Sternheimer Linear Response

Octopus can calculate dynamic polarizabilities and first-order hyperpolarizabilites in a linear-response scheme using the Sternheimer equation. It is also possible to calculate optical spectra with this technique, but it is slower than time-evolution.

Ground state

The first thing we will need for linear response is a Ground State calculation. Unlike the Casida approach, when using the Sterheimer equation you needn’t do a unoccupied-states calculation. To improve the convergence of the linear-response calculation, it is better to use tightly converged wavefunctions. For example, you can add these parameters to your gs calculation:

EigenSolverFinalTolerance = 1e-10
ConvRelDens = 1e-9

Input

The CalculationMode for polarizability calculations is em_resp. The main parameter you have to specify is the frequency of the perturbation, given by the EMFreqs block. You can also add an imaginary part to the frequency by setting the variable EMEta. Adding a small imaginary part is required if you want to get the imaginary part of the polarizability or to calculate polarizabilities near resonance; a reasonable value is 0.1 eV.

To get the hyperpolarizabilties, you also have to specify the variable EMHyperpol with the three coefficients with respect to the base frequency; the three values must sum to zero.

Output

After running, for each frequency in the input file, Octopus will generate a subdirectory under em_resp/ . In each subdirectory there is a file called alpha that contains the real part of the polarizability tensor $\alpha_{ij}$ and the average polarizability

$$ \bar{\alpha}=\frac13\sum_{i=1}^3\alpha_{ii}\, $$

The imaginary part $\eta$ is written to file eta . If $\eta > 0$, there is also a file called cross_section_tensor that contains the photo-absorption cross section tensor for that frequency, related to the imaginary part of the polarizability ($\sigma = \frac{4 \pi \omega}{c} \mathrm{Im} \alpha $).

The hyperpolarizability will be in a file called beta at the base frequency, containing all the 27 components and some reduced quantities:

$$ \beta_{||\,i} = \frac15 \sum_{j=1}^3(\beta_{ijj}+\beta_{jij}+\beta_{jji})\ . $$

Optionally, Born charges can also be calculated.

Finite differences

In this mode only static polarizability can be obtained. The calculation is done by taking the numerical derivative of the energy with respect to an external static and uniform electric field. To use this, run with ResponseMethod =finite_differences. Octopus will run several ground-state energy calculations and then calculate the polarizability using a finite-differences formula for the derivative. The results will be in the em_resp_fd directory. Hyperpolarizability and Born charges can also be calculated.