======= Projected density of states and Band structure for WO$_3$ =======
In this exercise, you will carry out Density Of States(DOS) and band structure calculation using K-point sampling for Cubic lattice WO$_3$. The reference DOS and band structure you can find in [[http://pubs.acs.org/doi/abs/10.1021/cm3032225|this paper]]
{{:exercises:2017_uzh_cmest:wo3.jpeg?1200|}}
====== Getting the PDOS ======
In the following exercise we are going to look at the density of states of WO3:
Similar to the previous exercise we write the coordinates in term of the unit cell:
&GLOBAL
PROJECT WO3-pdos
RUN_TYPE ENERGY
PRINT_LEVEL MEDIUM
&END GLOBAL
&FORCE_EVAL
METHOD Quickstep
&DFT
BASIS_SET_FILE_NAME BASIS_MOLOPT
POTENTIAL_FILE_NAME POTENTIAL
&POISSON
PERIODIC XYZ
&END POISSON
&SCF
SCF_GUESS ATOMIC
EPS_SCF 1.0E-6
MAX_SCF 300
ADDED_MOS 100
&DIAGONALIZATION
ALGORITHM STANDARD
EPS_ADAPT 0.01
&END DIAGONALIZATION
&SMEAR ON
METHOD FERMI_DIRAC
ELECTRONIC_TEMPERATURE [K] 300
&END SMEAR
&MIXING
METHOD BROYDEN_MIXING
ALPHA 0.2
BETA 1.5
NBROYDEN 8
&END MIXING
&END SCF
&XC
&XC_FUNCTIONAL PBE
&END XC_FUNCTIONAL
&END XC
&PRINT
&PDOS
# print all projected DOS available:
NLUMO -1
# split the density by quantum number:
COMPONENTS
&END
&END PRINT
&END DFT
&SUBSYS
&CELL
ABC [angstrom] 3.810000 3.810000 3.810000
PERIODIC XYZ
MULTIPLE_UNIT_CELL 2 2 2
&END CELL
&TOPOLOGY
MULTIPLE_UNIT_CELL 2 2 2
&END TOPOLOGY
&COORD
SCALED
W 0.0 0.0 0.0
O 0.5 0.0 0.0
O 0.0 0.5 0.0
O 0.0 0.0 0.5
&END
&KIND W
ELEMENT W
BASIS_SET DZVP-MOLOPT-SR-GTH
POTENTIAL GTH-PBE
&END KIND
&KIND O
ELEMENT O
BASIS_SET DZVP-MOLOPT-SR-GTH
POTENTIAL GTH-PBE
&END KIND
&END SUBSYS
&END FORCE_EVAL
The replication of the unit cell is necessary since the program samples only at the $\Gamma$ point unless instructed otherwise and we will otherwise do get a meaningful sampling of the density of states (e.g. the grid over the Brillouin Zone will be too coarse). Another option (which we will look into in the next exercise) is to sample over k-points instead.
What you will get in addition to the output file is a file named ''WO3_pdos-k1-1.pdos'' (to be precise, you will get one such file per atom kind but here we only have one, carbon) with a content similar to:
# Projected DOS for atomic kind W at iteration step i = 0, E(Fermi) = 0.144475 a.u.
# MO Eigenvalue [a.u.] Occupation s py pz px d-2 d-1 d0 d+1 d+2 f-3 f-2 f-1 f0 f+1 f+2 f+3
1 -2.621088 2.000000 0.87115225 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000256 0.00000000 0.00000256 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000
2 -2.621080 2.000000 0.85006340 0.00131878 0.00166813 0.00134861 0.00000000 0.00000000 0.00515023 0.00000000 0.00441289 0.00006412 0.00000000 0.00003847 0.00012977 0.00003934 0.00000000 0.00006557
[...]
The columns correspond to the orbitals present in the basis set (hence //projected// DOS). You would now do a convolution plot using a gaussian to get a smooth DOS
To the convoluted DOS, you may want to check [[http://wiki.wpi.edu/deskinsgroup/Density_of_States| this website]]. Here, it is provided two Python script to do the convolution.
Download two files [[http://wiki.wpi.edu/images//images/b/b7/Pdos.txt|pdos.py]] and [[http://wiki.wpi.edu/images//images/4/41/Get-smearing-pdos.txt| get-smearing-pdos.py]] to your folder. And execute the Python script using
python get-smearing-pdos.py file.pdos
Alternatively, you could also use the [[https://raw.githubusercontent.com/dev-zero/cp2k-tools/master/scripts/cp2k_pdos.py|Python script]] developed by Tiziano: python cp2k_pdos.py -h
Different $\sigma$ values give you different convolution, which mean the lineshape is different. A reasonable $\sigma$ value is required to get a good PDOS plot. When visualize the PDOS, only energy region close to the Fermi level is interesting. One need to adapt the xrange properly.
Please also note the unit of the energy, it is in $E_h$. When looking at DOS plots you may want to convert it to Electronvolt instead. In the convolution program, this has been done in the code.
While some of the new options to help with convergence are of numerical nature, [[howto:static_calculation#adding_smearing|the smearing is not]].
* Repeat the above calculation for the different multiple cells 3x3x3, 4x4x4
* Get the Total DOS and PDOS of O$_2p$ and W$_5d$ orbitals and compare to the literature value.
* Do you see why it is necessary to do the unit cell replication?
* What is the value of WO$_3$ band gap? Compare the plots for 3x3x3 and 4x4x4.
* .. which state ($s$, $p_x$, ..) is mainly responsible for that?
* Change the $\sigma$ (or $width$) value in convolution program, what effect does it have on the PDOS plot ?
====== Getting the band structure of WO$_3$ Lattice ======
To get the band structure for WO3, only a few changes are required compared to the previous example for [[PDOS|calculating the PDOS]]:
&GLOBAL
PROJECT WO3-kp-bs
RUN_TYPE ENERGY
PRINT_LEVEL MEDIUM
&END GLOBAL
&FORCE_EVAL
METHOD Quickstep
&DFT
BASIS_SET_FILE_NAME BASIS_MOLOPT
POTENTIAL_FILE_NAME POTENTIAL
&POISSON
PERIODIC XYZ
&END POISSON
&QS
EXTRAPOLATION USE_GUESS ! required for K-Point sampling
&END QS
&SCF
SCF_GUESS ATOMIC
EPS_SCF 1.0E-6
MAX_SCF 300
ADDED_MOS 2
&DIAGONALIZATION
ALGORITHM STANDARD
EPS_ADAPT 0.01
&END DIAGONALIZATION
&SMEAR ON
METHOD FERMI_DIRAC
ELECTRONIC_TEMPERATURE [K] 300
&END SMEAR
&MIXING
METHOD BROYDEN_MIXING
ALPHA 0.2
BETA 1.5
NBROYDEN 8
&END MIXING
&END SCF
&XC
&XC_FUNCTIONAL PBE
&END XC_FUNCTIONAL
&END XC
&KPOINTS
SCHEME MONKHORST-PACK 3 3 3
WAVEFUNCTIONS REAL
SYMMETRY .FALSE.
FULL_GRID .FALSE.
PARALLEL_GROUP_SIZE -1
&END KPOINTS
&PRINT
&BAND_STRUCTURE
ADDED_MOS 2
FILE_NAME WO3.bs
&KPOINT_SET
UNITS B_VECTOR
SPECIAL_POINT ??? #GAMA
SPECIAL_POINT ??? #X
SPECIAL_POINT ??? #M
SPECIAL_POINT ??? #GAMA
SPECIAL_POINT ??? #R
SPECIAL_POINT ??? #M
NPOINTS ???
&END
&END BAND_STRUCTURE
&END PRINT
&END DFT
&SUBSYS
&CELL
ABC [angstrom] 3.810000 3.810000 3.810000
PERIODIC XYZ
MULTIPLE_UNIT_CELL 1 1 1
&END CELL
&TOPOLOGY
MULTIPLE_UNIT_CELL 1 1 1
&END TOPOLOGY
&COORD
SCALED
W 0.0 0.0 0.0
O 0.5 0.0 0.0
O 0.0 0.5 0.0
O 0.0 0.0 0.5
&END
&KIND W
ELEMENT W
BASIS_SET DZVP-MOLOPT-SR-GTH
POTENTIAL GTH-PBE
&END KIND
&KIND O
ELEMENT O
BASIS_SET DZVP-MOLOPT-SR-GTH
POTENTIAL GTH-PBE
&END KIND
&END SUBSYS
&END FORCE_EVAL
At present, it is not possible to get the projected density of states when doing a K-Point calculation. The special points should be given in terms of the b-vectors.
Some notes on the input file:
* By specifying the ''KPOINT'' section you are enabling the K-Point calculation.
* While you could specify the K-Points directly, we are using the Monkhorst-Pack scheme [(http://journals.aps.org/prb/abstract/10.1103/PhysRevB.13.5188)] to generate them. The numbers following the keyword ''MONKHORST-PACK'' specify the tiling of the brillouin zone.
* After the basic calculation, CP2K calculates the energies along certain lines, denoted as ''KPOINT_SET'' (when you check [[https://manual.cp2k.org/trunk/CP2K_INPUT/FORCE_EVAL/DFT/PRINT/BAND_STRUCTURE/KPOINT_SET.html|the documentation]] you will note that this section can be repeated).
* The keyword ''NPOINTS'' specifies how many points (in the addition to the starting point) should be sampled between two special points.
* The ''SPECIAL_POINT'' keyword is used to specify the start-, mid- and endpoints of the line. Those points usually denote special points in the reciprocal lattice/unit cell, like the $\Gamma$, $M$ or $K$ point. You can find the definition for these in the appendix section of [[http://www.sciencedirect.com/science/article/pii/S0927025610002697|this paper]]. This keyword can also be specified multiple times, making it possible to directly get the band structure for a complete //path//.
You are encouraged to use [[ http://tools.materialscloud.org/seekpath|SeeK-path Tool]] when doing the sampling via K-Points to get a skeleton input file for CP2K with the important paths in the reciprocal space. Give SeeK-path the following as your XYZ file and specify a simple cubic cell with the lattice constant $a$ as specified below as well:
4
WO3; a=3.810000
W 0.000000 0.000000 0.000000
O 1.905000 0.000000 0.000000
O 0.000000 1.905000 0.000000
O 0.000000 0.000000 1.905000
Now, when you run this input file you will get in addition the the output file, a file named ''WO3.bs'' which will look similar to the following:
SET: 1 TOTAL POINTS: 26
POINT 1 ******** ******** ********
POINT 2 ******** ******** ********
POINT 3 ******** ******** ********
POINT 4 ******** ******** ********
POINT 5 ******** ******** ********
POINT 6 ******** ******** ********
Nr. 1 Spin 1 K-Point 0.00000000 0.00000000 0.00000000
20
-73.66652408 -38.53370023 -37.80464132 -37.79327769
-16.71308703 -16.11075946 -16.02553853 -1.43495530
-1.34739188 -1.33357408 0.37912017 0.38948689
0.39582882 0.40030859 0.46965212 0.47418816
2.60728842 2.62105342 3.16044140 6.99806305
Nr. 2 Spin 1 K-Point 0.00000000 0.10000000 0.00000000
20
-73.66647294 -38.53337818 -37.80859042 -37.79536623
-16.67479677 -16.09554462 -15.96731960 -1.68492873
-1.44087258 -1.34318045 0.09257368 0.13769271
0.21643888 0.38447849 0.44179002 0.45757924
2.61768501 3.02368022 3.51828287 7.06644645
[...]
For each set there is a block named ''SET'' with the special points listed as ''POINT'', followed by sub-blocks for each K-Point containing the energies for each MO.
Your tasks:
* Lookup the special points for the $\Gamma$, $X$,$M$,$R$ points in the [[http://pubs.acs.org/doi/abs/10.1021/cm3032225|mentioned paper]] (make sure you choose the right lattice). Calculate and plot the band structure for WO3 lattice along $\Gamma$, $X$,$M$,$\Gamma$,$R$,$M$ (you are free to decide whether to use multiple K-Point sets are multiple special points in a single set). Mark the special points. Choose an appropriate number of interpolation points to get a smooth plot.
* Compare your plot with plots from literature. What is different?
* How many orbital energies do you get and why? Try to change the input to get more unoccupied orbitals.
To convert the band structure file to a file which can be plotted directly, you can use the script ''cp2k_bs2csv.py'' from below, which when passed a band structure file ''WO3.bs'' as an argument will write files ''WO3.bs-set-1.csv'' for each set containing the K-Point coordinates and the energies in one line.
To plot the ''WO3.bs-set-1.csv'' file, you can either load it into $MATLAB$ or use $GNUPLOT$ command line.
gnuplot>set yrange [-8:14]
gnuplot>plot for [i=4:23] "WO3.bs.set-1.csv" u 0:i w l t ""
#!/usr/bin/env python
"""
Convert the CP2K band structure output to CSV files
"""
import re
import argparse
SET_MATCH = re.compile(r'''
[ ]*
SET: [ ]* (?P\d+) [ ]*
TOTAL [ ] POINTS: [ ]* (?P\d+) [ ]*
\n
(?P
[\s\S]*?(?=\n.*?[ ] SET|$) # match everything until next 'SET' or EOL
)
''', re.VERBOSE)
SPOINTS_MATCH = re.compile(r'''
[ ]*
POINT [ ]+ (?P\d+) [ ]+ (?P\S+) [ ]+ (?P\S+) [ ]+ (?P\S+)
''', re.VERBOSE)
POINTS_MATCH = re.compile(r'''
[ ]*
Nr\. [ ]+ (?P\d+) [ ]+
Spin [ ]+ (?P\d+) [ ]+
K-Point [ ]+ (?P\S+) [ ]+ (?P\S+) [ ]+ (?P\S+) [ ]*
\n
[ ]* (?P\d+) [ ]* \n
(?P
[\s\S]*?(?=\n.*?[ ] Nr|$) # match everything until next 'Nr.' or EOL
)
''', re.VERBOSE)
if __name__ == '__main__':
parser = argparse.ArgumentParser(description=__doc__)
parser.add_argument('bsfilename', metavar='bandstructure-file', type=str,
help="the band structure file generated by CP2K")
args = parser.parse_args()
with open(args.bsfilename, 'r') as fhandle:
for kpoint_set in SET_MATCH.finditer(fhandle.read()):
filename = "{}.set-{}.csv".format(args.bsfilename,
kpoint_set.group('setnr'))
set_content = kpoint_set.group('content')
with open(filename, 'w') as csvout:
print(("writing point set {}"
" (total number of k-points: {totalpoints})"
.format(filename, **kpoint_set.groupdict())))
print(" with the following special points:")
for point in SPOINTS_MATCH.finditer(set_content):
print(" {pointnr}: {a}/{b}/{c}".format(
**point.groupdict()))
for point in POINTS_MATCH.finditer(set_content):
results = point.groupdict()
results['values'] = " ".join(results['values'].split())
csvout.write("{a} {b} {c} {values}\n".format(**results))