View source for Manual:Linear Response

''This part of the code is quite recent and it is being actively developed, so not all the features might be available in the stable version.''

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

===== Ground state =====

The first thing we will need to do a linear response, it is a [[Manual:Ground State|Ground State]] calculation. Unlike other techniques, when using the Sterheimer equation you needn't do a unoccupied states calculation. To improve the convergency of the linear response calculation, it is better to use tightly converged wave functions. For example you can add this parameteres to your gs calculation:

<pre>
EigenSolverFinalTolerance = 1e-10
ConvAbsDens = 1e-9
</pre>


===== Input =====

The {{Variable|CalculationMode|Generalities}} for polarizability calculations is {{code| pol_lr}}. The main parameter you have to specify is the frequency of the perturbation, this is given by the {{Variable|PolFreqs|Linear_Response}} block. You can also add an imaginary part to the frequency by setting the variable {{Variable|PolEta|Linear_Response}}. 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 {{code|0.1 eV}}.

To get the hyperpolarizabilties, you also have to specify the {{Variable|PolHyper|Linear_Response}} with the three coefficients with respect to the base frequency, the three values must sum zero.

===== Output =====

After running, fr each frequency in the input file, {{octopus}} will generate a subdirectory under {{file|linear/}}. In each subdirectory there is a file called {{file|alpha}} that contains the real part of the polarizability tensor <math>\alpha_{ij}</math> and the average polarizability

<math>
\bar{\alpha}=\frac13\sum_{i=1}^3\alpha_{ii}\,,
</math>

there is also a file called {{file|cross_section_tensor}} that contains the photo-absorption cross section tensor for that frequency, that it is related to the imaginary part of the polarizability.

The hyperpolarizabilty will be in a file called {{file|beta}}, containing all the 27 components and some reduced quantities:

<math>
B_{//\,i} = \frac15 \sum_{j=1}^3(B_{ijj}+B_{jij}+B_{jji})\ .
</math>

==== 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 {{Variable|CalculationMode|Generalities}} {{code|{{eq}}pol_fd}}. {{octopus}} will run several ground state energy calculations (2 per dimension) and then calculate the polarizability using a finite differences formula for the derivative, the resulting tensor will be in the ''linear/polarizability_fd'' file.

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