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exercises:2015_ethz_mmm:monte_carlo_ice

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Properties of Ice from Monte Carlo Simulations

  1. Add the line PRINT_COORDS .FALSE. in the TMC section. It disables the output of the coordinate trajectory and thus avoids the files-size problem.
  2. Add the line RND_DETERMINISTIC 42 to the TMC section to choose a random number seed. You have to replace 42 with different values to obtain different samplings.

In this exercise we will use Monte Carlo sampling to calculate the dielectric constant and the thermal expansion coefficient of the disordered hexagonal phase (Ih) of water ice. In order to speed up the calculation we are going to use the simple SPC/E water model. For more information see: 10.1063/1.1568337. A DFT-version of the calculation can be found here: 10.1021/jp4103355.

You should run these calculations on 3 cores with bsub -n 3.

Task 1: Calculate the dielectric constant

The dielectric constant describes the response of a system to an external electric field. Within a linear response approximation the Kubo Formula is applicable. The Kubo Formula is an equation which expresses the linear response of an observable quantity due to a time-dependent perturbation. Hence, one can calculate the dielectric constant based on a sampling of the dipole moment.

The advantage of Monte Carlo is that we can employ special hydrogen reordering moves, which lead to an effective sampling of the dipole.

Run the input-file mc_exercise.inp. It will create a file tmc_trajectory_T200.00.dip_cl, which contains the dipole moment for each accepted step. Plot the histogram of the z-component of the dipole moment. The distribution should be symmetric, because the simulation cell is also symmetric in the z-direction.

The dielectric constant can then calculated from the dipole moments via: \begin{equation} \epsilon = 1 + \left(\frac{4 \pi}{3 \epsilon_0 V k_B T } \right ) \operatorname{Var}(M) \ , \end{equation}

where $M$ denotes the dipol moment of the entire simulation cell and $\operatorname{Var}(M)$ denotes the variance of the dipol moment of the sampling: \begin{equation} \operatorname{Var}(M) = (\langle M \cdot M\rangle - \langle M\rangle\langle M\rangle ) \ . \end{equation}

To simplify this task you can use the following python-script:

calc_dielectric_constant.py
#!/usr/bin/env python
 
import sys
import numpy as np
 
if(len(sys.argv) < 3):
    print "Usage: calc_dielectric_constant.py <temperature> <traj_1.dip_cl> ... <traj_N.dip_cl>"
    sys.exit()
 
T = float(sys.argv[1])
print "Temperature: %.2f K"%T
 
weights = []; M_vec = []; cell_vol = []
for fn in sys.argv[2:]:
    print "Reading: "+fn
    assert(fn.endswith(".dip_cl"))
    raw_data = np.loadtxt(fn) # [C*Angstrom]
    weights.append(raw_data[1:,0]-raw_data[:-1,0])
    M_vec.append(raw_data[:-1,1:4])
    cell_vol.append(np.loadtxt(fn.replace(".dip_cl", ".cell"), usecols=(10,)))
 
weights = np.concatenate(weights)
M_vec = np.concatenate(M_vec)
cell_vol = np.concatenate(cell_vol)
 
V = np.mean(cell_vol) # [Angstrom^3]
print "Total number of samples: %d"%np.sum(weights)
print "Mean cell volume: %.2f Angstrom^3"%V
 
M = np.sqrt(np.sum(np.square(M_vec), axis=1)) # take norm of dipol moment
var = np.average(np.square(M), weights=weights) - np.square(np.average(M, weights=weights))
 
epsilon_0 = 8.8541878176e-12 #[F/m] vacuum permittivity
e = 1.602176565e-19 # [C] elementary charge
kb = 1.3806488e-23 #[J/K] Boltzmann constant
angstrom2meter = 1e-10
s = (e*e*4*np.pi)/(3*V*kb*T*angstrom2meter*epsilon_0)
 
epsilon = 1 + s*var
print "Dielectric constant: %.2f"%epsilon
Before you can run the python-script you have to load a newer python-module and set the executable-bit:
you@brutusX ~$ module load new python/2.7.6
you@brutusX ~$ chmod +x calc_dielectric_constant.py
you@brutusX ~$ ./calc_dielectric_constant.py 200 tmc_trajectory_T200.00.dip_cl

Task 2: Gather more samplings

You can gather more samples by launching multiple independent runs in parallel with different random number seeds. The seed is given by the RND_DETERMINISTIC keyword. The gathered trajectories can then be analyzed collectively with the python-script:

you@brutusX ~$ ./calc_dielectric_constant.py 200 trj_1_T200.00.dip_cl trj_1_T200.00.dip_cl ...
You can also share trajectories with your colleges.

Task 3: Calculate the thermal expansion coefficient

Run the MC sampling again at temperature of 150K . Plot the cell volume of each sampling. Read off the converged cell volume and plot it vs. the temperature. Determine the expansion coefficient via a linear fit. Do you trust your result? Why (not)?

Required Files

Initial Coordinates

Main Input-File (run this)

mc_exercise.inp
&GLOBAL
  PROJECT H2O_MC
  PROGRAM TMC
  RUN_TYPE TMC
  PRINT_LEVEL LOW
  WALLTIME 1:00:00
&END GLOBAL
&MOTION
  &TMC
      RND_DETERMINISTIC 42 !<=== Change this number to obtain different samplings
      PRINT_COORDS .FALSE. !this avoids the file-size problem
      GROUP_CC_SIZE 0
      NUM_MC_ELEM 100000
      ENERGY_FILE_NAME H2O_ice.inp
      TEMPERATURE 200
      &MOVE_TYPE      MOL_TRANS
        SIZE          0.05
        PROB          3
      &END
      &MOVE_TYPE      ATOM_TRANS
        SIZE          0.01
        PROB          3
      &END
      &MOVE_TYPE      MOL_ROT
        SIZE          5
        PROB          3
      &END
      &MOVE_TYPE      PROT_REORDER
        PROB          5
      &END
      &MOVE_TYPE      VOL_MOVE
        SIZE          0.1
        PROB          1
      &END
      PRESSURE 0.01
      ESIMATE_ACC_PROB .TRUE.
      RESTART_OUT 0
      NUM_MV_ELEM_IN_CELL 5
      PRINT_ONLY_ACC .FALSE.
      &TMC_ANALYSIS
        CLASSICAL_DIPOLE_MOMENTS
	RESTART .FALSE.
        &CHARGE
           ATOM O
           CHARGE -0.8476
        &END CHARGE
        &CHARGE
           ATOM H
           CHARGE  0.4238
        &END CHARGE
      &END TMC_ANALYSIS
  &END TMC
&END MOTION

Auxiliary Input-File (do not run this directly)

H2O_ice.inp
&FORCE_EVAL
  METHOD FIST
  &MM
    &NEIGHBOR_LISTS
      GEO_CHECK OFF
      VERLET_SKIN 4.0
    &END NEIGHBOR_LISTS
    &FORCEFIELD
      &SPLINE
        EMAX_SPLINE 10.0
      &END
      &CHARGE
        ATOM O
        CHARGE -0.8476
      &END CHARGE
      &CHARGE
        ATOM H
        CHARGE 0.4238
      &END CHARGE
      &BOND
        ATOMS O  H
        K      [nm^-2kjmol] 502080.0 
        R0     [nm] 0.09572
        KIND   G87
      &END BOND
      &BEND
        ATOMS      H     O     H 
        THETA0     [deg] 104.500 
        K          [rad^2kjmol] 627.600
        KIND   G87
      &END BEND
      &NONBONDED
        &LENNARD-JONES 
    	  ATOMS O O
          EPSILON [kjmol] 0.650
          SIGMA [angstrom] 3.166
          RCUT 8.0
        &END LENNARD-JONES
      &LENNARD-JONES 
  	  ATOMS O H
          EPSILON 0.0
          SIGMA 3.1655 
          RCUT 8.0
        &END LENNARD-JONES
        &LENNARD-JONES 
  	  ATOMS H H 
          EPSILON 0.0
          SIGMA 3.1655 
          RCUT 8.0
        &END LENNARD-JONES
      &END NONBONDED
    &END FORCEFIELD
    &POISSON
      &EWALD
        EWALD_TYPE ewald
        ALPHA .45
        GMAX  19
      &END EWALD
    &END POISSON
  &END MM
  &SUBSYS
    &CELL
       ABC 13.5084309945 15.6001709945 14.7072509945
    &END CELL
    &TOPOLOGY
      COORD_FILE_NAME ./ice_ih_96.xyz
      COORD_FILE_FORMAT xyz
      CONNECTIVITY MOL_SET
      &MOL_SET
        &MOLECULE
          NMOL 96
          CONN_FILE_NAME topology_H2O.psf
        &END
      &END
    &END TOPOLOGY
  &END SUBSYS
&END FORCE_EVAL
&GLOBAL
&END GLOBAL

Topology File

topology_H2O.psf
 PSF

   1 !NTITLE
     Topology file for water

     3   !NATOM
       1 WAT     1 WAT  O    O          0.000000     15.999400       0
       2 WAT     1 WAT  H    H          0.000000      1.008000       0
       3 WAT     1 WAT  H    H          0.000000      1.008000       0

     2   !NBOND
       1       2       1       3     

     1   !NTHETA
       2       1       3 

     0   !NPHI

     0   !NIMPHI

     0   !NDON

     0   !NACC

     0   !NNB
exercises/2015_ethz_mmm/monte_carlo_ice.1423244958.txt.gz · Last modified: 2015/03/18 14:53 (external edit)