Valence electrons in clusters

For two and three dimensions, it provides a crude but useful picture for electronic states on surfaces or in crystals, respectively. Free motion within a spherical volume gives rise to eigenfunctions that are used in nuclear physics to describe the motions of neutrons and protons in nuclei. In the so-called shell model of nuclei, the neutrons and protons fill separate s, p, d, etc orbitals with each type of nucleon forced to obey the Pauli principle. These orbitals are not the same in their radial shapes as the s, p, d, etc orbitals of atoms because, in atoms, there is an additional radial potential V(r) = -Ze2/r present. However, their angular shapes are the same as in atomic structure because, in both cases, the potential is independent of 0 and (f>. This same spherical box model has been used to describe the orbitals of valence electrons in clusters of mono-valent metal atoms such as Csn, Cu , Na and their positive and negative ions. Because of the metallic nature of these species, their valence electrons are sufficiently delocalized to render this simple model rather effective (see T. P. Martin, T. Bergmann, H. Gohlich, and T. Lange, J. Phys. Chem. 95, 6421 (1991)). 

As pointed out above, there are two extreme situations involving localised and delocalised valence electrons in cluster physics at one end of the scale, the noble gases possess valence electrons which remain localised on individual atomic sites, while, at the other, the alkalis possess delocalised valence electrons which can wander over the cluster and resemble the conduction electrons of a metallic solid. 

Table 3.9. Number of Orbitals and Valence Electrons in Clusters ...

P4) is closely similar with P-P distances of 216 pm (smaller than for P4 itself, 221pm). Indeed, a whole series of complexes has now been established with the same structure-motif and differing only in the number of valency electrons in the cluster some of these are summarized in Table 13.11. The number of valence electrons in all these complexes falls in the range 30-34 as predicted by R. Hoffmann and his colleagues.Many other cluster types incorporating differing numbers of Group 15 and transition metal atoms are now known and have been fully reviewed. ... 

We have, however, made a careful definition of the term valence electrons ( the electrons that are most loosely bound see p. 269). We have also used carefully the term valence orbitals to mean the entire cluster of orbitals of about the same energy as those which are occupied by the valence electrons. In both of these uses, the word valence is used as an adjective. 

The number of valence electrons in niobium oxychloride clusters decreases as the number of oxide hgands increases (Table 6.4). The compound Cs2LuNbi Cli70 has the VEC of 16 as found in most chloride clusters, suggesting that the presence of one oxide ligand per cluster does not tip the balance between the Nb-Nb bonding and Nb-L anti-bonding contributions to the a2u state. The VEC of most clusters... 

Table 6.4 Number of valence electrons per cluster in niobium oxychlorides.

For more electropositive elements, which have an inferior number of valence electrons in the first place, and which in addition have to supply electrons to a more electronegative partner, the number of available electrons is rather small. They can gain electrons in two ways first, as far as possible, by complexation, i.e. by the acquisition of ligands and second, by combining their own atoms with each other. This can result in the formation of clusters. A cluster is an accumulation of three or more atoms of the same element or of similar elements that are directly linked with each other. If the accumulation of atoms yields a sufficient number of electrons to allow for one electron pair for every connecting line between two adjacent atoms, then each of these lines can be taken to be a 2c2e bond just as in a common valence bond (Lewis) formula. Clusters of this kind have been called electron precise. 

The next magic number for jellium clusters is 40. This is a particularly important magic number in cluster chemistry, since numerous 40 valence electron bare clusters with 9 to 11 vertices of the post-transition elements in Groups 13 to 15 are known as isolable species in intermetallics or salts with suitable counterions. Examples of such species include lun, Geg", and Big, all of which have been isolated in intermetallics (for Inn ) or as stable salts with suitable counterions (Geg" and Big ) and characterized by X-ray crystallography. 

SRPA has been already applied for atomic nuclei and clusters, both spherical and deformed. To study dynamics of valence electrons in atomic clusters, the Konh-Sham functional [14,15]was exploited [7,8,16,17], in some cases together with pseudopotential and pseudo-Hamiltonian schemes [16]. Excellent agreement with the experimental data [18] for the dipole plasmon was obtained. Quite recently SRPA was used to demonstrate a non-trivial interplay between Landau fragmentation, deformation splitting and shape isomers in forming a profile of the dipole plasmon in deformed clusters [17]. 

By analyzing the density matrix composition of planar and 3D structures of seven atom clusters (II and IV of Fig 1), calculated using scalar relativistic pseudo-potential at the GGA theory level, Fernandez and coworkers conclude that the planarity of An clusters is driven by the hybridization of the half-filled 6s orbital with the fully occupied 5d 2 orbital, which is favored by relativistic effects. Thus, the three valence electrons in the orbitals 6s and 5d 2, form a sticky-waist cylinder , where the cylinder is due to the almost filled s + d 2 hybrid, and the sticky-waist is due to the nearly half-filled s — d 2 hybrid orbital. 

R. Santra, J. Zobeley, L.S. Cederbaum, Electronic decay of valence holes in clusters and condensed matter, Phys. Rev. B 64 (2001) 245104. 

Exercise 4.5. Count the skeletal electron pairs (or cluster valence electrons) in the organometallic complex shown below at the left. Is there a relationship to cyclobutadiene iron tricarbonyl shown next to it ... 

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