Two topics will be covered in this lecture: (i) The visualization and
analysis of electronic motion by means of the so-called electron
localization function (ELF), and (ii) optimal control theory of
electronic dynamics. The ELF provides a way to visualize chemical bonds.
It is derived from the conditional probability of finding an electron in
the vicinity of a point r if one knows with certainty that there is
another electron with the same spin at r. The shape of the ELF (as
function of r) allows a topological classification of the different
types of chemical bonds . Here we generalize the ELF to the
time-dependent case . Two movies of the time-dependent ELF will be
presented, one that shows the formation and breaking of chemical bonds
in a proton-ethylene scattering process and another one that visualizes
a laser-induced π-π* transition in acetylene in a time-resolved
In the context of optimal control we first present two generalizations of the standard formulation  of optimal-control theory: The first generalization  allows the calculation of optimized pulses with
frequency constraints. The second generalization  achieves the optimization of time-dependent control targets. The latter allows one to drag the density of the system along a given trajectory, as shown in the figure above. Finally, some aspects of marrying optimal control theory with time-dependent density functional theory will be discussed.
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