dft_tools/doc/guide/dftdmft_selfcons.rst

.. _full_charge_selfcons:

Full charge self-consistency
============================

In order to do charge self-consistent calculations, we have to tell the band structure program about the
changes in the charge density due to correlation effects. The feedback of the charge density is generally
program-dependent and the procedure for running charge self-consistent calculations has to be adopted
accordingly for a given band structure program. Below we describe two implementations based on Wien2k
and VASP codes.

Wien2k + dmftproj
-----------------


.. warning::
  Before using this tool, you should be familiar with the band-structure package Wien2k, since
  the calculation is controlled by the Wien2k scripts! Be
  sure that you also understand how :program:`dmftproj` is used to
  construct the Wannier functions. For this step, see either sections
  :ref:`conversion`, or the extensive :download:`dmftproj manual<images_scripts/TutorialDmftproj.pdf>`.

In the following, we discuss how to use the
:ref:`TRIQS <triqslibs:installation>` tools in combination with the Wien2k program.

We can use the DMFT script as introduced in section :ref:`singleshot`,
with just a few simple modifications. First, in order to be compatible with the Wien2k standards,
the DMFT script has to be named :file:`case.py`, where `case` is the place holder name of the Wien2k
calculation, see the section :ref:`conversion` for details. We can then set the variable `dft_filename` dynamically::

  import os
  dft_filename = os.getcwd().rpartition('/')[2]

This sets the `dft_filename` to the name of the current directory. The
remaining part of the script is identical to
that for one-shot calculations. Only at the very end we have to calculate the modified charge density,
and store it in a format such that Wien2k can read it. Therefore, after the DMFT loop that we saw in the
previous section, we symmetrise the self energy, and recalculate the impurity Green function::

  SK.symm_deg_gf(S.Sigma,ish=0)
  S.G_iw << inverse(S.G0_iw) - S.Sigma_iw
  S.G_iw.invert()

These steps are not necessary, but can help to reduce fluctuations in the total energy.
Now we calculate the modified charge density::

  # find exact chemical potential
  SK.set_Sigma([ S.Sigma_iw ])
  chemical_potential = SK.calc_mu( precision = 0.000001 )
  dN, d = SK.calc_density_correction(filename = dft_filename+'.qdmft')
  SK.save(['chemical_potential','dc_imp','dc_energ'])

First we find the chemical potential with high precision, and after that the routine
``SK.calc_density_correction(filename)`` calculates the density matrix including correlation effects. The result
is stored in the file `dft_filename.qdmft`, which is later read by the Wien2k program. The last statement saves
the chemical potential into the hdf5 archive.

We need also the correlation energy, which we evaluate by the Migdal formula::

  correnerg = 0.5 * (S.G_iw * S.Sigma_iw).total_density()

Other ways of calculating the correlation energy are possible, for
instance a direct measurement of the expectation value of the
interacting Hamiltonian. However, the Migdal formula works always,
independent of the solver that is used to solve the impurity problem.
From this value, we subtract the double counting energy::

  correnerg -= SK.dc_energ[0]

and save this value in the file, too::

  if (mpi.is_master_node()):
    f=open(dft_filename+'.qdmft','a')
    f.write("%.16f\n"%correnerg)
    f.close()

The above steps are valid for a calculation with only one correlated atom in the unit cell, the most likely case
where you will apply this method. That is the reason why we give the index `0` in the list `SK.dc_energ`.
If you have more than one correlated atom in the unit cell, but all of them
are equivalent atoms, you have to multiply the `correnerg` by their multiplicity before writing it to the file.
The multiplicity is easily found in the main input file of the Wien2k package, i.e. `case.struct`. In case of
non-equivalent atoms, the correlation energy has to be calculated for
all of them separately and summed up.

As mentioned above, the calculation is controlled by the Wien2k scripts and not by :program:`python`
routines. You should think of replacing the lapw2 part of the Wien2k self-consistency cycle by

  |   `lapw2 -almd`
  |   `dmftproj`
  |   `python case.py`
  |   `lapw2 -qdmft`

In other words, for the calculation of the density matrix in lapw2, we
add the DMFT corrections through our python scripts.
Therefore, at the command line, you start your calculation for instance by:

  `me@home $ run -qdmft 1 -i 10`

The flag `-qdmft` tells the Wien2k script that the density
matrix including correlation effects is to be read in from the
`case.qdmft` file, and that you want the code to run on one computing
core only. Moreover, we ask for 10 self-consistency iterations are to be
done. If you run the code on a parallel machine, you can specify the
number of nodes to be used:

  `me@home $ run -qdmft 64 -i 10`

In that case, you will run on 64 computing cores. As standard setting,
we use `mpirun` as the proper MPI execution statement. If you happen
to have a different, non-standard MPI setup, you have to give the
proper MPI execution statement, in the `run_lapw` script (see the
corresponding Wien2k documentation).

In many cases it is advisable to start from a converged one-shot
calculation. For practical purposes, you keep the number of DMFT loops
within one DFT cycle low, or even to `loops=1`. If you encounter
unstable convergence, you have to adjust the parameters such as
the number of DMFT loops, or some mixing of the self energy to improve
the convergence.

In the section :ref:`DFTDMFTtutorial` we will see in a detailed
example how such a self-consistent calculation is performed from scratch.

VASP + PLOVasp
--------------

Unlike Wien2k implementation the charge self-consistent DMFT cycle in the
framework of PLOVasp interface is controlled by an external script. Because of
the specific way the DFT self-consistency is implemented in VASP the latter has
to run parallel to the DMFT script, with the synchronisation being ensured by a
lock file.

Once VASP reaches the point where the projectors are generated
it creates a lock file `vasp.lock` and waits until the lock file is
removed. The shell script, in turn, waits for the VASP process and once
the lock file is created it starts a DMFT iteration. The DMFT iteration
must finish by generating a Kohn-Sham (KS) density matrix (file `GAMMA`)
and removing the lock file. The VASP process then reads in `GAMMA`
and proceeds with the next iteration. PLOVasp interface provides a shell-script :program:`vasp_dmft` (in the triqs bin directory):
::
  vasp_dmft [-n <number of cores>] -i <number of iterations>  -j <number of VASP iterations with fixed charge density> [-v <VASP version>] [-p <path to VASP directory>] [<dmft_script.py>]

       If the number of cores is not specified it is set to 1 by default.
       Set the number of times the dmft solver is called with -i <number of iterations>

       Set the number of VASP iteration with a fixed charge density update
       inbetween the dmft runs with -j <number of VASP iterations with fixed charge density>

       Set the version of VASP by -v standard(default)/no_gamma_write to
       specify if VASP writes the GAMMA file or not.

       If the path to VASP directory is not specified it must be provided by a
       variable VASP_DIR.

       <dmft_script.py> must provide an importable function 'dmft_cycle()'
       which is invoked once per DFT+DMFT iteration. If the script name is
       omitted the default name 'csc_dmft.py' is used.

which takes care of the process management. The user must, however, specify a path to VASP code and provide the DMFT Python-script. See for an example :ref:`NiO CSC tutorial<nio_csc>`.

The user-provided script is almost the same as for Wien2k charge self-consistent
calculations with the main difference that its functionality (apart from the
lines importing other modules) should be placed inside a function `dmft_cycle()`
which will be called every DMFT cycle.

VASP has a special INCAR `ICHARG=5` mode, that has to be switched on to make VASP wait for the `vasp.lock` file, and read the updated charge density after each step. One should add the following lines to the `INCAR` file::

  ICHARG = 5
  NELM = 1000
  NELMIN = 1000

Technically, VASP runs with `ICHARG=5` in a SCF mode, and adding the DMFT
changes to the DFT density in each step, so that the full DFT+DMFT charge
density is constructed in every step. This is only done in VASP because only the
changes to the DFT density are read by VASP not the full DFT+DMFT density.
Moreover, one should always start with a converged `WAVECAR` file, or make sure,
that the KS states are well converged before the first projectors are created!
To understand the difference please make sure to read `ISTART flag VASP wiki
<https://www.vasp.at/wiki/index.php/ISTART>`_. Furthermore, the flags `NELM` and
`NELMIN` ensure that VASP does not terminate after the default number of
iterations of 60.


Elk
---------

The Elk CSC implementation is fairly similar to the Wien2k implementation. At the end of the :ref:`DMFT python script <SrVO3_elk>`, the density matrix in Bloch space needs to be calculated along with the correlation energy. This is written to DMATDMFT.OUT. An example of this (using the Migdal correlation energy formula) is given below::

  #output the density matrix for Elk interface
  dN, d = SK.calc_density_correction(dm_type='elk')
  #correlation energy via the Migdal formula
  correnerg = 0.5 * (S.G_iw * S.Sigma_iw).total_density()
  #subtract the double counting energy
  correnerg -= SK.dc_energ[0]
  #convert to Hartree
  correnerg = correnerg/SK.energy_unit
  #save the correction to energy
  if (mpi.is_master_node()):
    f=open('DMATDMFT.OUT','a')
    f.write("%.16f\n"%correnerg)
    f.close()

To read this into Elk and update the electron density, run task 808. So elk.in is amended with the following::

  task
   808

This solves the Kohn-Sham equations once with the updated electron density and outputs the new set of energy eigenvalues and wavefunctions. To start the next fully charge self-consistent DFT+DMFT cycle (FCSC), a new set of projectors need to be generated (using task 805) and the whole procedure continues until convergence. The Elk potential rms value for each FCSC DFT+DMFT cycle is given in DMFT_INFO.OUT. An extensive example for SrVO:math:`_3` can be found here: :ref:`Elk SVO tutorial <SrVO3_elk>`.

This FCSC method should be universal irrespective to what type of ground state calculation performed. However, not all types of ground state calculations have been tested.


Other DFT codes
---------------

The extension to other DFT codes is straightforward. As described
here, one needs to implement the correlated density matrix to be used
for the calculation of the charge density. This implementation will of
course depend on the DFT package, and might be easy to do or a quite
involved project. The formalism, however, is straight forward.