Shell effect

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]. 

Chambers C C, G D Hawkins, C J Cramer and D G Tmlilar 1996. Model for Aqueous Solvation Ba sed on Class IC Atomic Charges and First Solvation Shell Effects. Journal of Physical Chemistry 100 16385-16398. 

The effects of a rather distinct deformed shell at = 152 were clearly seen as early as 1954 in the alpha-decay energies of isotopes of californium, einsteinium, and fermium. In fact, a number of authors have suggested that the entire transuranium region is stabilized by shell effects with an influence that increases markedly with atomic number. Thus the effects of shell substmcture lead to an increase in spontaneous fission half-Hves of up to about 15 orders of magnitude for the heavy transuranium elements, the heaviest of which would otherwise have half-Hves of the order of those for a compound nucleus (lO " s or less) and not of milliseconds or longer, as found experimentally. This gives hope for the synthesis and identification of several elements beyond the present heaviest (element 109) and suggest that the peninsula of nuclei with measurable half-Hves may extend up to the island of stabiHty at Z = 114 andA = 184. 

There are two mechanisms in growth of a bubble in acoustic cavitation [14], One is coalescence of bubbles. The other is the gas diffusion into a bubble due to the area and shell effects described before. This is called rectified diffusion. 

Finally we address the issue of contributions. In our view it is unbalanced to concentrate on a converged treatment of electrostatics but to ignore other effects. As discussed in section 2.2, first-solvation-shell effects may be included in continuum models in terms of surface tensions. An alternative way to try to include some of them is by scaled particle theory and/or by some ab initio theory... 

In the development of solvation models, Cramer and Tmhalar have made several noteworthy contributions [8-11]. Most of the implicit solvation models do not include the effect of first solvation shell on the solute properties. This can be satisfactorily treated by finding the best effective radii within implicit models. In addition to the first-solvent-shell effects, dispersion interactions and hydrogen bonding are also important in obtaining realistic information on the solvent effect of chemical systems. 

It is evident that continuum models can be quite effective, for ionic solutes as well as for neutral ones. They also have the advantage of not being highly demanding in terms of computer resources. However a problem associated with these methods is posed by so-called first-solvation-shell effects 6 One aspect of this is the difficulty of properly accounting for specific types of solute-... 

The level of accuracy that can be achieved by these different methods may be viewed as somewhat remarkable, given the approximations that are involved. For relatively small organic molecules, for instance, the calculated AGsoivation is now usually within less than 1 kcal/mole of the experimental value, often considerably less. Appropriate parametrization is of key importance. Applications to biological systems pose greater problems, due to the size and complexity of the molecules,66 156 159 161 and require the use of semiempirical rather than ab initio quantum-mechanical methods. In terms of computational expense, continuum models have the advantage over discrete molecular ones, but the latter are better able to describe solvent structure and handle first-solvation-shell effects. 

The CASSCF/CASPT2 calculations were performed with an active space including the five nd, the (n + l)s, the three (n+ l)p orbitals, and a second set of nd orbitals to account for the double shell effect. The importance of including a second 3d shell in the active space was detected in an early study of the electronic spectrum of the nickel atom [2]. This had already been suggested from MRCI results [1]. The results obtained by RT at about the same time indicated that such effects are effectively accounted for when a method is used that includes cluster corrections to all orders, like the QCI method used by them [3]. This result will hold true also for the less approximate coupled cluster method CCSD(T). 

As a test of this procedure, we can obtain the energy which improves greatly the TF estimates and compares fairly closely to FLF results. As an example we show those for the Krypton atom. The energy obtained by the Hartree-Fock method is E p = 2752.06, whereas TF gives Epp = 3252.27, that is, a difference of 18.18%. In the present work we obtain an energy for the Krypton of Ep p = 2719.37, that is a 1.19% deviation. This is a general behaviour for all the atoms. Even for the atoms with few electrons we obtain the same difference with ELF, which is remarkable for a semiclasical model that employs average shell effects. For example for the Neon atom, we obtained Ep r = 125.893, while Epp = 128.547 (a difference of 2%). 

The TF and modified methods based on average shell effects does not reproduce fairly closely local properties like p(0). It diverges with TF and TFD and only after introducing gradient corrections, can we obtain at least a finite value. In the present work we have obtain results quite close to HFvalues (Table 1). As an example, in Table 2 we present the evolution of this value through the different theories in the case of Krypton. The improvement by the present approach is found to be large. 

Determination of the first-order shell-effect terms... 

Due to Heisenberg s uncertainty and Pauli s exclusion principles, the properties of a multifermionic system correspond to fermions being grouped into shells and subshells. The shell structure of the one-particle energy spectrum generates so-called shell effects, at different hierarchical levels (nuclei, atoms, molecules, condensed matter) [1-3]. 

The use of these two semiclassical levels of description - the statistical (e.g. TFW) model and the semiclassical (e g. WKB) treatment - of the one-body motion shows that the main, global or local, properties of a quantum system can be split into two terms the first, largest one is a smoothly varying term, where shell effects are averaged out, and the second one is an oscillating correction, which contains the information on the system specificity. The question now arises of the relevance and accuracy of such a decomposition. 

This approach to the averaging problem rests on the idea that shell effects in the ground-state energy stem from the shell structure of the one-particle energy-level distribution g(E) ... 

To summarize this subsection one can state that i) The role of second-order shell-effect terms can be determined by investigating the discrepancies between the discrete second derivatives of the functions 8E (Z) and 8 Ehfr(Z,Y(,). ii) For the second row of elements these terms determine the whole pattern of the oscillating part of the energy iii) They also play an important role in critical regions of shell filling (i.e., around shells fully, half, or irregularly filled). 

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