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208 lines
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ReStructuredText
208 lines
12 KiB
ReStructuredText
.. _Transport:
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Transport calculations
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============================
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Formalism
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---------
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The conductivity, the Seebeck coefficient and the electronic contribution to the thermal conductivity in direction :math:`\alpha\beta` are defined as [#transp1]_ [#transp2]_:
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.. math::
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\sigma_{\alpha\beta} = \beta e^{2} A_{0,\alpha\beta}
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.. math::
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S_{\alpha\beta} = -\frac{k_B}{|e|}\frac{A_{1,\alpha\beta}}{A_{0,\alpha\beta}},
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.. math::
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\kappa^{\text{el}}_{\alpha\beta} = k_B \left(A_{2,\alpha\beta} - \frac{A_{1,\alpha\beta}^2}{A_{0,\alpha\beta}}\right),
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in which the kinetic coefficients :math:`A_{n,\alpha\beta}` are given by
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.. math::
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A_{n,\alpha\beta} = N_{sp} \pi \hbar \int{d\omega \left(\beta\omega\right)^n f\left(\omega\right)f\left(-\omega\right)\Gamma_{\alpha\beta}\left(\omega,\omega\right)}.
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Here :math:`N_{sp}` is the spin factor and :math:`f(\omega)` is the Fermi function. The transport distribution :math:`\Gamma_{\alpha\beta}\left(\omega_1,\omega_2\right)` is defined as
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.. math::
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\Gamma_{\alpha\beta}\left(\omega_1,\omega_2\right) = \frac{1}{V} \sum_k Tr\left(v_{k,\alpha}A_{k}(\omega_1)v_{k,\beta}A_{k}\left(\omega_2\right)\right),
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where :math:`V` is the unit cell volume. In multi-band systems the velocities :math:`v_{k}` and the spectral function :math:`A(k,\omega)` are matrices in the band indices :math:`i` and :math:`j`.
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The frequency depended optical conductivity is given by
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.. math::
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\sigma(\Omega) = N_{sp} \pi e^2 \hbar \int{d\omega \Gamma_{\alpha\beta}(\omega+\Omega/2,\omega-\Omega/2)\frac{f(\omega-\Omega/2)-f(\omega+\Omega/2)}{\Omega}}.
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Prerequisites
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-------------
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First perform a standard :ref:`DFT+DMFT calculation <full_charge_selfcons>` for your desired material and obtain the
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real-frequency self energy.
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.. note::
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If you use a CT-QMC impurity solver you need to perform an **analytic continuation** of
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self energies and Green functions from Matsubara frequencies to the real-frequency axis!
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This packages does NOT provide methods to do this, but a list of options available within the TRIQS framework
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is given :ref:`here <ac>`. Keep in mind that all these methods have to be used very carefully. Especially for optics calculations
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it is crucial to perform the analytic continuation in such a way that the real-frequency self energy
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is accurate around the Fermi energy as low-energy features strongly influence the final results.
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Below will describe the prerequisites from the different DFT codes.
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Prequisites from Wien2k
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^^^^^^^^^^^^^^^^^^^^^^^
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Besides the self energy the Wien2k files read by the transport converter (:meth:`convert_transport_input <dft.converters.wien2k.Wien2kConverter.convert_transport_input>`) are:
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* :file:`.struct`: The lattice constants specified in the struct file are used to calculate the unit cell volume.
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* :file:`.outputs`: In this file the k-point symmetries are given.
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* :file:`.oubwin`: Contains the indices of the bands within the projected subspace (written by :program:`dmftproj`) for each k-point.
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* :file:`.pmat`: This file is the output of the Wien2k optics package and contains the velocity (momentum) matrix elements between all bands in the desired energy
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window for each k-point. How to use the optics package is described below.
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* :file:`.h5`: The hdf5 archive has to be present and should contain the dft_input subgroup. Otherwise :meth:`convert_dft_input <dft.converters.wien2k.Wien2kConverter.convert_dft_input>` needs to be called before :meth:`convert_transport_input <dft.converters.wien2k.Wien2kConverter.convert_transport_input>`.
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These Wien2k files are read and the relevant information is stored in the hdf5 archive by using the following::
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from triqs_dft_tools.converters.wien2k import *
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from triqs_dft_tools.sumk_dft_tools import *
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Converter = Wien2kConverter(filename='case', repacking=True)
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Converter.convert_transport_input()
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SK = SumkDFTTools(hdf_file='case.h5', use_dft_blocks=True)
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The converter :meth:`convert_transport_input <dft.converters.wien2k.Wien2kConverter.convert_transport_input>`
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reads the required data of the Wien2k output and stores it in the `dft_transp_input` subgroup of your hdf file.
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Wien2k optics package
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^^^^^^^^^^^^^^^^^^^^^
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The basics steps to calculate the matrix elements of the momentum operator with the Wien2k optics package are:
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1) Perform a standard Wien2k calculation for your material.
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2) Run `x kgen` to generate a dense k-mesh.
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3) Run `x lapw1`.
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4) For metals change TETRA to 101.0 in :file:`case.in2`.
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5) Run `x lapw2 -fermi`.
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6) Run `x optic`.
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Additionally the input file :file:`case.inop` is required. A detail description on how to setup this file can be found in the Wien2k user guide [#userguide]_ on page 166.
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The optics energy window should be chosen according to the window used for :program:`dmftproj`. Note that the current version of the transport code uses only the smaller
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of those two windows. However, keep in mind that the optics energy window has to be specified in absolute values and NOT relative to the Fermi energy!
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You can read off the Fermi energy from the :file:`case.scf2` file. Please do not set the optional parameter NBvalMAX in :file:`case.inop`.
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Furthermore it is necessary to set line 6 to "ON" and put a "1" in the following line to enable the printing of the matrix elements to :file:`case.pmat`.
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Prequisites from Elk
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^^^^^^^^^^^^^^^^^^^^
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The Elk transport converter (:meth:`convert_transport_input <dft.converters.elk.ElkConverter.convert_transport_input>`) reads in the following files:
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* `LATTICE.OUT`: Real and reciprocal lattice structure and cell volumes.
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* `SYMCRYS.OUT`: Crystal symmetries.
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* `PMAT.OUT`: Fortran binary containing the velocity matrix elements.
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* :file:`.h5`: The hdf5 archive has to be present and should contain the dft_input subgroup. Otherwise :meth:`convert_dft_input <dft.converters.elk.ElkConverter.convert_dft_input>` needs to be called before :meth:`convert_transport_input <dft.converters.elk.ElkConverter.convert_transport_input>`. It is recommended to call :meth:`convert_dft_input <dft.converters.elk.ElkConverter.convert_dft_input>` before :meth:`convert_transport_input <dft.converters.elk.ElkConverter.convert_transport_input>`.
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Except for `PMAT.OUT`, the other files are standard outputs from Elk's groundstate calculation and are used in :meth:`convert_dft_input <dft.converters.elk.ElkConverter.convert_dft_input>`. The `PMAT.OUT` file on the otherhand is generated by Elk by running **task 120**, see [#userguide2]_. Note that unlike in the Wien2k transport converter, the Elk transport converter uses the correlated band window stored in the `dft_misc_input` (which originates from running :meth:`convert_dft_input <dft.converters.elk.ElkConverter.convert_dft_input>`).
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These Elk files are then read and the relevant information is stored in the hdf5 archive by using the following::
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from triqs_dft_tools.converters.elk import *
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from triqs_dft_tools.sumk_dft_tools import *
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Converter = ElkConverter(filename='case', repacking=True)
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Converter.convert_transport_input()
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SK = SumkDFTTools(hdf_file='case.h5', use_dft_blocks=True)
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The converter :meth:`convert_transport_input <dft.converters.wien2k.Wien2kConverter.convert_transport_input>`
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reads the required data of the Elk output and stores it in the `dft_transp_input` subgroup of your hdf file.
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Using the transport code
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------------------------
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Once we have converted the transport data from the DFT codes (see above), we also need to read and set the self energy, the chemical potential and the double counting::
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with HDFArchive('case.h5', 'r') as ar:
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SK.set_Sigma([ar['dmft_output']['Sigma_w']])
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chemical_potential,dc_imp,dc_energ = SK.load(['chemical_potential','dc_imp','dc_energ'])
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SK.set_mu(chemical_potential)
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SK.set_dc(dc_imp,dc_energ)
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As next step we can calculate the transport distribution :math:`\Gamma_{\alpha\beta}(\omega)`::
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SK.transport_distribution(directions=['xx'], Om_mesh=[0.0, 0.1], energy_window=[-0.3,0.3],
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with_Sigma=True, broadening=0.0, beta=40)
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Here the transport distribution is calculated in :math:`xx` direction for the frequencies :math:`\Omega=0.0` and :math:`0.1`.
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To use the previously obtained self energy we set `with_Sigma` to **True** and the broadening to :math:`0.0`.
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As we also want to calculate the Seebeck coefficient and the thermal conductivity we have to include :math:`\Omega=0.0` in the mesh.
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Note that the current version of the code repines the :math:`\Omega` values to the closest values on the self energy mesh.
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For complete description of the input parameters see the :meth:`transport_distribution reference <dft.sumk_dft_tools.SumkDFTTools.transport_distribution>`.
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The resulting transport distribution is not automatically saved, but this can be easily achieved with::
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SK.save(['Gamma_w','Om_mesh','omega','directions'])
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You can retrieve it from the archive by::
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SK.Gamma_w, SK.Om_mesh, SK.omega, SK.directions = SK.load(['Gamma_w','Om_mesh','omega','directions'])
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Finally the optical conductivity :math:`\sigma(\Omega)`, the Seebeck coefficient :math:`S` and the thermal conductivity :math:`\kappa^{\text{el}}` can be obtained with::
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SK.conductivity_and_seebeck(beta=40)
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SK.save(['seebeck','optic_cond','kappa'])
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It is strongly advised to check convergence in the number of k-points!
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Example
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-------
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Here we present an example calculation of the DFT optical conductivity of SrVO3 comparing the results from the Elk and Wien2k inputs. The DFT codes used 4495 k-points in the
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irreducible Brillouin zone with Wannier projectors generated within a correlated energy window of [-8, 7.5] eV. We assume that the required DFT files have been read and saved by the TRIQS
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interface routines as discussed previously. Below is an example script to generate the conductivities::
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from sumk_dft_tools import *
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import numpy
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SK = SumkDFTTools(hdf_file=filename+'.h5', use_dft_blocks=True)
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#Generate numpy mesh for omega values
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om_mesh = list(numpy.linspace(0.0,5.0,51))
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#Generate and save the transport distribution
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SK.transport_distribution(directions=['xx'], Om_mesh=om_mesh, energy_window=[-8.0, 7.5], with_Sigma=False, broadening=-0.05, beta=40, n_om=1000)
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SK.save(['Gamma_w','Om_mesh','omega','directions'])
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#Generate and save conductivities
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SK.conductivity_and_seebeck(beta=40)
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SK.save(['seebeck','optic_cond','kappa'])
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The optic_cond variable can be loaded by using :meth:`SK.load` and then plotted to generate the following figure.
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.. image:: transport_plots/opt_comp.png
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:width: 700
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:align: center
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Note that the differences between the conductivities arise from the differences in the velocities generated in the DFT codes. The DMFT optical conductivity can easily be calculated by adjusting
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the above example script by setting `with_Sigma` to **True**. In this case however, the SK object will need the DMFT self-energy on the real frequency axis.
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References
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----------
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.. [#transp1] `V. S. Oudovenko, G. Palsson, K. Haule, G. Kotliar, S. Y. Savrasov, Phys. Rev. B 73, 035120 (2006) <http://link.aps.org/doi/10.1103/PhysRevB.73.0351>`_
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.. [#transp2] `J. M. Tomczak, K. Haule, T. Miyake, A. Georges, G. Kotliar, Phys. Rev. B 82, 085104 (2010) <https://link.aps.org/doi/10.1103/PhysRevB.82.085104>`_
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.. [#userguide] `P. Blaha, K. Schwarz, G. K. H. Madsen, D. Kvasnicka, J. Luitz, ISBN 3-9501031-1-2 <http://www.wien2k.at/reg_user/textbooks/usersguide.pdf>`_
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.. [#userguide2] `J. K. Dewhurst, S. Sharma, L. Nordstrom, F. Cricchio, O. Granas, and E. K. U. Gross, The Elk Code Manual <https://elk.sourceforge.io/elk.pdf>`_
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