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+====== Properties of Ice from Monte Carlo Simulations ======
+In this exercise we will use Monte Carlo sampling to calculate the [[wp>Relative_permittivity|dielectric constant]] of the disordered hexagonal phase (Ih) of water ice. \\
+In order to speed up the calculation, we are going to use the SPC/E water model.
+The model is a non polarizable forcefield model, with parameters for:
+  * Charge of the atomic species
+  * Harmonic O-H bond elongation $ U_{bond}= k_{bond}⋅(r-r_{eq})^2$
+  * Harmonic H-O-H angle bending $ U_{angle}= k_{angle}⋅(θ-θ_{eq})^2$
+  * Lennard Jones interaction between non bonded species (interaction between atoms belonging to different water molecules)
+ (A DFT-version of the calculation can be found here: [[doi>10.1021/jp4103355]]).
+===== Introduction =====
+Water molecules are arranged according to the so called //"ice rules"//:
+  - Water molecules are present as neutral H2O
+  - Each molecule makes four hydrogen bonds with its four nearest neighbors, two as a hydrogen bond donor and two as an acceptor.
+All the possible proton arrangements in the ice strucrure are not energetically equivalent and we sample and weight them with a Monte Carlo approach. For this purpose, a specific algorithm was created (see: [[doi>10.1063/1.1568337]]), that allows to employ special proton reordering moves, which lead to an effective sampling of the ice dipole (N.B: different proton arrangement = different dipole). \\
+The dielectric constant of a system describes its response to an external electric field.
+If the dipole moment is properly sampled, one can compute the dielectric constant of ice, by applying the //Kubo Formula//. This is valid in the approximation that the response of the system to the time-dependent perturbation (the field) is linear. \\
+The dielectric constant can then be 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 dipole moment of the entire simulation cell and $\operatorname{Var}(M)$ denotes the variance of the dipole moment of the sampling:
+\begin{equation}
+\operatorname{Var}(M) = (\langle M \cdot M\rangle - \langle M\rangle\langle M\rangle ) \ .
+\end{equation}
+===== Task 1: Calculate the dielectric constant from one simulation =====
+      * As usual, download the 4 required files (at the end of this page: input, auxiliary input, ice coordinates and topology) in the same directory. \\
+      * 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.
+<note tip>
+You should run these calculations on 3 cores with : \\
+ ''bsub -W 01:00 -n 3 mpirun cp2k.popt -i mc_exercise.inp -o mc_exercise.out''
+</note>
+    * 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.
+  * Compute the dielectric constant of ice. To simplify this task you can use the following python-script:
+<code python 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
+</code>
+<note tip>
+Before you can run the python-script you have to load a newer python-module and set the executable-bit:
+<code>
+you@eulerX ~$ module load python/2.7.6
+you@eulerX ~$ chmod +x calc_dielectric_constant.py
+you@eulerX ~$ ./calc_dielectric_constant.py 200 tmc_trajectory_T200.00.dip_cl
+</code>
+</note>
+===== 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:
+<code>
+you@eulersX ~$ ./calc_dielectric_constant.py 200 trj_1_T200.00.dip_cl trj_1_T200.00.dip_cl ...
+</code>
+<note tip>
+You can also share trajectories with your colleges.
+</note>
+===== Required Files =====
+==== Initial Coordinates ====
+This file contains the initial ice coordinates.
+{{ice_ih_96.xyz.gz| Download here}}
+==== Main Input-File (run this) ====
+This file specifies the type of algorithm do adopt and the simulation parameters.
+<code - mc_exercise.inp>
+&GLOBAL
+  PROJECT H2O_MC
+  PROGRAM TMC         ! Tree Monte Carlo algorithm
+  RUN_TYPE TMC
+  PRINT_LEVEL LOW     ! Low amount of information written in the output
+  WALLTIME 1:00:00    ! The simulation will last one hour and then will stop
+&END GLOBAL
+&MOTION
+  &TMC
+      RND_DETERMINISTIC 42 !<=== Change this number to obtain different samplings
+      PRINT_COORDS .FALSE. !this avoids the printing of all coordinates and file-size problems
+      GROUP_CC_SIZE 0
+      NUM_MC_ELEM 100000
+      ENERGY_FILE_NAME H2O_ice.inp ! refers to the auxiliary input for the force specification
+      TEMPERATURE 200
+      &MOVE_TYPE      MOL_TRANS       ! specifies "proton moves" for better sampling
+        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.        !accuracy parameters, do not change
+      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
+</code>
+==== Auxiliary Input-File (do NOT run this directly) ====
+This auxiliary input specifies the system properties
+<code - 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
+</code>
+==== Topology File ====
+This file specifies the topology of the system (which atoms, bonds, angles in the water molecule). It allows the program to distinguish which atoms belong to which water molecule, and therefore discriminate between inter and intramolecular interactions.
+<code - topology_H2O.psf>
+ PSF
+!NTITLE
+     Topology file for water
+   !NATOM
+WAT     1 WAT  O    O          0.000000     15.999400       0
+WAT     1 WAT  H    H          0.000000      1.008000       0
+WAT     1 WAT  H    H          0.000000      1.008000       0
+   !NBOND
+       2       1       3
+   !NTHETA
+       1       3
+   !NPHI
+   !NIMPHI
+   !NDON
+   !NACC
+   !NNB
+</code>