Electronic Transitions in Different Redox States of Trinuclear 5,6,11,12,17,18-Hexaazatrinaphthylene-Bridged Titanium Complexes: Spectroelectrochemistry and Quantum Chemistry

Multinuclear transition metal complexes bridged by ligands with extended π-electronic systems show a variety of complex electronic transitions and electron transfer reactions. While a systematic understanding of the photochemistry and electrochemistry has been attained for binuclear complexes, much less is known about trinuclear complexes such as hexaphenyl-5,6,11,12,17,18-hexaazatrinaphthylene-tristitanocene [(Cp₂Ti)₃HATN(Ph)₆]. The voltammogram of [(Cp₂Ti)₃HATN(Ph)₆] shows six oxidation and three reduction waves. Solution spectra of [(Cp₂Ti)₃HATN(Ph)₆] and of the electrochemically formed oxidation products show electronic transitions in the UV, visible and the NIR ranges. Density functional theory (DFT) and linear response time-dependent DFT show that the three formally titanium(II) centers transfer an electron to the HATN ligand in the ground state. The optically excited transitions occur exclusively between ligand-centered orbitals. The charged titanium centers only provide an electrostatic frame to the extended π-electronic system. Complete active self-consistent field (CASSCF) calculation on a structurally simplified model compound, which considers the multi-reference character imposed by the three titanium centers, can provide an interpretation of the experimentally observed temperature-dependent magnetic behavior of the different redox states of the title compound in full consistency with the interpretation of the electronic spectra.     

Short interval asymptotics for a class of arithmetic functions

Summary We provide a general asymptotic formula which permits applications to sums like <InlineEquation ID=IE"1"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"2"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"3"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"4"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"5"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"6"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"7"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"8"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"9"><EquationSource Format="TEX"><![CDATA[$]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>$]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation> \sum_{x< n\le x+y} \big(d(n)\big)^2, \quad \sum_{x< n\le x+y} d(n^3),\quad \sum_{x< n\le x+y}\big(r(n)\big)^2, \quad \sum_{x< n\le x+y}r(n^3), $$ where $d(n)$ and $r(n)$ are the usual arithmetic functions (number of divisors, sums of two squares), and $y$ is small compared to~$x$.     

Regularity for minimizers of functionals with p – q growth

We prove higher integrability for the gradient of vector-valued minimizers of some integral functionals with p – q growth. Received February 9, 1998     

C imaging by double resonance scalar-coupling editing

The two-dimensional FT Imaging of ¹³C-glucose obtained with twin spin-echo double resonance sequence is presented. The images have been obtained by a doubly tuned surface coil on samples containing water and ¹³C-enriched glucose in water. It is shown, both theoretically and experimentally, that the whole editing capability of the twin spin-echo double resonance imaging sequence is also preserved in the presence of the radiofrequency field inhomogeneity produced by the surface coil. As in an efficient selective irradiation method, the enhancement in the signal-to-noise ratio with respect to the direct ¹³C detection, depends on the number of protons J-coupled to ¹³C.     

The use of Raman microscopy and laser desorption ionization mass spectrometry in the examination of synthetic organic pigments in modern works of art

Synthetic organic pigments are widely used in modern and contemporary works of art. They have been examined by a variety of techniques including spectroscopic methods such as Fourier transform infrared spectroscopy, Raman, and X‐ray powder diffraction as well as chromatographic or mass spectrometric techniques such as pyrolysis‐gas chromatography/mass spectrometry and laser desorption ionization mass spectrometry (LDI–MS). Often, a combination of techniques has been used to examine these pigments. This paper describes use of Raman spectroscopy to create a database of colorants including two pigments not previously reported, PO1 and PO2. Then, using Raman spectroscopy in combination with LDI–MS, samples from modern works of art by artists including Mark Rothko, Barnett Newman, and José de Rivera were examined in order to identify the pigments present. One finding was that Rothko used a variety of red pigments over the course of his career including PR11 which has not been previously reported in artwork, and PO2 found with its positional isomer PR1. Knowledge of the colorants serves to inform conservators about display and treatment decisions. Published 2014. This article is a U.S. Government work and is in the public domain in the USA.     

Diophantine equations with products of consecutive members of binary recurrences

We prove a finiteness result for the number of solutions of a Diophantine equation of the form \(u_n u_{n+1}\cdots u_{n+k}\pm 1 =\pm u_m^2\), where \(\{ u_n\}_{n\ge 1}\) is a binary recurrent sequence whose characteristic equation has roots which are real quadratic units.     

Stability properties of an inverse parabolic problem with unknown boundaries

We treat the stability issue for an inverse problem arising from non-destructive evaluation by thermal imaging. We consider the determination of an unknown portion of the boundary of a thermic conducting body by overdetermined boundary data for a parabolic initial-boundary value problem. We obtain that when the unknown part of the boundary is a priori known to be smooth, the data are as regular as possible and all possible measurements are taken into account, the problem is exponentially ill-posed. Then, we prove that a single measurement with some a priori information on the unknown part of the boundary and minimal assumptions on the data, in particular on the thermal conductivity, is enough to have stable determination of the unknown boundary. Given the exponential ill-posedness, the stability estimate obtained is optimal. AMS 2000 Mathematics Subject Classification. Primary 35R30, Secondary 35B60, 33C90     

On the Dirichlet and Neumann Evolution Operators in ℝ d +

We prove some uniform and pointwise gradient estimates for the Dirichlet and the Neumann evolution operators \(G_{\mathcal {D}}(t,s)\) and \(G_{\mathcal {N}}(t,s)\) associated with a class of nonautonomous elliptic operators (t) with unbounded coefficients defined in I× \(\mathbb{R}_{+}\) (where I is a right-halfline or I=?). We also prove the existence and the uniqueness of a tight evolution system of measures \(\left \{\mu _{t}^{\mathcal {N}}\right \}_{t \in I}\) associated with \(G_{\mathcal {N}}(t,s)\) , which turns out to be sub-invariant for \(G_{\mathcal {D}}(t,s)\) , and we study the asymptotic behaviour of the evolution operators \(G_{\mathcal {D}}(t,s)\) and \(G_{\mathcal {N}}(t,s)\) in the L ^p -spaces related to the system \(\left \{\mu _{t}^{\mathcal {N}}\right \}_{t \in I}\) .