Angular resolved photoelectron spectra of metal clusters have been experimentally measured for the first time only recently. These measurements have been performed systematically for sodium clusters in a broad range of cluster sizes. This work attracted a lot of attention and was reported practically at all major international cluster conferences because it revealed a very non-trivial behavior of the angular anisotropy parameter with respect to photon energy and provided a method for probing the angular momentum character of the valence orbitals of free nanoclusters. Initial attempts to explain these observations within single particle approximations fail completely. In this Letter we present a consistent many-body theory for the description of angular resolved photoelectron spectra of metal clusters. Jellium model formalism is employed. Our calculations demonstrate the dominant role of the many-body effects in the formation of angular distributions of photoelectrons emitted from sodium clusters and are in a good agreement with experimental data reported in. The concrete comparison of theory and experiment has been performed for the photoionization of $Na_7^{-}$ and $Na_{19}^{-}$ anions being characterized by the entirely closed shells of delocalized electrons.
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Angular resolved photoelectron spectra of metal clusters have been experimentally measured for the first time only recently. These measurements have been performed systematically for sodium clusters in a broad range of cluster sizes. This work attracted a lot of attention and was reported practically at all major international cluster conferences because it revealed a very non-trivial behavior of the angular anisotropy parameter with respect to photon energy and provided a method for probing the angular momentum character of the valence orbitals of free nanoclusters. Initial attempts to explain these observations within single particle approximations fail completely. In this Letter we present a consistent many-body theory for the description of angular resolved photoelectron spectra of metal clusters. Jellium model formalism is employed. Our calculations demonstrate the dominant role of the many-body effects in the formation of angular distributions of photoelectrons emitted from sodiu
tant achievements have been made. Thus, the plasmon resonances have been observed in small metal clusters of nanometer size for different materials, e.g. Na, K, Mg, in fullerenes and in many other cases of study, for references see [2]. The structure of these plasmon resonances have been studied in detail. It was established that patterns of plasmon resonances in many cases are determined by the deformation parameter of the cluster. This result is very important because it provides a method of monitoring the cluster size and its shape.
A comprehensive review of the results in this field can be found in [3].
The detail geometrical structure of mass selected clusters has been also studied by means of photoelectron spectroscopy and the density functional theory (DFT). Thus, for sodium cluster anions with up to 57 atoms the geometrical structures have been determined with high accuracy from the comparison of DFT calculations results with measured photoelectron spectra [4,5].
The measurement of photoabsorption and photoionization of small mass selected clusters is not a trivial task. Until recently the measurement was fulfilled only for the total photoionization or photoabsorption cross sections of various cluster targets. In the recent work [1] the first measurement of angular resolved photoelectron spectra of sodium clusters has been reported. These experiments have been performed for negatively charged sodium ions (anions) in a broad range of cluster sizes 3 ≤ N ≤ 147 [1]. These experiments allowed to probe the angular momenta of single electron orbitals in sodium metal clusters. In the work [1] it was also demonstrated that simple models based on single electron treatment of the photoionization process fail to describe the angular anisotropy of photoelectrons emitted in the process of photoionization of cluster anions.
In this Letter we present a consistent many-body theory for the description of angular resolved photoelectron spectra of metal clusters. As a case of study sodium cluster anions have been chosen. The Hartree-Fock (HF) approximation has been used as a single particle theoretical framework. Many-electron correlations have been accounted for within the Random Phase Approximation with Exchange (RPAE). The results of calculations allow one to conclude that many-electron correlations play the very essential role in the formation of angular distributions of photoelectrons in the process of photoionization of metal clusters. This effect is a consequence of the large dynamic polarizability of metal cluster targets, being entirely determined by many-electron correlations in the vicinity of the plasmon resonance frequencies.
In this Letter we consider the most characteristic case of a spherically symmetric cluster target and apply the jellium model for the description of the photoionization process. This model proved to be well applicable for the description of electronic structure [8] and collision processes involving metal clusters and fullerenes [2,3]. Cross sections of the collision processes are very sensitive to the correct accounting for many-electron correlations, see [2,3] and references therein. In this Letter we limit our description by the cluster targets possessing the spherical symmetry, which corresponds to the case of the so-called magic clusters, i.e. clusters with the entirely closed electronic shells. For the concrete analysis and illustration we have chosen the Na - 7 and Na - 19 magic clusters. Let us note here also that the angular anisotropy of photoelectrons emitted in the process of photoionization of atomic negative ions has been investigated in sufficient detail both theoretically and experimentally (see, for example, review [6]). It was demonstrated that the parameter of angular anisotropy β [7] is very sensitive to accounting for the manyelectron correlation effects. Thus, it is not a surprise the failure of single particle approaches reported in [1] for the case of the photoionization of cluster targets.
The characteristic features of small metallic clusters, like the shell structure of delocalized electrons or plasmon excitations, can be well understood in terms of quantum motion of the delocalised valence electrons moving in the field created by themselves and the positively charged ionic core [8]. This concept, known as the jellium model for metallic clusters, can be also utilized for the description of a metallic cluster anions.
In this work for the elucidation of the role of many-electron correlations in the formation of angular distributions of photoelectrons we have addressed to the most characteristic example. Thus, we considered the magic cluster anions possessing the spherical symmetry due to the entire closure of their electronic shells. In this case the complicated ionic structure can be reduced with the sufficient accuracy to the uniformed spherically symmetric positive charge distribution, named below also as ionic core.
Within the Hartree-Fock (HF) app
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