Certifying the Potential Energy Landscape
[Technical Report, Research Report]
North Carolina State University
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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 employ Smales -theory to stationary points, providing a certification that serves 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 -theory can be used to certify all the known minima and transition states of Lennard-Jones LJN atomic clusters for N 7, . . . , 14.
- Numerical Mathematics
- Operations Research