parameterization of a reactive force ﬁeld using a monte ... · where innovation starts...

Where innovation starts

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.


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.


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

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

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.

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

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 (100 * σi) : 7.0 kcal/mol

DFT


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 (100 * σi): 10.7 kcal/mol

DFT


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!!

parameterization of a reactive force ﬁeld using a monte ... · where innovation starts...

Documents