tutorial 2: fese and optimization of the structure

6/27/2019 DMFT_W2K Tutorial 1: FeSe

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Tutorial 2: FeSe and optimization of the structureTutorial 2.a. : Preparing the jobThis tutorial is on FeSe, one of the Fe superconductors. We will first give tutorial on computing a single DFT+EDMFT calculation. In the later sections we will show howto optimize the structure. FeSe cristalizes in P4/nmm with lattice parameter a=7.121433 a.u.. In our structure file, two iron atoms are at position (1/4,3/4,0) and at (3/4,1/4,0), while Se atoms are at(3/4,3/4,1-z) and (1/4,1/4,z), respectively. According to experiment, the internal parameter z is close to 0.27 (see PRB 79, 014522 (2009)) while in DFT is 0.24 (GGA) or0.23 (LDA). In this tutorial we want to see how much dynamic correlations on iron change this parameter.

The correlations are on Fe ion, which contains around 6 electrons. These electrons are forming fluctuating spin, which is at finite temperature relatively poorly screened inthe solid. While the charge fluctuations are very well screened and behave just as in a Cu metal, the orbital fluctuations are somewhat slower and result in enhanced orbitalsusceptibility. However, the spin degrees of freedom are really slow due to Hund's coupling, which penalizes low spin configurations, therefore spins and very poorlyscreened, hence the Fermi liquid scales can become very small, and the material is a poor metal. This physics was called Hund's metal physics and was first discoveredhere. We start by performing DFT calculation to get good charge density. We create a directory with the name FeSe and copy the FeSe.struct file into it. [Note that Wien2krequires to call directory with the same name as the structure file]. We choose to start with z=0.26 (which is slightly smaller than experimental value), in order to beslightly away from the theoretical minimum. We initialized the DFT by

x init_lapw

and choose the default options, except we will use LDA instead of GGA because LDA+DMFT gives better energies than GGA+DMFT. This is because in LDA theenergies and the band structure are more consistent than in GGA. One way to understand this is to realize that both LDA and GGA give too broad bands and quasiparticle-renormalization Z=1. Therefore both should overbind, and consequently give too small lattice parameters in moderately correlated systems. As is well known, LDA indeedgives too small volumes, while GGA tends to give too large volumes in weakly to moderately correlated materials. As DMFT is expected to reduce binding and increasevolumes in most material (as Z is always < 1), the LDA+DMFT is expected to be universally applicable, while GGA+DMFT is definitely wrong in moderately correlatedsystems, and experience shows that it does not get better in strongly correlated systems. In order to correct energies by DMFT, it is thus better to start with a consistenttheory, such as LDA.

In version 14, we will choose the following options:

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: 5 SELECT ENERGY : -6 Ctrl-X-C continue with kgen : c Ctrl-X-C Ctrl-X-C NUMBER OF K-POINTS : 500 Shift of k-mesh allowed. Do you want to shift: (0=no, 1=shift): 1 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!.

Next, run the DFT calculation using the command

run_lapw

[Of course Wien2k can also be run in parallel, and the Wien2k manual discusses this option in detail.] In DMFT calculation, one needs converged frequency dependentlocal Green's function, hence more k-points is 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 with shifted mesh. It is also advisable to increase the reciprocal cutoff of the DFT basis. Namely, in LAPW we form the basis with the basisfunctions that have envelope form of plane waves. The most oscillating plane wave in the basis is chosen according to criteria K*RMT < RKmax, where RMT is thesmallest muffin-thin radius, and RKmax is a parameter in FeSe.in1, by default 7. We will change it to 9, so that the basis becomes much more precise (but calculation isslower). If you are limited with computer power, you might want to keep RKmax=7.

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+DMFT calculation).

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http://journals.aps.org/prb/abstract/10.1103/PhysRevB.79.014522

http://www.physics.rutgers.edu/~haule/papers/nmat3120.pdf

http://stacks.iop.org/1367-2630/11/i=2/a=025021

http://hauleweb.rutgers.edu/tutorials/files2a/FeSe.struct



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:

There are 4 atoms in the unit cell: 1 Fe 2+ 2 Fe 2+ 3 Se 2- 4 Se 2- Specify correlated atoms (ex: 1-4,7,8): 1-2 You have chosen the following atoms to be correlated: 1 Fe 2+ 2 Fe 2+ Do you want to continue; or edit again? (c/e): c For each atom, specify correlated orbital(s) (ex: d,f): 1 Fe 2+: d 2 Fe 2+: d You have chosen to apply correlations to the following orbitals: 1 Fe 2+-1 d 2 Fe 2+-2 d Do you want to continue; or edit again? (c/e): c Specify qsplit for each correlated orbital (default = 0): Qsplit Description ------ ------------------------------------------------------------ 2 real harmonics basis, no symmetry, except spin (up=dn) ------ ------------------------------------------------------------ 1 Fe 2+-1 d: 2 2 Fe 2+-2 d: 2 Do you want to continue; or edit again? (c/e): c Specify projector type (default = 2): ------ ------------------------------------------------------------ Projector Drscription 5 fixed projector, which is written to projectorw.dat. You can generate projectorw.dat with the tool wavef.py ------ ------------------------------------------------------------ > 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): 1-2 Each set of equivalent correlated problems are listed below: 1 (Fe 2+1 d) (Fe 2+2 d) are equivalent. 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

The init_dmft.py script generates two necessary input files, case.indmfl and case.indmfi. They connect the solid and the impurity with the DMFT engine, respectively. Letus take a closer look at the header of the FeSe.indmfl file:

19 44 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) 2 # number of correlated atoms 1 1 0 # iatom, nL, locrot 2 2 1 # L, qsplit, cix 2 1 0 # iatom, nL, locrot 2 2 2 # 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 19-44. For each k-point we willuse 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 ink-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, here two irons. The forth line specifies which atom (from case.struct) is correlated. These are the first two Fe. 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=2" which we chose during initialization. Each correlated orbital-set gets a unique index during initialization. The

first and second Fe atoms were given index "cix=1" and "cix=2", respectively. For each correlated set, a more detailed specification is needed, which is given in theremaining 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 ================ 2 5 5 # Number of independent kcix blocks, max dimension, max num-independent-components 1 5 5 # cix-num, dimension, num-independent-components #---------------- # Independent components are -------------- 'z^2' 'x^2-y^2' 'xz' 'yz' 'xy' #---------------- # Sigind follows -------------------------- 1 0 0 0 0 0 2 0 0 0 0 0 3 0 0 0 0 0 4 0 0 0 0 0 5 #---------------- # 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 linespecifies the dimension of the first correlated block (here "d" has dimension 5), and the number of orbital components which differ.

The transformation matrix from spheric harmonics to the DMFT basis is given next, and shows how these orbitals are defined: It is the expansion of each of these orbitalsin 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.

The same information (except the first line above) is repeated once more for the second Fe atom.

This file FeSe.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 2 0 0 0 0 0 3 0 0 0 0 0 4 0 0 0 0 0 5

Notice that it contains only one impurity problem, while FeSe.indmfl contains two correlated blocks. Due to the symmetry, the index table ("Sigind") relates bothcorrelated blocks with the impurity problem, hence we do not need to run the impurity solver twice. 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 aDMFT calculation. These include the .struct file as well as the various Wien2K input files such as .inm, .in1, etc. The charge density file, .clmsum, is also copied. This isall 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+EDMFT. Its content is:

solver = 'CTQMC' # impurity solver DCs = 'nominal' # double counting scheme 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 = 50 # 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 = 2e-6 # the charge density precision to stop the LDA+DMFT run ec = 2e-6 # the energy precision to stop the LDA+DMFT run recomputeEF = 1 # Recompute EF in dmft2 step. If recomputeEF = 2, it tries to find an insulating gap. 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" : [5.0 , "# Coulomb repulsion (F0)"], "J" : [0.8 , "# Coulomb repulsion (F0)"], "CoulombF" : ["'Ising'" , "# Can be set to 'Full'"], "beta" : [50 , "# Inverse temperature"], "svd_lmax" : [25 , "# We will use SVD basis to expand G, with this cutoff"], "M" : [5e6 , "# Total number of Monte Carlo steps"], "mode" : ["SH" , "# We will use self-energy sampling, and Hubbard I tail"], "nom" : [200 , "# Number of Matsubara frequency points sampled"], "tsample" : [400 , "# How often to record measurements"], "GlobalFlip" : [1000000 , "# How often to try a global flip"], "warmup" : [3e5 , "# Warmup number of QMC steps"], "nf0" : [6.0 , "# Nominal occupancy nd for double-counting"], }

Note that the order of parameters is not important. Note also that this file is written in python syntax, hence it could be executed as python script. This could be used tocheck for possible errors by executing python params.dat. If no error is reported, the file is syntactically correct.

Below we explain the meaning of the variables

http://hauleweb.rutgers.edu/tutorials/files2a/params.dat



solver = 'CTQMC'

The impurity solver will be Continous Time Quantum Monte Carlo, as implemented here.

DCs = 'nominal' # nominal double-counting

In order to relax the structure, we choose nominal double-counting here, because it is much faster and quite accurate. For example, using DC='nominal' in relaxedstructure, we will get nd=6.28 while using DC='exact' we get nd=6.25.

max_dmft_iterations = 1 max_lda_iterations = 100 finish = 50

Impurity solver is slower than DFT part, hence we will solve one impurity, and converge DFT in each global loop. The maximum number of global loops is set to50. In this picture

we are choosing a single iteration for the red part, maximum 100 in the grey, and 50 iterations of both.

ntail = 300

The number of Matsubara points, distributed in logarithmic mesh, to describe the tail of Greens functions.

cc = 2e-6 # the charge density precision to stop the LDA+DMFT run ec = 2e-6 # the energy precision to stop the LDA+DMFT run

The DFT loop will stop after it reaches the above given limit (max_lda_iterations). The loop will also stop when both the charge and energy are sufficientlyconverged (difference of charge less than cc, and energy less then ec).

recomputeEF = 1

For metals, we must always re-calculate the chemical potential, so that the solid is charge neutral.

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

This is important only when svd_lmax=0 and we sample directly in Matsubara basis. See Tutorial on MnO for more explanation.

"iparams0=...."

For each impurity problem (in this case we have a single impurity) we need a python dictionary, containing parameters for the impurity solver. When more thansingle impurity is needed, one needs to specify "iparams1", "iparams2", ...

"U" : [5.0]

The value of the on-site Coulomb repulsion (Slater F0). Note that in the charge self-consistent DMFT, one needs larger value of U than it is required for downfoldedDMFT calculations, because part of the on-site screening (hybridization screening by many itinerant degrees of freedom) is explicitly taken into account in the suchDMFT calculation. We will calculate the value of U in later tutorial.

"J" : [0.8]

The Hund's coupling. Internally, we use Slater F2 and F4, but here we rather give more commonly used parameter "J", which is related to F2 = 14./1.625 J, and F4 =0.625 * 14./1.625 * J. Notice that "J" can also be a list of two numbers, in which case F2=14./1.625 J[0] and F4 = 0.625 * 14./1.625 * J[1]. For more in-depthdiscussion, see the notes on the Coulomb repulsion.

"CoulombF" : ["'Ising'"]

Here we use the density-density form of the Coulomb repulsion, which speeds up the calculation considerably. However, the Fermi liquid scale is different in thedensity-density and rotationally invariant calculation. We will adress this latter.

"beta" : [50. ]

Inverse temperature in eV. beta=50.0/eV means that temperature about 232 Kelvins.

"svd_lmax" : [25 ],

Instead of sampling self-energy in Matsubara basis, we will project it to the SVD basis, and determine 25 coefficients of expansion. This should give precision betterthan 1e-7.

"M" : [10e6]

Number of Monte Carlo steps carried out on each processor. Notice that the total number of Monte Carlo steps in M*Ncores.

"mode" : ["SH" ]

We will calculate self-energy from the two particle Green's function, as explained here. The tail will be computed by the Hubbard I approximation.

"nom" : [200]

http://www.physics.rutgers.edu/~haule/papers/qmc.pdf

http://hauleweb.rutgers.edu/tutorials/CoulombUexplain.html

http://hauleweb.rutgers.edu/tutorials/files00/HB2_documentation.html



We calculate a finite number of Matsubara frequency points with sampling, while the rest are approximated by an analytic tail, which is much more stable. Typically,we need around nom ~ 3-4*beta points, which corresponds to an energy scale between 6*pi-8*pi ~ 22eV scale, outside which we can expect the analytic tail,computed up to the second order, to be an excellent approximation.

"nf0" : [6.0 ]

The nominal occupancy of the Fe-d shell, which is needed for double-counting. It is evaluated according to the formula:

and

"Ncout" : [1000000]

How often to print information of ctqmc solver to imp.?/nohup_imp.out.

"GlobalFlip" : [1000000 , "# How often to try a global flip"] "warmup" : [3e5 , "# Warmup number of QMC steps"], "tsample" : [400 , "# How often to record measurements"],

Every "GlobalFlip"=1000000 MC steps we try a global flip, where we try to flip all up spin into down spins. We skip first "warmup" MC steps before we startmeasuring observables. Observables are being measured every "tsample" MC steps. Note that it is a good idea to look into imp.0/nohup_imp.out.000 to see howmuch time is spend in measuring the self-energy as compared to accepting the step. It is also a good idea to compare tsample with the Acceptance ratio printed inthe same file.

There is only one more file that we need: A blank self energy (sig.inp) file. Generate it by the command

szero.py

Tutorial 2.b. : Running the jobNow we are ready to submit the job. The executable script is

run_dmft.py

which calls all other executable. To make the code parallel, we need to prepare at least one file mpi_prefix.dat, but it is better two prepare also mpi_prefix.dat2. The firstcontains the parallel command used by ctqmc code, and the second is used by the dmft1 and dmft2 step. An example of creating such command is given byecho "mpirun -np $NSLOTS" > mpi_prefix.datorecho "mpiexec -port $port -np $NSLOTS -machinefile $TMPDIR/machines" > mpi_prefix.dat

but the precise form of the command depends on the supercomputer and its environment. 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

open_mp 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 dmft1, and dmft2 partof 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 environment dependent, and therefore this script might not work on each system. Before submitting the job, make sure that the computing notes have the following environmental variable specified: $WIENROOT, $WIEN_DMFT_ROOT, $SCRATCH.Also make sure that $WIENROOT and $WIEN_DMFT_ROOT are in $PATH as well as $WIEN_DMFT_ROOT is in $PYTHONPATH. Typically, the .bashrc will containthe 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

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 FeSe #$ -pe mpi2 250 #$ -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="FeSe" # 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,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* find . -size 0|xargs rm

The metallic FeSe is a bit harder to converge than the insulating MnO, as it typically takes a few more dmft steps. The impurity and local occupancy could be plotted frominfo.iterate file (column 9 and 10). By executing analizeInfo.py, we get a printout of only dmft steps, and if we plot the 9-th and 10-th column, we should see somethinglike that:

After 30 or so dmft iterations (300 dft iterations), the occupancy converged to roughly 6.3.

If we let the job run for another 20 or so iterations (finish=50), we can get much better estimation of the total energy and free energy. By averaging over last 15 steps, i.e.,steps 35-50, we can estimate the free energy of the system, which should be around -14802.540 (for RKmax=7) or -14802.570 (for RKmax=8) and statistical noise 1e-4Ry (roughly 1meV) using 250 core computer.

Note that this is not the entire free energy yet. To compute the entire free energy, we need to subtract the entropy of the impurity, multiplied by temperature. We will learnhow to compute entropy of the impurity in a subsequent tutorial, but for now it suffices to say that this term is very small at normal temperature as entropy is limited to asmall fraction of log(2) in metallic systems and hence the term T*Simp is smaller than a meV.

Tutorial 2.c.: Computing the Forces on atomsWe want to optimize the position of Se atom, therefore we want to see how large is force on Se atom. In order to compute forces on all atoms, we need to change only asingle variable in FeSe.in2. [We can change that at the beginning of the run, or, during the run, before it converges, so that we have some statistics for the forces.] The firstline in FeSe.in2 file should start with FOR rather than TOT, i.e,

# FeSe.in2, line 1 : FOR (TOT,FOR,QTL,EFG,FERMI)

After this change, FeSe.scf should contain a printout of the total force, for example:

TOTAL FORCE IN mRy/a.u. = |F| Fx Fy Fz with/without FOR in case.in2 :FOR001: 1.ATOM 0.000000 0.000000 0.000000 0.000000 total forces :FOR002: 2.ATOM 19.938644 0.000000 0.000000 -19.938644 total forces TOTAL FORCE WITH RESPECT TO GLOBAL CARTESIAN COORDINATES: :FCA001: 1.ATOM 0.000000 0.000000 0.000000 total forces :FCA002: 2.ATOM 0.000000 0.000000 -19.938644 total forces TOTAL FORCE WITH RESPECT TO THE GLOBAL COORDINATE SYSTEM: :FGL001: 1.ATOM 0.000000000 0.000000000 0.000000000 total forces :FGL002: 2.ATOM 0.000000000 0.000000000 -19.938644357 total forces

This format is identical to the format in Wien2k, and it says that the Se atom has a force in z-direction of -19.94mRy/a.u. (for different RKmax, it might be somewhatdifferent value). We can grep the force at each step by

grep ':FGL002' FeSe.scf

and its plot should look like



It shows the force on Se atom at each dft iteration. It is remarkable how stable is the force. It converges to 19.9 mRy/a.u. as soon as the occupancy is converged, and itsnoise is only around 0.1mRy/a.u. (when using 240 cores).

Tutorial 2.d.: Structural relaxationOnce the calculation is converged, we can start structural relaxation. We can either resubmit the job again, or, just change a few variables while running the code.

1. First, we need to make sure that FeSe.in2 file starts with FOR, as described above, so that the forces are computed inside dmft2 step.

2. Next, make sure that max_lda_iterations in params.dat file is large enough, for example 100, so that dft will successfully relax structure at each dmft step.3. Finally, we will change the FeSe.inm file, so that it starts with mixing method MSR1a, i.e.,

# FeSe.inm, line 1: MSR1a 0.0 YES (BROYD/PRATT, extra charge (+1 for additional e), norm)

In the next iteration, we should start seeing the following lines in FeSe.scf file:

:APOS001 X=0.25000000 Y=0.75000000 Z=0.00000000 : DX= 0.00 DY= 0.00 DZ= 0.00 TOT= 0.00 mau :APOS002 X=0.75000000 Y=0.75000000 Z=0.72948443 : DX= 0.00 DY= 0.00 DZ= -21.28 TOT= 21.28 mau :FRMSA: (mRyd/au) 73.175 103.485 RMS (au) 1.50E-02 MAX 2.13E-02 :F-condition (mRyd/au) 2.000 F

It prints the current positions of the two inequivalent atoms, and the attempted move of each. The third line shows the current condition to stop the relaxation of crystalstructure. When that happens, the code continues to converge the charge at fixed crystal structure, so that at the end charge is very well converged.

The condition for stopping relaxation is the last number (2.000) and is called TRUST in wien2k. The code will attempt structural relaxation as long as forces are aboveTRUST. If all forces are smaller than TRUST for three consecutive dft iterations, and at the same time the charge and energy convergence are better than force_Rho_diffand force_Ene_diff, the structural relaxation is switched off.

We can change TRUST in FeSe.inM file (first line, second number). Let's reduce it to 0.5, because in such simples cases as FeSe we can achieve much better convergence.

We could also reduce force_Rho_diff, which is by default 0.01, or force_Ene_diff, which is 0.001. Both parameters can be added in params.dat file. Let's add parameter

force_Rho_diff = 1e-4

to params.dat file.

After some time passed, we can check how is the structural relaxation progressing:

grep ':FRM' FeSe.scf

should look like:

:FRMSA: (mRyd/au) 658.782 931.658 RMS (au) 2.47E-03 MAX 3.50E-03 :F-condition (mRyd/au) 2.000 F :FRMSA: (mRyd/au) 622.141 879.841 RMS (au) 2.41E-03 MAX 3.41E-03 :F-condition (mRyd/au) 2.000 F :FRMSA: (mRyd/au) 599.243 847.458 RMS (au) 5.76E-02 MAX 8.15E-02 :F-condition (mRyd/au) 2.000 F :FRMSA: (mRyd/au) 73.175 103.485 RMS (au) 1.50E-02 MAX 2.13E-02 :F-condition (mRyd/au) 2.000 F :FRMSA: (mRyd/au) 22.986 32.507 RMS (au) 3.19E-03 MAX 4.50E-03 :F-condition (mRyd/au) 2.000 F :FRMSA: (mRyd/au) 14.839 20.985 RMS (au) 2.63E-03 MAX 3.72E-03 :F-condition (mRyd/au) 2.000 F :FRMSA: (mRyd/au) 4.120 5.826 RMS (au) 2.07E-03 MAX 2.92E-03 :F-condition (mRyd/au) 2.000 F :FRMSA: (mRyd/au) 1.355 1.916 RMS (au) 9.79E-03 MAX 1.38E-02 :F-condition (mRyd/au) 2.000 F :FRMSA: (mRyd/au) 9.552 13.509 RMS (au) 1.60E-02 MAX 2.26E-02 :F-condition (mRyd/au) 2.000 F :FRMSA: (mRyd/au) 5.092 7.201 RMS (au) 1.24E-02 MAX 1.75E-02 :F-condition (mRyd/au) 2.000 F :FRMSA: (mRyd/au) 7.584 10.725 RMS (au) 2.91E-03 MAX 4.11E-03 :F-condition (mRyd/au) 2.000 F :FRMSA: (mRyd/au) 3.919 5.542 RMS (au) 4.41E-03 MAX 6.23E-03 :F-condition (mRyd/au) 2.000 F :FRMSA: (mRyd/au) 0.782 1.105 RMS (au) 4.38E-03 MAX 6.20E-03 :F-condition (mRyd/au) 2.000 F :FRMSA: (mRyd/au) 0.759 1.073 RMS (au) 6.89E-03 MAX 9.75E-03 :F-condition (mRyd/au) 2.000 F :FRMSA: (mRyd/au) 0.727 1.028 RMS (au) 6.85E-03 MAX 9.68E-03 :F-condition (mRyd/au) 2.000 F :FRMSA: (mRyd/au) 0.089 0.126 RMS (au) 5.45E-03 MAX 7.71E-03 :F-condition (mRyd/au) 2.000 F :FRMSA: (mRyd/au) 0.649 0.918 RMS (au) 1.87E-03 MAX 2.65E-03 :F-condition (mRyd/au) 2.000 F :FRMSA: (mRyd/au) 1.281 1.811 RMS (au) 2.39E-03 MAX 3.38E-03 :F-condition (mRyd/au) 2.000 F :FRMSA: (mRyd/au) 0.475 0.671 RMS (au) 2.51E-03 MAX 3.55E-03 :F-condition (mRyd/au) 2.000 F :FRMSA: (mRyd/au) 0.991 1.402 RMS (au) 1.37E-03 MAX 1.94E-03 :F-condition (mRyd/au) 2.000 T :FRMSA: (mRyd/au) 0.056 0.079 RMS (au) 1.14E-04 MAX 1.61E-04 :F-condition (mRyd/au) 2.000 T :FRMSA: (mRyd/au) 0.046 0.066 RMS (au) 9.53E-05 MAX 1.35E-04 :F-condition (mRyd/au) 2.000 T :FRMSA: (mRyd/au) 0.005 0.007 RMS (au) 1.51E-05 MAX 2.14E-05 :F-condition (mRyd/au) 2.000 T :FRMSA: (mRyd/au) 0.425 0.600 RMS (au) 1.06E-05 MAX 1.50E-05 :F-condition (mRyd/au) 2.000 T :FRMSA: (mRyd/au) 0.153 0.216 RMS (au) 1.14E-05 MAX 1.61E-05 :F-condition (mRyd/au) 2.000 T :FRMSA: (mRyd/au) 3.312 4.684 RMS (au) 2.50E-04 MAX 3.53E-04 :F-condition (mRyd/au) 2.000 F :FRMSA: (mRyd/au) 13.764 19.465 RMS (au) 1.90E-04 MAX 2.68E-04 :F-condition (mRyd/au) 2.000 F :FRMSA: (mRyd/au) 7.423 10.498 RMS (au) 2.81E-03 MAX 3.98E-03 :F-condition (mRyd/au) 2.000 F :FRMSA: (mRyd/au) 0.956 1.352 RMS (au) 3.48E-05 MAX 4.92E-05 :F-condition (mRyd/au) 2.000 T :FRMSA: (mRyd/au) 1.352 1.912 RMS (au) 2.41E-03 MAX 3.41E-03 :F-condition (mRyd/au) 2.000 F :FRMSA: (mRyd/au) 1.693 2.394 RMS (au) 3.51E-03 MAX 4.96E-03 :F-condition (mRyd/au) 2.000 F :FRMSA: (mRyd/au) 2.032 2.874 RMS (au) 4.66E-03 MAX 6.60E-03 :F-condition (mRyd/au) 2.000 F :FRMSA: (mRyd/au) 0.223 0.315 RMS (au) 7.32E-03 MAX 1.04E-02 :F-condition (mRyd/au) 2.000 F :FRMSA: (mRyd/au) 1.224 1.731 RMS (au) 8.78E-03 MAX 1.24E-02 :F-condition (mRyd/au) 2.000 F :FRMSA: (mRyd/au) 0.098 0.139 RMS (au) 2.79E-03 MAX 3.94E-03 :F-condition (mRyd/au) 2.000 F :FRMSA: (mRyd/au) 0.804 1.136 RMS (au) 7.93E-04 MAX 1.12E-03 :F-condition (mRyd/au) 2.000 T :FRMSA: (mRyd/au) 0.001 0.001 RMS (au) 1.27E-04 MAX 1.79E-04 :F-condition (mRyd/au) 2.000 T :FRMSA: (mRyd/au) 0.149 0.211 RMS (au) 1.80E-04 MAX 2.55E-04 :F-condition (mRyd/au) 2.000 T :FRMSA: (mRyd/au) 0.001 0.001 RMS (au) 4.60E-05 MAX 6.50E-05 :F-condition (mRyd/au) 2.000 T :FRMSA: (mRyd/au) 0.010 0.014 RMS (au) 1.59E-05 MAX 2.25E-05 :F-condition (mRyd/au) 2.000 T

We have TRUST at 2.0, and force_Rho_diff = 1e-4. The forces are very large at the beginning, but they quickly drop, and somewhere in the middle of the above output, thestop_condition is satisfied 6 times (line ends with "T"). Consequently, the relaxation was stopped at that point, and the charge density was converged to higher precision.But once the charge is converged, the impurity solver was called again, and it updated the self-energy. At this point, the structural relaxation is attempted again, and theforce is again large (but considerably smaller than in the first step).

We can also print the current crystal structure parameter by

grep ':APOS002' FeSe.scf



As the printed parameter is 1-z, we want to print z directly, which can be achieved with a bit of transformation

grep ':APOS002' FeSe.scf| sed 's/Z=/ /'|strans '$j++' '1-#4'

The plot of this z-parameter might look like

We started off with z=0.26, somewhat too small parameter. After a few hundred dft steps (30 dmft steps), we confidently converged to a value close to z=0.27, which is inmuch better agreement with experiment then GGA result (0.24). Of course the precise value of z depends on the value of Hund's coupling and also a bit on RKmax, and itcould be slightly increased by increasing J or decreased by decreasing J. For physically relevant Hund's couplings (0.7-0.9eV), the z parameter is in the range 0.265-0.28.

2.e. Spectral function plotsNext we wanted to plot the spectral function. First we need self-energy on the real axis, therefore we will use once more the maximum entropy method on auxiliaryGreen's function (as explained in a previous tutorial), to analytically continue the self-energy. To reduce the MC noise, we average the self energy (sig.inp.*) files from thelast few steps by

saverage.py saverage.py sig.inp.2[5-9].1

The output is written to sig.inpx. [We could append -o option, if we wanted to change the sig.inpx name.] Next, create a new directory (lets call it maxent), and copy theaveraged 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' : 0.005, # 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 moredense 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, as explained in a previous tutorial.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 all orbitals have Fermi-liquid like behaviour at low energy, with substantial mass enhancement. The most correlated is the "xy" orbital. Notice also that therotationally invariant calculation (CoulombF="Full") would give somewhat more coherent self-energy at low frequency, and should be preferred method of calculatingspectra. We will show this below. 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, andwhile it the new directory, use

http://hauleweb.rutgers.edu/tutorials/files2a/maxent_params.dat



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 isrun on the real axis (0) or the imaginary axis (1). Change it to 0. The header of FeSe.indmfl file should now look like this

19 42 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) 2 # number of correlated atoms

Then, inside the new directory run

x lapw0 -f FeSe x_dmft.py lapw1 x_dmft.py dmft1

Before you execute these lines, you might want to set OMP_NUM_THREADS to a large number (but smaller then the number of cores on your node), such as 4, toparallelize 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 Fe-d (column 3 and 4) are stored in FeSe.cdos:

Next we create a desired k-path in the Brillouin zone. One can do that with the xcrysden tool, or, by editing a script klist3_generate.py. Let's create a path from Gamma, toX and to M and back to Gamma with 240 points in total with the name FeSe.klist_band.

Next, adjust the frequency range in FeSe.indmfl file. Suppose that we want to plot spectra in the window between -1 eV to 1 eV with 200 frequency points. The header fileof FeSe.indmfl should then be corrected to

19 42 1 5 # hybridization band index nemin and nemax, renormalize for interstitials, projection type 0 0.025 0.025 200 -1.000000 1.000000 # matsubara, broadening-corr, broadening-noncorr, nomega, omega_min, omega_max (in eV)

Notice that we changed only the last three numbers of the second row.

Next, in directory with real axis self-energy, run lapw on these k-points by executing

x_dmft.py lapw1 --band x_dmft.py dmftp

The first line computes the Kohn-Sham eigenstates on the new k-points from FeSe.klist_band. The second line computes the frequency dependent eigenvalues, and storesthem to eigvals.dat. Now we can display the spectra by executing

wakplot.py 1.0

Note that you might need to adjust brightness by reducing the argument to a number between 0.0-1.0, so that the excitations are clearly seen in the picture.

The spectra is extremely incoherent here, so that all excitations are washed out. This is because we used quite high temperature, and partially also because we useddensity-density form of the interaction.

2.e.a. Low temperature spectra and Orbitally resolved spectra

Next, we will reduce the temperature a bit, to recover the coherence of the spectral function. We will also use the rotationally invariant Coulomb interaction, which shouldgive better results (less scattering at low energy, but higher effective masses). Note that the rotationally invariant Coulomb interaction never leads to spin-freezing at zerotemperature. Only when the Ising interaction is used, one can get spin-freezing down to lowest temperatures.

To proceed, we will create a new directory, and while in that directory, copy results from previous dmft run into this new directory, i.e.,

http://hauleweb.rutgers.edu/tutorials/files2a/FeSe.klist_band



dmft_copy.py <dmft_previous_run>

We will then change the following parameters in params.dat file:

"CoulombF" : ["'Full'" , "# Can be set to 'Full'"], "beta" : [100 , "# Inverse temperature"], "nom" : [400 , "# Number of Matsubara frequency points sampled"], "M" : [20e6 , "# Total number of Monte Carlo steps"], "mode" : ["SH" , "# We will use self-energy sampling, and Hubbard I tail"], "PChangeOrder" : [0.97 , "# How often to add/rm versus move a kink"], "warmup" : [1e6 , "# Warmup number of QMC steps"],

and resubmit the job. The crucial parameters to change are CoulombF and beta, however, to converge easier, we have adjusted several other parameters. We increasednom, because we increased beta, and we want to keep similar frequency cutoff for sampled self-energy (w=(2*nom+1)*pi/beta). We also increased the number of MCsteps, because the acceptance rate is much smaller when Full interaction is used. We also changed mode to sample the Green's function rather than the self-energy. Thelatter still has some issues when the full Coulomb repulsion is used. In addition, we notice that using the default value for PChangeOrder=0.9, the acceptance ratio foradding/removing kinks is very small (less than 1%) while the acceptance for moving a kink is above 20%. For better statistics, it is a good idea to more often try to add/rma kink. We also notice that the trial time is very short compared to acceptance time, hence we will not increase much the trial time by modifying PChangeOrder. We alsoincreased the warmup, because the first DMFT step will produce very poor self-energy as it starts without a good starting configuration. In the next DMFT step, we canreduce warmup for an order of magnitude.

We will also change the mixing method back to MSR1 in FeSe.inm file, because the structure will not change appreciably anymore. Once we converge the solution withthe full Coulomb repulsion, using previously determined optimized structure, the force should still be appreciably smaller than our previous cutoff (0.5mRy/a.u.). The first iteration is not very good, because we are starting without status.xxx files, therefore the impurity problem starts from atomic configuration. But nevertheless weachieve rapid convergence of the self-energy. After a few dmft steps, we can also increase slightly the number of MC steps, to have better statistics:

"M" : [30e6 , "# Total number of Monte Carlo steps"]

The results are converged within 5-10 DMFT steps. To have some confidence in convergence, we will let the code run for a few more iterations. Here is the plot of totalenergy and free energy as a function of dmft iterations (up to 30 dmft iterations):

We have some MC noise, but the results seem converged after just two iterations. To obtain this plot (printing only the dmft steps), remember to run analizeInfo.py script.

However, to be confident in the convergence, it is important to also check how well is the self-energy converged. Here is the plot obtained on 250 cores:

Clearly, the convergence was not good until iteration 5 or so. But after iteration 10, the results are converged within the statistical noise, in particular at low energy, wheremost of the physics is. Next we will perform the analytic continuation of the self-energy. First average self-energy over a few iterations to improve statistics:

saverage.py sig.inp.2?.1 sig.inp.30.1

The averaged self-energy sig.inpx should look like that:



Then copy sig.inpx into a new directory for analytic continuation, and copy maxent_params.dat from previous analytic continuation. Finally, run

maxent_run.py sig.inpx

to obtain Sig.out, which should look like that:

On this scale, the self-energy does not look very different from the previous self-energy at higher temperature (and Ising-type interaction), however, the imaginary part atzero goes to zero now, so that there is very little scattering left at low frequency. Note that this scattering rate can be read off from the imaginary axis data (sig.inpx) above. We will now repeat several steps from above, to obtain the density of states on the real axis:

create a new directory, for example on_real_axiswhile in the new director, copy converged results into the new directory by

dmft_copy.py ../

in the new directory, overwrite the imaginary axis self-energy with the real axis counterpart

cp ../maxent/Sig.out sig.inp

Change the second line of FeSe.indmfl file, so that the first flag (matsubara) is set to 0. Also change the frequency range for plotting to -1.0 to 1.0, which we willneed in the next step. The second line should look like:

0 0.025 0.025 200 -1.000000 1.000000 # matsubara, broadening-corr, broadening-noncorr, nomega, omega_min, omega_max (in eV)

Obtain DOS by running lapw0, lapw1, and dmft1 steps:

x lapw0 -f FeSe x_dmft.py lapw1 x_dmft.py dmft1

Not surprisingly, the DOS looks quite similar to previous results,

except that the quasiparticle peak is considerably sharper.

Next we copy FeSe.klist_band from previous calculation into the current directory, and re-run Kohn-Sham diagonalization on the selected k-path by

x_dmft.py lapw1 --band

followed by computation of the DMFT eigenvalues

x_dmft.py dmftp

Finally display results by



wakplot.py 0.1

Notice that the intensity (0.1) should be tuned until a nice plot like that

is obtained.

Now we want to resolve the spectra in its orbital components, i.e., xz, yz, and xy orbital. To do that, we will use another executable, which can print out both theeigenvalues and the eigenvectors of the Dyson equation.

First, we will print DMFT projector, which relates the Fe-orbitals with the Kohn-Sham orbitals.

x_dmft.py dmftu -g --band

Note that -g stands here for printing only the specified k-points in the klist, but not their star (equivalent k-points). Moreover --band switch takes FeSe.klist_band ratherthan FeSe.klist.

The self-energy on real axis, with the desired frequency range, should normally be prepared here. However, since we already run x_dmft.py dmftp above, we should havefiles like sig.inp1_band already present.

Next, we will prepare an input file dmftgke.in, which should look like:

e # mode [g/e]: we use mode to compute eigenvalues and eigenvectors BasicArrays.dat # filename for projector 0 # matsubara FeSe.energy # LDA-energy-file, case.energy(so)(updn) FeSe.klist_band # k-list FeSe.rotlm # for reciprocal vectors Udmft.0 # filename for projector 0.0025 # gamma for non-correlated 0.0025 # gammac sig.inp1_band sig.inp2_band # self-energy name, sig.inp(x) eigenvalues.dat # eigenvalues UR.dat # right eigenvector in orbital basis UL.dat # left eigenvector in orbital basis -2. # emin for printed eigenvalues 2. # emax for printed eigenvalues

Note that this input file is a normal text file read by fortran, and the order in which parameters appear is important here. Most of the parameters are self-explanatory. Thefiles like BasicArrays.dat and Udmft.0 and FeSe.rotlm were obtained before when printing projector (x_dmftp.py dmftu). Further FeSe.energy file was obtained by runningx_dmft.py lapw1. Furthermore sig.inp1_band and sig.inp2_band were obtained before when running x_dmft.py dmftp step. Note that the number of files in this line (linenumber 10) needs to match the Number of independent kcix blocks from FeSe.indmfl file. In the case of FeSe, we have two Fe-atoms, and for each we needs to specify theself-energy, although the two self-energies are equal. Finally, the output eigenvalues will be written to eigenvalues.dat and the left and the right eigenvectors will appear inUL.dat UR.dat, respectively. Note that we will print only the projection of the eigenvectors to the orbital basis.

The last two lines contain the range in which the eigenvalues/eigenvectors will be printed. Once you have dmftgke.in file, you can execute

dmftgk dmftgke.in

which prints out the eigenvalues and eigenvectors. You should see that eigenvalues.dat, UL.dat and UR.dat appeared.

Finally, copy the python script

wakplot_sophisticated.py

to your current directory, and edit it. Scroll down to the main part of the script, which should look something like this

if __name__ == '__main__': aspect_scale=0.3 small = 1e-5 DY = 0.0 # intensity = [3.,3.,3.,1.] COHERENCE=True orb_plot=[1,1,2,2,1,1,0,0,0,0,0,0,0,0] #2,2,2,2,2,2] # should have integers 0,1,2 for red,green,blue #orb_plot=[] # plotting unfolded band structure, but all orbits with hot map

To produce similar plot as before, tune the first number in the intensity list to similar number then you used above in wakplot.py. For example, setting

intensity = [0.1,3.,3.,1.]

will make the plot brighter. If you have much larger number of k-points than frequency points, you might also need to tune aspect_scale, to make plot more/less elongated.For example

aspect_scale=1.



will plot similar plot as wakplot.py above.

To display the plot, run

./wakplot_sophisticated.py eigenvalues.dat

Once we have a nice aspect ratio plot, we start adding orbital components. First we will make two symbolic links to our eigenvectors

ln -s UL.dat UL.dat_ ln -s UR.dat UR.dat_

This is because UL.dat_ might be in general different than UL.dat if we were attempting to unfold to a different size Brillouin zone. We will not do that here. Next we will tune orb_plot (in wakplot_sophisticated.py) so that different orbitals are plotted with desired color. We have three color options, i.e., red, green, blue, which

will be assigned numbers 0,1,2. We also have 10 orbitals in total, which corresponds to 5 orbitals of the first Fe atom and 5 of the second Fe atom, which appear in thefollowing order [z^2, x^2-y^2, xz, yz, xy, z^2, x^2-y^2, xz, yz, xz]. This is of course specified in FeSe.indmfl file. We will assign red color to xy orbital, green to xz/yz orbitaland blue to z^2 and x^2-y^2.

# z2, x2, xz, yz, xy, z2, x2, xz, yz, xy orb_plot=[ 2, 2, 1, 1, 0, 2, 2, 1, 1, 0]

The intensity now stands for relative intensity of the three colors (RGB) and transparency (alpha). Lest set it to:

intensity = [3.,3.,3.,1.]

to get

You might notice that the k-points are now displayed with Greek symbol Gamma, rather than the text GAMMA. You can achieve that by replacing word GAMMA inFeSe.klist_band with $\Gamma$, and M by $M$,....

As one can see, the inside hole pocket at Gamma is almost entirely of xz/yz character, the second pocket has a small admixture of z^2/x^2-y^2 to the xz/yz. The outsidehole pocket at Gamma is almost entirely of xy character when it crosses the Fermi level, but it mixes with z^2/x^2-y^2 at x-point. The electron pockets are very yellow,hence equal mixture of xz/yz and xy character.

If we want to make the xy orbital a bit stronger, we might want to increase the first component of intensity, or better, decrease the second and third component of intensity.Furthermore, if we want to make the non-Fe orbitals more transparent, we might want to increase the last number for transparency. For example, the list

# red,green,blue,transparent intensity = [3., 2., 2., 3.]

gives the following plot:

tutorial 2: fese and optimization of the structure

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