automatic construction of ab initio potential energy surfaces interpolative moving least squares...

Automatic Construction of Ab Initio Potential Energy Surfaces

Interpolative Moving Least Squares (IMLS) Fitting of Ab Initio Data for Constructing Global Potential Energy

Surfaces for Spectroscopy and Dynamics

Donald L. ThompsonUniversity of Missouri – Columbia

Richard Dawes, Al Wagner,& Michael Minkoff

Fourth International meeting : "Mathematical Methods for Ab Initio Quantum Chemistry"

13-14 November 2008Laboratoire J.A. Dieudonné

CNRS et Université de Nice - Sophia-Antipolis

Potential Energy Surfaces

Basis for quantum and classical dynamics, spectroscopy Electronic structure calculations can provide accurate energies (even gradients and Hessians) – but at a high cost (Highly accurate energy calculations for a single geometry can take hours or days)

We want to: Generate accurate global PESs fit to a minimum number (100’s – 1000’s) of ab initio points Make ab initio dynamics feasible for the highest levels of quantum chemistry methods (for which gradients may not be directly available)

As “blackbox” as possible

Requirements:: Minimize number of ab initio points Minimal human effort and cost of fitting Low-cost accurate evaluations

Our approach:Interpolating Moving Least Squares (IMLS) Much cheaper than high-level quantum chemistry Doesn’t need gradients, but can use gradients and Hessians Can use high-degree polynomials

How to make efficient and practical: Optimally place minimum number of points Weight functions Reuse fitting coefficients (store local expansions) Use zeroth-order PES and fit difference Other techniques

Least-Squares FittingLeast-Squares FittingUsual applications are Usual applications are for data with statisticalfor data with statisticalerrors, but errors, but trends that that follow known follow known functional forms.functional forms.

1-D morse function: five data points

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Bond distance

En

erg

yFitting data

exact function

Fitting ab initio energies

Ab initiob initio energies do not have random errors energies do not have random errors A PES does not have a precisely known functional form A PES does not have a precisely known functional form the energy points lie on a surface the energy points lie on a surface of unknown shapeof unknown shape Thus, fit with a general basisThus, fit with a general basisset (e.g., polynomials) set (e.g., polynomials) Basis functions ~ the “true”Basis functions ~ the “true”function provides a more compactfunction provides a more compactrepresentationrepresentation

Weighted least squares equationsWeighted least squares equations

V fitted (z) j1

m

aiT (z) bi(z)

D[V fitted (z)] w i

j1

N

(z)[V (z(i)) V fitted (z(i))]2

BTW(z) B a(z) = BTW(z)V

)()()(

)()()(

)()()(

)()(2

)(1

)2()2(2

)2(1

)1()1(2

)1(1

nm

nn

m

m

bbb

bbb

bbb

zzz

zzz

zzz

B

W=1 gives standardleast squares

We use standardroutines

Weighted vs. standard least squaresWeighted vs. standard least squares1st-degree std least squares vs IMLS

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exact function

std least squares

IMLS interpolation

2nd-degree std least squares vs IMLS

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std least squares

IMLS interpolation

Standard, first degreefit to the 5 points

IMLS, first degree

IMLS fits perfectlyat each point

Standard, second degree

IMLS, second degree

First Degree

Second Degree

Optimum Point Placement We want to do the fewest number of ab initio calculations

A non-uniform distribution of points is best

We can use the fact that IMLS fits perfectly at each point

to determine where to place points for the most accurate

fit using the fewest possible points Use fits of different degree IMLS fits

1-D morse function: five data points

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erg

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exact function

Illustrate for 1-D Morse potential 5 “seed” points

Automatic Point Placement: 1-D IllustrationAutomatic Point Placement: 1-D Illustration

Squared difference surface indicates point where data is required

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exact function

2nd-degree IMLS

3rd-degree IMLS

squared difference

Start with 5 uniformly placed pointsFit with 2nd & 3rd degree IMLSAdd new point where they differ the most

Squared difference indicates where new points are needed

Automatic Point Placement Point Placement


-20

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Ene

rgy

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2nd-degree IMLS

3rd-degree IMLS

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rgy

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2nd-degree IMLS

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squared difference


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2nd-degree IMLS

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gy

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2nd-degree IMLS

3rd-degree IMLS

squared difference

1 new point added5 initial points

2 new points added 3 new points added

Density adaptive weight functionDensity adaptive weight function

))

)(/(()( 2)(

2

piid

zz

i id

zzeZw

i

10 5 0 5 100

2000

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6000

8000

1 104

Automatic pointplacement will generate a nonuniform densityof points.

Thus, we use aflexible, density-dependentweight function

High Dimensional Model Representation(HDMR) basis set

...),()()...,(,

)2(,

)1(21

jijiji

iiiN QQVQVQQQV

• Can represent high dimensional function through an Can represent high dimensional function through an expansion of lower order termsexpansion of lower order terms

• Can also use full dimensional expansion but restrict the Can also use full dimensional expansion but restrict the order of terms differentlyorder of terms differently

• Evaluation scales as NMEvaluation scales as NM22. HDMR greatly reduces M.. HDMR greatly reduces M.

• This also reduces the number of points required.This also reduces the number of points required.

Accurate PESs from Low-Density DataAccurate PESs from Low-Density Data

Initial testing for 3-D: HCN-HNC

We used the global PES fit to ab initio points by van Mourik et al.* as a source for (cheap) points. Saves time obtaining points Allows extensive error analyses

We fit using (12,9,7) HDMR basis: 1-coordinate term truncated at 12th degree 2-coordinate term truncated at 9th degree 3-coordinate term truncated at 7th degree180 basis functions

* T. van Mourik, G. J. Harris, O. L. Polyansky, J. Tennyson, A. G. Császár, and P. J. Knowles, J. Chem. Phys. 115, 3706 (2001).

Error as function of automatically selected data points

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120 160 200 240 280 320

Number of points with Hessians

Err

or

(kca

l/m

ol)

RMS succ. orders IMLS

RMS error IMLS

Mean succ. orders IMLS

Mean error IMLS

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RMS error IMLS

Mean succ. orders IMLS

Mean error IMLS

3-D HCN:HNC

Automatic surface generationUsing (12,9,7) & (11,8,6) bases

Data Points:van Mourik et al. PES

Seed points: Start with 4, 6, & 8 for r, R & cosθ Energy cutoff: 100 kcal/mol

RMS

Mean

Successive Order: Solid True Error: Open

The difference in successiveorders follows closely thetrue error.

Thus, adding points based ondifference criteria results inconverged true error

Convergence rate dependence on basis set: HCN

Power-law convergence of 3-D PES (HCN)

0.00001

0.0001

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1

100 1000 10000

Number of points

RM

S e

rro

r (k

cal/m

ol)

5th-degree

4th-degree

6th-degree

7th-degree

8th-degree

HDMR (12,9,7)

Number of Points

RM

S E

rror

(k

cal/

mol

)

Obeys power law over 3 orders of magnitude

Accuracy follows Farwig’s* formula for power-law Accuracy follows Farwig’s* formula for power-law convergence convergence Linear on log-log plot with slope ~(n+1)/D, Linear on log-log plot with slope ~(n+1)/D, where n = degree of basiswhere n = degree of basis

* R. Farwig, J. Comput. Appl. Math. 16, 79 (1986); Math. Comput. 46, 577 (1986).

8th degree &HDMR (12,9,7)both have ~ 180fcts., but HDMRconverges faster

Cutting cost: Local IMLS

• Cost of evaluation scales as NM2 for standard IMLS (N=# ab initio points, M=# basis functions)• High-degree standard IMLS is too costly to use directly, thus we use local-IMLS: Local approximants (polynomials) of the

potential near data points are calculated using IMLS (expensive) & the interpolated value is taken to be a weighted sum of them

• In standard IMLS they are recomputed at each evaluation point In standard IMLS they are recomputed at each evaluation point (very accurate, but too costly)(very accurate, but too costly)

• The coefficients are generally slowly varyingThe coefficients are generally slowly varying• In the L-IMLS approach coefficients are computed & stored at a In the L-IMLS approach coefficients are computed & stored at a

relatively small number of pointsrelatively small number of points• Evaluations are low cost weighted interpolations between stored Evaluations are low cost weighted interpolations between stored

pointspoints

Overcoming scaling problem for automatic point selection• We get high accuracy & low cost with high-degree L-IMLS But must find optimum place to add each We get high accuracy & low cost with high-degree L-IMLS But must find optimum place to add each ab initioab initio

pointpoint• Trivial in 1-DTrivial in 1-D

as shownas shown

earlierearlier

• With L-IMLS the functions whose maxima we seek are continuously globally defined as are their gradientsWith L-IMLS the functions whose maxima we seek are continuously globally defined as are their gradients• So, define negative of the squared-difference surfaceSo, define negative of the squared-difference surface• We can use efficient minimization schemes such as conjugate gradient to find local minimaWe can use efficient minimization schemes such as conjugate gradient to find local minima

• Difference between successive orders of IMLSDifference between successive orders of IMLS• Can also use variance of weighted contributions to interpolated value with local IMLSCan also use variance of weighted contributions to interpolated value with local IMLS

• Grid or random search scales very poorly with dimensionGrid or random search scales very poorly with dimension


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En

erg

yFitting data

exact function

2nd-degree IMLS

3rd-degree IMLS

squared difference

Method schematicMethod schematic

Compute Ab Initio Data

Compute L-IMLS

Automatic Data Point Location

Add New Ab Initio Data

Test Fitting Error Statistics Write PES Data, Terminate

Seed Grid

Read Input

Automated PES fitting in 3-D: HCN-HNCAutomatic surface generation:HCN

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1000

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Number of points

RM

S e

rro

r (k

cal/m

ol)

)

Val only

Val+grad

Val+grad+Hess

(12,9,7)

Basis set not well supported

Spectroscopic accuracy To less than 1 cm-1

within 792 pts with Hessians or 1000 pts with gradients

The PES is fit up to 100 kcal/mol

~ cm-1

828318

223

Used 30 random starting points for minimizations

HDMR (12,9,7)

For 0.1 kcal/mol

But we can do even betterDiscussed below

Dynamic Basis Procedure

Avoids including points in the seed data that are not optimally located

Start with very small initial grid of points &use automatic surface generation with a small basis, successively increasing the basis as points are added

Automated Dynamic Basis: 6-D (HOOH)

Automatic surface generation: HOOH

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1000

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RM

S e

rro

r (k

ca

l/m

ol)

Val only

Val+grad

Val+grad+Hess

(10,7,5,4)

(6,3)

(7,4)(8,5,3)

(9,6,4)

Dynamic basis

Fit up to 100 kcal/mol

Fit to analylic H2O2 PES*

* B. Kuhn et al. J. Chem. Phys. 111, 2565 (1999).

114

164

754

RMS error based onrandomly selectedtest points

A min. of 591 pts. would be needed if we started with the(10,7,5,4) basis.We started with 108.Convergence alsomuch faster

Spectroscopic Accuracy: 9-D (CH4)

Test Case: Schwenke & Partridge PES: a least squares fit to ~8000 CCSD(T)/cc-pVTZ ab initio data over therange 0-26,000 cm-1

We fit the range 0-20,000 cm-1 (57.2 kcal/mol). Energies & gradients only (Hessians data not cost effective as shown earlier) Bond distances Exploited permutation symmetry Dynamic basis procedure

D. W. Schwenke & H. Partridge, Spectrochim Acta Part A 57, 887 (2001)

Automated PES fitting in 9-D (CH4)Automatic surface generation: CH4

9D dynamic basis

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Mea

n e

rro

r (k

cal/m

ol)

Val only

Val + grad

(7,4)

(8,5,3)

(9,6,4)

(6,3)

(9,6,4)

With 1552 pts. the E onlyRMS error is 0.41 kcal/mol& including gradientsbrings it down to 0.32 kcal/mol.

The RMS error for the Schwenke-Partridge PES(based on 8000 pts) is~0.35 kcal/mol

The IMLS fitting is essentially automatic, little human effort, and no priorknowledge of the topology

9,6,4,4

9,6,4,4

A General 3-Atom IMLS-QC CodeA General 3-Atom IMLS-QC Code

• Input fileInput file• Accuracy targetAccuracy target• Energy rangeEnergy range• Basis setBasis set• Number of seed points and coordinate rangesNumber of seed points and coordinate ranges• Type of coordinates, Jacobi, valence, bond Type of coordinates, Jacobi, valence, bond

distancesdistances• Generates input files for Gaussian, MolPro, and Generates input files for Gaussian, MolPro, and

Aces IIAces II• Energies only or energies & gradientsEnergies only or energies & gradients

A New PES for the Methylene RadicalA New PES for the Methylene Radical

We have generated a spectroscopically accurate PES for CH2 forenergies up to 20,000 cm-1 (216 vibrational states).

CASSCF calculations in valence coordinates.

Vibrational levels were computed using a discrete variable representation(DVR) method.

DVR typically requires 10’s of thousands of ab initio points. For abenchmark we performed a DVR calculation using ab initio calculations at all 22,400 DVR points.

Singlet Methylene: fit to energies and gradientsSinglet Methylene: fit to energies and gradients

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an

an

d R

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err

ors

(c

m-1

)

mean est. error

RMS est. error

true mean error

true RMS error

291 355 435259

CASSCF calculation in valence coordinates. Energy range of 20000 cm-1.Estimated error vs. true error (sets of 500 random ab initio calcs).True error (RMS and mean) are sub-wavenumber using 355 points.

Black: estimated errors

Red: true errors

True and estimatederrors are in nearperfect agreement

Singlet Methylene Vibrational Levels: Singlet Methylene Vibrational Levels: Discrete Variable Representation (DVR) CalculationDiscrete Variable Representation (DVR) Calculation

Absolute errors for 216 vibrational levels (below 20,000 cm-1). Variational vibrational calculations were performed using DVR and a PES fitted with a mean estimated error of 2.0 cm-1

Exact levels were benchmarked by a DVR calculation using ab initio calculations at all 22,400 DVR points.

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Plot of absolute errors for 216 vibrational levels (below 20,000 cm-1). Variational vibrational calculations were performed using a DVR and fitted PESs with mean estimated errors of 0.5 cm-1

Exact levels were benchmarked by a DVR calculation using ab initio calculations at all 22,400 DVR points.


Singlet Methylene Vibrational Levels: ComparisonsSinglet Methylene Vibrational Levels: Comparisons

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Absolute errors for 216 vibrational levels (below 20,000 cm-1). Variational vibrational calculations were performed using a DVR and PES fitted with mean estimated errors of 0.33 cm-1

Exact levels were benchmarked by a DVR calculation using ab initio calculations at all 22,400 DVR points. Mean and maximum errors for levels computed with this PES are 0.10 and 0.41 cm-1.

Singlet Methylene Vibrational Levels: ComparisonsSinglet Methylene Vibrational Levels: Comparisons

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IMLS & Classical TrajectoriesIMLS & Classical TrajectoriesPreliminary EffortsPreliminary Efforts

Two difference approaches:

IMLS-accelerate direct dynamics

Dynamics Driven Fitting

(both under development)

In both cases IMLS “intercepts” ab initio PES calls & the electronic structure code is called only if necessary (based on error estimate)

Accelerated Direct DynamicsAccelerated Direct DynamicsTest case: HONO Test case: HONO cis-transcis-trans isomerization isomerization

Trajectories were initiated with 8 quanta in the HON bend to cause Trajectories were initiated with 8 quanta in the HON bend to cause rapid IVR & then isomerization (Want rapid exploration of rapid IVR & then isomerization (Want rapid exploration of configuration space)configuration space)

Integration stepsize: 0.05 fsIntegration stepsize: 0.05 fs

Trajectories were stopped once they spent 3 times the period of the Trajectories were stopped once they spent 3 times the period of the torsion mode in the range of the torsion mode in the range of the transtrans torsion angle or violated energy torsion angle or violated energy conservation criterionconservation criterion

Used HF/cc-pVDZ – want fast Used HF/cc-pVDZ – want fast ab initioab initio calculation to test the method calculation to test the method• IMLS “intercepts” direct dynamics IMLS “intercepts” direct dynamics ab initioab initio PES calls. Electronic PES calls. Electronic

structure code is called only if necessary (based on error estimate) structure code is called only if necessary (based on error estimate) Data collection trajectories are moved back in time if the rare event of Data collection trajectories are moved back in time if the rare event of adding new ab initio data occursadding new ab initio data occurs

Accelerated direct dynamics with IMLS: HONO

Speedup

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PE

S e

va

lua

tio

ns

speedup 25.2

speedup 18.1

speedup 7.4

speedup 76.3

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12

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speedup

Ma

x E

ne

rgy

dri

ft (

kc

al/m

ol-

ps

)

(10,7,5,5) basis of 651 functions Values and gradients usedThe fit began after 25 ab initio "seed" points were generated

10-2

10-5

Factor of ~20 speed upwith 0.06 drift in total energy

Speedup depends on error tolerance

7.6 evaluations per ab initio callfor 10-5 error tolerance

76.3 evaluations per ab initio callfor 10-2 error tolerance

Dynamics Driven Fitting: HONO Dynamics Driven Fitting: HONO cis-transcis-trans isomerization rateisomerization rate

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PE

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0.1 kcal/mol max error





A series of sets of trajectories, with various energy conservation limits, are used to explore configuration space.

Accelerated direct dynamics: HONO Accelerated direct dynamics: HONO cis-transcis-trans isomerization rateisomerization rate Rate calculation

y = 1.2491x

R2 = 0.9911

0

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0.6

0.8

1

1.2

1.4

1.6

1.8

0 0.2 0.4 0.6 0.8 1 1.2 1.4

t (ps)

-ln

(Nt/

N0

)

IMLS tolerance (kcal/mol)

k (ps-1) Ab initio points PES calls

0.1 0.97 1867 ~107

0.3 0.99 1020 ~107 0.5 1.02 807 ~107 1.0 1.11 428 ~107 2.0 1.02 362 ~107 3.0 1.02 261 ~107 4.0 1.25 229 ~107 5.0 1.28 221 ~107

Results for PESs fit with 8 differentmaximum error tolerances

Concluding CommentsConcluding Comments

• IMLS allows automated generation of PESs for various IMLS allows automated generation of PESs for various applicationsapplications• SpectroscopySpectroscopy• DynamicsDynamics

• Flexible fits to energies, energies and gradients, or higher Flexible fits to energies, energies and gradients, or higher derivatives…derivatives…

• Interfaced to general classical trajectory code: GenDynInterfaced to general classical trajectory code: GenDyn• Interfaced to electronic structure codesInterfaced to electronic structure codes

• Gaussian, Molpro, Aces IIGaussian, Molpro, Aces II• Robust, efficient, practical methods that assures fidelity toRobust, efficient, practical methods that assures fidelity to the the ab initioab initio data data

automatic construction of ab initio potential energy surfaces interpolative moving least squares...

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