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: 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.
This is useful to speed up the calculation Download here
$ gunzip t_c_g.dat.gz tio2_pbe-restart.wfn.gz
&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 ./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
# 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
#
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
.cif = crystallographic information file
#====================================================================== # 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