The bending-mode potential energy surface of a triatomic molecule

PHYSICS · Multi-part · Fermi University Physics · Computational Physics · Scientific programming for physicists · Scientific programming for physicists

Scientific Computing for Physicists, Ch Minimisation with guarantees, with constraints and with randomness

Problem

The bending-mode potential energy surface of a triatomic molecule, near its equilibrium geometry, has Hessian eigenvalues

\lambda_{\max} = 850\,\epsilon/\sigma^2

and

\lambda_{\min} = 2\,\epsilon/\sigma^2

in Lennard-Jones reduced units. ((a)) Compute the condition number

\kappa = \lambda_{\max}/\lambda_{\min}

. ((b)) Compute the guaranteed per-step factor of exact-line-search steepest descent,

\left[\dfrac{\kappa-1}{\kappa+1}\right]^2

. ((c)) Estimate the number of steps needed to reduce

f - f^*

by four decades (a factor of

10^{-4}

), assuming this worst-case factor applies every step.

Browse the problem library