# Open SourceMolecular Dynamics

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exercises:2014_ethz_mmm:monte_carlo_ice

# 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)
print "Temperature: %.2f K"%T

weights = []; M_vec = []; cell_vol = []
for fn in sys.argv[2:]:
assert(fn.endswith(".dip_cl"))
weights.append(raw_data[1:,0]-raw_data[:-1,0])
M_vec.append(raw_data[:-1,1:4])

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

### 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
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 