^{1}, Yuli Liu

^{1}, Utpal Sarkar

^{1}and Paul W. Ayers

^{1,a)}

### Abstract

The number of the potential energy calculations required by the quadratic string method (QSM), and the fast marching method (FMM) is significantly reduced by using Shepard interpolation, with a moving least squares to fit the higher-order derivatives of the potential. The derivatives of the potential are fitted up to fifth order. With an error estimate for the interpolated values, this moving least squares enhanced Shepard interpolation scheme drastically reduces the number of potential energy calculations in FMM, often by up 80%. Fitting up through the highest order tested here (fifth order) gave the best results for all grid spacings. For QSM, using enhanced Shepard interpolation gave slightly better results than using the usual second order approximate, damped Broyden-Fletcher-Goldfarb-Shanno updated Hessian to approximate the surface. To test these methods we examined two analytic potentials, the rotational dihedral potential of alanine dipeptide and the reaction of methyl chloride with fluoride.

The authors thank the Chemistry Department of McMaster University, the Hamilton Foundation, and the Canada Research Chairs for financial support. P.W.A. thanks Deborah Crittenden for helpful discussions on Shepard interpolation and for suggesting that IMLS be used to approximate derivatives.

I. INTRODUCTION

II. THE FAST MARCHING AND STRING METHOD

III. INTERPOLATION METHOD

IV. APPLICATIONS

A. The four-well potential

B. Müller–Brown potential

C. Alanine dipeptide

D. methyl chloride reaction

V. SUMMARY

### Key Topics

- Interpolation
- 56.0
- Potential energy surfaces
- 6.0
- Peptides
- 4.0
- Surface dynamics
- 4.0
- Chemical reactions
- 2.0

## Figures

The four-well potential with . Starting from minimum and ending at minimum with . The black curve is the backtrace path. The dots show points where the energy and gradient are calculated by FMM explicitly; at all other points the interpolated energy is used.

The four-well potential with . Starting from minimum and ending at minimum with . The black curve is the backtrace path. The dots show points where the energy and gradient are calculated by FMM explicitly; at all other points the interpolated energy is used.

The MB potential with . The starting point for the FMM is , with backtrace path (black curve) starting from the minimum at . The dots show points where the energy and gradient are calculated. For this example with a large , most points are calculated.

The MB potential with . The starting point for the FMM is , with backtrace path (black curve) starting from the minimum at . The dots show points where the energy and gradient are calculated. For this example with a large , most points are calculated.

Log of the convergence to the correct barrier height for a 15 point path on the M-B potential using different orders of the Shepard interpolation and the damped BFGS Hessian update. The maximum of the cubic spline interpolated energy was used to compare against the exact energy .

Log of the convergence to the correct barrier height for a 15 point path on the M-B potential using different orders of the Shepard interpolation and the damped BFGS Hessian update. The maximum of the cubic spline interpolated energy was used to compare against the exact energy .

AM1 potential energy surface for alanine dipeptide. Small dark regions are areas where GAUSSIAN03 had trouble converging to the correct energy. For FMM, , , , and . FMM was started from , axial. The QSM path is given, as an initial guess, the shortest linear path between the endpoints, using eight points and the initial trust radius set to 11.5°. The QSM path after ten steps is shown with stars . The backtrace path is the black curve.

AM1 potential energy surface for alanine dipeptide. Small dark regions are areas where GAUSSIAN03 had trouble converging to the correct energy. For FMM, , , , and . FMM was started from , axial. The QSM path is given, as an initial guess, the shortest linear path between the endpoints, using eight points and the initial trust radius set to 11.5°. The QSM path after ten steps is shown with stars . The backtrace path is the black curve.

The energy profile for alanine dipeptide (inset molecule), with the energy values obtained by the Shepard interpolation from the backtrace path. The sharp point at an arc length of about 80° is due to an ill converged result from GUASSIAN03 close to the path.

The energy profile for alanine dipeptide (inset molecule), with the energy values obtained by the Shepard interpolation from the backtrace path. The sharp point at an arc length of about 80° is due to an ill converged result from GUASSIAN03 close to the path.

The alanine dipeptide barrier from the, cubic spline interpolated, energy profile of the QSM path using eight points.

The alanine dipeptide barrier from the, cubic spline interpolated, energy profile of the QSM path using eight points.

Methyl chloride reaction with . The path with large points is the QSM path after five iterations. The solid black curve is the backtrace path and the dots are the points calculated with GAUSSIAN03 (Ref. 47). Here with endpoints at and . The -axis is the Cl–C distance and the -axis is the C–F distance.

Methyl chloride reaction with . The path with large points is the QSM path after five iterations. The solid black curve is the backtrace path and the dots are the points calculated with GAUSSIAN03 (Ref. 47). Here with endpoints at and . The -axis is the Cl–C distance and the -axis is the C–F distance.

## Tables

The efficiency of interpolation with the four-well potential for different orders of Shepard interpolation and different grid spacings. “Total” is the total number of FMM points used while “interpolated” is the total number of points where interpolation was accurate enough to be used. For this table .

The efficiency of interpolation with the four-well potential for different orders of Shepard interpolation and different grid spacings. “Total” is the total number of FMM points used while “interpolated” is the total number of points where interpolation was accurate enough to be used. For this table .

The efficiency of interpolation with the MB potential. The same labeling is used as in Table I. Here .

The efficiency of interpolation with the MB potential. The same labeling is used as in Table I. Here .

Article metrics loading...

Full text loading...

Commenting has been disabled for this content