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.     

Protonation study of peroxynitric acid and peroxynitrous acid

The equilibrium structures and harmonic vibrational frequencies of peroxynitric acid (HOONO(2)) and seven structures of protonated peroxynitric acid, along with peroxynitrous acid (HOONO) and its 12 protonated peroxynitrous acid structures, have been investigated using several ab initio and density functional methods. The ab initio methods include second-order Moller-Plesset perturbation theory, quadratic configuration interaction, including single and double excitations theory (QCISD), and the QCISD(T) methods, which incorporate a perturbational estimate of the effects of connected triple excitation. The Becke three-parameter hybrid functional combined with Lee, Yang, and Parr correlation function is the density functional method used. The lowest energy form of protonated peroxynitric acid is a complex between H(2)O(2) and NO(+) rather than between H(2)O and NO(2) (+). For peroxynitrous acid, a complex between H(2)O(2) and NO(2) (+) is found to be the lowest energy structure. The ab initio proton affinity (PA) of HOONO and HOONO(2) is predicted to be 182.1 and 175.1 kcal mol(-1), respectively, at the QCISD(T)/6-311++G(3df,3pd) level of theory. The results are contrasted with an earlier study on nitrous acid, and is shown that peroxynitric acid and peroxynitrous acid have a smaller PA than nitrous acid.