Single Particle Picture

The most simple model, being a crude approximation, is the single particle picture.

The main assumption is that the incoming photon only interacts with one single electron in the cluster and completely transfers its energy to it. The electron gets detached if the photon energy hν is higher than its binding energy EBind. Besides that it is assumed that the electron does not interact with its neighbors, neither in the bound state nor while leaving the cluster. The kinetic energy E_kin of the photoelectron is given by

E_kin=hν−E_Bind. (3.1)

A photoelectron spectrum displays the kinetic energy distribution of the electrons and corresponds to a an averaging over many ionization processes. For electrons originating from different orbitals, various peaks with different kinetic energies can be seen. Figure 3.5 shows this correlation schematically. The relative peak intensi-ties indicate the probabiliintensi-ties of the different detachment processes and depend on the number of electrons occupying the orbital, the symmetry of the orbital and the photon energy [191]. In this respect, PES directly maps the electronic structure up to a maximum binding energy, which corresponds to the difference between pho-ton energy and electron affinity. The single particle model fails to explain the line broadening sketched in figure 3.5. However, this can be made up within the many particle model (see page 35).

2The ionization potential of an atom or molecule is the minimum energy required to remove one of its electrons

HOMO-LUMO gap

E

E

E

E

HOMO LUMO

Figure 3.5: Scheme for photodetachment in the single particle picture. The orbitals of the particle below the vacuum level E_vac are displayed. The additional electron of the anion occupies the LUMO of the neutral. A photon having the energy hν detaches an electron from its bound state above E_vac [192].

The single particle picture can be used both for neutrals and for anions. However, figure 3.5 shows a special property of PES on anions: If for a neutral the highest occupied molecular orbital (HOMO) is completely filled, which corresponds to a shell closing (see section 3.1.1), the additional electron of the anion has to occupy the lowest unoccupied molecular orbital (LUMO)³. Analogous to atom, molecular and nuclear physics, a shell closing usually corresponds to a stable particle. In case of a closed shell species (for the neutral), in the PES spectrum of the anion a peak having high kinetic energy appears, since the relative weakly bound electron from the LUMO of the neutral is detached. The energy difference between this peak and the next, lower energetic feature corresponds to the energy difference between HOMO and LUMO of the neutral, which is referred to as HOMO-LUMO gap. This gap corresponds to the lowest possible excitation energy of the neutral and can be associated with the band gap of a bulk semiconductor. The size of this gap is an

3here the notation refers to the neutral

3.2 Photoelectron Spectroscopy

important parameter correlated with the stability and chemical reactivity of a clus-ter [190].

The above considerations hold under the assumption of identical geometries for anion and neutral. If the geometries differ, the above conclusions do not hold exactly anymore, since changes in the electronic structure are much faster than changes in geometry (see Born-Oppenheimer approximation in section 3.2.3). It is important to note that the PES spectrum yields information about the electronic states of the neutral, but in the geometry of the anion.

In an accurate description, the photoemission process is a transition from the initial state E_i^N having N electrons to the final state E_f^N−1 with N −1 electrons [187], which yields a kinetic energy for the photoelectrons of

E_kin=hν−(E_f^N−1−E_i^N). (3.2) The single particle model and the simplified equation3.1 are based on Koopmans theorem [193], which neglects the effects of the emitted electron on the final state and the influence on the remaining electrons. Hence, there are features in the spectra, which cannot be explained within the single particle picture, since they originate from processes which take place during the detachment process. To obtain better results, the following final state effects can be taken into account to amend this simple model [194, 35]:

• Relaxation

The charge state of the particle increases due to the detachment of an electron, which yields an increase of the binding energy of all orbitals. This relaxation energy is transferred to the outgoing electron and increases its kinetic energy.

• Multiplet Splitting

In the final state, the remaining electrons can couple their spins and angular momenta to different total spins and angular momenta. Depending on the energy of the final state, the kinetic energy of the photoelectron can differ by the energy differences of the possible final states, so that multiple peaks might appear in the spectrum.

• “Shake Up” Process

In this case, the photon interacts with more than one electron: During the detachment process, another bound electron might be excited into a previously unoccupied state, which leads to a detached electron having a kinetic energy lowered by the excitation energy.

• Configuration Interaction

Especially the emission of electrons from deeper lying orbitals can disturb the

remaining electrons so strongly, that the single particle orbitals of the final state do not have much similarity with the ones from the initial state. In that case, the features in the PES spectrum cannot be related anymore to the emission from occupied orbitals of the particle in the initial state and the final state has to be described by a linear combination of many initial state configurations.

Multiplet splitting and shake up processes have been observed e.g. for PES on Ag_n^- clusters [35]. In all four processes, not only the photon and the emitted electron but also the remaining electrons play an important role, which cannot be accounted for in the single particle picture. Therefore, a model including all valence electrons with spins and angular momenta has to be used for obtaining a more detailed inter-pretation of the spectra.