theoretical chemical dynamics studies for elementary combustion reactions donald thompson, gia...

Theoretical Chemical Dynamics Studies for Elementary Combustion ReactionsDonald Thompson, Gia Maisuradze, Akio Kawano, Yin Guo, Oklahoma State University

Advanced Software for the Calculation of Thermochemistry, Kinetics, and DynamicsStephen Gray, Ron Shepard, Al Wagner, Mike Minkoff, Argonne National Laboratory

• Interpolating Moving Least-Squares Method (IMLS): Potential Energy Surface (PES) Fitting Project

Outline

• Motivation

• Method

• Applications

Motivation

Potential Energy Surface (PES)• electronic energy of a molecular system as a function of molecular coordinates• hypersurface of the internal degrees of freedom fit to values calculated at discrete geometries by expensive electronic structure calculations

We seek to develop an automatic PES generator for both:- structured applications e.g., define PES everywhere below 50 kcal/mol above an equilibrium position- dynamic applications e.g., on-the-fly calculation of PES for trajectory studies

Automatic PES generation: Given seed points on the PES, computer alone determines• whether new points are necessary to refine the PES to the input accuracy• what the geometries are where the new points will be calculated

Fitting Technique• not high performance computing• directs high performance computational electronic structure calculations

Current most popular PES fitting method: Modified Shepard

• Higher degree IMLS fits are more accurate in value,derivative => Systematic exploration of IMLS for higher degrees

• unModified Shepard = 0th degree Interpolative Moving Least Squares (IMLS) fit => fits values of PES only => poor derivative properties

• Modified Shepard fits value, gradient, and hessian => accurate derivatives

Motivation

HN2 example33 equally spaced points

Solid line = exact derivativeSpiked line = shepard derivative

Method:

• We want V(x) when we know • {V(xi) | i = 1,…,N} calc. ab. initio points• b(x) basis: e.g., b(x) = x

= cos(x)= e-x

• Fit by IMLS of degree m?

• V(x) = j=0m aj(x) b(x)j

where aj from weighted least squares fit to {V(xi)}

• Fit by Taylor Series of degree m?

• V(x) = j=0m aj b(x)j

where aj from least squares fit to {V(xi)}

weights = wi(x,xi), e.g, = e-∆x2/[∆xn + ] where ∆x = x-xi

Method:

• a obeys B(xi)T W(x,xi) B(xi) a(x) = B(xi)T W(x,xi) V(xi)

• final IMLS fit: V(x) = j=0m aj(x) b(x)j

=> non-linear fit

• SVD solution method is best:- more stable- allows reduction in parameters if justified by data

N

m+1

weights basis unknown aj {V(xi)}

• Shepard fit on V (not ∂V/∂x or ∂2V/∂x2) = IMLS fit for m=0

Method:

• ∂V(x)/∂x = j=0m [aj(x) jb(x)j-1∂b(x)/∂x + ∂aj(x)/∂x b(x)]

• ∂a/∂x obeys:

B(xi)T W(x,xi) B(xi) ∂a(x)/∂x = B(xi)T ∂W(x,xi)/∂x [V(xi)- B(xi)a(x)]

same left hand sideas equation for a(x)

unique right hand side

=> reuse decomposition of left hand side

=> direct derivatives (no finite differences)

• Shepard has poor derivative properties because 0th IMLS => b(x)0 or derivative of basis does not contribute

=> only ∂aj(x)/∂x contribute ---sensitive to weights

Method:

Automatic PES generation

• IMLS strategy- reasonable weights mean IMLS fits of all degrees are very close to PES at all ab initio points- away from ab initio points, different degree IMLS differ=> let max. difference locate next ab initio point=> let minimization of max. difference end generation

• Given some seed ab initio points, can fit method determine:

- where to pick next ab initio points- when current fit is converged to a input accuracy

Morse Oscillator (MO)

1D slice of HN2 spline PES by Koizumi et al.

100

kcal

/mo

l ra

ng

e

Results: 1D Applications

MO exampleEqually spaced points

IMLS degree0

1

234567 9

8

cubic spline

HN2 exampleEqually spaced points

IMLS degree0

1

9

8

2 734 5 6


RMS error in fitting values

• compact fit capable of very high accuracy

• increasing degree generally increases accuracy

• oscillatory behavior at high degree degrades fit

• non-linear fit => third degree better than cubic spline

MO exampleEqually spaced points

IMLS degree0

1

23

4567 9

8

cubic spline

HN2 exampleEqually spaced points

IMLS degree0

9

1 82 3 4 5 6 7


RMS error in fitting derivatives

• 0th degree (i.e., Shepard) improves poorly with more points

• higher degrees have qualitatively improved accuracy


Automatic PES generation:1D Morse Oscillator Example

• IMLS degrees for 17 points• max differences where there are no points

• contrast of

automatic PES generation: 5 seed points + a point at a time where degree difference is maximum

to

repeated halving of grid increment

0.1

1.0

10.0

2 4 6 8 10 30

rms

err

or

(kc

al/

mo

l)

# Ab initio points

Color Code IMLS: First DegreeSecond DegreeThird Degree

dashed linesfor

automatic surface generation:

solid linesfor

repeated grid doubling

5 seed points +First Degree: FD-ZD differenceSecond Degree: SD-FD differenceThird Degree: TD-SD difference

-5.0

0.0

5.0

10.0

15.0

2 3 4 5

zerolineZero DegreeFirst DegreeSecond DegreeThird Degree

Fit

tin

g-E

rro

r(R

) (

kca

l/m

ol)

R (ao)

Results: HOOH 6-D Applications

• Tom Rizzo’s 6D HOOH PES

• Coordinate representation in terms of 6 interatomic distancesROH, RO’H’, ROH’, RO’H, RHH’, ROO’

• Ab initio sampling- 89 points in the vicinity of HOOH minimum,

HOOH hindered rotation barrier, HO--OH reaction path- augmented by Monte Carlo (MC) or Grid sampling up to 100 kcal

* MC: (EMS or Random or Combination (EMS+Random)* Grid: Ri = fni Ro

i for i = 1,6 where f>1 determines increment

• RMS error by MC or Grid: Sampling method matters

RandomEMS

COMB

GRID

100

101

100.0 1000.0 10000.0

Modified ShepardIMLS (second degree)

rms

err

or

(kca

l/m

ol)

# Ab initio points

HOOH 6D(Rizzo Group Surface)

Fit V everywhere V < 100 kcal/mole

100

101

100.0 1000.0 10000.0

Modified Shepard (²V)IMLS (second degree) (²V)

rms

err

or

(kca

l/m

ol)

# Ab initio points


Fit V-Vo everywhere V < 100 kcal/mole

EMS-EMS EMS-EMS

Fitting to Differences:

• Develop a qualitative fit Vo Apply IMLS to V-Vo

• HOOH example - simple functional form - 89 predetermined ab initio

+ 100 random ab initio pts.

Results: HOOH 6-D Applications

Fitting directly to PES:

100

101

100.0 1000.0 10000.0

FD-IMLS (GRID-GRID)SD-IMLS (GRID-GRID)TD-IMLS (GRID-GRID)SD-IMLS no x termsSD-IMLS no x terms without rxn coordTD-IMLS no x termsTD-IMLS no x terms without rxn coord

rms

err

or

(kca

l/m

ol)

# Ab initio points



Variation in IMLS degree and cross terms

100

101

100.0 1000.0 10000.0

EMS-EMS

COMB-COMB

GRID-GRID

rms

err

or

(kca

l/m

ol)

# Ab initio points



Variation in matched ab initio and RMS error sampling

Results: 6-D Applications

• sampling techniques make noticeable differences in rms error

• higher degree usually implies higher accuracy

• fit cross terms uncoupled to reaction coordinate have negligible effects

• fit cross terms coupled to reaction coordiate have noticeable effects

100

101

100.0 1000.0 10000.0

FD-IMLS (GRID-GRID)SD-IMLS (GRID-GRID)TD-IMLS (GRID-GRID)SD-IMLS auto-gen(489)SD-IMLS auto-gen(889)TD-IMLS auto-gen(489)TD-IMLS auto-gen(889)

rms

err

or

(kca

l/m

ol)

# Ab initio points



Effect of Automatic Surface Generation


Automatic PES Generation

• substantially improves accuracy

• works well with modest numbers of seed points

• 1 kcal/mol accuracy for 1000 points => 3.2 points/dimension in a 6D grid


Rate constant convergence:

• data point selection - 5 points on reaction path - 20 points near HOOH equilibrium - extra points randomly selected

• fit: fourth degree with only 6 cross terms

• trajectories: - 500 for each case - zero angular momentu

• results: - rates from trajectories converge much faster than rms error on the surface

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

0.0

0.5

1.0

1.5

2.0

200 400 600 800 1000 1200

rms

err

or

(kc

al/

mo

l)

rate

(1/p

s)

# ab initio points

rate on analytic surface

Results: up to 15D Applications

Model variable dimensional PES

V(x1,x2,x3,…) = VEckart(x1) + iNDOF{VMO(x1,xi)}

where VEckart(x1) =>

where VMO(x1,xi) =>

productsreactants

x1 = rxn path

rxn barrier

xi = deviation off rxn path

Local Diss. energy(x1 dependent)

width (i dependent)

0 -

VMO

number of degrees of freedom

Parameter values:• 10 kcal/mol barrier for thermoneutral reaction• 100 kcal/mol global dissociation energy• MO width chosen randomly within a range

Fit constraints:• fit V < 40 kcal/mol• know turning points at 40 kcal/mol for all xi

Local Diss. Energy + VEckart = fixed global Diss. Energy


0.5

1.0

1.5

2.0

2.5

3.0

-1

0

1

2

3

4

3 6 9 12 15

rms

err

or

(kc

al/

mo

l)

(# o

f fitted

po

ints

)/dim

en

sio

n

dimension

(65)

(305)

(725)

(1325)(# of fitted points)

(2105)

• point selection: - on single diagonal (…,xi,…) i = 1,N - on double diagonal (…,xi,…,yi,…) i = 1, N - points accepted if V < Vmax

• basis set: Third Degree IMLS without cross terms

• Results: - reasonable accuracy - very few points (uniform grid would have very few points per dimension)


Effect of cutoff: weight = 0 if weight/max-weight < input limit

- cutoff: • reduces effective # of points • time/evaluation goes linearly • at extremes, increases error effective # of points

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

10-15 10-13 10-11 10-9 10-7 10-5

<e

ffe

cti

ve

# o

f p

oin

ts>

/(to

tal

# o

f p

oin

ts)

rms

-err / [rm

s-e

rr for w

(min

)/w(m

ax

)=1

.e-1

5]

weight(min)/weight(max)

color code: 3d 6d 9d 12d 15d

10-3

10-2

10-1

100

101

103 104 105 106 107

tim

e/e

va

lua

tio

n

<effective # of points>(basis fcn.)2

yfit

= 7.1123e-07 x.9844

time/evaluationfrom

3d - 15d calculations

Conclusions

IMLS: is interesting• PROs

- non-linear, flexible, easy extension of Shepard- gradients and hessian not necessary but can be used- efficient direct derivatives- compact, black box code for any dimension PES • user cleverness in basis selection- automatic point selection encouraging- sensitivity to weight selection seems minor

• CONs- least squares evaluation every time- every ab initio point “touched” every evaluation unless weight-based screening of points

• Future- perfect “black box” code- develop parallel IMLS drivers for

electronic structure and trajectory automatic surface generation (collaboration with other SciDAC efforts)

theoretical chemical dynamics studies for elementary combustion reactions donald thompson, gia...

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