Electronic rearrangement motions

A formulation of electronic rearrangement in quantum molecular dynamics has been based on the Liouville-von Neumann equation for the density matrix. Introducing an eikonal representation, it naturally leads to a general treatment where Hamiltonian equations for nuclear motions are coupled to the electronic density matrix equations, in a formally exact theory. Expectation values of molecular operators can be obtained from integrations over initial conditions. 

In a 7r electron system the orbitals of an aromatic positive ion are similar to the corresponding orbitals of the neutral molecule. In contrast, in small molecules electronic rearrangement following excitation is often sufficiently important that changes in nuclear geometry, correlation energy, etc., are all essential to the correct interpretation of the excitation phenomenon. Because of the similarity in the orbital systems of neutral and positive ion aromatic compounds, we shall assume that it is possible to describe the photoionization of an aromatic molecule within the framework of a one-electron model. Given that the n and a electrons are describ-able by a set of separable equations of motion, we need consider only the initial and final orbitals of the most weakly bound electron to determine the ionization cross section near to the threshold of ionization. 

Photostimulated molecular motion is an important photophysical phenomenon frequently exploited in molecular switches. The molecular electronic rearrangements accompanying optical excitation may stimulate nuclear rearrangement of the excited species. Like electron and energy transfer, such processes compete with radiative events and therefore reduce the measured lifetime and quantum yield of emission. The most important nuclear rearrangements in supramolecular species are proton transfer and photoisomerization. 

Oscillations in time of quantal states are usually much faster than those of the quasiclassical variables. Since both degrees of freedom are coupled, it is not efficient to solve their coupled differential equations by straightforward time step methods. Instead it is necessary to introduce propagation procedures suitable for coupled equations with very different time scales short for quantal states and long for quasiclassical motions. This situation is very similar to the one that arises when electronic and nuclear motions are coupled, in which case the nuclear positions and momenta are the quasiclassical variables, and quantal transitions lead to electronic rearrangement. The following treatment parallels the formulation introduced in our previous review on this subject [13]. Our procedure introduces a unitary transformation at every interval of a time sequence, to create a local interaction picture for propagation over time. 

With the new density matrices known up to time tj, it is possible to advance the quasiclassical positions and momenta by integrating their Hamilton equations. This completes a cycle which can be repeated to advance to a later time t2- This sequence, based on relaxing the density matrix for fixed nuclei and then correcting it to account for quasiclassical motions, has been called the relax-and-drive procedure, and has been numerically implemented in several applications involving electronic rearrangement in atomic collisions [8]. 

To illustrate the molecular basis for this selectivity consider the molecular orbitals of butadiene and cyciobutene which are involved in the electronic rearrangement. These are illustrated in Fig. 6.7 their relative energies are tii < 7t2< ti < and a < n < n < a. Now consider disrotatory motion around the C—C bonds (rotation in the opposite direction) as shown in Fig. 6.8. The two orbitals and must correlate with two orbitals of cyciobutene. One of these is obviously n since both 7ti and TTg are in phase and therefore bonding between Cg and Cg the other is a since in both 7ti and Jig disrotatory motion creates a bonding (T-orbital between Cj and C4. Similar arguments can be used to establish the pair-... 

The Franck-Condon principle says that the intensities of die various vibrational bands of an electronic transition are proportional to these Franck-Condon factors. (Of course, the frequency factor must be included for accurate treatments.) The idea was first derived qualitatively by Franck through the picture that the rearrangement of the light electrons in die electronic transition would occur quickly relative to the period of motion of the heavy nuclei, so die position and iiioiiientiim of the nuclei would not change much during the transition [9]. The quaiitum mechanical picture was given shortly afterwards by Condon, more or less as outlined above [10]. 

In this chapter we will illustrate examples of three families of molecular-level devices (i) devices for the transfer of electrons or electronic energy, (ii) devices capable of performing extensive nuclear motions, often called molecular-level machines, and (Hi) devices whose function implies the occurrence of both electronic and nuclear rearrangements. Most of the examples that will be illustrated refer to devices studied in our laboratories. 

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