Finite-Dimensional Simple Leibniz Pairs and Simple Poisson Modules

Simple modules over the Leibniz pairs are studied. Simple Poisson modules over Poisson algebras of the semisimple associative algebra structure are determined and they are nothing but simple bimodules over simple associative algebras with standard noncommutative Poisson algebra structure.     

Generalized characters of the symmetric group

Normalized irreducible characters of the symmetric group S(n) can be understood as zonal spherical functions of the Gelfand pair (S(n)×S(n),diagS(n)). They form an orthogonal basis in the space of the functions on the group S(n) invariant with respect to conjugations by S(n). In this paper we consider a different Gelfand pair connected with the symmetric group, that is an “unbalanced” Gelfand pair (S(n)×S(n−1),diagS(n−1)). Zonal spherical functions of this Gelfand pair form an orthogonal basis in a larger space of functions on S(n), namely in the space of functions invariant with respect to conjugations by S(n−1). We refer to these zonal spherical functions as normalized generalized characters of S(n). The main discovery of the present paper is that these generalized characters can be computed on the same level as the irreducible characters of the symmetric group. The paper gives a Murnaghan-Nakayama type rule, a Frobenius type formula, and an analogue of the determinantal formula for the generalized characters of S(n).     

Lewis acid promoted reactions of γ,γ-dialkoxyallylic zirconium species with various carbonyl compounds

The reactions of γ,γ-dialkoxyallylic zirconium species with carbonyl compounds in the presence of Lewis acid are reported. The reactivity of γ,γ-dialkoxyallylic zirconium species and reaction pathway were strongly dependent on the structure and electrostatic nature of the carbonyl compounds.     

How electrons guard the space: shape optimization with probability distribution criteria

Efficient formulas for computing the probabilities of finding exactly electrons in an arbitrarily chosen volume ³ for Hartree–Fock wavefunctions are presented. These formulas allow the use of shape optimization techniques, such as level set methods, for optimizing with respect to various criteria involving such probabilities. The criterion defined as the difference between the Hartree–Fock and the independent-particle model probabilities of finding electrons in stresses the quantum effects due to the Pauli principle. We have implemented a 2D level set method for optimizing this criterion in order to study spatial separation of electron pairs in linear molecules. The method is described and the illustrative example of the BH molecule is reported.Contribution to the Jacopo Tomasi Honorary Issue     

Chiral Recognition in π Complexes of Alkenes,Aldehydes, and Ketones with Transition Metal Lewis Acids; Development of a General Model for Enantioface Binding Selectivities

Prochiral alkenes, aldehydes, and ketones constitute the most frequently used starting materials for enantioselective organic syntheses. Protocols often involve chiral binding agents or Lewis acids that can give two diastereomeric adducts, the ratios of which are measures of chiral recognition. With π adducts, the diastereomers differ in the enantioface of the C?C or O?C group bound to the Lewis acid. This review provides the first comprehensive analysis of such equilibria and related binding phenomena with chiral transition metal Lewis acids. An extensive body of data from the authors' laboratory for complexes of the pyramidal rhenium fragment [(η⁵?C₅H₅)Re(No)(PPh₃)]⁺ ( I ) affords particular insight. Literature data for other complexes are also summarized. A general model for chiral recognition based upon the relative steric properties of four quadrants is presented. This enables binding selectivities to be individually and rationally optimized for different classes of ligands. Electronic effects are also identified and correlated with specific structural properties. Relationships between binding equilibria, reactivity, and product configurations are discussed.     

Electronic effect on the regioselectivity in the ring opening of para-substituted phenyloxiranes by acetylides

The electronic effect on the regioselectivity in the alkynylation of phenyloxiranes was investigated using three kinds of metal acetylides. BF₃ mediated lithium acetylide provided either the α- or β-alkynylated products by controlling the effect of the para-substituents of the phenyloxiranes. LiClO₄ mediated lithium acetylide and titanium acetylide, on the other hand, afforded predominantly the β- and α-products, respectively.     

Direct conversion of tert-butyl 2-hydroxyalkyl sulfides to 1,3-oxathiolanes

tert-Butyl 2-hydroxyalkyl sulfides, prepared by reaction of epoxides with 2-methylpropane-2-thiol, are converted directly to 1,3-oxathiolanes upon treatment with pivalaldehyde and boron trifluoride diethyl etherate in the presence of thioanisole.     

Diastereoselective addition of organolithiums to 1,3-oxazolidines complexed with aluminum tris(2,6-diphenylphenoxide) (ATPH)

1,3-Oxazolidines were easily obtained by condensation of N-substituted (R)-phenylglycinol with aldehydes. Addition of organolithium reagents to 1,3-oxazolidines by complexation with the bulky Lewis acid aluminum tris(2,6-diphenylphenoxide) (ATPH) readily produced the corresponding chiral amines with good yield and high diastereoselectivity. The configuration of the new stereogenic center was shown to be opposite to that of adducts obtained for the same 1,3-oxazolidines using Grignard reagents. The best diastereoselectivity was achieved using N-isopropyl-1,3-oxazolidines. The mechanism of addition was deduced by determining the stereochemistry of the iminium-aluminum complex by NOE experiments.