Time-Dependent Electron Localization Function - Fachbereich ...

3 Optimal control
It is more important to have beauty in one’s equations than to
have them fit experiment . . . It seems that if one is working from
the point of view of getting beauty in one’s equations, and if one
has a really sound insight, one is on a sure line of progress.
If there is not complete agreement between the results of one’s
work and experiment, one should not allow oneself to be too
discouraged, because the discrepancy may well be due to minor
features that are not properly taken into account and that will
get cleared up with further developments of the theory.
— Paul Dirac, 1902–84
We are interested in maximizing the transfer of population to a particular molecular
state, such as the π ∗ state as depicted in the previous chapter (section 2.4.1). This
state optimization can be used not only to stimulate chemical reactions, but also to
trigger molecular switches. In this chapter, we concentrate on the HOMO–LUMO
transition of lithium fluoride, which bears some of the hallmarks needed for transport.
After a short introduction, we describe the used algorithm in section 3.2, which
is based on the idea to maximize a suitable functional. In section 3.3 we look at the
actual implementation of this algorithm for molecules having cylindrical symmetry.
This encompasses the discretization and the time-propagation. Section 3.4 contains
the results obtained for lithium fluoride and section 3.5 contains the conclusion and
an outlook.
3.1 Introduction
In subjects reaching from mathematics, engineering and physics to chemistry and
economics optimal control theories (OCT) are used. In physics, such theories are
applied to prepare quantum bits (qbits), align and orient molecules, select reaction
pathways, increase the yield of chemical reactions or to control molecular transport.
Several optimal control techniques are used [36], among them are genetic alogrithms,
feedback-control of experimental systems [37–38], and ab-initio, functional based
methods [39–40]. Coming from a density-functional theory background, we focus on
the last method in this chapter.
3.2 Algorithm
Since we want to use a laser for optimal control, we assume that the Hamilton
operator is of the following form
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