%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 }