Tutorial:Sternheimer linear response

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The Sternheimer approach to perturbation theory allows efficient calculations of linear and non-linear response properties.[1] 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

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

Radius = 10
Spacing = 0.65
ConvRelDens = 1e-6

We use a tighter setting on SCF convergence (ConvRelDens) which will help the ability of the Sternheimer calculation to converge numerically, and we increase a bit the size of the box as response calculations tend to require more space around the molecule than ground-state calculations to be converged.[2]

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]:

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

We are using the default symmetric QMR linear solver. In the run, you will see calculations for each frequency for the x, y, and z directions, showing SCF iterations, each having linear-solver iterations for the individual states' , labelled by the k-point/spin (ik) and state (ist). The negative state indices listed indicate response for . The norm of , the number of linear-solver iterations (iter), and the residual are shown for each.

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).


  1. 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)
  2. F. D. Vila, D. A. Strubbe, Y. Takimoto, X. Andrade, A. Rubio, S. G. Louie, and J. J. Rehr, Basis set effects on the hyperpolarizability of CHCl3: Gaussian-type orbitals, numerical basis sets and real-space grids, J. Chem. Phys. 133 034111 (2010)

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