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Parameterization of a reactive force field using a MonteCarlo algorithm

Eldhose Iype

Asst. Prof.Department of Chemical Engineering,

BITS Pilani Dubai Campus

[email protected]

Eldhose Iype (BITS Pilani Dubai Campus) Parameterization of a reactive force field using a Monte Carlo algorithm 1 / 18

Who am I?

Asst. Prof. (Currently)

Department of Chemical Engineering,

BITS Pilani Dubai Campus

Post-doc (2014-2016)

Simulation of self-replenishing hydrophilic polyurethane coatings.

Eindhoven University of Technology

PhD (2014)

Computational modelling of hydration/dehydration behaviour ofenergy storage materials.

Eindhoven University of Technology

Masters in Chemical Engineering (2008)

Indian Institute of Technology, Madras

Bachelors in Chemical Engineering (2006)

Kerala University


Molecular dynamics

Force Field

Energy ,E = E (r1, r2, · · · rN)

Force,F = −∇E (r1, r2, · · · rN)Eldhose Iype (BITS Pilani Dubai Campus) Parameterization of a reactive force field using a Monte Carlo algorithm 3 / 18

Reax 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


What 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


Parameterization of ReaxFF force field

Quantum Chemical (DFT) data is used to parameterize the force field

A training data set is prepared which contains the followinginformations

Atomic charges (Mulliken)Equilibrium bond lengthsEquilibrium bond anglesTorsion anglesEnergies of the DFT optimized geometriesHeat of formation

Error in the force field is then calculated

Err(p1, p2, · · · pn) =n∑

i=1

[xi ,QM − xi ,ReaxFF

σi

]2


Challenges with parametrization of ReaxFF

Multiple parameters need to be optimized simultaneouslyShould not be computationally expensiveShould not get stuck in any local minimaWe do not know 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


Metropolis Monte Carlo algorithm

Start from any random point in space and calculate the initial energy,Eold

Propose a new move in any direction (random(−1.0, 1.0) ∗ dmax ),here dmax is the step size.

Calculate the new energy of the particle, Enew

Calculate the difference in the Energies, ∆E = Enew − Eold

Accept the move with a probability given by,P = min [1, exp (−β∆E )], where β = 1

kB T

Repeat the algorithm


Monte Carlo sampling in a double well potential

U(x) = x4 − 0.75x2 + 0.01x

The distribution has a global maximum near the left well.

−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

−2.5 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 2.50

100

200

300

400

500

600


MCFF-Optmizer

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


Error 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)


Comparison 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


Comparison of the error values

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


Technical details

Maximum step size (dmax )

Maximum change in parameter value at every stepIt is different for each parameter in a force fieldSmaller step size results in more acceptance and vice versa, for a giventemperature

Acceptance rate

The ratio of accepted moves to the total number of movesSet an acceptance rate of your choice (typically, I use around 20%)Lower, the better (but computationally expensive)

Cooling rate

Slow cooling rate is goodAt any temperature, the acceptance rate is maintained by scaling up ordown dmax valueSo ideally, one would like to have a slow cooling rate and lowacceptance rate. But this come at a cost.


Challenges

Bounds for the parameters

Taken from the values of each parameter from a set of 11 force fieldsThis would mean, two different elements have same bounds for a givenparameterNo way to monitor if the optimized value for the parameter isphysically meaningful


Energies 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/mol)

Geometry number

Errrms : 9.8 kcal/mol

DFT

ReaxFF


Equation 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)


Equation 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



Binding 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



Examples of works based on MCFF-optimizer

J. Phys. Chem. C. 2014. 118. 6882-6886

ACS Catal. 2015. 5. 596601

PCCP 2016. 18. 15838


Conclusion

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 the globalminima in the parameter space.

The method is efficient and cheaper

One does not need a good starting point

Acknowledgement

dr. M. (Markus) Hutter, dr. A.P.J. (Tonek) Jansen, dr. Adri van Duin

SCM, NCF-SARA, EGS, NSCCS


Thank You!!


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