27r-electron system

Roberts and coworkers, investigating the ionization reactions of cyclobutene derivatives, found that the resulting cyclobutenyl ions were unusually stable10-113. They suggested that rather than regarding these ions as simple allyl cations, their properties were consistent with a C(1 ),C(3) interaction and cyclic delocalization of the 7t-electrons. As such, these 27r-electron systems were considered to be the homoaromatic counterparts of the well established, aromatic cyclopropenium ions39. 

Vinylpyrroles can also react as 27r-electron systems to yield the expected heterocyclic derivatives through the cyclization of the vinyl group with 1,3-dipolar reagents (80H(14)185). 

In general terms, the Diels-Alder reaction is one between a conjngated diene (a drr-electron system) and a compound containing a donble bond (a 27r-electron system) called a dienophile (diene + philein, (jreek to love). The prodnct of a Diels-Alder reaction is often called an adduct. 

In structures (31) and (32) corresponding to minima on the PES [81JOM(219)279 86MI6], the atoms of Li and Be accept two electrons from the 7r-system i.e., they are 27r-electron molecules... 

The 2,6-dimesitylphenyl substituent is also extremely sterically demanding, and reduction of (2,6-Mes2QH3)GaCl2 with Na yields Na2[(2,6-Mes2C6H3)3Ga3] the [(2,6-MeS2C6H3)3Ga3] anion possesses the cyclic structure (18.17) and is a 27r-electron aromatic system. 

Woodward—Hoffmann orbital symmetry rules can be applied to the charged systems as well. The conversion of a cyclopropyl cation to an allylic cation is the simplest one, which involves only 27r-electrons (Figure 2.13). This is an electrocyclic reaction of (4n + 2) type (n = 0) and should, therefore, be a disrotatory process. 

A most unusual result was encountered when the irradiation of 243 was carried out in pentane in the presence of both methanol and methyl trifluoro-acetate (excess).Under these conditions, the only cycloadducts observed consisted of a mixture of the two stereoisomers of 3-oxazoline, 260. This result suggests that 2-phenylazirinylidene (244) reacts with methanol to give mainly azirine 258, which is subsequently converted to nitrile ylide 259 (and thus cycloadduct 260) on further irradiation. The formation of 245 has been explained in terms of a photoinduced methoxy migration of 258 which competes with C-C bond cleavage of the azirine ring. Ciabattoni and Cabell have previously reported that 2-chloro-2/f-azirines undergo ready isomerization at room temperature via a 27r-electron azacyclopropenyl cation. A similar mechanism would rationalize the apparent photoconversion of azirines 258 and 245. The results obtained with this spiro azirine system indicate that cycloelimination of ethylene from 243 is much more efficient than C-C bond scis-... 

By construction, LSD is exact for a uniform density, or more generally for a density that varies slowly over space [6]. More precisely, LSD should be valid when the length scale of the density variation is large in comparison with length scales set by the local density, such as the Fermi wavelength 27r/fcF or the screening length l/feg. This condition is rarely satisfied in real electronic systems, so we must look elsewhere to understand why LSD works. 

Hiickel pointed out that, on the basis of molecular orbital theory, monocyclic conjugated polymethines have filled shells of tt-electrons when the number of TT-electrons is An + 2, where n is an integer. These systems may be expected to be stable. The rule may be illustrated by reference to Fig. 2.1. If = 0, then a system with 27r-electrons should be stable. Such a situation is found in the cyclopropenyl positive ion, which has been isolated as the hexachloroanti-monate. For n = the prediction is that the cyclopentadienyl anion, benzene and the cycloheptatrienyl (tropylium) cation are stable. This is certainly in accord with experience. The stability of benzene is well known, the cydo-pentadienyl anion is readily formed by the action of potassium metal on cyclopentadiene, and the cycloheptatrienyl cation is one of the most stable carbonium ions known. Huckel s rule also predicts that some of the larger cyclic conjugated systems are stable, e.g. those with 10,14 and 18 rr-electrons. However, the situation is complicated by steric problems (see for example Garratt, 1971) and need not be considered further here. 

Currently available numerical results indicate that the one-dimensional heUum atom is completely chaotic. The best-known semiclassical quantization procedure for completely chaotic systems is Gutzwiller s trace formula (see Section 4.1.3), which is based on classical periodic orbits. Therefore we search for simple periodic orbits of the one-dimensional he-hum atom. Since a two-electron orbit is periodic if the orbits ni t), 0i t)) and (ri2(t), 2( )) of the first and second electron have a common period, the periodic orbits of the one-dimensional model can be labelled with two integers, m and n, which count the 27r-multiplicity of the angle variables 0i and 02 after completion of the orbit. Therefore, if for some periodic orbit... 

For the nonuniform electron gas at a metal surface, the Slater potential has an erroneous asymptotic behavior both in the classically forbidden region as well as in the metal bulk. In the vacuum region, the Slater potential has the analytical [10] asymptotic structure [35,51] V r) = — Xs(p)/x, with the coefficient otsiP) defined by Eq. (103). In the metal bulk this potential approaches [35] a value of ( — 1) in units of (3kp/27r) instead of the correct Kohn-Sham value of ( — 2/3). Further, in contrast to finite systems, the Slater potential V (r) and the work W,(r) are not equivalent [31, 35, 51] asymptotically in the classically forbidden region. This is because, for asymptotic positions of the electron in the vacuum, the Fermi hole continues to spread within the crystal and thus remains a dynamic charge distribution [34]. 

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