Part 1: Melting silicon¶

1 Solid cubic-diamond silicon$\uparrow$

By the end of this tutorial, you will be able to:

• state what ab-initio molecular dynamics (MD) refers to
• create supercells and handle the Crystallographic Information File (CIF) format using pymatgen
• tell the difference between vasp_gam and vasp_std
• distinguish and plot different energies in the context of MD calculations over time steps using py4vasp
• plot MD trajectories using py4vasp

Perform an ab-initio MD simulation for cubic-diamond (cd) silicon for 90fs with 64 atoms in a canonical ensemble using the Nosé-Hoover thermostat at 2000K.

In MD simulations, the motion of atoms (or molecules) at a specific temperature is simulated by means of the classical equation of motion. In other words, each iteration simulates a time step, where atoms are treated as classical particles subject to forces as in Newton's second law. When these forces are computed quantum mechanically using ab-initio methods, one speaks of ab-initio MD. To employ the canonical ensemble, or NVT ensemble, the calculation must be done at constant number of particles (N), constant volume (V) and constant temperature (T).

To include the effect of temperature, some kind of thermostat needs to be included. Methods to achieve that involve modifying the equations of motion either by introducing stochastic or deterministic terms through additional dynamical variables, which mimic the action of a heat bath in a real thermostat. The Nosé-Hoover thermostat corresponds to the latter. A possible deficiency of the Nosé-Hoover thermostat is the lack of ergodicity in small or stiff systems, for instance in the simulation of a single butane molecule, but it is perfectly suitable for the present example.

In order to learn more about MD algorithms in VASP and how the effect of temperature is included by means of the Nosé-Hoover thermostat in this calculation, read the linked VASP Wiki articles.

1.2 Input¶

The input files to run this example should be prepared at $TUTORIALS/md-part1/e01_solid-cd-Si. The conventional unit cell for the cd silicon structure is provided as a Crystallographic Information File (CIF). CIFs are the standard for crystallographic data exchange prescribed by the International Union of Crystallography. To create a POSCAR file based on a CIF you can use the Python Materials Genomics (pymatgen) package, which is an open-source Python library for materials analysis. Many MD calculations are done with a supercell spanning multiple conventional unit cells. This is, to capture all lattice vibrations that affect the essential dynamics of the system. In practice, it is expedient to carefully check the convergence of the quantity of interest with respect to the cell size. In pymatgen, you can easily read CIFs, create supercells, and write the appropriate POSCAR file as you can see in the code cell below. Execute the code! In [2]: from pymatgen.core import Structure my_struc = Structure.from_file("./e01_solid-cd-Si/Si_mp-149_conventional_standard.cif") # make a 2x2x2 supercell my_struc.make_supercell(2) # write supercell to POSCAR format with specified filename my_struc.to(fmt="poscar", filename="./e01_solid-cd-Si/POSCAR") # A POSCAR file will appear at$TUTORIALS/md-part1/e01_solid-cd-Si
# You may need to refresh the file browser to see it


Why do we use a supercell to perform MD simulations?

The size of the supercell imposes a limit on the maximum wavelength of lattice vibrations. The supercell used in an MD simulation should be large enough to account for all vibration modes with significant contribution to the specific quantity of interest to be computed in MD. This can be estimated, e.g., from an appropriate phonon calculation, or from a series of MD simulations with different supercell sizes.

Furthermore, in calculations considering for instance an absorbate-substrate problem, or simulations of gases and liquids, the size of the unit cell should be large enough to remove unphysical interactions between atoms and their periodic images. Note that, the same holds also for relaxations of such systems.

In summary, for your MD simulation, you should choose a supercell large enough to ensure an ergodic simulation and capture all long-wavelength vibrations of your system.

POSCAR

Click to see POSCAR file generated from CIF!
Si64
1.0
10.937456 0.000000 0.000000
0.000000 10.937456 0.000000
0.000000 0.000000 10.937456
Si
64
direct
0.125000 0.375000 0.125000 Si
0.125000 0.375000 0.625000 Si
0.125000 0.875000 0.125000 Si
0.125000 0.875000 0.625000 Si
0.625000 0.375000 0.125000 Si
0.625000 0.375000 0.625000 Si
0.625000 0.875000 0.125000 Si
0.625000 0.875000 0.625000 Si
0.000000 0.000000 0.250000 Si
0.000000 0.000000 0.750000 Si
1.000000 0.500000 0.250000 Si
1.000000 0.500000 0.750000 Si
0.500000 0.000000 0.250000 Si
0.500000 0.000000 0.750000 Si
0.500000 0.500000 0.250000 Si
0.500000 0.500000 0.750000 Si
0.125000 0.125000 0.375000 Si
0.125000 0.125000 0.875000 Si
0.125000 0.625000 0.375000 Si
0.125000 0.625000 0.875000 Si
0.625000 0.125000 0.375000 Si
0.625000 0.125000 0.875000 Si
0.625000 0.625000 0.375000 Si
0.625000 0.625000 0.875000 Si
1.000000 0.250000 0.000000 Si
1.000000 0.250000 0.500000 Si
1.000000 0.750000 0.000000 Si
1.000000 0.750000 0.500000 Si
0.500000 0.250000 0.000000 Si
0.500000 0.250000 0.500000 Si
0.500000 0.750000 1.000000 Si
0.500000 0.750000 0.500000 Si
0.375000 0.375000 0.375000 Si
0.375000 0.375000 0.875000 Si
0.375000 0.875000 0.375000 Si
0.375000 0.875000 0.875000 Si
0.875000 0.375000 0.375000 Si
0.875000 0.375000 0.875000 Si
0.875000 0.875000 0.375000 Si
0.875000 0.875000 0.875000 Si
0.250000 0.000000 0.000000 Si
0.250000 0.000000 0.500000 Si
0.250000 0.500000 0.000000 Si
0.250000 0.500000 0.500000 Si
0.750000 0.000000 0.000000 Si
0.750000 0.000000 0.500000 Si
0.750000 0.500000 1.000000 Si
0.750000 0.500000 0.500000 Si
0.375000 0.125000 0.125000 Si
0.375000 0.125000 0.625000 Si
0.375000 0.625000 0.125000 Si
0.375000 0.625000 0.625000 Si
0.875000 0.125000 0.125000 Si
0.875000 0.125000 0.625000 Si
0.875000 0.625000 0.125000 Si
0.875000 0.625000 0.625000 Si
0.250000 0.250000 0.250000 Si
0.250000 0.250000 0.750000 Si
0.250000 0.750000 0.250000 Si
0.250000 0.750000 0.750000 Si
0.750000 0.250000 0.250000 Si
0.750000 0.250000 0.750000 Si
0.750000 0.750000 0.250000 Si
0.750000 0.750000 0.750000 Si


INCAR

SYSTEM = cd Si

! ab initio
ISMEAR = 0        ! Gaussian smearing
SIGMA  = 0.1      ! smearing in eV

LREAL  = Auto     ! projection operators in real space

ALGO   = VeryFast ! RMM-DIIS for electronic relaxation
PREC   = Low      ! precision
ISYM   = 0        ! no symmetry imposed

! MD
IBRION = 0        ! MD (treat ionic dgr of freedom)
NSW    = 30       ! no of ionic steps
POTIM  = 3.0      ! MD time step in fs

MDALGO = 2        ! Nosé-Hoover thermostat
SMASS  = 1.0      ! Nosé mass

TEBEG  = 2000     ! temperature at beginning
TEEND  = 2000     ! temperature at end
ISIF   = 2        ! update positions; cell shape and volume fixed

KPOINTS

Gamma-point only
0
Monkhorst Pack
1 1 1
0 0 0

POTCAR

Pseudopotentials of Si.

Check the tags set in the INCAR file!

The first six tags, ISMEAR, SIGMA, LREAL, ALGO, PREC and ISYM, concern the computation of the Kohn–Sham (KS) orbitals.

The remaining tags specify the settings for the MD simulation. IBRION = 0 switches on MD. Then, NSW and POTIM set the number of ionic updates and the step size. For MD, the step size is a unit of time given in fs.

MDALGO sets the thermostat. We use the Nosé-Hoover thermostat, where the flow of energy between the physical system and the heat reservoir is regulated by the thermal inertia, or Nosé mass, SMASS. To realize an NVT ensemble, the temperature at the beginning and at the end, i.e., TEBEG and TEEND, are equal and the volume is kept fixed with ISIF = 2.

The KPOINTS file specifies a single $\vec{k}$ point: The so-called 𝝘 point at $\vec{k}=(0,0,0)$. This is enough, because you are using a large supercell.

1.3 Calculation¶

Open a terminal, navigate to this example's directory and run VASP by entering the following:

cd \$TUTORIALS/md-part1/e01_*
mpirun -np 2 vasp_gam


This calculation is done using vasp_gam, which got its name from the fact that it can only perform calculations at the 𝝘 point. Underneath that implies that the KS orbitals can be considered to be real valued. That means if you run the same calculation with vasp_gam and vasp_std, vasp_gam is faster.

Let us have a closer look at the standard output (stdout), that is printed to the terminal during the calculation. Note that every run is different because it starts from a different random seed.

Click to see the stdout!
 running on    2 total cores
distrk:  each k-point on    2 cores,    1 groups
distr:  one band on    1 cores,    2 groups
vasp.6.3.0 16May21 (build Oct 27 2021 15:52:58) gamma-only

POSCAR found type information on POSCAR Si
POSCAR found :  1 types and      64 ions
scaLAPACK will be used
LDA part: xc-table for Pade appr. of Perdew
POSCAR, INCAR and KPOINTS ok, starting setup
FFT: planning ...
prediction of wavefunctions initialized - no I/O
entering main loop
N       E                     dE             d eps       ncg     rms          rms(c)
RMM:   1     0.155222068701E+04    0.15522E+04   -0.44526E+04   162   0.508E+02
RMM:   2     0.297589951477E+03   -0.12546E+04   -0.12725E+04   162   0.145E+02
RMM:   3    -0.664488972878E+02   -0.36404E+03   -0.43415E+03   162   0.787E+01
RMM:   4    -0.220663864053E+03   -0.15421E+03   -0.16002E+03   162   0.560E+01
RMM:   5    -0.287411476155E+03   -0.66748E+02   -0.65206E+02   162   0.316E+01
RMM:   6    -0.320343488885E+03   -0.32932E+02   -0.28274E+02   162   0.212E+01
RMM:   7    -0.334632556790E+03   -0.14289E+02   -0.12331E+02   162   0.124E+01
RMM:   8    -0.340563651457E+03   -0.59311E+01   -0.53511E+01   162   0.792E+00
RMM:   9    -0.344491928264E+03   -0.39283E+01   -0.38231E+01   349   0.472E+00
RMM:  10    -0.344908101129E+03   -0.41617E+00   -0.41412E+00   352   0.165E+00
RMM:  11    -0.344951509187E+03   -0.43408E-01   -0.32463E-01   344   0.581E-01
RMM:  12    -0.344962136179E+03   -0.10627E-01   -0.10202E-01   336   0.281E-01    0.358E+01
RMM:  13    -0.340710112381E+03    0.42520E+01   -0.23514E+00   417   0.230E+00    0.221E+01
RMM:  14    -0.338768179258E+03    0.19419E+01   -0.54002E+00   463   0.369E+00    0.108E+00
RMM:  15    -0.338786847044E+03   -0.18668E-01   -0.10702E-01   324   0.600E-01    0.444E-01
RMM:  16    -0.338800009272E+03   -0.13162E-01   -0.16991E-02   327   0.294E-01    0.381E-01
RMM:  17    -0.338801421308E+03   -0.14120E-02   -0.26217E-03   321   0.108E-01    0.696E-02
RMM:  18    -0.338801478964E+03   -0.57656E-04   -0.60546E-04   262   0.494E-02
1 T=  2000. E= -.32251463E+03 F= -.33880148E+03 E0= -.33880148E+03  EK= 0.16287E+02 SP= 0.00E+00 SK= 0.00E+00
bond charge predicted
N       E                     dE             d eps       ncg     rms          rms(c)
RMM:   1    -0.337852804095E+03    0.94862E+00   -0.22457E+01   324   0.780E+00    0.189E+00
RMM:   2    -0.338087873331E+03   -0.23507E+00   -0.24923E+00   325   0.306E+00    0.108E+00
RMM:   3    -0.338132790582E+03   -0.44917E-01   -0.50218E-01   340   0.124E+00    0.110E+00
RMM:   4    -0.338121036810E+03    0.11754E-01   -0.79021E-02   324   0.547E-01    0.579E-01
RMM:   5    -0.338129671306E+03   -0.86345E-02   -0.76437E-02   324   0.388E-01    0.314E-01
RMM:   6    -0.338123062866E+03    0.66084E-02   -0.16448E-02   326   0.207E-01    0.114E-01
RMM:   7    -0.338122886896E+03    0.17597E-03   -0.90935E-03   323   0.119E-01    0.334E-02
RMM:   8    -0.338122942074E+03   -0.55178E-04   -0.10506E-03   316   0.619E-02    0.244E-02
RMM:   9    -0.338122995490E+03   -0.53416E-04   -0.72833E-04   288   0.348E-02
2 T=  1916. E= -.32252355E+03 F= -.33812300E+03 E0= -.33812299E+03  EK= 0.15599E+02 SP= -.49E-06 SK= 0.61E-12
bond charge predicted
prediction of wavefunctions
N       E                     dE             d eps       ncg     rms          rms(c)
RMM:   1    -0.336180152440E+03    0.19428E+01   -0.58207E-01   324   0.139E+00    0.654E-01
RMM:   2    -0.336188071907E+03   -0.79195E-02   -0.84588E-02   339   0.505E-01    0.141E-01
RMM:   3    -0.336189675117E+03   -0.16032E-02   -0.16716E-02   353   0.208E-01    0.133E-01
RMM:   4    -0.336189894771E+03   -0.21965E-03   -0.26384E-03   332   0.918E-02    0.929E-02
RMM:   5    -0.336189974660E+03   -0.79889E-04   -0.63462E-04   285   0.448E-02
3 T=  1685. E= -.32249392E+03 F= -.33618997E+03 E0= -.33618915E+03  EK= 0.13725E+02 SP= -.32E-01 SK= 0.27E-02
bond charge predicted
prediction of wavefunctions
N       E                     dE             d eps       ncg     rms          rms(c)
RMM:   1    -0.333466462434E+03    0.27234E+01   -0.11737E+00   324   0.186E+00    0.677E-01
RMM:   2    -0.333487410180E+03   -0.20948E-01   -0.21196E-01   349   0.731E-01    0.231E-01
RMM:   3    -0.333490239144E+03   -0.28290E-02   -0.30702E-02   343   0.293E-01    0.176E-01
RMM:   4    -0.333490732641E+03   -0.49350E-03   -0.60607E-03   341   0.135E-01    0.933E-02
RMM:   5    -0.333490872875E+03   -0.14023E-03   -0.13599E-03   307   0.636E-02    0.379E-02
RMM:   6    -0.333490903027E+03   -0.30152E-04   -0.32459E-04   233   0.298E-02
4 T=  1373. E= -.32245824E+03 F= -.33349090E+03 E0= -.33348022E+03  EK= 0.11179E+02 SP= -.18E+00 SK= 0.37E-01
...
...
...
28 T=  2564. E= -.32254692E+03 F= -.30541712E+03 E0= -.30535338E+03  EK= 0.20878E+02 SP= -.38E+02 SK= 0.13E+00
bond charge predicted
prediction of wavefunctions
N       E                     dE             d eps       ncg     rms          rms(c)
RMM:   1    -0.304962686456E+03    0.45437E+00   -0.50291E-01   338   0.931E-01    0.244E-01
RMM:   2    -0.304975571260E+03   -0.12885E-01   -0.13325E-01   371   0.442E-01    0.982E-02
RMM:   3    -0.304977016808E+03   -0.14455E-02   -0.15036E-02   370   0.179E-01    0.880E-02
RMM:   4    -0.304977314818E+03   -0.29801E-03   -0.32635E-03   345   0.764E-02    0.645E-02
RMM:   5    -0.304977364930E+03   -0.50112E-04   -0.63532E-04   268   0.359E-02
29 T=  2426. E= -.32253591E+03 F= -.30497736E+03 E0= -.30491565E+03  EK= 0.19755E+02 SP= -.38E+02 SK= 0.28E+00
bond charge predicted
prediction of wavefunctions
N       E                     dE             d eps       ncg     rms          rms(c)
RMM:   1    -0.304475187728E+03    0.50213E+00   -0.47067E-01   340   0.882E-01    0.228E-01
RMM:   2    -0.304487073861E+03   -0.11886E-01   -0.12380E-01   370   0.429E-01    0.918E-02
RMM:   3    -0.304488454010E+03   -0.13801E-02   -0.14617E-02   368   0.176E-01    0.790E-02
RMM:   4    -0.304488726979E+03   -0.27297E-03   -0.30125E-03   343   0.756E-02    0.581E-02
RMM:   5    -0.304488773682E+03   -0.46703E-04   -0.56686E-04   269   0.352E-02
30 T=  2262. E= -.32252795E+03 F= -.30448877E+03 E0= -.30442641E+03  EK= 0.18424E+02 SP= -.37E+02 SK= 0.41E+00
Information: wavefunction orthogonal band  154  0.8863
bond charge predicted
prediction of wavefunctions
wavefunctions rotated
writing wavefunctions


After each electronic relaxation, the final line summarizes:

tag meaning
T The instantaneous temperature.
E The total energy E including the potential energy F of the ionic degree of freedom, the potential energy SP and kinetic energy SK of the Nose Hoover thermostat, and the kinetic energy of the ionic motion EK. It is called ETOTAL in the OUTCAR file.
F The total free energy of the DFT calculation considering the artificial electronic temperature introduced by the smearing factor SIGMA. In fact, from the view point of MD, this is the potential energy of the ionic degree of freedom. It is called TOTEN in the OUTCAR file.
E0 The total energy of the DFT calculation obtained by subtracting the entropy term and letting SIGMA go to zero for the DFT total free energy F.
EK The kinetic energy of the ionic motion, called EKIN in the OUTCAR file..
SP The potential energy of the Nosé-Hoover thermostat, called ES in the OUTCAR file.
SK The kinetic energy of the Nosé-Hoover thermostat, called EPS in the OUTCAR file.

Why is the temperature not constant in every step?

That is the instantaneous temperature and not the observable ensemble average. Note that the idea of a constant temperature calculation is not that the instantaneous temperature is constant in every time step, but that the observable temperature, i.e., the ensemble average of the temperature is constant. You can find the value of the mean temperature <T/S>/<1/S> in the OUTCAR file. In the thermodynamic limit, for sufficiently large number of atoms the fluctuations of the instantaneous temperature would also vanish.
import py4vasp