Atomic-shell closing

See text. The first two columns give the numbers of metal atoms at which electronic shell closings have been observed in experiment for Cs-covered C o and for pure alkali metal clusters, respectively. The columns on the right list the number of electrons required for shell closings in an infinitely deep potential well with and without a central barrier. The numbers in the different columns are mainly arranged in a manner to show correlations. 

Figure 6. Wolfgang Pauli s discovery of the exclusion principle led to his development of a fourth quantum number to describe the electron. At the time, it was known that each successive electron shell in an atom could contain % 8, 18. .. 2nz electrons (where n is the shell number), and Pauli s fourth number made it possible to explain this. When an electron s first quantum number is one, the second and third must be zero, leaving two possibilities for the fourth number Thus the first shell can contain only two electrons. At = 2, there are four possible combinations of the second and third numbers, each of which has two possible fourth numbers. Thus the second shell closes when it contains eight electrons.

At this stage, the valence shells for the two oxygen atoms are closed, but the sulfur atom is two electrons short of a complete octet. If we complete the octet for sulfur by converting a lone pair of electrons on the right hand side oxygen atom into a sulfur-to-oxygen n-bond, we end up generating the resonance contributor (A) shown below ... 

Atoms or ions with completely filled orbitals have J equal to zero, which means that atoms with closed shells have no magnetic moment. The only atoms that display a magnetic moment are those with incompletely filled shells. These are particularly found in the transition metals, with incompletely filled d shells, and the lanthanides and actinides, which have incompletely filled/shells. 

Figure 11.5 Covalent bonding, (a) shows two isolated hydrogen atoms coming together to form a covalently bonded (di)hydrogen molecule, (b) shows a simple model of the bonding in a dihydrogen molecule, with the single Is orbital electron from each atom being shared by the molecule, to give each atom a closed shell.

Removal of an electron from a molecule can formally be considered to occur at a a-bond, a 7i-bond or at a lone electron pair with the a-bond being the least favorable and the lone electron pair being the most favorable position for charge-localization within the molecule. This is directly reflected in the lEs of molecules (Table 2.1). Nobel gases do exist as atoms having closed electron shells and there-... 

In this review we will give an overview of the properties (asymptotics, shell-structure, bond midpoint peaks) of exact Kohn-Sham potentials in atomic and molecular systems. Reproduction of these properties is a much more severe test for approximate density functionals than the reproduction of global quantities such as energies. Moreover, as the local properties of the exchange-correlation potential such as the atomic shell structure and the molecular bond midpoint peaks are closely related to the behavior of the exchange-correlation hole in these shell and bond midpoint regions, one might be able to construct... 

Let us now discuss the correlation effects on the atomic shell structure. We plot in Fig. 7 some of the described potentials for the case of the beryllium atom. The exact exchange-correlation potential v c is calculated from an accurate Cl (Configuration Interaction) density using the procedure described in [20]. The potentials Vx, and u" , are calculated within the optimized potential model (OPM) [21,40,41] and are probably very close to their exact values which can be obtained from the solution for of the OPM integral equation [21,40,41] by insertion of the exact Kohn-Sham orbitals instead of the OPM... 

Lines, corresponding to different transitions from initial states with vacancy in the shells with the same n, compose a series of spectra, e.g. K-, L-, M-series etc. Main diagram lines correspond to electric dipole ( 1) transitions between shells with different n. The lines of 2-transitions also belong to diagram lines. Selection rules of 1-radiation as well as the one-particle character of the energy levels of atoms with closed shells and one inner vacancy cause, as a rule, a doublet nature of the spectra, similar to optical spectra of alkaline elements. X-ray spectra are even simpler than optical spectra because their series consist of small numbers of lines, smaller than the number of shells in an atom. The main lines of the X-ray radiation spectrum, corresponding to transitions in inner shells, preserve their character also for the case of an atom with open outer shells, because the outer shells hardly influence the properties of inner shells. 

There exist a number of methods to account for correlation [17, 45, 48] and relativistic effects as corrections or in relativistic approximation [18]. There have been numerous attempts to account for leading radiative (quantum-electrodynamical) corrections, as well [49, 50]. However, as a rule, the methods developed are applicable only for light atoms with closed electronic shells plus or minus one electron, therefore, they are not sufficiently general. 

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