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Automatic Construction of Ab Initio Potential Energy Surfaces

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,

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Automatic Construction of Ab Initio Potential Energy Surfaces

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  1. 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 Fourth International meeting : "Mathematical Methods for Ab Initio Quantum Chemistry" 13-14 November 2008Laboratoire J.A. DieudonnéCNRS et Université de Nice - Sophia-Antipolis

  2. 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

  3. 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

  4. 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 ~ the “true” function provides a more compact representation Fitting ab initio energies

  5. Weighted least squares equations W=1 gives standard least squares We use standard routines BTW(z) B a(z) = BTW(z)V

  6. 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

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

  8. Automatic Point Placement: 1-D Illustration Start 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

  9. 5 initial points 1 new point added Automatic Point Placement 2 new points added 3 new points added

  10. Density adaptive weight function Automatic point placement will generate a nonuniform density of points. Thus, we use a flexible, density-dependent weight function

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

  12. 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).

  13. 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

  14. 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 fcts., but HDMR converges faster Number of Points * R. Farwig, J. Comput. Appl. Math. 16, 79 (1986); Math. Comput. 46, 577 (1986).

  15. 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

  16. 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

  17. Method schematic

  18. 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

  19. 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

  20. 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).

  21. 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)

  22. 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

  23. A General 3-Atom IMLS-QC Code • 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

  24. A New PES for the Methylene Radical We have generated a spectroscopically accurate PES for CH2 for energies 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 a benchmark we performed a DVR calculation using ab initio calculations at all 22,400 DVR points.

  25. Singlet Methylene: fit to energies and gradients True and estimated errors are in near perfect agreement Black: estimated errors Red: true errors 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.

  26. 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.

  27. 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.

  28. Singlet Methylene Vibrational Levels: Comparisons 2.0 cm-1 mean estimated error 0.5 cm-1 mean estimated error

  29. 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.

  30. Singlet Methylene Vibrational Levels: Comparisons 2.0 cm-1 mean estimated error 0.33 cm-1 mean estimated error

  31. IMLS & Classical TrajectoriesPreliminary 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)

  32. Accelerated Direct DynamicsTest case: HONO cis-trans isomerization • Trajectories were initiated with 8 quanta in the HON bend to cause rapid IVR & then isomerization (Want rapid exploration of configuration space) Integration stepsize: 0.05 fs  Trajectories were stopped once they spent 3 times the period of the torsion mode in the range of the trans torsion angle or violated energy conservation criterion • Used HF/cc-pVDZ – want fast ab initio calculation to test the method • IMLS “intercepts” direct dynamics ab initio PES calls. Electronic structure code is called only if necessary (based on error estimate) Data collection trajectories are moved back in time if the rare event of adding new ab initio data occurs

  33. Accelerated direct dynamics with IMLS: HONO (10,7,5,5) basis of 651 functions Values and gradients used The fit began after 25 ab initio "seed" points were generated 10-2 10-5 Speedup depends on error tolerance 7.6 evaluations per ab initio call for 10-5 error tolerance 76.3 evaluations per ab initio call for 10-2 error tolerance Factor of ~20 speed up with 0.06 drift in total energy

  34. Dynamics Driven Fitting: HONO cis-trans isomerization rate A series of sets of trajectories, with various energy conservation limits, are used to explore configuration space.

  35. Accelerated direct dynamics: HONO cis-trans isomerization rate Results for PESs fit with 8 different maximum error tolerances

  36. 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 general classical trajectory code: GenDyn • Interfaced to electronic structure codes • Gaussian, Molpro, Aces II • Robust, efficient, practical methods that assures fidelity to the ab initio data

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