Subshell model

The spherical shell model can only account for tire major shell closings. For open shell clusters, ellipsoidal distortions occur [47], leading to subshell closings which account for the fine stmctures in figure C1.1.2(a ). The electron shell model is one of tire most successful models emerging from cluster physics. The electron shell effects are observed in many physical properties of tire simple metal clusters, including tlieir ionization potentials, electron affinities, polarizabilities and collective excitations [34]. 

The incompletely filled d-subshell is responsible for the wide range of colors shown by compounds of the d-block elements. Furthermore, many d-metal compounds are paramagnetic (see Box 3.2). One of the challenges that we face in this chapter is to build a model of bonding that accounts for color and magnetism in a unified way. First, though, we consider the physical and chemical properties of the elements themselves. 

The nature of the radial wave functions thus leads us to the following interpretation1 of the subshells of the shell model ... 

The Structural Basis of the Magic Numbers.—Elsasser10 in 1933 pointed out that certain numbers of neutrons or protons in an atomic nucleus confer increased stability on it. These numbers, called magic numbers, played an important part in the development of the shell model 4 s it was found possible to associate them with configurations involving a spin-orbit subsubshell, but not with any reasonable combination of shells and subshells alone. The shell-model level sequence in its usual form,11 however, leads to many numbers at which subsubshells are completed, and provides no explanation of the selection of a few of them (6 of 25 in the range 0-170) as magic numbers. 

According to the latest atomic model, the electrons in an atom are located in various energy levels or shells that are located at different distances from the nucleus. The lower the number of the shell, the closer to the nucleus the electrons are found. Within the shells, the electrons are grouped in subshells of slightly different energies. The number associated with the shell is equal to the number of subshells found at that energy level. For example, energy... 

The two eflPects above constitute what is called central field covalency since they aflFect both the a and the tt orbitals on the metal to the same extent. There is also, of course, symmetry restricted covalency which acts difiFerently on metal orbitals of diflFerent symmetries. This type of covalency shows up in optical absorption spectra as differences in the values of Ps and p -, as compared with 35. The first two s refer to transitions within a given symmetry subshell while 635 refers to transitions between the two subshells. This evidence of covalency almost of necessity forces one to admit the existence of chemical bonds since it is difficult to explain on a solely electrostatic model. The expansion of the metal orbitals can be caused either by backbonding to vacant ligand orbitals, or it may be a result of more or less extensive overlap of ligand electron density in the bond region. Whether or not this overlap density can properly be assigned metal 3d character is what we questioned above. At any... 

Dipole oscillator strengths form important input into all stopping models based on Bethe or Bohr theory. Emphasis has frequently been on total /-values which show only little sensitivity to the specific input. More important are differential oscillator-strength spectra, in particular at projectile speeds where inner-shell excitation channels are closed. Spectra bundled into principal or subshells [60] are sufficient for many purposes, but the best available tabulations are based on analysis of optical data rather than on theory, and such data are unavailable for numerous elements and compounds [61]. 

Note carefully that each shell has been divided into a series of finer shells known as subshells. Each subshell corresponds to a specific orbital type. The four of the seventh shell, for example, includes the 7s orbital, the 5/orbitals, the 6z/orbitals, and the 7p orbitals. Gallium is larger than zinc because it has an electron in three subshells of the fourth shell, while zinc has electrons only in the first inner two subshells of the fourth shell. Thus, what you see here is a refinement on the model presented in Section 5.7. Don t worry about fully understanding this refinement. Rather, better that you understand that all conceptual models are subject to refinement. We chose the level of refinement that best suits our needs. 

The VSEPR approach is largely restricted to Main Group species (as is Lewis theory). It can be applied to compounds of the transition elements where the nd subshell is either empty or filled, but a partly-filled nd subshell exerts an influence on stereochemistry which can often be interpreted satisfactorily by means of crystal field theory. Even in Main Group chemistry, VSEPR is by no means infallible. It remains, however, the simplest means of rationalising molecular shapes. In the absence of experimental data, it makes a reasonably reliable prediction of molecular geometry, an essential preliminary to a detailed description of bonding within a more elaborate, quantum-mechanical model such as valence bond or molecular orbital theory. 

Here, Iq is the electron affinity of C60 (Iq = 2.65 eV [50]). Thus, the 5-potential model ignores the finite thickness nature of the carbon cage within the model, A = 0. Furthermore, in the framework of this model, the size of the embedded atom ra is considered to be so small, compared to the size of C60, that the ground state electronic wavefunctions of the embedded atom coincide exactly with those for a free atom. In other words, the model assumes no interaction between the ground state encaged atom and the carbon cage at all. Therefore, the model is applicable only to the deep inner subshells of the encaged atom. As for the carbon atoms from... 

In the framework of the A-potential model, combined with the frozen-cage approximation, the problem is solved simply. Namely, HF wavefunctions and energies of the encaged atom, solutions of the extended to encaged atoms Hartree-Fock equations (2), must be substituted into corresponding formulae for the photoionization of an nl subshell of the free atom, Equations (18)-(26), thereby turning them into formulae for the encaged atom (to be marked with superscript " A") rrni(o>) —> a A(co), Pni(fi>) Yni o>) - and 8ni((o) - 8 A(co). This accounts... 

The dependence of confinement resonances on quantum numbers of the ionized subshell nl can be illustrated by the 5-potential model calculated data [34] for the Ne Is and 2s photoionization cross sections from Ne C60, see Figure 5. 

For outer subshells of the encaged atom, the ionization thresholds of which vary from a few eV to a few tens eV, the dynamical-cage model is required. The photoionization cross section of the encaged atom in the dynamical-cage approximation will be marked with a tilde sign 5 s and... 

Figure 18 Calculated [33] RPAE results for the Xe 5s photoionization cross section of Xe Cgo obtained in the A-potential model at the frozen-cage approximation level. (a) o 1" A iro), complete RPAE calculation accounting for interchannel coupling between photoionization transitions from the Xe 4d10, 5s2 and 5p6 subshells (b) 5 A ( >), the same as in (a) but with the 4d - f, p transitions being replaced by those of free Xe, for comparison purposes (c) o AA(

Figure 28 Relativistic RPAE calculated results [30] of the 6s dipole photoelectron angular distribution parameter of Hg at two different levels of truncation with regard to RRPA interchannel coupling (a) including channels from the 6s2 subshell alone, Aa, and (b) including channels from the 6s2 and 5d10 subshells of d>Hg, as in Figure 27. Confinement effects were accounted for in the A-potential model at the frozen-cage approximation level.

Finally, because models such as the IBM consider only valence particles—those beyond the nearest closed shell or, in some instances, subshell—researchers are interested in mapping, experimentally, the locations of shells and subshells into regions far from stability. An improved knowledge of shell and subshell gaps at the extremes of nuclear stability will provide important benchmarks for testing nuclear models. 

Quasi-relativistic ab initio core model potential calculations (SCF level) the (n - l)d subshell is included in the valence space (i.e. 14-valence electrons). The reaction energies do not include ZPE from Reference 103. 

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