It is highly desirable for numerical approximations to stationary points for a potential energy landscape to lie in the corresponding quadratic convergence basin. However, it is possible that an approximation may lie only in the linear convergence basin, or even in a chaotic region, and hence not converge to the actual stationary point when further optimization is attempted. Proving that a numerical approximation will quadratically converge to the associated stationary point is termed certification. Here, we apply Smale's α-theory to stationary points, providing a certification serving as a mathematical proof that the numerical approximation does indeed correspond to an actual stationary point, independent of the precision employed. As a practical example, employing recently developed certification algorithms, we show how the α-theory can be used to certify all the known minima and transition states of Lennard-Jones LJN atomic clusters for N = 7,..., 14.
%0 Journal Article
%1 mehta:171101
%A Mehta, Dhagash
%A Hauenstein, Jonathan D.
%A Wales, David J.
%D 2013
%I AIP
%J The Journal of Chemical Physics
%K chemistry energy mathematics potential root stability surface
%N 17
%P 171101
%R 10.1063/1.4803162
%T Communication: Certifying the potential energy landscape
%U http://link.aip.org/link/?JCP/138/171101/1
%V 138
%X It is highly desirable for numerical approximations to stationary points for a potential energy landscape to lie in the corresponding quadratic convergence basin. However, it is possible that an approximation may lie only in the linear convergence basin, or even in a chaotic region, and hence not converge to the actual stationary point when further optimization is attempted. Proving that a numerical approximation will quadratically converge to the associated stationary point is termed certification. Here, we apply Smale's α-theory to stationary points, providing a certification serving as a mathematical proof that the numerical approximation does indeed correspond to an actual stationary point, independent of the precision employed. As a practical example, employing recently developed certification algorithms, we show how the α-theory can be used to certify all the known minima and transition states of Lennard-Jones LJN atomic clusters for N = 7,..., 14.
@article{mehta:171101,
abstract = {It is highly desirable for numerical approximations to stationary points for a potential energy landscape to lie in the corresponding quadratic convergence basin. However, it is possible that an approximation may lie only in the linear convergence basin, or even in a chaotic region, and hence not converge to the actual stationary point when further optimization is attempted. Proving that a numerical approximation will quadratically converge to the associated stationary point is termed certification. Here, we apply Smale's α-theory to stationary points, providing a certification serving as a mathematical proof that the numerical approximation does indeed correspond to an actual stationary point, independent of the precision employed. As a practical example, employing recently developed certification algorithms, we show how the α-theory can be used to certify all the known minima and transition states of Lennard-Jones LJN atomic clusters for N = 7,..., 14.},
added-at = {2013-05-25T19:00:42.000+0200},
author = {Mehta, Dhagash and Hauenstein, Jonathan D. and Wales, David J.},
biburl = {https://www.bibsonomy.org/bibtex/2aad29f44f1573ffc19a86693f50e2d26/drmatusek},
doi = {10.1063/1.4803162},
eid = {171101},
interhash = {9590f8c85dab5bae2f03077b21e82663},
intrahash = {aad29f44f1573ffc19a86693f50e2d26},
journal = {The Journal of Chemical Physics},
keywords = {chemistry energy mathematics potential root stability surface},
month = may,
number = 17,
numpages = {4},
pages = 171101,
publisher = {AIP},
timestamp = {2013-09-05T03:26:48.000+0200},
title = {Communication: Certifying the potential energy landscape},
url = {http://link.aip.org/link/?JCP/138/171101/1},
volume = 138,
year = 2013
}