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Table of Contents
How to Converge the CUTOFF and REL_CUTOFF
Introduction
QUICKSTEP, as with nearly all ab initio Density Functional Theory
simulation packages, requires the use of a real-space (RS)
integration grid to represent certain functions, such as the
electron density and the product Gaussian functions. QUICKSTEP
uses a multi-grid system for mapping the product Gaussians onto the
RS grid(s), so that wide and smooth Gaussian functions are mapped
onto a coarser grid than narrow and sharp Gaussians. The electron
density is always mapped onto the finest grid.
Choosing a fine enough integration grid for a calculation is crucial in obtaining meaningful and accurate results. In this tutorial, we will show the reader how to systematically find the correct settings for obtaining a sufficiently fine integration grid for his/her calculation.
This tutorial assumes the reader already has some knowledge of how
to perform a simple energy calculation using QUICKSTEP (this can
be found in tutorial: Calculating Energy and Forces using Quickstep).
A completed example from an earlier calculation can be obtained from the file converging_grid.tgz that comes with this tutorial. The calculations were carried out using CP2K version 2.4.
''QUICKSTEP'' Multi-Grid
Before we go through the input file, it is worthwhile to explain
how the multi-grid is constructed in QUICKSTEP, and how the
Gaussians are mapped onto the different grid levels. Hopefully this
will offer the reader a clear picture of how the key control
parameters affect the grids, and thus the overall accuracy of a
calculation.
All multi-grid related settings for a calculation is controlled via
keywords in MULTIGRID subsection of DFT subsection in
FORCE_EVAL. The number of levels for the multi-grid is defined by
NGRIDS, and by default this is set to 4. The keyword CUTOFF
defines the planewave cutoff (default unit is in Ry) for the
finest level of the multi-grid. The higher the planewave cutoff,
the finer the grid. The corresponding planewave cutoffs for the
subsequent grid levels (from finer to coarser) are defined by the
formula:
\begin{equation*}
E^i_{\mathrm{cut}} = \frac{E_{\mathrm{cut}}^1}
{\alpha^{(i-1)}}
\end{equation*}
where \(\alpha\) has a default value of 3.0, and since CP2K
versions 2.0, can be configured by the keyword
PROGRESSION_FACTOR. Therefore, the higher the value of CUTOFF
the finer grid for all multi-grid levels.
Having constructed the multi-grid, QUICKSTEP then needs to map
the Gaussians onto the grids. The keyword REL_CUTOFF controls
which product Gaussians are mapped onto which level of the
multi-grid. CP2K tries to map each Gaussian onto a grid such
that the number of grid points covered by the Gaussian—no matter
how wide or narrow—are roughly the same. REL_CUTOFF defines the
planewave cutoff of a reference grid covered by a Gaussian with
unit standard deviation (\(e^{\vert\vec{r}\vert^2}\)). A Gaussian
is mapped onto the coarsest level of the multi-grid, on which the
function will cover number of grid points greater than or equal to
the number of grid points \(e^{\lvert\vec{r}\rvert^2}\) will cover on
a reference grid defined by REL_CUTOFF.
Therefore, the two most important keywords effecting the
integration grid and the accuracy of a calculation are CUTOFF and
REL_CUTOFF. If CUTOFF is too low, then all grids will be coarse
and the calculation may become inaccurate; and if REL_CUTOFF is
too low, then even if you have a high CUTOFF, all Gaussians will
be mapped onto the coarsest level of the multi-grid, and thus the
effective integration grid for the calculation may still be too
coarse.
Example: Bulk Si with 8 atoms in a cubic cell
We demonstrate the process using an example based on Bulk Si with 8 atoms in a face centred cubic unit cell.
Template Input File
To systematically find the best CUTOFF and REL_CUTOFF values which are sufficient for a given accuracy (say, \(10^{-6}\) Ry in total energy), we need to perform a series of single point energy calculations. It is much easier to use a set of scripts that can automate this process.
To do this, we first write a template input file: template.inp,
as shown below:
&GLOBAL
PROJECT Si_bulk8
RUN_TYPE ENERGY
PRINT_LEVEL MEDIUM
&END GLOBAL
&FORCE_EVAL
METHOD Quickstep
&DFT
BASIS_SET_FILE_NAME BASIS_SET
POTENTIAL_FILE_NAME GTH_POTENTIALS
&MGRID
NGRIDS 4
CUTOFF LT_cutoff
REL_CUTOFF LT_rel_cutoff
&END MGRID
&QS
EPS_DEFAULT 1.0E-10
&END QS
&SCF
SCF_GUESS ATOMIC
EPS_SCF 1.0E-6
MAX_SCF 1
ADDED_MOS 10
CHOLESKY INVERSE
&SMEAR ON
METHOD FERMI_DIRAC
ELECTRONIC_TEMPERATURE [K] 300
&END SMEAR
&DIAGONALIZATION
ALGORITHM STANDARD
&END DIAGONALIZATION
&MIXING
METHOD BROYDEN_MIXING
ALPHA 0.4
BETA 0.5
NBROYDEN 8
&END MIXING
&END SCF
&XC
&XC_FUNCTIONAL PADE
&END XC_FUNCTIONAL
&END XC
&END DFT
&SUBSYS
&KIND Si
ELEMENT Si
BASIS_SET SZV-GTH-PADE
POTENTIAL GTH-PADE-q4
&END KIND
&CELL
SYMMETRY CUBIC
A 5.430697500 0.000000000 0.000000000
B 0.000000000 5.430697500 0.000000000
C 0.000000000 0.000000000 5.430697500
&END CELL
&COORD
Si 0.000000000 0.000000000 0.000000000
Si 0.000000000 2.715348700 2.715348700
Si 2.715348700 2.715348700 0.000000000
Si 2.715348700 0.000000000 2.715348700
Si 4.073023100 1.357674400 4.073023100
Si 1.357674400 1.357674400 1.357674400
Si 1.357674400 4.073023100 4.073023100
Si 4.073023100 4.073023100 1.357674400
&END COORD
&END SUBSYS
&PRINT
&TOTAL_NUMBERS ON
&END TOTAL_NUMBERS
&END PRINT
&END FORCE_EVAL
We go through this input file quickly. Readers who have gone
through the tutorial on how to perform a simple static energy and
force calculation using QUICKSTEP should have no trouble in
understanding most parts the above input.
Some noticeable settings are:
&GLOBAL PROJECT Si_bulk8 RUN_TYPE ENERGY PRINT_LEVEL MEDIUM &END GLOBAL
The keyword RUN_TYPE is set to ENERGY, this tells CP2K to only
calculate the energies of the system, forces will not be
calculated. Since we are only interested in the convergence of the
integration grid, just looking at the total energy usually suffices;
and since we will be performing a series of computations, the
cheaper each run is the better. We set PRINT_LEVEL to MEDIUM, so
that the information about how many Gaussian functions are mapped
onto which grid are printed. We need this information to analyse the
suitability of the chosen REL_CUTOFF value.
The most important part in the template input is:
&MGRID NGRIDS 4 CUTOFF LT_cutoff REL_CUTOFF LT_rel_cutoff &END MGRID
The symbols LT_cutoff and LT_rel_cutoff are markers, which
the automated scripts will search for and replace with the relevant
values. The default units for both CUTOFF and REL_CUTOFF are
Ry.
In SCF subsection, we have set
MAX_SCF 1
So that no self-consistent loops will be performed. This is okay for checking the integration grid, because irrespective of self-consistency, grid settings with fine enough meshes should give consistent energies.
Converging ''CUTOFF''
We start by setting REL_CUTOFF to a relatively high number, and systematically vary CUTOFF. Setting REL_CUTOFF to 60 Ry is usually sufficient for most calculations, and in any case this will be checked later when we vary REL_CUTOFF.
Generating Inputs
We want to perform a series of calculations, with CUTOFF ranging
from 50 Ry to 500 Ry in steps of 50 Ry. From experience, the
desired CUTOFF for an accuracy of \(10^{-6}\) Ry for the total
energy should be well within this range. To do this, we first need
to make sure the basis and pseudopotential parameter files
BASIS_SET and GTH_POTENTIALS are in the working directory
together with template.inp, then one can write a bash script,
such as the file cutoff_inputs.sh shown below:
#!/bin/bash cutoffs="50 100 150 200 250 300 350 400 450 500" basis_file=BASIS_SET potential_file=GTH_POTENTIALS template_file=template.inp input_file=Si_bulk8.inp rel_cutoff=60 for ii in $cutoffs ; do work_dir=cutoff_${ii}Ry if [ ! -d $work_dir ] ; then mkdir $work_dir else rm -r $work_dir/* fi sed -e "s/LT_rel_cutoff/${rel_cutoff}/g" \ -e "s/LT_cutoff/${ii}/g" \ $template_file > $work_dir/$input_file cp $basis_file $work_dir cp $potential_file $work_dir done
The user should remember to set the permission of the new script file to be executable:
chmod u+x ./cutoff_inputs.sh
Entering the command line
./cutoff_inputs.sh
generates directories cutoff_50Ry, cutoff_100Ry, …,
each containing BASIS_SET, GTH_POTENTIALS and an input file
Si_bulk8.inp, which is exactly the same as template.inp, except
that REL_CUTOFF is set to 60, and CUTOFF is set to the respective
values in the range between 50 Ry and 500 Ry.
Running Calculations
With the input files generated and checked, the next step is to
run them. A bash script such as cutoff_run.sh shown below does
the job:
#!/bin/bash cutoffs="50 100 150 200 250 300 350 400 450 500" cp2k_bin=cp2k.popt input_file=Si_bulk8.inp output_file=Si_bulk8.out no_proc_per_calc=2 no_proc_to_use=16 counter=1 max_parallel_calcs=$(expr $no_proc_to_use / $no_proc_per_calc) for ii in $cutoffs ; do work_dir=cutoff_${ii}Ry cd $work_dir if [ -f $output_file ] ; then rm $output_file fi mpirun -np $no_proc_per_calc $cp2k_bin -o $output_file $input_file & cd .. mod_test=$(echo "$counter % $max_parallel_calcs" | bc) if [ $mod_test -eq 0 ] ; then wait fi counter=$(expr $counter + 1) done wait
The above script is slightly complex, because it allows several
jobs to run in parallel. Setting the variable cp2k_bin defines
the path to the CP2K binary. In this case, the parallel version
cp2k.popt is found in the system PATH. no_proc_per_calc sets
the number of MPI processes to be used in parallel for each
job. no_proc_to_use sets the total number of processors to be
used for running all of the jobs. In the above example, the jobs
are run on a 24 core local workstation, a total of 16 cores are
used for performing the CUTOFF convergence test calculations,
and 2 cores are used for each calculation. This means up to 8 jobs
will run in parallel, until the jobs are exhausted from the list
given in cutoffs.
The reader can write their own script where they see fit, and if he/she just want the jobs to run in serial, then there is no need for this complexity.
Again
chmod u+x ./cutoff_run.sh
followed by
./cutoff_run.sh &
runs the calculations in the background. This calculation only took a couple of minutes to complete on our local workstation.
Analysing Results
After all of the calculations have finished, all the information
about total energies and distribution of Gaussians on the
multi-grid are written in the Si_bulk8.out files in each job
directories.
The total energy can be found in the section of the output shown
below (in this example from cutoff_100Ry/Si_bulk8.out):
SCF WAVEFUNCTION OPTIMIZATION
Step Update method Time Convergence Total energy Change
------------------------------------------------------------------------------
Trace(PS): 32.0000000000
Electronic density on regular grids: -31.9999999980 0.0000000020
Core density on regular grids: 31.9999999944 -0.0000000056
Total charge density on r-space grids: -0.0000000036
Total charge density g-space grids: -0.0000000036
1 NoMix/Diag. 0.40E+00 0.4 1.10090760 -32.3804557631 -3.24E+01
1 NoMix/Diag. 0.40E+00 0.4 1.10090760 -32.3804557631 -3.24E+01
*** SCF run NOT converged ***
Electronic density on regular grids: -31.9999999980 0.0000000020
Core density on regular grids: 31.9999999944 -0.0000000056
Total charge density on r-space grids: -0.0000000036
Total charge density g-space grids: -0.0000000036
Overlap energy of the core charge distribution: 0.00000000005320
Self energy of the core charge distribution: -82.06393942512820
Core Hamiltonian energy: 16.92855916540793
Hartree energy: 42.17635056223367
Exchange-correlation energy: -9.42142606564066
Electronic entropic energy: 0.00000000000000
Fermi energy: 0.00000000000000
Total energy: -32.38045576307407
Regexp search
"^[ \t]*Total energy:"
will find the relevant line.
Similarly, information on distribution of Gaussians on the multi-grid can be found in the section:
------------------------------------------------------------------------------- ---- MULTIGRID INFO ---- ------------------------------------------------------------------------------- count for grid 1: 2720 cutoff [a.u.] 50.00 count for grid 2: 5000 cutoff [a.u.] 16.67 count for grid 3: 2760 cutoff [a.u.] 5.56 count for grid 4: 16 cutoff [a.u.] 1.85 total gridlevel count : 10496
which tells us that for CUTOFF of 100 Ry and REL_CUTOFF of 60
Ry, 2720 product Gaussians has been distributed to grid level 1,
the finest level, 5000 for level 2, 2760 for level 3 and 16 for
level 4, the coarsest. The planewave cutoff for each multi-grid
level can be read from the right-hand-side columns. Here [a.u.]
means the Hartree energy unit, 1 Ha = 2 Ry.
It is much easier if we can gather all the information together
into one file, which allows us to plot the results. This can be
done, again, by using a simple script. cutoff_analyse.sh shown
below is such an example:
#!/bin/bash cutoffs="50 100 150 200 250 300 350 400 450 500" input_file=Si_bulk8.inp output_file=Si_bulk8.out plot_file=cutoff_data.ssv rel_cutoff=60 echo "# Grid cutoff vs total energy" > $plot_file echo "# Date: $(date)" >> $plot_file echo "# PWD: $PWD" >> $plot_file echo "# REL_CUTOFF = $rel_cutoff" >> $plot_file echo -n "# Cutoff (Ry) | Total Energy (Ha)" >> $plot_file grid_header=true for ii in $cutoffs ; do work_dir=cutoff_${ii}Ry total_energy=$(grep -e '^[ \t]*Total energy' $work_dir/$output_file | awk '{print $3}') ngrids=$(grep -e '^[ \t]*QS| Number of grid levels:' $work_dir/$output_file | \ awk '{print $6}') if $grid_header ; then for ((igrid=1; igrid <= ngrids; igrid++)) ; do printf " | NG on grid %d" $igrid >> $plot_file done printf "\n" >> $plot_file grid_header=false fi printf "%10.2f %15.10f" $ii $total_energy >> $plot_file for ((igrid=1; igrid <= ngrids; igrid++)) ; do grid=$(grep -e '^[ \t]*count for grid' $work_dir/$output_file | \ awk -v igrid=$igrid '(NR == igrid){print $5}') printf " %6d" $grid >> $plot_file done printf "\n" >> $plot_file done
Type
chmod u+x ./cutoff_analyse.sh
and then run it using
./cutoff_analyse.sh
will produce a file named cutoff_data.ssv, which looks like:
# Grid cutoff vs total energy
# Date: Mon Jan 20 21:20:34 GMT 2014
# PWD: /home/tong/tutorials/converging_grid/sample_output
# REL_CUTOFF = 60
# Cutoff (Ry) | Total Energy (Ha) | NG on grid 1 | NG on grid 2 | NG on grid 3 | NG on grid 4
50.00 -32.3795329864 5048 5432 16 0
100.00 -32.3804557631 2720 5000 2760 16
150.00 -32.3804554850 2032 3016 5432 16
200.00 -32.3804554982 1880 2472 3384 2760
250.00 -32.3804554859 264 4088 3384 2760
300.00 -32.3804554843 264 2456 5000 2776
350.00 -32.3804554846 56 1976 5688 2776
400.00 -32.3804554851 56 1976 3016 5448
450.00 -32.3804554851 0 2032 3016 5448
500.00 -32.3804554850 0 2032 3016 5448
The data shows that given the REL_CUTOFF value of 60 Ry, setting CUTOFF to 250 Ry and above would give an error in total energy less than \(10^{-8}\) Ha. The reader may also notice that as CUTOFF increases, the number of Gaussians being assigned to the finest grids decreases. Therefore, simply increasing CUTOFF without increasing REL_CUTOFF may eventually lead to a slow convergence in energy, as more and more Gaussians get pushed to coarser grid levels, negating the increase in CUTOFF.
In this example, the test results point to 250 Ry as a good choice for CUTOFF, as the total energy is converged, and the distribution of Gaussian functions on the grids are reasonable: it is the lowest cutoff energy where the finest grid level is used, but at the same time with the majority of the Gaussians on the coarser grids.
Converging ''REL_CUTOFF''
In the next step, we vary the value of REL_CUTOFF while keeping CUTOFF fixed at 250 Ry.
Generating Inputs
For the energy convergence test with varying REL_CUTOFF, we
follow a similar procedure as that for CUTOFF. Using the same
template input file template.inp, we can write a script called
rel_cutoff_inputs.sh:
#!/bin/bash rel_cutoffs="10 20 30 40 50 60 70 80 90 100" basis_file=BASIS_SET potential_file=GTH_POTENTIALS template_file=template.inp input_file=Si_bulk8.inp cutoff=250 for ii in $rel_cutoffs ; do work_dir=rel_cutoff_${ii}Ry if [ ! -d $work_dir ] ; then mkdir $work_dir else rm -r $work_dir/* fi sed -e "s/LT_cutoff/${cutoff}/g" \ -e "s/LT_rel_cutoff/${ii}/g" \ $template_file > $work_dir/$input_file cp $basis_file $work_dir cp $potential_file $work_dir done
Setting the permission for the script to “executable”, and running
it produces directories rel_cutoff_10Ry, rel_cutoff_20Ry, …,
each containing files BASIS_SET, GTH_POTENTIALS and an input
Si_bulk8.inp, which is identical to template.inp, except that
CUTOFF is set to 250, and REL_CUTOFF is set to 10, 20, …,
100 respectively.
Running Calculations
Again to run the calculations, we can use the script
rel_cutoff_run.sh, as shown below:
#!/bin/bash rel_cutoffs="10 20 30 40 50 60 70 80 90 100" cp2k_bin=cp2k.popt input_file=Si_bulk8.inp output_file=Si_bulk8.out no_proc_per_calc=2 no_proc_to_use=16 counter=1 max_parallel_calcs=$(expr $no_proc_to_use / $no_proc_per_calc) for ii in $rel_cutoffs ; do work_dir=rel_cutoff_${ii}Ry cd $work_dir if [ -f $output_file ] ; then rm $output_file fi mpirun -np $no_proc_per_calc $cp2k_bin -o $output_file $input_file & cd .. mod_test=$(echo "$counter % $max_parallel_calcs" | bc) if [ $mod_test -eq 0 ] ; then wait fi counter=$(expr $counter + 1) done wait
In the above example, again, we have used 16 cores in total, and with each job using 2 MPI processes. To run the jobs, use:
./rel_cutoff_run.sh &
Analysing Results
Total energies and distribution of Gaussian functions on the multi-grid are obtained the same way from the results as that for the CUTOFF calculations.
To put all of the results from the REL_CUTOFF calculations in
one place, we can make some minor modifications to
cutoff_analyse.sh and save it as rel_cutoff_analyse.sh:
#!/bin/bash rel_cutoffs="10 20 30 40 50 60 70 80 90 100" input_file=Si_bulk8.inp output_file=Si_bulk8.out plot_file=rel_cutoff_data.ssv cutoff=250 echo "# Rel Grid cutoff vs total energy" > $plot_file echo "# Date: $(date)" >> $plot_file echo "# PWD: $PWD" >> $plot_file echo "# CUTOFF = ${cutoff}" >> $plot_file echo -n "# Rel Cutoff (Ry) | Total Energy (Ha)" >> $plot_file grid_header=true for ii in $rel_cutoffs ; do work_dir=rel_cutoff_${ii}Ry total_energy=$(grep -e '^[ \t]*Total energy' $work_dir/$output_file | awk '{print $3}') ngrids=$(grep -e '^[ \t]*QS| Number of grid levels:' $work_dir/$output_file | \ awk '{print $6}') if $grid_header ; then for ((igrid=1; igrid <= ngrids; igrid++)) ; do printf " | NG on grid %d" $igrid >> $plot_file done printf "\n" >> $plot_file grid_header=false fi printf "%10.2f %15.10f" $ii $total_energy >> $plot_file for ((igrid=1; igrid <= ngrids; igrid++)) ; do grid=$(grep -e '^[ \t]*count for grid' $work_dir/$output_file | \ awk -v igrid=$igrid '(NR == igrid){print $5}') printf " %6d" $grid >> $plot_file done printf "\n" >> $plot_file done
Making the script executable, and running the script using
./rel_cutoff_analyse.sh
produces the following results written in file
rel_cutoff_data.ssv:
# Rel Grid cutoff vs total energy
# Date: Mon Jan 20 00:45:14 GMT 2014
# PWD: /home/tong/tutorials/converging_grid/sample_output
# CUTOFF = 250
# Rel Cutoff (Ry) | Total Energy (Ha) | NG on grid 1 | NG on grid 2 | NG on grid 3 | NG on grid 4
10.00 -32.3902980020 0 0 2032 8464
20.00 -32.3816384686 0 264 4088 6144
30.00 -32.3805115576 0 2032 3016 5448
40.00 -32.3805116025 56 1976 3016 5448
50.00 -32.3804555002 264 2456 5000 2776
60.00 -32.3804554859 264 4088 3384 2760
70.00 -32.3804554859 1880 2472 3384 2760
80.00 -32.3804554859 1880 2472 3384 2760
90.00 -32.3804554848 2032 3016 5432 16
100.00 -32.3804554848 2032 3016 5432 16
The results show that as one increases the value of REL_CUTOFF, more Gaussians get mapped onto the finer grids. The error in total energy reduces to less than \(10^{-8}\) Ha when REL_CUTOFF is greater or equal to 60 Ry. The results thus indicate that 60 Ry is indeed a suitable choice for the value of REL_CUTOFF.
So finally we conclude that the setting
&MGRID CUTOFF 250 REL_CUTOFF 60 &END MGRID
is sufficient for a calculation with the required accuracy.