^{1,a)}and Eric Vanden-Eijnden

^{2,b)}

### Abstract

The string method originally proposed for the computation of minimum energy paths (MEPs) is modified to find saddle points around a given minimum on a potential energy landscape using the location of this minimum as only input. In the modified method the string is evolved by gradient flow in path space, with one of its end points fixed at the minimum and the other end point (the climbing image) evolving towards a saddle point according to a modified potential force in which the component of the potential force in the tangent direction of the string is reversed. The use of a string allows us to monitor the evolution of the climbing image and prevent its escape from the basin of attraction of the minimum. This guarantees that the string always converges towards a MEP connecting the minimum to a saddle point lying on the boundary of the basin of attraction of this minimum. The convergence of the climbing image to the saddle point can also be accelerated by an inexact Newton method in the late stage of the computation. The performance of the numerical method is illustrated using the example of a 7-atom cluster on a substrate. Comparison is made with the dimer method.

We are grateful to Weinan E for helpful discussions. The work of Ren was in parts supported by Singapore A*STAR SERC “Complex Systems Programme” grant R-146-000-171-305 (Project No. 1224504056) and A*STAR SERC PSF grant R-146-000-173-305 (Project No. 1321202071). The work of Vanden-Eijnden was in parts supported by NSF (Grant No. DMS07-08140) and ONR (Grant No. N00014-11-1-0345).

I. INTRODUCTION

II. THE STRING METHOD

III. THE CLIMBING STRING METHOD FOR SADDLE POINT SEARCH

A. Step 1: Evolution of the string

B. Step 2: Imposing the monotonic-energy constraint

C. Step 3: Reparametrization of the string

IV. ACCELERATION OF THE CONVERGENCE BY INEXACT NEWTON METHOD

V. NUMERICAL EXAMPLES

A. Performance of the climbing string method

B. Performance of the inexact Newton method

C. Performance of the complete algorithm

VI. CONCLUSION

### Key Topics

- Interpolation
- 8.0
- Boundary value problems
- 5.0
- Potential energy surfaces
- 4.0
- Acceleration measurement
- 3.0
- Atomic and molecular clusters
- 3.0

## Figures

The configuration of the seven-atom cluster on a substrate at the minimum of the potential energy.

The configuration of the seven-atom cluster on a substrate at the minimum of the potential energy.

Snapshots of the climbing string at different times. The string is projected onto the *xy* plane. The surface is the energy . (a) The initial string; (b) the string at a time when the potential energy along the string becomes non-monotone; (c) the string after truncation at the point where the potential energy attains its first maxima; (d) the converged string and the saddle point located by the final point of the string.

Snapshots of the climbing string at different times. The string is projected onto the *xy* plane. The surface is the energy . (a) The initial string; (b) the string at a time when the potential energy along the string becomes non-monotone; (c) the string after truncation at the point where the potential energy attains its first maxima; (d) the converged string and the saddle point located by the final point of the string.

The five saddle points (filled circles) and their corresponding convergence region in the 3D problem. The minimum is shown as the filled square near the center of figure. The lines are the level curves of the energy .

The five saddle points (filled circles) and their corresponding convergence region in the 3D problem. The minimum is shown as the filled square near the center of figure. The lines are the level curves of the energy .

Convergence history of the inexact Newton method to different saddle points and for different choice of the forcing parameter η: η = 0.1 (circles), η = 0.01 (stars), and η = 0.001 (squares). *N* _{ Newton } is the number of force evaluations. Each cluster corresponds to one inexact Newton iteration.

Convergence history of the inexact Newton method to different saddle points and for different choice of the forcing parameter η: η = 0.1 (circles), η = 0.01 (stars), and η = 0.001 (squares). *N* _{ Newton } is the number of force evaluations. Each cluster corresponds to one inexact Newton iteration.

Histogram of the energy barriers at the saddle points obtained from 100 runs using the climbing string method (upper panel, *N* = 20) and the dimer method (lower panel). The initial data were obtained by randomly displacing the minimum by Δ*x* = 0.5. The climbing string method locates a saddle point of low energy with higher probability. Other runs with fewer images along the string exhibit similar behavior.

Histogram of the energy barriers at the saddle points obtained from 100 runs using the climbing string method (upper panel, *N* = 20) and the dimer method (lower panel). The initial data were obtained by randomly displacing the minimum by Δ*x* = 0.5. The climbing string method locates a saddle point of low energy with higher probability. Other runs with fewer images along the string exhibit similar behavior.

## Tables

Performance of the accelerated climbing string method for the example of 7-atom cluster in the 525d space. The performance of the dimer method is shown for comparison. The data are based on 100 runs from different initial data, which are prepared by randomly displacing the minimum by Δ*x*. *N* is the number of images along the string, ϱ is the ratio of the number of runs that converged to a saddle point directly connected to the minimum, and *n* _{ s } is the number of different saddle points obtained in these runs. *N* _{ string } is the average number of steps (i.e., the number of force evaluations per image) in the climbing string method, *N* _{ Newton } is the average number of force evaluations in the inexact Newton method, is the total number of force evaluations. *N* _{ dimer } and *N* _{ relax } are the average number of force evaluations in the dimer method and in the relaxation step, respectively.

Performance of the accelerated climbing string method for the example of 7-atom cluster in the 525d space. The performance of the dimer method is shown for comparison. The data are based on 100 runs from different initial data, which are prepared by randomly displacing the minimum by Δ*x*. *N* is the number of images along the string, ϱ is the ratio of the number of runs that converged to a saddle point directly connected to the minimum, and *n* _{ s } is the number of different saddle points obtained in these runs. *N* _{ string } is the average number of steps (i.e., the number of force evaluations per image) in the climbing string method, *N* _{ Newton } is the average number of force evaluations in the inexact Newton method, is the total number of force evaluations. *N* _{ dimer } and *N* _{ relax } are the average number of force evaluations in the dimer method and in the relaxation step, respectively.

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