Uppsala University Department of Information Technology

Technical Report 2009-022

A Fourier-Coefficient Based Solution of an Optimal Control Problem in Quantum Chemistry

Katharina Kormann, Sverker Holmgren, and Hans O. Karlsson

September 2009

We consider an optimal control problem for the time-dependent Schrödinger equation modeling molecular dynamics. Given a molecule in its ground state, the interaction with a tuned laser pulse can result in an excitation to a state of interest. By these means, one can optimize the yield of chemical reactions. The problem of designing an optimal laser pulse can be posed as an optimal control problem. We reformulate the optimization problem by Fourier-transforming the electric field of the laser and narrow the frequency band. In this way, we reduce the dimensionality of the control variable. This allows for storing an approximate Hessian and, thereby, we can solve the optimization problem with a quasi-Newton method. Such an implementation provides superlinear convergence. We show computational results for a Raman-transition example and give numerical evidence that our algorithm can outperform the standard Krotov-like method which does not employ approximative second derivatives. end{abstract}

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