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Parameterization of areactive force field using aMonte Carlo algorithm
Eldhose Iype ([email protected])
November 19, 2015
2/1Thermochemical energy storage
MgSO4.xH
2O +Q MgSO
4+xH
2O MgSO
4+xH
2O MgSO
4.xH
2O +Q
Energy storage density of thermochemical materials is about 10 timeshigher than that of water.
2/1Thermochemical energy storage
MgSO4.xH
2O +Q MgSO
4+xH
2O MgSO
4+xH
2O MgSO
4.xH
2O +Q
Dehydration
Hydration
Problem: Changes in the crystallinity of the material, Slow kinetics,Reusability etc.
2/1Thermochemical energy storage
MgSO4.xH
2O +Q MgSO
4+xH
2O MgSO
4+xH
2O MgSO
4.xH
2O +Q
Dehydration
Hydration
◮ Aim:• To study hydration and dehydration reactions• To characterize the structural changes• To understand the mechanism of water release during dehydration
3/1Molecular dynamics
◮ Atoms are assumed to be point mass particles which obey Newton’slaws of motion
◮ All particles interact with each other through some potential,U(r1, r2, · · · rN)
◮ Force acting on any particle at any time is calculated as,F = −∇U(r1, r2, · · · rN)
◮ Positions are updated by integrating the equation of motion, F = ma
Force Field
Energy ,E = E (r1, r2, · · · rN)
Force,F = −∇E (r1, r2, · · · rN)
4/1Condensed water between two Pt slabs
Details: NVT ensemble, ρwater = 1001.22kg/m3
4/1Condensed water between two Pt slabs
Details: NVT ensemble, ρwater = 1001.22kg/m3
5/1Two phase water between platinum slabs
Details: NVT ensemble, ρwater = 385.08kg/m3
5/1Two phase water between platinum slabs
Details: NVT ensemble, ρwater = 385.08kg/m3
6/1ReaxFF force field
Esystem =EvdWaals + ECoulomb + Ebond + Eval + Etors
+ Eover + Eunder + EH−bond + Elp
+ Econj + Epen + Ecoa + EC2 + Etriple
Characteristics of ReaxFF
◮ Dynamic charges are calculated using EEM
◮ van Der Waal’s interaction is calculated using a Morse-type potential
◮ Energy surface is made continuous◮ Connected interactions include
• Bonded interaction (two body)• Valence angle interaction (three body)• Torsion interaction (four body)• Hydrogen bond interaction
6/1ReaxFF force field
Esystem =EvdWaals + ECoulomb + Ebond + Eval + Etors
+ Eover + Eunder + EH−bond + Elp
+ Econj + Epen + Ecoa + EC2 + Etriple
Characteristics of ReaxFF
◮ Dynamic charges are calculated using EEM
◮ van Der Waal’s interaction is calculated using a Morse-type potential
◮ Energy surface is made continuous◮ Connected interactions include
• Bonded interaction (two body)• Valence angle interaction (three body)• Torsion interaction (four body)• Hydrogen bond interaction
7/1What makes ReaxFF reactive?
◮ ReaxFF calculates bond order between every pair of atoms
◮ Bond order is a function of distance of separation
◮ Every connected interaction is made a function of this bond order
◮ Thus all the bonds become dynamic
BO′
ij =BO′σij + BO
′πij + BO
′ππij
=exp
[
Pbo1 ·
(
rij
rσo
)Pbo2
]
+ exp
[
Pbo3 ·
(
rij
rπo
)Pbo4
]
+ exp
[
Pbo5 ·
(
rij
rππo
)Pbo6
]
8/1Bond energy in Reax force field
Esystem =EvdWaals + ECoulomb + Ebond + Eval + Etors
+ Eover + Eunder + EH−bond + Elp
+ Econj + Epen + Ecoa + EC2 + Etriple
Bond energy
Ebond =−Dσe · BO
σij · exp
[
Pbe1
(
1 −(
BOσij
)Pbe2
)]
−Dπe · BO
πij − D
ππe · BO
ππij
9/1Econj and Ecoa in ReaxFF
10/1Epen for valence angle in ReaxFF
11/1EC2 for valence angle in ReaxFF
12/1DFT introduction
First Hohenberg-Kohn Theorem
vext(r) ⇔| Ψ0〉 ⇔ n0(r) = 〈Ψ0 | n(r) | Ψ0〉
The theorem states that, one has a one-to-one correspondence betweenthe external potential Vext in the Hamiltonian, the (non-degenerate)ground state | Ψ0〉 resulting from the Schrödinger equation and theassociated ground state (electron) density n0.
12/1DFT introduction
First Hohenberg-Kohn Theorem
vext(r) ⇔| Ψ0〉 ⇔ n0(r) = 〈Ψ0 | n(r) | Ψ0〉
The theorem states that, one has a one-to-one correspondence betweenthe external potential Vext in the Hamiltonian, the (non-degenerate)ground state | Ψ0〉 resulting from the Schrödinger equation and theassociated ground state (electron) density n0.
13/1DFT introduction
E [n] = 〈Ψ[n] | H | Ψ[n]〉
◮ Thus, the many-body problem of N electrons with 3N spatialcoordinates is reduced to a problem involving only 3 spatialcoordinates.
◮ The second theorem states that the ground state correponds to thedensity which minimizes the total energy of the system.
E0 < E [n]
13/1DFT introduction
E [n] = 〈Ψ[n] | H | Ψ[n]〉
◮ Thus, the many-body problem of N electrons with 3N spatialcoordinates is reduced to a problem involving only 3 spatialcoordinates.
◮ The second theorem states that the ground state correponds to thedensity which minimizes the total energy of the system.
E0 < E [n]
14/1Parameterization of ReaxFF force field
◮ Quantum Chemical (DFT) data is used to parameterize the forcefield
◮ A training data set is prepared which contains the followinginformations
• Atomic charges (Mulliken)• Equilibrium bond lengths• Equilibrium bond angles• Torsion angles• Energies of the DFT optimized geometries• Heat of formation
◮ Error in the force field is then calculated
Err(p1, p2, · · · pn) =
n∑
i=1
[
xi ,QM − xi ,ReaxFF
σi
]2
14/1Parameterization of ReaxFF force field
◮ Quantum Chemical (DFT) data is used to parameterize the forcefield
◮ A training data set is prepared which contains the followinginformations
• Atomic charges (Mulliken)• Equilibrium bond lengths• Equilibrium bond angles• Torsion angles• Energies of the DFT optimized geometries• Heat of formation
◮ Error in the force field is then calculated
Err(p1, p2, · · · pn) =
n∑
i=1
[
xi ,QM − xi ,ReaxFF
σi
]2
14/1Parameterization of ReaxFF force field
◮ Quantum Chemical (DFT) data is used to parameterize the forcefield
◮ A training data set is prepared which contains the followinginformations
• Atomic charges (Mulliken)• Equilibrium bond lengths• Equilibrium bond angles• Torsion angles• Energies of the DFT optimized geometries• Heat of formation
◮ Error in the force field is then calculated
Err(p1, p2, · · · pn) =
n∑
i=1
[
xi ,QM − xi ,ReaxFF
σi
]2
15/1Parabolic search algorithm
Err(p1, p2, · · · pn) =
n∑
i=1
[
xi ,QM − xi ,ReaxFF
σi
]2
1170
1172
1174
1176
1178
1180
1.44 1.46 1.48 1.5 1.52 1.54 1.56 1.58 1.6 1.62 1.64 1.66
Err
or in
the
forc
e fi
eld
Trial values for a paramter
Error
16/1Drawbacks of parabolic search algorithm
◮ Only one parameter is searched at a time◮ The procedure has to repeated over several rounds◮ It will only find a local minimum◮ Needs a good starting point
Shapes of typical ReaxFF Error surface
20000
40000
60000
80000
100000
120000
140000
160000
2.4 2.5 2.6 2.7 2.8 2.9 3 3.1
Err
or
σ-bond radius of S-atom
Error vs σ-bond radius
24500
24600
24700
24800
24900
25000
25100
25200
25300
25400
25500
25600
-0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5
Err
or
EEM hardness for Mg-atom
Error vs EEM hardness
0
1e+06
2e+06
3e+06
4e+06
5e+06
6e+06
7e+06
8e+06
9e+06
1e+07
2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3
Err
or
vdw parameter for S-O interaction
Error vs vdw parameter
23000
24000
25000
26000
27000
28000
29000
30000
31000
32000
2.4 2.5 2.6 2.7 2.8 2.9 3 3.1 3.2
Err
or
vdw radius of Mg-atom (rvdw)
Error vs rvdw
17/1A double well potential with a linear term
U(x) = x4 − 0.75x2 + 0.01x
−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1−0.15
−0.1
−0.05
0
0.05
0.1
0.15
0.2
0.25
17/1Equilibrium distribution of states
The distribution has a global maximum near the left well.
−2.5 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 2.50
100
200
300
400
500
600
18/1Metropolis Monte Carlo (MMC) method
◮ Calculate the error of the starting force field, Errold
◮ Make a new proposition (move) for the parameters
◮ Calculate the error of the new force field, Errnew
◮ Calculate the difference in the Error, i.e.
∆Err = Errnew − Errold
◮ Accept the move with a probability given by,
P = min [1, exp (−β∆Error)] ,where, β =1
kBT
◮ Repeat the algorithm
19/1Error vs β for a Simulated Annealing run
0
1e+06
2e+06
3e+06
4e+06
5e+06
6e+06
7e+06
1e-05 0.0001 0.001
Err
or
β
Error vs β =1/(KBT)
19/1Comparison between MMC and ParSearch
For five random starting force fields
0
0.2
0.4
0.6
0.8
1
1.2
0 200 400 600 800 1000
Err
or r
atio
(E
rror
MM
C/E
rror
ParS
earc
h)
Iteration
Trial 1Trial 2Trial 3Trial 4Trial 5
20/1Comparison of the error values
Table : Initial and final errors of the five simulations. The average < Error >
and the coefficient of variation, std(sigma)<Error>
, of the final errors are shown.
Trial Errorinitial × 1E − 6 Errorfinal × 1E − 6
ParSearch MMC1 4.3 0.9 0.312 1.1 0.4 0.443 8.1 2.5 0.314 2.0 1.4 0.265 4.3 1.3 0.42
< Error > 1.3 0.35std(Error)<Error>
0.6 0.21
21/1Energies of the hydrates of MgSO4.xH2O
x ranging from 0 to 6
-3000
-2500
-2000
-1500
-1000
-500
0 1 2 3 4 5 6 7 8 9
Energy (kca
l/m
ol)
Geometry number
Errrms : 9.8 kcal/mol
DFT
ReaxFF
21/1Equation of state for MgSO4
-3450
-3440
-3430
-3420
-3410
-3400
-3390
-3380
230 240 250 260 270 280 290 300 310
Energy (kcal/mol)
Volume
ErrrmsReaxFF: 4.8 kcal/mol
ErrrmsReaxFF (100 * σi): 4.5 kcal/mol
DFT
ReaxFFReaxFF (100 * σi)
21/1Equation of state for MgSO4.7H2O
-13140
-13130
-13120
-13110
-13100
-13090
-13080
860 880 900 920 940 960 980 1000
Ener
gy (
kca
l/m
ol)
Volume
ErrrmsReaxFF: 3.4 kcal/mol
ErrrmsReaxFF (100 * σi) : 7.0 kcal/mol
DFT
ReaxFFReaxFF (100 * σi)
21/1Binding energy
Binding energy of one water molecule on the (100) surface of MgSO4
-3400
-3350
-3300
-3250
-3200
2 2.5 3 3.5 4 4.5 5 5.5
Energy (kcal/mol)
Distance
Binding Energy with one water molecule MgSO4
ErrrmsReaxFF: 18.0 kcal/mol
ErrrmsReaxFF (100 * σi): 10.7 kcal/mol
DFT
ReaxFFReaxFF (100 * σi)
22/1Hydrogen bonds in MgSO4 hydrates
0
100
200
300
400
500
600
0 0.5 1 1.5 2 2.5
Num
ber
of e
scap
ed w
ater
mol
ecul
es
Time (ns)
9.75mbar-noHB19.5mbar-noHB
39mbar-noHB9.75mbar19.5mbar
39mbar
The hydrogen bonds slows down the kinetics of dehydration as can be seen from
the two sets of dehydration curves (one with hydrogen bonds and the other
without hydrogen bonds) in the figure.
23/1Dissaperance of step-edge sites
The movement of step-edge sites in Co-nanoparticles are studied using ReaxFF.
24/1Conclusion
◮ Metropolis MC algorithm is used to parameterize the ReaxFF forcefield.
◮ The method shows good improvement over the traditionaloptimization scheme.
◮ The stochastic nature of the method allows one to arrive at theglobal minima in the parameter space.
25/1
Thank You!!