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 dmftproj is used to construct the Wannier functions. For this step, see either sections Supported interfaces, or the extensive dmftproj manual.

In the following, we discuss how to use TRIQS in combination with the Wien2k program.

We can use the DMFT script as introduced in section Single-shot DFT+DMFT, with just a few simple modifications. First, in order to be compatible with the Wien2k standards, the DMFT script has to be named case.py, where case is the place holder name of the Wien2k calculation, see the section Supported interfaces 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 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 DFT+DMFT tutorial: Ce with Hubbard-I approximation 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 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 NiO CSC tutorial.

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 and returns both the correlation energy and the SumK object.

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
IMIX=1
BMIX=0.5
AMIX=0.02

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. Here, we also adjust the mixing, since iterations become quickly unstable for insulating or charge ordered solutions. Also note, that in each DAV step you still have to calculate the projectors, recalculate the chemical potential, and update the GAMMA file. See the triqs_dft_tools.converters.plovasp.sc_dmft() script for details.

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. Furthermore, the flags NELM and NELMIN ensure that VASP does not terminate after the default number of iterations of 60.

For more detailed and fine grained methods to run Vasp in CSC also on clusters see the methods implemented in solid dmft.

Elk

The Elk CSC implementation is fairly similar to the Wien2k implementation. At the end of the DMFT python script, 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: Elk SVO tutorial.

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.