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Tutorial 1: MnO

Tutorial 1.a. : Preparing the input files


We begin by a simple example of a charge transfer insulator: MnO. It cristalizes in the rock salt structure (two interpenetrating cubic FCC str.) with lattice parameter 8.401155a a.u.. The Mn ion contains around 5 electrons, which form the high spin state. The ground state has a large gap, which is usually categorized as the charge transfer gap. We will treat dynamically the Mn-d electrons.

The first step is to perform a DFT calculation to get a good charge density. (We assume that the user has some familiarity with the Wien2K code and we give only a very brief introductions to the DFT part.)

First create a directory with the name MnO and copy the MnO.struct file into it. [Note that Wien2k requires to call directory with the same name as the structure file]. The structure file contains the details of the crystal structure in the format that Wien2K uses. Initialize the DFT calculation by using init_lapw and then choosing the default options. In version 14, one needs to type the following sequence of commands:
Enter reduction in %:   0
Use old or new scheme (o/N) : N
Do you want to accept these radii; .... (a/d/r) : a
specify nn-bondlength factor: 2
Ctrl-X-C
continue with sgroup : c
Ctrl-X-C
continue with symmetry : c
Ctrl-X-C
continue with lstart : c
specify switches for instgen_lapw (or press ENTER): ENTER
SELECT XCPOT: 13
SELECT ENERGY : -6
Ctrl-X-C
continue with kgen : c
Ctrl-X-C
Ctrl-X-C
NUMBER OF K-POINTS : 500
Ctrl-X-C
continue with dstart or execute kgen again or exit (c/e/x) : c
Ctrl-X-C
do you want to perform a spinpolarized calculation : n
Here Ctrl-X-C stands for breaking out of your editor. Ctrl-XC is used in emacs and similar editors, but in vi, one should replace Ctrl-X-C by :q!.

These are pretty much the default values that Wien2K uses. We chose to work with the PBE functional (note that total energies are better with LDA+DMFT functional, while spectra is almost identical when using PBE or LDA), and use a k-point grid of 500 points (in the whole Brillouin zone). Note that spin-orbit interaction can be safely ignored as there are no heavy ions in the system.

Next, run the DFT calculation using the command
run_lapw
In DMFT calculation, one needs converged frequency dependent local Green's function, hence more k-points are typically needed. We will increase the number of k-points to 2000 in this tutorial. To do that, run
x kgen
and specify 2000 k-points. Next, rerun wien2k on this k-mesh, i.e., run_lapw. (This step is not essential as the DFT charge will not be very precise anyway for our DFT+Embedded-DMFT calculation).

Once the DFT calculation is done, we have a good charge density to start a DMFT calculation. The next step is to initialize the DMFT calculation. For this, use the script init_dmft.py and choose the following options
Specify correlated atoms (ex: 1-4,7,8):  1

Do you want to continue; or edit again? (c/e): c

For each atom, specify correlated orbital(s) (ex: d,f):
  1 Mn: d

Do you want to continue; or edit again? (c/e): c

Specify qsplit for each correlated orbital (default = 0):
  1  Mn-1 d: 7

Do you want to continue; or edit again? (c/e): c

Specify projector type (default = 2): 5

Do you want to continue; or edit again? (c/e): c

Do you want to group any of these orbitals into cluster-DMFT problems? (y/n): n

Enter the correlated problems forming each unique correlated
problem, separated by spaces (ex: 1,3 2,4 5-8): 1

Do you want to continue; or edit again? (c/e): c

Range (in eV) of hybridization taken into account in impurity
problems; default -10.0, 10.0: <ENTER>

Perform calculation on real; or imaginary axis? (r/i): i

Ctrl-X-C

Is this a spin-orbit run? (y/n): n

Ctrl-X-C
Now we want to explain the options and their choice:

The init_dmft.py script generates two mandatory input files, case.indmfl and case.indmfi, and also projectorw.dat when fixed projector=5 is used. These files connect the solid and the impurity with the DMFT engine, respectively. Let us take a closer look at the header of the case.indmfl file:

5 15 1 5                              # hybridization band index nemin and nemax, renormalize for interstitials, projection type
1 0.025 0.025 200 -3.000000 1.000000  # matsubara, broadening-corr, broadening-noncorr, nomega, omega_min, omega_max (in eV)
1                                     # number of correlated atoms
1     1   0                           # iatom, nL, locrot
  2   7   1                           # L, qsplit, cix
The first line specifies the hybridization window. We set it to "(-10eV,10eV)" around EF, which it turns out, contains bands numbered from 5-15. For each k-point we will use the same bands, i.e., we will follow the bands with the projector.
The same line also specified the projector type, which is 5.
The second line contains a switch for the real-axis or imaginary axis calculation. If we require the local green's function or density of states on real axis, we will set the switch "matsubara" to 0. Here, we will work on imaginary axis, hence "matsubara" is set to 1. The second line also contains information on small lorentzian broadening in k-point summation. The last three numbers are used for plotting the spectral functions, to which we will return latter.
The third line specifies the number of correlated atoms.
The forth line specifies which atom (from case.struct) is correlated (Note that each atom in the structure file counts, even when several atoms in structure files are equivalent). The forth line also specifies how many orbital indeces (different L's) we will use for projection, and calculation of local green's function and partial dos. The last index "locrot" is used for rotation of local axis, and is not needed here (set to 0).
The fifth line specifies orbital momentum "l=2", "qsplit=7" which we chose during initialization. Each correlated orbital-set gets a unique index during initialization. This correlated set was given index "cix=1". For each correlated set, a more detailed specification is needed, which is given in the remaining of "case.indmfl" file.


The specification of correlated set is given in the second part of case.indmfl:
#================ # Siginds and crystal-field transformations for correlated orbitals ================
1     5   2       # Number of independent kcix blocks, max dimension, max num-independent-components
1     5   2       # cix-num, dimension, num-independent-components
#---------------- # Independent components are --------------
'eg' 't2g' 
#---------------- # Sigind follows --------------------------
1 0 0 0 0
0 1 0 0 0
0 0 2 0 0
0 0 0 2 0
0 0 0 0 2
#---------------- # Transformation matrix follows -----------
 0.00000000 0.00000000   0.00000000 0.00000000   1.00000000 0.00000000   0.00000000 0.00000000   0.00000000 0.00000000 
 0.70710679 0.00000000   0.00000000 0.00000000   0.00000000 0.00000000   0.00000000 0.00000000   0.70710679 0.00000000 
 0.00000000 0.00000000   0.70710679 0.00000000   0.00000000 0.00000000  -0.70710679 0.00000000   0.00000000 0.00000000 
 0.00000000 0.00000000   0.00000000 0.70710679   0.00000000 0.00000000   0.00000000 0.70710679   0.00000000 0.00000000 
-0.70710679 0.00000000   0.00000000 0.00000000   0.00000000 0.00000000   0.00000000 0.00000000   0.70710679 0.00000000 

The first line specifies the number of all correlated blocks, the dimension of the largest correlated block, and number of orbital components which differ. The second line specifies the dimension of the first correlated block (here "d" has dimension 5), and the two eg and the three t2g components are treated as degenerate.

The transformation matrix from spheric harmonics to the DMFT basis is given in the end, and shows how these orbitals are defined: It is the expansion of each of these orbitals in spherical harmonics. The columns correspond to the following quantum numbers (m_l=-2,-1,0,1,2) and rows to ('z^2','x^2-y^2','xz','yz','xy'). For example, the 'z^2' orbital, which is expanded in the first line, is equal to the m=0 spherical harmonic, whereas the yz orbital on the forth line is an equal superposition of m=1 and m=-1, multiplied by i (note that two consecutive real numbers define one complex number). Note that the transformation matrix must be unitary.

This file case.indmfl can be edited in text editor and can be adjusted if necessary.

The second file, created at the initialization, is case.indmfi and contains the following lines:
1   # number of sigind blocks
5   # dimension of this sigind block
1 0 0 0 0
0 1 0 0 0
0 0 2 0 0
0 0 0 2 0
0 0 0 0 2
In this simplest case, where we have a single impurity problem, no new information is contained in case.indmfi file, just the "Sigind" block from case.indmfl is repeated. We will show more complex use of this file later.



Now we want to start DMFT with the minimal number of necessary files. To do this, create a new folder (with any name), and when in that folder, use the command
dmft_copy.py <dft_results_directory>
where <dft_results_directory> is the directory where the output of the DFT calculation with DMFT initialization is. This command copies the necessary files to start a DMFT calculation. These include the MnO.struct file as well as the various Wien2K input files such as MnO.inm, MnO.in1, etc. The charge density file, MnO.clmsum, is also copied. This is all we need from the initial Wien2K run, and now we proceed to create the rest of the files, needed by the DMFT calculation.

Copy the params.dat file. This file contains information about the impurity solver and additional parameteres, which appear in DFT+Embedded-DMFT. It is written in Python format, and hence any Python syntax is accepted. For checking the correct syntax, one can execute it as python script. Its content is:
solver         =  'CTQMC'   # impurity solver

max_dmft_iterations = 1     # number of iteration of the dmft-loop only
max_lda_iterations  = 100   # number of iteration of the LDA-loop only
finish              = 30    # number of iterations of full charge loop (1 = no charge self-consistency)

ntail          =  300       # on imaginary axis, number of points in the tail of the logarithmic mesh

cc             =  5e-6      # the charge density precision to stop the LDA+DMFT run
ec             =  5e-6      # the energy precision to stop the LDA+DMFT run
recomputeEF    =  0         # Recompute EF in dmft2 step. If recomputeEF = 0, EF is fixed. Good for insulators.

DCs            =  'exactd'  # exact DC with dielectric model  

wbroad         =  0.0       # broadening of sigma on the imaginary axis
kbroad         =  0.0       # broadening of sigma on the imaginary axis

# Impurity problem number 0
iparams0={"exe"                : ["ctqmc"               , "# Name of the executable"],
          "U"                  : [9.0                   , "# Coulomb repulsion (F0)"],
          "J"                  : [1.14                  , "# Coulomb J"],
          "CoulombF"           : ["'Ising'"             , "# Density-density form. Can be changed to 'Full' "],
          "beta"               : [38.68                 , "# Inverse temperature"],
          "svd_lmax"           : [25                    , "# We will use SVD basis to expand G, with this cutoff"],
          "M"                  : [5e6                   , "# Total number of Monte Carlo steps per core"],
          "mode"               : ["GH"                  , "# We will use Greens function sampling, and Hubbard I tail"],
          "nom"                : [100                   , "# Number of Matsubara frequency points sampled"],
          "tsample"            : [100                   , "# How often to record measurements"],
          "GlobalFlip"         : [500000                , "# How often to try a global flip"],
          "warmup"             : [1e5                   , "# Warmup number of QMC steps"],
          "nf0"                : [5                     , "# Nominal valence we expect. Used only for starting DC."],
          }
Note that this input file is a python script, and the correcteness of its syntax can be checked with Python interpreter by tying python params.dat. Below we explain the meaning of the variables

There is only one more file that we need: A blank self energy (sig.inp) file. Generate it by the command
szero.py
One can add parameters to the script, like
szero.py -e 38.22 -T 0.0258531540847983
in which case we set starting double-counting to 38.22 and temperature to 0.0258531540847983eV. If we do not specify these values, "szero.py" will read params.dat file, and extract this information from "beta" and "nf0". Note that if one start with a very wrong double-counting energy, one will need very many DMFT iterations before reaching the correct physical regime. For the initial guess, we read "nf0" from the params file, and we use Anisimov's approximation for the double-counting, given by VDC=U*(nf0-1/2)-J/2*(nf0-1).

Tutorial 1.b. : Submitting the job

Now we are ready to run DFT+Embedded-DMFT calculation. In the simplest case, we can just run the script
run_dmft.py
However, single CPU will take a really long time and the statistics will be too bad. One typically needs at least 100 cores (but more is better -- this jobs can easily scale to 100,000 cores) to achieve good statistics. [The DFT part is most efficient when the number of cores is commensurate with the number of k-points]. Note also that the parameter "M" in params file specifies the number of Monte Carlo steps per core, therefore more cores are available, better is the result (but it is not faster in execution).

To submit a job to a parallel computer, we need to prepare a submit script. This script must create a file "mpi_prefix.dat", which contains the command for mpi parallel execution. This command however depends very much on the operating system and environment of a supercomputer. On many systems, this command is called "mpirun", and in this case, the submit script would contain
echo "mpirun -np $NSLOTS" > mpi_prefix.dat
where $NSLOTS stands for the number of available cores. On some systems, the command might be:
echo "mpiexec -port $port -np $NSLOTS -machinefile $TMPDIR/machines" > mpi_prefix.dat
We can also tune the number of OpenMP threads per mpi node, which can be achieved by giving an environmental variable OMP_NUM_THREADS to mpi command. In mpich, for example, one could set
echo "mpirun -np $NSLOTS -env OMP_NUM_THREADS 1" > mpi_prefix.dat
and in openmpi, one would set
echo "mpirun -np $NSLOTS -x OMP_NUM_THREADS=1" > mpi_prefix.dat
When running the CTQMC solver, it is advisable to use one thread per mpi core (OMP_NUM_THREADS=1). For lapw1, dmft1 and dmft2 one should divide k-points over available cores, and if there are more cores available than there is k-points, one can set OMP_NUM_THREADS so that ncores ~ nkpt*OMP_NUM_THREADS. To distinguish between the two different mpirun commands, we can set-up mpi_prefix.dat2 in addition to mpi_prefix.dat. The former is used for dmft1, dmft2 while the latter is used for CTQMC.

Finally, once we created "mpi_prefix.dat" file, we should run the DFT+Embedded-DMFT script by
$WIEN_DMFT_ROOT/run_dmft.py > nohup.dat 2>&1
Note that the python script is not run with mpi. However, the impurity solver, and other dmft steps will be executed in parallel using command specified in "mpi_prefix.dat".

As discussed above, if many more CPU-cores are available than the number of k-points, we can optimize the execution further. Namely, a single k-point can be executed on many cores using OpenMP instructions. To use this feature, one needs to specify "mpi_prefix.dat2" in addition to "mpi_prefix.dat" file. The former is than used in the dft part of the loop, while the latter is used by the impurity solver. Clearly, "mpi_prefix.dat2" should specify the mpi command, which is started on a subset of machines. Therefore "-np" should be smaller than "$NSLOTS" and "OMP_NUM_THREADS" should be larger than 1.
We provide a script that attempts to do that, hence you might want to insert the following line between "....> mpi_prefix.dat" and "run_dmft.py":
$WIEN_DMFT_ROOT/createDMFprefix.py $JOBNAME.klist mpi_prefix.dat > mpi_prefix.dat2
where "$JOBNAME" should be set to wien2k case. Unfortunately mpi execution is computer environment dependent, and therefore this script might not run everywhere.

Before submitting the job, make sure that the computing notes have the following environmental variable specified: $WIENROOT, $WIEN_DMFT_ROOT, $SCRATCH. If lapw1 and lapwso are executed through x_dmft.py script, then these variables need to be set only on the master node. Otherwise all compute nodes need to have these variables set.
Also make sure that $WIEN_DMFT_ROOT is in $PYTHONPATH on the master node. It is advisable to add $WIENROOT and $WIEN_DMFT_ROOT to the $PATH, although it is not mandatory. Typically, the .bashrc (or its equivalent) should contain the following lines:

export WIENROOT=<wien-instalation-dir>
export WIEN_DMFT_ROOT=<dir-containing-dmft-executables>
export SCRATCH="."
export PATH=$PATH:$WIENROOT:$WIEN_DMFT_ROOT
export PYTHONPATH=$PYTHONPATH:$WIEN_DMFT_ROOT

Note that SCRATCH does not need to point to the current directory, instead it could be more efficient to have it set to SCRATCH=/tmp/ or SCRATCH=/scratch/, so that the large vector files are written locally for each process.

It might be necessary to repeat these commands (seeting variables) also in the submit script. This of ocurse depends on the system.

Here we paste an example of the submit script for SUN Grid Engine , but different system will require different script.

#!/bin/bash
########################################################################
# SUN Grid Engine job wrapper
########################################################################
#$ -N MnO
#$ -pe mpi2  216
#$ -q <your_queue>
#$ -j y
#$ -M <email@your_institution>
#$ -m e
########################################################################
source $TMPDIR/sge_init.sh
########################################################################
export WIENROOT=<your_w2k_root>
export WIEN_DMFT_ROOT=<your_dmft_root>
export PATH=$PATH:$WIENROOT:$WIEN_DMFT_ROOT
export PYTHONPATH=$PYTHONPATH:.:$WIEN_DMFT_ROOT
export SCRATCH='.'

JOBNAME="MnO"   # must be equal to case name

mpi_prefix="mpiexec  -np $NSLOTS  -machinefile $TMPDIR/machines -port $port -env OMP_NUM_THREADS 1 -envlist SCRATCH,WIEN_DMFT_ROOT,PYTHONPATH,WIENROOT" 
echo $mpi_prefix > mpi_prefix.dat

$WIEN_DMFT_ROOT/createDMFprefix.py $JOBNAME.klist mpi_prefix.dat > mpi_prefix.dat2

$WIEN_DMFT_ROOT/run_dmft.py > nohup.dat 2>&1

rm -f $JOBNAME.vector*
For PBE system, the script might look like
#!/bin/bash
#PBS -l walltime=12:00:00,nodes=72:ppn=16
#PBS -N  MnO
#PBS -j oe

export WIENROOT=<your_w2k_root>
export WIEN_DMFT_ROOT=<your_dmft_root>
export PYTHONPATH=$PYTHONPATH:.:$WIEN_DMFT_ROOT
export SCRATCH='.'

cd $PBS_O_WORKDIR
echo "mpirun -np 1152 -x OMP_NUM_THREADS=1" > mpi_prefix.dat
echo "mpirun -np 72   -x OMP_NUM_THREADS=16" > mpi_prefix.dat2

$WIEN_DMFT_ROOT/run_dmft.py >& nohup.dat

rm -f $JOBNAME.vector*


Tutorial 1.c. : Monitoring the job

While the code is running, there are multiple files that you can check to see how it is going. One is the ':log' file, which serves exactly the same purpose as in Wien2K. It lists the individual modules that are run:
Thu Mar 23 22:57:43 EDT 2017> (x) -f MnO lapw0
Thu Mar 23 22:57:45 EDT 2017>     lapw1
Thu Mar 23 22:57:46 EDT 2017>     dmft1
Thu Mar 23 22:57:47 EDT 2017>     impurity
Thu Mar 23 23:00:13 EDT 2017>     dmft2
Thu Mar 23 23:00:14 EDT 2017> (x) -f MnO lcore
Thu Mar 23 23:00:15 EDT 2017> (x) -f MnO mixer
Thu Mar 23 23:00:16 EDT 2017> (x) -f MnO lapw0
Thu Mar 23 23:00:37 EDT 2017>     lapw1
Thu Mar 23 23:00:38 EDT 2017>     dmft2
Thu Mar 23 23:00:41 EDT 2017> (x) -f MnO lcore
Thu Mar 23 23:00:41 EDT 2017> (x) -f MnO mixer
Thu Mar 23 23:00:43 EDT 2017> (x) -f MnO lapw0
Thu Mar 23 23:00:45 EDT 2017>     lapw1
Thu Mar 23 23:00:47 EDT 2017>     dmft2
Thu Mar 23 23:00:48 EDT 2017> (x) -f MnO lcore
Thu Mar 23 23:00:51 EDT 2017> (x) -f MnO mixer
The code starts by running the lapw0 and lapw1 modules of Wien2K to calculate the vector files, etc; but then, dmft1 and dmft2 steps are inserted. Impurity is running in-between dmft1 and dmft2 step. The dmft2 step replaces the lapw2 step of Wien2k.

Since DFT part is here very fast (much faster than impurity solver), we used several DFT step (up to 100) in combination with a single dmft step. Therefore after dmft1, one can see several blocks of the following cycle repeated: dmft2,lcore,mixer,lapw0,lapw1. This is the charge loop. The reason to do this is to obtain a well converged DFT run between each dmft1 step, since dmft1 is the most time consuming part (as can be seen in the timing information above). Whether the charge density is well converged can be checked by grep'ing ':CHARGE' in the .dayfile:
grep ':CHARGE' MnO.dayfile
Which will print
:CHARGE convergence:  0.3650051
:CHARGE convergence:  0.320649
:CHARGE convergence:  0.1980197
:CHARGE convergence:  0.1882957
:CHARGE convergence:  0.1454709
:CHARGE convergence:  0.0432447
:CHARGE convergence:  0.0190978
:CHARGE convergence:  0.0055925
:CHARGE convergence:  0.0048235
:CHARGE convergence:  0.0024782
:CHARGE convergence:  0.0006552
:CHARGE convergence:  0.0001947
:CHARGE convergence:  0.0002805
:CHARGE convergence:  2.05e-05
:CHARGE convergence:  1.16e-05
:CHARGE convergence:  1.69e-05
:CHARGE convergence:  1.84e-05
:CHARGE convergence:  1.4e-06

Note that the charge converges at fixed self-energy, but when the self-energy is updated (after dmft1 step) the charge is again non-converged. Towards the end of the run (after about 20 cycles), the self enery is also converged and so the jump in the charge after each dmft1 is much less significant:
0  :CHARGE convergence: 0.3650051 
1  :CHARGE convergence: 0.320649 
2  :CHARGE convergence: 0.1980197 
3  :CHARGE convergence: 0.1882957 
4  :CHARGE convergence: 0.1454709 
5  :CHARGE convergence: 0.0432447 
6  :CHARGE convergence: 0.0190978 
7  :CHARGE convergence: 0.0055925 
8  :CHARGE convergence: 0.0048235 
9  :CHARGE convergence: 0.0024782 
10 :CHARGE convergence: 0.0006552 
11 :CHARGE convergence: 0.0001947 
12 :CHARGE convergence: 0.0002805 
13 :CHARGE convergence: 2.05e-05 
14 :CHARGE convergence: 1.16e-05 
15 :CHARGE convergence: 1.69e-05 
16 :CHARGE convergence: 1.84e-05 
17 :CHARGE convergence: 1.4e-06 
  ....
  ....
51 :CHARGE convergence: 0.0046878 
52 :CHARGE convergence: 0.0041113 
53 :CHARGE convergence: 0.00243 
54 :CHARGE convergence: 0.0013541 
55 :CHARGE convergence: 0.0006717 
56 :CHARGE convergence: 0.000342 
57 :CHARGE convergence: 0.0001287 
58 :CHARGE convergence: 3.47e-05 
59 :CHARGE convergence: 1.37e-05 
60 :CHARGE convergence: 3.6e-06 
61 :CHARGE convergence: 0.002555 
62 :CHARGE convergence: 0.0022443 
63 :CHARGE convergence: 0.0013392 
64 :CHARGE convergence: 0.0007663 
65 :CHARGE convergence: 0.0003645 
66 :CHARGE convergence: 0.0002563 
67 :CHARGE convergence: 7.28e-05 
68 :CHARGE convergence: 1.09e-05 
69 :CHARGE convergence: 9.6e-06 
70 :CHARGE convergence: 6.1e-06 
71 :CHARGE convergence: 6e-07
....
....
135 :CHARGE convergence: 0.0002954 
136 :CHARGE convergence: 0.0002592 
137 :CHARGE convergence: 0.0001538 
138 :CHARGE convergence: 0.0001346 
139 :CHARGE convergence: 5.24e-05 
140 :CHARGE convergence: 2.72e-05 
141 :CHARGE convergence: 6.8e-06 
142 :CHARGE convergence: 3.2e-06 
143 :CHARGE convergence: 0.0003925 
144 :CHARGE convergence: 0.0003348 
145 :CHARGE convergence: 0.0001878 
146 :CHARGE convergence: 0.0001563 
147 :CHARGE convergence: 4.16e-05 
148 :CHARGE convergence: 2.61e-05 
149 :CHARGE convergence: 1.05e-05 
150 :CHARGE convergence: 2.8e-06 

Another important file to check is the info.iterate, which contains information about the chemical potential, the number of electrons on the V ion, etc:
  #   #.   #           mu          Vdc            Etot     Ftot+T*Simp     Ftot+T*Simp       n_latt        n_imp      Eimp[0]     Eimp[-1] 
  0   0.   0     6.531263    37.772879    -2467.797924    -2467.812926    -2467.796784     4.832377     5.013534     0.054065    -0.045105 
  1   0.   1     6.531263    37.772879    -2467.795198    -2467.839362    -2467.794737     4.856478     5.013534     0.054065    -0.045105 
  2   0.   2     6.531263    37.772879    -2467.790068    -2467.919012    -2467.791341     4.928763     5.013534     0.054065    -0.045105 
  3   0.   3     6.531263    37.772879    -2467.805037    -2468.148462    -2467.807211     5.119680     5.013534     0.054065    -0.045105 
  4   0.   4     6.531263    37.772879    -2467.806312    -2468.041367    -2467.803524     4.975357     5.013534     0.054065    -0.045105 
  5   0.   5     6.531263    37.772879    -2467.797455    -2468.058270    -2467.797604     4.989835     5.013534     0.054065    -0.045105 
  6   0.   6     6.531263    37.772879    -2467.795467    -2468.037324    -2467.795559     4.964081     5.013534     0.054065    -0.045105 
  7   0.   7     6.531263    37.772879    -2467.795670    -2468.039345    -2467.795727     4.967218     5.013534     0.054065    -0.045105 
  8   0.   8     6.531263    37.772879    -2467.795453    -2468.037709    -2467.795553     4.968465     5.013534     0.054065    -0.045105 
  9   0.   9     6.531263    37.772879    -2467.795395    -2468.034744    -2467.795412     4.970982     5.013534     0.054065    -0.045105 
 10   0.  10     6.531263    37.772879    -2467.794961    -2468.029487    -2467.794965     4.974418     5.013534     0.054065    -0.045105 
 11   0.  11     6.531263    37.772879    -2467.794815    -2468.027650    -2467.794804     4.974379     5.013534     0.054065    -0.045105 
 12   0.  12     6.531263    37.772879    -2467.794816    -2468.027676    -2467.794805     4.974472     5.013534     0.054065    -0.045105 
 13   0.  13     6.531263    37.772879    -2467.794813    -2468.027656    -2467.794803     4.974440     5.013534     0.054065    -0.045105 
 14   0.  14     6.531263    37.772879    -2467.794813    -2468.027658    -2467.794803     4.974441     5.013534     0.054065    -0.045105 
 15   0.  15     6.531263    37.772879    -2467.794813    -2468.027657    -2467.794803     4.974440     5.013534     0.054065    -0.045105 
 16   0.  16     6.531263    37.772879    -2467.794813    -2468.027655    -2467.794803     4.974440     5.013534     0.054065    -0.045105 
 17   1.   0     6.531263    37.940316    -2467.791665    -2467.796323    -2467.793875     5.101315     5.033536    -0.240213    -0.362603 
 18   1.   1     6.531263    37.940316    -2467.791016    -2467.781532    -2467.792627     5.089714     5.033536    -0.240213    -0.362603 
 19   1.   2     6.531263    37.940316    -2467.789431    -2467.737004    -2467.789185     5.054946     5.033536    -0.240213    -0.362603 
 20   1.   3     6.531263    37.940316    -2467.787966    -2467.669527    -2467.784865     5.006160     5.033536    -0.240213    -0.362603 
 21   1.   4     6.531263    37.940316    -2467.786916    -2467.648442    -2467.783808     5.018529     5.033536    -0.240213    -0.362603 
 22   1.   5     6.531263    37.940316    -2467.785769    -2467.646216    -2467.783363     5.028555     5.033536    -0.240213    -0.362603 
 23   1.   6     6.531263    37.940316    -2467.785657    -2467.641236    -2467.783057     5.024862     5.033536    -0.240213    -0.362603 
 24   1.   7     6.531263    37.940316    -2467.785486    -2467.638166    -2467.782886     5.025596     5.033536    -0.240213    -0.362603 
 25   1.   8     6.531263    37.940316    -2467.785492    -2467.638303    -2467.782892     5.025582     5.033536    -0.240213    -0.362603 
 26   1.   9     6.531263    37.940316    -2467.785489    -2467.638261    -2467.782888     5.025539     5.033536    -0.240213    -0.362603 
 27   1.  10     6.531263    37.940316    -2467.785488    -2467.638250    -2467.782887     5.025538     5.033536    -0.240213    -0.362603 
 28   1.  11     6.531263    37.940316    -2467.785489    -2467.638264    -2467.782888     5.025541     5.033536    -0.240213    -0.362603 
 29   2.   0     6.531263    37.919732    -2467.784434    -2467.789381    -2467.784203     5.024368     5.031140    -0.077242    -0.205319 
 30   2.   1     6.531263    37.919732    -2467.784453    -2467.790595    -2467.784272     5.025296     5.031140    -0.077242    -0.205319 
 31   2.   2     6.531263    37.919732    -2467.784511    -2467.794236    -2467.784480     5.028077     5.031140    -0.077242    -0.205319 
....
....
153  14.   6     6.531263    37.919319    -2467.784809    -2467.786146    -2467.784766     5.031462     5.031090    -0.090639    -0.218392 
154  14.   7     6.531263    37.919319    -2467.784806    -2467.786104    -2467.784764     5.031478     5.031090    -0.090639    -0.218392 
155  14.   8     6.531263    37.919319    -2467.784806    -2467.786091    -2467.784763     5.031482     5.031090    -0.090639    -0.218392 
156  14.   9     6.531263    37.919319    -2467.784805    -2467.786087    -2467.784763     5.031482     5.031090    -0.090639    -0.218392 
157  15.   0     6.531263    37.917886    -2467.784779    -2467.789688    -2467.784788     5.031145     5.030921    -0.085861    -0.213710 
158  15.   1     6.531263    37.917886    -2467.784780    -2467.789750    -2467.784791     5.031194     5.030921    -0.085861    -0.213710 
159  15.   2     6.531263    37.917886    -2467.784784    -2467.789936    -2467.784802     5.031343     5.030921    -0.085861    -0.213710 
160  15.   3     6.531263    37.917886    -2467.784789    -2467.790227    -2467.784820     5.031569     5.030921    -0.085861    -0.213710 
161  15.   4     6.531263    37.917886    -2467.784792    -2467.790226    -2467.784820     5.031496     5.030921    -0.085861    -0.213710 
162  15.   5     6.531263    37.917886    -2467.784793    -2467.790225    -2467.784820     5.031493     5.030921    -0.085861    -0.213710 
163  15.   6     6.531263    37.917886    -2467.784793    -2467.790227    -2467.784821     5.031488     5.030921    -0.085861    -0.213710 
164  15.   7     6.531263    37.917886    -2467.784794    -2467.790230    -2467.784821     5.031487     5.030921    -0.085861    -0.213710 

The second column counts the number of DMFT steps (impurity solver), the third counts charge steps at each DMFT steps, next columns shows the chemical potential, the fifth column shows the double-counting potential, the sixth column shows total energy, followed by two different estimates of the free energy. The seventh column estimate of the free energy is not precise, instead the eight column should be used. [Notice that all quantities, except the total energy and the free energy are give in eV. The total energies and free energies are give in Ry, to be compatible with wien2k]. The ninth and tenth column show the impurity occupation computed from the solid (n_latt) and from the impurity (n_imp). The last two columns show the impurity levels.

The chemical potential in this insulating materials is fixed. The impurity occupation, computed from the solid, is quite off at the beginning, however, it becomes close to nominal 5.0 is just a few iterations. When the calculation is converged, n_latt and n_imp should match.

Very many steps appear in info.iterate file, and sometimes we just want to print one line per each dmft steps (the step at which the charge is converged, and just before the self-energy is updated). This would correspond to step 16,28,... in the above example.
We can execute
analizeInfo.py
to list only these dmft steps. We can then plot the lattice and impurity occupation, which should look like this:
Within 6 dmft iterations, the charge is converged to 5.032, and we can now increase the precision of the Monte Carlo steps. For example, increase M in params.dat file for factor of 4-5 to have much more precise statistics in the last few iterations.

Notice that each variable in params.dat file can be changed during the run, and it will be updated during the run.

Within 10 or so dmft iterations (around 120 charge iterations), the total energy and the free energy are converged within 1meV (using 216 cores for impurity solver), as seen here:
Notice that total energy (or free energy) would converge better only if Monte Carlo statistics is increase, as resulting noise is due to MC noise.

Other files to monitor the run include
dmft_info.out      -- prints what is being executed, and what were parameters
MnO.scf                -- like in dft, contains energies and forces and lots of other info
dmft1_info.out         -- progress of dmft1 step
MnO.outputdmf1         -- log file of the dmft1 step. Contains the density matrix in :NCOR and the matrix of impurity levels.
dmft2_info.out         -- progress of dmft2 step
MnO.outputdmf2         -- log file of the dmft2 step (similar to MnO.output2 from lapw2); contains energies and forces.
MnO.dlt1*              -- the hybridization function
MnO.gc1*               -- the local green's function
imp.0/ctqmc.log*       -- log file of the impurity solver
imp.0/nohup_imp.out.xxx -- progress of the impurity solver for each mpi process
Delta.inp*             -- input hybridization function
imp.0/histogram.dat    -- the histogram of the perturbation order, i.e., how many kinks does the time evolution contain


Even though the impurity occupations and total energy converged, we should check the spectral functions to decide if the run is properly converged. In a metal, it is best to look at the self-energy (impurity self-energies are at : imp.0/Sig.out*, while the lattice self-energy, which is just a combination of impurity ones, are at: sig.inp*).
In a Mott insulator, the self-energy on real axis has a pole inside the gap, and therefore its shape on imaginary axis is very sensitive to the relative position of the chemical potential an the pole. If one accidentally hits the pole by placing mu on it, the self-energy on imaginary axis diverges at zero frequency. If however the chemical potential is slightly moved away from the pole, self-energy just becomes large at finite frequencies, but its imaginary part vanishes at zero. This is because on real axis the imaginary part vanishes inside the gap, except for the pole. The real part is large, but not diverging, except when chemical potential is at the pole. The shape of the self-energy on imaginary axis is therefore not very illustrative.

We will therefore rather concentrate on the Green's function, of which the imaginary part needs to vanish at zero frequency (both on the real and the imaginary axis). Here is the plot of the impurity Green's function with iterations:
We see that the green's function is metallic only at the first iteration, while at the second iteration is almost converged. Such rapid convergence is here due to the robustness of the gap in the high-spin state on Mn.

We should always check that the impurity and the local green's function of the solid match. This is a very non-trivial check of the correctness of the calculation, as the algorithm equates only the self-energy at each iteration, while the Green's functions differ until the self-consistency is achieved. We see that at the six-th dmft iteration, the two Green's functions match very well:
Notice that the last Green's functions are named imp.0/Gf.out and MnO.gc1, while each previous iteration is saved under the same name, with consecutive number attached to it. The first number stands for the number of the outside loop, while the second number enumerates the steps inside the dmft loop (red loop in the figure above). Since we have max_dmft_iterations=1, the second number is always 1. The first number would go from 1 to the parameter finish from params.dat file, or less, if the convergence is achieved earlier

We see that for practical purposes, the convergence is achieved around 6-7th iteration, and the next steps are within the statistical noise identical. The job would not finish though, because by default (to be cautious) the convergence criteria are set very high. It is therefore desirable to kill the job, once the overal convergence is obviously achieved.

Tutorial 1.d. : Postprocessing, analytic continuation, DOS

At this point, we can proceed to the next step of plotting spectra on the real axis.

Once the calculation is done on the imaginary axis, we need to do analytical continuation to obtain the self energy on the real axis. For this, we first take average of the sig.inp files from the last few steps (in order to reduce the noise) by
saverage.py saverage.py sig.inp.1[0-5].1
The output is written to sig.inpx. We could append -o option, if we wanted to change this name. All python scripts include a short help, so by executing for example
saverage.py --help
it prints the options for this script.

Next, create a new directory (lets call it maxent), and copy the averaged self energy sig.inpx into it. Also copy the maxent_params.dat file, which contains the parameters for the analytical continuation process:
params={'statistics': 'fermi', # fermi/bose
        'Ntau'      : 300,     # Number of time points
        'L'         : 20.0,    # cutoff frequency on real axis
        'x0'        : 0.005,   # low energy cut-off
        'bwdth'     : 0.004,   # smoothing width
        'Nw'        : 300,     # number of frequency points on real axis
        'gwidth'    : 2*15.0,  # width of gaussian
        'idg'       : 1,       # error scheme: idg=1 -> sigma=deltag ; idg=0 -> sigma=deltag*G(tau)
        'deltag'    : 0.004,   # error
        'Asteps'    : 4000,    # anealing steps
        'alpha0'    : 1000,    # starting alpha
        'min_ratio' : 0.001,   # condition to finish, what should be the ratio
        'iflat'     : 1,       # iflat=0 : constant model, iflat=1 : gaussian of width gwidth, iflat=2 : input using file model$
        'Nitt'      : 1000,    # maximum number of outside iterations
        'Nr'        : 0,       # number of smoothing runs
        'Nf'        : 40,      # to perform inverse Fourier, high frequency limit is computed from the last Nf points
  }
Perhaps the most important is mesh on the real axis, which is given by the upper cutoff L, the low energy cutoff x0 and Nw points. The mesh is non-uniform and more dense at low energy (tan-mesh). The MC error is here set to 0.004, and can be estimated from the variation in sig.inp.? files. Notice that we do not continue the self-energy, which would have very large error, and would not be stable to continue analytically. We will analytically continue an auxiliary function, which is constructed as

and then perform Kramers-Kronig on imaginary part of this function:

and finally invert it to solve for the self-energy on the real axis

The script which does that is called maxent_run.py. We will thus run
maxent_run.py sig.inpx
which uses the maximum entropy method to do analytical continuation. This creates the self energy Sig.out on the real axis. It should be similar to this plot:
Notice that both orbitals have Mott-type gap, as there is a pole in the gap. The sizes of the two gaps are different, with the eg orbitals having somewhat larger (4.8eV) and t2g smaller (2.7eV) gap.

Now we need to make one last dmft1 calculation in order to obtain the Green's function and DOS on the real axis. Create a new directory inside your output directory, and while it the new directory, use
dmft_copy.py ../
to copy necessary files from the output of the DMFT run to the new directory. Also copy the analytically continued self energy Sig.out to sig.inp in this new directory:
cp ../maxent/Sig.out sig.inp
Since we need to run the code on the real axis, we need to change the NiO.indmfl file. The first number on the second line of NiO.indmfl file determines whether the code is run on the real axis (0) or the imaginary axis (1). Change it to 0. The header of MnO.indmfl file should now look like
5 15 1 5                              # hybridization band index nemin and nemax, renormalize for interstitials, projection type
0 0.025 0.025 200 -3.000000 1.000000  # matsubara, broadening-corr, broadening-noncorr, nomega, omega_min, omega_max (in eV)
1                                     # number of correlated atoms
1     1   0                           # iatom, nL, locrot
  2   7   1                           # L, qsplit, cix
Then, inside the new directory run
x lapw0 -f  MnO
x_dmft.py lapw1
x_dmft.py dmft1
Before you might want to set OMP_NUM_THREADS to a large number (but smaller then the number of cores on your node), such as 10, to parallelize some loops with open_mp.

Once the run is done, we now have the densities of states on the real axis. Both the total (column 2) and partial Mn-d (column 3) are stored in MnO.cdos:

To obtain this plot, we increased the number of k-points to 10,000. The procedure is the same as above: rerun x kgen -f MnO and when asked enter 10000. Then rerun lapw1 and dmft1 step.
Notice that the first excitation into the valence band has roughly 1/3 of d-character, and 2/3 of the oxygen character. Notice also that this is not the real lower Hubbard band within our description, as U is too large (9eV). This peak is than due to screening of the Mn large spin by oxygen degrees of freedom, and is a type of Kondo effect. It is interesting to point out that even in an insulator the Kondo effect is not completely absent, as such screening occurs at the first possible excitation. In literature, such effect is also called Zhang-Rice singlet, as it was first discussed in cuprates by Zhang & Rice.

Lets note that the first excitation into the conduction band is of itinerant character, mostly 4s character, hence the scattering rate for this state is very small, and this state has very high mobility despite the correlated nature of the gap. Due to finite k-point mesh, this DOS appears somewhat spiky.

We can also take a look at the imaginary part of the green's function. Recall that the imaginary part of the green's function, divided by -pi, gives the DOS. So, plotting the even-numbered columns in the MnO.gc1 one, we can get the DOS of individual d orbitals (eg and t2g). The plot is
Notice that column 3 is the eg, and column 5 the t2g DOS. In this plot it is even more clear that the lower Hubbard band is below -4eV, and that the first valence excitation of the eg-states is split away, making a singlet between the oxygen states and the two eg orbitals. Notice that it is exactly this type of screening (hybridization screening) which reduces the low energy Coulomb interaction, and therefore if one simulates a Hubbard model for MnO, one needs to take much smaller U. Within our embedded method, the Coulomb repulsion will thus always be large compared to model calculations, as such screening very efficiently reduces interaction in the solid.

Note that the imaginary part of green's function and DOS use slightly different normalization, therefore the sum of all green's functions will typically produce slightly larger DOS than is computed in case.cdos. This is because Green's functions are computed with normalized projectors within a mufin-thin sphere (the integral of spectral function is by construction exactly unity), while partial dos usually uses unnormalized projector.

Another quantity of physical interest is the impurity hybridization function. It is stored in the file MnO.dlt1, and its imaginary part, which has the units of eV, looks like the following:


The large peak below the Fermi level signals the hybridization of eg states with the oxygen p states, as expected on the basis of previous discussion.