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ABSTRACT:We present here a systematic study by quantum mechanical methods of structural and energetic properties for a series of substitutions by alkyl groups of the hydrogens in hydrogen peroxide. The emphasis is on the torsion around the peroxidic bond, which leads to the chirality changing stereomutation. The dihedral angle dependence of the geometrical features and of the dipole moment is discussed with reference to previous experimental and theoretical information, and with respect to the preceding paper on hydrogen peroxide (Maciel et al., Chem Phys Lett, 2006, 432, 383). This information is of interest for chiral separation experiments as well as in view of a possible dynamical mechanism for chirality exchange by molecular collisions. The cis and trans barriers appear to vary remarkably upon substitution by alkyl groups (methyl, ethyl, n-and iso-propyl, sec-and tert-butyl hydroperoxides), the most important property being their geometrical dimensions. As the latter increase, tendency for the equilibrium configuration towards the trans structure increases, so that the trans barrier becomes negligible for dimethyl and diethyl peroxides and for n-and iso-butyl hydroperoxides, giving essentially achiral molecules. For the chiral ones (HOOH, CH 3 OOH, and C 2 H 5 OOH) torsional level energies and eigenfunctions are calculated and their distribution as a function of temperature determined. Their use is exemplified by a calculation of the dipole moment of hydrogen peroxide at room temperature, reconciling previous disagreement between theory and experiment.

The mathematical apparatus of quantum-mechanical angular momentum (re)coupling, developed originally to describe spectroscopic phenomena in atomic, molecular, optical and nuclear physics, is embedded in modern algebraic settings which emphasize the underlying combinatorial aspects. SU (2) recoupling theory, involving Wigner's 3nj symbols, as well as the related problems of their calculations, general properties, asymptotic limits for large entries, plays nowadays a prominent role also in quantum gravity and quantum computing applications. We refer to the ingredients of this theory -and of its extension to other Lie and quantum group-by using the collective term of 'spin networks'.Recent progress is recorded about the already established connections with the mathematical theory of discrete orthogonal polynomials (the so-called Askey Scheme), providing powerful tools based on asymptotic expansions, which correspond on the physical side to various levels of semi-classical limits. These results are useful not only 1 in theoretical molecular physics but also in motivating algorithms for the computationally demanding problems of molecular dynamics and chemical reaction theory, where large angular momenta are typically involved. As for quantum chemistry, applications of these techniques include selection and classification of complete orthogonal basis sets in atomic and molecular problems, either in configuration space (Sturmian orbitals) or in momentum space. In this paper we list and discuss some aspects of these developments -such as for instance the hyperquantization algorithm-as well as a few applications to quantum gravity and topology, thus providing evidence of a unifying background structure.

ABSTRACT:This article describes a direct method for the exact computation of 3nj symbols from the defining series, and continues discussing properties and asymptotic formulas focusing on the most important case, the 6j symbols or Racah coefficients. Relationships with families of hypergeometric orthogonal polynomials are presented and the asymptotic behavior is studied to account for some of the most relevant features, both from the viewpoints of the basic geometrical significance and as a source of accurate approximation formulas, such as those due to Ponzano and Regge and Schulten and Gordon. Numerical aspects are specifically investigated in detail, regarding the relationship between functions of discrete and of continuous variables, exhibiting the transition in the limit of large angular momenta toward both Wigner's reduced rotation matrices (or Jacobi polynomials) and harmonic oscillators (or Hermite polynomials).

In view of the particular attention recently devoted to hindered rotations, we have tested reduced kinetic energy operators to study the torsional mode around the O-O bond for H(2)O(2) and for a series of its derivatives (HOOCl, HOOCN, HOOF, HOONO, HOOMe, HOOEt, MeOOMe, ClOOCl, FOOCl, FOOF, and FOONO), for which we had previously determined potential energy profiles along the dihedral ROOR(') angle [R,R(')=H,F,Cl,CN,NO,Me (=CH(3)), Et (=C(2)H(5))]. We have calculated level distributions as a function of temperature and partition functions for all systems. Specifically, for the H(2)O(2) system we have used two procedures for the reduction in the kinetic energy operator to that of a rigid-rotor-like one and the calculated partition functions are compared with previous work. Quantum partition functions are evaluated both by quantum level state sums and by simple classical approximations. A semiclassical approach, using a linear approximation of the classical path and a quadratic Feynman-Hibbs approximation of Feynman path integral, introduced in previous work and here applied to the torsional mode, is shown to greatly improve the classical approximations. Further improvement is obtained by the explicit introduction of the dependence of the moment of inertia from the torsional angle. These results permit one to discuss the characteristic time for chirality changes for the investigated molecules either by quantum mechanical tunneling (dominating at low temperatures) or by transition state theory (expected to provide an estimate of racemization rates in the high energy limit).