Closed-shell structure

I he results of their calculations were summarised in two rules. The first rule states that at least one isomer C with a properly closed p shell (i.e. bonding HOMO, antibonding I. U.MO) exists for all n = 60 - - 6k (k = 0,2,3,..., but not 1). Thus Qg, C72, Cyg, etc., are in lhi-< group. The second rule is for carbon cylinders and states that a closed-shell structure is lound for n = 2p(7 - - 3fc) (for all k). C70 is the parent of this family. The calculations Were extended to cover different types of structure and fullerenes doped with metals. 

In addition to diamond and amorphous films, nanostructural forms of carbon may also be formed from the vapour phase. Here, stabilisation is achieved by the formation of closed shell structures that obviate the need for surface heteroatoms to stabilise danghng bonds, as is the case for bulk crystals of diamond and graphite. The now-classical example of closed-shell stabilisation of carbon nanostructures is the formation of C o molecules and other Fullerenes by electric arc evaporation of graphite [38] (Section 2.4). 

The results are critically dependent on the level of theory. However, a stepwise mechanism with closed shell structures along the reaction path was found to be lower in energy than a concerted reaction. An all-cw conformer of 172 is reported to be a transition state rather than an intermediate. Similarities of the conformational isomers of the intermediate 2-butenedithial 172 with the dinitrosoethylenes discussed in Section IV,c are evident. 3,6-Diamino-substituted dithiins are predicted to be more stable in the open-chain bisthioamide structure [95JST51]. The... 

Scheme 3). The qualitative energy levels (Scheme 4) show the number of valence electrons necessary to obtain closed-shell electronic structures. Each orbital in the. y-orbital set is assumed to be occupied by a pair of electrons since the 5-orbital energies are low and separate from those of the p-orbital ones, especially for heavy atoms. The total number of valence electrons for the closed-shell structures... 

The i-orbital array of three and four-membered rings is of the Hiickel conjugation. (Scheme 2). The splitting patterns of the orbital energy levels (Scheme 3) show that the total number of valence electrons for the closed-shell structures is 4Af + 2 for the three- N= 0) and four-membered rings (N= 0, 1). 

The Mg + dicadon [42] with AN+2 (N= 1) valence electrons has a stable structure in agreanent with the rule, but this is a local energy minimum. The linear structure is more stable because it minimizes the Coulomb repulsion. This is in contrast to the tetrahedral stmcture of the Li dication with two electrons (N= 0). The six electron systems caimot form closed-shell structures in the tetrahedron, but the two electron systems can do. 

In general, linear 7r-electron systems with Z1r = 2N electrons at the lowest energy levels have closed-shell singlet states while cyclic systems reach closed shell structures only when ZT = 4N + 2. Cyclic 7r-electron systems with Zn 4N + 2 will therefore exhibit multiplet ground states according to Hund s rules, and should be chemically reactive because of the unpaired electrons. Hiickel s rule that predicts pronounced stability for so-called aromatic ring systems with 4jV + 2 7r-electrons is based on this shell structure. The comparison with cyclic systems further predicts that ring closure of linear 7r-electron systems should be exothermic by an amount... 

The import of diabatic electronic states for dynamical treatments of conical intersecting BO potential energy surfaces is well acknowledged. This intersection is characterized by the non-existence of symmetry element determining its location in nuclear space [25]. This problem is absent in the GED approach. Because the symmetries of the cis and trans conformer are irreducible to each other, a regularization method without a correct reaction coordinate does not make sense. The slope at the (conic) intersection is well defined in the GED scheme. Observe, however, that for closed shell structures, the direct coupling of both states is zero. A configuration interaction is necessary to obtain an appropriate description in other words, correlation states such as diradical ones and the full excited BB state in the AA local minimum cannot be left out the scheme. 

The metal clusters in HFeCo3(CO)g(P(OMe)3)3 and H3Ni4(Cp)4 contain different numbers of electrons. The former cluster is a closed-shell structure (60 electrons) while the latter contains 63 electrons and is paramagnetic with S... 

The selection rules (3.76)—(3.78) were derived without reference to the form of the radial functions hence they are valid for any one-particle central-field problem. For example, they hold well for the sodium valence electron, which moves outside a closed-shell structure. 

Figure 12 shows a selection of narrow distributions, indicating weak interaction.21 As expected, weak interaction arises in cases where the target atom has a closed-shell structure. Wide distributions, indicating strong interaction, are shown in Fig. 13.21 The distribution for the system He(23S)-Hg—which has an intermediate width, indicating a well depth of Z) 0.085 eV—is shown in both figures for comparison.

So far we have only discussed attractive effects. The origin of the repulsive force between molecules having a closed shell structure is easy to visualise but complicated to describe in a quantitative manner. When... 

Hiickel noted that if electron pairs are filled into the energy-level pattern 32 or 33, a closed-shell structure (all electrons paired) will result only when the total number of pairs is odd (total number of electrons = 4re + 2, = 0, 1, 2,...) if the number of pairs is even (total number of electrons = 4re, re = 0, 1, 2,...),... 

The fine-structure constant a indicates that first-order perturbation theory has been applied the linear dependence on the photon energy Eph is due to the length form of the dipole operator used in equ. (2.1), and the wavenumber k compensates the 1 /k which appears if the absolute squared value of the continuum wavefunction is used (see equ. (7.29)). The summations over the magnetic quantum numbers M, of the photoion and ms of the photoelectron s spin are necessary because no observation is made with respect to these substates. Due to the closed-shell structure of the initial state with f — 0 and M = 0, the averaging over the magnetic quantum numbers M simply yields unity and is omitted. 

As seen from Table 10.2 (Entries 8 vs. 7), the improvement of SL-BOVB compared with L-BOVB amounts to 2 kcal/mol. Practically, the SL-BOVB wave function can be calculated directly in the form of Equation 10.6, but an alternative way may be preferred, for the sake of a faster convergence. Thus, instead of using the open-shell formulation of the orbital splitting, the calculations describe each ionic structure as a combination of two closed-shell structures, as in Equation 10.7. 

Stemheimer antishielding factor is given where meaningful calculations have been carried out for ions with closed-shell structure e.g. Al3+, Cr. Where possible given for a lattice,8 and if not for free ions. 

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