Theoretical Chemical Dynamics Studies for Elementary Combustion Reactions Donald Thompson, Gia Maisuradze, Akio Kawano, Yin Guo, Oklahoma State University Advanced Software for the Calculation of Thermochemistry, Kinetics, and Dynamics Stephen 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 = 0 th 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 HN 2 example 33 equally spaced points Solid line = exact derivative Spiked line = shepard derivative

Method: We want V(x) when we know {V(x i ) | 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=0 m a j (x) b(x) j where a j from weighted least squares fit to {V(x i )} Fit by Taylor Series of degree m? V(x) =  j=0 m a j b(x) j where a j from least squares fit to {V(x i )} weights = w i (x,x i ), e.g, = e -  ∆x 2 /[∆x n +  ] where ∆x = x-x i

Method: a obeysB(x i ) T W(x,x i ) B(x i ) a(x) = B(x i ) T W(x,x i ) V(x i ) final IMLS fit:V(x) =  j=0 m a j (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 ∂ 2 V/∂x 2 ) = IMLS fit for m=0

Method: ∂V(x)/∂x =  j=0 m [a j (x) jb(x) j-1 ∂b(x)/∂x + ∂a j (x)/∂x b(x)] ∂a/∂x obeys: B(x i ) T W(x,x i ) B(x i ) ∂a(x)/∂x = B(x i ) T ∂W(x,x i )/∂x [V(x i )- B(x i )a(x)] same left hand side as 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 0 th IMLS => b(x) 0 or derivative of basis does not contribute => only ∂a j (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 HN 2 spline PES by Koizumi et al. 100 kcal/mol range Results: 1D Applications

MO example Equally spaced points IMLS degree cubic spline HN 2 example Equally spaced points IMLS degree Results: 1D Applications 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 example Equally spaced points IMLS degree cubic spline HN 2 example Equally spaced points IMLS degree Results: 1D Applications RMS error in fitting derivatives 0th degree (i.e., Shepard) improves poorly with more points higher degrees have qualitatively improved accuracy

Results: 1D Applications 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

Results: HOOH 6-D Applications Tom Rizzo’s 6D HOOH PES Coordinate representation in terms of 6 interatomic distances ROH, 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: R i = f ni R o i for i = 1,6 where f>1 determines increment RMS error by MC or Grid: Sampling method matters Random EMS COMB GRID

EMS-EMS Fitting to Differences: Develop a qualitative fit Vo Apply IMLS to V-Vo HOOH example - simple functional form - 89 predetermined ab initio random ab initio pts. Results: HOOH 6-D Applications Fitting directly to PES:

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

Results: 6-D Applications 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

Results: 6-D Applications 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: for each case - zero angular momentu results: - rates from trajectories converge much faster than rms error on the surface

Results: up to 15D Applications Model variable dimensional PES V(x 1,x 2,x 3,…) = V Eckart (x 1 ) +  i NDOF {V MO (x 1,x i )} where V Eckart (x 1 ) => where V MO (x 1,x i ) => products reactants x 1 = rxn path rxn barrier x i = deviation off rxn path Local Diss. energy (x 1 dependent) width (i dependent) 0 - V MO 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 x i Local Diss. Energy + V Eckart = fixed global Diss. Energy

Results: up to 15D Applications point selection: - on single diagonal (…,x i,…) i = 1,N - on double diagonal (…,x i,…,y i,…) 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)

Results: up to 15D Applications 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

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)