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A climbing string method for saddle point search
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Image of FIG. 1.
FIG. 1.

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

Image of FIG. 2.
FIG. 2.

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.

Image of FIG. 3.
FIG. 3.

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 .

Image of FIG. 4.
FIG. 4.

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.

Image of FIG. 5.
FIG. 5.

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 Δx = 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.


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Generic image for table
Generic image for table
Generic image for table
Generic image for table
Table I.

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 Δx. 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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752b84549af89a08dbdd7fdb8b9568b5 journal.articlezxybnytfddd
Scitation: A climbing string method for saddle point search