Rational tetrahedra with edges in geometric progression

This paper discusses tetrahedra with rational edges forming a geometric progression, focussing on whether they can have rational volume or rational face areas. We examine the 30 possible configurations of such tetrahedra and show that no face of any of these has rational area. We show that 28 of these configurations cannot have rational volume, and in the remaining two cases there are at most six possible examples, and none have been found.     

Preparation and characterization of M2(SeAr′)6 and mixed ligand M2(OR)2(SeAr′)4 species (M = Mo, W)

Toluene solutions of M₂(NMe₂)₆ (M = Mo, W) react with mesitylene selenol (Ar′SeH) to give M₂(SeAr′) ₆ complexes. MO₂(OR)₆ (R = ^tBu, CH₂^tBu) react with excess> 6 fold) Ar′SeH to give Mo₂ (SeAr′)₆, whilst W₂(OR)₆(py)₂ (R = ⁱPr, CH₂^tBu) react with excess (> 6 fold) Ar′SeH to give W₂(OR)₂(SeAr′)₄. Reaction of MO₂(OPrⁱ)₆ with Ar′SeH produces Mo₂(OPrⁱ)₂ (SeAr′)₄ which crystallizes in two different space groups. These areneselenato complexes are air-stable and insoluble in common organic solvents. X-ray crystallographic studies revealed that the Mo₂(SeAr′)₆ and W₂(SeAr′)₆ compounds are isostructural in the solid state and adopt ethane-like staggered configurations with the following important structural parameters, M---M (W---W/Mo---Mo) 2.3000(11)/2.2175(13) Å, M---Se 2.430 (av.)/2.440 (av.) Å, M---M---SE 97.0° (av.)°. In the solid state W₂(OⁱPr)₂(SeAr′)₄ adopts the anti-configuration with crystallographically imposed C_i symmetry and W---W 2.3077(7) Å, W---Se 2.435 (av.) Å, W---O 1.858(6) Å; W---W---SE 100.27(3)°, 93.8(3)° and W---W---O 108.41(17)°. Mo₂(OPrⁱ)₂(SeAr′) ₄ crystallizes in both P and A2/a space groups in which the molecules are isostructural with each other and the tungsten analogue. Important bond lengths and angles are Mo---Mo 2.180(24) Å, Mo---Se 2.432(av.) Å, Mo---O 1.872(9) Å, Mo---Mo---Se 99.39(9)°, 94.71(8)°, Mo---Mo---O 107.55(28)°.     

A double expansion method for calculating molecular properties

Using a form of double perturbation theory we present results of calculations on the ground state energies for the diatomic systems H₂ ⁺, HHe²⁺, H₂ and HHe⁺. For the homonuclear systems the energy values are unexpectedly good. For the heteronuclear cases the results are very encouraging and lead to optimism about the application of the method to larger systems which contain one heavy atom.     

Ring-opening polymerization of l-lactide by organotin(IV)alkoxides,R2Sn(OPr)2: Estimation of the activation parameters

The ring-opening polymerization of l-lactide, l-LA, to give poly-l-lactide by R₂Sn(OPrⁱ)₂ compounds, where R = Buⁿ and p-XC₆H₄ (X = CF₃, F, H, Me and OMe) has been studied in benzene over a temperature range. There is a relatively small variation in ΔH^≠ as a function of R with all the values falling within the range 11 ± 2 kcal mol⁻¹. The entropy of activation, ΔS^≠, is consistently large and negative, −50 ± 5 eu, supporting the view that the ring-opening event, the enchainment step involves a highly ordered transition state. The crystal and molecular structures of the compounds Ph₂Sn(OPrⁱ)₂, (p-FC₆H₄)₂Sn(OPrⁱ)₂ and (p-Me₂NC₆H₄)₃SnOPrⁱ are also reported. While the latter compound is monomeric in the solid state the former are both dimeric with a pair of bridging OPrⁱ ligands.