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Automatic Construction of Ab Initio Potential Energy Surfaces

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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. Thompson

University of Missouri – Columbia

Richard Dawes, Al Wagner,

& Michael Minkoff

IMA Workshop: Chemical Dynamics: Challenges and Approaches

January 12-16, 2009

University of Minnesota

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 of the fitting methods:

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 Fitting

Usual applications are

for data with statistical

errors, but trends that follow known functional forms.

- Ab initio energies do not have random errors
- A PES does not have a precisely known functional form
- the energy points lie on a surface
- of unknown shape
- Thus, fit with a general basis

set (e.g., polynomials)

- Basis functions that ~ the “true”

function provides a more compact

representation

Fitting ab initio energies

Weighted Least Squares Equations

W=1 gives standard

least squares

We use standard

routines

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

Weighted vs. standard least squares

Standard, first degree

fit to the 5 points

First Degree

IMLS, first degree

IMLS fits perfectly

at each point

Second Degree

Standard, second degree

IMLS, 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 for automatic

point placement

Illustrate for 1-D

Morse potential

5 “seed” points

Automatic Point Placement: 1-D Illustration

Consider starting with 5 uniformly placed points

Fit with 2nd & 3rd degree IMLS

Add new point where they differ the most

Squared difference indicates where

new points are needed

Density adaptive weight function

Automatic point

placement will

generate a

nonuniform density

of points.

Thus, we use a

flexible,

density-dependent

weight function

High Dimensional Model Representation(HDMR) basis set

- Can represent high dimensional function through an expansion of lower order terms
- Can also use full dimensional expansion but restrict the order of terms differently
- Evaluation scales as NM2. HDMR greatly reduces M.
- HDMR reduces the number of points required.

Accurate 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 degree

180 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

Seed points: Start with 4, 6, & 8 for r, R & cosθ

Energy cutoff: 100 kcal/mol

Data Points:

van Mourik et al. PES

3-D HCN:HNC

Automatic surface generation

Using (12,9,7) & (11,8,6)

bases

Successive Order: Solid

True Error: Open

The difference in successive

orders follows closely the

true error.

Thus, adding points based on

difference criteria results in

converged true error

RMS

Mean

Convergence rate dependence on basis set: HCN

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

Obeys power law over 3 orders of magnitude

RMS Error (kcal/mol)

8th degree &

HDMR (12,9,7)

both have ~ 180

functions, but

HDMR

converges faster

Number of Points

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

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 (very accurate, but too costly)
- The coefficients are generally slowly varying
- In the L-IMLS approach coefficients are computed & stored at a relatively small number of points
- Evaluations are low cost weighted interpolations between stored points

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

ab initio point

- Trivial in 1-D

as shown earlier

- With L-IMLS the functions whose maxima we seek are continuously globally defined as are their gradients
- So, define negative of the squared-difference surface
- We can use efficient minimization schemes such as conjugate gradient to find local minima
- Difference between successive orders of IMLS
- Can also use variance of weighted contributions to interpolated value with local IMLS
- Grid or random search scales very poorly with dimension

Automated PES fitting in 3-D: HCN-HNC

Used 30 random starting points for minimizations

Basis set not well supported

Spectroscopic accuracy

To less than 1 cm-1

within 792 pts with

Hessians or 1000 pts with gradients

223

828

318

For 0.1 kcal/mol

~cm-1

HDMR (12,9,7)

But we can do even better

Discussed below

The PES is fit up to 100 kcal/mol

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

to support the larger basis.

Automated Dynamic Basis: 6-D (HOOH)

Fit to analylic

H2O2 PES*

Dynamic basis

Fit up to 100 kcal/mol

A min. of 591 pts.

would be needed if

we started with the

(10,7,5,4) basis.

We started with 108.

Convergence also

much faster

164

114

754

RMS error based on

randomly selected

test points

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

Spectroscopic Accuracy: 9-D (CH4)

Test Case: Schwenke & Partridge PES: a least squares fit

to ~8000 CCSD(T)/cc-pVTZ ab initio data over the

range 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)

With 1552 pts. the E only

RMS error is 0.41 kcal/mol

& including gradients

brings 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 prior

knowledge of the topology

9,6,4,4

9,6,4,4

CASSCFPES for the CH2Testing Fitting Accuracy

Generated a spectroscopically accurate PES for CH2 for

energies up to 20,000 cm-1

CASSCF/aug-cc-pVDZ calculations in valence coordinates.

216 vibrational levels were computed using a discrete

variable representation (DVR) method.

For a benchmark we performed a “direct” DVR calculation

using ab initio calculations at all 22,400 DVR points.

CH2: fit to energies and gradients

True and estimated

errors are in near

perfect agreement

Black: estimated errors

2.0 cm-1

Red: true errors

0.5 cm-1

0.33 cm-1

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.

CH2: Comparison of fits with valence and bond distance coordinates to energies and gradients

Valence coordinates

Bond

Distances

Stoppedat 500 pts.

Mean error ~2.8 cm-1

Singlet Methylene Vibrational Levels: Discrete 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.

CASSCF (full valence) with aug-cc-pVDZ

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

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: Comparisons

2.0 cm-1 mean

estimated error

0.5 cm-1 mean

estimated error

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

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: Comparisons

2.0 cm-1 mean

estimated error

0.33 cm-1 mean

estimated error

A New PES Highly Accurate PES for the CH2

- A series of sub-wavenumber PESs fit using automatic surface generator
- Separate surfaces for
- MRCI/avdz, MRCI/avtz, MRCI/avqz, MRCI+Q/avdz, MRCI+Q/avtz,
- MRCI+Q/avqz,CCSD(T)(AE)/acvtz, CCSD(T)(FC)/acvtz
- Used DVR to determine best PES based on comparisons with experiment.
- generated a spectroscopically accurate PES for CH2 for energies up to
- 20,000 cm-1
- MRCI/CBS+C-V+DW(+Q)
- [Multi-reference configuration interaction (MRCI) with complete basis
- extrapolation (CBS) and CCSD(T) based Core-Valence (CV) correction.
- Davidson correction to MRCI added with dynamic weighting based on
- small basis Full CI calculations]

CH2 Vibrational Frequencies

Errors relative to experimental values

Expt: Gu et al. J. Mol. Str. 2000, 517, 247.

Slices though IMLS-based PESs for CH2

Scan of one C-H bond distance with the other bond distance

and angle held fixed at equilibrium values.

HCN-HCN: Highly Accurate PESs

- We have generated a spectroscopically accurate PES for HCN-HNC for energies up to 35,000 cm-1
- Vibrational levels were computed using a discrete variable representation (DVR) method.
- Details coming soon!

Concluding Comments

IMLS allows automated generation of PESs for various applications

- Spectroscopy
- Dynamics

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

Interfaced to electronic structure codes

- Gaussian, Molpro, Aces II

Robust, efficient, practical methods that assures fidelity to

the ab initio data

General PES fitter for 3-atom systems

Interfaced to general classical trajectory code: GenDyn

A General 3-Atom PES Fitter

- Input file
- Accuracy target
- Energy range
- Basis set
- Number of seed points and coordinate ranges
- Type of coordinates, Jacobi, valence, bond distances
- Generates input files for Gaussian, MolPro, and Aces II
- Energies only or energies & gradients

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