Chaotic Expansion of Elements of the Universal Enveloping Algebra of a Lie Algebra Associated with a Quantum Stochastic Calculus

The functional Ito formula, in the form df() = f( + d ) –f(),is formulated and proved in the context of a Lie algebra L associatedwith a quantum (non-commutative) stochastic calculus. Here fis an element of the universal enveloping algebra U of L, andf() + d() – f() is given a meaning using the coproductstructure of U even though the individual terms of this expressionhave no meaning. The Ito formula is equivalent to a chaoticexpansion formula for f() which is found explicitly. 1991 MathematicsSubject Classification: primary 81S25; secondary 60H05; tertiary18B25.     

Stereochemistry of 1,2-elimination reactions at the E2-E1cB interface--tert-butyl 3-tosyloxybutanoate and its thioester

Experimental data on the stereoselectivity of base-catalyzed 1,2-elimination reactions that produce conjugated carbonyl compounds are scarce in spite of the importance of these reactions in organic and biochemistry. As part of a comprehensive study in this area, we have synthesized stereospecifically-deuterated beta-tosyloxybutanoate esters and thioesters and studied the stereoselectivity of their elimination reactions under non-ion pairing conditions. With the availability of both the (2R*,3R*) and (2R*,3S*) diastereomers the innate stereoselectivity could be determined unambiguously. (1)H and (2)H NMR data show that these substrates produce 5-6% syn elimination, the usual amount for acyclic substrates undergoing E2 reactions. Contrary to earlier suggestions, activation by a carbonyl group has virtually no influence upon the stereoselectivity. Elimination of the (2R*,3R*) diastereomer of the beta-tosyloxyester and thioester produces 21-25% of the (Z)-alkene, much more than observed with a poorer beta-nucleofuge. A relatively large amount of (Z)-alkene product seems to be a good marker for an E2 pathway, in which the transition state is E1cB-like, rather than an E1cB(irrev) mechanism. Syn KIE values were higher than those for anti elimination for the esters as well as the thioesters. Experimental challenges to the synthesis of stereospecifically-deuterated beta-tosyloxyesters are discussed.     

Microwave spectra and molecular structures of (Z)-pent-2-en-4-ynenitrile and maleonitrile.

Accurate equilibrium structures have been determined for (Z)-pent-2-en-4-ynenitrile (8) and maleonitrile (9) by combining microwave spectroscopy data and ab initio quantum chemistry calculations. The microwave spectra of 10 isotopomers of 8 and 5 isotopomers of 9 were obtained using a pulsed nozzle Fourier transform microwave spectrometer. The ground-state rotational constants were adjusted for vibration-rotation interaction effects calculated from force fields obtained from ab initio calculations. The resultant equilibrium rotational constants were used to determine structures that are in very good agreement with those obtained from high-level ab initio calculations (CCSD(T)/cc-pVTZ). The geometric parameters in 8 and 9 are very similar; they also do not differ significantly from the all-carbon analogue, (Z)-hex-3-ene-1,5-diyne (7), the parent molecule for the Bergman cyclization. A small deviation from linearity about the alkyne and cyano linkages is observed for 7-9 and several related species where accurate equilibrium parameters are available. The data on 7-9 should be of interest to radioastronomy and may provide insights on the formation and interstellar chemistry of unsaturated species such as the cyanopolyynes.