Generalized Complex Spherical Harmonics, Frame Functions, and Gleason Theorem

Consider a finite dimensional complex Hilbert space \cH, with dim(\cH) \geq 3, define \bS(\cH):= \{x\in \cH \:|\: ||x||=1\}, and let \nu_\cH be the unique regular Borel positive measure invariant under the action of the unitary operators in \cH, with \nu_\cH(\bS(\cH))=1. We prove that if a complex frame function f : \bS(\cH)\to \bC satisfies f \in \cL^2(\bS(\cH), \nu_\cH), then it verifies Gleason's statement: There is a unique linear operator A: \cH \to \cH such that $f(u) =$ for every u \in \bS(\cH). $A$ is Hermitean when $f$ is real. No boundedness requirement is thus assumed on $f$ {\em a priori}.Comment: 9 pages, Accepted for publication in Ann. H. Poincar\'

‘User-friendly’ primary phosphines and an arsine: synthesis and characterization of new air-stable ligands incorporating the ferrocenyl group

Reaction of FcCH₂CH₂P(O)(OH)₂ or FcCH₂P(O)(OH)(OEt) [Fc=Fe(η⁵-C₅H₄)(η⁵-C₅H₅)] with excess CH₂N₂ followed by reduction with Me₃SiCl–LiAlH₄ gives the air-stable primary phosphines FcCH₂CH₂PH₂ and the previously reported analogue FcCH₂PH₂ in high yields. Reduction of 1,1′-Fc′[CH₂P(O)(OEt)₂] [Fc′=Fe(η⁵-C₅H₄)₂] and 1,2-Fc″[CH₂P(O)(OEt)₂] [Fc″=Fe(η⁵-C₅H₅)(η⁵-C₅H₃)] similarly gives the new primary phosphines 1,1′-Fc′(CH₂PH₂)₂ and 1,2-Fc″(CH₂PH₂)₂, respectively. The arsine FcCH₂CH₂AsH₂, which is also air-stable, has been prepared by reduction of the arsonic acid FcCH₂CH₂As(O)(OH)₂ using Zn/HCl. An X-ray structure has been carried out on the arsine, which is only the second structure determination of a free primary arsine. The molybdenum carbonyl complex [1,2-Fc″(CH₂PH₂)₂Mo(CO)₄] was prepared by reaction of the phosphine with [Mo(CO)₄(pip)₂] (pip=piperidine), and characterized by a preliminary X-ray structure determination. However, the same reaction of 1,1′-Fc′(CH₂PH₂)₂with [Mo(CO)₄(pip)₂] gave [1,1′-Fc′(CH₂PH₂)₂Mo(CO)₄] and the dimer [1,1′-Fc′(CH₂PH₂)₂Mo(CO)₄]₂, characterized by electrospray mass spectrometry. 1,1′-Fc′[CH₂PH₂Mo(CO)₅]₂ and 1,2-Fc″[CH₂PH₂Mo(CO)₅]₂ were likewise prepared from the phosphines and excess [Mo(CO)₅(THF)]

Ferrocenyl hydroxymethylphosphines (η⁵-C₅H₅)Fe[η⁵⁻C₅H₄P(CH₂OH)₂] and 1,1′-[Fe{η⁵-C₅H₄P(CH₂OH)₂}₂] and their chalcogenide derivatives

The ferrocenyl hydroxymethylphosphines FcP(CH₂OH)₂ [Fc=(η⁵-C₅H₅)Fe(η⁵-C₅H₄)] and 1,1′-Fc′[P(CH₂OH)₂]₂ [Fc′=Fe(η⁵⁻C₅H₄)₂] were prepared by reactions of the corresponding primary phosphines FcPH₂ and 1,1′-Fc′(PH₂)₂ with excess aqueous formaldehyde. The crystal structure of FcP(CH₂OH)₂ was determined and compared with the known ferrocenyl hydroxymethylphosphine FcCH₂P(CH₂OH)₂. The chalcogenide derivatives FcP(E)(CH₂OH)₂ and 1,1′-Fc′[P(E)(CH₂OH)₂]₂ (E=O, S, Se) were prepared and fully characterised. Crystal structure determinations on FcP(O)(CH₂OH)₂ and FcP(S)(CH₂OH)₂ were performed, and the hydrogen-bonding patterns are compared with related compounds. The sulfide shows no hydrogen-bonding involving the phosphine sulfide group, in contrast to other reported ferrocenyl hydroxymethylphosphine sulfides. The platinum complex cis-[PtCl₂{FcP(CH₂OH)₂}₂] was prepared by reaction of 2 mol equivalents of FcP(CH₂OH)₂ with [PtCl₂(1,5-cyclo-octadiene)], and was characterised by 31P-NMR spectroscopy and negative ion electrospray mass spectrometry, which gave a strong [M+Cl]⁻ ion

Derivation of tropospheric methane from TCCON CH₄ and HF total column observations

The Total Carbon Column Observing Network (TCCON) is a global ground-based network of Fourier transform spectrometers that produce precise measurements of column-averaged dry-air mole fractions of atmospheric methane (CH₄). Temporal variability in the total column of CH₄ due to stratospheric dynamics obscures fluctuations and trends driven by tropospheric transport and local surface fluxes that are critical for understanding CH₄ sources and sinks. We reduce the contribution of stratospheric variability from the total column average by subtracting an estimate of the stratospheric CH₄ derived from simultaneous measurements of hydrogen fluoride (HF). HF provides a proxy for stratospheric CH₄ because it is strongly correlated to CH₄ in the stratosphere, has an accurately known tropospheric abundance (of zero), and is measured at most TCCON stations. The stratospheric partial column of CH₄ is calculated as a function of the zonal and annual trends in the relationship between CH₄ and HF in the stratosphere, which we determine from ACE-FTS satellite data. We also explicitly take into account the CH₄ column averaging kernel to estimate the contribution of stratospheric CH₄ to the total column. The resulting tropospheric CH₄ columns are consistent with in situ aircraft measurements and augment existing observations in the troposphere

Herschel/HIFI detections of hydrides towards AFGL 2591: Envelope emission versus tenuous cloud absorption

The Heterodyne Instrument for the Far Infrared (HIFI) onboard the Herschel Space Observatory allows the first observations of light diatomic molecules at high spectral resolution and in multiple transitions. Here, we report deep integrations using HIFI in different lines of hydrides towards the high-mass star forming region AFGL 2591. Detected are CH, CH^+, NH, OH^+, H_2O^+, while NH^+ and SH^+ have not been detected. All molecules except for CH and CH^+ are seen in absorption with low excitation temperatures and at velocities different from the systemic velocity of the protostellar envelope. Surprisingly, the CH(J_(F,P) = 3/2_(2,−) − 1/2_(1,+)) and CH^+(J = 1−0, J = 2−1) lines are detected in emission at the systemic velocity. We can assign the absorption features to a foreground cloud and an outflow lobe, while the CH and CH^+ emission stems from the envelope. The observed abundance and excitation of CH and CH^+ can be explained in the scenario of FUV irradiated outflow walls, where a cavity etched out by the outflow allows protostellar FUV photons to irradiate and heat the envelope at larger distances driving the chemical reactions that produce these molecules

Herschel-HIFI detections of hydrides towards AFGL 2591 (Envelope emission versus tenuous cloud absorption)

The Heterodyne Instrument for the Far Infrared (HIFI) onboard the Herschel Space Observatory allows the first observations of light diatomic molecules at high spectral resolution and in multiple transitions. Here, we report deep integrations using HIFI in different lines of hydrides towards the high-mass star forming region AFGL 2591. Detected are CH, CH+, NH, OH+, H2O+, while NH+ and SH+ have not been detected. All molecules except for CH and CH+ are seen in absorption with low excitation temperatures and at velocities different from the systemic velocity of the protostellar envelope. Surprisingly, the CH(JF,P = 3/2_2,- - 1/2_1,+) and CH+(J = 1 - 0, J = 2 - 1) lines are detected in emission at the systemic velocity. We can assign the absorption features to a foreground cloud and an outflow lobe, while the CH and CH+ emission stems from the envelope. The observed abundance and excitation of CH and CH+ can be explained in the scenario of FUV irradiated outflow walls, where a cavity etched out by the outflow allows protostellar FUV photons to irradiate and heat the envelope at larger distances driving the chemical reactions that produce these molecules.Comment: Accepted for publication in Astronomy and Astrophysics (HIFI first results issue