Chemistry

All manipulations were carried out with the exclusion of oxygen and moisture using flamed Schlenk-type glassware and high vacuum (10-5 Torr) techniques. For storage of air-sensitive materials, a nitrogen-filled "Vacuum Atmospheres" or "Mbraun" glove box with a medium capacity recirculator (1-2 ppm O 2) was used. The gases (argon, nitrogen, ethylene) were purified by passage through a MnO oxygen removal column and a Davison 4 Ǻ molecular sieves column. Analytically pure solvents were distilled under argon from potassium benzophenone ketyl (tetrahydrofuran), Na/K alloy (diethylether, hexane) Na (toluene). All solvents for vacuum line manipulations were stored in vacuum over Na/K alloy. Methylalumoxane cocatalyst (Witco) was prepared from a 30% suspension in toluene by vacuum evaporation of the solvent at 25 o C/10-5 Torr. NMR spectra were recorded on Bruker AM-200 and AM-400 spectrometers or on Bruker Avance 300 and 500 MHz spectrometers. Chemical shifts for 1 H-NMR and 13 C-NMR were referenced to internal solvent resonances and reported relative to SiMe 4 .

The synthesis of mixed (aryloxo)(imidazolin-2-iminato)titanium dichloride complexes is presented. Full characterization of the complexes, including solid-state X-ray diffraction analysis of four complexes, is reported. Their catalytic activity in the polymerization of ethylene, propylene, 1-hexene, and 1-octene was studied. The different substituents at the aryloxo moiety were found to have a substantial impact on the reactivity of the complexes, indicating the need for a balance of steric and electronic effects. The complex (1,3-di-tertbutylimidazolin-2-iminato)(2,6-di-tert-butyl-4-methylphenoxide)titanium dichloride (1) was found to be the most active in the polymerization of ethylene, whereas the complex (1,3-di-tert-butylimidazolin-2-iminato)-(2,6-di-tert-butylphenoxide)titanium dichloride (2) was found to be the most active in the polymerization of propylene. Interestingly, all complexes were more active in the polymerization of propylene than in that of ethylene.

Explicit expressions for the Temperley-Lieb-Martin algebras, i.e., the quotients of the Hecke algebra that admit only representations corresponding to Young diagrams with a given maximum number of columns (or rows), are obtained, making explicit use of the Hecke algebra representation theory. Similar techniques are used to construct the algebras whose representations do not contain rectangular subdiagrams of a given size.

The asymptotic form of bound-state wave functions is derived by analytic continuation of asymptotic scattering-state wave functions. The result is also rigorously derived by using an approach that is independent of scattering theory. One aspect of the result is that the N electron wave function becomes the lowest accessible exact wave function for the remaining N -1 electrons when one electron is far away from all the nuclei. This shows that the recently developed extended Koopmans' procedures are in principle exact for e first ionization energy.

The normal ordering formulae for powers of the boson number operatorn are extended to deformed bosons. It is found that for the "M-type" deformed bosons, which satisfy aa † − qa † a = 1, the extension involves a set of deformed Stirling numbers which replace the Stirling numbers occurring in the conventional case. On the other hand, the deformed Stirling numbers which have to be introduced in the case of the "P-type" deformed bosons, which satisfy aa † − qa † a = q −n , are found to depend on the operatorn. This distinction between the two types of deformed bosons is in harmony with earlier observations made in the context of a study of the extended Campbell-Baker-Hausdorff formula.

An algorithm is formulated for the construction of many-particle permutational symmetry adapted functions in hyperspherical coordinates. A recursive procedure is proposed, introducing hyperspherical coefficients of fractional parentage (hscfps). These coefficients are the eigenvectors of the transposition class sum of the symmetric group in an appropriate basis. Only the matrix element of the transposition of the last two particles has to be calculated in each step. This matrix element is obtained by using the hscfps calculated in the preceding step as well as the Raynal-Revai and the T coefBcients. The results are applicable to the study of the atomic, molecular, and nuclear few-body problem.

A conjecture concerning the construction of explicit expressions for the central characters of the symmetric group S, is presented. The expression for the central characters corresponding to a class with a given cycle structure (l)"(2)" . (n)'" is a polynomial in the symmetric power sums over the "contents" of the Young diagram specifying the irreducible representation of interest. Each term in this polynomial is a product of symmetric power sums multiplied by a polynomial in n. Both the degrees of the polynomials in n and the powers of the various symmetric power sums appearing in each term are specified by an algorithm involving sets of partitions of integers associated with the lengths of the nontrivial cycles specifying the class of interest and combinations thereof. The coefficients of the polynomials in n are obtained by solving a system of linear equations which arise from the evaluation of the proposed expression for the central characters with respect to Young diagrams whose number of boxes is less than EYE, if,, and equating to zero. 0166-218X/96/$15.00 0 1996 Elsevier Science B.V. All rights reserved SSDI 0166-218X(95)00016-X

An algorithm for the construction of non-spurious harmonic oscillator wave functions with arbitrary permutational symmetry is formulated by adapting a procedure recently developed for building general single shell wave functions. The harmonic oscillator functions, expressed in Jacobi coordinates, are calculated recursively using a new type of harmonic oscillator coefficients of fractional parentage. These coefficients are the eigenvectors of the two-cycle class operator of the permutation group in the appropriate basis. The matrix elements of the class operators are evaluated by using a specific version of the harmonic oscillator brackets. The significance of this procedure to atomic, molecular and nuclear physics is pointed out. The presently proposed method is expected to be computationally very efticient.

A reformulation of the Hylleraas-Undheim-MacDonald variational procedure for excited states, involving the application of Lowdin's supersecular equation to a variational problem concerning the ground state of an appropriate super-Hamiltonian, is proposed. Some formal applications are presented.

The use of Orbital Correspondence Analysis in Maximum Symmetry (OCAMS) for determining the allowed course of a reaction is described in terms of the familiar "A, S" notation. Several well known thermal and photochemical reactions are used as examples : Polyene cyclization; the fragmentation and dimerization of cyclobutadiene; [.4 + A2] cycloaddition; the isomerization of Dewar-benzene. The relation of OCAMS to the classical correlation procedures of Woodward and Hoffmann and of Longuet-Higgins and Abrahamson is brought out, as are the sources of the occasionally contradictory predictions of OCAMS and of the classical procedures.

The bound and resonance states for a family of piecewise linear potentials are obtained by the complex coordinate method. The complex spectra were found to be weakly affected by isochronous transformations of the potential, i.e. transformations which keep the classical periods of bound trajectories and the classical delay times of scattering trajectories invariant.