All 36 exactly solvable solutions of eigenvalues for nuclear electric quadrupole interaction Hamiltonian and equivalent rigid asymmetric rotor with expanded characteristic equation listing

This paper derives all 36 analytical solutions of the energy eigenvalues for nuclear electric quadrupole interaction Hamiltonian and equivalent rigid asymmetric rotor for polynomial degrees 1 through 4 using classical algebraic theory. By the use of double-parameterization the full general solution sets are illustrated in a compact, symmetric, structural, and usable form that is valid for asymmetry parameter $\eta \in \left({- \infty , + \infty}\right)$ . These results are useful for code developers in the area of Perturbed Angular Correlation (PAC), Nuclear Quadrupole Resonance (NQR) and rotational spectroscopy who want to offer exact solutions whenever possible, rather that resorting to numerical solutions. In addition, by using standard linear algebra methods, the characteristic equations of all integer and half-integer spins I from 0 to 15, inclusive are represented in a compact and naturally parameterized form that illustrates structure and symmetries. This extends Nielson’s?[1] listing of characteristic equations for integer spins out to I?=?15, inclusive.     

Tb3N@C84: an improbable, egg-shaped endohedral fullerene that violates the isolated pentagon rule

The structure of isomer 2 of Tb3N@C84 has been determined through single-crystal X-ray diffraction on Tb3N@C84.NiII(OEP).2(C6H6). The carbon cage has a distinct egg shape due to the presence of a single pair of fused pentagons at one apex of the molecule. Thus, although 24 IPR structures are available to the C84 cage, Nature utilizes one of the 51 568 isomeric structures that do not conform to the IPR for this unusual molecule. The Tb3N portion of isomer 2 of Tb3N@C84 is strictly planar. One Tb atom is nestled within the fold of the fused pentagons, while the other Tb atoms are disordered over four pairs of sites.     

Incorporating a flexible crown ether into neutral discrete self-assemblies driven by metal coordination

Driven by metal coordination, a flexible crown ether was self-assembled into neutral discrete complexes with two organoplatinum acceptors. Specifically, a [1 + 1]-assembled structure was separated from self-assemblies with different stoichiometries by vapor diffusion of acetone into their dichloromethane solution. The formation of these crown ether-incorporated self-assemblies was confirmed by NMR, elemental analysis, and X-ray single-crystal analysis.     

A closed molecular cube and an open book: two different products from assembly of the same metal salt and bridging ligand

Reaction of the bis-bidentate bridging ligand L1 with Co(ClO4)2 or Zn(BF4)2 affords a mixture of complexes [M8(L1)12]X16 and [M6(L1)9]X12 having the same metal : ligand ratio: the former is a molecular cube with a metal ion at each vertex and a bridging ligand spanning each edge, whereas the latter has a metal framework like that of an 'open book' containing cross-linked double helical metal-ligand subunits.     

Synthesis of a library of xylogluco-oligosaccharides for active-site mapping of xyloglucan endo-transglycosylase

Complex oligosaccharides containing alpha-D-xylosyl-(1-->6)-beta-D-glucosyl residues and unsubstituted beta-(1-->4)-linked D-glucosyl units were readily synthesized using enzymatic coupling catalyzed by the Cel7B E197A glycosynthase from Humicola insolens. Constituting this library required four key steps: (1) preparing unprotected building blocks by chemical synthesis or enzymatic degradation of xyloglucan polymers; (2) generating the donor synthon in the enzymatic coupling by temporarily introducing a lactosyl motif on the 4-OH of the terminal glucosyl units of the xylogluco-oligosaccharides; (3) synthesizing the corresponding alpha-fluorides, followed by their de-O-acetylation and the glycosynthase-catalyzed condensation of these donors onto various acceptors; and (4) enzymatically releasing lactose or galactose from the reaction product, affording the target molecules in good overall yields. These complex oligosaccharides proved useful for mapping the active site of a key enzyme in plant cell wall biosynthesis and modification: the xyloglucan endo-transglycosylase (XET). We also report some preliminary enzymatic results regarding the efficiency of these compounds.     

Vernier‐Templated Synthesis,Crystal Structure,and Supramolecular Chemistry of a 12‐Porphyrin Nanoring

Vernier templating exploits a mismatch between the number of binding sites in a template and a reactant to direct the formation of a product that is large enough to bind several template units. Here, we present a detailed study of the Vernier‐templated synthesis of a 12‐porphyrin nanoring. NMR and small‐angle X‐ray scattering (SAXS) analyses show that Vernier complexes are formed as intermediates in the cyclo‐oligomerization reaction. UV/Vis/NIR titrations show that the three‐component assembly of the 12‐porphyrin nanoring figure‐of‐eight template complex displays high allosteric cooperativity and chelate cooperativity. This nanoring–template 1:2 complex is among the largest synthetic molecules to have been characterized by single‐crystal analysis. It crystallizes as a racemate, with an angle of 27° between the planes of the two template units. The crystal structure reveals many unexpected intramolecular C?H???N contacts involving the tert‐butyl side chains. Scanning tunneling microscopy (STM) experiments show that molecules of the 12‐porphyrin template complex can remain intact on the gold surface, although the majority of the material unfolds into the free nanoring during electrospray deposition.     

A ratio limit theorem for the finite nonhomogeneous Markov chains

A new simple proof concerning the structure of the tail σ-field of a nonhomogeneous Markov chain as well as a result regarding the asymptotic behaviour of the ratio between the transition probabilities and the absolute probabilities of the chain are given. It is shown that the latter result can be used to characterize the atomic sets of the tail σ-field.     

A central limit theorem for “critical” first-passage percolation in two dimensions

Summary. Consider (independent) first-passage percolation on the edges of ℤ ² . Denote the passage time of the edge e in ℤ ² by t(e), and assume that P{t(e) = 0} = 1/2, P{0<t(e)<C ₀ } = 0 for some constant C ₀ >0 and that E[t ^δ (e)]<∞ for some δ>4. Denote by b _0,n the passage time from 0 to the halfplane {(x,y): x ≧ n}, and by T( 0 ,nu) the passage time from 0 to the nearest lattice point to nu, for u a unit vector. We prove that there exist constants 0<C ₁ , C ₂ <∞ and γ _n such that C ₁ ( log n) ^1/2 ≦γ _n ≦ C ₂ ( log n) ^1/2 and such that γ _n ⁻¹ [b _0,n −Eb _0,n ] and (√ 2γ _n ) ⁻¹ [T( 0 ,nu) − ET( 0 ,nu)] converge in distribution to a standard normal variable (as n →∞, u fixed). A similar result holds for the site version of first-passage percolation on ℤ ² , when the common distribution of the passage times {t(v)} of the vertices satisfies P{t(v) = 0} = 1−P{t(v) ≧ C ₀ } = p _c (ℤ ² , site ) := critical probability of site percolation on ℤ ² , and E[t ^δ (u)]<∞ for some δ>4. Received: 6 February 1996 / In revised form: 17 July 1996