CP2K Open Source Molecular Dynamics

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====== TiO$_2$ Band Gap as a function of %hfx ======

One problem with standard DFT is that correlation effects can lead to errors in evaluating certain system properties, such as the band gap of semiconductors. More information can be found here: [[doi>10.1126/science.1158722]]

A common approach to correct for electronic self-interaction is the //ad hoc// mixing of a fraction of the exact Hartree-Fock exchange. The goal of this exercise is to identify the needed amount of exact Hartree-Fock exchange (%hfx) to correctly reproduce the anatase TiO$_2$ experimental band gap (3.2 eV). To do so, you will need to run at least 4 single point calculations on bulk TiO$_2$ with varying amount of exact exchange. 
 
Report the result as follows:
^%hfx^E$_{gap}$^
| 0  | ... |
| 10 | ... |
| .. | ... |
Then plot E$_{gap}$  as a function of %hfx. By interpolating the data on the plot, the needed amount of hfx to correctly reproduce the experimental band gap can be determined.

<note tip>
Hybrid calculations can be fairly expensive, because they scale with $\mathcal{O}(N^4)$. Therefore, you should run these jobs on 16 nodes. To further decrease the cost of the calculation, you can restart from previously generated wave-function files (*-RESTART.wfn). At the end of this exercise, a readily optimized wave-function is provided to you.
</note>

===== Questions =====
  - What is the relation between %hfx and $E_\text{gap}$?
  - What is the correct amount of exchange needed to reproduce the experimental band gap?

===== Required files =====
==== Parameters for Truncated Coulomb Potential ====
{{t_c_g.dat.gz| Download here}}
  
==== Restart wave-function ====
 This is useful to speed up the calculation
 {{tio2_pbe-restart.wfn.gz| Download here}}

<note> These files are compressed with gzip. To unpack them run:
<code>
$ gunzip t_c_g.dat.gz tio2_pbe-restart.wfn.gz
</code>
</note>

==== Input File ====
<code - anatase_25hfx.inp>

&GLOBAL
  PROJECT anatase_25hfx         ! Project name, remember to change it for each calculation 
   RUN_TYPE ENERGY       
&END GLOBAL

&FORCE_EVAL
  METHOD Quickstep
  &DFT
   ! external data-files for basis-set, pseudo-potentials and wave-function to restart from
   BASIS_SET_FILE_NAME ./BASIS_TiO
   POTENTIAL_FILE_NAME ./POTENTIALS_TiO
   RESTART_FILE_NAME   ./tio2_pbe-restart.wfn
   
    &MGRID              ! section required to define the cutoff of the grids in the program
      CUTOFF 400        ! for this system the default value is too small 
    &END MGRID          ! and can lead to non-physical results
    
    &PRINT              ! section required to obtain the HOMO-LUMO gap.  
       &MO_CUBES
       WRITE_CUBE .FALSE. ! no cube file is generated
       NHOMO 1            ! but we require 1 HOMO and 1 LUMO  in the output 
       NLUMO 1            ! so that we get the band gap 
       &END
    &END
    
    &SCF                   ! ensures convergence of the SCF cycle
      EPS_SCF 1.0E-6        
      SCF_GUESS RESTART    ! here we specify to restart from an external wave-function (name specified above)
      MAX_SCF 40
      &OUTER_SCF
        EPS_SCF 1.0E-6
        MAX_SCF 10
      &END
      &OT
         PRECONDITIONER FULL_ALL
         ENERGY_GAP 0.01
      &END OT
    &END SCF
    
    &XC                       ! this is the section to define the electronic exchange  
      &XC_FUNCTIONAL          ! our functional is hybrid 
         &PBE                 ! it has 75% of PBE 
           SCALE_X 0.75
           SCALE_C 1.0
         &END
         &PBE_HOLE_T_C_LR
           SCALE_X 0.25       ! + 25% of truncated PBE0 functional - that includes exact hfx
           CUTOFF_RADIUS 3.5  ! that has interaction truncated at 3.5 A from the atomic core 
         &END
      &END XC_FUNCTIONAL
      &HF
        FRACTION 0.25         ! this is the hfx section. The amount of hfx must be consistent with above
        &SCREENING            ! Screening of the electronic repulsion up to the given threshold. This section is needed
          EPS_SCHWARZ 1.0E-6
          SCREEN_ON_INITIAL_P TRUE  ! having an external wave-function, a preliminary screening
        &END                        ! can be performed to speed up calculations
        &MEMORY
          MAX_MEMORY 800            ! uses some memory to store data and not recompute each step
        &END
        &INTERACTION_POTENTIAL      ! Sets up interaction potential between the two regions
          POTENTIAL_TYPE TRUNCATED  ! the potential is  truncated 
          CUTOFF_RADIUS 3.5         ! at 3.5 A (see above)
          T_C_G_DATA ./t_c_g.dat    ! external file with parameters needed to truncate the potential
        &END
      &END
     &END
  &END DFT
  &SUBSYS
    &CELL
       ABC 3.7842 3.7842 9.5146    ! anatase crystal unit cell
       MULTIPLE_UNIT_CELL  2 2 1   ! a 2x2x1 system is required to get realistic results
    &END CELL
    &TOPOLOGY
            MULTIPLE_UNIT_CELL 2 2 1 ! a 2x2x1 system is required to get realistic results. This has to be repeated here.
            COORD_FILE_FORMAT CIF    ! specifies the type and name of coordinate file
            COORD_FILE_NAME tio.cif  
    &END
    &KIND O                          ! external basis and pseudo-potentials for Ti and O 
      BASIS_SET cpFIT3
      POTENTIAL GTH-PBE-q6
    &END KIND
    &KIND Ti
      BASIS_SET FIT
      POTENTIAL GTH-PBE-q12
    &END KIND
  &END SUBSYS
&END FORCE_EVAL
</code>

==== Basis Set ====
<code - BASIS_TiO>

# O
O cpFIT3
 5
 1  0  0  1  1
 0.27061 1.0
 1  0  0  2  1
 0.88493 1.0
 8.50409 -0.23485406547006335010
 1  1  1  1  1
 0.31040 1.0
 1  1  1  2  1
 1.38256 1.0
 6.08264 0.34108818521609718388
 1  2  2  1  1
 1.00000000          1.00000000

#Ti
Ti FIT
           11
            2           0           0           1           1
    4.0993528      0.99999994
            2           0           0           1           1
    1.3253083      0.99999994
            2           0           0           1           1
   0.54136300      0.99999994
            2           0           0           1           1
   0.10175999      0.99999994
            2           1           1           1           1
    8.9996204      0.99999994
            2           1           1           1           1
    1.6365473      0.99999994
            2           1           1           1           1
   0.54614919      0.99999994
            2           2           2           1           1
    3.6633232      0.99999994
            2           2           2           1           1
   0.88037348      0.99999994
            2           2           2           1           1
   0.23061574      0.99999994
            2           3           3           1           1
   0.49442869      0.99999994

</code>

==== Pseudo-Potential ====
<code - POTENTIALS_TiO>
#
O GTH-PBE-q6
    2    4
     0.24455430    2   -16.66721480     2.48731132
    2
     0.22095592    1    18.33745811
     0.21133247    0
#
Ti GTH-PBE-q12
    4    6    2
     0.38000000    2     8.71144218    -0.70028677
    3
     0.33777078    2     2.57526386     3.69297065
                                       -4.76760461
     0.24253135    2    -4.63054123     8.87087502
                                      -10.49616087
     0.24331694    1    -9.40665268

</code>

==== Anatse Crystal Structure ====

''.cif'' = crystallographic information file

<code - tio.cif>
#======================================================================

# ANATASE CRYSTAL DATA

#----------------------------------------------------------------------

data_phase_1


_pd_phase_name                         'Ti O2'
_cell_length_a                         3.78920
_cell_length_b                         3.78920
_cell_length_c                         9.53700
_cell_angle_alpha                      90
_cell_angle_beta                       90
_cell_angle_gamma                      90
_symmetry_space_group_name_H-M         'I 41/a m d'
_symmetry_Int_Tables_number            141

loop_
_symmetry_equiv_pos_as_xyz
   'x, y, z'
   '-x+1/2, -y+1/2, z+1/2'
   '-y, x+1/2, z+1/4'
   'y+1/2, -x, z+3/4'
   '-x+1/2, y, -z+3/4'
   'x, -y+1/2, -z+1/4'
   'y+1/2, x+1/2, -z+1/2'
   '-y, -x, -z'
   '-x, -y+1/2, -z+1/4'
   'x+1/2, y, -z+3/4'
   'y, -x, -z'
   '-y+1/2, x+1/2, -z+1/2'
   'x+1/2, -y+1/2, z+1/2'
   '-x, y, z'
   '-y+1/2, -x, z+3/4'
   'y, x+1/2, z+1/4'
   'x+1/2, y+1/2, z+1/2'
   '-x, -y, z'
   '-y+1/2, x, z+3/4'
   'y, -x+1/2, z+1/4'
   '-x, y+1/2, -z+1/4'
   'x+1/2, -y, -z+3/4'
   'y, x, -z'
   '-y+1/2, -x+1/2, -z+1/2'
   '-x+1/2, -y, -z+3/4'
   'x, y+1/2, -z+1/4'
   'y+1/2, -x+1/2, -z+1/2'
   '-y, x, -z'
   'x, -y, z'
   '-x+1/2, y+1/2, z+1/2'
   '-y, -x+1/2, z+1/4'
   'y+1/2, x, z+3/4'

loop_
   _atom_site_label
   _atom_site_occupancy
   _atom_site_fract_x
   _atom_site_fract_y
   _atom_site_fract_z
   _atom_site_adp_type
   _atom_site_U_iso_or_equiv
   _atom_site_type_symbol
   Ti         1.0     0.000000      0.000000      0.000000     Uiso  0.008060 Ti
   O          1.0     0.000000      0.000000      0.207900     Uiso  0.012040 O


</code>