An approach to creating novel anions: silylated triel compounds with –CH2– and –O– linkers, [InCl2{O(HO)Si(t‐Bu)2}]2 and Li[B(CH2SiMe3)4]

Bis[μ‐di‐tert‐butyl(hydroxy)silanolato]bis[chloridoindium(III)], [In₂(C₈H₁₉O₂Si)₂Cl₄], (I), is a centrosymmetric two‐centre indium complex featuring a system of three annulated four‐membered rings; the structure is the first example of an In₂O₂ ring which is annulated with two Si—O units to form a ring system composed of three rings. The coordination environment of the In centres is a distorted trigonal bipyramid. The crystal packing of (I) is characterized by chains of molecules connected by O—H...Cl hydrogen bonds. The crystal of (I) was a nonmerohedral twin. There is no known example of an In₂O₂ ring in which the In atoms carry any two halogen ligands. The structure of tetrakis(tetrahydrofuran)lithium tetrakis[(trimethylsilyl)methyl]borate, [Li(C₄H₈O)₄](C₁₆H₄₄BSi₄), (II), is composed of discrete cations and anions. The coordination geometries of the Li and B centres is tetrahedral. The cations and anions lie in planes parallel to the ab plane. There are no short contacts between the cations and anions. Compound (II) is the first example of a B centre bonded to four –CH₂Si units.     

Mehler's formula and functional calculus Dedicated to Professor Jean-Yves Chemin on the Occasion of His 60th Birthday

We show that Mehler's formula can be used to handle several formulas involving the quantization of singular Hamiltonians. In particular, we diagonalize in the Hermite basis the Weyl quantization of the characteristic function of several domains of the phase space.     

Four (2,2′‐bipyridyl)(ferrocenyl)boronium derivatives

Crystal structures are reported for four (2,2′‐bipyridyl)(ferrocenyl)boronium derivatives, namely (2,2′‐bipyridyl)(ethenyl)(ferrocenyl)boronium hexafluoridophosphate, [Fe(C₅H₅)(C₁₇H₁₅BN₂)]PF₆, (Ib), (2,2′‐bipyridyl)(tert‐butylamino)(ferrocenyl)boronium bromide, [Fe(C₅H₅)(C₁₉H₂₂BN₃)]Br, (IIa), (2,2′‐bipyridyl)(ferrocenyl)(4‐methoxyphenylamino)boronium hexafluoridophosphate acetonitrile hemisolvate, [Fe(C₅H₅)(C₂₂H₂₀BN₃O)]PF₆·0.5CH₃CN, (IIIb), and 1,1′‐bis[(2,2′‐bipyridyl)(cyanomethyl)boronium]ferrocene bis(hexafluoridophosphate), [Fe(C₁₇H₁₄BN₃)₂](PF₆)₂, (IVb). The asymmetric unit of (IIIb) contains two independent cations with very similar conformations. The B atom has a distorted tetrahedral coordination in all four structures. The cyclopentadienyl rings of (Ib), (IIa) and (IIIb) are approximately eclipsed, while a bisecting conformation is found for (IVb). The N—H groups of (IIa) and (IIIb) are shielded by the ferrocenyl and tert‐butyl or phenyl groups and are therefore not involved in hydrogen bonding. The B—N(amine) bond lengths are shortened by delocalization of π‐electrons. In the cations with an amine substituent at boron, the B—N(bipyridyl) bonds are 0.035 (3) Å longer than in the cations with a methylene C atom bonded to boron. A similar lengthening of the B—N(bipyridyl) bonds is found in a survey of related cations with an oxy group attached to the B atom.     

Fast rotating Bose-Einstein condensates in an asymmetric trap

We investigate the effect of the anisotropy of a harmonic trap on the behaviour of a fast rotating Bose-Einstein condensate. This is done in the framework of the 2D Gross-Pitaevskii equation and requires a symplectic reduction of the quadratic form defining the energy. This reduction allows us to simplify the energy on a Bargmann space and study the asymptotics of large rotational velocity. We characterize two regimes of velocity and anisotropy; in the first one where the behaviour is similar to the isotropic case, we construct an upper bound: a hexagonal Abrikosov lattice of vortices, with an inverted parabola profile. The second regime deals with very large velocities, a case in which we prove that the ground state does not display vortices in the bulk, with a 1D limiting problem. In that case, we show that the coarse grained atomic density behaves like an inverted parabola with large radius in the deconfined direction but keeps a fixed profile given by a Gaussian in the other direction. The features of this second regime appear as new phenomena.