# Tutorial:Sternheimer linear response

The Sternheimer approach to perturbation theory allows efficient calculations of linear and non-linear response properties. The basis of this method, just as in standard perturbation theory, is to calculate the variation of the wave-functions under a given perturbing potential. The advantage of the method is that the variations are obtained by solving the linear equation

that only depends on the occupied states instead of requiring an (infinite) sum over unoccupied states. In the case of (time-dependent) density functional theory the variation of the Hamiltonian includes a term that depends on the variation of the density, so this equation must be solved self-consistently.

To run a Sternheimer calculation with Octopus, the only previous calculation you need is a ground-state calculation. For this tutorial we will use a water molecule, with this basic input file for the ground state:

CalculationMode = gs

%Coordinates
'O'  |  0.000000  | -0.553586  |  0.000000
'H'  |  1.429937  |  0.553586  |  0.000000
'H'  | -1.429937  |  0.553586  |  0.000000
%



Note that we will use the default spacing, but we will increase a bit the size of the box as response calculations tend to require more space around the molecule than ground-state calculations.

After the ground-state calculation is finished, we change the run mode to em_resp, to run a calculation of the electric-dipole response:

CalculationMode = em_resp


Next, to specify the frequency of the response we use the EMFreqs block; in this case we will use three values 0.00, 0.15 and 0.30 [Ha]:

%EMFreqs
3 | 0.0 | 0.3
%


and we will also specify a small imaginary part to the frequency of 0.1 [eV], which avoids divergence on resonance:

EMEta = 0.1/27.211383


and set the linear solver to be used:

LinearSolver = bicgstab


In the run, you will see calculations for each frequency for the x, y, and z directions. The negative state indices listed indicate response for . At the end, you will have a directory called em_resp containing a subdirectory for each frequency calculated, each in turn containing eta (listing = 0.1 [eV]), alpha (containing the real part of the polarizability tensor), and cross_section (containing the cross-section for absorption, based on the imaginary part of the polarizability).

For more information, see Xavier Andrade, Silvana Botti, Miguel Marques and Angel Rubio, Time-dependent density functional theory scheme for efficient calculations of dynamic (hyper)polarizabilities, J. Chem. Phys 126 184106 (2007)

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