Liquid Si - MLFF
Task
Generating a machine learning force field for liquid Si.
Input
POSCAR
- In this example we start from a 64 atom super cell of diamond-fcc Si (the same as in this example: Liquid Si - Standard MD:
Si cubic diamond 2x2x2 super cell of conventional cell
5.43090000000000
2.00000000 0.00000000 0.00000000
0.00000000 2.00000000 0.00000000
0.00000000 0.00000000 2.00000000
Si
64
Direct
0.00000000 0.00000000 0.00000000
0.50000000 0.00000000 0.00000000
0.00000000 0.50000000 0.00000000
0.50000000 0.50000000 0.00000000
0.00000000 0.00000000 0.50000000
0.50000000 0.00000000 0.50000000
0.00000000 0.50000000 0.50000000
0.50000000 0.50000000 0.50000000
0.37500000 0.12500000 0.37500000
0.87500000 0.12500000 0.37500000
0.37500000 0.62500000 0.37500000
0.87500000 0.62500000 0.37500000
0.37500000 0.12500000 0.87500000
0.87500000 0.12500000 0.87500000
0.37500000 0.62500000 0.87500000
0.87500000 0.62500000 0.87500000
0.00000000 0.25000000 0.25000000
0.50000000 0.25000000 0.25000000
0.00000000 0.75000000 0.25000000
0.50000000 0.75000000 0.25000000
0.00000000 0.25000000 0.75000000
0.50000000 0.25000000 0.75000000
0.00000000 0.75000000 0.75000000
0.50000000 0.75000000 0.75000000
0.37500000 0.37500000 0.12500000
0.87500000 0.37500000 0.12500000
0.37500000 0.87500000 0.12500000
0.87500000 0.87500000 0.12500000
0.37500000 0.37500000 0.62500000
0.87500000 0.37500000 0.62500000
0.37500000 0.87500000 0.62500000
0.87500000 0.87500000 0.62500000
0.25000000 0.00000000 0.25000000
0.75000000 0.00000000 0.25000000
0.25000000 0.50000000 0.25000000
0.75000000 0.50000000 0.25000000
0.25000000 0.00000000 0.75000000
0.75000000 0.00000000 0.75000000
0.25000000 0.50000000 0.75000000
0.75000000 0.50000000 0.75000000
0.12500000 0.12500000 0.12500000
0.62500000 0.12500000 0.12500000
0.12500000 0.62500000 0.12500000
0.62500000 0.62500000 0.12500000
0.12500000 0.12500000 0.62500000
0.62500000 0.12500000 0.62500000
0.12500000 0.62500000 0.62500000
0.62500000 0.62500000 0.62500000
0.25000000 0.25000000 0.00000000
0.75000000 0.25000000 0.00000000
0.25000000 0.75000000 0.00000000
0.75000000 0.75000000 0.00000000
0.25000000 0.25000000 0.50000000
0.75000000 0.25000000 0.50000000
0.25000000 0.75000000 0.50000000
0.75000000 0.75000000 0.50000000
0.12500000 0.37500000 0.37500000
0.62500000 0.37500000 0.37500000
0.12500000 0.87500000 0.37500000
0.62500000 0.87500000 0.37500000
0.12500000 0.37500000 0.87500000
0.62500000 0.37500000 0.87500000
0.12500000 0.87500000 0.87500000
0.62500000 0.87500000 0.87500000
KPOINTS
- We will start with a single k point in this example:
K-Points 0 Gamma 1 1 1 0 0 0
INCAR
#Basic parameters ISMEAR = 0 SIGMA = 0.1 LREAL = Auto PREC = FAST ALGO = FAST ISYM = -1 NELM = 100 EDIFF = 1E-4 LWAVE = .FALSE. LCHARG = .FALSE. #Parallelization of ab initio calculations NCORE = 2 #MD IBRION = 0 MDALGO = 2 ISIF = 2 SMASS = 1.0 TEBEG = 2000 NSW = 10000 POTIM = 3.0 #Machine learning paramters ML_FF_LMLFF = .TRUE. ML_FF_ISTART = 0 ML_FF_NWRITE = 2 ML_FF_EATOM = -.70128086E+00
- The user should be familiar at this step how to run a basic molecular dynamics calculations. If not please go through the example here: Liquid Si - Standard MD.
- Machine learning is switched on by setting the following tag: ML_FF_LMLFF=.TRUE..
- By setting the tag ML_FF_ISTART to zero learning from scratch is selected.
- The flag ML_FF_NWRITE=2 selects a more verbose output where the error on energies, forces and stress are output to the ML_LOGFILE file. This setting is very handy to check the accuracy of the force field.
- The tag ML_FF_EATOM=-.70128086E+00 sets the atomic reference energy for each species. How to obtain that energy is explained below.
Calculation
Reference energy
Before the force field for liquid Si can be calculated, the atomic energy of a single Si atom in a large enough box has to be calculated. For that the following steps have to be done:
- Create a new directory Si_ATOM by typing mkdir Si_ATOM and go to that directory cd Si_ATOM.
- Create an INCAR file with the following parameters:
#Basic parameters ISMEAR = 0 SIGMA = 0.1 LREAL = Auto PREC = FAST ALGO = FAST ISYM = 0 NELM = 100 EDIFF = 1E-4 LWAVE = .FALSE. LCHARG = .FALSE. ISPIN = 2
- Create a POSCAR file with a single atom in a large enough box (the box should be orthorombic to have enough degrees of freedom for electronic relaxation):
Si atom
1.00090000000000
12.00000000 0.00000000 0.00000000
0.00000000 12.01000000 0.00000000
0.00000000 0.00000000 12.02000000
Si
1
Direct
0.00000000 0.00000000 0.00000000
- Create a KPOINTS file with a single k point (or copy the one delivered with this example):
test 0 0 0 Gamma 1 1 1 0 0 0
- Copy the POTCAR from the previous directory to this directory (cp ../POTCAR).
- Run the calculation and look at the total energy in the OUTCAR file (or OSZICAR). AFter that switch back to the original directory. That energy will be used for the ML_FF_EATOM tag. If multiple atom types are present in the structure than this step has to be repeated for each atom type separately and the reference energies are provided as a list after the ML_FF_EATOM tag, where the ordering of the energies corresponds to the ordering of the elements in the POTCAR file.
Creating the liquid structure
We will start this example from a perfect super cell of crystalline silicon (fcc diamond structure) containing 64 atoms. The temperature is set to 2000 K so that when an MD is run the melting should occur relatively fast. We will do the melting with on-the-fly learning. This will greatly accelerate the melting since after some time most of the ab initio calculations are skipped and the very fast force field takes over. This calculation will be executed for 10000 steps with a step size of 3 fs.
Please run now the calculation.
After running the calculation we should obtain a fairly good liquid in the CONTCAR file.
As a side effect we have also learned a force field, but with maybe a quite bad trajectory at the beginning. Usually it is more systematic to learn on the pure structures. In our case this means to use the CONTCAR file obtained after the melting. Also the force field was only learned using a single k point. To obtain a better accuracy we will use more k points. So after this step we will start the learning from scratch using the CONTCAR obtained so and using more accurate parameters.
But before we do that we will take a look at the accuracy of the force obtained now.
The main output files for the machine learning are:
- ML_LOGFILE: This contains the main output of the machine learning. Since we use ML_FF_NWRITE=2 the error on energy, forces and stress of the force field compared to ab initio is also written out on this file for every step.
- ML_ABNCAR: This contains the ab initio data used for the learning. It will be needed for continuation runs as ML_ABCAR.
- ML_FFNCAR: This contains the regression results (weights, parameters, etc.). It will be needed for continuation runs as ML_FFCAR.
In the ML_LOGFILE we will look for the last entry on the error of energy, forces and stress, Bayesian error and spilling factor, which should look similar to this (since we run a molecular dynamics calculation in parallel on different computers actual results can deviate a little):
====================================================================================================
Information on error estimations
----------------------------------------------------------------------------------------------------
Error in energy (eV atom^-1): 0.013195
Error in force (eV Angst^-1): 0.238029
Error in stress (kB): 4.671078
Bayesian error (eV Angst-1): 0.074626 0.118880
Spilling factor (-): 0.006025 0.020000
====================================================================================================
The first entry of the Bayesian error is the estimated error from our model. The second entry is the error threshold. In our case it is newly determined during the calculations. The first and second entry for the spilling factor are the calculated spilling factor and the threshold, respectively. The threshold for the spilling factor is usually always kept constant during the calculations.
Next we will look at the accuracy of structural properties of the force field. For the we first run a 3000 fs molecular dynamics calculation with and without the force field starting from the new CONTCAR file. First we save that new CONTCAR file to
cp CONTCAR POSCAR.T2000_relaxed
Now do the following steps to run the force field calculation:
- Copy POSCAR.T2000_relaxed to POSCAR.
- Copy ML_ABNCAR to ML_ABCAR.
- Copy ML_FFNCAR to ML_FFCAR.
- Change INCAR tags:
- Change ML_FF_ISTART=0 to ML_FF_ISTART=2: This will switch of learning and only use the machine-learned force field.
- Change NSW=10000 to NSW=1000.
- Run calculation.
- Copy XDATCAR to XDATCAR.MLFF_3ps.
Now do the following steps to run the ab initio reference calculation:
- Copy POSCAR.T2000_relaxed to POSCAR.
- Change ML_FF_LMLFF=.TRUE. to {TAG|ML_FF_LMLFF}}=.FALSE.: This will turn off the machine learning completely.
- Run caclulation.
- Copy XDATCAR to XDATCAR.AI_3ps.
To analyze the pair correlation function use the PERL script pair_correlation_function.pl:
#!/usr/bin/perl
use strict;
use warnings;
#configuration for which ensemble average is to be calculated
my $confmin=1; #starting index of configurations in XDATCAR file for pair correlation function
my $confmax=20000; #last index of configurations in XDATCAR file for pair correlation function
my $confskip=1; #stepsize for configuration loop
my $species_1 = 1; #species 1 for which pair correlation function is going to be calculated
my $species_2 = 1; #species 2 for which pair correlation function is going to be calculated
#setting radial grid
my $rmin=0.0; #minimal value of radial grid
my $rmax=10.0; #maximum value of radial grid
my $nr=300; #number of equidistant steps in radial grid
my $dr=($rmax-$rmin)/$nr; #stepsize in radial grid
my $tol=0.0000000001; #tolerance limit for r->0
my $z=0; #counter
my $numelem; #number of elements
my @elements; #number of atoms per element saved in list/array
my $lattscale; #scaling factor for lattice
my @b; #Bravais matrix
my $nconf=0; #number of configurations in XDATCAR file
my @cart; #Cartesian coordinates for each atom and configuration
my $atmin_1=0; #first index of species one
my $atmax_1; #last index of species one
my $atmin_2=0; #first index of species two
my $atmax_2; #last index of species two
my @vol; #volume of cell (determinant of Bravais matrix)
my $pi=4*atan2(1, 1); #constant pi
my $natom=0; #total number of atoms in cell
my @pcf; #pair correlation function (list/array)
my $mult_x=1; #periodic repetition of cells in x dimension
my $mult_y=1; #periodic repetition of cells in y dimension
my $mult_z=1; #periodic repetition of cells in z dimension
my @cart_super; #Cartesian cells over multiple cells
my @vec_len; #Length of lattice vectors in 3 spatial coordinates
#my $ensemble_type="NpT"; #Set Npt or NVT. Needs to be set since both have different XDATCAR file.
my $ensemble_type="NVT"; #Set Npt or NVT. Needs to be set since both have different XDATCAR file.
my $av_vol=0; #Average volume in cell
#reading in XDATCAR file
while (<>)
{
chomp;
$_=~s/^/ /;
my @help=split(/[\t,\s]+/);
$z++;
if ($z==2)
{
$lattscale = $help[1];
}
if ($z==3)
{
$b[$nconf+1][1][1]=$help[1]*$lattscale;
$b[$nconf+1][1][2]=$help[2]*$lattscale;
$b[$nconf+1][1][3]=$help[3]*$lattscale;
}
if ($z==4)
{
$b[$nconf+1][2][1]=$help[1]*$lattscale;
$b[$nconf+1][2][2]=$help[2]*$lattscale;
$b[$nconf+1][2][3]=$help[3]*$lattscale;
}
if ($z==5)
{
$b[$nconf+1][3][1]=$help[1]*$lattscale;
$b[$nconf+1][3][2]=$help[2]*$lattscale;
$b[$nconf+1][3][3]=$help[3]*$lattscale;
}
if ($z==7)
{
if ($nconf==0)
{
$numelem=@help-1;
for (my $i=1;$i<=$numelem;$i++)
{
$elements[$i]=$help[$i];
$natom=$natom+$help[$i];
}
}
}
if ($_=~m/Direct/)
{
$nconf=$nconf+1;
#for NVT ensemble only one Bravais matrix exists, so it has to be copied
if ($ensemble_type eq "NVT")
{
for (my $i=1;$i<=3;$i++)
{
for (my $j=1;$j<=3;$j++)
{
$b[$nconf][$i][$j]=$b[1][$i][$j];
}
}
}
for (my $i=1;$i<=$natom;$i++)
{
$_=<>;
chomp;
$_=~s/^/ /;
my @helpat=split(/[\t,\s]+/);
$cart[$nconf][$i][1]=$b[1][1][1]*$helpat[1]+$b[1][1][2]*$helpat[2]+$b[1][1][3]*$helpat[3];
$cart[$nconf][$i][2]=$b[1][2][1]*$helpat[1]+$b[1][2][2]*$helpat[2]+$b[1][2][3]*$helpat[3];
$cart[$nconf][$i][3]=$b[1][3][1]*$helpat[1]+$b[1][3][2]*$helpat[2]+$b[1][3][3]*$helpat[3];
}
if ($ensemble_type eq "NpT")
{
$z=0;
}
}
last if eof;
}
if ($confmin>$nconf)
{
print "Error, confmin larger than number of configurations. Exiting...\n";
exit;
}
if ($confmax>$nconf)
{
$confmax=$nconf;
}
for (my $i=1;$i<=$nconf;$i++)
{
#calculate lattice vector lengths
$vec_len[$i][1]=($b[$i][1][1]*$b[$i][1][1]+$b[$i][1][2]*$b[$i][1][2]+$b[$i][1][3]*$b[$i][1][3])**0.5;
$vec_len[$i][2]=($b[$i][2][1]*$b[$i][2][1]+$b[$i][2][2]*$b[$i][2][2]+$b[$i][2][3]*$b[$i][2][3])**0.5;
$vec_len[$i][3]=($b[$i][3][1]*$b[$i][3][1]+$b[$i][3][2]*$b[$i][3][2]+$b[$i][3][3]*$b[$i][3][3])**0.5;
#calculate volume of cell
$vol[$i]=$b[$i][1][1]*$b[$i][2][2]*$b[$i][3][3]+$b[$i][1][2]*$b[$i][2][3]*$b[$i][3][1]+$b[$i][1][3]*$b[$i][2][1]*$b[$i][3][2]-$b[$i][3][1]*$b[$i][2][2]*$b[$i][1][3]-$b[$i][3][2]*$b[$i][2][3]*$b[$i][1][1]-$b[$i][3][3]*$b[$i][2][1]*$b[$i][1][2];
$av_vol=$av_vol+$vol[$i];
}
$av_vol=$av_vol/$nconf;
#choose species 1 for which pair correlation function is going to be calculated
$atmin_1=1;
if ($species_1>1)
{
for (my $i=1;$i<$species_1;$i++)
{
$atmin_1=$atmin_1+$elements[$i];
}
}
$atmax_1=$atmin_1+$elements[$species_1]-1;
#choose species 2 to which paircorrelation function is calculated to
$atmin_2=1;
if ($species_2>1)
{
for (my $i=1;$i<$species_2;$i++)
{
$atmin_2=$atmin_2+$elements[$i];
}
}
$atmax_2=$atmin_2+$elements[$species_2]-1;
#initialize pair correlation function
for (my $i=0;$i<=($nr-1);$i++)
{
$pcf[$i]=0.0;
}
# loop over configurations, make histogram of pair correlation function
for (my $j=$confmin;$j<=$confmax;$j=$j+$confskip)
{
for (my $k=$atmin_1;$k<=$atmax_1;$k++)
{
for (my $l=$atmin_2;$l<=$atmax_2;$l++)
{
if ($k==$l) {next};
for (my $g_x=-$mult_x;$g_x<=$mult_x;$g_x++)
{
for (my $g_y=-$mult_y;$g_y<=$mult_y;$g_y++)
{
for (my $g_z=-$mult_y;$g_z<=$mult_z;$g_z++)
{
my $at2_x=$cart[$j][$l][1]+$vec_len[$j][1]*$g_x;
my $at2_y=$cart[$j][$l][2]+$vec_len[$j][2]*$g_y;
my $at2_z=$cart[$j][$l][3]+$vec_len[$j][3]*$g_z;
my $dist=($cart[$j][$k][1]-$at2_x)**2.0+($cart[$j][$k][2]-$at2_y)**2.0+($cart[$j][$k][3]-$at2_z)**2.0;
$dist=$dist**0.5;
#determine integer multiple
my $zz=int(($dist-$rmin)/$dr+0.5);
if ($zz<$nr)
{
$pcf[$zz]=$pcf[$zz]+1.0;
}
}
}
}
}
}
}
#make ensemble average, rescale functions and print
for (my $i=0;$i<=($nr-1);$i++)
{
my $r=$rmin+$i*$dr;
if ($r<$tol)
{
$pcf[$i]=0.0;
}
else
{
$pcf[$i]=$pcf[$i]*$av_vol/(4*$pi*$r*$r*$dr*(($confmax-$confmin)/$confskip)*($atmax_2-$atmin_2+1)*($atmax_1-$atmin_1+1));#*((2.0*$mult_x+1.0)*(2.0*$mult_y+1.0)*(2.0*$mult_z+1.0)));
}
print $r," ",$pcf[$i],"\n";
}
Obtain the pair correlation function from the previously saved XDATCAR files:
perl pair_correlation_function.pl XDATCAR.MLFF_3ps > pair_MLFF_3ps.dat perl pair_correlation_function.pl XDATCAR.AI_3ps > pair_AI_3ps.dat
Plot the pair correlation functions using the following command:
gnuplot -e "set terminal jpeg; set xlabel 'r(Ang)'; set ylabel 'PCF'; set style data lines; plot 'pair_MLFF_3ps.dat', 'pair_AI_3ps.dat' " > PC_MLFF_vs_AI_3ps.jpg
The pair correlation functions obtained that way should look similar like that:
We see that pair correlation is quite well reproduced although the error on the force is a little bit large. Nevertheless we just used this learning to very create a liquid structure very fast, since creating a liquid from a perfect crystal can be a time consuming task.
Learning of more accurate force field
After creating the liquid we will restart the learning but with better ab initio parameters and already from a somewhat not so badly equilibrated liquid. Bad ab initio parameters can really limit the accuracy of the learning or even inhibit a proper learning.
We do the following steps to restart learning:
- Copy POSCAR.T2000_relaxed to POSCAR.
- Change INCAR tags:
- Set ML_FF_LMLFF=.FALSE. back to ML_FF_LMLFF=.TRUE..
- Set ML_FF_ISTART=2 back to ML_FF_ISTART=0.
- Set NSW=1000 back to NSW=10000.
- Optionally add k point parallelization wth the KPAR tag (if you are not familiar with this tag leave it be).
- Increase the k mesh in the KPOINTS file. The new file looks like:
test 0 0 0 Gamma 2 2 2 0 0 0
- Run calculation using the standard version of VASP (usually vasp_std).