} \sum_{kl} Z^*_{ik} \Phi^{-1}_{kl} Z^*_{lj} }
The ionic contributions to the elastic tensor Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): {\displaystyle C^{\text{ion}}_{ik}= \sum_{kl} \Xi_{ij} \Phi^{-1}_{jk} \Xi_{kl} }
The ionic contributions to the piezoelectric tensor Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): {\displaystyle e^{\text{ion}}_{ij}= \sum_{kl} Z^*_{ij} \Phi^{-1}_{jk} \Xi_{kl} }