An insight into the aromaticity of fullerene anions: experimental evidence for diamagnetic ring currents in the five-membered rings of C(60)(6-) and C(70)(6-)

Reduction of the two "closed" [6,6] methanofullerenes, [6,6]C(61)H(2) (1) and [6,6]C(71)H(2) (5), to the corresponding hexaanions with lithium metal causes the bridgehead-bridgehead bonds to open, at least partially, and this change gives rise to diamagnetic ring currents in the resulting homoconjugated six-membered rings (6-MRs). These new ring currents shield the overlying hydrogen atoms on the methylene bridge and induce upfield shifts of 1.60 and 0.11 ppm in their (1)H NMR resonances, respectively. Analogous reduction of the already "open" [5,6]methanofullerenes, [5,6]C(61)H(2) (2) and [5,6]C(71)H(2) (3 and 4), only slightly enhances the shielding of the hydrogen atoms over the homoconjugated 6-MRs (upfield shifts of 0.13, 0.68, and 0.14 ppm, respectively) but leads to exceptionally strong diamagnetic ring currents in the homoconjugated five- membered rings (5-MRs), as evidenced by dramatic shielding of the hydrogen atoms situated over them (upfield shift of 5.01, 6.78, and 1.63 ppm, respectively). The strongest shielding is seen for the hydrogen atom sitting over the 5-MR at the pole of C(71)H(2)(6)(-) (delta = -0.255 ppm) indicating that the excess charge density is concentrated at the poles.     

Dimension theory and superpositions of continuous functions

The main result of this paper is the following: IfX is a compact two dimensional metric space, and {φ _i} _i ^{= 1/4} are four functions inC(X), then there exists a functionf inC(X) which cannot be represented in the form: $$f(x) = \sum\limits_{i = 1}^4 {g_\iota (\varphi _i (x))} $$ , with $$g_\iota \in C(R)$$ .     

Dense subgroups ofC(K)-Stone-Weierstrass-type theorems for groups

LetB be a subgroup ofC(K) which separates points and contains the constants. An elementh∈C(R) operates onB iff∈B implies thath°f∈B. An elementh∈C(R) is condensing if its operation onB implies the density ofB inC(K). Similar notation applies to subgroupsB ofC(K, G) whereG is a metrizable group. We study the setD(G) of condensing functions inC(G, G) whenG is the additive group of a real Banach space and in particular whenG=R ⁿ.     

Two helium atoms inside fullerenes: probing the internal magnetic field in C60(6-) and C70(6-)

A 3He NMR resonance of C606- containing He is assigned to He2@C606-, thus showing that C60 can also accommodate two helium atoms. The ratio of the di-helium compound relative to the mono- is 1:200, 10 times lower than the equivalent counterpart of C70. The 3He NMR chemical shift of He2@C606- is 0.093 ppm downfield from the already known resonance of He@C606-. In the reduced endohedral mono- and di-helium C70, the 3He NMR chemical shift of He2@C706- is 0.154 ppm upfield from the peak of He@C706-.     

Uniform separation of points and measures and representation by sums of algebras

LetX andY _i, 1 ≦i ≦k, be compact metric spaces, and letρ _i:X →Y _i be continuous functions. The familyF={ρ _i} _i ^1/k is said to be ameasure separating family if there exists someλ > 0 such that for every measureμ inC(X)*, ‖μ ^o ρ _i ^?1 ‖ ≧λ ‖μ ‖ holds for some 1 ≦i ≦k.F is auniformly (point) separating family if the above holds for the purely atomic measures inC(X)*. It is known that fork ≦ 2 the two concepts are equivalent. In this note we present examples which show that fork ≧ 3 measure separation is a stronger property than uniform separation of points, and characterize those uniformly separating families which separate measures. These properties and problems are closely related to the following ones: letA ₁,A ₂, ...,A _k be closed subalgebras ofC(X); when isA ₁ +A ₂ + ... +A _k equal to or dense inC(X)?     

Anions and Polyanions of Oligoindenopyrenes: Modes of Electron Delocalization and Dimerization

A series of pyrene‐based polycyclic aromatic compounds, indeno[cd]pyrene, diindeno[cd,fg]pyrene, diindeno[cd,jk]pyrene, tris‐(tert‐butylindeno[cd,fg,jk])pyrene, and tetrakis‐(tert‐butylindeno[cd,fg,jk,mn])pyrene, were reduced with alkali metals in [D₈]tetrahydrofuran, and the resulting anions were studied by NMR spectroscopy. It was found that the diatropic character of the dianions obtained depends on the number of annulated indeno groups. When one such group is present, a paratropic dianion is obtained, which is similar to the dianion of the parent pyrene; the effect, however, is weak. When more indeno groups are annulated, the dianions become diatropic owing to the greater number of five‐membered rings that can acquire aromatic character as a result of reduction. The ¹H NMR chemical shifts of tetrakis‐(tert‐butylindeno[cd,fg,jk,mn])pyrene in the neutral state show an interesting dependence on concentration that reflects an association of the molecules in solution by π stacking. This phenomenon was not observed for the reduced species. The trianion radicals of tris‐(tert‐butylindeno[cd,fg,jk])pyrene and tetrakis‐(tert‐butylindeno[cd,fg,jk,mn])pyrene undergo reductive dimerization and form bilayered hexaanions.     

A fixed point theorem for a class of star-shaped sets inc 0

A subsetK ofc ₀ is coordinatewise star-shaped (c.s.s.) if there exists a center pointx ∈K such that fory ∈K andz ∈c ₀, ifz is coordinatewise betweenx andy thenz ∈K. We prove that a weakly compact c.s.s. subset ofc ₀ has the fixed point property for nonexpansive mappings and that a fixed point for such a mapping can be obtained in a constructive manner. Research of the first two authors was partially supported by NSF Grant MCS78-01344 and of the last author by MCS78-01501.