# CP2K Open Source Molecular Dynamics

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exercises:common:bs

# Band structure of WO$_3$ Lattice

In this exercise, we will carry out band structure calculation using K-point sampling for Cubic lattice WO$_3$. The reference DOS and band structure you can find in this paper

K-point sampling only support GGA functionals.

Band structure calculations using high-level electronic structure theory, such as hybrid functionals are not available in CP2K.

To get the band structure for WO3, only a few changes are required compared to the previous example for calculating the PDOS:

WO3-bs.inp
&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

&DIAGONALIZATION
ALGORITHM STANDARD
&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
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 1) 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 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 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 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:
WO3-cubic.xyz
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.

• Lookup the special points for the $\Gamma$, $X$,$M$,$R$ points in the 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 ""

plt_band.py
import numpy as np
import matplotlib.pyplot as plt

file=np.genfromtxt('WO3.bs')

bs = open ('WO3.bs','r')
num_points, num_k_points, num_bands = int(line[3]),int(line[6]),int(line[8])
sp=[""]*num_points
i=0
for i in range(num_points):
sp[i] = line[7]

new=np.resize(file,(num_k_points*2,num_bands,3))
ener=new[:,:,1:2]
ener=np.resize(new[:,:,1:2],(num_k_points*2,num_bands)).transpose()
num_homo = len(new[:,:,2][1][new[:,:,2][1]==1])
num_lumo = len(new[:,:,2][1][new[:,:,2][1]==0])

for i in range(num_homo):
plt.plot(ener[i,::2],color='k')
for i in range(num_lumo):
plt.plot(ener[i+num_homo,::2],color='k',linestyle='-')

#plt.ylim([-4,6])
plt.xlim([0,num_k_points-1])
plt.xticks(np.linspace(0,num_k_points-1,num_points),sp)
plt.ylabel("Energy (eV)")
plt.show()