Communication: Newton homotopies for sampling stationary points of potential energy landscapes

Mehta, Dhagash; Chen, Tianran; Hauenstein, Jonathan D.; Wales, David J.

doi:10.1063/1.4896657

Cited by 20 publications

(32 citation statements)

References 62 publications

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“…PRL 117, 028301 (2016) P H Y S I C A L R E V I E W L E T T E R S week ending 8 JULY 2016 028301-4 kinetic transition rates. We also plan to develop more specific algorithms to locate local and global minima by exploiting small-world properties [86][87][88]. Analyzing the appropriately weighted and directed networks of free energy minima [89] and transition states may provide additional insight into the Thomson problem.…”

Section: Prl 117 028301 (2016) P H Y S I C a L R E V I E W L E T T Ementioning

confidence: 99%

Kinetic Transition Networks for the Thomson Problem and Smale’s Seventh Problem

Mehta

Chen

et al. 2016

Phys. Rev. Lett.

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Readers may view, browse, and/or download material for temporary copying purposes only, provided these uses are for noncommercial personal purposes. Except as provided by law, this material may not be further reproduced, distributed, transmitted, modied, adapted, performed, displayed, published, or sold in whole or part, without prior written permission from the American Physical Society.Additional information: Use policyThe full-text may be used and/or reproduced, and given to third parties in any format or medium, without prior permission or charge, for personal research or study, educational, or not-for-prot purposes provided that:• a full bibliographic reference is made to the original source • a link is made to the metadata record in DRO • the full-text is not changed in any way The full-text must not be sold in any format or medium without the formal permission of the copyright holders.Please consult the full DRO policy for further details. The Thomson problem, arrangement of identical charges on the surface of a sphere, has found many applications in physics, chemistry and biology. Here, we show that the energy landscape of the Thomson problem for N particles with N ¼ 132, 135, 138, 141, 144, 147, and 150 is single funneled, characteristic of a structure-seeking organization where the global minimum is easily accessible. Algorithmically, constructing starting points close to the global minimum of such a potential with spherical constraints is one of Smale's 18 unsolved problems in mathematics for the 21st century because it is important in the solution of univariate and bivariate random polynomial equations. By analyzing the kinetic transition networks, we show that a randomly chosen minimum is, in fact, always "close" to the global minimum in terms of the number of transition states that separate them, a characteristic of small world networks.

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Section: Prl 117 028301 (2016) P H Y S I C a L R E V I E W L E T T Ementioning

confidence: 99%

Kinetic Transition Networks for the Thomson Problem and Smale’s Seventh Problem

Mehta

Chen

et al. 2016

Phys. Rev. Lett.

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“…However, most numerical methods have shortcomings of their own and are not guaranteed to find all the solutions. In particular, the Newton-Raphson method breaks down at singular solutions, 22,23 and the gradient-square minimization method 24,25 may give numerous spurious solutions, which are not physically relevant. 18,26,27 Other more sophisticated approaches based on the Broyden-Fletcher-Goldfarb-Shanno (BFGS) algorithm, 28,29 and eigenvector following 18,26 coupled with single-and double-ended searches, 30-37 may be better alternatives.…”

Section: Introductionmentioning

confidence: 99%

Exploring the potential energy landscape of the Thomson problem via Newton homotopies

Mehta

Chen

Morgan³

et al. 2015

The Journal of Chemical Physics

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Locating the stationary points of a real-valued multivariate potential energy function is an important problem in many areas of science. This task generally amounts to solving simultaneous nonlinear systems of equations. While there are several numerical methods that can find many or all stationary points, they each exhibit characteristic problems. Moreover, traditional methods tend to perform poorly near degenerate stationary points with additional zero Hessian eigenvalues. We propose an efficient and robust implementation of the Newton homotopy method, which is capable of quickly sampling a large number of stationary points of a wide range of indices, as well as degenerate stationary points. We demonstrate our approach by applying it to the Thomson problem. We also briefly discuss a possible connection between the present work and Smale's 7th problem.

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“…(3 ′′ ). 38,43,113 Future progress in this field will likely be tied to improvements in numerical efficiency; in this regard, approaches employing targeted saddle-point searches 68,114 seem promising.…”

Section: A Which Bond Breaks First?mentioning

confidence: 99%

Perspective: Mechanochemistry of biological and synthetic molecules

Makarov

2016

The Journal of Chemical Physics

116

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Coupling of mechanical forces and chemical transformations is central to the biophysics of molecular machines, polymer chemistry, fracture mechanics, tribology, and other disciplines. As a consequence, the same physical principles and theoretical models should be applicable in all of those fields; in fact, similar models have been invoked (and often repeatedly reinvented) to describe, for example, cell adhesion, dry and wet friction, propagation of cracks, and action of molecular motors. This perspective offers a unified view of these phenomena, described in terms of chemical kinetics with rates of elementary steps that are force dependent. The central question is then to describe how the rate of a chemical transformation (and its other measurable properties such as the transition path) depends on the applied force. I will describe physical models used to answer this question and compare them with experimental measurements, which employ single-molecule force spectroscopy and which become increasingly common. Multidimensionality of the underlying molecular energy landscapes and the ensuing frequent misalignment between chemical and mechanical coordinates result in a number of distinct scenarios, each showing a nontrivial force dependence of the reaction rate. I will discuss these scenarios, their commonness (or its lack), and the prospects for their experimental validation. Finally, I will discuss open issues in the field.

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Communication: Newton homotopies for sampling stationary points of potential energy landscapes

Cited by 20 publications

References 62 publications

Kinetic Transition Networks for the Thomson Problem and Smale’s Seventh Problem

Kinetic Transition Networks for the Thomson Problem and Smale’s Seventh Problem

Exploring the potential energy landscape of the Thomson problem via Newton homotopies

Perspective: Mechanochemistry of biological and synthetic molecules

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