Second virial coefficients of asymmetric top molecules

A short self-contained derivation is given for the second virial coefficient B2(T) of a gas consisting of identical interacting asymmetric rigid rotors. The resulting expression is correct through variant Planck's over h2. First, the canonical partition function is derived by means of an variant Planck's over h expansion of exp[-H/(k(B)T)] due to Friedmann [Adv. Chem. Phys. 4, 225 (1962)]. The present work applies angular momentum operators and known facts from angular momentum theory. It is considerably more accessible than Friedmann's exposition, which is not based on angular momentum operators, but instead on explicit derivatives with respect to Euler angles. The partition function obtained from the variant Planck's over h expansion is applied to the derivation of an expression for B2(T) that is identical in appearance to the expression for symmetric rotors of T Pack [J. Chem. Phys. 78, 7217 (1983)]. The final equation in this work is valid for rigid rotors of any symmetry.     

Zur Photochemie von Allylaryläthern. 55. Mitteilung über Photoreaktionen

The photochemical reactions of different allyl aryl ethers (Scheme 3) were investigated in hydrocarbons (Chap. 3.1) and in alcoholic solvents (Chap. 3.2). The composition of the photoproducts depended very much on the nature of the solvent. Irradiation (3–95 h) of different methyl substituted allyl aryl ethers ( 1, 3, 5, 7 and 11 ) with a low pressure mercury lamp (λ_Emiss. = 254 nm; 6 or 15 Watt) under argon (quartz vessel) resulted in the formation of 2-, 3– and 4-substituted phenols, dienones and products of consecutive reactions (Tables 1–4 and 6). The results suggested that all products were formed by homolytic cleavage of the C? O bond in the singlet state of the ethers to intermediate radical-geminates (Scheme 5) followed by radical recombination of the two fragments. No products were formed by concerted processes (Table 5, Schemes 5 and 6). Upon irradiation of allyl aryl ethers lacking alkyl substituents at position 4 ( 1 and 5 ) in protic solvents, mainly 2- and 4-allylated phenols were obtained (Tables 1 and 4); 3-allylated phenols were formed only in small amounts (0.02%). However, in aromatic hydrocarbons or cyclohexane 3-allylated phenols were obtained from 1 , 5 and 11 in significant amounts (3–11%; Tables 1, 4 and 6). E.g., upon irradiation of allyl-2,6-dimethyl-2,4-cyclohexadien-1-one ( 6 ) besides 3- and 4-allyl-2, 6-dimethyl-phenol ( 23 and 24 ). Irradiation of 5 in methanol afforded 23 and 6 only in traces, whereas 24 was the main product.     

A note on the importance of d orbitals in the calculation of effective interactions and the excitation spectra of atoms and molecules

The focal point of our discussion is the examination of truncated basis sets used in obtaining an accurate first principles clculation of the effective valence shell Hamiltonian by the canonical transformation-cluster expansion approasch. Subsequent diagonalization of this effecitve valence shell hamiltonian yields the valence shell transition energies. A detailed analysis of numerical results obtained using a number of different basis sets of hydrogen-like orbitals together with rigorous symmetry arguments celarly demonstrates the special role played by d orbitals in computing the ³P → ¹D transition energy in carbon. The failure of early attempts to calculate the effective Hamiltonian for ethylene from first principles is examined in the light of recent ab initio calculations on ethylene involving d orbitals and the computations reported in this paper. We conclude that accurate calculations of the effective valence shell Hamiltonian for molecules must consider d orbitals in the excited orbital basis set.     

Heats of adsorption of linear CO species adsorbed on the Au degrees and Ti+delta sites of a 1% Au/TiO2 catalyst using in situ FTIR spectroscopy under adsorption equilibrium

The heats of adsorption of two linear CO species adsorbed on the Au degrees particles (denoted L(Au degrees)) and on the Ti(+delta) sites (denoted L(Ti+delta)) of a 1% Au/TiO(2) catalyst are determined as the function of their respective coverage by using the AEIR procedure (adsorption equilibrium infrared spectroscopy) previously developed. Mainly, the evolutions of the IR band area of each adsorbed species (2184 cm(-1) for L(Ti+delta) and at 2110 cm(-1) for L(Au degrees)) as a function of the adsorption temperature T(a), at a constant CO adsorption pressure P(CO), provide the evolutions of the coverages theta(LTi+delta) and theta(LAu degrees ) of each adsorbed CO species with T(a) in isobar conditions that give the individual heats of adsorption. It is shown that they linearly vary from 74 to 47 kJ/mol for L(Au degrees ) and from 50 to 40 kJ/mol for L(Ti+delta) at coverages 0 and 1, respectively. These values are consistent with literature data on model Au particles and TiO(2). In particular, it is shown that the mathematical formalism supporting the AEIR procedure can be applied to literature data on Au-containing solids (single crystals and model particles).     

Periodic solutions of dry friction problems

Dry friction problems lead to discontinuous differential equations, e.g. to

Processes of flats induced by higher dimensional processes III

Stationary processes of k-flats in ^d can be thought of as point processes on the Grassmannian _k ^d of k-dimensional subspaces of ^d . If such a process is sampled by a (d–k+ j)-dimensional space F, it induces a process of j-flats in F. In this work we will investigate the possibility of determining the original k-process from knowledge of the intensity measures of the induced j-processes. We will see that this is impossible precisely when 1<k<d–1 and j=0,...,2[r/2]–1, where r is the rank of the manifold _k ^d . We will show how the problem is equivalent to the study of the kernel of various integral transforms, these will then be investigated using harmonic analysis on Grassmannian manifolds.The research of the first and third authors was supported in part by NSF grants DMS-9207019 and DMS-9304284. The research of the second author was supported in part by NFR contract number R-RA 4873-306 and the Swedish Academy of Sciences.     

η5-cyclopentadienyl-η5-syn-1-methylpentadienylrhodium and iridium cations - preparation and reactions with nucleophiles

η⁵-cyclopentadienyl-η⁵-syn-1-methylpentadienyliridium(I) hexafluorophosphate, a novel complex containing an open chain pentadienyl ligand, has been prepared $\text{via}$ protonation of η⁵-cyclopentadienyl-η⁴-hexa-2,4-dien-1-ol iridium. It and the corresponding rhodium salt are attacked by methoxide ion at the central 3-position of the dienyl group to give 1,4-diene complexes. This contrasts with nucleophilic attack on pentadienyltricarbonyliron cations which occurs at the terminal positions of the dienyl group.